Recognized as an
American National Standard (ANSI)
IEEE Std 399-1997
IEEE Recommended Practice for
Industrial and Commercial
Power Systems Analysis
Sponsor
Power Systems Engineering Committee
of the
Industrial and Commercial Power Systems Department
of the
IEEE Industry Applications Society
Approved 16 September 1997
IEEE Standards Board
Approved 28 April 1998
American National Standards Institute
Abstract: This Recommended Practice is a reference source for engineers involved in
industrial and commercial power systems analysis. It contains a thorough analysis of the power
system data required, and the techniques most commonly used in computer-aided analysis, in
order to perform specific power system studies of the following: short-circuit, load flow, motorstarting, cable ampacity, stability, harmonic analysis, switching transient, reliability, ground
mat, protective coordination, dc auxiliary power system, and power system modeling.
Keywords: cable ampacity, dc power system studies, ground mat studies, harmonic analysis,
load flow studies, motor-starting studies, power system analysis, power system modeling,
power system studies, protective coordination studies, reliability studies, short-circuit studies,
stability studies, switching transient studies.
Grateful acknowledgment is made to the following organization for having granted permission to reprint
illustrations in this document as listed below:
The General Electric Company, Schenectady, NY, for Figures 16-2, 16-4, 16-6, and 16-7.
First Printing
August 1998
SH94571
The Institute of Electrical and Electronics Engineers, Inc.
345 East 47th Street, New York, NY 10017-2394, USA
Copyright © 1998 by the Institute of Electrical and Electronics Engineers, Inc.
All rights reserved. Published 1998. Printed in the United States of America
ISBN 1-55937-968-5
No part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise,
without the prior written permission of the publisher.
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Introduction
(This introduction is not a part of IEEE Std 399-1997, IEEE Recommended Practice for Industrial and
Commercial Power Systems Analysis.)
This Recommended Practice, commonly known as the “Brown Book,” is intended as a
practical, general treatise on power system analysis theory and as an engineer’s reference
source on the techniques that are most commonly applied to the computer-aided analysis of
electric power systems in industrial plants and commercial buildings. The Brown Book is a
useful supplement to several other power system analysis texts that appear in the references
and bibliography subclauses of the various chapters of this book. The Brown Book is both
complementary and supplementary to the rest of the Color Book series.
One new and important chapter has been added: Chapter 16, entitled “DC auxiliary power
system analysis.” All the other chapters in this new edition have been revised and updated—
in some cases quite substantially—to reflect current technology.
To many members of the working group who wrote and developed this Recommended
Practice, the Brown Book has become a true labor of love. The dedication and support of
each individual member is clearly evident in every chapter of the Brown Book. These
individuals deserve our many thanks for their excellent contributions.
iv
The Brown Book Working Group for the 1997 edition had the following membership:
L. Guy Jackson, Chair
Chapter 1:
Introduction—L. Guy Jackson, Chair; George A. Terry
Chapter 2:
Applications of power system analysis—L. Guy Jackson, Chair;
George A. Terry
Chapter 3:
Analytical procedures—M. Shan Griffith, Chair; Anthony J. Rodolakis
Chapter 4:
System modeling—Stephen S. Miller, Co-Chair; Mark Halpin; Co-Chair;
Matt McBurnett; Anthony J. Rodolakis; Michael S. Tucker
Chapter 5:
Computer solutions and systems—Glenn E. Word, Chair; Anthony J. Rodolakis
Chapter 6:
Load flow studies—Chet E. Davis, Co-Chair; James W. Feltes, Co-Chair;
Mark Halpin; Anthony J. Rodolakis
Chapter 7:
Short-circuit studies—Anthony J. Rodolakis, Chair; William M. Hall;
Mark Halpin; Michael E. Lick; Matt McBurnett; Conrad St. Pierre
Chapter 8:
Stability studies—Wei-Jen Lee, Co-Chair; Mark Halpin, Co-Chair;
Matt McBurnett; Anthony J. Rodolakis
Chapter 9:
Motor-starting studies—M. Shan Griffith, Co-Chair; Mike Aimone, Co-Chair;
Anthony J. Rodolakis
Chapter 10: Harmonic analysis studies—Suresh C. Kapoor, Chair; M. Shan Griffith;
Mark Halpin
Chapter 11: Switching transient studies—Carlos B. Pinheiro, Chair
Chapter 12: Reliability studies—Michael R. Albright, Chair
Chapter 13: Cable ampacity studies—Farrokh Shokooh, Chair
Chapter 14: Ground mat studies—M. Shan Griffith, Chair; Anthony J. Rodolakis
Chapter 15: Coordination studies—A. Elizabeth Ronat, Chair; Mike Aimone;
Michael E. Lick; John F. Witte
Chapter 16: DC auxiliary power system analysis—Kenneth Fleishcher, Co-Chair;
Scott Munnings, Co-Chair; Ajkit K. Hiranandani; Gene A. Poletto
Others who contributed to the development of this document are as follows:
J. J. Dia
C. R. Heising
A. D. Patton
David Shipp
George W. Walsh
Erzhuan Zhou
v
The following persons were on the balloting committee:
Mike A. Aimone
Michael R. Albright
Robert J. Beaker
Reuben F. Burch IV
Chet Davis
James W. Feltes
Landis H. Floyd
Jerry M. Frank
Dan Goldberg
M. Shan Griffith
William M. Hall
Mark S. Halpin
L. Guy Jackson
Suresh C. Kapoor
vi
Don O. Koval
Wei-Jen Lee
Michael E. Lick
Matt McBurnett
Richard H. McFadden
Reg Mendis
Steve S. Miller
Daleep Mohla
William J. Moylan
R. Scott Munnings
Andrew T. Morris
Ed Palko
Gene A. Poletto
Brian Rener
Rasheek Rifaat
Milton D. Robinson
Anthony Rodolakis
A. Elizabeth Ronat
Donald R. Ruthman
Vincent Saporita
Lynn F. Saunders
Stan Shilling
David Shipp
Farrokh Shokooh
Conrad R. St. Pierre
Erzhuan Zhou
Donald W. Zipse
When the IEEE Standards Board approved this standard on 16 September 1997, it had the
following membership:
Donald C. Loughry, Chair
Richard J. Holleman, Vice Chair
Andrew G. Salem, Secretary
Clyde R. Camp
Stephen L. Diamond
Harold E. Epstein
Donald C. Fleckenstein
Jay Forster*
Thomas F. Garrity
Donald N. Heirman
Jim Isaak
Ben C. Johnson
Lowell Johnson
Robert Kenelly
E.G. “Al” Kiener
Joseph L. Koepfinger*
Stephen R. Lambert
Lawrence V. McCall
L. Bruce McClung
Marco W. Migliaro
Louis-François Pau
Gerald H. Peterson
John W. Pope
Jose R. Ramos
Ronald H. Reimer
Ingo Rüsch
John S. Ryan
Chee Kiow Tan
Howard L. Wolfman
*Member Emeritus
Also included are the following nonvoting IEEE Standards Board liaisons:
Satish K. Aggarwal
Alan H. Cookson
Paula M. Kelty
IEEE Standards Project Editor
National Electrical Code and NEC are both registered trademarks of the National Fire Protection Association, Inc.
vii
Contents
Chapter 1
Overview.................................................................................................................................. 1
1.1 Scope and general information ................................................................................... 1
1.2 History of power system studies ................................................................................. 1
1.3 Applying power system analysis techniques to industrial and
commercial power systems ......................................................................................... 2
1.4 Purposes of this Recommended Practice .................................................................... 2
1.5 References ................................................................................................................... 5
Chapter 2
Applications of power system analysis.................................................................................... 7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
Introduction ................................................................................................................. 7
Load flow analysis ...................................................................................................... 7
Short-circuit analysis................................................................................................... 8
Stability analysis ......................................................................................................... 8
Motor-starting analysis ............................................................................................... 8
Harmonic analysis....................................................................................................... 9
Switching transients analysis .................................................................................... 10
Reliability analysis .................................................................................................... 10
Cable ampacity analysis............................................................................................ 10
Ground mat analysis.................................................................................................. 11
Protective device coordination analysis .................................................................... 11
DC auxiliary power system analysis ......................................................................... 12
Chapter 3
Analytical procedures ............................................................................................................ 13
3.1 Introduction ............................................................................................................... 13
3.2 Fundamentals ............................................................................................................ 14
3.3 Bibliography.............................................................................................................. 40
Chapter 4
System modeling.................................................................................................................... 43
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Introduction ............................................................................................................... 43
Modeling ................................................................................................................... 43
Review of basics ....................................................................................................... 44
Power network solution ............................................................................................ 49
Impedance diagram ................................................................................................... 53
Extent of the model ................................................................................................... 54
Models of branch elements ....................................................................................... 55
Power system data development ............................................................................... 71
Models of bus elements............................................................................................. 80
ix
4.10 References ................................................................................................................. 99
4.11 Bibliography............................................................................................................ 100
Chapter 5
Computer solutions and systems.......................................................................................... 103
5.1
5.2
5.3
5.4
Introduction ............................................................................................................. 103
Numerical solution techniques................................................................................ 104
Computer systems ................................................................................................... 122
Bibliography............................................................................................................ 129
Chapter 6
Load flow studies................................................................................................................. 133
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Introduction ............................................................................................................. 133
System representation ............................................................................................. 134
Input data................................................................................................................. 137
Load flow solution methods.................................................................................... 140
Load flow analysis .................................................................................................. 149
Load flow study example ........................................................................................ 151
Load flow programs ................................................................................................ 162
Conclusions ............................................................................................................. 162
Chapter 7
Short-circuit studies ............................................................................................................. 165
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Introduction and scope ............................................................................................ 165
Extent and requirements of short-circuit studies..................................................... 166
System modeling and computational techniques .................................................... 168
Fault analysis according to industry standards ....................................................... 172
Factors affecting the accuracy of short-circuit studies............................................ 179
Computer solutions ................................................................................................. 182
Example .................................................................................................................. 187
References ............................................................................................................... 203
Bibliography............................................................................................................ 206
Chapter 8
Stability studies.................................................................................................................... 209
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
x
Introduction ............................................................................................................. 209
Stability fundamentals............................................................................................. 209
Problems caused by instability................................................................................ 216
System disturbances that can cause instability........................................................ 216
Solutions to stability problems................................................................................ 217
System stability analysis ......................................................................................... 218
Stability studies of industrial power systems .......................................................... 223
Summary and conclusions ....................................................................................... 228
Bibliography ............................................................................................................ 229
Chapter 9
Motor-starting studies .......................................................................................................... 231
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
Introduction ............................................................................................................. 231
Need for motor-starting studies............................................................................... 231
Recommendations ................................................................................................... 235
Types of studies ...................................................................................................... 237
Data requirements ................................................................................................... 238
Solution procedures and examples.......................................................................... 241
Summary ................................................................................................................. 259
References ............................................................................................................... 263
Bibliography............................................................................................................ 263
Chapter 10
Harmonic analysis studies.................................................................................................... 265
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
Introduction ............................................................................................................. 265
Background ............................................................................................................. 266
Purpose of harmonic study...................................................................................... 267
General theory......................................................................................................... 268
System modeling..................................................................................................... 276
Example solutions ................................................................................................... 290
Remedial measures ................................................................................................. 302
Harmonic standards................................................................................................. 307
References ............................................................................................................... 309
Bibliography............................................................................................................ 309
Chapter 11
Switching transient studies .................................................................................................. 313
11.1
11.2
11.3
11.4
11.5
11.6
Power system switching transients ......................................................................... 313
Switching transient studies...................................................................................... 338
Switching transients—field measurements ............................................................. 359
Typical circuit parameters for transient studies ...................................................... 363
References ............................................................................................................... 367
Bibliography............................................................................................................ 367
Chapter 12
Reliability studies................................................................................................................. 375
12.2
12.3
12.4
12.5
12.6
Definitions............................................................................................................... 375
System reliability indexes ....................................................................................... 377
Data needed for system reliability evaluations ....................................................... 377
Method for system reliability evaluation ................................................................ 378
References ............................................................................................................... 380
xi
Chapter 13
Cable ampacity studies ........................................................................................................ 383
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
Introduction ............................................................................................................. 383
Heat flow analysis ................................................................................................... 384
Application of computer program........................................................................... 386
Ampacity adjustment factors .................................................................................. 388
Example .................................................................................................................. 399
Conclusion .............................................................................................................. 403
References ............................................................................................................... 404
Bibliography............................................................................................................ 404
Chapter 14
Ground mat studies .............................................................................................................. 407
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
14.10
14.11
Introduction ............................................................................................................. 407
Justification for ground mat studies ........................................................................ 407
Modeling the human body ...................................................................................... 407
Traditional analysis of the ground mat ................................................................... 410
Advanced grid modeling ......................................................................................... 415
Benchmark problems .............................................................................................. 418
Input/output techniques........................................................................................... 420
Sample problem ...................................................................................................... 420
Conclusion .............................................................................................................. 420
Reference ................................................................................................................ 423
Bibliography............................................................................................................ 424
Chapter 15
Coordination studies ............................................................................................................ 429
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
Introduction ............................................................................................................. 429
Basics of coordination............................................................................................. 430
Computer programs for coordination...................................................................... 435
Common structure for computer programs ............................................................. 436
How to make use of coordination software............................................................. 441
Verifying the results................................................................................................ 443
Equipment needs ..................................................................................................... 443
Conclusion .............................................................................................................. 444
Bibliography............................................................................................................ 444
Chapter 16
DC auxiliary power system analysis.................................................................................... 445
16.1
16.2
16.3
16.4
xii
Introduction ............................................................................................................. 445
Purpose of the recommended practice .................................................................... 445
Application of dc power system analysis................................................................ 445
Analytical procedures ............................................................................................. 446
16.5
16.6
16.7
16.8
16.9
System modeling..................................................................................................... 446
Load flow/voltage drop studies............................................................................... 461
Short-circuit studies ................................................................................................ 464
International guidance on dc short-circuit calculations .......................................... 466
Bibliography............................................................................................................ 466
Index .................................................................................................................................... 469
xiii
IEEE Recommended Practice for Industrial and
Commercial Power Systems Analysis
Chapter 1
Overview
1.1 Scope and general information
This Recommended Practice, commonly known as the IEEE Brown Book, is published by
the Institute of Electrical and Electronics Engineers, Inc. (IEEE) as a reference source to give
plant engineers a better understanding of the purpose for and techniques involved in power
system studies. The IEEE Brown Book can also be a helpful reference source for system and
data acquisition for engineering consultants performing necessary studies prior to designing a
new system or expanding an existing power system. This Recommended Practice will help
ensure high standards of power system reliability and maximize the utilization of capital
investment.
The IEEE Brown Book emphasizes up-to-date techniques in system studies that are most
applicable to industrial and commercial power systems. It complements the other IEEE Color
Books, and is intended to be used in conjunction with, not as a replacement for, the many
excellent texts available in this field.
The IEEE Brown Book was prepared on a voluntary basis by engineers and designers functioning as a Working Group within the IEEE, under the Industrial and Commercial Power
Systems Department of the Industry Applications Society.
1.2 History of power system studies
The planning, design, and operation of a power system requires continual and comprehensive
analyses to evaluate current system performance and to establish the effectiveness of alternative plans for system expansion.
The computational work to determine power flows and voltage levels resulting from a single
operating condition for even a small network is all but insurmountable if performed by
manual methods. The need for computational aids led to the design of a special purpose
analog computer (ac network analyzer) as early as 1929. It provided the ability to determine
flows and voltages during normal and emergency conditions and to study the transient
behavior of the system resulting from fault conditions and switching operations.
The earliest application of digital computers to power system problems dates back to the late
1940s. Most of the early applications were limited in scope because of the small capacity of
the punched card calculators in use during that period. Large-scale digital computers became
Copyright © 1998 IEEE. All rights reserved.
1
IEEE
Std 399-1997
CHAPTER 1
available in the mid-1950s, and the initial success of load flow programs led to the
development of programs for short-circuit and stability calculations.
Today, the digital computer is an indispensable tool in power system planning, in which it is
necessary to predict future growth and simulate day-to-day operations over periods of twenty
years or more.
1.3 Applying power system analysis techniques to industrial and
commercial power systems
As computer technology has advanced, so has the complexity of industrial and commercial
power systems. These power systems have grown in recent decades with capacities far
exceeding that of a small electric utility system.
Today’s intensely competitive business environment forces plant or building management
personnel to be very aware of the total owning cost of the power distribution system.
Therefore, they demand assurances of maximum return on all capital investments in the
power system. The use of digital computers makes it possible to study the performance of
proposed and actual systems under many operating conditions. Answers to many questions
regarding impact of expansion on the system, short-circuit capacity, stability, load
distribution, etc., can be intelligently and economically obtained.
1.4 Purposes of this Recommended Practice
1.4.1 Why a study?
As is stated in Chapter 2, the planning, design, and operation of industrial and commercial
power systems require several studies to assist in the evaluation of the initial and future
system performance, system reliability, safety, and the ability to grow with production and/or
operating requirements. The studies most likely to be needed are load flow studies, cable
ampacity studies, short-circuit studies, coordination studies, stability studies, and routine
motor-starting studies. Additional studies relating to switching transients, reliability,
grounding, harmonics, and special motor-starting considerations may also be required. The
engineer in charge of system design must decide which studies are needed to ensure that the
system will operate safely, economically, and efficiently over the expected life of the system.
A brief summary of these studies is presented in Chapter 2, along with a discussion pertaining to data collection and preparation of the data files required to perform the desired study
on a digital computer.
1.4.2 How to prepare for a power system study
For a plant engineer to solve a power system analysis problem, he or she must be thoroughly
familiar with the fundamentals of power electrical engineering. He or she can then analyze
2
Copyright © 1998 IEEE. All rights reserved.
OVERVIEW
IEEE
Std 399-1997
the problem, prepare the necessary equivalent circuits, and obtain appropriate system data
before using a computer program to perform repetitive calculations. Failure to use a valid
analytical procedure to establish a sound basic approach to the problem could lead to
disastrous consequences in both the design and operation of a system. Furthermore, a basic
understanding of power engineering is essential to correctly interpret the results of computer
calculations. It is important to emphasize the need for a thorough foundation and base of
experience in power system engineering in addition to modern, effective computer software.
Power system analysis engineering software is an excellent tool for studying power systems,
but it cannot be used as a substitute for knowledge and experience. Chapter 3 offers an excellent review of the most essential fundamentals in a system study.
To set up a computer program for system analysis, certain basic data must be gathered with
accuracy and proper presentation. The extent of system representation, restrictions in terms of
nodes (buses) and branches (lines and transformers), balanced three-phase network and
single-line diagram, impedance diagram, etc., are all important inputs to a meaningful system
study. Chapter 4 deals with system modeling and data requirements to illustrate how these
basic data for a study can be properly prepared or organized.
Once the basic preparations are completed, the next step is to look for an actual computer
program. Programs are available—written for personal computers (PCs)—that allow inhouse calculation for the studies outlined in this standard. Chapter 5 discusses basic computation methods, various types of computer systems and their requirements, and availability of
commercial computing services and their capabilities.
1.4.3 The most common system studies
The following chapters address the most common studies for the design or operation of an
industrial or commercial power system:
Chapter 6, Load flow studies
Chapter 7, Short-circuit studies
Chapter 8, Stability studies
Chapter 9, Motor-starting studies
Chapter 10, Harmonic analysis studies
Chapter 11, Switching transient studies
Chapter 12, Reliability studies
Chapter 13, Cable ampacity studies
Chapter 14, Ground mat studies
Chapter 15, Coordination studies
Chapter 16, DC auxiliary power system analysis
The purpose of each study and what can be achieved by it are briefly explained in each chapter.
Figure 1-1 is a typical composite single-line diagram for a large industrial power system that
is used in Chapters 6 through 15.
Copyright © 1998 IEEE. All rights reserved.
3
IEEE
Std 399-1997
CHAPTER 1
Figure 1-1—Single-line diagram of typical industrial power system
for load flow study example
4
Copyright © 1998 IEEE. All rights reserved.
OVERVIEW
IEEE
Std 399-1997
After studying these chapters, an engineer should be better equipped to prepare necessary
data and criteria for a specific computer study. The study can be performed in-house or by an
outside consultant. There is a growing number of consulting firms that specialize in
performing system studies at a reasonable cost.
Studying these chapters will provide the basic understanding of the studies needed to
coordinate the data and criteria for specific studies and will also serve as a reference to those
analysts for whom studies are a principal activity.
1.5 References
This chapter shall be used in conjunction with the following publications:
IEEE Std 141-1993, IEEE Recommended Practice for Electric Power Distribution for
Industrial Plants (IEEE Red Book).1
IEEE Std 142-1991, IEEE Recommended Practice for Grounding of Industrial and
Commercial Power Systems (IEEE Green Book).
IEEE Std 241-1990, IEEE Recommended Practice for Electric Power Systems in
Commercial Buildings (IEEE Gray Book).
IEEE Std 242-1986 (Reaff 1991), IEEE Recommended Practice for Protection and
Coordination of Industrial and Commercial Power Systems (IEEE Buff Book).
1IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
Copyright © 1998 IEEE. All rights reserved.
5
Chapter 2
Applications of power system analysis
2.1 Introduction
The planning, design, and operation of industrial and commercial power systems require
engineering studies to evaluate existing and proposed system performance, reliability, safety,
and economics. Studies, properly conceived and conducted, are a cost-effective way to prevent surprises and to optimize equipment selection. In the design stage, the studies identify
and avoid potential deficiencies in the system before it goes into operation. In existing systems, the studies help locate the cause of equipment failure and misoperation, and determine
corrective measures for improving system performance.
The complexity of modern industrial power systems makes studies difficult, tedious, and
time-consuming to perform manually. The computational tasks associated with power
systems studies have been greatly simplified by the use of digital computer programs. Sometimes, economics and study requirements dictate the use of an analog computer—a transient
network analyzer (TNA)—which provides a scale model of the power system.
2.1.1 Digital computer
The digital computer offers engineers a powerful tool to perform efficient system studies.
Computers permit optimal designs at minimum costs, regardless of system complexity.
Advances in computer technology, like the introduction of the personal computer with its
excellent graphics capabilities, have not only reduced the computing costs but also the
engineering time needed to use the programs. Study work formerly done by outside
consultants can now be performed in-house. User-friendly programs utilizing interactive
menus, online help facilities, and a graphical user interface (GUI) guide the engineer through
the task of using a digital computer program.
2.1.2 Transient network analyzer (TNA)
The TNA is a useful tool for transient overvoltage studies. The use of microcomputers to
control and acquire the data from the TNA allows the incorporation of probability and
statistics in switching surge analysis. One of the major advantages of the TNA is that it allows
for quick reconfiguration of complex systems with immediate results, avoiding the relatively
longer time associated with running digital computer programs for these systems.
2.2 Load flow analysis
Load flow studies determine the voltage, current, active, and reactive power and power factor
in a power system. Load flow studies are an excellent tool for system planning. A number of
operating procedures can be analyzed, including contingency conditions, such as the loss of a
generator, a transmission line, a transformer, or a load. These studies will alert the user to
Copyright © 1998 IEEE. All rights reserved.
7
IEEE
Std 399-1997
CHAPTER 2
conditions that may cause equipment overloads or poor voltage levels. Load flow studies can
be used to determine the optimum size and location of capacitors for power factor
improvement. Also, they are very useful in determining system voltages under conditions of
suddenly applied or disconnected loads. The results of a load flow study are also starting
points for stability studies. Digital computers are used extensively in load flow studies due to
the complexity of the calculations involved.
2.3 Short-circuit analysis
Short-circuit studies are done to determine the magnitude of the prospective currents flowing
throughout the power system at various time intervals after a fault occurs. The magnitude of
the currents flowing through the power system after a fault vary with time until they reach a
steady-state condition. This behavior is due to system characteristics and dynamics. During
this time, the protective system is called on to detect, interrupt, and isolate these faults. The
duty imposed on this equipment is dependent upon the magnitude of the current, which is
dependent on the time from fault inception. This is done for various types of faults (threephase, phase-to-phase, double-phase-to-ground, and phase-to-ground) at different locations
throughout the system. The information is used to select fuses, breakers, and switchgear ratings in addition to setting protective relays.
2.4 Stability analysis
The ability of a power system, containing two or more synchronous machines, to continue to
operate after a change occurs on the system is a measure of its stability. The stability problem
takes two forms: steady-state and transient. Steady-state stability may be defined as the ability of a power system to maintain synchronism between machines within the system following relatively slow load changes. Transient stability is the ability of the system to remain in
synchronism under transient conditions, i.e., faults, switching operations, etc.
In an industrial power system, stability may involve the power company system and one or
more in-plant generators or synchronous motors. Contingencies, such as load rejection,
sudden loss of a generator or utility tie, starting of large motors or faults (and their duration),
have a direct impact on system stability. Load-shedding schemes and critical fault-clearing
times can be determined in order to select the proper settings for protective relays.
These types of studies are probably the single most complex ones done on a power system. A
simulation will include synchronous generator models with their controls, i.e., voltage
regulators, excitation systems, and governors. Motors are sometimes represented by their
dynamic characteristics as are static var compensators and protective relays.
2.5 Motor-starting analysis
The starting current of most ac motors is several times normal full load current. Both
synchronous and induction motors can draw five to ten times full load current when starting
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them across the line. Motor-starting torque varies directly as the square of the applied voltage. If the terminal voltage drop is excessive, the motor may not have enough starting torque
to accelerate up to running speed. Running motors may stall from excessive voltage drops, or
undervoltage relays may operate. In addition, if the motors are started frequently, the voltage
dip at the source may cause objectionable flicker in the lighting system.
By using motor-starting study techniques, these problems can be predicted before the
installation of the motor. If a starting device is needed, its characteristics and ratings can be
easily determined. A typical digital computer program will calculate speed, slip, electrical
output torque, load current, and terminal voltage data at discrete time intervals from locked
rotor to full load speed. Also, voltage at important locations throughout the system during
start-up can be monitored. The study can help select the best method of starting, the proper
motor design, or the required system design for minimizing the impact of motor starting on
the entire system.
2.6 Harmonic analysis
A harmonic-producing load can affect other loads if significant voltage distortion is caused.
The voltage distortion caused by the harmonic-producing load is a function of both the
system impedance and the amount of harmonic current injected. The mere fact that a given
load current is distorted does not always mean there will be undue adverse effects on other
power consumers. If the system impedance is low, the voltage distortion is usually negligible
in the absence of harmonic resonance. However, if harmonic resonance prevails, intolerable
harmonic voltage and currents are likely to result.
Some of the primary effects of voltage distortion are the following:
a)
b)
c)
Control/computer system interference
Heating of rotating machinery
Overheating/failure of capacitors
When the harmonic currents are high and travel in a path with significant exposure to parallel
communication circuits, the principal effect is telephone interference. This problem depends
on the physical path of the circuit as well as the frequency and magnitude of the harmonic
currents. Harmonic currents also cause additional line losses and additional stray losses in
transformers.
Watthour meter error is often a concern. At harmonic frequencies, the meter may register
high or low depending on the harmonics present and the response of the meter to these
harmonics. Fortunately, the error is usually small.
Analysis is commonly done to predict distortion levels for addition of a new harmonicproducing load or capacitor bank. The general procedure is to first develop a model that can
accurately simulate the harmonic response of the present system and then to add a model of
the new addition. Analysis is also commonly done to evaluate alternatives for correcting
problems found by measurements.
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Only very small circuits can be effectively analyzed without a computer program. Typically, a
computer program for harmonic analysis will provide the engineer with the capability to
compute the frequency response of the power system and to display it in a number of useful
graphical forms. The programs provide the capability to predict the actual distortion based on
models of converters, arc furnaces, and other nonlinear loads.
2.7 Switching transients analysis
Switching transients severe enough to cause problems in industrial power systems are most
often associated with inadequate or malfunctioning breakers or switches and the switching of
capacitor banks and other frequently switched loads. The arc furnace system is most
frequently studied because of its high frequency of switching and the related use of capacitor
banks.
By properly using digital computer programs or a TNA, these problems can be detected early
in the design stage. In addition to these types of switching transient problems, digital computer programs and the TNA can be used to analyze other system anomalies, such as lightning arrester operation, ferroresonance, virtual current chopping, and breaker transient
recovery voltage.
2.8 Reliability analysis
When comparing various industrial power system design alternatives, acceptable system
performance quality factors (including reliability) and cost are essential in selecting an
optimum design. A reliability index is the probability that a device will function without
failure over a specified time period. This probability is determined by equipment maintenace
requirements and failure rates. Using probability and statistical analyses, the reliability of a
power system can be studied in depth with digital computer programs.
Reliability is most often expressed as the frequency of interruptions and expected number of
hours of interruptions during one year of system operation. Momentary and sustained system
interruptions, component failures, and outage rates are used in some reliability programs to
compute overall system reliability indexes at any node in the system, and to investigate
sensitivity of these indexes to parameter changes. With these results, economics and
reliability can be considered to select the optimum power system design.
2.9 Cable ampacity analysis
Cable ampacity studies calculate the current-carrying capacity (ampacity) of power cables in
underground or above ground installations. This ampacity is determined by the maximum
allowable conductor temperature. In turn, this temperature is dependent on the losses in the
cable, both I 2R and dielectric, and thermal coupling between heat-producing components and
ambient temperature.
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The ampacity calculations are extremely complex. This is due to many considerations, some
examples of which are heat transfer through the cable insulation and sheath, and, in the case
of underground installations, heat transfer to duct or soil as well as from duct bank to soil.
Other considerations include the effects of losses caused by proximity and skin effects. In
addition, depending on the installation, the cable-shielding system may introduce additional
losses. The analysis involves the application of thermal equivalents of Ohm’s and Kirchoff’s
laws to a thermal circuit.
2.10 Ground mat analysis
Under ground-fault conditions, the flow of current will result in voltage gradients within and
around the substation, not only between structures and nearby earth, but also along the
ground surface. In a properly designed system, this gradient should not exceed the limits that
can be tolerated by the human body.
The purpose of a ground mat study is to provide for the safety and well-being of anyone that
can be exposed to the potential differences that can exist in a station during a severe fault. The
general requirements for industrial power system grounding are similar to those of utility
systems under similar service conditions. The differences arise from the specific
requirements of the manufacturing or process operations.
Some of the factors that are considered in a ground-mat study are the following:
a)
b)
c)
d)
e)
Fault-current magnitude and duration
Geometry of the grounding system
Soil resistivity
Probability of contact
Human factors such as
1) Body resistance
2) Standard assumptions on physical conditions of the individual
2.11 Protective device coordination analysis
The objective of a protection scheme in a power system is to minimize hazards to personnel
and equipment while allowing the least disruption of power service. Coordination studies are
required to select or verify the clearing characteristics of devices such as fuses, circuit
breakers, and relays used in the protection scheme. These studies are also needed to
determine the protective device settings that will provide selective fault isolation. In a
properly coordinated system, a fault results in interruption of only the minimum amount of
equipment necessary to isolate the faulted portion of the system. The power supply to loads in
the remainder of the system is maintained. The goal is to achieve an optimum balance
between equipment protection and selective fault isolation that is consistent with the
operating requirements of the overall power system.
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Short-circuit calculations are a prerequisite for a coordination study. Short-circuit results
establish minimum and maximum current levels at which coordination must be achieved and
which aid in setting or selecting the devices for adequate protection. Traditionally, the coordination study has been performed graphically by manually plotting time-current operating
characteristics of fuses, circuit breaker trip devices, and relays, along with conductor and
transformer damage curves—all in series from the fault location to the source. Log-log scales
are used to plot time versus current magnitudes. These “coordination curves’’ show
graphically the quality of protection and coordination possible with the equipment available.
They also permit the verification/confirmation of protective device characteristics, settings,
and ratings to provide a properly coordinated and protected system.
With the advent of the personal computer, the light-table approach to protective device
coordination is being replaced by computer programs. The programs provide a graphical
representation of the device coordination as it is developed. In the future, computer programs
are expected to use expert systems based on practical coordination algorithms to further assist
the protection engineer.
2.12 DC auxiliary power system analysis
The need for direct current (dc) power system analysis of emergency standby power supplies
has steadily increased during the past several years in data processing facilities, long distance
telephone companies, and generating stations.
DC emergency power is used for circuit breaker control, protective relaying, inverters,
instumentation, emergency lighting, communications, annunciators, fault recorders, and auxiliary motors. The introduction of computer techniques to dc power systems analysis has
allowed a more rapid and rigorous analysis of these systems compared to earlier manual
techniques.
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Copyright © 1998 IEEE. All rights reserved.
Chapter 3
Analytical procedures
3.1 Introduction
With the development of the digital computer and advanced computer programming
techniques, power system problems of the most complex types can be rigorously analyzed.
Previously, solutions were usually only approximate. Limitations and errors were introduced
by the many simplifying assumptions necessary to permit classical longhand calculating
procedures. For progress to be realized in using the computer for power system analysis, it
has been necessary for the creators of power system analysis computer programs to thoroughly understand the application of basic analytical solution methods that apply. It is also
important for those concerned with assembling and preparing data for input to a power system analysis computer program and those interpreting and applying results generated by such
a program to understand the application of analytical solution methods.
This chapter identifies and documents the basic analytical solution methods that are valid for
determining the voltage and current relationships that exist during various power system network events and operating conditions. These basic analytical solution methods are demonstrated in cases where they are not self-evident. Finally, critical restraints that must be
respected to avoid serious error in applying analytical solution methods will be discussed.
Whether a power system analysis problem is to be solved directly or by a computer program,
proper application of sound analytical solution methods is essential for three reasons. First,
accuracy of the solution to each individual problem being considered will be directly
affected. Second, and perhaps the most important because of the significant expense
involved, accuracy of the solution determines the validity and effectiveness of any remedial
measures suggested. Finally, extension of erroneous results to related problems or to what
appears to be a trivial modification of the original problem, possibly in combination with
other misapplied or misunderstood techniques, can lead to a compounding of initial error and
a progression of incorrect conclusions.
The most common causes of errors in circuit analysis work are the following:
a)
Failure to use a valid analytical procedure because the analyst is unaware of its
existence or applicability
b)
Careless or improper use of “cookbook” methods that have neither a factual basis,
nor support in the technical literature, nor a valid place in the electrical engineering
discipline
c)
Improper use of a valid solution method due to application beyond limiting boundary
restraints or in combination with an inaccurate simplifying assumption
Many situations occur in industrial and commercial power systems that illustrate some or all
of these common causes of error, as well as the resulting evils. Any problem investigated as a
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part of the general types of power system analysis studies covered in other sections of this
recommended practice and described as follows would qualify.
—
—
—
—
—
—
—
—
—
Short-circuit studies
Load analysis studies
Load flow studies
Stability studies
Motor-starting studies
Harmonic studies
Reliability studies
Ground mat studies
Switching transient studies
3.2 Fundamentals
The following list identifies the more important analytical solution methods that are either
available as, or are the basis for, valid techniques in solving power system network circuit
problems:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
Linearity
Superposition
Thevenin and Norton equivalent circuits
Sinusoidal forcing function
Phasor representation
Fourier representation
Laplace transform
Single-phase equivalent circuit
Symmetrical component analysis
Per unit method
Rigorous treatment of these analytical techniques is available in several circuit analysis texts
(Beeman [B1]1, Close [B3], Hayt and Kemmerly [B7], Stevenson, Jr. [B14], Wagner and
Evans [B15], Weedy [B16]) and is beyond the scope of this discussion. In the following subclauses of this chapter, however, a brief qualitative explanation of each principle is presented,
along with a review of major benefits and restraints associated with the use of each principle.
3.2.1 Linearity
Probably the simplest concept of all, linearity is also one of the most important because of its
influence on the other principles. Linearity is best understood by examination of Figure 3-1.
The simplified network represented by the single-impedance element Z in Figure 3-1(a) is
linear for the chosen excitation and response functions, if a plot of response magnitude
1The
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numbers in brackets correspond to those of the bibliography in 3.3.
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ANALYTICAL PROCEDURES
(a)
(b)
Figure 3-1—Linearity
(current) versus source excitation magnitude (voltage) is a straight line. This is the situation
shown for case A in Figure 3-1(b). When linearity exists, the plot applies either to the steadystate value of the excitation and response functions or to the instantaneous value of the
functions at a specific time.
When linear dc circuits are involved, the current doubles if the voltage is doubled. The same
holds for linear ac circuits if the frequency of the driving voltage is held constant. In a similar
manner, it is possible to predict the response of a constant impedance circuit (that is, constant
R, L, and C elements) to any magnitude of dc source excitation or fixed frequency sinusoidal
excitation based on the known response at any other level of excitation. For the chosen
excitation function of voltage and the chosen response function of current, both dotted curves
B and C are examples of the response characteristic of a nonlinear element.
With the circuit element represented by any of the response curves shown in Figure 3-1
(including the linear element depicted by curve A), the circuit will, in general, become nonlinear for a different response function—for example, power. If, for example, the element was
a constant resistance (which would have a linear voltage-current relationship), the power
dissipated would increase by a factor of 4 if voltage were doubled (P = I 2R).
An important limitation of linearity, therefore, is that it applies only to responses that are
linear for the circuit conditions described (that is, a constant impedance circuit will yield a
current that is linear with voltage). This restraint must be recognized in addition to the
previously mentioned limitations of constant source excitation frequency for ac circuits and
constant circuit element impedances for ac or dc circuits. Excitation sources, if not
independent, must be linearly dependent. This restraint forces a source to behave just as
would a linear response (which, by definition, is also linearly dependent).
3.2.2 Superposition
This very powerful principle is a direct consequence of linearity and can be stated as follows:
In any linear network containing several dc or fixed frequency ac excitation sources
(voltages), the total response (current) can be calculated by algebraically adding all the
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individual responses caused by each independent source acting alone, i.e., all other sources
inactivated (voltage sources shorted by their internal impedances, current sources opened).
An example that illustrates this principle is shown in Figure 3-2. The equation written is for
the sum of the currents from each individual source V1 and V2. Although Figure 3-2 also
illustrates a way this principle might actually be used, more often its main application is in
support of other calculating methods. The only restraint associated with superposition is that
the network should be linear. All limitations associated with linearity apply.
Figure 3-2—Superposition
The nonapplicability of superposition is why all but the very simplest nonlinear circuits are
almost impossible to analyze using hand calculations. Although most real circuit elements are
nonlinear to some extent, they can often be accurately represented by a linear approximation.
Solutions to network problems involving such elements can be readily obtained.
Problems involving complex networks having substantially nonlinear elements can
practically be solved only through the use of certain simplification procedures, or through the
adjustment of calculated results to correct for nonlinearity. But both of these approaches can
potentially lead to significant inaccuracy. Tiresome iterative calculations performed in an
instant by the digital computer make more accurate solutions possible when the nonlinear
circuit elements can be described mathematically.
3.2.3 Thevenin and Norton equivalent circuits
The Thevenin equivalent circuit is a powerful analysis tool based on the fact that any active
linear network, however complex, can be represented by a single voltage source, VOC, equal
to the open-circuit voltage across any two terminals of interest, in series with the equivalent
impedance, ZEQ, of the network viewed from the same two terminals with all sources in the
network inactivated (voltage sources shorted by their internal impedances, current sources
opened). Validity of this representation requires only that the network be linear. Existence of
linearity is a necessary restraint. Application of the Thevenin equivalent circuit can be appreciated by referring to the simple circuit of Figure 3-2 and developing the Thevenin equivalent
for the network with the switch in the open position as illustrated in Figure 3-3.
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ANALYTICAL PROCEDURES
(a)
(b)
Figure 3-3—Thevenin equivalent
After connecting the 6 Ω load to the Thevenin equivalent network by closing the switch, the
solution for IL is the same as before, 1 A. Use of the simple Thevenin equivalent shown for
the entire left side of the network makes it easy to examine circuit response as the load
impedance value is varied.
The Thevenin equivalent circuit solution method is equally valid for complex impedance
circuits. It is the type of representation shown in Figure 3-3 that is the basis for making per
unit short-circuit calculations, although the actual values for the source voltage and branch
impedances would be substantially different from those used in this case. (The circuit
property of linearity would, incidentally, allow them to be scaled up or down.) The network
shown in Figure 3-3(a), with the 6 Ω resistance shorted and the other resistances visualized as
reactances, might well serve as an oversimplified representation of a power system about to
experience a bolted fault with the closing of the switch.
The V1 branch of the circuit would correspond to the utility supply while the V2 branch might
represent a large motor running unloaded, immediately adjacent to the fault bus, and highly
idealized so as to have no rotor flux leakage. For such a model, the 5 V source corresponds to
the pre-fault, air-gap voltage behind a stator leakage (subtransient) reactance of 3 Ω [B4]. In
a more realistic situation where rotor leakage is evident, a model that accurately describes the
V2 branch in detail before and after switch closing is much more difficult to develop, because
the air-gap voltage decreases (exponentially) with time and varies (linearly) with the steady-
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state rms magnitude of the motor stator current following application of the fault. The problem of accounting for motor internal behavior is avoided altogether by use of a Thevenin
equivalent. This permits the V2 branch to be represented by the apparent motor impedance
effective at the time following switch closure. In shunt with the equivalent impedance for the
remainder of the network, the Thevenin equivalent impedance, ZEQ, for the motor (at any
point in time of interest) is simply connected in series with the pre-fault open-circuit voltage,
VOC, to obtain the corresponding current response to switch closing.
The current response obtained in each branch of a network using a Thevenin equivalent
circuit solution represents the change of current in that branch. The actual current that flows
is the vector sum of currents before and after the particular switching event being considered.
See Figure 3-4.
(a)
(b)
Figure 3-4—Current flow of a Thevenin equivalent representation
In Figure 3-4(a), the current flowing in the V2 branch circuit is shown to be 1/3 A. A more
detailed representation of the Thevenin equivalent circuit previously examined in Figure 3-3
is shown in Figure 3-4(b). Here, the solution for the same current IV 2 is determined by
subtracting the current flowing in the V2 branch prior to closing the switch [5/6 A from
inspection of the circuit in Figure 3-3(a)] from the current IV 2 = 1/2 A, calculated to be
flowing in the Thevenin equivalent for this V2 branch.
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ANALYTICAL PROCEDURES
In the branch of the circuit defined by the switch itself, the change of current due to closing is
normally the response of interest. This means the solution to the Thevenin equivalent is sufficient. The resultant current in the other branches, however, cannot be determined by the solution to the Thevenin equivalent network alone.
In the case where the V2 branch represents a motor switched onto a bolted fault, the motor
contribution is the locked-rotor current minus the pre-fault current as illustrated in Figure 3-5
and not just the locked-rotor current as it is so often carelessly described.
Figure 3-5—Fault flow
As a rule, this effect is never as significant as the example suggests, even when the motor is
loaded prior to the fault; the load current is much smaller than the locked-rotor current and
almost 90° out of phase with it.
A Norton equivalent circuit, which can be developed directly from the Thevenin equivalent
circuit, consists of a current source of magnitude, VOC / ZEQ, to account for the voltage
sources in each separate branch of the network in parallel with the same equivalent
impedance for that branch, ZEQ. This representation of circuits can be useful in power system
analysis work if some of the sources are true current sources, as is often the case when
performing harmonic studies.
3.2.4 Sinusoidal forcing function
It is a most fortunate truth in nature that the excitation sources (driving voltage) for electrical
networks, in general, have a sinusoidal character and can be represented by a sine wave plot
of the type illustrated in Figure 3-6.
There are two important consequences of this circumstance. First, although the response
(current) for a complex R, L, C network represents the solution to at least one second-order
differential equation, the result will also be a sinusoid of the same frequency as the excitation
and different only in magnitude and phase angle. The relative character of the current with
respect to the voltage for simple R, L, and C circuits is also shown in Figure 3-6.
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Figure 3-6—Sinusoidal forcing function
The second important concept is that when the sine waveshape of current is forced to flow in
a general impedance network of R, L, and C elements, the voltage drop across each element
will always exhibit a sinusoidal shape of the same frequency as the source. The sinusoidal
character of all the circuit responses makes the application of the superposition technique to a
network with multiple sources surprisingly manageable. The necessary manipulation of the
sinusoidal terms is easily accomplished using the laws of vector algebra, which evolve from
the next technique to be reviewed.
The only restraint associated with the use of the sinusoidal forcing function concept is that
the circuit must be comprised of linear elements, that is, R, L, and C are constant as current or
voltage varies.
3.2.5 Phasor representation
Phasor representation allows any sinusoidal forcing function to be represented as a phasor in
a complex coordinate system as shown in Figure 3-7.
As indicated, the expression for the phasor representation of a sinusoid can assume any of the
following shorthand forms:
Exponential: E e jθ
Rectangular: E cos θ + jE sin θ
Polar: E ∠θ
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Figure 3-7—Phasor representation
For most calculations, it is more convenient to work in the frequency domain where any angular velocity associated with the phasor is ignored, which is equivalent to assuming the coordinate system rotates at a constant angular velocity of ω. The impedances of the network can
likewise be represented as phasors using the vectorial relationships shown. As illustrated, the
circuit responses (current) can be obtained through the simple vector algebraic manipulation of
the quantities involved. The need for solving complex differential equations to determine the
circuit responses is completely eliminated. The restraints that apply are as follows:
a)
b)
c)
Sources must all be sinusoidal
Frequency must remain constant
Circuit R, L, and C elements must remain constant (that is, linearity must exist)
3.2.6 Fourier representation
This powerful tool allows any nonsinusoidal periodic forcing function, of the type plotted in
Figure 3-8, to be represented as the sum of a dc component and a series (infinitely long, if
necessary) of ac sinusoidal forcing functions. The ac components have frequencies that are
integral multiples of the periodic function fundamental frequency. The general mathematical
form of the Fourier series is also shown in Figure 3-8.
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Figure 3-8—Fourier representation
The importance of the Fourier representation is immediately apparent. The response to the
original driving function can be determined by first solving for the response to each Fourier
series component forcing function and summing all the individual solutions to find the total
superposition. Since each component response solution is readily obtained, the most difficult
part of the problem becomes the determination of the component forcing function. The
individual harmonic voltages can be obtained, occasionally in combination with numerical
integration approximating techniques through several well-established mathematical
procedures. Detailed discussion of their use is better reserved for the many excellent texts
(Close [B3], Hayt and Kemmerly [B7]) that treat the subject.
There are several abstract mathematical conditions that must be satisfied to use a Fourier
representation. The only restraints of practical interest to the power systems analyst are that
the original driving function must be periodic (repeating) and the network must remain linear.
3.2.7 Laplace transform
In the solution of circuit transients by classical methods, the models of circuit elements are
represented with sets of differential equations. In addition, for a specific problem, a set of
initial conditions must be known in order to solve the differential equations for the unknown
quantity. An alternative technique for solving a transient problem is by the use of the Laplace
transform. The proper use of this technique eliminates the need for the solution of the
differential equations and simplifies all mathematical manipulations to elementary algebra.
It is helpful to keep in mind that the concept of mathematical transformations to simplify the
solution to a problem is not new. For example, the mathematical operations of multiplication
and division are transformed into the simpler operations of addition and subtraction by means
of the logarithm transform. Once the addition/subtraction is performed, the solution to the
problem is obtained by using the inverse transform, or antilog operation. The transformation
is designed to create a new “domain” where the mathematical manipulations are easier to
carry out. Once the unknown is found in the new domain, it can be inverse-transformed back
to the original domain.
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In circuit analysis, the Laplace transform is used to transform the set of differential equations
from the time domain (t) to a set of algebraic equations in the new domain called the complex
frequency domain or, alternatively, the s-domain. The Laplace transform of a function is
given by the expression
L{ f (t)} =
∞
∫0
f (t)e
– st
dt
(3-1)
where the symbol L { f ( t ) } is read as “the Laplace transform of f (t).”
The Laplace transform is also denoted by the notation F (s), that is,
F (s) = L{ f (t)}
(3-2)
This notation emphasizes that once the above integral has been evaluated, the resulting
expression is a function of s. Since the exponent of the e in Equation (3-1) must be
dimensionless, s must have the units of reciprocal time, hence the use of the alternate terms
“frequency domain” and “s-domain” to describe the realm of the transformed function.
It can be shown that the “transformation” (or, more briefly, “transform”) described by
Equation (3-1) has special mathematical properties. Given an original expression involving
both an unknown function (i.e., current, voltage, etc.), and operations on that function (i.e.,
derivatives, integrals, etc.), the s-domain expression that results when each term is
transformed according to Equation (3-1) can be manipulated by ordinary algebraic procedures to yield a solution for the unknown function. The solution for the unknown function in
the s-domain can then be transformed back to the time domain to produce the desired result.
These mathematical methods can be used to greatly simplify the solution of complex differential equations.
The solution of a system problem involving a linear expression can then be determined in
four simple steps, as follows:
a)
Formulate the differential time domain equations for the particular expression, which
may contain terms like
dv
Ri ( t ), C ------,
dt
t
∫ 0 i ( t ) d t , etc.
(3-3)
b)
Find the Laplace transform of the terms in the differential equation, including any
initial conditions, according to the definition of the Laplace transform or using
Laplace transform tables and equivalent circuit tables such as that shown in Table 3-1
and Table 3-2, respectively.
c)
Solve the transform for the unknown variable. The form of the s function should be
manipulated into a form similar to those available in tables of Laplace transform pairs.
d)
From a table, find the inverse Laplace transform of the unknown.
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Table 3-1—Laplace transform pairs
where
δ(t) is called the unit impulse function defined as
∞
δ(t) = 0 for t ≠ 0 and s –∞ δ ( t )dt = 1
u(t) is called the unit step function defined as
u(t) = 1 t ≥ 0
0 t <0
An alternate approach is to interchange steps a) and b) above. In this case, the network is first
transformed into the s-domain so that all the advantages of impedance and admittance operations and the use of network solution techniques such as mesh/nodal analysis and Thevenin/
Norton equivalent circuits become available.
Applying this transformation to all elements in the system under study, the result will be a
network consisting of s-domain equivalent elements. Then, usual circuit analysis will provide
the quantities of interest in the s-domain. Finally, from a table of inverse Laplace transforms,
the quantity of interest in the time domain can be obtained. In subsequent paragraphs, we will
discuss the Laplace transform and its application to the solution of transient problems.
3.2.7.1 Transient analysis by Laplace transforms
The transient analysis of a circuit using Laplace transforms is a straightforward application of
elementary algebra. Both the excitation sources and the impedance elements of the system
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Table 3-2—s-domain equivalent circuits
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CHAPTER 3
under study must be replaced with their s-domain equivalent circuits. Table 3-1 contains
Laplace transform pairs for commonly used excitation sources. The Laplace transform of the
driving voltages and/or currents in the studied circuit must be determined for use in the
s-domain equivalent circuit. Using Table 3-2, one of two possible s-domain equivalent circuits must be chosen for each impedance element in the circuit. Once the s-domain network
has been developed, the circuit equations can be solved algebraically.
The following examples will illustrate the procedure. They are simple and could very well be
solved with less sophistication.
3.2.7.1.1 RL and RC transients
The study of RL transients will begin with a series RL network with a constant (step function)
voltage source V(t) applied to the circuit through the closing of a switch. The network is illustrated in Figure 3-9.
Figure 3-9—RL network
According to Kirchoff’s voltage law, at t = 0+ the equation describing the circuit is
di
Vu ( t ) = Ri ( t ) + L ----dt
(3-4)
Converting the terms of the above equation into the s-domain, according to either Table 3-1
or Table 3-2, yields
V
–
---- = RI ( s ) + sLI ( s ) – LI ( 0 )
s
26
(3-5)
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IEEE
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ANALYTICAL PROCEDURES
Since there could be no current flowing prior to the closing of the switch, the last term on the
right can be ignored (initial conditions equal to zero). Solving for the current according to
3.2.7 step c), the result is
V
1
V 1
1
I ( s ) = ---- -------------------- = ---- --- – -----------L
R
s
R
R
s + --s s + ---
L
L
(3-6)
The transient response is readily obtained from Table 3-1 as
V
–t R ⁄ L
i ( t ) = ---- [ 1 – e
]
R
(3-7)
The current i(t) is shown graphically in Figure 3-10.
Figure 3-10—Current response (Equation [3-7])
The next circuit to be examined is the RC network depicted in Figure 3-11, already drawn in
the symbols of the Laplace transform. The equation describing the circuit is
–
I (s) V c(0 )
V
---- = RI ( s ) × --------- – --------------sC
s
s
Copyright © 1998 IEEE. All rights reserved.
(3-8)
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CHAPTER 3
Assuming there is no initial charge in the capacitor [Vc (0–) = 0] and solving for the current,
yields
1
V
I ( s ) = ---- ----------------------R s + 1 ⁄ RC
(3-9)
Again, using the appropriate inverse Laplace transform from Table 3-1, the transient response
in the time domain is
V –t R ⁄ C
i ( t ) = ---- e
R
(3-10)
Figure 3-11—RC network
The current response of the RC circuit is shown graphically in Figure 3-12.
If the quantity of interest was the voltage across the capacitor, then from Ohm’s law we
would have
V c ( s ) = I ( s )Z ( s )
(3-11)
Where I(s) is defined by Equation (3-9) and Z(s) is the capacitor impedance, namely
–1
Z ( s ) = -----sC
28
(3-12)
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Figure 3-12—Current response (Equation [3-10])
Since, in a series circuit, the current is common to all elements, we can substitute Equation
(3-12) into Equation (3-11). After solving for I(s), we substitute the result into Equation
(3-9). Rearranging the terms, we obtain
V
1
1
1
V c ( s ) = -------- ------------------------------- = V --- – ----------------------RC s ( s + 1 ⁄ RC )
s s + 1 ⁄ RC
(3-13)
From Table 3-1, the voltage across the capacitor in the time domain is
V c(t ) = V (1 – e
– t ⁄ RC
)
(3-14)
The capacitor voltage described by Equation (3-14) is shown graphically in Figure 3-13.
3.2.7.1.2 Circuits involving all three circuit elements
In cases involving more advanced circuits, the next step is the combination of all three
components, R, L, and C, in a common network. This approach is considered in Chapter 11 of
this book, where additional circuits with higher degrees of difficulty are investigated using
the same basics presented here.
3.2.8 Single-phase equivalent circuit
The single-phase equivalent circuit is a powerful tool for simplifying the analysis of balanced
three-phase circuits, yet its restraints are probably most often disregarded. Its application is
best understood by examining a three-phase diagram of a simple system and its single-phase
equivalent, as shown in Figure 3-14. Also illustrated is the popular single-line diagram
representation commonly used to describe the same three-phase system on engineering
drawings.
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CHAPTER 3
Figure 3-13—Voltage response
If a three-phase system has a perfectly balanced symmetrical source excitation (voltage) and
load, as well as equal series and shunt system and line impedances connected to all three
phases [see Figure 3-14(a)], imagine a conductor (shown as a dotted line) carrying no current
connected between the effective neutrals of the load and the source. Under these conditions,
the system can be accurately described by either Figure 3-14(b) or Figure 3-14(c).
The single-phase equivalent circuit is particularly useful since the solution to the classical
loop equations is much easier to obtain than for the more complicated three-phase network.
To determine the complete solution, it is only necessary to realize that the other two phases
will have responses that are shifted by 120° and 240° but are otherwise identical to the
reference phase.
Anything that upsets the balance of the network renders the model invalid. A subtle way this
might occur is illustrated in Figure 3-15.
If the switching devices operate independently in each of the three poles, and for some reason
the device in phase A becomes opened, the balance or symmetry of the circuit is destroyed.
Neither the single-phase equivalent nor the single-line diagram representation is valid. Even
though the single-phase and the single-line diagram representations would imply that the load
has been disconnected, it continues to be energized by single-phase power. This can cause
serious damage to motors and result in unacceptable operation of certain load apparatus.
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ANALYTICAL PROCEDURES
A
C
B
(a) Three-phase diagram
(b) Single-phase equivalent
(c) Single-line impedance diagram
Figure 3-14—A balanced three-phase system with load connected prior to fault
More importantly, if only one switching device operates in response to a fault condition in the
same phase, as depicted at location X, the system sources would continue to supply fault
current from the other unopened phases through the impedance of the load. The throttling
effect of the normally substantial load impedance, possibly in combination with additional
arc impedance, can reduce the level of the current to a point where detection may not occur in
phases B and C. Needless to say, substantial damage can result before the fault finally burns
enough to involve the other phases directly and accomplish complete interruption. Meanwhile, both of the single-line representations fail to recognize the problem, and in fact, suggest that the condition has been safely disconnected. Therefore, the restraints of this
calculating aid are as follows:
a)
Symmetry of the electrical system, including all switching devices and applied load.
b)
Any of the other previously described restraints that apply to the analytical technique
being used in combination with the single-phase equivalent.
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CHAPTER 3
(a) Three-phase diagram
(b) Single-phase equivalent
(c) Single-line impedance diagram
Figure 3-15—A balanced three-phase system with load connected
after application of phase A line-ground fault
3.2.9 Symmetrical component analysis
This approach comes to the analyst’s rescue when he or she is confronted with an unbalance,
the most common circuit condition that invalidates the single-phase equivalent circuit solution method. The symmetrical component analysis allows the response to any unbalanced
condition in a three-phase power system to be investigated and correctly synthesized by the
sum of the responses to as many as three separate balanced system conditions.
The application of an unbalanced set of voltage phasors, such as those displayed in Figure
3-16, to a balanced downstream load is the sum of the responses of the balanced components
that vectorially add to form the original unbalanced set.
Similar conclusions apply when the voltages are balanced, but the connected phase
impedances and/or loads and the line currents are unbalanced. Here, the unbalanced current
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ANALYTICAL PROCEDURES
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Figure 3-16—Symmetrical component analysis
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CHAPTER 3
phasors are the sum of up to three balanced sets that flow through the balanced system
impedances on either or both sides of the unbalance, producing voltage drops that satisfy the
needs of the applied voltages and the boundary conditions at the point of unbalance.
The mathematical expression for three unbalanced phasors as a function of the balanced
phasor components is as follows:
A = A0 + A1 + A2
B = B0 + B1 + B2
C = C0 + C1 + C2
The positive (1), negative (2), and zero (0) sequence vector components of any phase always
have the angular relationship with respect to one another as described by the vector diagram
and defined by the identities of Figure 3-16.
The operator a causes a counterclockwise rotation through an angle of 120° and is defined as
a = 1 ∠120° = – 0.5 + j0.866
These are assigned a counterclockwise direction of rotation in the time domain as illustrated.
In the space domain, the negative sequence phasors will produce exactly the same results as a
set of equal magnitude phasors that are displaced from one another by 120° and that rotate
clockwise with time.
The proof that a set of N unbalanced vectors can be completely represented by N sets of
balanced vectors is seldom presented in texts dealing with the subject of symmetrical
components. The derivation of symmetrical component analysis is most meaningful when
applied to three-phase systems, that is, systems where N = 3. First, it is postulated that it
might be possible to describe three arbitrary but defined vectors, A, B, and C, by summing
their respective symmetrical components (as previously shown). Using the vector identities
of Figure 3-16, the set of equations for the unbalanced vectors, A, B, and C, can be simplified
into the linear set of three equations involving three unknowns as follows:
A = A0 + A1 + A2
2
B = A0 + a A1 + a A2
C = A0 + a A1 + a A2
Rearranging, as shown in the following three equations, produces three independent equations for three unknowns (A0, A1, and A2), which is everything required to uniquely and com-
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ANALYTICAL PROCEDURES
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Std 399-1997
pletely describe A0, A1, and A2, and therefore substantiate their existence. The corresponding
B and C phase components are defined by the relationships shown in Figure 3-16.
1
A 0 = --- ( A + B + C )
3
1
2
A 1 = --- ( A + aB + a C )
3
1
2
A 2 = --- ( A + a B + aC )
3
The merit of the symmetrical component analysis is that a relatively complicated and often
unwieldy problem can be solved by simply vectorially summing the solution to no more than
three balanced, independent networks. The three independent networks are referred to as the
positive, negative, and zero sequence networks and are illustrated in Figure 3-17 for the system shown in Figure 3-15 prior to application of the line to ground fault. It should be noted in
these diagrams that the system voltage source (EL-N) and the load pre-fault open-circuit voltage (VD) are both balanced, positive sequence voltages, and they appear only in the positive
sequence network. Also, the impedance elements the load (RD, XD) and the remainder of the
system in each network are, as discussed in 3.2.3, intended to represent the Thevenin
equivalent impedances at the point in time of interest after the (unbalanced) event.
Typically, the three impedance networks are symbolically represented in shorthand fashion
by an empty block diagram for the phase most definitive of the condition being studied up to
the unbalance or other point of interest, as shown in Figure 3-18. Here, the final interconnections of the networks are shown that satisfy the necessary boundary conditions describing
the system at the point of concern in this case, the line-to-ground fault on phase A. In all
cases, the analyst can fill the block diagrams with the proper sources and impedances, including loads, in each sequence network, properly connect any fault or other impedance involved
with the imbalance, and solve the single-phase loop equations. The three current or voltage
sequence responses are produced in each of the respective networks, which then add vectorially to produce the resultant responses in the phase represented. The other phase responses
then can be obtained by adding vectorially the individual sequence solutions shifted by the
appropriate multiple of 120°.
One curious and often confounding feature of this solution procedure is that the phase in the
system that usually provides the best, and sometimes the only, approach to the solution for an
unbalance is the one least actively involved in the event. The unbalance illustrated in Figure
3-16 is one such example where the solution is obtained through an analysis of the
nonconducting phase A. A double line-to-ground fault is another example where examination
of the unfaulted phase gives the most direct access to the network solution. With practice
evaluating the more commonly encountered unbalances, the analyst can quickly discern the
appropriate network interconnection by inspection of the boundary conditions for the event.
Several reference texts (Beeman [B1], Electrical Transmission and Distribution Reference
Book [B4], and Wagner and Evans [B15]) provide convenient diagrams, tables, and other
information showing the network interconnections that must be used to solve for the
responses to many system unbalances, as well as certain balanced conditions.
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CHAPTER 3
N
Neutral
Load
X S+
E L-N
X L+
R L+
R D+
X D+
VD
F
(a) Positive sequence network (+)
N
Neutral
Load
X S–
X L–
R L–
R D–
X D–
F
(b) Negative sequence network (–)
N
Neutral
Load
X S0
R L0
X L0
R D0
X D0
F
(c) Zero sequence network (0)
Figure 3-17—Symmetrical component sequence networks for system
in Figure 3-15(a) prior to line-ground fault (F) connection
The symmetrical component analysis always involves the use of superposition as well as
most of the other procedures previously discussed. The restraints that apply to these other
procedures, therefore, must also govern the use of the symmetrical component analysis. In
addition, due to mutual phase winding coupling and other effects, the impedance displayed
by electrical machines will be different when excited by the different sequence sources.
Hence, the per phase impedance of circuit elements within the positive, negative, and zero
sequence networks will, in general, be different from each other. Currents flowing in the zero
sequence network are in phase and do not sum to zero as do the positive and negative
sequence currents. Zero sequence currents must therefore flow through the ground circuit and
are influenced by any impedance in this circuit path. When harmonic excitation sources are
present (requiring the use of the Fourier representation), special care must be exercised in
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IEEE
Std 399-1997
ANALYTICAL PROCEDURES
+
N
E A+
F
–
EA
N
–
F
0
N
E A0
F
Boundary Conditions
EA = 0 (= EA+ + EA– + EA0)
IB
IC
=0
=0
(IA+ = IA– = IA0 = 1/3 IA)
Figure 3-18—Interconnection of Figure 3-17 sequence networks to
satisfy Phase A boundary conditions after line-ground fault application
treating the sequence networks. Starting with the fundamental, the harmonic terms
progressively shift from the positive to the negative to the zero sequence networks, and then
the process repeats.
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 3
3.2.10 Per-unit method
The per-unit method of calculation and its close companion, the percentage method, are well
documented in Beeman [B1], Stevenson, Jr. [B14], and Weedy [B16] and are generally wellknown. As a result, they will only be mentioned in passing here.
Fundamentally, the per-unit method and the percentage method amount to a shorthand
calculating procedure for which all equivalent system and circuit impedances are converted to
a common kVA and kVbase. This permits the ready combination of circuit elements in a
network where different system voltages are present without the need to convert impedances
each time responses are to be determined at a different voltage level.
Associated with each impedance element and its kVA base is a line-to-line kV base (usually
the nominal line voltage at which the element is connected to the system), along with the
resulting base impedance and base current related by the following expressions:
a)
Three-phase network
kVA base
I base = ----------------------3 kV base
2
kV base
Z base = ------------------- × 1000
kVA base
kVA base =
b)
3 kV base I base
Single-phase network
kVA base
I base = -----------------kV base
2
Z base
kV base
= ------------------- × 1000
kVA base
kVA base = kV base I base
NOTE—Each of the above relationships involving kVAbase are sometimes expressed in terms of the
corresponding MVAbase, where MVAbase = kVAbase /1000. This results in the convenient expression of
kV 2base /MVAbase for Zbase for both three-phase and single-phase networks.
To illustrate the use of the per-unit method, consider the example in Figure 3-19. The first step
in a per-unit calculation is the arbitrary selection of the system base kVA. Second, the choice of
base kV must be made at one voltage level from which the base value at the other voltage level
is dictated by the turns ratios of the transformers in the network. For the circuit in Figure
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Copyright © 1998 IEEE. All rights reserved.
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ANALYTICAL PROCEDURES
(a) Classical ohmic representation
(b) Per-unit representation
Figure 3-19—Per-unit method at different voltage levels
3-19(a), the base values selected are 5000 kVA and 1.0 kV on the primary of the 10:1 transformer. The resultant base voltage on the secondary of the transformer is 1.0 kV/10 or 0.1 kV.
The base impedance can next be calculated (both on the primary and secondary levels) and the
per-unit values of the primary and secondary impedances can be determined as shown.
Once the per-unit impedance and excitation source values have been determined, the circuit
can be simplified as shown in Figure 3-19(b). A key advantage of the per-unit method, the
transformer turns ratio becomes 1:1, thereby effectively removing it from the calculations as
Copyright © 1998 IEEE. All rights reserved.
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IEEE
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CHAPTER 3
(c) Simplified per-unit representation
Figure 3-19—Per-unit method at different voltage levels (Continued )
modeled in Figure 3-19(c). Working with the circuit in Figure 3-19(c), the primary current
(Ip) and secondary current (Is) are the same and can be easily calculated by simply applying
Ohm’s law. The last step in the procedure is to determine the actual current in amperes by
multiplying each per-unit value by the base current on the primary and secondary levels.
Although the per-unit values calculated for Ip and Is are equal, the base currents are different,
and therefore the solutions expressed in amperes are different. Since the per-unit method of
calculation is based on the existence of linearity and is always used in combination with one
or more of the other principles, it is necessary to observe all of the associated restraints
discussed earlier as they apply.
3.3 Bibliography
Additional information may be found in the following sources:
[B1] Beeman, D. (Ed.), Industrial Power Systems Handbook, New York: McGraw-Hill, 1955,
Chapter 2.
[B2] Blackburn, J. L., Symmetrical Components for Power Systems Engineering, Marcel
Dekker, Inc., NYC, 1993.
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ANALYTICAL PROCEDURES
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Std 399-1997
[B3] Close, C. M., The Analysis of Linear Circuits, New York: Harcourt, Brace and World,
Inc., 1969.
[B4] Electrical Transmission and Distribution Reference Book, East Pittsburgh, PA: Westinghouse Electric Corporation, 1964, Chapters 2 and 6.
[B5] Fitzgerald, A. E., and Kingsley, Jr., Charles, Electric Machinery, New York: McGrawHill, 1961.
[B6] Goldman, S., Laplace Transform Theory and Electrical Transients, New York: Dover
Publications, 1966.
[B7] Hayt, Jr., W. H., and Kemmerly, J. E., Engineering Circuit Analysis, New York:
McGraw-Hill, 1962.
[B8] IEEE Std 142-1991, IEEE Recommended Practice for Electric Power Distribution for
Industrial Plants (IEEE Red Book), Chapter 4.2
[B9] Miller, R., Algebraic Transient Analysis, San Francisco: Reinhart Press, 1971.
[B10] Mohamed, E. El-Hawary, Electrical Power Systems, IEEE Press, 1995.
[B11] Nilsson, J. W., Electric Circuits, Reading: Addison-Wesley Publishing Co., 1984.
[B12] Nilsson, J. W., Electric Circuits, Reading: Addison-Wesley Publishing Co., 1987.
[B13] Puchstein, A. F., and Lloyd, T. C., Alternating Current Machines, New York: John
Wiley and Sons, Inc., 1947.
[B14] Stevenson, Jr., W. D., Elements of Power System Analysis, New York: McGraw-Hill,
1982.
[B15] Wagner, C. F., and Evans, R. D., Symmetrical Components, New York: McGraw-Hill,
1933.
[B16] Weedy, B. M., Electric Power Systems, New York: John Wiley and Sons, Inc., 1972,
Chapter 2.
2IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
Copyright © 1998 IEEE. All rights reserved.
41
Chapter 4
System modeling
4.1 Introduction
This chapter addresses the following questions:
a)
b)
c)
How can each component or each group of components of an industrial or commercial
power system be represented so that an analysis of the system performance can be made?
Which of the several possible representations, or models, of the system is best suited
to meet the objectives of a given study?
What mathematical expressions will describe the characteristics of each system element so that it can be quantified and programmed for computer input?
There is an infinite number of possible power system configurations and a large variety of
study types. Consequently, standards cannot be established to dictate specific models for all
specific circumstances. This text will therefore serve as a guide to help the reader make
judicious trade-offs in selecting models for a study.
Derivations and proofs of mathematical expressions will not be given. References should be
used for such purposes. However, several fundamental relationships of electrical and
mechanical quantities will be mentioned in the text. This should save the reader the time
needed to locate them in textbooks or handbooks. It should also help refresh the memories of
those who have not been exposed to power studies or academic activities for a long time.
The material is intended to be as basic and simple as permitted by the subject. The emphasis
is on the correlation of real-life systems with the abstraction of mathematics in order to
facilitate computer simulations of these systems.
4.2 Modeling
Scale modeling of power systems as a means of analyzing their performance is impractical.
However, scale models of certain mechanical components of power systems are used to
evaluate their characteristics. This is often the case with hydraulic sections of hydroelectric
plants, such as turbine runners, spiral cases, gates, draft tubes, etc. Nonetheless, much
expertise is required to establish scaling and normalization factors, to construct the model, to
gather meaningful data by measurement, and to interpret and extrapolate the results.
Digital computers can be programmed to solve large numbers of simultaneous equations
quickly and inexpensively and handle the algebra of large matrices. This makes them
particularly well suited for applications in power system analysis. An immense variety of
programs have been written to study an ever-increasing number of problems in the electrical
field. These programs are usually set up to receive the problem information in the form of
numbers rather than analog settings, thus forcing the power system analyst to model the
system quantitatively. The data input portions of these programs are typically written in
general, nonspecific terms. This exposes the analyst to a choice of program features and
alternatives that require decisions to be made every step of the way. Finally, the programs are
often structured to handle extensive power systems (3000 bus programs are not uncommon).
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 4
4.3 Review of basics
Power network elements may be classified in two categories, passive elements and active
elements.
4.3.1 Passive elements
The passive elements comprise such components as transmission lines, transformers,
reactors, and capacitors. They will, in general, be regarded as linear and will be modeled by
one or more of the following electrical quantities:
Name
Symbol
Unit
resistance
R
ohm
inductance
L
henry
capacitance
C
farad
The voltage across and the current through the element will be governed by these
relationships:
v = Ri
v
i = --R
di
v = L ----dt
1
i = --L
1
v = ---- ∫ i d t
C
dv
i = C -----dt
(4-1)
∫ vdt
(4-2)
(4-3)
where the lowercase letters represent the time-varying functions of voltage and current. In dc
circuits under steady-state conditions, these equations will reduce to
44
V = RI
V
I = ---R
V = 0
di
since ---- = 0
dt
I = 0
dv
since ----- = 0
dt
(4-4)
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SYSTEM MODELING
In ac circuits with sinusoidal wave shapes, the equations become
V = RI
V
I = ---R
V = jX L I
where X L = 2πfL is the inductive reactance
(4-5)
V = – jX C I
1
where X C = ------------- is the capacitive reactance
2πfC
(4-6)
The capital letters for voltages and currents represent their rms values, f is the frequency in
hertz, and j is the 90° operator (= – 1 ). Inverting and combining these elements in series or
parallel will define the set of quantities in Table 4-1.
Table 4-1—Equation references for conductance, susceptance,
impedance and admittance
Name
Symbol
Unit
Defining expression
conductance
G
S (siemens)
1/R
inductive susceptance
B
S (siemens)
1/X
capacitive susceptance
B
S (siemens)
1/X
impedance
Z
Ω (ohms)
(R + jX)
admittance
Y
S (siemens)
(G + jB)
It should be noted here that it is customary in ac power circuits to use the R, X, and Z
quantities for the series (line) elements and the G, B, and Y quantities for the shunt (line to
neutral) elements. Note also that Z and Y are complex quantities that can be expressed in the
rectangular form above or the polar form Z = Z ∠θ or Y = Y ∠θ . Most computer programs accept the Z and Y values in the rectangular form.
A final remark concerns the sign ahead of the reactances and susceptances. The four diagrams
in Figure 4-1 are self-explanatory. The wise analyst will verify the program instructions to
make sure that the computer will interpret the input data properly.
Copyright © 1998 IEEE. All rights reserved.
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IEEE
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CHAPTER 4
Figure 4-1—Equivalent circuit diagrams showing sign convention
4.3.2 Active elements
The active elements of a power system comprise such components as motors, generators,
synchronous condensers, and other loads such as furnaces, adjustable speed drives, etc. The
active elements will be regarded as nonlinear, although some of the components may behave
linearly under certain circumstances.
One or more of the parameters of a model of an active element will vary as a function of time,
phase angle, frequency, speed, etc.
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SYSTEM MODELING
The four expressions for power quantities given in Table 4-2 can be used to model nonlinear
elements. Given any two of the four values, the remaining two can be defined. Power can also
be expressed in polar form: S = S ∠θ which yields these relationships: PF = cos θ, P = S
cos θ, and Q = S sin θ. Note that the magnitude of the complex power must be used in the
previous equations and in the relations of table below.
Note that the signs of P or Q may be positive or negative. By convention, the positive sign of
Q is used for inductive loads; that is, the current will lag the voltage applied to a load that
consumes vars. In this sense, it is said that capacitors generate vars (current leads the voltage)
and that induction motors absorb vars.
Table 4-2—Four defining expressions for power quantities
Name
Symbol
Unit
complex power
S
voltampere (VA)
active power
P
watt (W)
reactive power
Q
var
power factor
PF
per unit (pu)
Defining expression
S = P + jQ
2
2
2
2
P =
S –Q
Q =
S –P
PF = P/S
By convention also, the sign of P is positive for a load that consumes energy or a source that
generates energy. Thus a load with a negative sign for P could be used to represent a
generator, and vice versa for a motor. It should also be noted that the expressions above are
appropriate for fundamental frequency applications only; modifications are required for
application in harmonic analyses.
Some of the power system components can best be expressed in terms of current or voltage.
For instance, an infinite bus may be specified by a voltage source of constant magnitude and
angle, and a particular load may be described as a constant current element. The current and
voltage quantities may be complex numbers, in which case they have to be described in terms
of a reference vector that may be a voltage or current quantity.
This introduces the phase angle concept:
I = I ∠θ = I x + jI y = I ( cos θ + j sin θ )
V = V ∠θ = V x + jV y = V ( cos θ + j sin θ )
where the x axis of the coordinate system is taken as the reference shown in Figure 4-2.
Copyright © 1998 IEEE. All rights reserved.
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IEEE
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CHAPTER 4
Figure 4-2—Vector diagram
So far, the review has included most of the quantities used in the type of studies that require
steady-state network solutions at a single point in time. It should be noted that multiple
applications, each appropriate for a different point in time, of these steady-state concepts may
be used to analyze certain system conditions that are time-varying (e.g., fault calculations).
Some studies, such as motor-starting and transient stability, require that a complete period of
time be covered to assess the effects of a disturbance on system performance. This requirement
that a certain period of time be considered introduces the need for mechanical quantities.
The fundamental quantities of mechanics are space, matter, and time. Other mechanical
quantities are derived from these three. Some of the derived quantities officially recognized
for electrical engineering work have been tabulated in Table 4-3, which shows the MKS
system of units and the defining equations.
Table 4-3—Fundamental equations for translation and rotation
Name
Symbol
Unit
Defining expression
length
l
meter (m)
–
mass
m
kilogram (kg)
–
time
t
second (s)
–
v
m/s
Fundamental
Translation
velocity
48
dl
v = ----dt
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SYSTEM MODELING
Table 4-3—Fundamental equations for translation and rotation (Continued)
Name
Symbol
Unit
Defining expression
acceleration
a
m/s2
force
F
newton (N)
F = ma
work
W
joule (J)
W = ∫ Fdl
power
P
watt (W)
dW
P = -------dt
momentum
M'
N/s
M ' = mv
2
d l
a = ------2dt
Rotation
radius
r
m
–
circular arc
s
m
–
moment of inertia
I
kg⋅m2
I = ∫ r dm
angle
θ
radian (rad)
θ = s⁄r
angular velocity
ω
rad/s
dθ
ω = -----dt
angular acceleration
α
rad/s2
d θ
α = --------2
dt
torque
T
N⋅m
T = rF
work
W
J
W = ∫ T dθ
power
P
W
P = Tω
angular momentum
M
J·s/rad
M = Iω
2
2
4.4 Power network solution
Before dealing with the detailed models of power system components, it is important to
review what constitutes the solution of a network. From an electric circuits point-of-view, it
can be said that a network is solved if all the bus voltages and the relative phase angles
between these voltages are known. In this case, solved means that all other network voltages,
currents, and power flows can be calculated from the known bus voltages using simple
algebraic equations. This, of course, requires that the impedances between the buses be
known. Note, however, that the solution for the set of bus voltages can be a formidable problem, and the methods used to obtain this solution vary depending on the type of analysis.
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 4
Consider for instance Figure 4-3, which shows a small simplified section of the typical plant
from the one-line diagram (Figure 1-1) in Chapter 1.
Figure 4-3—Simplified one-line diagram
To demonstrate typical calculations, assume that the voltages at buses 02:69-02, 04:MILL-2,
and 16:T9 PRI (called A, B, and C, respectively, to simplify notation) are known. The
impedances of the transformer T-2 and the cable from 04: MILL-2 (B) to 16: T9 PRI (C) are
given on Figure 4-3.
These data are summarized in Figure 4-4 and are listed as follows:
V 02:69 – 2 = V A = 69.00 kV ∠0 °
V 04:MILL – 2 = V B = 13.60 kV ∠– 1.6 °
V 16:T9 PRI = V C = 13.57 kV ∠– 1.82 °
Z T – 2 = Z AB = 1.49 + j25.35 Ω (at 69 kV) = 0.06 + j1.01 Ω (at 13.8 kV)
Z CABLE = Z BC = 0.05 + j0.04 Ω
50
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Std 399-1997
Figure 4-4—Impedance diagram with numerical data
The current from bus A to bus B can be found by
I AB = ( V A – V B ) ⁄ ( Z AB 3 )
69
69 000 ∠0 ° – ---------- 13 600 ∠– 1.6 °
13.8
= -------------------------------------------------------------------------------3 ( 1.49 + j25.35 )
= 49.07 ∠– 25.03 ° A (at 69 kV)
(4-7)
By definition, the power flow from bus A to bus B, as measured at bus A, is
S AB =
3V A × Î AB (the caret on Î means conjugate)
(4-8)
= ( 3 × 69 ∠0 ° ) ( 49.07 ∠+25.03 ° )
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CHAPTER 4
= 5864.4 ∠25.03 °kVA
= ( P AB + jQ AB )
= 5313.68 + j2481.19 kVA
The current IAB on the 13.8 kV side is equal to that at the 69 kV side multiplied by the transformer turns ratio:
69
I AB = 49.07 ∠– 25.03° ----------
13.8
= 245.37 ∠– 25.03 °(at 13.8 kV)
By definition, the power flow from bus B to bus A, as measured at bus B, is
S BA =
3V B × ( Î BA )
= ( 3 × 13.60 ∠– 1.6 ° ) × ( – 245.37 ∠25.03 ° )
= – 5779.9 ∠23.43° kVA
= P BA + jQ BA
= – 5303.34 – j2298.25 kVA
The transformer real and reactive power losses can now be found by adding the power from A
to B and that from B to A:
S losses = S AB + S BA
= ( 5313.68 + j2481.19 ) + ( – 5303.34 – j2298.25 )
= 10.34 + j182.95 kVA
The losses are 10.34 kW and 182.95 kvar between buses A and B. Figure 4-5 summarizes
these results. This example illustrates that once the bus voltages are known, the remaining
calculations are straightforward.
52
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Figure 4-5—Current and complex power flows
The problem of analyzing even a modest size system involves the determination of the bus
voltages. The loads are known but they are often a nonlinear function of the applied voltage.
In order to find the bus voltages, one has to resort to a cut-and-try iterative method. The
computer is an effective tool for this method because it can complete the set of calculations
shown above much faster than if done by hand.
4.5 Impedance diagram
Several things remain to be said about Figures 4-3 through 4-5:
a)
All three diagrams show a single-phase equivalent of the three-phase system. The
conditions for this equivalence to be true were covered in Chapter 3.
b)
Bus and line designators have been shown on all three diagrams. The impedance
diagram of Figure 4-4 is the rigorously correct way to represent graphically
resistances and reactances. However, it is felt that drawing the graphical symbols of
resistances, inductances, and capacitances is superfluous since the expressions for
impedances along side a straight line sufficiently describe the line elements. Rough
drawings, such as that in Figure 4-3, showing buses, lines, generators, and loads, can
be duplicated for multiple use as an impedance diagram and as a flow diagram. It
should be noted that these diagrams are working tools and as such do not require
standardization. However, the analyst should adopt a method suitable for keeping
track of masses of data, for even small system studies require and generate a large
amount of information.
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 4
c)
In power systems analysis, the term bus does not always have the meaning
understood by a plant electrician, for instance. The analyst calls a bus any point of the
system where voltages are calculated. The term is interchangeable with node.
Fictitious buses may be introduced on the network to obtain voltage solutions at
certain points of interest. An example of this may be a 150 mi transmission line
broken down in 5 sections of 30 mi (that is, a bus introduced every 1/5 of the length)
in order to avoid the complicated but exact model of the long line. Conversely, a
single bus (node) in the model may represent what the electrician would call several
buses. Examples might included “buses” connected with normally closed breakers or
very short lengths of cables.
d)
In the same vein, branches are used to represent types of different equipment
connecting buses (nodes). In general, a branch is any element between two nodes. For
example, the transformer data will be entered on a computer input document called
“branch data.” In many cases, the terms line and branch are used interchangeably.
4.6 Extent of the model
4.6.1 General
No rigid rules can be established on how much of a power system should be modeled for a
given study. System analysts have to exercise judgment and develop a feel for this as they
gain experience. In general, the objectives of the study should always be kept in focus to
avoid unnecessary effort that could be spent modeling certain aspects of the system that are
not pertinent to the analysis to be conducted.
4.6.2 Utility supplied systems
A large number of industrial and commercial establishments are supplied by stiff utility
systems. Stiffness is a relative function of the size of the plant load and local generation. If
the external power system or utility is large compared with that of the plant, disturbances
within the plant do not affect the voltage at the point of connection. In such a case, the utility
system is said to be an infinite system. The connection point will be an infinite bus.
This concept can be extended within the plant electrical distribution system when studies are
concerned with small areas electrically remote from the utility supply. Conversely, sections of
utility systems may require modeling in cases where this stiffness does not exist. It is,
therefore, important that a sound knowledge of the utility supply systems be acquired before
proceeding with the studies.
4.6.3 Isolated systems
Determining whether an isolated system should be modeled in full or in part is easier than with
a complex interconnected system. These systems are usually relatively small and, as such,
could be represented fully for most kinds of studies. The extra effort of gathering a set of data
for the entire system, even though a smaller section would suffice, will not be lost since the
additional data will likely be used in some future study. The nature of an isolated system is
such that a modification or a disturbance is more apt to be felt throughout the system.
54
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IEEE
Std 399-1997
Figure 4-6—Equivalent circuit of short conductor
4.7 Models of branch elements
4.7.1 Lines
Four parameters affect the performance of the conductors connecting a source to a load:
series resistance, series inductance, shunt capacitance, and shunt conductance. These
parameters affect both overhead lines and cables and the same calculation procedures are
used to determine parameter values in each case. For this reason, it is common to use the
word line to represent both overhead lines and cables. Note that certain assumptions that are
valid for widely spaced conductors used in overhead lines may not be valid for the much
closer conductor arrangements found in cables. Conductor length, however, affects overhead
lines and cables in the same way.
A short conductor can be modeled as in Figure 4-6. A line can be considered as many short
conductors placed in series to yield the model of Figure 4-7. The individual lengths of each
conductor could be made shorter thus increasing the number of these conductors for a given
length of line. Continuing this process to the limit defines the model called the distributedparameter line model. This model has been reduced to the equivalent circuit shown in
Figure 4-8, where the series branch is defined by
sinh γ l
Z ' = Z C --------------γl
(4-9)
and the shunt branches by
Y C tanh ( γ l ⁄ 2 )
Y'
----- = ------ --------------------------γl ⁄ 2
2
2
1 ( cosh γ l – 1 )
= ------ ----------------------------ZC
sin h γ l
Copyright © 1998 IEEE. All rights reserved.
(4-10)
55
IEEE
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CHAPTER 4
(a) Line with distributed constants
Z' =
Y' =
Zc
Yc
(b) Long line equivalent circuit
Figure 4-7—Equivalent line models
Two figures of merit appear in these equations:
ZC =
γ =
z ⁄ y , defined as the characteristic or surge impedance of the line
yz , defined as the propagation constant
(4-11)
(4-12)
Both ZC and γ are complex numbers. The propagation constant γ can be expressed in the
rectangular form:
γ = α + jβ
(4-13)
This defines
α
is the attenuation constant
β
is the phase constant (in radians)
The other variables are
l
r
x
g
56
is the total length of line,
is the conductor effective resistance in ohms per unit of length,
is the conductor series inductive reactance in ohms per unit of length
x = 2πfL ,
is the shunt conductance to neutral in siemens per unit of length,
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SYSTEM MODELING
is the shunt capacitive susceptance in siemens per unit of length
b = 2πfC ,
L
is the conductor total inductance in henrys per unit of length,
C
is the conductor shunt capacitance in farads per unit of length,
z = r + jx is the series impedance in ohms per unit of length,
y = g + jb is the shunt admittance to neutral in siemens per unit of length,
Z = zl
is the total series impedance of line in ohms,
Y = yl
is the total shunt admittance of line to neutral in siemens per unit of length,
sinh, cosh, tanh are hyperbolic functions.
b
These functions of complex numbers can be evaluated by using the following relationships:
sinh ( γl ) = sinh ( αl + jβl ) = sinh ( αl ) cos ( βl ) + j cosh ( αl ) sin ( βl )
(4-14)
cosh ( γl ) = cosh ( αl ) cos ( βl ) + jsinh ( α l ) sin ( β l )
(4-15)
sinh X
tanh X = ---------------cosh X
(4-16)
αl
– αl
e –e
sinh ( αl ) = ---------------------2
αl
(4-17)
– αl
e +e
cosh ( αl ) = ---------------------2
(4-18)
The surge impedance ZC is approximately 400 Ω for typical single circuit overhead power
lines, while typical cable circuits have a surge impedance of 30–40 Ω. The distributed
constants model is valid for short or long lines at power or communication frequencies.
4.7.1.1 Long lines
Power frequency overhead lines in excess of 150 mi should be represented by the distributed
constants model reduced to an equivalent π as shown in Figure 4-7 (b).
The shunt conductance g may be neglected because the dielectric is air (a good dielectric) and
the conductor spacing is large. However, if the corona losses are important, they may be
represented by g. Computer programs will readily accept the data for Z ' expressed in the
rectangular form, that is, the equivalent series resistance R ' and the equivalent series inductive
reactance X '. It should be noted that even though g is neglected (g = 0), a nonzero value of G '
will appear in the equivalent circuit of Figure 4-8 (b). The values of G '/2 and B '/2 (in
siemens) may have to be modified to MW or Mvar to suit the program input requirements.
(The computer treats them as constant impedance loads.) If the computer requires a line
charging Mvar, the value B ' and not B '/2 must be used to calculate
2
Mvar = 3 ( kV ln ) × B '
Copyright © 1998 IEEE. All rights reserved.
(4-19)
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CHAPTER 4
The program will internally assign half of that value to the bus at each end of the line.
The value of G'/2 can be modified to
G'
2
MW = 3 ( kV ln ) × -----2
(4-20)
and input as a constant impedance bus load.
The voltage (kV) is the line-to-ground voltage corresponding to the base voltage used in the
program input document.
4.7.1.2 Medium lines
In the range of approximately 50–150 mi (80–240 km), little accuracy is lost by simplifying
Equation (4-9) and Equation (4-10) to
Z' = Z
Y
Y'
----- = --2
2
sinh ( γl )/γl = 1
Thus, neglecting the products sinh ( γl )/γl and tanh ( γl/2 )/ ( γl/2 ) yields the model of Figure
4-8 (a), called the nominal π circuit. In this model, the shunt branches are purely capacitive
(no conductance).
The nominal π circuit can be thought of as being formed by the process described at the
beginning of 4.7.1, except that the unit length of conductor is increased (instead of
decreased), and the circuit is made symmetrical (bilaterally). This results in the constants
being lumped by an approximation process. The nominal T circuit is formed the same way,
except that all the shunt constants are lumped together compared with the π circuit where all
series constants are lumped together.
Use of the nominal T model [Figure 4-8 (b)] is not popular since it requires addition of a fictitious bus in the middle of the line. Entering data into the program for the nominal π circuit
follows the same requirements as for the long line model.
4.7.1.3 Short lines
For overhead lines shorter than 50 mi, neglecting the shunt capacitance in the models
presented earlier will not greatly affect the results of load flow, short-circuit, or stability
calculations. This yields the model of Figure 4-9.
58
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SYSTEM MODELING
(a) Nominal π
(b) Nominal T
Figure 4-8—Medium line equivalent circuits
Figure 4-9—Short line equivalent circuit
4.7.2 Cables
The overhead line models are equally applicable to cables. Though the resistances are
substantially the same, the relative values of reactances are vastly different. Table 4-4
compares two cases, one at 69 kV, the other at 13.8 kV. The cable inductive reactance is about
1/4 that of the line; but the capacitive reactance is 30–40 times that of the line.
This comparison suggests that, for fundamental frequency, the medium line model, the
nominal π, should be used for cables in the order of one mile in length (approximately 1/40 of
50 mi). The shorter the cable run, the better the accuracy when using this model.
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 4
Table 4-4—Comparison of overhead lines and cable conductors
Values in Ω/mile (Ω/km) for 500 kcmil Cu conductors
69 kV
Overhead linea
13.8 kV
Cableb
Overhead linec
Cabled
Resistance
0.134 (0.083)
0.134 (0.083)
0.134 (0.083)
0.134 (0.083)
Inductive
resistance
0.695 (0.432)
0.175 (0.109)
0.613 (0.381)
0.146 (0.091)
Capacitive
reactance
0.005 (0.008) × 106 0.162 (0.026) × 106 0.003 (0.005) × 106 0.142 (0.229) × 106
aOpen-wire equilateral conductor spacing of 8 ft
bThree-conductor oil-filled paper-insulated cable
cOpen-wire equilateral conductor spacing of 4 ft
dThree-conductor oil-filled paper-insulated cable
rated 69 kV
rated 15 kV
It is doubtful that any medium voltage system will have feeder lengths requiring
representation of the capacitive reactance.
4.7.3 Determination of constants
Electrical conductor characteristics are available from numerous sources and need not be
repeated here. A few general comments are appropriate.
4.7.3.1 Resistance
The effective resistance of the conductors should be used. The effective resistance takes into
account the conductor:
a)
b)
c)
d)
e)
f)
Material
Size
Shape
Temperature
Frequency
Environment
Copper and aluminum are the most used conductor materials for lines and cables. Soft
annealed chemically pure copper has 100% conductivity (IACS standards).
This is equivalent to 875.2 Ω for a 1-mile-long round wire weighing one pound at 20 °C. All
other materials can have their conductivities expressed as a percentage of the standard, a few
of which are listed in Table 4-5.
60
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SYSTEM MODELING
Table 4-5—Conductor data
Material
Conductivity
Application
Copper
soft
100
cable construction
soft, tinned
93.15 to 97.3
cable construction
hard, shape
98.4
bus bar
hard, round
97.0
overhead line conductors
61.0
cables, bars, tubes
5005-H19
53.5
overhead line conductors
6201-T81
52.5
overhead line conductors
Siemens-Martin
12.0
shield wire
high-strength
10.5
shield wire
9.4
shield wire
Aluminum
Aluminum alloys
Galvanized steel
extra-high-strength
The conductor resistance will vary with temperature according to the following formula:
R2 = R1 [ 1 + α ( t 1 – t 2 ) ]
(4-21)
where
R2
R1
is the resistance at temperature t2
is the resistance at temperature t1
Ω
is the temperature coefficient per degree at temperature t1
At 20 °C, the coefficient α per degree Celsius is as follows:
—
—
—
Copper: 0.00393
Aluminum: 0.00403
Galvanized steel:
Siemens-Martin: 0.0039
High-strength: 0.0035
Extra-high-strength: 0.0032
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It is not possible to predict the exact operating conductor temperature if the conductor current
is not known. The analyst has the choice of either estimating the conductor temperature or
assuming the worst case, which, in some studies, might be the maximum allowable
temperature of the cable. Other studies might require that the minimum conductor
temperature be used for the worst case.
The ac resistance of conductors is higher than the dc resistance due to skin effects. The effect
is more pronounced as the conductor cross section or the operating frequency increases.
Conductor data tables usually include ac resistances at power frequencies. The skin effect is a
major factor in the design of high-current (several thousand amperes) ac bus systems, such as
those for electric furnaces.
The flux established by alternating current in a conductor may link other conductors or
metallic masses in its proximity thus generating voltages in those parts. These voltages may
cause currents to flow through closed circuits and thus cause I2R losses other than those of the
conductor itself. These losses can be represented as an additional component of resistance in
series with the conductor resistance. The reader should consult the Electrical Transmission
and Distribution Reference Book [B5]1 and Puchstein and Lloyd [B20] for information on
this subject.
4.7.3.2 Inductive reactance
The inductive reactance of a circuit has two components: that due to its own circuit (self) and
that due to other circuits in its vicinity (mutual). The inductance of a conductor also has two
components: that caused by the current in itself, and that caused by the currents in other
conductors of the same circuit. Finally, the inductance of a conductor due to its own current is
divided in two parts: the first part considers the flux internal to the conductor; the second part
considers the flux external to the conductor. This last division has been modified to simplify
tables of conductor characteristics.
There are two types of tables commonly used to determine the inductive reactance of a
conductor. One table lists the conductor inductive reactance Xs at one foot spacing even if the
actual spacing is larger or smaller than one foot. A second table, valid for any type or size of
conductor, lists spacing factors Xd which, when added to the one foot (0.3048 m) reactance,
will give the correct total reactance for the given circuit conductor spacing.
The spacing factor table is calculated from the following equation:
X d = 4.657 × 10
–3
× f × log GMD
Ω
X d = --------------------------------------------( conductor × mile )
1The
62
(4-22)
numbers in brackets correspond to those of the bibliography in 4.11.
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SYSTEM MODELING
where
GMD is the geometric mean distance of the conductors
For three conductors spaced d1, d2, d3,
GMD =
d1 × d2 × d3
Note that in Equation (4-22) a GMD smaller than 1 yields a negative spacing factor.
Cables in steel conduit exhibit higher reactances than those in free air. The calculations are
too complex to develop by hand; hence, the tables in Chapter 1 of the Industrial Power Systems Handbook [B16] should be used for estimating purposes.
4.7.3.3 Shunt capacitive reactance
Capacitive reactance can be determined in a similar fashion. Conductor tables give the value
of reactance Xs at one foot spacing. A spacing factor Xd is added to Xs to yield the total
capacitive reactance of the conductor. Spacing factor tables are calculated from
4.099
–6
X' d = ------------- × 10 log GMD
f
(4-23)
= Ω – mile/conductor
The capacitive reactance of shielded cables is determined from
6
1.79G × 10
X' C = ----------------------------f ×k
(4-24)
= Ω – mile/conductor
where
G
k
f
is the geometric factor
is the dielectric constant of cable insulation
is the frequency
2r
G = 2.303 log ----d
(4-25)
where
r
d
is the inside diameter of shield
is the outside diameter of conductor
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CHAPTER 4
Typical values of k are 6.0 for rubber, 5.0 for varnished cambric, 2.6 for polyethylene, and 3.7
for paper.
4.7.4 Reactors
Reactors are used as branch elements in the following applications:
a)
To limit current during fault conditions;
b)
To buffer cyclic voltage fluctuations caused by repetitive loads (in conjunction with
condensers);
c)
To limit motor-starting currents.
Reactors are modeled as impedances consisting of an inductive reactance in series with a
resistance expressed as R + jX. Manufacturers’ design or test data should be obtained for
existing installations.
4.7.5 Capacitors
Series capacitors are sometimes used on transmission and distribution lines to compensate for
the inductive reactance drop or to improve the system stability by increasing the amount of
power that can be transmitted on tie lines. They are represented by a negative reactance of the
form 0 – jX, in series with the line impedance.
For capacitors specified in microfarads per phase, the reactance may be expressed in the
general form:
6
10
X = ------------- = Ω ⁄ ( per phase )
2πfC
(4-26)
When specified in kvars per phase (QC), the capacitor voltage rating (VC) must also be known
to calculate the following:
2
VC
3
X = ------- × 10 = Ω ⁄ ( per phase )
QC
(4-27)
It should be noted that the series capacitor voltage rating is a function of the amount of
compensation of the design and will generally be a fraction of the system line-to-neutral
voltage. The application of series capacitors should always be accompanied by thorough
studies, because they can contribute to destructive overvoltage and ferroresonance conditions.
64
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SYSTEM MODELING
4.7.6 Transformers
4.7.6.1 Two-circuit transformers
The equivalent circuit of a transformer is shown in Figure 4-10(a). The dashed rectangle
represents an ideal voltage transformation ratio ns /np = N, where ns and np are the number of
turns of the secondary, and primary windings, respectively. Rp and Rs are the effective
resistances of the windings; Xp and Xs are the leakage reactances. G0, the shunt conductance,
models the iron losses that remain constant when the transformer is energized at rated voltage
and BM, the shunt inductive susceptance, is equivalent to the quadrature magnetizing current
at no load.
(a)
(b)
Figure 4-10—Two-winding transformer equivalent circuits
It can be demonstrated that Figure 4-10 (a) is equivalent to Figure 4-10 (b). In the latter, the
secondary resistance and reactance have been reflected to the primary side of the ideal
transformer by multiplication with the inverse of the square of the turns ratio N. This circuit
can be approximated by moving the shunt branch and combining the primary and secondary
impedances as shown on Figure 4-11 (a). The model can be further simplified by eliminating
the shunt branch completely as shown in Figure 4-11 (b). In many types of studies, the
resistance RT , being small with respect to XT , is also neglected thus reducing the model of the
transformer to a single series reactance.
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(a)
(b)
Figure 4-11—Two-winding transformer approximate equivalent circuits
For existing equipment, the nameplate specifies an impedance ZT and the transformation ratio.
An assumption may be made that XT ≅ ZT so that the single series reactance model may be used.
Use of Figure 4-11 (b) model requires that an estimate of R be made from typical data (IEEE
Std C37.010-19792), and a value for XT calculated from
X =
2
Z –R
2
Transformer test data will usually be sufficient to allow calculation of all the parameters for the
circuits of Figure 4-10 (a), Figure 4-11 (a), and Figure 4-11 (b). When maximum accuracy is
needed, the effective resistance RT should include the winding resistances corrected for the
operating temperature and another series resistance to account for stray losses (see McFarland
and Van Nostrand [B19] and Puchstein and Lloyd [B20]). The model of Figure 4-10 necessitates the creation of a fictitious bus for entry of the shunt admittance data in the program.
2Information
66
on references can be found in 4.10.
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IEEE
Std 399-1997
4.7.6.2 Transformer taps
Thus far, only single ratio transformers have been considered. In real life, transformers have
taps, normally on the high-voltage windings, to provide a voltage ratio best suited to the
power system. The taps may be changeable automatically under load (LTC transformer) or
fixed (manually changed when de-energized).
The resistances and leakage reactances of the tapped windings are slightly different at
different taps. This may be ignored if the correct values are not known. On the other hand,
transformer test data may specify impedance values for the taps, in which case these values
should be used. The main effect of changing taps is the change of voltage ratio and therefore
the change of voltage base for which the impedance diagram should be prepared. This will be
described in more detail in 4.8.
The analyst should pay particular attention to the specific requirements of programs for specifying taps. For instance, the tap value 1.05 per unit (105%), interpreted as an additional 5% to
the voltage ratio, yields opposite results if applied to the opposite side of the transformer.
Once the data is entered for a given condition and a certain voltage on a specified bus is
required, the computer program will, upon request, automatically adjust the taps and modify
the system impedances as necessary for the new turns ratio.
4.7.6.3 Three-circuit transformers
In general, three-circuit transformers require models that have three ports or connections to
the power system, instead of two. A three-port model is a necessity if each transformer circuit
is attached to a different power system circuit. However, for hand calculations and most
computer programs, the only model available is the two-port model of Figure 4-11 (b). The
goal then is to construct a model that predicts the relevant behavior of the three-winding
transformer from the available (and manageable for hand calculations) two-port models.
Test data for three-circuit transformers include short- and open-circuit tests. Open-circuit
tests are made by opening all the circuits and applying a voltage to one of the circuits.
Though it is possible to consider these magnetizing effects in a model of the transformer, the
usual practice is to ignore them. Exceptions to this practice involve zero sequence models for
certain types of transformers and specialized studies that involve phenomena for which the
magnetizing effect is important. For the purposes of this development, magnetizing effects
will be ignored.
Short-circuit tests involve shorting one circuit of a transformer and applying a voltage to
another circuit sufficient to give rated current in the circuit pair. The third circuit is usually,
but not always, open. Measurements taken in these tests allow a short-circuit test impedance
to be calculated for each circuit pair. In the subsequent discussion, we will assume that such a
test impedance is available for each pair and that the test is made with the third circuit open.
Using notation that corresponds to the winding labeling of IEEE Std C57.12.80-1978, these
three values of test impedances may be labeled ZHX, ZHY, and ZXY. A positive sequence test
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for ZHX is shown schematically in Figure 4-12 (a). The dashed box surrounding the circuit
can be thought of as the transformer enclosure.
(a) Schematic of a positive sequence test
(b) Simplified wye model
Figure 4-12—Three-circuit transformer and approximate equivalent circuits
Though any three-port network made up of at least three independent two-port branch models
will suffice, the smallest number of elements with the required three degrees of freedom is
three. The two possible connections of these elements are shown in Figure 4-12 (b) and
Figure 4-12 (c). Note that the ideal transformer of Figure 4-11 (b) has been omitted.
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(c) Simplified delta model
Figure 4-12—Three-circuit transformer and approximate
equivalent circuits (Continued)
Elimination of the ideal transformer is made possible by calling the voltages at the terminal
nodes 1.0 per unit and assigning base voltages of EH, EX, and EY. The usual choice is the tee
model of Figure 4-12 (b). One of the advantages of the tee model is that the relationship
between the test impedances and the model parameters is linear and may be easily derived.
By keeping one winding of the model open, shorting another, and applying a voltage to the
third, the relationship between the test values and the model parameters is easily shown to be
ZHX = Zh + Zx
(4-28)
ZHY = Zh + Zy
(4-29)
ZXY = Zx + Zy
(4-30)
Be sure that the test values are on the same kVA base. Frequently, test values are given on the
winding kVA base. Because the test values are usually known but the model parameters are
not, the inverse of these equations is desired. In per unit, they are
1
Z h = --- ( Z HX + Z HY – Z XY )
2
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(4-31)
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1
Z x = --- ( Z HZ + Z HY – Z HY )
2
(4-32)
1
Z y = --- ( Z HX + Z HY – Z HY )
2
(4-33)
Note that any of these impedances, including the resistive part, may be negative.
Occasionally, it is desirable to use the delta model of Figure 4-12 (c). While the model
parameters may be derived directly from the test values, this calculation involves the solution
of nonlinear equations.
It is easier to derive the tee model and then use a standard wye-to-delta conversion to find the
delta model parameters. The equations necessary for this are as follows:
Z hZ x + Z xZ y + Z yZ h
Z hx = ------------------------------------------------Zy
(4-34)
Z hZ x + Z xZ y + Z yZ h
Z hy = ------------------------------------------------Zx
(4-35)
Z hZ x + Z xZ y + Z yZ h
Z xy = ------------------------------------------------Zh
(4-36)
It should be carefully noted that the test and the model parameters are not the same: ZHX is
not equal to Zhx , etc.
4.7.6.4 Phase-shifting transformers
Consider again the example in 4.4. Assume this time that the voltage at bus B has the same
magnitude but that the angle is shifted back another 2° by the winding arrangement within the
transformer, and that the rest of the information given in Figure 4-5 is the same as before. The
new impedance diagram will appear as shown in Figure 4-13 (a). Now solve as before with
VB = 13.60° ∠–1.6°–2.0°. The results are shown in Figure 4-13 (b). Note that the phase shift
of a mere 2° has caused the real power from A to B to jump from 5 314 to 11 803 kW, but the
reactive power decreased from 2 481 to 1 605 kvar.
This example illustrates that the main purpose of phase-shifting transformers to control the
flow of real power between two buses.
These transformers usually have load tap changing mechanisms that can vary the phase angle
between primary and secondary, either automatically or manually. Thus, computer programs
are set up to vary the amount of angle shift (plus or minus), within limits specified in the
input, to achieve the desired amount and direction of real power flow.
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SYSTEM MODELING
1.49 + j 25.35 Ω
(a) Impedance diagram
(b) Flow diagram
Figure 4-13—Impedance and flow diagrams
The data required by computer programs will generally include the following:
a)
b)
c)
d)
e)
Center (0° shift) position impedance
Positive limit position impedance
Negative limit position impedance
Angle shift interval between taps
Number of taps
Program subroutines allow the computer to automatically estimate the values of the
impedances at the intermediate taps.
4.7.6.5 Other transformer models
The foregoing discussion has been intended to give the reader an introduction to the subject
of transformer modeling. It is far from being complete. The Electrical Transmission and Distribution Reference Book [B5] contains a considerable number of models that may be useful.
4.8 Power system data development
4.8.1 Per-unit representations
In power system calculations, variables are routinely expressed using the per-unit system
instead of actual quantities such as ohms, volts, etc. While both representations will yield
identical results, the per-unit method is generally preferred for hand calculations because it
will do the job much more conveniently, especially if the system being studied has several
different voltage levels. Also, the impedances of electric apparatus are usually given in per
unit or percent by manufacturers.
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The per-unit value of any quantity is its ratio to the chosen base quantity of the same
dimensions, expressed as a dimensionless number. For example, if base voltage is taken to be
4.16 kV, voltages of 3 740 V, 4 160 V, and 4 330 V will be 0.9, 1.00, and 1.04, respectively,
when expressed in per unit on the given base voltage. The chosen base voltage, 4.16 kV, is
referred to as base voltage, 100% voltage, or unit voltage.
There are four base quantities in the per-unit system: base kVA, base volts, base ohms, and base
amperes. They are so related that the selection of any two of them determines the base values of
the remaining two. It is a common practice to assign study base values to kVA and voltage. Base
amperes and base ohms are then derived for each of the voltage levels in the system.
In power system studies, the base voltage is usually selected as the nominal system voltage at one
point of the system, such as the voltage rating of a generator or the nominal voltage at the delivery point of the utility supply. The base kVA can be similarly taken as the kVA rating of one of
the predominant pieces of system equipment, such as a generator or a transformer; but usual
practice is to choose a convenient round number such as 10 000 for the base kVA. The latter
selection has some advantage of commonality when several studies are made, while the former
choice means that at least one significant component will not have to be converted to a new base.
The basic relationships for the electrical per unit quantities are as follows:
actual volts
per-unit volts = --------------------------base volts
(4-37)
actual amperes
per-unit amperes = ----------------------------------base amperes
(4-38)
actual ohms
per-unit ohms = ---------------------------base ohms
(4-39)
For three-phase systems, the nominal line-to-line system voltages are normally used as the
base voltages. The base kVA is assigned the three-phase kVA value. The derived values of the
remaining two quantities are as follows:
base kVA
base amperes = ---------------------------3 base kV
(4-40)
2
( base kV )
base ohms = -------------------------base MVA
(4-41)
It is convenient in practice to convert directly from ohms to per-unit ohms without first
determining base ohms according to the following expression:
ohms × base MVA
per-unit ohms = -------------------------------------------2
( base kV )
72
(4-42)
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SYSTEM MODELING
For a three-phase system, the impedance is in ohms per phase, the base kVA is the threephase value, and the base kV is line-to-line.
Where two or more systems with different voltage levels are interconnected through
transformers, the kVA base is common for all systems; but the base voltage of each system is
determined by the turns ratio of the transformer connecting the systems, starting from the one
point for which the base voltage has been declared. Base ohms and base amperes will thus be
correspondingly different for systems of different voltage levels.
Once the system quantities are expressed as per-unit values, the various systems with
different voltage levels can be treated as a single system during the solution for the unknown
variables. Ideal transformers are typically eliminated from the circuit model by a prudent
choice of voltage bases. Only when reconverting the per-unit values to actual voltage and
current values is it necessary to recall that different base voltages exist throughout the
system.
When impedance values of devices are expressed in terms of their own kVA and voltage
ratings, which differ from the base values of a circuit, it is necessary to refer these values
to the system base values. This typically happens due to equipment kVA bases being different from the chosen system kVA base. In such cases, the per-unit impedance of the device
must be changed to either a new base kVA or new base voltage, or both, by the following
equation:
2
( base kV 1 ) base kVA 2
per-unit Z 2 = per-unit Z 1 × ----------------------------2- -------------------------( base kV 2 ) base kVA 1
(4-43)
where subscripts 1 and 2 refer to the equipment and system base conditions, respectively.
4.8.2 Applications example
A section of the power system shown in Figure 1-1 of Chapter 1 has been repeated in Figure
4-14 to illustrate the per-unit system.
The steps in reducing the data to per unit are as follows:
a)
b)
Select base power: S = 10 000 kVA
Determine base voltages
1) Select bus 02:69-2 nominal voltage of 69 kV as base kV at this bus
2) Calculate base voltages at other system levels
13.8
Bus 04:MILL-2: kV base = 69.0 × ---------- = 13.8 kV
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Figure 4-14—Single-line diagram for example application of per-unit system
0.48
Bus 21:T9 SEC: kV base = 13.8 × ---------- = 0.48 kV
13.8
2.4
Bus 20:T8 SEC: kV base = 13.8 × ---------- = 2.4 kV
13.8
c)
Calculate base impedances using Equation (4-41)
1) 69 kV system:
2
3
69 × 10
Z base = ---------------------- = 476.1 Ω
10 000
2)
13.8 kV system:
2
3
13.8 × 10
Z base = --------------------------- = 19.044 Ω
10 000
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SYSTEM MODELING
3)
2.4 kV system:
2
3
2.4 × 10
Z base = ------------------------ = 0.576 Ω
10 000
4)
0.48 kV system:
2
3
0.48 × 10
Z base = --------------------------- = 0.023 Ω
10 000
d)
Calculate base currents using Equation (4-40)
1) 69 kV system:
10 000
I base = ------------------------ = 83.67 A
3 × 69.0
2)
13.8 kV system:
10 000
I base = ------------------------ = 418.37 A
3 × 13.8
3)
2.4 kV system:
10 000
I base = --------------------- = 2405.63 A
3 × 2.4
4)
0.48 kV system:
10 000
I base = ------------------------ = 12028.13 A
3 × 0.48
e)
f)
Summarize the base data in Table 4-6.
Convert transformer impedances to the new base using Equation (4-43).
1)
T-2:
2
0.4698 + j7.9862 69 10
Z = ----------------------------------------- × -------2- × ------ = 0.0031 + j0.0532 per unit
100
69 15
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Table 4-6—Summary of base values for example system
Base values
Bus(es)
Power (kVA)
Voltage (kV)
Impedance (Ω)
02:69-2
10 000
69.0
476.10
04:MILL-2, 08:FDR L,
16:T9 PRI, 15:FDR I
10 000
13.8
21:T9 SEC
10 000
20:T8 SEC
10 000
Current (A)
83.67
19.044
418.37
0.48
0.023
12 028.13
2.4
0.576
2 405.63
or
2
0.4698 + j7.9862 13.8 10
Z = ----------------------------------------- × ------------2 × ------ = 0.0031 + j0.0532 per unit
100
13.8 15
2)
T-8:
2
10
0.4568 + j5.481 13.8
Z = -------------------------------------- × ------------2 × ---------- = 0.0122 + j0.1462 per unit
3.75
100
13.8
or
2
0.4568 + j5.481 2.4
10
Z = -------------------------------------- × ---------2 × ---------- = 0.0122 + j0.1462 per unit
100
3.75
2.4
3)
T-9:
2
1.1277 + j5.6383 13.8
10
Z = ----------------------------------------- × ------------2 × ---------- = 0.1504 + j0.7518 per unit
100
13.8 0.75
or
2
1.1277 + j5.6383 0.48
10
Z = ----------------------------------------- × ------------2 × ---------- = 0.1504 + j0.7518 per unit
100
0.75
0.48
g)
76
Calculate line impedance in ohms
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Std 399-1997
SYSTEM MODELING
The cables in the example system are 3/C, copper cables, paper insulated, shielded
conductors with the data given in Table 4-7. Shunt capacitive reactance is neglected due to the
relatively short lengths of the cables.
Table 4-7—Cable data for example system
Length
(ft)
Cable
number
510
Bus
R
(Ω/1 000 ft)
XL
(Ω/1 000 ft)
XC
(Ω/1 000 ft)
From
To
1
04:MILL-2
08:FDR L
0.0284
0.0344
neglect
1187
2
04:MILL-2
16:T9 PRI
0.0441
0.0367
neglect
980
3
04:MILL-2
15:FDR I
0.0441
0.0367
neglect
1)
Cable #1:
510
R = 0.0284 × ------------ = 0.0145 Ω
1000
510
X L = 0.0344 × ------------ = 0.0175 Ω
1000
2)
Cable #2:
1187
R = 0.0441 × ------------ = 0.0523 Ω
1000
1187
X L = 0.0367 × ------------ = 0.0436 Ω
1000
3)
Cable #3:
980
R = 0.0441 × ------------ = 0.0432 Ω
1000
980
X L = 0.0367 × ------------ = 0.036 Ω
1000
h)
Calculate line/cable impedances in per unit with Equation (4-42)
1)
Cable #1:
0.0445 + j0.0175
Z = ----------------------------------------- = 0.00076 + j0.00092 per unit
19.044
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2)
CHAPTER 4
Cable #2:
0.0523 + j0.0436
Z = ----------------------------------------- = 0.00275 + j0.00229 per unit
19.044
3)
Cable #3:
0.0432 + j0.036
Z = -------------------------------------- = 0.00227 + j0.00189 per unit
19.044
i)
Calculate X'd of the synchronous machine in per unit with Equation (4-43)
1)
Synchronous motor on bus 08:FDR L:
2
10 000
13.8
X' d = j0.2 × ------------2 × ------------------------- = j0.1778 per unit
000 ⁄ 0.8
9
13.8
j)
Calculate the utility system impedance in per unit
1)
at bus 02: 69-2:
10
Z = ( 0.876 + j6.1655 ) × -------2- = 0.0018 + j0.013 per unit
69
The per-unit data and the base voltages have been transferred to the impedance diagram of
Figure 4-15 in readiness for the preparation of computer input document and as a record of
the basic information for the study. The previous calculation method illustrates the use of the
per-unit system by selection of base voltages on the different parts of the system using
transformer turns ratios. While this method is straightforward for hand calculations, it can
lead to errors in interpretation of results when used in computer studies where nominal
voltages in different parts of the systems are not related by the transformer turns ratios. In
computer studies, it is usually preferred to select the base voltages for each part of the system
to be equal to the nominal voltage without specific regard for transformer turns ratios. Any
difference between the transformer turns ratio and the ratio of the bases selected for the buses
on each side of the transformer is accounted for by using an off-nominal (not equal to 1.0)
turns ratio in the computer program model of the transformer. It is customary to represent the
off-nominal ratio as N:1, where N is associated with the high-voltage winding. This approach
to selecting base voltages allows transformer taps to be incorporated without recalculating the
entire set of base values at each affected voltage level.
The use of nominal voltages as base voltages has the following advantages:
—
78
The chance of errors in interpretation of results is reduced; it is clear to all what 1.0
per unit means.
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SYSTEM MODELING
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Std 399-1997
Figure 4-15—Impedance diagram showing results of per-unit conversions
—
It is much easier to spot abnormally high or low voltages. One only has to scan for
voltages outside a certain range, say 0.95 to 1.05 per unit.
—
Changes in network configuration will not require changes in network impedances.
For example, if two parts of the 13.8 kV system are modeled using base voltages of 14.427
and 14.0 kV due to different transformer turns ratios, the study of the outage of one
transformer and the connection of the two systems via a normally open breaker could not be
performed until the impedances of one of the systems was converted to the base of the other.
The user’s manual of most computer programs will contain program-specific information on
the modeling of transformer taps and off-nominal ratios.
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CHAPTER 4
4.9 Models of bus elements
4.9.1 Loads in general
Power system loads may be classified by one or a combination of the following types, to
account for their voltage dependence:
Constant power S
Constant impedance Z
Constant current I
Figure 4-16 shows their respective power/voltage relationships.
Figure 4-16—Effect of voltage variations for three types of loads
A single expression,
V k
S
--- = ----- ,
V i
S i
(4-44)
can represent the three load types by making k = 0 for constant power, k = 1 for constant
current, and k = 2 for constant impedance loads. Si is the initial power at Vi, the initial
voltage; S is the power at voltage V.
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A more general expression can be formulated by expanding Equation (4-44) for the real and
reactive power:
V k2
V k1
P + jQ = P i ----- + jQ i -----
V i
V i
(4-45)
The subscript i has the same meaning as in Equation (4-44). The exponents k1 and k2 could be
different and nonintegers.
A load or group of loads could also be expressed in a more restrictive way by
P + jQ =
V 2
V
V 2
V
A + B ----- + C ----- P i + j D + E ----- + F ----- Q i
V i
V i
Vi
Vi
(4-46)
again to reflect voltage dependency. In this case A, B, C and D, E, F represent fractions of P
and Q, respectively. The sums A + B + C and D + E + F must equal 1.
In stability studies, frequency, like voltage, may become an important factor in the modeling
of loads. Linear frequency dependence would take the following form:
P + jQ = ( 1 + G ∆ f )P i + j ( 1 + H ∆ f )Q i
(4-47)
where G and H are the fractions of Pi and Qi , respectively, being affected by the frequency
deviation from the steady-state frequency.
The problem of assigning correct values to the constants (k1, k2, A through H ) is very difficult
when studying utility type systems because the nature of the load is not known accurately.
Additionally, the tasks of simulating the load in all its details would require a computer
program of such size and cost that the effort might be prohibitive (see Adler and Mosher
[B1], IEEE Committee Report [B14], Ilicito, Ceyhan, and Ruckstuhl [B15], and Kent et al.
[B17]).
Industrial and commercial power systems are relatively modest in size. Moreover, bus loads
are often arranged in such a way that grouping by type is easy to do, thus facilitating the
preparation of computer input data and offering the possibility of combining large sections of
the system to reduce the overall sizes of the study sections.
4.9.2 Induction motors
The induction motor equivalent circuit is shown in Figure 4-17 (see Fitzgerald, Kingsley, Jr.,
and Umons [B6], McFarland and Van Nostrand [B19], and Puchstein and Lloyd [B20]). The
solution of this circuit for values of slip s at a given input voltage and frequency will yield,
among other performance characteristics, three sets of curves: torque, current, and power factor, typified in Figure 4-18, Figure 4-19, and Figure 4-20.
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Figure 4-17—Induction motor equivalent circuit
If the three curves are known, the shaft output power can be calculated for any speed using
the slip and torque with the following equation:
Po
τ = 7.05 × ---------------------( 1 – s )N s
(4-48)
where
τ
Po
is the torque in lb-ft
is the output power in watts
Ns
is the synchronous speed in rev/min
Input power can also be calculated using the current and power factor curves for the desired
voltage.
S = 3VI ( cos θ + j sin θ )
(4-49)
θ = arc cos PF
(4-50)
where
V
is the line-to-neutral voltage
In some cases, the equivalent circuit constants will be known along with the input or output
power. The analytical solution, then, consists of finding the correct value of slip that will
match the specified power. A cut-and-try method may be used here; but this would prove to
be a tedious process if solved by hand because of the repeated reduction of the complex
network for each trial of slip value. Computers excel at iterative methods, and for this reason
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programs will generally be written to accept this kind of input. The equivalent circuit model
finds its major applications in motor starting and stability studies.
The curves of Figures 4-18 through 4-20 are also used in motor-starting studies. The computer input data will consist of the motor equivalent circuit and/or sets of values of torque,
current, and power factor taken at different intervals of slip along these curves. The program
will normally require a similar torque-speed curve for the load. The motor and load moments
of inertia will complement the description of the mechanical system.
Figure 4-18—Induction motor torque versus speed
NOTE—All inertias must be calculated on the same base. In addition, inertias of couplings, pulleys, and
flywheels should also be considered.
The equivalent circuit constants can be calculated from motor test data or obtained from the
manufacturer (calculated or test values). IEEE Std 112-1996, McFarland and Van Nostrand
[B19], and Puchstein and Lloyd [B20] describe the methods used for the calculations from
test data.
4.9.2.1 Constant load model
It may be appropriate in some studies to represent a motor as a constant kVA load:
S = P + jQ
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Figure 4-19—Induction motor current versus speed
Figure 4-20—Induction motor power factor versus speed
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where P and Q are constant. This requires that the shaft horsepower (BHP), nameplate voltage, motor efficiency η, and power factor be known or estimated and substituted in the following equations:
BHP
P ( kW ) = 0.746 × ----------η
(4-51)
θ = arc cos PF
(4-52)
Q ( kvar ) = P tan θ
(4-53)
Both the efficiency and power factor are functions of motor voltage and percent load. An
example is shown in Figure 4-21. Consequently, if the objectives of the study warrant, it may
be necessary to estimate the voltage in order to determine appropriate values for τ and PF.
Manufacturers publish motor characteristics (current, efficiency, power factor) at various
loadings for a wide spectrum of motor sizes and types. This model is generally applicable in
load flow studies or other studies in which the effect of model simplification is of secondary
importance. It should be emphasized that the reactive power, Q, is positive for an induction
motor, even if the motor is operating in the generating mode, that is, with a negative slip.
Figure 4-21—Effect of voltage variations on typical induction motor
characteristics
4.9.2.2 Models for short-circuit studies
In this area of power system analysis, standards specify the model to be used for motors to
account for their contributions to short-circuit currents (see IEEE Std 112-1996 and Huening,
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Jr. [B7]). The model for a motor is invariably a voltage source in series with an impedance as
in Figure 4-22.
The voltage source is justified by the fact that at the very instant of a short-circuit, the flux
that exists at the air gap cannot change instantly. The energy that this flux represents has to
dissipate in the form of current through resistance. This current flows in the electrical circuits
linked by that flux, and is limited by the inductance of these circuits. The stator winding,
being linked by that flux and having inductance, will therefore act as a limited source of
current to the network to which it is connected. The flux will decay rapidly because there is
no source of current to maintain it. Thus, the time constant will be in the order of a few
cycles.
Figure 4-22—Model of induction motor for short-circuit study
The value of impedance in series with the voltage source is nearly the same as that of the
impedance which limits the locked rotor current when voltage is applied to the motor at rest.
Accordingly, the impedance in Figure 4-22 can be calculated from the following:
V
R + jX = --------------3I LR
(4-54)
where
V
ILR
is the motor nameplate line-to-line voltage and
is the motor locked rotor current
The resistance R is usually small relative to the reactance X. Moreover, short-circuit currents
have low power factors. These two factors justify neglecting R in those short-circuit studies
where only the symmetrical current magnitude is required.
When the short-circuit study objective is to check circuit breaker capabilities, it is necessary
to consider different conditions such as first cycle duty, momentary duty, interrupting duty,
and breaker speed, in order to determine an appropriate value of motor series impedance that
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will account for the decay in the current that it contributes to a fault. This is spelled out in
IEEE Std C37.010-1979 and takes the form of impedance multiplying factors that vary as a
function of time from the moment a fault occurs. The rationale for neglecting small motor
contributions is based on their electrical remoteness from the power circuit breakers and their
very short time constants.
4.9.2.3 Constant impedance model
It is sometimes desirable to determine the system voltages at the instant a motor at rest is
energized. The appropriate model is the equivalent circuit given earlier with the slip s equal to
1, which reduces the circuit to one with a constant impedance. It is generally simplified to a
single impedance to ground and calculated from Equation (4-54).
4.9.3 Synchronous machines
Synchronous machine models vary tremendously with the type of study. In this subclause, the
text will include both the simpler model for steady-state conditions and the more complicated
models for transient conditions.
4.9.3.1 Steady-state models
4.9.3.1.1 Generators
Once a power system has settled down after a change of any kind, generator output voltages
will have been automatically brought back to the desired output values by the voltage
regulators. The prime mover governor will also have taken a steady position to maintain the
generator speed, consistent with the rest of the system, and to supply the amount of power
programmed by its setting.
The generator output voltage will be a function of the field current. Its output power will be a
function of the mechanical power applied to its shaft, which will also determine the rotor
angle that the field poles maintain relative to the revolving magnetic field in the stator.
Thus, in a load flow study for steady-state conditions, the generator can be modeled as a
constant voltage source and a scheduled amount of kW. The system operating conditions may
demand that a generator voltage output be adjusted automatically so that it supplies an
amount of reactive power such that a bus, somewhere on the system, maintains a specified
voltage. In this situation, the analyst specifies a range of reactive power (kvar) within which
the machine will operate. The generator voltage will be adjusted by the program.
Other possibilities are
a)
b)
To specify a voltage and angle and let the kW and kvar fall where they may.
To specify fixed kW and kvar and let the voltage fall where it may.
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The last alternative suggests that a generator may be considered analytically as a negative
constant power load. Many computer programs will accept negative power signs. Thus
negative kW input to the bus load data would model a generator.
4.9.3.1.2 Synchronous condenser
One difference between this machine and a generator is that the condenser consumes a small
amount of real power corresponding to its losses. It will generally be equipped with a voltage
regulator similar to a generator and have its reactive power output, specified within certain
limits, adjusted by the computer program to maintain a specified voltage at its own terminals
or elsewhere in the system.
4.9.3.1.3 Synchronous motors
Synchronous motors may or may not be equipped with regulators to control their excitation.
Those equipped with regulators may control voltage, power factor, reactive power, or even
current at their terminals or elsewhere. The real power in kW will be a function of the load
driven by the motor, and will not be adjustable for the given set of conditions under study.
The analyst may, therefore, resort to modeling the synchronous motors as negative generators
with reactive power limits, if a voltage or current control device is supplied. In the case of
power factor or reactive power regulators, the specified kW will define a value of kvar using
Equations (4-55) through (4-57).
For motors with fixed excitation and a given fixed load, vee-curves are often used to calculate
the equivalent reactive power. Another option is to calculate (approximately) the reactive
power using given manufacturer’s data and detailed formulas. Figure 4-23 shows the veecurve for a particular motor. The reactive power is calculated as follows:
a)
b)
c)
d)
Draw a vertical line for the fixed excitation current
Read the line current (armature current) I at the intersection of the load line
representing the motor running load and the field current line
Estimate the motor terminal voltage (line to line)
Calculate the reactive power
kvar =
2
2
3 ( V ln I l) – kW 3φ
(4-55)
Alternately, if the power factor curves are shown on the graph, replace above steps c) and d)
with the following:
a)
b)
88
Read the power factor at the intersection of the load line and I line
Calculate the reactive power
kvar = kW tan θ
(4-56)
θ = arc cos PF
(4-57)
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SYSTEM MODELING
Figure 4-23—Vee-curves: Synchronous motor, 2000 hp, 4000 V,
180 r/min, 0.8 lead power factor
4.9.3.2 Short-circuit models
The current contributed to a fault by a synchronous machine varies with time, from a high
initial value to a moderate final steady-state value. Equations (4-58), (4-59), and (4-60) depict
this variation of current as a function of time for a three-phase short circuit at the terminals of
a machine operating initially unloaded.
–t
--------
I ac
–t
---------
1 T ''
1
1 T'
1
1
= E ------ + -------- – ------ e d + --------- – -------- e d
X '' d X ' d
X d X ' d X d
(4-58)
–t
-----
2E cos α T
I dc = ------------------------ e a
X '' d
IT =
I
2
ac
+I
2
dc
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The rms value of the total current IT is made up of two components:
a)
A power frequency ac component Iac, the rms value of which decreases with time
t, in accordance with Equation (4-58). This component is called the symmetrical
current.
b)
A dc component Idc, which decreases with time in accordance with Equation (4-59).
The initial magnitude of Idc is a function of the angle α of the voltage wave at which the short
circuit occurred. It determines the amount of offset of the current wave and is proportional to
the rate of change of voltage at the instant of short circuit. The offset is maximum at α = 0
because the rate of change of voltage is maximum when the voltage is 0. The offset is 0 at α
= 90° of change is 0 at the positive or negative peak voltage values.
In the above equations,
E
X d, X ' d, X '' d
is the open circuit voltage
are the direct-axis synchronous, transient, and subtransient reactances,
respectively
T d, T ' d, T '' d
are the armature and the direct-axis transient and subtransient short-circuit
time constants, respectively
For typical values of reactances and time constants, and with the maximum offset condition
(cos α = 1), the equations will reduce to the following:
At time t = 0 (under subtransient conditions)
E
I '' ac = --------X '' d
(4-61)
2E
I '' dc = -----------X '' d
(4-62)
E
E
I ' T = --------- 1 + 2 = --------- 3
X '' d
X '' d
(4-63)
At time t = ∞
E
I ac = -----Xd
I dc = 0
E
I T = I ac = -----Xd
90
(4-64)
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Because T ' d is much larger than T '' d , there is a time t ' smaller than T ' d and larger
than T '' d , that will make e –t ' ⁄ T 'd approximately equal to 1 and e –t ' ⁄ T ''d approximately equal
to 0. In this case, the original equations reduce to the following:
E
I ' ac = -------X 'd
(4-65)
–t ' ⁄ T '
a
2Ee
I ' dc = ---------------------------X '' d
(4-66)
Should there be an impedance ZL = RL + jXL between the machine terminal (still at no load)
and the point of fault, Equations (4-61), (4-64), and (4-65) will become
E
I '' ac = -----------------------------------------R L + j ( X L + X '' d )
(4-67)
E
I ' ac = ----------------------------------------RL + j ( X L + X 'd )
(4-68)
E
I ac = --------------------------------------RL + j ( X L + X d )
(4-69)
The armature time constant Ta will be shortened appreciably by addition of resistance RL in
the external circuit. So Equation (4-66) will become
–t ⁄ T '
a
2Ee
I ' dc = --------------------------X '' ( T 'a < T a )
(4-70)
The offset current will decay more rapidly the farther away (electrically) the machine is from
the point of fault.
The voltage E, in all the above equations, is equal to the terminal voltage Vt , since it was
assumed that the machine was carrying no load before the short circuit. If the machine was
carrying a current IL before the short circuit, the voltage E will be different in each equation,
to satisfy prefault conditions. For the case of a generator, the voltages in Equations (4-61),
(4-64), and (4-65) will be
E '' = V t + I L X '' d
(4-71)
E ' = V t + I L X 'd
(4-72)
E = V t + ILXd
(4-73)
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respectively. These voltages have been called voltage behind subtransient reactance (E"),
voltage behind transient reactance (E'), and voltage behind synchronous reactance (E). It is
not practical, in short-circuit studies, to calculate the system currents for the entire period—
from the time of fault to the time that the current reaches a steady-state value. The normal
procedure is to solve the network at times t = 0 or t = t', or both, using the models of Figure
4-25. Network solutions at t = α are meaningless since the machine field excitation has likely
changed by that time.
Depending on the study objectives, the effect of the offset current Idc may or may not be
important. In power circuit breaker applications, however, it is a very important
consideration. To obviate the difficulties in solving Equation (4-66), the breaker standards
specify multipliers for the X"d and X'd current components. These are functions of machine
type and of the time from the inception of the short circuit. IEEE Std 122-1991 lists those
multipliers, gives examples of their use, and expands on this important aspect of short-circuit
studies.
The models of Figures 4-24 and 4-25 are also applicable to synchronous motors and synchronous condensers (using motor convention for the Kirchhoff’s Voltage Law equations), the
difference being that the E", E', and E voltages are calculated with the following:
E '' = V t – I L X '' d
(4-74)
E ' = V t – I L X 'd
(4-75)
E = V t – ILXd
(4-76)
(a)
(b)
Figure 4-24—Models of synchronous machines for short-circuit studies
4.9.3.3 Stability models
In stability studies, it is necessary to solve the electrical networks at a series of intervals
starting from the time of inception of a disturbance and continuing for as long as necessary to
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Figure 4-25—General model for ac machines in short-circuit studies
establish a trend in the system performance. This may cover a period shorter than 1 s for
transient stability or several seconds for dynamic stability studies.
4.9.3.3.1 Classical model
The classical model for a transient stability study consists of a simple constant voltage source
behind a constant transient reactance as in Figure 4-24 (b). This model neglects the following
factors by assuming the following:
a)
b)
c)
d)
e)
The shaft mechanical power remains constant
Field flux linkages remain constant
Damping is nonexistent
The constant voltage and reactance are not affected by speed variations
The rotor mechanical angle coincides with the phase angle of the internal voltage
Ignoring dynamic stability effects sometimes associated with high-speed exciter systems, a
system found stable under these assumptions will likely be stable if any or all of the above
factors are taken into consideration. However, due to exciter effects and in cases where the
above analysis suggests instability, there is a need to account for the neglected factors along
with regulator action in certain circumstances. The various models, in increasing degree of
sophistication, will be presented with simple explanations, if possible, or without
explanations, in which case references will be given for the reader to consult.
4.9.3.3.2 The H constant
Stability studies are concerned with relative speed variations of rotating masses. The kinetic
energy of a rotating mass, using units of Table 4-7, is as follows:
1
1 2
KE = --- I ω = --- Mω J
2
2
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The rotating mass associated with a generator includes the rotor, shaft, coupling, turbine, and
exciter, if the rotating type is used. Since this mass is designed to rotate at a fixed
(synchronous) speed, the stored energy, at synchronous speed, of a given machine is used as a
constant. This constant is usually normalized by defining a quantity H that expresses the
stored energy per unit of machine rated power. If kinetic energy is in megajoules (MJ) and the
rated power S in megavoltamperes (MVA), the H constant will be
J
KE
H = -------- = -------VA
S
Since 1 joule (J) is equal to 1 watt-second (Ws), the H constant is sometimes stated in equivalent kilowatt-second (kWs) per kVA units.
To calculate the H constant, use the following:
2
2
–3
( WR ) ( r ⁄ min ) × 10
H = 0.231 -------------------------------------------------------MVA
(4-78)
where WR2 is the moment of inertia in pounds-feet squared and r/min is the speed in
revolutions per minute. The H constant values will fall within the narrow range of
approximately 1 to 15, irrespective of the size of the machines.
4.9.3.3.3 Stability model variations
In the discussion of 4.9.3.2, only the direct-axis parameters were considered on the basis that
short circuits produce currents of low power factor (quadrature currents predominate). This
assumption may not be acceptable for disturbances considered in stability analysis.
Therefore, additional synchronous machine parameters are required to more accurately
model the behavior and account for the differences in the magnetic construction types, such
as salient poles, smooth rotor, laminated rotor, solid iron rotor, with or without dampers.
Quadrature-axis reactances and open-circuit time constants are defined for that purpose.
Chapter 1 of Kimbark [B18] is especially recommended as a clear and basic text on the
subject.
The classical model may be improved one step by taking account of the variation of machine
impedance with time from its initial value, X'd, to a steady-state value of Xd . The variation
will be an exponential described by a time constant T'do (transient, open-circuit time
constant). The three parameters Xd, X'd and T'do will ignore the major effect of dampers.
Another improvement will involve adding the effect of dampers that predominate during fast
changing conditions, that is, the subtransient state. The additional parameters X"d and T"do
will take care of this effect.
The saliency of the rotor will be represented by the quadrature-axis parameters, Xq, X'q, and
X"q and the associated time constants T'qo and T"qo.
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Adding the parameters X'q and T'qo to those of the first improved model will increase the
accuracy in the case of a generator with a solid iron rotor. But, as before, the damping effect
will have been mostly neglected. The solid iron rotor will be fully represented by all directand quadrature-axis, synchronous, transient, and subtransient reactances and associated time
constants.
The transient quadrature-axis reactance of salient pole machines has the same value as the
equivalent synchronous reactance. Thus salient pole machines can be fully modeled as the
solid iron rotor machine by omission of the X'q and T'qo parameters.
4.9.3.3.4 Exciter models
Various IEEE committees have developed a number of models to represent excitation
systems for stability studies. These models have been updated and published as IEEE Std
421.5-1992. Both this standard and its predecessors (IEEE Committee Reports [B8] and
[B11]) should be consulted for their description of excitation system models for stability
studies and for their tutorial value.
Figure 4-26 shows a standard model that is available with some minor modifications in IEEE
Std 421.5-1992 and IEEE Committee Reports [B8] and [B11]. This model is listed as type
DC1A, DC1, and type 1, respectively, in each of the aforementioned references.
Figure 4-26—IEEE type 1 excitation system
IEEE Committee Report [B13] discusses the transfer function blocks and their practical interpretation as well as other topics related to system response. It is vital to recognize that excitation system models relate the output signal (the synchronous machine field current) to the
error signal, which is most commonly the difference between a reference voltage and the
synchronous machine terminal voltage. Any transfer function that succeeds in giving the
proper relationship is satisfactory for the purpose of a system-wide stability study.
Consequently, there are many models with widely varying parameters that will give identical
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results. This has the important ramification that typical data for excitation systems are
difficult to define. One result is that IEEE Std 421.5-1992 gives sample data and carefully
eschews the use of the word typical.
If typical data must be employed, the following guidelines need to be carefully observed:
—
If you have data for a particular model, use that model in your study. Do not attempt
to apply data for one model to a different model. For example, do not use data for
a type 1 model from IEEE Committee Report [B11] in a model from IEEE Std
421.5-1992. Many generators and their excitation systems are older than the respective standards. Consequently, unless someone has made tests explicitly to define
parameters for one of the models in the newer standards, you should use the older
model for which you have data.
—
Notwithstanding the above, if you are able, select a model that corresponds to the
type of excitation system that you are to model. In general, there are three distinctive
types of excitation systems that are identified on the basis of their excitation power
source. These are type DC, which utilizes a direct current generator with a
commutator as the source of excitation power; type AC, which uses an alternator and
either a stationary or rotating rectifier; and type ST, which uses transformers or
auxiliary windings and rectifiers. If possible, or particularly if tests are made, use a
model corresponding to the kind of excitation system that is to be modeled.
—
Beware of the per-unit system. The standard for excitation models calls for the voltage
base of the excitation system to be defined so that one per-unit exciter output voltage is
that voltage required to produce rated generator voltage on the generator air gap line.
Strictly speaking, the parameters for a particular excitation system cannot be defined
without knowing the parameters of the synchronous machine it supplies. Most typical
data are given assuming that the full-load field voltage (Efdfl) will be 3.0 per unit. If it
is not, the numerical data should be corrected. You should also be careful because different manufacturers use a different base for the voltage amplifier block. While either
system works, the numerical values for gains and limits vary widely. Unless you
understand the basis used, do not combine data from different sources.
—
Certain parameters such as feedback gains are not inherent in the equipment, but
represent user-definable gains that are set to maintain excitation system or power
system stability. These can often be set by assuming midpoint operation or zero gain
during normal operations. However, beware that if you make this assumption, you
may be misrepresenting some parameters that perhaps have been incorrectly set in
the field, with potentially grave implications for the operation of the power system.
If you must use typical data, an excellent reference is Anderson and Fouad [B2].
Many models have a provision for modeling the saturation that occurs when the excitation
current exceeds a certain level. Figure 4-27 shows the standard representation for loaded
saturation representation. Computer programs usually account for the related nonlinearity of
the air-gap voltage and field current from input data representing two points on the saturation
curve defined by the equation in the figure. Refer to the program instructions for information
on which two points to use.
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Figure 4-27—Saturation curves
Note also that certain models in the newer standard (all type AC except AC5A) require the
use of the no-load saturation curve.
4.9.3.3.5 Prime movers and governor models
Basic models for speed-governing systems and turbines in power system stability studies
have been presented in an IEEE Committee Report [B12]. As mentioned earlier, the models
are in the form of block diagrams with transfer functions describing the system components’
performance. Two more papers, Eggenberger [B4] and Ramey and Skooglund [B21] cover
some of the basics and will help the novice understand the relationships between the physical
elements and the transfer functions.
Typical parameter values are also available in these references. Of course, analysts will be
well advised to seek from the manufacturer the data applicable to their equipment before
compromising with typical data.
4.9.4 Miscellaneous bus element models
4.9.4.1 Lighting and electric heating
Lighting and electric heating often constitute a large section of a plant load, particularly in
commercial buildings. This type of load can be modeled as constant admittance as suggested
by Figures 4-10, 4-11, and 4-12 of the Industrial Power Systems Handbook [B16]. The
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constant admittance model would seem appropriate for fluorescent and mercury vapor lighting as well as for incandescent lighting. In utility type networks where substations are
equipped with voltage regulators, lighting and heating can be represented as constant P + jQ.
To calculate the admittances, determine the watts P and vars Q at rated voltage V and solve
P + jQ
Y ( seimens ) = --------------2
E
The fluorescent and mercury vapor lighting power factors will be determined from
manufacturers’ data in order to calculate the vars Q. Incandescent lights and electric heating
will have unity power factors.
4.9.4.2 Electric furnaces
In load flow studies, this load will usually be represented by constant power that will reflect a
desired controlled operating condition to be analyzed. In the case of short-circuit and stability
studies, the electric furnaces may behave like a constant impedance load. It is unlikely that
automatic load tap changers and electrode position controls will have had time to change
from the prefault condition to the end of the period covered in transient stability studies. This
may very well be the case also in dynamic stability studies extending to several seconds.
Because furnace loads can be a large section of a plant load, it is important that the constant
impedance model be specified in stability problems because of its damping effect on the
power system.
4.9.4.3 Shunt capacitors
Banks of capacitors are used extensively to correct power factors and, as a result, improve
voltage regulation at the point of connection. They are modeled simply by a constant shunt
capacitive susceptance.
jB = j2π f C
where C is in farads, f is in hertz, and B is in siemens.
Often the capacitors will be specified in kvars. The rated voltage of the bus to which the
capacitors are connected must be known whether the susceptance or the equivalent kvar value
at the base voltage is used in the study. Remember that kvars vary proportionally with the
square of the voltage for a constant susceptance.
4.9.4.4 Shunt reactors
These are seldom used in industrial power systems. A constant admittance would be the
correct model for reactors that do not saturate. The same voltage considerations as in the
previous paragraph would apply.
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4.10 References
This chapter shall be used in conjunction with the following publications. When the following standards are superseded by an approved revision, the revision shall apply.
IEEE Std 58-1978, IEEE Standard Induction Motor Letter Symbols.3
IEEE Std 86-1987, IEEE Recommended Practice: Definitions of Basic Per-Unit Quantities
for AC Rotating Machines.4
IEEE Std 112-1996, IEEE Standard Test Procedure for Polyphase Induction Motors and
Generators.5
IEEE Std 115-1995, IEEE Guide: Test Procedures for Synchronous Machines, Part I—
Acceptance and Performance Testing, Part II—Test Procedures and Parameter Determination
for Dynamic Analysis.
IEEE Std 122-1991, IEEE Recommended Practice for Functional Performance
Characteristics of Control Systems for Steam Turbine Generator Units.
IEEE Std 421.5-1992, IEEE Recommended Practice for Excitation Systems for Power System Stability Studies.
IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in
Stability Analysis.
IEEE Std C37.010-1979 (Reaff 1988), Application Guide for AC High-Voltage Circuit
Breakers Rated on a Symmetrical Current Basis.
IEEE Std C37.04-1979 (Reaff 1989), IEEE Standard Rating Structure for AC High-Voltage
Circuit Breakers Rated on a Symmetrical Current Basis.
IEEE Std C57.12.80-1978 (Reaff 1992), Terminology for Power and Distribution
Transformers.
3IEEE
Std 58-1978 has been withdrawn; however, copies can be obtained from Global Engineering, 15 Inverness
Way East, Englewood, CO 80112-5704, USA, tel. (303) 792-2181.
4IEEE Std 86-1987 has been withdrawn; however, copies can be obtained from Global Engineering, 15 Inverness
Way East, Englewood, CO 80112-5704, USA, tel. (303) 792-2181.
5IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
Copyright © 1998 IEEE. All rights reserved.
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4.11 Bibliography
Additional information may be found in the following sources:
[B1] Adler, R. B., and Mosher, C. C., “Steady State Power Characteristics for Power System
Loads,” IEEE Paper 70CP 706-PWR, 1970.
[B2] Anderson, P. M., and Fouad, A. A., Power System Control and Stability, New Jersey:
IEEE Press, 1994 (revised printing).
[B3] Aluminum Electrical Conductor Handbook, The Aluminum Association, 750 Third
Ave., New York, 1971.
[B4] Eggenberger, M. A., “A simplified analysis of the no-load stability of mechanical–
hydraulic speed control systems for steam turbines,” ASME Paper 60-WA-34.
[B5] Electrical Transmission and Distribution Reference Book, East Pittsburgh, PA,
Westinghouse Electric Corporation, 1964.
[B6] Fitzgerald, A. E., Kingsley, Jr., C., and Umans, S. D., Electric Machinery, Fifth Edition,
New York, McGraw-Hill, 1990.
[B7] Huening, Jr., W. C., “Interpretation of new American national standards for power
circuit breaker applications,” IEEE Transactions on Industry and General Applications,
Sept./Oct. 1969.
[B8] IEEE Committee Report, “Excitation system models for power system stability studies,”
IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 2, Feb. 1981.
[B9] IEEE Committee Report, “Current usage and suggested practices in power system
stability simulations for synchronous machines,” IEEE Transactions on Energy Conversion,
vol. EC-1, Mar. 1986, pp. 77–93.
[B10] IEEE Committee Report, “Dynamic models for fossil fueled steam units in power system studies,” IEEE Transactions on Power Systems, vol. 6, no. 2, May 1991, pp. 753–761.
[B11] IEEE Committee Report, “Computer representation of excitation systems,” IEEE
Transactions on Power Apparatus and Systems, June 1968.
[B12] IEEE Committee Report, “Dynamic models for steam and hydro turbines in power system studies,” IEEE Transactions on Power Apparatus and Systems, Nov./Dec. 1973.
[B13] IEEE Committee Report, “Excitation system dynamic characteristics,” IEEE
Transactions on Power Apparatus and Systems, Jan./Feb. 1973.
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[B14] IEEE Committee Report, “System load dynamics—simulation effects and
determination of load constants,” IEEE Transactions on Power Apparatus and Systems, Mar./
Apr. 1973.
[B15] Ilicito, F., Ceyhan, A., and Ruckstuhl, G., “Behavior of loads during voltage dips
encountered in stability studies—field and laboratory tests,” IEEE Transactions on Power
Apparatus and Systems, Nov./Dec. 1972.
[B16] Industrial Power Systems Handbook, Beeman, D., editor, New York: McGraw-Hill,
1955.
[B17] Kent, M. H., Schmus, W. R., McCrackin, F. A., and Wheeler, L. M., “Dynamic modeling of loads in the stability studies,” IEEE Transactions on Power Apparatus and Systems,
May 1969.
[B18] Kimbark, E. W., Power System Stability: Synchronous Machines, vol. 3, New Jersey:
IEEE Press, 1995 (revised printing).
[B19] McFarland, T. C., and Van Nostrand, D., Alternating-Current Machines, New York:
John Wiley & Sons, 1948.
[B20] Puchstein, A. F., and Lloyd, T. C., Alternating-Current Machines, New York: John
Wiley & Sons, 1974.
[B21] Ramey, D. G., and Skooglund, J. W., “Detailed hydrogovernor representation for
system stability studies,” IEEE Transactions on Power Apparatus and Systems, Jan. 1970.
[B22] Standard Handbook for Electrical Engineers, Fink and Beatty, editors, New York:
McGraw-Hill, 12th edition.
[B23] Stevenson, Jr., W. D., Elements of Power System Analysis, Third Edition, New York:
McGraw-Hill, 1975.
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Chapter 5
Computer solutions and systems
5.1 Introduction
Serious power system analysis work requires the use of computers and specialized programs.
Hand calculations are suitable for estimating the operating characteristics of a few individual
circuits; but accurate calculation of voltages, power flows, or short-circuit currents throughout an industrial or commercial power system would be impractical without the use of computer programs. The various types of power system studies, as described in the following
chapters generally involve developing mathematical models in terms of algebraic or differential equations and then solving the equations for numerous design conditions. Computeraided analysis permits the engineer to focus on power system operation rather than on
numerical manipulations.
Success in selecting and applying computer techniques requires the engineer to be familiar
with the power system problem as well as many computer hardware and software considerations. The purpose of this chapter is to acquaint prospective users of computer methods in
power system analysis with the available types of hardware and software resources. In particular, commonly used numerical methods are introduced to provide insight into how computer
programs manipulate large quantities of power system data and determine accurate solutions
for various kinds of equations. This chapter is not intended to explain in detail how to write
computer programs for power system analysis. For those so inclined, numerous books are
devoted to power system analysis programming (see Arrillaga, Arnold, and Harker [B2],1
Heydt [B18], Kusic [B19], Stagg and El-Abiad [B32], Wallach [B36]).
Digital computers were first used for small power system problems in the late l940s; but it was
not until the availability of large-scale computers in the mid 1950s that power system analysis
programs became really useful. Before this time, system studies were performed using
special-purpose analog computers, called network analyzers or network calculators, which
were introduced in 1929 (Gonen [B12]). Power system studies were limited to hand calculations, assisted by slide rule, before 1929. The development of digital computer technology has
been remarkable. Currently, it is common to find computers many times more powerful than
the room-filling machines of the 1950s on the desks of individual engineers. Steady advancements have also been made in numerical methods and computer programming.
The advancements in computer technology, particularly as reflected in shrinking computing
costs, have brought many engineers into direct contact with computers for the first time. Most
find the experience rewarding, if approached with patience. Computers and power system
analysis programs have become more readily available and easier to use. Unfortunately, it is
still easy to get erroneous results from computer programs. Technology has done little to
invalidate the maxim, “garbage in, garbage out.” Data used in a study must be carefully
assembled and checked for input errors. The impact of errors in any assumptions made by the
1The
numbers in brackets correspond to those of the bibliography in 5.4
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user or by the program (e.g., default values) should be considered. Modeling and solution
techniques should be understood. In general, it is important to exercise engineering judgment
when reviewing computer results and avoid the tendency to blindly accept numbers on a
printout.
5.2 Numerical solution techniques
Computer programs for power system analysis use efficient numerical methods that permit a
standardized step-by-step approach to setting up and solving equations. Methodical procedures
are required in general-purpose computer programs designed for solving complex engineering
problems. Such programs adapt, within limits, to the size of the problem at hand. For example,
a load flow analysis program might be designed to handle systems with only two buses or as
many as 1000. Matrix methods are essential for achieving this flexibility and orderliness.
5.2.1 Matrix algebra fundamentals
A matrix is a rectangular array of numbers arranged in rows and columns enclosed by square
brackets or large parentheses. The numbers in a matrix, called elements, may be real or complex. The dimension of a matrix is expressed as m × n (read “m by n”) if the matrix has m
rows and n columns. If a matrix has only one row, it is called a row matrix or row vector.
Similarly, a matrix with a single column is known as a column matrix or column vector. A
matrix is often denoted by a capital, bold-faced letter such as A, and an element in the matrix
is represented as aij, where i is the row number and j is the column number. For example,
A=
a 11 a 12
…
a 1n
a 21 a 22
…
a 2n
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
a m1 a m2 … a mn
If m = n, the matrix is called a square matrix of order n. The elements of a square matrix for
which i = j are called diagonal elements. The other elements in a square matrix are referred to
as off-diagonal elements. If corresponding off-diagonal elements are equal (i.e., aij = aji), the
matrix is a symmetric matrix. For example,
3
6
2
7
6
5
1
5
2
1
4
8
7
5
8
9
is a symmetric matrix of order 4 (dimension 4 × 4).
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Two matrices are equal if they have the same dimension and the corresponding elements in
each matrix are equal. If the rows and columns of a matrix A are interchanged, the resulting
matrix is called the transpose of A and is denoted by AT. If A is a symmetric matrix, then
A = AT.
Addition and subtraction of matrices are valid only for matrices of the same dimension. The
sum, A + B = C, is carried out for each element in C as ci j = ai j + bi j. The order of addition
does not matter (i.e., A + B = B + A). Similarly, the difference, A – B = C, is determined by
calculating each element in C as ci j = ai j – bi j.
Multiplication of a matrix by a number (or scalar) is valid for any matrix. The product, kA =
B, is found by multiplying each element in A by the number k (i.e., bi j = kai j). Multiplication
of two matrices, AB = C, is valid only if the number of columns in A equals the number of
rows in B. If A is a matrix of dimension m × n and B is a matrix of dimension n × p, then the
product C is a matrix of dimension m × p. Each element of C is calculated as follows:
ci j = ai1 b1j + ai2 b2j + … + ain bnj
For example, if
A=
1 2 3
4 5 6
7 8 9
and B =
u + 2v + 3w
AB = 4u + 5v + 6w
7u + 8v + 9w
u x
v y
w z
then
x + 2y + 3z
4x + 5y + 6z = C
7x + 8y + 9z
If the product AB is valid, then BA is valid only if A and B are square matrices of the same
order. In general, the products AB and BA are not equal. An exception is when B is equal to
the unit matrix U. The unit matrix is a square matrix of the same order as A in which all diagonal elements equal 1 and all off-diagonal elements are 0. In this case, AU = UA = A. For
example:
83 10 = 10 83 = 83
52 01
01 52
52
The only form of division defined in matrix algebra is division of a matrix by a scalar k. This
is equivalent to multiplication of a matrix by the reciprocal, or inverse, of the number k –1.
Given the matrix expression AX = B in which the elements of X are unknown, a solution X =
A–1B may exist where A–1 is the inverse of the square matrix A. If A–1 exists, it is a square
matrix of the same order as A, and it satisfies the condition A–1 A = AA–1 = U, where U is the
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unit matrix. This definition implies a method for determining the inverse of a matrix. For
example:
If A = 8 3 let A–1 =
52
w y
x z
Then AA–1 = 8w + 3x 8 y + 3 z = 1 0
5w + 2 x 5 y + 2z
01
The resulting equations 8w + 3x = 1
5w + 2x = 0
8y + 3z = 0
5y + 2z = 1
have the solution w = 2, x = –5, y = –3, z = 8.
Therefore A–1 =
2 –3
–5 8
Matrix inversion can be used to solve simultaneous algebraic equations. This will be
described later along with a more practical method for finding the inverse of a matrix.
An operation that is useful for simplifying computations with a large matrix or for showing
the specific structure of a matrix is called partitioning. A partitioned matrix is divided by
horizontal and vertical lines into submatrices that are treated as single elements in addition,
subtraction, and multiplication. For example, the 3 × 3 matrix A shown below can be partitioned into four submatrices A1, A2, A3, and A4.
1 2 3
A= 4 5 6
7 8 9
=
A1 A2
A3 A4
A column vector B can be partitioned to facilitate multiplication by A.
B=
3
2
1
AB =
=
B1
B2
A1 A2 B1
A3 A4 B2
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= C =
C1
C2
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The submatrix C2, which consists of a single element for this partitioning, can be determined
as follows:
C2 = A3B1 + A4B2 = [7 8] 3 + [9] [1]
2
= 21 + 16 + 9 = 46
The original matrices and corresponding submatrices must be compatible for multiplication,
addition, or subtraction if one of these operations is to be performed on partitioned matrices.
Special techniques for finding the transpose and the inverse of a matrix by operating on its
submatrices are also available (Stagg and El-Abiad [B32]).
Only the fundamental definitions in matrix algebra have been presented here. Comprehensive
information is available in numerous books on mathematics and electrical circuits (Groza and
Shelley [B15], Spiegel [B31], Stagg and El-Abiad [B32], Stevenson [B34]).
5.2.2 Power system network matrices
The matrices used in computer programs for several types of power system analysis are based
on the mesh-current and node-voltage analysis methods described in introductory electrical
circuit theory texts (Edminister [B8], Hayt and Kemmerly [B17], Scott [B29]). In power systems work, the terms loop and bus are frequently used in place of mesh and node, respectively. The simple, three-bus power system shown in the single-line diagram of Figure 5-1
will be used to explain these methods. Resistance is neglected in the discussion in order to
keep the arithmetic easier to follow, although computer programs are normally written to use
both resistance and reactance in calculations.
Figure 5-1—Single-line diagram
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A single-phase equivalent impedance diagram is the basis for the loop current analysis
method. Generators and motors are represented as series reactances and emfs connected to
the neutral bus. The values of the series reactances will vary depending on whether a transient
or steady-state analysis is required. A per unit impedance diagram for the system of Figure
5-1 is shown in Figure 5-2. Three loop currents, I1, I2, and I3, are shown circulating clockwise
within each loop. Kirchoff’s voltage law, which states that the sum of the potential differences around a closed circuit loop is 0, is applied to the loop indicated by I1. This can be
repeated for the other two loops, but the equations can be expressed directly in standard
matrix form. The voltage matrix VLOOP includes the total emf rise around each loop in the
direction assumed positive for the loop current. The impedance matrix ZLOOP consists of
self-impedances and mutual impedances. The impedances Z11, Z22, and Z33 are the selfimpedances that equal the sum of the impedances in loops 1, 2, and 3, respectively. The other
impedances are the mutual impedances that equal the negative of the impedance common to
two loops. For example, Z23 = Z32 = – j2.5, the negative of the impedance common to loops 2
and 3. If the direction assumed positive for one of the currents had been counterclockwise,
then the loop currents would flow in the same direction through the common impedance, and
the mutual impedance would be positive. It is important to realize that the voltage, current,
and impedance subscripts refer to loops, not buses.
Figure 5-2—Impedance diagram and mesh (loop) current analysis
An analysis based on the node-voltage method directly references bus quantities. Most power
system analysis programs work with bus, rather than loop, quantities. Since the node-voltage
method uses admittances instead of impedances, the impedance diagram in Figure 5-2 has
been converted to the admittance diagram in Figure 5-3. Voltage sources with series impedances are converted to equivalent current sources with shunt impedances using well-known
methods. The shunt impedances equal the series impedances, and currents are calculated as
the voltages divided by the series impedances (i.e., I1 = VA /j0.1, I2 = VB /j1, I3 = VC /j2.5 +
VD /j1). Individual admittances are calculated as reciprocals of the corresponding impedances. Kirchoff’s current law, which states that the sum of the currents entering a node equals
the sum of the currents leaving the node, can now be applied to bus 1. This is shown in
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Figure 5-3. Bus voltages V1, V2, and V3 are with respect to the neutral reference bus. Currents
leaving a node are expressed as the potential difference across a branch multiplied by the
admittance of the branch. The resulting current equations can be written in a standard form
suitable for matrix methods. The bus admittance matrix YBUS consists of self-admittances
and mutual admittances. The self-admittances, Y11, Y22, and Y33, are each equal to the sum of
the admittances connected to the node identified by the double subscripts. The mutual admittances are each equal to the negative of the admittance between the two nodes indicated by
the subscripts (e.g., Y12 = Y21 = j10).
Figure 5-3—Admittance diagram and node (bus) voltage analysis
The admittances in the bus admittance matrix are not reciprocals of the impedances in the
loop impedance matrix, nor are the two matrices inverses of each other. However, the bus
impedance matrix ZBUS is defined to be the inverse of the bus admittance matrix YBUS. The
diagonal elements of ZBUS are called driving-point impedances, and the off-diagonal elements are known as transfer impedances. Although the bus impedance matrix cannot be
determined from a simple impedance diagram, efficient algorithms for developing ZBUS
directly from a list of impedance elements are available (Brown, Person, Kirchmayer, and
Stagg [B3], Stagg and El-Abiad [B32]). The bus impedance matrix is widely used in shortcircuit calculation programs. The bus admittance matrix is often used in programs for load
flow analysis.
Certain properties of power system network matrices are exploited to improve the efficiency
of computer programs (Heydt [B18]). The fact that impedance and admittance matrices are
symmetric allows a reduction in computer memory for this data of nearly 50%. Only the
diagonal elements and the elements above the diagonal need be stored since Yji = Yij and Zji =
Zij. Symmetry may be compromised if phase-shifting transformers are modeled. Another
matrix property that can be used to reduce storage requirements is known as sparsity. A
matrix is sparse if most of its elements are 0. The bus admittance matrix is usually very sparse
since each bus is connected through a branch (nonzero mutual admittance) to only a few
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other buses. A row or column of YBUS contains 3 to 4 nonzero entries, on average. Consequently, at least 96% of the elements in the bus admittance matrix for a typical 100 bus system are 0. The bus impedance matrix, in contrast, frequently contains no zero elements.
Programs which take advantage of sparsity may store only the nonzero elements along with
the row-column locations of those entries. Also, numerical techniques have been developed
that are especially effective in performing operations on sparse matrices.
Matrix partitioning is also useful for reducing computer storage and processing requirements.
One particularly powerful application of partitioning is network reduction. Buses that are not
connected to current sources can be grouped together in submatrices within the expression
IBUS + YBUS VBUS. These buses can be eliminated from the expression through a straightforward series of matrix operations (Stevenson [B34]).
There is often a trade-off of increased processing time associated with techniques to reduce
computer memory requirements. Extra steps may be required to locate an element to be used
in a calculation if it is not stored in standard matrix form. However, some memory reduction
schemes are commonly used in power system analysis programs in order to maximize the
number of buses that can be handled on a given size computer.
5.2.3 Solution of simultaneous algebraic equations
Modeling of many problems in science, engineering, or business results in systems of simultaneous algebraic equations that must be solved for several unknowns. Simultaneous equations typically exist in cases of interdependent quantities. For example, the current that one
conductor in an underground duct bank can carry is dependent on the heat produced by the
current flowing in all of the conductors in the duct bank. Similarly, all the voltages and currents of a power system are mutually dependent. The development of the equations and the
associated matrix representation (i.e., IBUS = YBUS VBUS) for modeling this relationship has
been described. The solution of these equations is the basis for load flow analysis as
described in Chapter 6.
Simultaneous algebraic equations may be classified as linear or nonlinear. In linear equations, the sum of one or more variables of degree 1, each multiplied by a constant coefficient,
is equal to a constant. Nonlinear equations may contain variables raised to powers other than
1, division of an expression by a variable, multiplication of two or more variables, or expressions with nonalgebraic functions of variables (e.g., trigonometric). In this context, a linear
equation is not just the equation of a line in the xy plane. Systems of linear equations are
common in many kinds of analysis even though, by definition, they represent a small subset
of all types of equations. Sometimes a linear system of equations becomes nonlinear due to
constraints placed on the solution. For example, in the load flow problem, real and reactive
power for each bus, rather than bus current, are typically known. When the required substitution (I = (P–jQ)/V*) is made, the equations become nonlinear.
Techniques for solving simultaneous algebraic equations may be described as either direct or
iterative (Stagg and El-Abiad [B32]). Direct methods are applicable to linear systems of
equations. Iterative methods may be employed to solve either linear or nonlinear systems of
equations. Sometimes a combination of direct and iterative methods is used. Regardless of
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method, some type of computer-determined solution becomes almost essential for systems of
more than three or four equations.
5.2.3.1 Direct methods
A direct method determines a solution in a predictable number of arithmetic operations. Such
methods are also called “exact” since all steps are carried out according to strict mathematical
definitions without assuming values for any unknowns. Unfortunately, these methods can be
far from exact unless measures are taken to minimize the cumulative effect of round-off
errors.
Several direct methods, useful for solving systems of linear equations, involve the use of
determinants. A determinant is similar to a matrix in that it consists of entries arranged in
rows and columns. A determinant is denoted by vertical bars instead of the brackets or parentheses used to indicate a matrix. The main difference between a determinant and a matrix is
that a single value can be calculated for a determinant, while the number of entries in a matrix
cannot be changed. A determinant is defined based on a system of two linear equations as
follows (Spiegel [B31]):
a1x + b1y = c1
a2x + b2y = c2
This system of equations, representing two lines in the xy plane, can be solved simultaneously by various algebraic manipulations to find the unknowns x and y, which are the coordinates where the two lines intersect.
c1 b2 – b1 c2
x = ---------------------------a1 b2 – b1 a2
a1 c2 – c1 a2
y = ---------------------------a1 b2 – b1 a2
Given the definition of a determinant of order 2,
a b = ad – bc
c d
we can write the solutions for x and y in determinant form.
c1 b1
c2 b2
x = -------------------a1 b1
a2 b2
a1 c1
a2 c2
y = -------------------a1 b1
a2 b2
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Examination of these expressions shows that the denominator for both x and y is a determinant containing the coefficients of x and y in the original system of equations. The numerator
for x is in the same form, except the coefficients of x are replaced by the constants c1 and c2.
Similarly, to write the numerator of y, the coefficients of y are replaced by the constants. This
procedure, known as Cramer’s rule, is applicable to systems of any number of linear
equations.
To evaluate a determinant of order greater than 2, some additional definitions are required.
Given a square matrix A of order n, there is an associated determinant, det(A). Taking any
element aij in A, a new determinant of order (n – 1) can be obtained by removing row i and
column j. This new determinant is called the minor of aij. If the minor of aij is multiplied by
(–1)i +j, the result is called the cofactor of the element aij and is denoted Aij. The value of a
determinant can be calculated by adding the products aij Aij for all the elements in any row or
column. This rule, called the Laplace expansion, is illustrated below to find the determinant
associated with the bus admittance matrix developed in Figure 5-3.
det (Y) = j
– 20 10 0
10 – 15 4
0 4 – 5.4
If this is expanded in terms of cofactors for the first column, the result is as follows:
det ( Y ) = – j20 ( – 1 )
1+1
– 15
4
4
2 + 1 10
+ j10 ( – 1 )
– 5.4
4
0
+0
– 5.4
= – j20 ( 81 – 16 ) – j10 ( – 54 – 0 ) = – j760
Note that the 0 in the first column that was picked for expansion eliminated the need to
evaluate the third determinant. A technique for systematically introducing zeros by algebraic
manipulations is known as the Gauss elimination method (Buchanan and Turner [B4],
Burden and Faires [B5], Gerald and Wheatley [B10], Stagg and El-Abiad [B32]). This
technique can also be applied directly to a system of linear equations. The result is a
coefficient matrix with all diagonal elements equal to 1 and all elements below the diagonal
equal to 0. The solution can then be determined by a simple process of back substitution. The
Gauss elimination method is much more efficient computationally than Cramer’s rule. For
example, solving a set of 10 simultaneous equations requires about 70 million multiplications
and divisions using the Laplace expansion compared to only about 400 multiplications and
divisions using Gaussian elimination (Gerald and Wheatley [B10]). An example solution
using the Gauss elimination method is included at the end of this section.
Elimination techniques are based on several valid operations for systems of linear equations:
(1) any equation may be multiplied by a constant; (2) any equation may be replaced by its
sum with another equation; and (3) the order of the equations may be changed. Computer
programs reorder rows to avoid dividing by 0 and to minimize round-off errors in
calculations. This technique is known as pivoting. Variations on the Gauss elimination
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method, including LU decomposition, have been developed to reduce computer storage
requirements and processing time. The LU decomposition method begins by transforming the
coefficient matrix A into the product of two matrices L and U, where L is a lower triangular
matrix with 0’s above the diagonal and U is an upper triangular matrix with 1’s on the diagonal and 0’s below the diagonal (Gerald and Wheatley [B10]).
Matrix inversion was previously described as a useful technique for solving systems of linear
equations. However, the method shown for inverting a matrix only generates more equations
that must be solved simultaneously using another method. Now that determinants and cofactors have been introduced, an effective matrix inversion procedure can be presented. If A is a
square matrix and det(A) is not zero (i.e., A is nonsingular), then a unique inverse A–1 exists
that can be expressed as follows:
A
–1
T
[ A ij ]
= ----------------det ( A )
where
[Aij] is the matrix of the cofactors of the corresponding elements aij in A.
[Aij]T indicates its transpose (i.e., its rows and columns are interchanged).
To illustrate this method, a system of equations will be solved by matrix inversion. The basis
of the method will be clarified by solving the same equations by Cramer’s rule. The system of
equations
8x1 + 3x2 = 14
5x1 + 2x2 = 9
can be expressed in matrix form AX = B as
8 3 x 1 = 14
9
5 2 x2
The solution matrix X can be determined as X = A–1B:
1+1
(2)
[ A ij ] = ( – 1 )
2+1
( –1 )
(3)
T
det ( A ) =
83
52
( –1 )
1+2
(5)
( –1 )
2+2
(8)
T
=
2 –3
–5 8
= (8)(2) – (3)(5) = 1
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A
–1
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T
[ A ij ]
= ----------------- = 2 –3
det ( A )
–5 8
Notice that A–1 matches the matrix determined previously using the fundamental definition
of an inverse matrix. Now the solution of the equations can be determined by matrix
multiplication.
x1
x2
=
2 – 3 14 = ( 2 ) ( 14 ) + ( – 3 ) ( 9 ) = 1
–5 8 9
( – 5 ) ( 14 ) + ( 8 ) ( 9 )
2
Using Cramer’s rule:
14 3
9 2
1
( 14 ) ( 2 ) – ( 3 ) ( 9 )
x 1 = ------------------- = ----------------------------------------- = --- = 1
1
(8)(2) – (3)(5)
83
52
8 14
5 9
2
( 8 ) ( 9 ) – ( 14 ) ( 5 )
x 2 = ------------------- = ----------------------------------------- = --- = 2
1
1
1
The similarity in the calculations for the two methods shows the common basis for the
techniques.
To solve the equations using the Gauss elimination method, augment the coefficient matrix A
with B.
8 3 14
5 2 9
Create a 0 in the first column of the second row by multiplying the first row by 5/8 and
subtracting the result from the second row.
8 3
14
0 1⁄8 1⁄4
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Create 1’s on the diagonal of the coefficient matrix by multiplying the first row by 1/8 and the
second row by 8.
1 0.375 1.75
0
1
2
Therefore, x2 = 2 and x1 = 1.75 – (0.375)(2) = 1.
Additional direct methods for solving systems of linear equations are described in
bibliographic references (Buchanan and Turner [B4], Burke and Faires [B5], Gerald and
Wheatley [B10], Heydt [B18], Maron [B21], Press, Flannery, Teukolsky, and Vetterling
[B26], Stagg and El-Abiad [B32]).
5.2.3.2 Iterative methods
Iterative methods for solving linear and non-linear simultaneous algebraic equations are sometimes called “approximate.” Generally, however, a solution determined by an iterative method
can be as accurate as required. Also, round-off errors tend to be corrected from one iteration to
the next. Useful iterative methods should have the following features (Grove [B14]):
a)
b)
c)
A means for making a satisfactory first guess
A means for systematically improving on previous approximations
A criterion for stopping the iterations when sufficient accuracy has been achieved
The number of successive approximations, or iterations, needed to arrive at a solution of the
desired accuracy cannot be predicted in advance. Depending on the nature of the equations,
the choice of initial approximations, and the specific solution technique, the successive
approximations may converge to an accurate solution quickly or slowly. Sometimes each
iteration will cause the approximations to oscillate or become less accurate (i.e., diverge). For
some types of problems, one solution technique may fail consistently, while another may
provide reliable results. Two commonly used iterative methods, Gauss-Seidel and NewtonRaphson, are described here in general terms. These methods are widely used in programs for
load flow analysis. Specific application considerations and comparison information for these
methods, as applied to load flow analysis, are included in Chapter 6.
The Gauss-Seidel iterative method is a simple substitution technique in which a variable calculated using one equation is substituted in the following equations to calculate other variables. The calculations are repeated until the results match the previous iteration within a
specified tolerance. To implement the Gauss-Seidel method, each equation is rewritten to isolate a different unknown variable. This is shown below for a system of three linear equations.
a11 x1 + a12 x2 + a13 x3 = b1
a21 x1 + a22 x2 + a23 x3 = b2
a31 x1 + a32 x2 + a33 x3 = b3
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The variable x1 can be isolated in the first equation and x2 and x3 in the second and third
equations, respectively. Provided that the equations are processed sequentially in the order
listed, we can also assign a superscript of k or k + 1 to indicate the iteration number of the
variable being calculated or the variable used in a calculation.
x1k+1 = 1/a11 (b1 – a12 x2k – a13x3k)
x2k+1 = 1/a22 (b2 – a21 x1k+1 – a23 x3k)
x3k+1 = 1/a33 (b3 – a31 x1k+1 – a32 x2k + 1)
The iteration process begins with calculations for iteration number 1 (i.e., k = 0). An initial
guess for a variable is used if its iteration number is 0. In calculating x2 for the first iteration,
the value just calculated for x1 is used; but a calculation for x3 has not yet been made so its
initial value is used. Unless the nature of the problem provides a means of guessing initial
values, the following is generally used:
xi0 = bi /aii
for i = 1, 2, …, n
When x3 is calculated, an iteration is finished. If the changes in the variables from the previous iteration are within the specified tolerance, the procedure is stopped. Otherwise, k is
incremented by 1, and calculations for another iteration are performed.
Although this procedure may seem involved, it is quite easy to implement in a computer program. A simple BASIC language program for solving the same system of equations previously solved using direct methods is shown in Figure 5-4. Comments are included in the
program listing to clarify the main functions. The program prints iteration number, x1, and x2,
after each iteration. The beginning and ending sections of the printout are also shown in
Figure 5-4. Changes in x1 and x2 of less than 0.00001 between iterations indicate that the program has converged to a solution of sufficient accuracy. Notice that 151 iterations were
required before the solution converged. Acceleration factors are often used to speed up convergence with the Gauss-Seidel method. This is done by replacing each new value of xi with
the following accelerated value:
accelerated xik + 1 = xik + a (xik +1 – xik)
where
acceleration factor
a = 1.0 to 1.8
i = 1, 2, …, n
When this was added to the program of Figure 5-4, the optimum acceleration factor was
found to be 1.61, which reduced the required number of iterations to 27. Factors greater than
1.8 caused the solution to diverge. The optimum acceleration factor will vary depending on
the nature of the equations. The value 1.6 is commonly used in Gauss-Seidel load flow
solutions.
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Figure 5-4—Computer program using Gauss-Seidel method
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The characteristics of the Newton-Raphson iterative method are quite different from those of
the Gauss-Seidel method. While the Gauss-Seidel method tends to require a large number of
simple iterations, the Newton-Raphson method needs relatively few iterations; but the calculations needed to complete an iteration are extensive. The Newton-Raphson method is only
useful for systems of nonlinear equations since solution of a linear system of equations by
another method is part of each Newton-Raphson iteration.
Since the Newton-Raphson method is also applicable to one equation in one variable, a
graphical development of the procedure based on Figure 5-5 is helpful (Grove [B14]). Given
an equation in the form f (x) = 0, we can graph the function as y = f (x). The solution of the
equation is the x-coordinate where the curve crosses the x-axis (i.e., y = 0). If an initial guess
at the solution of x0 is made, the corresponding y-coordinate can be calculated as y0 = f (x0).
A line tangent to the curve can be drawn at (x0, y0). The x-coordinate (x1) at which this line
crosses the x-axis will be a closer approximation to the actual solution. Analytically, we can
take the equation of the tangent line and solve it to find the point (x1, 0):
df
y – y 0 = ------ ( x – x 0 )
dx 0
df
0 – f ( x 0 ) = ------ ( x 1 – x 0 )
dx 0
df
where -----dx
0
= the derivative (slope) of the function f(x) with respect to x evaluated at the
point (x0, y0)
To show the similarity of this equation to that presented later for the case of two equations in
two unknowns, it can be rewritten as follows:
df
f ( x 0 ) + ------ ( x 1 – x 0 ) = 0
dx 0
The function and its derivative can now be evaluated for x = x0 to calculate an improved
approximation of the solution:
f ( x0 )
x 1 = x 0 – ------------df
-----dx 0
This process can be repeated to find x2, the next approximation.
This method has been developed based on a graphical approach; but the same result can be
obtained by expanding f (x) as a Taylor series about x0 and truncating the series after the first
derivative term. This approach is required to rigorously develop the Newton-Raphson method
for multiple equations in multiple unknowns. Here, we will take the case of two equations in
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Figure 5-5—Newton-Raphson method to solve an equation
of the form f(x) = 0
two unknowns and simply write the Newton-Raphson equations using the similarity to the
case of one equation in one unknown as justification. Given a system of two equations in the
form
f (x, y) = 0
g (x, y) = 0
and an initial guess at the solution (x0, y0), the following linear system of equations can be
solved simultaneously to find an improved approximation (x1, y1):
∂f
∂f
f ( x 0, y 0 ) + ------ ( x 1 – x 0 ) + ----- ( y 1 – y 0 ) = 0
∂y 0
∂x 0
∂g
∂g
g ( x 0, y 0 ) + ------ ( x 1 – x 0 ) + ------ ( y 1 – y 0 ) = 0
∂x 0
∂y 0
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These equations contain additional terms because of the second variable; but the form of the
equations matches that for the case of one equation in one unknown. Partial derivatives are
required since we are now dealing with functions of more than one variable. Although the equations look formidable, everything reduces to simple numbers except the variables x1 and y1.
Iterative methods provide the only means for solving many types of simultaneous algebraic
equations. Additional information on this subject may be found in numerous texts about
numerical methods (Arbenz and Wohlhauser [B1], Grove [B14], Maron [B21], Press, Flannery, Teukolsky, and Vetterling [B26], Stagg and El-Abiad [B32]).
5.2.4 Solution of differential equations
The dynamic performance of a physical system is described by one or more differential equations. The physical system may be mechanical, electrical, fluidal, thermal, or a combination
of these [B4]). For example, in a complex power system transient stability study, differential
equations describing synchronous machine rotor positions and speeds, internal voltages,
exciter control systems, and prime mover speed governors are solved simultaneously (Stagg
and El-Abiad [B32]).
The Laplace transform technique is useful for evaluating the dynamics of a system since easily
manipulated algebraic equations are substituted for the more difficult differential equations
(Dorf [B7], Edminister [B8], Hayt and Kemmerly [B17], Rainville and Bedient [B28], Scott
[B29], Spiegel [B31]). The technique permits a complex system to be divided into smaller elements, each described by a transfer function. The transfer function of a system, or part of a system, is defined as the Laplace transform of its output variable divided by the Laplace transform
of its input variable. The transfer function for each part of a system may be represented within
a block of a diagram like that shown for an exciter control system in Figure 5-6. Numerous
block diagram transformations are available to reduce such a diagram to a single transfer function (Dorf [B7]). For a computer program solution, however, the individual transfer functions
are usually replaced by differential equations that are solved simultaneously. For example, the
differential equation for the first block in Figure 5-6 can be determined as follows:
E
1
-----2- = -----------------E 1 1 + sT R
Cross-multiplying yields the frequency-domain equation:
1
sE 2 = ------ ( E 1 – E 2 )
TR
Since the Laplace operator s can be treated as the differential operator d/dt, we can write the
time-domain differential equation as follows:
dE 2 1
--------- = ------ ( E 1 – E 2 )
TR
dt
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Figure 5-6—Block diagram of an exciter control system
First-order differential equations of this type can be solved at desired time increments using
a mathematical approach similar to that used in the graphical development of the Newton-Raphson equation. A differential equation of order n can be represented as n first-order
equations so that the methods described below can be used. For each time increment, the
value of the dependent variable is determined. The calculated values approximate the smooth
curve, which is the actual solution of the differential equation. For example, if we know E1
and E2 at time zero, we can calculate the slope (dE2 /dt) of a line tangent to the solution curve
at time zero. If the time increment ∆t is sufficiently small, we can assume that a point on the
tangent line also falls on the solution curve. Therefore, a new value of E2 after the first time
increment can be calculated as follows:
1
0 dE 2
E 2 = E 2 + --------- ∆t
dt 0
This value for E2 may be used in a similar manner to calculate a value for E2 after the second
time interval. This method for solving a differential equation is known as Euler’s method.
Euler’s method is not recommended for serious use because more accurate methods are available. The modified Euler method uses an iteration scheme to find the slope of the tangent line
at the end of each time interval. The average slope over the time interval is used, instead of
the beginning slope, to calculate each new solution. One of the most commonly used methods
for the numerical solution of differential equations is the fourth-order Runge-Kutta method.
In this method, the slope is evaluated four times during each interval: once at the initial point,
twice at trial midpoints, and once at a trial end point. These four slope values are used in a
special formula to calculate the average slope over the time interval without using iteration.
More information on these and other methods for solving differential equations is available in
several texts (see Press, Flannery, Teukolsky, and Vetterling [B26], Rainville and Bedient
[B28], Stagg and El-Abiad [B32]).
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5.3 Computer systems
A few of the numerical techniques used in power system analysis programs have been
described in order to eliminate some of the mystery in how computers generate answers to
problems too big to tackle manually. Many other aspects of computer programs, or software,
should be understood before selecting or using computer techniques. The computer itself and
peripheral equipment (i.e., hardware) must be considered in conjunction with software. A
program is designed to be used on a particular type of computer, although alternate versions
for different computers are sometimes available. Factors important in choosing the most
appropriate hardware and software for power system analysis are described in this section.
The improvements in computer technology continue to be rapid and dramatic. For this
reason, the discussion will be kept somewhat general. Still, some of the information will
probably seem antiquated in a few years.
5.3.1 Computer terminology
The following is a brief, informal glossary of terms frequently used in discussing hardware
and software (Markowsky [B20], Somerson [B30], Stallings [B33]):
ASCII: A 7-bit digital code used by many computers, including most personal computers, to
represent 128 alphanumeric and device control characters. ASCII stands for American Standard Code for Information Interchange. Many computers use an eighth bit to provide 128
more nonstandard special characters (e.g., graphics).
binary: A number system using only the digits 0 and 1 (also called base 2). Binary is
employed in digital computers since the use of off and on to represent 0 and 1 is easily
implemented in electronic circuits.
bit: An acronym for binary digit (i.e., a single 0 or 1). Usually, bit is abbreviated as lowercase
b.
bug: An error in a program or a hardware flaw.
bus: In a computer, a group of conductors carrying related signals between devices. Bus
width, expressed in bits (typically 8, 16, 32, or 64), affects the processing speed of a computer (wider = faster).
byte: A string of 8 bits treated as a unit. One byte is required to store one character. Usually,
byte is abbreviated as uppercase B.
central processing unit (CPU): The part of a computer that fetches and executes instructions. It consists of an arithmetic and logic unit (ALU), a control unit, and a number of registers for storing data currently being processed.
copy protection: A method for preventing a program from being used on more computers
than it was purchased for.
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cursor: A pointer that appears on a monitor, usually to show where characters will appear as
they are typed on the keyboard. In a graphical user interface (GUI) the cursor may be moved
with a mouse (or other pointing device), and its shape may change when moved from a text
input position to a menu selection.
default: A value or option assumed by a program when none is specified by the user.
disk (or disc): A revolving, flat, circular plate coated with magnetic material upon which
data is recorded in concentric tracks. A “hard” or “fixed” disk holds large quantities of data
and is permanently installed. A “floppy” disk or diskette is removable and stores a much
smaller volume of data on a thin, flexible magnetic disk inside a flexible or rigid cover.
expert system: A computer application that solves problems by inference using the knowledge of human experts coded in the form of rules, logic, or cases. An expert system is a form
of Artificial Intelligence (AI). Expert systems may be used in power system analysis software
for data checking, case selection, and analysis of results (Mitsche [B22], Venkata, Sumic,
Vadari, and Liu [B35]).
graphical user interface (GUI): typically utilizes displays with windows, dialog boxes,
pull-down menus, selection boxes, diagrams, and pictures to simplify and enhance communication with a software user. The user generally operates a mouse to move the cursor and make
selections when using a GUI (Chan [B6]).
K: In reference to data storage, used as a prefix meaning 1024 (210). For example, 2 KB
(kilobytes) is 2048 bytes. If K is used alone in this context, bytes are implied.
20
M: In reference to data storage, used as a prefix meaning 1 048 576 (2 ). For example,
10 MB (megabytes) is 10 485 760 bytes. If M is used alone in this context, bytes are implied.
microprocessor: A single integrated circuit, or “chip,” which performs the functions of the
central processing unit (CPU). A microprocessor is used as the main “engine” in personal
computers and many other computers. Microprocessor type is a key factor determining the
processing power of such computers. A related factor is the microprocessor clock speed (in
megahertz [MHz]) that sets the rate at which internal functions are performed.
modem: Acronym for modulator/demodulator. Converts digital signals to analog and vice
versa, so that two computers can communicate over a telephone line.
monitor: A video display (often containing a cathode ray tube [CRT]) which is used to display program output and data entered by the user on the keyboard. A wide range of types,
from monochrome to high-resolution color, are available.
operating system: Software that controls the execution of other programs and the interaction
with disk drives, printers, and other devices. Operating systems maybe described as single
user, multitasking, or multiuser.
pel: See: pixel.
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pixel: A picture element or dot used to make up the display on a monitor. The more pixels
that a monitor displays, the higher its resolution will be. Syn: pel.
RAM: Random access memory that can be read from and written to by a user program. The
amount of memory required by a program is an important factor for matching hardware to
software.
ROM: Read-only memory that can only be read from. Usually, ROM contains programs that
control certain basic computer functions. A program stored in ROM is called firmware (software directly implemented in hardware).
terminal: A keyboard and monitor for accessing a remote computer. Also called video display terminal (VDT), or simply cathode ray tube (CRT). A teletypewriter can also be considered a terminal.
word: The natural unit of memory in a computer consisting of a series of bits treated as a unit
and stored at one memory address. The number of bits in a word generally matches the internal bus width.
5.3.2 Computer hardware
There is probably no area of technology that changes as quickly or as dramatically as that
related to computer hardware. In particular, the steadily increasing power of desktop
computers has changed many basic ideas about data processing. Even the categories of
computer types have blurred as computers of one type begin to match, or even exceed, the
capabilities associated with a more powerful category of computers (Gleason [B11], Grant,
Laskowski, and Weekley [B13], Murphy [B23], Pollack [B25]). Because of these advancements, even the smallest engineering organizations can now afford some kind of genuinely
useful computer equipment. In fact, the need to remain competitive pushes most companies
toward expanded computer usage in all areas. The requirements of key software to be implemented should be checked carefully when selecting hardware. In addition, current computer
periodicals should be reviewed for help in selecting hardware.
There may still be valid reasons, however, for not using an in-house computer for all power
system studies. For example, the purchase cost for some complicated analysis programs (e.g.,
transient stability) may make use of a program from a computer time-sharing service more
economical for infrequent studies (e.g., one per year or fewer). Even in this case, a personal
computer with a modem would typically be used to access the program running on the timesharing computer. Computer time-sharing was a big business in the early 1980s; but the availability of software and inexpensive hardware has forced many such companies to disband.
Several companies, however, still offer time-sharing programs for power system analysis.
Also, comprehensive studies can be performed by engineering consultants, either as part of
an expansion project design contract or as a stand-alone service. Since certain studies are
already required for significant expansion projects, it is a good time to request a thorough
review of the plant distribution system. A stand-alone study by a consultant is justified when
qualified plant engineering personnel are unavailable. Such studies are more economical if
the client assists in developing the input data.
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There are many available options to consider if in-house hardware is to be used for power
system studies. If the prospective user is already doing other work on a computer, then
obtaining suitable analysis software for the existing equipment would be a natural first
approach. It’s best not to jump to this conclusion, however, since analysis programs may be
much more effective with a faster processor, higher resolution color monitor for graphics, and
better printing/plotting equipment. Generally, hardware can be obtained either for lease or for
purchase, depending on company financial considerations. Although distinctions are
sometimes vague, four types of computer systems suitable for engineering applications are
described below. In order of increasing cost (although there is some overlap), the categories
are personal computer, engineering workstation, server, and mainframe. A fifth category,
supercomputer, is not described here since these processors, the fastest and most expensive
number crunchers available, are not necessary for analyzing even the largest industrial or
commercial power systems. Supercomputers can be effectively applied, however, for
modeling interconnected utility systems with thousands of buses.
a)
Personal computers, or PCs, have been available since the late 1970s. Personal computers are described as desktop computers, although some PC system units are placed
on the floor next to a desk supporting a monitor and keyboard. Also, notebook-size
PCs weighing only a few pounds are now available that have as much processing
power as full-size units.
Until 1984, PCs were used mainly for simple, isolated tasks like word processing and
spreadsheet calculations. Since then, more powerful personal computers have been
introduced that can be used effectively for power system analysis. More advanced
microprocessors, faster clock speeds, and math coprocessor chips (which work with
the microprocessor to speed up arithmetic) have been responsible for this improvement in processing power.
As performance improves, personal computers are also being connected through
local area networks (LANs) to PC file servers and larger computers to share expensive peripherals and to access common databases. PCs can also be used simply as terminals to access programs running on servers and mainframes. Advanced graphics
capabilities suitable for demanding computer-aided design and drafting (CADD) are
available for use with PCs. Newer operating systems have been implemented which
allow a PC to perform several tasks at the same time, support a graphical user interface, and handle programs that need large amounts of memory. These improvements
are bringing top-of-the-line personal computers into the same class as engineering
workstations.
b)
An engineering workstation can be described as a high-powered, multitasking, networked, desktop computer with large memory, designed for scientific, engineering,
and CADD applications. This description is essentially the same as for high-end PCs.
Also, the cost of some engineering workstations is dropping into the personal computer range. Terms like “personal workstation” are being used to describe the apparent merging of the two categories.
c)
The term server is now generally used to describe a multiuser computer that maintains a database, manages files, or runs applications. The processing for some applications can be split between the server and a PC or workstation. PCs or workstations
are the “clients” in this architecture known as client/server. The term server has
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largely replaced the term minicomputer, which was first used in the 1960s to describe
a class of computer that was small enough to fit into an office, yet capable of serving
about five people working at terminals. The capabilities of these computers increased
to support a few dozen users. When more advanced computers using 32-bit words
were introduced in the late 1970s, the term super-minicomputer was used to distinguish them from 16-bit minicomputers.
d)
A mainframe is a large, general-purpose computer supporting perhaps hundreds of
users, sometimes in worldwide locations. Other computers, also called mainframes,
are no more powerful than large servers. Again, there is a blurring of the distinction
in the categories. Mainframes and large servers are supported by an information systems staff. Usually, there is some type of intracompany accounting and billing for the
use of time on these types of computers.
5.3.3 Power system analysis software
Programs for power system analysis are available for all of the computer types described in
the previous section. Usually, a program for a larger computer will cost more than an
equivalent program for a smaller computer. This is not unreasonable because development
costs are usually more and because more people can use the one copy of the program
installed on a central mainframe or server. Alternatively, a company can develop its own
analysis software. In-house development was more common before a wide range of PC-based
software became available. Development usually cannot be justified when a usable program
is available for purchase, since, even with qualified engineering and programming personnel,
development costs will probably be at least ten times the purchase cost. Available programs
for power system analysis are frequently discussed and advertised in periodicals (see Electrical Construction & Maintenance [B9], IEEE Computer Applications in Power [B13], Plant
Engineering [B27]).
A careful evaluation is always required before purchasing any kind of engineering software.
A trial use period for the complete program is preferred over use of a limited demonstration
disk. The following points should be considered (Oliver [B24], Plant Engineering [B27]):
a)
Type of computer required. Manufacturer, model, amount of RAM needed, graphics
board and monitor, and operating system should be checked.
b)
Program accuracy. A description of benchmark tests to verify program results should
be available. The user should also run the program for a case previously solved by
another program or method.
c)
Vendor. The vendor should be investigated to determine length of time in business,
ability to support the program, and policy on program upgrades.
d)
Documentation. The user’s manual should be clear, concise, and thorough.
e)
Limitations. The program should model all items required for a particular power system including a sufficient number of buses, local generation, three-winding transformers, load tap changing transformers, industry standard adjustment factors for
short-circuit currents, etc.
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f)
Ease of use. “User friendly” programs can be operated without constant reference to
the user’s manual. Programs with on-screen help provide input assistance when
requested. Menu-driven programs allow selection of the desired program option from
a displayed list. In many cases, a mouse may be used to select menu options and to
move to desired input fields.
g)
Input. Full screen input on labeled screens as shown in Figure 5-7 is preferred over the
80 column input file approach of Figure 5-8. Programs that use input screens, menus,
and user prompts are called interactive programs. Batch programs work using an input
data file without interacting with the user during execution. Both Figure 5-7 and
Figure 5-8 show data entry methods for exciter control system data like that shown in
Figure 5-6. A program with a graphical user interface (GUI) might display the diagram
for the exciter control system and allow input of the constants by using a mouse to
select each block in the diagram. An example screen from a program with a GUI is
shown in Figure 5-9. Ideally, input data is available in a common database for use by a
family of analysis programs (e.g., load flow, short circuit). The input data should be
easy to edit after each run in order to make changes for different design cases. Data
should be validated, to the extent possible, when entered. The program should allow
input in the form normally available (e.g., transformer impedance in percent on kVA
base).
File: EXAMPLE1
Machine (Stability) Editor
Date: MAR 30, 95
Synchronous Generator
Machine #
101
Type
4
Tdo"
0.002
Name
Gen. #1
Xd
110.00
Tdo'
5.600
Bus #
1
Xq"
12.00
Tqo"
0.002
Rated MVA
35.300
Xq'
23.00
Tqo'
3.700
Rated kV
13.800
Xq
108.00
H
1.200
X/R
89.50
X1
11.00
% Damp
5.000
Xd"
12.800
S100
1.070
% Bus P
100.0
Xd'
18.600
S120
1.180
% Bus Q
100.0
VRmax
7.500
Bmin
-1.00
T4
0.020
VRmin
-5.30
C
40.00
TI
6.800
SEmax
1.650
D
0.025
TD
2.000
SE .75
1.130
Kpow
0.025
TF
2.000
Efdmax
7.500
KQ
0.003
Tdsty
0.005
Amax
4.000
KE
1.000
TP
5.000
Amin
-3.00
Ctrl Bus
3
TQ
0.011
Mode
Isoch
X
0.000
Tf
0.400
Ki
4.000
Y
0.050
Tcr
0.010
MAX
1.000
Z
0.000
Tcd
0.100
MIN
-0.07
a
1
Ttd
0.020
Tem.Ctrl
Yes
b
0.050
T
0.250
Acc.Ctrl
Yes
c
1
Tt
450.0
Kf
0
Tr
1000
Exciter
Type
HPC
Bmax
4.000
TE
0.510
Governor
Type
GTH
Governor mode of operation; Droop or Isochronous - Press SPACEBAR to toggle.
F3:Calculator
F6:Dyn/Constant Z
F7:Lib/Typ Data
F8:Find
F9:Return
Figure 5-7—Full-screen data input
Copyright © 1998 IEEE. All rights reserved.
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Figure 5-8—Forms for 80-column file input
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Figure 5-9—Graphical user interface (GUI)
h)
Output. The output report should list all input data in a readable manner so that the
user can verify the basis for the run at a later date. The pages in each section of a
solution report should be in a logical, visually pleasing format with appropriate titles.
Some programs have the ability to plot results on the system single-line diagram.
The chapters that follow discuss many aspects of software that are specific to particular
studies.
5.4 Bibliography
Additional information may be found in the following sources:
[B1] Arbenz, K., and Wohlhauser, A., Advanced Mathematics for Practicing Engineers, Norwood, MA: Artech House, 1986.
[B2] Arrillaga, J., Arnold, C. P., and Harker, B. J., Computer Modeling of Electrical Power
Systems, New York: John Wiley & Sons, 1983.
[B3] Brown, H. E., Person, C. E., Kirchmayer, L. K., and Stagg, G. W., “Digital calculation
of three-phase short circuits by matrix method,” Transactions of the AIEE, vol. 79, pt. III, pp.
1277–1281, 1960.
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[B4] Buchanan, J. L., and Turner, P. R., Numerical Methods and Analysis, New York:
McGraw-Hill, 1992.
[B5] Burden, R. L., and Faires, J. D., Numerical Analysis, Boston: Prindle, Weber & Schmidt,
1985.
[B6] Chan, S., “Interactive Graphics Interface for Power System Network Analysis,” IEEE
Computer Applications in Power, IEEE Power Engineering Society, Jan. 1990, pp. 34–38.
[B7] Dorf, R. C., Modern Control Systems, Reading, MA: Addison-Wesley, 1974.
[B8] Edminister, J. A., Theory and Problems of Electric Circuits, Schaum’s Outline Series,
New York: McGraw-Hill, 1965.
[B9] Electrical Construction & Maintenance, New York, McGraw-Hill (monthly).
[B10] Gerald, C. F., and Wheatley, P. O., Applied Numerical Analysis, Reading, MA: Addison-Wesley, 1984.
[B11] Gleason, B., “A New Culture—32-bit Technology on Your Desk,” Computer, IEEE
Computer Society, July 1988, pp. 74–76.
[B12] Gonen, T., Modern Power System Analysis, New York: John Wiley & Sons, 1988.
[B13] Grant, I. S., Laskowski, T. F., and Weekley, A. R., “Small computer capabilities vs.
large power-planning program requirements,” IEEE Computer Applications in Power, IEEE
Power Engineering Society, Jan. 1988, pp. 34–36.
[B14] Grove, W. E., Brief Numerical Methods, Englewood Cliffs, NJ: Prentice-Hall, 1966.
[B15] Groza, V. S., and Shelley, S., Precalculus Mathematics, New York: Holt, Rinehart and
Winston, 1972.
[B16] Hashemi, N., Love, D. J., and Tajaddodi, F. Y., “A Relational Database Approach to
Design of Power Plant and Large Industrial Electrical Facilities,” 1987 I & CPS Conference
Paper ICPSD-87-01, presented in Nashville, TN, May 5, 1987.
[B17] Hayt, Jr., W, H., and Kemmerly, J. E., Engineering Circuit Analysis, New York:
McGraw-Hill, 1978.
[B18] Heydt, G. T., Computer Analysis Methods for Power Systems, New York: Macmillan,
1986.
[B19] Kusic, G. L., Computer Aided Power System Analysis, Englewood Cliffs, NJ: PrenticeHall, 1986.
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[B20] Markowsky, G., A Comprehensive Guide to the IBM Personal Computer, Englewood
Cliffs, NJ: Prentice-Hall, 1984.
[B21] Maron, W. J., Numerical Analysis, New York: Macmillan, 1987.
[B22] Mitsche, J. V., “Stretching the Limits of Power System Analysis,” IEEE Computer
Applications in Power, IEEE Power Engineering Society, Jan. 1993, pp. 16–21.
[B23] Murphy, E. E., “Minisuper, Supermini: What’s in an Adjective?,” IEEE Spectrum, May
1989, p. 22.
[B24] Oliver, W. K., “Evaluating CAE Software,” Hydrocarbon Processing, Feb. 1985.
[B25] Pollack, A., “Sun, Apple on Collision Course,” Houston Chronicle, Mar. 12, 1989.
[B26] Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., Numerical Recipes in C, Cambridge, England: Cambridge University Press, 1988.
[B27] “Problem Solving Software,” Plant Engineering, Cahners, June 9, 1988, pp. 62–102.
[B28] Rainville, E. D., and Bedient, P. E., Elementary Differential Equations, New York:
Macmillan, 1974.
[B29] Scott, R. E., Elements of Linear Circuits, Reading, MA: Addison-Wesley, 1965.
[B30] Somerson, P., PC Magazine DOS Power Tools, New York: Bantam Books, 1988.
[B31] Spiegel, M. R., Theory and Problems of Advanced Mathematics for Engineers and
Scientists, Schaum’s Outline Series, New York: McGraw-Hill, 1971.
[B32] Stagg, G. W., and El-Abiad, A. H., Computer Methods in Power System Analysis, New
York: McGraw-Hill, 1968.
[B33] Stallings, W., Computer Organization and Architecture, New York: Macmillan, 1987.
[B34] Stevenson, Jr., W. D., Elements of Power System Analysis, New York: McGraw-Hill,
1982.
[B35] Venkata S., Sumic, Z., Vadari, S., and Liu, C., “Applying AI systems in the T&D
arena,” IEEE Computer Applications in Power, IEEE Power Engineering Society, April 1993,
pp. 29–34.
[B36] Wallach, Y., Calculations and Programs for Power System Networks, Englewood
Cliffs, NJ: Prentice-Hall, 1986.
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CHAPTER 5
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Chapter 6
Load flow studies
6.1 Introduction
One of the most common computational procedures used in power system analysis is the load
flow calculation. The planning, design, and operation of power systems require such
calculations to analyze the steady-state (quiescent) performance of the power system under
various operating conditions and to study the effects of changes in equipment configuration.
These load flow solutions are performed using computer programs designed specifically for
this purpose. The basic load flow question is this: Given the load power consumption at all
buses of a known electric power system configuration and the power production at each
generator, find the power flow in each line and transformer of the interconnecting network
and the voltage magnitude and phase angle at each bus.
Analyzing the solution of this problem for numerous conditions helps ensure that the power
system is designed to satisfy its performance criteria while incurring the most favorable
investment and operation costs.
Some examples of the uses of load flow studies are to determine the following:
—
—
—
—
—
—
—
Component or circuit loadings
Steady-state bus voltages
Reactive power flows
Transformer tap settings
System losses
Generator exciter/regulator voltage set points
Performance under emergency conditions
Modern systems are complex and have many paths or branches over which power can flow.
Such systems form networks of series and parallel paths. Electric power flow in these
networks divides among the branches until a balance is reached in accordance with
Kirchoff’s laws.
Computer programs to solve load flows are divided into two types—static (offline) and
dynamic (real time). Most load flow studies for system analysis are based on static network
models. Real time load flows (online) that incorporate data input from the actual networks are
typically used by utilities in automatic Supervisory Control And Data Acquisition (SCADA)
systems. Such systems are used primarily as operating tools for optimization of generation,
var control, dispatch, losses, and tie line control. This discussion is concerned with only static
network models and their analysis.
Because the load flow problem pertains to balanced, steady-state operation of power systems,
a single-phase, positive sequence model of the power system is used. Three-phase load flow
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analysis software is available; but it is not normally needed for routine industrial power
system studies.
A load flow calculation determines the state of the power system for a given load and
generation distribution. It represents a steady-state condition as if that condition had been
held fixed for some time. In actuality, line flows and bus voltages fluctuate constantly by
small amounts because loads change constantly as lights, motors, and other loads are turned
on and off. However, these small fluctuations can be ignored in calculating the steady-state
effects on system equipment.
As the load distribution, and possibly the network, will vary considerably during different
time periods, it may be necessary to obtain load flow solutions representing different system
conditions such as peak load, average load, or light load. These solutions will be used to
determine either optimum operating modes for normal conditions, such as the proper setting
of voltage control devices, or how the system will respond to abnormal conditions, such as
outages of lines or transformers. Load flows form the basis for determining both when new
equipment additions are needed and the effectiveness of new alternatives to solve present
deficiencies and meet future system requirements.
The load flow model is also the basis for several other types of studies such as short-circuit,
stability, motor starting, and harmonic studies. The load flow model supplies the network data
and an initial steady-state condition for these studies.
6.2 System representation
Utility and industrial plant electrical systems can be extensive. A simplified visual means of
representing the complete system is essential to understanding the operation of the system
under its various possible operating modes. The system single-line diagram serves this
purpose. The single-line diagram consists of a drawing identifying buses and interconnecting
lines. Loads, generators, transformers, reactors, capacitors, etc., are all shown in their
respective places in the system. It is necessary to show equipment parameters as well as their
relationship to each other. Figure 6-1 is a single-line diagram of the sample industrial plant
system that will be used later to illustrate some aspects of load flow studies. It shows the
operating condition to be studied in terms of which breakers are open or closed.
Buses may be named, numbered, or both. Interconnecting lines are usually shown with their
impedance values entered or cross-referenced with tables of values. Equipment associated
with a bus is shown connected to that bus. For instance, generators are shown connected to
their bus with their equipment parameters specified, as illustrated in Figure 6-2. Similarly, a
load is shown connected to bus 49 in Figure 6-3. Motor loads are often indicated separately to
aid in their modeling in short circuit and other studies. Each line originates on a bus and terminates on a different bus, as depicted in Figure 6-3. For example, a line runs from bus 3
(MILL-1) to bus 5 (FDR F) and is shown to be 325 ft long, 3 conductor, 250 kcmil cable.
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135
Figure 6-1—Single-line diagram of typical industrial power system for load flow study example
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Figure 6-2—Bus and generator representation
Figure 6-3—Representation of loads, lines, and transformers
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Transformers, like lines, are shown between two buses with the primary connected to one bus
and the secondary to the other. Information to convey an off-nominal turns ratio should be
given when applicable.
Drawing format will vary depending on the computer programs used and the preference of
the users, but the single-line diagram should give the necessary network information in a
clear, concise manner. The transfer of this data to the load flow program for analysis is
discussed in the next section.
6.3 Input data
The system information, shown on the single-line diagram, defines the system configuration
and the location and size of loads, generation, and equipment. It is organized into a list of
data that defines the mathematical model for each power system component and how the
components are connected together. The preparation of this data file is the foundation of all
load flow analysis, as well as other analysis requiring the network model, such as shortcircuit and stability analysis. It is therefore essential that the data preparation be performed in
a consistent, thorough manner. Data values must be as accurate as possible. Rounding off, or
not including enough decimal places in certain parameters, can lead to erroneous results.
Influential parameters must not be ignored.
The details of the models of the system components are addressed in Chapter 4. In this subclause, data organization is shown in general terms and some comments are given on data
preparation. The data is divided into the following categories, as this organization is typical of
most load flow analysis software: system data, bus data, generator data, branch data, and
transformer data. In order to illustrate the approach required when using a typical program,
the details of one such system will be described.
6.3.1 System data
As noted in Chapter 4, most load flow programs perform their calculations using a per unit
representation of the system rather than working with volts, amperes, and ohms. The input of
data to the program may either be in per unit or in physical units, depending on program
convention. Converting the system data to a per unit representation requires the selection of a
base kVA and base voltage. Selection of the base kVA and base voltage specifies the base
impedance and current.
The system data specifies the base kVA (or MVA) for the entire system. A base kVA of
10 000 kVA (10 MVA) is often used for industrial studies. For utility systems, the accepted
convention is a base of 100 MVA.
The base kV is chosen for each voltage level. Selecting the nominal voltage to be the base
voltage simplifies the analyses and reduces the chance of errors in interpretation of results.
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6.3.2 Bus data
The bus data describes each bus and the load and shunts connected to that bus. The data
includes the following:
—
—
—
—
—
—
—
Bus number
Bus name
Bus type
Load
Shunt
Per unit voltage and angle
Bus base kV
The bus number is normally the primary index to the information about the bus. For example,
it is used to define the line connections in the line data and will be used to get output about a
bus during program execution. The bus name is normally used only for informational
purposes, allowing the user to give a descriptive name to the bus to make program output
more easily understood. Some programs allow the use of the bus name as the primary index.
The bus type is a program code to allow the program to properly organize the buses for load
flow solution. This organization varies among programs and may be handled internally by the
program. Typically, the four bus types are as follows:
a)
b)
c)
d)
Load buses
Generator buses
Swing buses
Disconnected buses
The terms “load” bus and “generator” bus should not be taken literally. A load bus is any bus
that does not have a generator. A load bus need not have load; it may simply be an
interconnection point for two or more lines. A generator bus could also have load connected
to it. The “swing” or “slack” bus is a special type of generator bus that is needed by the
solution process. The swing generator adjusts its scheduled power to supply the system MW
and Mvar losses that are not otherwise accounted for. This is explained in more detail in the
section on load flow solution. There is normally only one swing bus. In utility studies, a large
generator is picked as the swing bus. In industrial studies, the utility supply is usually
represented as the swing bus. A “disconnected” bus is a bus that is temporarily deenergized. It
is not included in the load flow solution and must not have in-service lines connected to it.
The data is retained so the bus could be reenergized (connected to the system) later in the
studies.
Load is normally entered in MW and Mvar at nominal voltage. Normally, the load is treated
as a constant MVA, that is, independent of voltage. In some cases, a constant current or
constant impedance component of load will also be entered so that the load is a function of
voltage as explained in 4.9. Shunts generally are entered in Mvar at nominal voltage. Care
must be taken to ensure the proper sign convention is used to distinguish reactors from
capacitors as defined for the particular load flow program being used.
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The bus base kV is often entered to allow output reports to show voltages in kV and currents
in amperes.
6.3.3 Generator data
Generator data is entered for each generator in the system including the system swing
generator. The data defines the generator power output and how voltage is controlled by the
generator. The data items normally entered are as follows:
—
—
—
—
—
Real power output in MW
Maximum reactive power output in Mvar (i.e., machine maximum reactive limit)
Minimum reactive power output in Mvar (i.e., machine minimum reactive limit)
Scheduled voltage in per unit
Generator in-service/out-of-service code
Other data items that may be included are the generator MVA base and the generator’s
internal impedance for use in short-circuit and dynamic studies. The use of the above data
items to determine the generator voltage and reactive power output is discussed in 6.4. Some
programs may allow a generator to regulate a remote bus voltage.
6.3.4 Branch data
Data is also entered for each branch in the system. Here the term “branch” refers to all
elements that connect two buses including transmission lines, cables, series reactors, series
capacitors, and transformers. The data items include the following:
—
—
—
—
—
—
Resistance
Reactance
Charging susceptance (shunt capacitance)
Line ratings
Line in-service/out-of-service code
Line-connected shunts
As described in 4.7, lines are represented by a π model with series resistance and reactance
and one-half of the charging susceptance placed on each end of the line. The resistance,
reactance, and susceptance are usually input in either per unit or percent, depending on
program convention.
Line rating are normally input in amperes or MVA. Current ratings can be converted to MVA
with the formula:
3 × kV BASE × rating A
rating MVA = -----------------------------------------------------1000
(6-1)
A series reactor, series capacitor, or transformer would not have a charging susceptance term.
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The modeling of the charging susceptance is often ignored for short overhead lines and
industrial plant systems.
6.3.5 Transformer data
Additional data is required for transformers. This can either be entered as part of the branch
data or as a separate data category depending on the particular load flow program being used.
This additional data usually includes the following:
—
—
—
—
—
Tap setting in per unit
Tap angle in degrees
Maximum tap position
Minimum tap position
Scheduled voltage range with tap step size or a fixed scheduled voltage using a
continuous tap approximation
The last three data items are needed only for load tap changing (LTC) transformers that automatically vary their tap setting to control voltage on one side of the transformer.
The organization of transformer tap data requires an understanding of the tap convention used
by the load flow program to ensure the representation gives the correct boost or buck in
voltage. Transformers whose rated primary or secondary voltages do not match the system
nominal (base kV) voltages on the terminal buses will require an off-nominal tap
representation in the load flow (and possibly require corresponding adjustment of the
transformer impedance).
6.4 Load flow solution methods
The computational task of determining power flows and voltages for even a small network for
a given system condition is formidable. Solution of large networks for many system
conditions as required for analysis of present-day power systems requires sophisticated
computational tools.
The first load flow solution device was a special purpose analog computer called the ac
network analyzer developed in the late 1920s. Power system networks under study were
represented by an equivalent, scaled-down network. The device allowed the analysis of a
variety of operating conditions and expansion plans. However, setup time was long. Due to
the large amount of hardware involved, only about 50 network analyzers were operational by
the mid-1950s.
Digital computers began to emerge in the late 1940s as computational tools. By the mid1950s, large-scale digital computers of sufficient speed and size to handle the requirements of
a power system network calculation were available. Parallel to the hardware development,
algorithms to efficiently solve the network equations were developed. Ward and Hale
developed a successful load flow program using a modified Newton iterative procedure in
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1956 [B5].1 The application of the Gauss-Seidel iteration algorithm followed soon after.
Research in algorithms continued and the Newton-Raphson method was introduced in the
early 1960s [B4]. Considerable research has been performed in the interim years to improve
the performance of these algorithms, making them more robust, able to handle additional
power system components, and allowing much larger network sizes.
6.4.1 Problem formulation
The load flow calculation is a network solution problem. The voltages and currents are
related by the following equation:
[ I ] = [Y ][V ]
(6-2)
where
[I ]
[V ]
[Y]
is the vector of total positive sequence currents flowing into the network nodes (buses)
is the vector of positive sequence voltages at the network nodes (buses)
is the network admittance matrix
Equation (6-2) is a linear algebraic equation with complex coefficients. If either [I] or [V]
were known, the solution for the unknown quantities could be obtained by application of
widely used numerical solution techniques for linear equations.
Partly because of tradition and partly because of the physical characteristics of generation
and load, the terminal conditions at each bus are normally described in terms of active and
reactive power (P and Q). The bus current at bus i is related to these quantities as follows:
( P i + jQ i )*
I i = -------------------------V i*
(6-3)
where * designates the conjugate of a complex quantity. Combining Equations (6-2) and
(6-3) yields
P – jQ
---------------- = [ Y ] [ V ]
V*
(6-4)
Equation (6-4) is nonlinear and cannot be readily solved by closed-form matrix techniques.
Because of this, load flow solutions are obtained by procedures involving iterative
techniques.
6.4.2 Iterative solution algorithms
Since the original technical papers describing digital load flow solution algorithms appeared
in the mid-1950s, a seemingly endless collection of iterative schemes has been developed and
1 The
numbers in brackets preceded by the letter B correspond to those of the bibliography in 6.10.
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reported. Many of these are variations of one or the other of two basic techniques that are in
widespread use by the industry today: the Gauss-Seidel technique and the Newton-Raphson
technique. The preferred techniques used by most commercial load flow software are variations of the Newton technique.
All of these techniques solve bus equations in admittance form, as described in the previous
section. This system of equations has gained widespread application because of the simplicity
of data preparation and the ease with which the bus admittance matrix can be formed and
changed in subsequent cases.
In a load flow study, the primary parameters are as follows:
P
Q
|V |
δ
is the active power into the network
is the reactive power into the network
is the magnitude of bus voltage
is the angle of bus voltage referred to a common reference
In order to define the load flow problem to be solved, it is necessary to specify two of the four
quantities at each bus. For generating units, it is reasonable to specify P and |V | because these
quantities are controllable through governor and excitation controls, respectively. For loads,
one generally specifies the real power demand P and the reactive power Q. Since there are
losses in the transmission system and these losses are not known before the load flow solution
is obtained, it is necessary to retain one bus where P is not specified. At this bus, called a
swing bus, |V | as well as δ are specified. Since δ is specified (that is, held constant during the
load flow solution), it is the reference angle for the system. The swing bus is therefore also
called the reference bus. Since the real power, P, and reactive power, Q, are not specified at
the swing bus, they are free to adjust to “cover” transmission losses in the system.
Table 6-1 summarizes the standard electrical specifications for the three bus types. The
classifications “generator bus” and “load bus” should not be taken as absolute. There will, for
example, be occasions where a pure load bus may be specified by P and |V |.
Bus specification is the tool with which the engineer manipulates the load flow solution to
obtain the desired information. The objective of the load flow solution is to determine the two
quantities at each bus that are not specified.
The generator specification of holding the bus voltage constant and calculating the reactive
power output will be overridden in the load flow solution if the generator reactive output
reaches its maximum or minimum var limit. In this case, the generator reactive power will be
held at the respective limit, and the bus voltage will be allowed to vary.
6.4.3 Gauss-Seidel iterative technique
Descriptions of load flow solution techniques can become rather complicated, due more to
the notation required for complex arithmetic rather than the basic concepts of the solution
method. In the following sections, therefore, the basic techniques are developed by
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LOAD FLOW STUDIES
Table 6-1— Load flow bus specifications
Bus type
P
Q
Load
✓
✓
Generator or
synchronous
condenser
✓
✓
V
δ
Comments
Usual load representation
✓
when
Q– <Qg <Q+
Generator or synchronous condenser
(P=0) with var limits
Q– = minimum var limit
Q+ = maximum var limit
✓
when
Qg <Q–
or
Qg >Q+
V is held as long as Qg is within
limit
✓
Swing
✓
“Swing bus” must adjust net power to
hold voltage constant (essential for
solution).
NOTES
1—Quantities checked are the bus boundary conditions.
2— [ P, δ ], [ Q, V ], and [ Q, δ ] combinations are generally not used.
considering their application to a dc circuit. Applications to ac problems are then a natural
extension of the dc problem.
The Gauss-Seidel solution algorithm, although not the most powerful, is the easiest to
understand. The performance of the Gauss-Seidel technique will be illustrated using the
direct current circuit shown in Figure 6-4.
Bus 3 is a load bus with specified per unit power. Bus 2 is a generator bus with power
specified, and bus 1 is the swing bus with voltage specified. The voltages V2 and V3 are
sought. From these, the branch flows can be calculated.
The system equations on an admittance basis are from Equation (6-2).
I1
V1
I2
Y 11 Y 12 Y 13
= Y 21 Y 22 Y 23
I3
Y 31 Y 32 Y 33
V3
V2
(6-5)
The terms of the admittance matrix are easily determined from the circuit [2], [3], [4], [7].
The off-diagonal term Yij is the negative of the line admittance between bus i and bus j.
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Figure 6-4—Three-bus dc network
–1
Y ij = -----Z ij
(6-6)
The diagonal terms are the sum of the admittances of the lines leaving a bus plus the
admittance of the bus shunt plus one-half of the charging admittance for each connected line.
The Y matrix is very sparse (few nonzero elements), so special matrix techniques are often
used to minimize computer storage requirements. From Equation (6-5),
I 2 = Y 21 V 1 + Y 22 V 2 + Y 23 V 3
(6-7)
1
V 2 = -------- [ I 2 – ( Y 21 V 1 + Y 23 V 3 ) ]
Y 22
(6-8)
or
Substituting
I 2 = P2 / V 2
144
(6-9)
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LOAD FLOW STUDIES
1 P
V 2 = -------- -----2- – ( Y 21 V 1 + Y 23 V 3 )
Y 22 V 2
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Std 399-1997
(6-10)
This is a nonlinear equation in V2.
For bus 3, a similar procedure yields
1 –P
V 3 = -------- --------3- – ( Y 31 V 1 + Y 32 V 2 )
Y 33 V 3
(6-11)
where the negative sign on P3 is from the load sign convention.
Equations (6-10) and (6-11) are in a form convenient for the application of the Gauss-Seidel
iterative solution technique. The steps in this procedure are as follows:
a)
b)
c)
d)
Step 1: Assign an estimate of V2 and V3 (for example, V2 = V3 = 1.0). Note that V1 is
fixed.
Step 2: Compute a new value for V2 using the initial estimates for V2 and V3 [see
Equation (6-10)].
Step 3: Compute a new value for V3 using the initial estimate for V3 and the just
computed value for V2 [see Equation (6-11)].
Step 4: Repeat b) and c) using the latest computed voltages V2 and V3 until the
solution is reached. One complete computation of V2 and V3 is one iteration.
The computed voltages are said to converge when, for each iteration, they come closer and
closer to the actual solution satisfying the network equations. Since the computer time
increases linearly with the number of iterations, it is necessary to have the computer program
make a check after each iteration and decide whether the last computed voltages are
sufficiently close to the true solution or whether further computations are required. The
criterion specifying the desired accuracy is called the “convergence criterion.”
A reliable convergence criterion is the power mismatch check. Based on the last computed
voltage solution, the sum of the power flows (real and reactive) on all lines connected to the
bus and to the bus shunt is compared with the specified bus real and reactive power. The
difference, which is the power mismatch, is a measure of how close the computed voltages
are to the true solution. The power mismatch tolerance is generally specified in the range of
0.01 to 0.0001 p.u. on the system MVA base.
A different convergence check evaluates the maximum change in any bus voltage from one
iteration to the next. A solution with desired accuracy is assumed when the change is less than
a specified small value, for example, 0.0001 p.u.
A voltage check is dependent on the rate of convergence and is thus less reliable than the
power mismatch check. However, the voltage check is much faster (computationally, on a
digital computer) than the power mismatch check and since the power mismatch will be large
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until the voltage change is quite small, one may economically use a procedure where
computation of mismatch is avoided until a small amount of voltage change occurs.
While the solution for a dc circuit was described, solution of an ac circuit would be very
similar. For the three-bus example, voltage magnitude and angle at bus 1, generator power
and bus voltage at bus 2, and real and reactive load power at bus 3 would be specified. The
load flow solution would determine the voltage angle and generator reactive power output of
bus 2 and the voltage magnitude and angle at bus 3.
The ac version of Equations (6-9) and (6-10) can be obtained from Equation (6-4) as follows:
Vi
(m)
1 P i – jQ i
= ------ -------------------–
Y ii V * ( m – 1 )
i
i–1
∑ Y ik V k
(m)
k=1
N
–
∑
Y ik V k
k = i+1
( m – 1 )
i = 1, 2, …, N – 1
(6-12)
where
N
m
i and k
V and Y
V*
is the number of buses in the system, and the swing bus is bus N
is the present iteration number
are bus indexes
are complex voltage and admittance, respectively
is the complex conjugate of V
6.4.4 Newton-Raphson iterative technique
Not all load flow problems can be solved efficiently using the Gauss-Seidel technique. For
some problems, this scheme converges rather slowly. For others, it does not converge at all.
Problems that cannot be solved using the Gauss-Seidel technique may often be solved using
the Newton-Raphson technique.
This approach utilizes the partial derivatives of the load flow relationships to estimate the
changes in the independent variables required to find the solution. In general, the NewtonRaphson technique achieves convergence using fewer iterations than the Gauss-Seidel
technique. However, the computational effort per iteration is somewhat greater.
To apply the Newton-Raphson technique to the three-bus example in Figure 6-4, the bus
powers are expressed as nonlinear functions of the bus voltage.
P 1 = V 1 ( Y 11 V 1 + Y 12 V 2 + Y 13 V 3 )
P 2 = V 2 ( Y 21 V 1 + Y 22 V 2 + Y 23 V 3 )
(6-13)
P 3 = V 3 ( Y 31 V 1 + Y 32 V 2 + Y 33 V 3 )
Small changes in bus voltages (∆V) will cause corresponding, small changes in bus powers
(∆P). A linearized approximation to the power change as a function of voltage changes can be
obtained as follows:
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∆P 1
∂P ∂P ∂P
--------1- --------1- --------1∂V 1 ∂V 2 ∂V 3
∆V 1
∆P 2
∂P ∂P ∂P
= --------2- --------2- --------2∂V 1 ∂V 2 ∂V 3
∆V 2
∆P 3
∂P ∂P ∂P
--------3- --------3- --------3∂V 1 ∂V 2 ∂V 3
∆V 3
(6-14)
or symbolically:
[ ∆P 1 ] = [ J ] [ ∆ V ]
where [J ], the Jacobian matrix, contains the partial derivatives of power with respect to
voltages for a particular set of voltages, V1, V2, and V3, that is, the partial derivations of Equation (6-13). When one or more of the voltages changes substantially, a new Jacobian matrix
must be computed.
In the load flow problem, V1 is specified; that is, (V1 = 0. Also, since ∆P1 does not enter the
computations explicitly, Equation (6-14) may be reduced to
∆P 2
∆P 3
=
∂P ∂P
--------2- --------2∂V 2 ∂V 3
∆V 2
∂P ∂P
--------3- --------3∂V 2 ∂V 3
∆V 3
(6-15)
Changes in V2 and V3 due to changes in P2 and P3 are obtained by inverting [J ] to obtain
–1
[ ∆V ] = [ J ] [ ∆P ]
(6-16)
The Newton-Raphson load flow solution method is then as follows:
a)
b)
c)
Step 1: Assign estimates of V2 and V3 (for example, V2 = V3 = 1.0).
Step 2: Compute P2 and P3 from Equation (6-13).
Step 3: Compute the differences (∆P) between computed and specified powers:
∆P 2 = P 2 – P′2
∆P 3 = P 3 – P′3
(6-17)
where the “prime” indicates specified value.
d)
Step 4: Since ∆P ≠ 0 is caused by errors in the voltages, it seems that the voltages
should be incorrect by an amount that is closely approximated by ∆V as evaluated
from Equation (6-16).
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Therefore, the new estimate for the bus voltages is
V2
V3
=
new
V2
V3
– J
old
–1
∆P 2
(6-18)
∆P 3
This is the basic equation in the Newton-Raphson method. The negative sign is due to
the way ∆P was defined.
e)
Step 5: Recompute and “invert” the Jacobian matrix using the last computed voltages
and compute the new estimate for the voltages using Equations (6-17) and (6-18).
Repeat this procedure until ∆P2 and ∆P3 are less than a small value (convergence
criterion).
Digital computer programs solving large power system load flows do not explicitly compute
the Jacobian inverse. Rather, the voltage correction ∆V is obtained by a numerical technique
known as Gaussian elimination. This technique is much faster and requires much less storage
than matrix inversion.
The convergence of the Newton-Raphson technique is not asymptotic as was the case with
the Gauss-Seidel iterative scheme. The convergence is very rapid for the first few iterations
and slows as the solution is neared.
For the ac load flow solution, the Jacobian matrix may be arranged as follows:
∆P
∆Q
=
J1 J2
∆δ
J3 J4
∆V
(6-19)
where the complex bus voltage is written in polar coordinates, |V| ∠ δ . The Jacobian matrix
can be arranged in many different ways to fit the particular programming techniques selected.
An approximation to the Newton-Raphson formulation can be obtained by observing that, for
a small change in the magnitude of bus voltage ∆|V |, the real power, P, does not change
appreciably. Similarly, for a small change in bus voltage phase angle ∆δ, the reactive power,
Q, does not change very much. Thus, in Equation (6-19):
[ J 2] = [∂P / ∂ V ] ≅ 0
(6-20)
[ J 3 ] = [ ∂ Q / ∂δ ] ≅ 0
(6-21)
This allows Equation (6-19) to be “decoupled” into the following form:
[ ∆P ] = [ J 1 ] [ ∆δ ]
(6-22)
[ ∆Q ] = [ J 4 ] [ ∆ V ]
(6-23)
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Note that these two equations can be solved independently and sequentially, thereby reducing
the storage and solution time requirements compared to using the full Jacobian. The
decoupled Newton-Raphson technique may be used in applications where computational
speed is important and the starting solution is close to the actual solution. This situation often
occurs where a series of contingencies are being investigated about a previously solved
reference case. However, the decoupled technique does not work well for systems with high
branch resistance to reactance ratios, such as often found in industrial systems.
6.4.5 Comparison of load flow solution techniques
The techniques described in the previous subclauses are the basic load flow solution techniques.
There are many variations and improvements to these techniques that have been developed and
incorporated into load flow programs to improve the starting or convergence characteristics.
Although it is useful to understand how load flow solution techniques work, it is more
important to understand the characteristics they exhibit. Because their convergence
characteristics are dependent upon network, load, and generator conditions, each of the
iterative techniques discussed has its own strengths and weaknesses.
Gauss-Seidel methods generally exhibit poor convergence characteristics when compared to
Newton methods and thus are no longer widely used for load flow studies. Most of the
research into load flow solution techniques has centered on Newton methods. Variations of
the Newton methods have been developed to overcome the weaknesses of the original
methods, especially the ability to converge from a poor initial voltage estimate.
The modified Newton methods employed by commercial load flow programs combine good
convergence characteristics and solution algorithm robustness. Details on such algorithms are
available in the references.
6.5 Load flow analysis
A load flow solution determines the bus voltages and the flows in all branches for a given set
of conditions. A load flow study is a series of such calculations made when certain equipment
parameters are set at different values, or circuit configuration is changed by opening or
closing breakers, adding or removing a line, etc. Load flow studies are performed to check the
operation of an existing system under normal or outage conditions, to see if the existing system is capable of supplying planned additional loads, or to check and compare new
alternatives for system additions to supply new load or improve system performance.
Generally, the study engineer has a predefined set of criteria that the system must meet. These
include the following:
—
—
—
Voltage criteria, such as defined in IEEE Std 141-19932
Flows on lines and transformers must be within defined thermal ratings
Generator reactive outputs must be within the limits defined by the generator
capability curves
2Information
on references can be found in 6.9.
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The voltage criteria are usually divided into an acceptable voltage range for normal
conditions and a wider range of acceptable voltage under outage conditions. The thermal
criteria for lines and transformers may also have such a division, allowing for a temporary
overload capability due to the thermal time constant of the equipment or additional forced
cooling capabilities of transformers.
A study normally begins with the preparation of base cases to represent the different
operating modes of the system or plant. The operating condition normally chosen is
maximum load. (Here maximum load refers to the maximum amount of coincident loaded,
not the sum of all the loads. See 4.9 for an explanation of load diversity and load modeling.)
When maximum load occurs at different times on different parts of the system, several base
cases may be needed. The base cases should represent realistic operating conditions.
Abnormal conditions and worst-case scenarios will be addressed later in the study.
The base cases are analyzed to determine if voltages and flows are within acceptable ranges.
Sample outputs are shown in 6.6. If voltages or overload problems are noted, system changes
can be made to the load flow data and the case resolved to see if the changes are effective in
remedying the problem. To remedy low-voltage problems, possible changes include the
following:
Change in transformer tap positions
—
—
—
—
—
Increase in generator schedule voltage
Addition of shunt capacitors
System reconfiguration to shift load to less heavily loaded lines
Disconnection of shunt reactors
Addition of lines or transformers
To remedy heavy line or transformer loadings, most of the same remedies apply. In general,
the first two of the above remedies will not help heavy loadings due to large real power (watt)
flows. Real power flows from the generators to the loads. Real power flow is determined by
the phase angle of the supply bus leading the phase angle of the load bus, with voltage
magnitudes having a secondary effect. However, reactive power flow is primarily determined
by the voltage magnitude with reactive power flowing from the higher voltage bus to the
lower voltage bus. Real and reactive power flow, being primarily influenced by different
constraints, can flow in different directions on the same line.
Transformer off-nominal taps can change the relative relationship of the voltage on the
primary and secondary bus and thus can change the reactive power flow, while the real power
flow is largely unaffected by a change in tap position. When the base case voltages and flows
are in the desired range, the system must be examined to check operation under abnormal
conditions (contingency analysis). These conditions include the following:
—
—
—
—
150
Loss of a transmission line or cable
Loss of a transformer
Loss of a generator
Abnormal supply conditions
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When the load flow model is changed, for example, to represent a line outage, a new solution
is obtained. The voltage and flows are checked against their respective criterion. If necessary,
further system changes are made to correct the problems noted. In contingency analysis, it is
important to note that several outages may cause system problems; but the different remedies
applied may not help equally for all outages. To minimize the number and cost of the
remedies, it is necessary to choose those remedies that have the most beneficial effects for the
most outages. The load flow analysis is used to design a system that has a good voltage
profile and acceptable line loadings during normal operation and that will continue to operate
acceptably when one or more lines become inoperative due to line damage, lightning strokes,
failure of transformers, etc. Performing a series of load flow cases and analyzing the results
provides operating intelligence in a short time that might take years of actual operating
experience to obtain.
In addition to the benefits described above, a study of reactive power flows on the branches
can lead to reduced line losses and improved voltage distribution. Reduction in kVA demand
due to power factor correction can lead to lower utility bills for an industrial plant. The size
and placement of power factor correction capacitors and the setting of generator scheduled
voltages and transformer tap positions can be studied with load flows.
Knowledge of branch flows supplies the protection engineer with requirements for proper
relay settings. The load flow studies can also provide data for automatic load and demand
control, if needed.
The load flow is also used to check the effects of future load growth and the effectiveness of
planned additions. These studies are performed in the same way as studies of the present
system. The future loads are determined and entered into the model. Base case conditions are
studied and additions made, if necessary, to get the system to meet the performance criteria.
Then outage conditions are studied and again system changes may be required. Studies of
future systems vary in that there are usually more alternative ways of solving the problems
encountered. The load flow is the tool that allows the alternatives to be compared in terms of
their effectiveness under normal and contingency conditions. Coupled with other studies as
well as cost and reliability data, the results lead to the selection of the best alternative.
6.6 Load flow study example
To illustrate the use of a load flow program, a typical industrial plant will be studied. The
single-line diagram of the plant electrical system was shown previously in Figure 6-1.
The first step in performing a load flow study is the preparation of the input data file as
explained in 6.3. This data will be input into the load flow program and the network solved.
The input data for the sample system is shown in Figure 6-5. Data is given in terms of both
physical equipment parameters and per unit.
For existing systems, the network configuration, load, and generation are often chosen to
match a known operating condition so results can be compared to values known from
operating experience to help validate the model. The base case represents the system in the
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Figure 6-5—Input data for sample system
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Figure 6-5—Input data for sample system (Continued)
LOAD FLOW STUDIES
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Figure 6-5—Input data for sample system (Continued)
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normal operating mode supplying present maximum loads. Most load flow programs have
data checking and analysis routines to help find data input errors. These include a check of
the network topology to see that all in-service buses are connected to the swing bus and range
checking of certain data items to flag uncharacteristic values. A fundamental check of the
base case is to examine the ability of the load flow solution to converge. As noted in 6.4,
convergence should lead to a very small amount of MW and Mvar mismatch on every bus,
where the mismatch is simply the sum of all the powers or reactive powers entering the bus.
The mismatch should ideally equal zero to satisfy Kirchoff’s laws; however, a small
mismatch is acceptable provided its percentage is small in comparison to the total bus load as
a small amount of mismatch will not adversely affect the accuracy of the calculated bus
voltages. If the load flow solution cannot approach this point for a known normal operating
condition, then a problem in the system model is indicated. Scanning of the load flow output
to see buses with large values of mismatch or abnormal voltages will often help find the
problem area. The problem could be incomplete or inconsistent data. Short low-impedance
lines in close proximity with long lines may make convergence difficult. Very low-impedance
lines will likely cause convergence problems unless the load flow program contains special
logic in the solution techniques to handle them. Engineering judgment is needed to determine
whether it is more appropriate to model these elements explicitly or to lump them with
adjacent elements.
Most load flow programs will have the ability to take a solved load flow base case and store
all the necessary data including the solution in a file on the computer disk. This will allow
easy retrieval of the base case to incorporate future changes or to perform outage condition
studies.
One page of the solved load flow output for the sample base case is shown in Figure 6-6 in
tabular form. For each bus, the bus voltage magnitude and angle are given. The voltage
magnitude may be given as per unit or kV (in this case, both are listed). Each line going from
that bus to another bus is listed, giving the MW and Mvar flow (or kW and kvar) on the line
out of the “from” bus. A negative flow means the flow is coming into the “from” bus. For
transformers, the tap is also listed. If there is significant mismatch on the bus, it will also be
listed. Different programs will use somewhat different formats; but they will present basically
the same information.
A more concise and usually more informative method of presenting load flow results is to
display them graphically on the system single-line diagram. System flows can be quickly
analyzed from this visual presentation that relates system configuration, operating conditions,
and equipment parameters.
Figure 6-7 displays the base case load flow results in graphical form. This figure shows the
voltages on all buses and flows on all lines. As well as this output data, the system
configuration is shown clearly: which buses are supplied by each feeder, loads being
modeled, generator output, transformer tap ratios, and shunt capacitor values.
Most commercial load flow programs will generate such drawings. As noted before, output
format will vary. In Figure 6-7, the line flows are shown near one of the buses and arrows
indicate the direction of the MW and the Mvar flow.
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Figure 6-6—Sample load flow output
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Figure 6-7—Example system base case load flow output
LOAD FLOW STUDIES
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Load flow output as shown in Figure 6-7 provides an excellent opportunity to document study
results. Case titles or text on the drawing should indicate the system condition being
analyzed. Comments can be entered as to good or poor conditions, or both, that exist in
circuit parameters or configuration. It may be desirable to list corrective action taken for the
next load flow run, which will hopefully improve the operation.
6.6.1 Analysis of sample system
Now that a load flow has been run for the base condition, what has been learned about the
system, and what can be done to improve its operation? Analysis of load flow output shows
the following:
a)
b)
c)
d)
Voltage at load buses are generally somewhat low, averaging around 0.97 per unit.
The two plant generators at buses 4 and 50 supply about 87% of the real power load,
with the remainder coming from the utility.
The reactive power requirements of the system are also supplied primarily from the
plant generators. The reactive power loading on the two generators is large while only
a small amount of reactive power comes from the utility.
Loadings on all transformers and cables are within normal operating capabilities.
Conditions in this base case load flow are thus acceptable, although they may not be optimal.
The next step would be to see if improvements could be made and to check to see the effects
of outages.
One of the most critical contingencies will be outage of one of the plant’s generators. Figure
6-8 shows the load flow solution following outage of the generator on bus 50 (along with its
auxiliary loads on bus 51). Voltages on load buses fed from bus 3 all drop to 0.93 to 0.94 per
unit.
One remedy to these low voltages would be to improve the base case voltage profile. Two
methods which could be used are changing the tap position on transformers supplied with
off-nominal tap capability or changing the voltage control setpoints of the two generators. To
improve voltage profiles following the outage of the generators, the transformer tap changes
would be more effective.
Figure 6-9 shows the base case load flow with the following transformer tap changes:
Nominal transformer voltage
ratio
Revised off-nominal tap
position
Per unit tap
13.8 kV / 4160 V
13.46 kV / 4160 V
0.975
13.8 kV / 2400 W
13.46 kV / 2400 V
0.975
13.8 kV / 480 V
13.46 kV / 480 V
0.975
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Figure 6-8—Outage of generator on bus 50
LOAD FLOW STUDIES
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Figure 6-9—Base case after transformer tap changes
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Note that the voltages at the load buses are now all near unity. With these changes, voltages
following the outage of the generator on bus 50 all remain above 0.95 per unit.
The taps on the utility tie transformers were also changed from the nominal 69 kV/13.8 kV to
67.3 kV/13.8 kV or a per unit tap position of 0.975. As the generators hold the main mill
buses (3 and 4) near 1.0 per unit, these tap changes do not affect the load bus voltages in the
base case condition. However, the change in tap on the supply transformers has a very
significant change in the reactive power supply to the plant. The reactive output of the plant
generators drops to a small value and almost all of the reactive power needs of the plant come
from the utility. This may not necessarily be an improvement; its benefit depends on the
demand/power factor terms in the rate structure with the utility. However, the effect of
transformer tap positions on reactive power flow is clearly demonstrated.
Another remedy that could be analyzed is changes to the generator voltage setpoints, which
will affect both the plant voltage profile and also the relative amount of reactive power
supplied by each generator and the system. The effect on real power flows would be minor.
Some of the reactive power needs of the plant could also be supplied by capacitors. Analysis
of the proper size and placement of capacitors for power factor improvement is explained in
IEEE Std 141-1993. The load flow would demonstrate the effectiveness of the proposed
capacitor additions.
Any changes made must be checked to see if they result in acceptable conditions for different
operating conditions, such as light load in the plant or the utility system supply voltage being
higher or lower than the nominal voltage modeled in the base case.
A couple of other examples of plant conditions that could be analyzed are outage of one of
the main supply transformers with the closing of the breaker between buses 3 and 4 and the
alternate supply of buses 12, 17, and 22 from bus 10. The load flow would be checked to
ensure both acceptable voltage levels and that loadings on equipment were within their
capabilities. In this manner, the load flow allows the analysis of outages to see if the system
will operate in an acceptable (although possibly degraded) condition or if curtailment of the
plant’s operations would be necessary, and allows alternatives for system improvement to be
tested.
Analysis of the system load flow outputs after each set of changes results in the system being
gradually tuned to obtain the most efficient and reliable operation. Experience with load
flows improves the engineer’s ability to make corrections with a minimum number of load
flow solutions. However, it is stressed that any change affects the whole system, and a cure at
one spot can create unexpected problems at another location in the system. For this reason, it
is better not to make too many changes in a single run as the effects on the system may be
difficult to understand. Each change case should be documented showing changes made and
results obtained in order to keep future changes consistent with improving the system.
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6.7 Load flow programs
Many load flow programs are presently available, both in the public domain and as
commercial products. These programs may be designed to run on large mainframe
computers, medium-sized minicomputers, or on microcomputers, such as personal
computers. The various load flow programs differ in the ease of use, program documentation,
program sophistication and, of course, cost.
Most load flow programs now operate in an interactive mode, either prompting the user for
input and direction in basically a question and answer mode, or using a GUI where the user
directs the operation of the program by selecting functions with a “mouse,” i.e., a “windows”
interface. This approach is much more efficient from an engineering standpoint than the previous batch-job-oriented programs.
There is a wide range in the level of sophistication in the available programs. There is a
corresponding range in the level of user need for this sophistication. The more sophisticated
programs may contain several load flow solution techniques allowing for easier solution of a
wide range of problems, more data checking activities to help in debugging data input errors,
more data handling activities to ease changes to system data or configuration, graphic display
of load flow results, ability to handle much larger networks, the modeling of additional power
system components, and the incorporation of additional control functions into the solution
techniques. For example, load flows are available that will handle 50 000 buses, incorporate
such power system elements as HVDC lines and phase shifting transformers, and perform
utility area interchange control as part of the solution process. Although essential for large
utility systems, a high level of sophistication may not be needed by industrial engineers to
analyze their systems.
Since the same network data is used for load flow, short-circuit, transient stability, motor
starting, and harmonic studies, it is useful to select a program package that integrates these
calculations so that they all use the same load flow network data. Of course, all of the above
study capabilities will not be needed by all users.
As an alternative to in-house use of a load flow program, there are consultants who can do the
analysis and present the user with a complete report including a technical analysis of the
computer output and suggestions and advice on system improvement. Even in this study
mode, it is to the user’s advantage to understand the data requirements and how a study is
performed.
6.8 Conclusions
It should be evident to designers and operators of industrial plant electrical systems, as well
as to utility system engineers, that a tool that predicts the performance of their electrical systems under various operating conditions that can actually be encountered before these conditions occur is of value. Such a tool can be used to prevent expensive outages, damaged
equipment, and possibly even loss of life. The load flow solution provides a means to study
systems under real or hypothetical conditions. The solution results should be evaluated and
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analyzed with respect to optimum present and future operation. This leads to a diagnosis of
the system as it exists. The analysis can also point the way to improved operation and provide
a meaningful basis for future system planning.
6.9 References
This chapter shall be used in conjunction with the following publications:
IEEE Std 141-1993, IEEE Recommended Practice for Electric Power Distribution for Industrial Plants (IEEE Red Book).3
6.10 Bibliography
Additional information may be found in the following sources:
[B1] Brown, H. E., Solution of Large Networks by Matrix Methods, New York, NY: John
Wiley and Sons, 1975.
[B2] Stagg, G. W., and El-Abiad, A. H., Computer Methods in Power System Analysis, New
York, NY: McGraw-Hill, 1968.
[B3] Stevenson, W. D., Elements of Power System Analysis, New York, NY: McGraw-Hill,
1975.
[B4] Tinney, W. F., and Hart, C. E., “Power flow solution by Newton’s method,” IEEE
Transactions on Power Apparatus and Systems, vol. PAS-86, pp. 1444–1460, Nov. 1967.
[B5] Ward, J. B., and Hale, H. W., “Digital Computer Solution of Power Flow Problems,”
AIEE Transactions, Part III—Power Apparatus and Systems, vol. 75, pp. 398–404, June
1956.
[B6] Wood, A. J., and Wollenberg, B. F., Power Generation, Operation and Control, New
York, NY: John Wiley and Sons, 1984.
3IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
Copyright © 1998 IEEE. All rights reserved.
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Chapter 7
Short-circuit studies
7.1 Introduction and scope
Electrical power systems are, in general, fairly complex systems composed of a wide range of
equipment devoted to generating, transmitting, and distributing electrical power to various
consumption centers. The very complexity of these systems suggests that failures are
unavoidable, no matter how carefully these systems have been designed. The feasibility of
designing and operating a system with zero failure rate is, if not unrealistic, economically
unjustifiable. Within the context of short-circuit analysis, system failures manifest themselves
as insulation breakdowns that may lead to one of the following phenomena:
—
—
—
—
—
Undesirable current flow patterns
Appearance of currents of excessive magnitudes that could lead to equipment damage and downtime
Excessive overvoltages, of the transient and/or sustained nature, that compromise the
integrity and reliability of various insulated parts
Voltage depressions in the vicinity of the fault that could adversely affect the operation of rotating equipment
Creation of system conditions that could prove hazardous to personnel
Because short circuits cannot always be prevented, we can only attempt to mitigate and to a
certain extent contain their potentially damaging effects. One should, at first, aim to design
the system so that the likelihood of the occurrence of the short circuit becomes small. If a
short circuit occurs, however, mitigating its effects consists of a) managing the magnitude of
the undesirable fault currents, and b) isolating the smallest possible portion of the system
around the area of the mishap in order to retain service to the rest of the system. A significant
part of system protection is devoted to detecting short-circuit conditions in a reliable fashion.
Considerable capital investment is required in interrupting equipment at all voltage levels that
is capable of withstanding the fault currents and isolating the faulted area. It follows, therefore, that the main reasons for performing short-circuit studies are the following:
—
—
—
Verification of the adequacy of existing interrupting equipment. The same type of
studies will form the basis for the selection of the interrupting equipment for system
planning purposes.
Determination of the system protective device settings, which is done primarily by
quantities characterizing the system under fault conditions. These quantities also
referred to as “protection handles,” typically include phase and sequence currents or
voltages and rates of changes of system currents or voltages.
Determination of the effects of the fault currents on various system components such
as cables, lines, busways, transformers, and reactors during the time the fault persists.
Thermal and mechanical stresses from the resulting fault currents should always be
compared with the corresponding short-term, usually first-cycle, withstand capabilities of the system equipment.
Copyright © 1998 IEEE. All rights reserved.
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—
—
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Assessment of the effect that different kinds of short circuits of varying severity may
have on the overall system voltage profile. These studies will identify areas in the
system for which faults can result in unacceptably widespread voltage depressions.
Conceptualization, design and refinement of system layout, neutral grounding, and
substation grounding.
Compliance with codes and regulations governing system design and operation, such as the
National Electrical Code® (NEC®) (NFPA 70-1996) [B6],1 article 110-9.
It is not the intention of this chapter to provide details on system modeling and computational
procedures under fault conditions, since these topics are exhaustively treated in many comprehensive textbooks and articles (see bibliography) and other IEEE publications such as
IEEE Std 141-1993,2 IEEE Std 241-1990, and IEEE Std 242-1986. Rather, the intention is to
address the subject of short-circuit analysis within the context of three-phase industrial and
commercial power systems by focusing on the following:
a)
b)
c)
d)
e)
f)
The concerns and fundamental phenomena associated with short-circuit studies
Viable computational approaches and some aspects of system modeling
Various factors affecting the results and accuracy of short-circuit studies
The salient principles, methodologies, and computational procedures suggested by
the North American IEEE and ANSI C37 standards, specifically, ANSI C37.06-1979,
ANSI C37.06-1987, IEEE Std C37.010-1979, IEEE Std C37.5-1979, and IEEE Std
C37.13-1990). Reference will, however, be made to the international standard, IEC
60909 (1988), because
1) It features several significant conceptual and computational deviations from the
C37 standards, and
2) Equipment designed and built according to European specifications is being sold
in North America, which should be analyzed on the equipment-tested ratings.
Computer-based solutions and related aspects of software dedicated to computerized
fault analysis
The outline of a typical short-circuit study procedure through an example
7.2 Extent and requirements of short-circuit studies
Short-circuit studies are as necessary for any power system as other fundamental system
studies such as power flow studies, transient stability studies, harmonic analysis studies, etc.
Short-circuit studies can be performed at the planning stage in order to help finalize the
system layout, determine voltage levels, and size cables, transformers, and conductors. For
existing systems, fault studies are necessary in the cases of added generation, installation of
extra rotating loads, system layout modifications, rearrangement of protection equipment,
verification of the adequacy of existing breakers, relocation of already acquired switchgear in
order to avoid unnecessary capital expenditures, etc. “Post-mortem” analysis may also
1The
numbers in brackets correspond to those of the bibliography in 7.9.
on references can be found in 7.8.
2Information
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involve short-circuit studies in order to duplicate the reasons and system conditions that led to
the system’s failure.
The requirements and extent of a short-circuit study will depend on the engineering objectives sought. In fact, these objectives will dictate what type of short-circuit analysis is
required. The amount of data required will also depend on the extent and the nature of the
study. The great majority of short-circuit studies in industrial and commercial power systems
address one or more of the following four kinds of short circuits:
—
—
—
—
Three-phase fault. May or may not involve ground. All three phases shorted together.
Single line-to-ground fault. Any one, but only one, phase shorted to ground.
Line-to-line fault. Any two phases shorted together.
Double line-to-ground fault. Any two phases connected together and then to ground.
These types of short circuits are also referred to as “shunt faults,” since all four exhibit the
common attribute of being associated with fault currents and MVA flows diverted to paths
different from the prefault “series” ones.
Three-phase short circuits often turn out to be the most severe of all. It is thus customary to
perform only three phase-fault simulations when seeking maximum possible magnitudes of
fault currents. However, important exceptions do exist. For instance, single line-to-ground
short-circuit currents can exceed three-phase short-circuit current levels when they occur in
the vicinity of
—
—
—
—
A solidly grounded synchronous machine
The solidly grounded wye side of a delta-wye transformer of the three-phase core
(three-leg) design
The grounded wye side of a delta-wye autotransformer
The grounded wye, grounded wye, delta-tertiary, three-winding transformer
For systems where any one or more of the above conditions exist, it is advisable to perform a
single line-to-ground fault simulation. The fact that medium- and high-voltage circuit breakers have 15% higher interrupting capabilities for single line-to-ground faults should be taken
into account, if elevated single line-to-ground fault currents are found. Line-to-line or double
line-to-ground fault studies may also be required for protective device coordination requirements. It should be noted that, since only one phase of the line-to-ground fault can experience
higher interrupting requirements, the three-phase fault will still contain more energy because
all three phases will experience the same interrupting requirements.
Other types of fault conditions that may be of interest include the so-called “series faults”
(Anderson [B1]) and pertain to one of the following types of system unbalances:
—
—
—
One line open. Any one of the three phases may be open.
Two lines open. Any two of the three phases may be open.
Unequal impedances. Unbalanced line impedance discontinuity.
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The term “series faults” is used because the above unbalances are associated with a redistribution of the prefault load current. Series faults are of interest when assessing the effects of
snapped overhead phase wires, failures of cable joints, blown fuses, failure of breakers to
open all poles, inadvertent breaker energization across one or two poles and other situations
that result in the flow of unbalanced currents.
7.3 System modeling and computational techniques
7.3.1 AC and dc decrement
The basic physical phenomena that determine the magnitude and duration of the short-circuit
currents are
a)
b)
c)
The behavior of the rotating machinery in the system
The electrical proximity of the rotating machinery to the short-circuit location
The fact that the prefault system currents cannot change instantaneously, due to the
significant system inductance
The first two can be conceptually linked to the ac decrement, while the third, to the dc
decrement.
7.3.1.1 AC decrement and rotating machinery
AC decrement is characterized by the fact that the magnetic flux trapped in the windings of
the rotating machinery cannot change instantaneously (constant flux theorem). This gradual
change is a function of the nature of the magnetic circuits involved. That is why synchronous
machines, under short-circuit conditions, feature different flux variation patterns as compared
to induction machines. The flux dynamics dictates that the short-circuit current decays with
time until a steady-state value is reached. Computationally convenient machine models,
widely used for short-circuit studies, picture rotating machines as constant voltages behind
time-varying impedances, as outlined in IEEE Std 141-1993 and IEEE Std 242-1986. For
modeling purposes, these impedances increase in magnitude from the minimum post fault
subtransient value X "d, to the relatively higher transient value X 'd , and finally reach the even
higher steady-state value Xd, assuming the fault persists long enough (in reality it is the voltage that decays). The rate of increase of machine reactances is different for synchronous generators/motors and induction motors, with the latter increasing more rapidly than the former.
This modeling framework is fundamental in properly determining the symmetrical rms values of the short-circuit currents furnished by the rotating equipment for a short circuit anywhere in the system.
7.3.1.2 DC decrement and system impedances
DC decrement is also characterized by the fact that because the prefault system current cannot change instantaneously, a significant unidirectional component may be present in the fault
current depending on the exact instant of the occurrence of the short circuit (Anderson [B1],
Blackburn [B3], Roeper [B8], Stevenson [B10]). This unidirectional current component,
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often referred to as dc offset, decays with time exponentially. The rate of decay is closely
related to the system reactances and resistances. Despite the fact that this decay is relatively
rapid, the dc component could last long enough to be sensed by the interrupting equipment,
particularly when rapid fault clearing is very desirable to maintain system stability or prevent
the damaging thermal and mechanical effects of the short-circuit currents. Total fault currents
interrupted by circuit breakers must take into account this unidirectional component, particularly for shorter interrupting times as clearly outlined in IEEE Std C37.010-1979, IEEE Std
C37.13-1990, and IEEE Std C37.5-1979. The same component is equally important when
assessing the ability of a circuit breaker to close against or withstand short-circuit currents.
Fault currents containing high dc offsets, often present no zero crossings in the first few
cycles immediately after fault initiation and are particularly onerous to the circuit breakers of
large generators.
7.3.2 System modeling requirements
Industrial and commercial power systems are normally multimachine systems with many
motors and possibly more than one generator, all interconnected through transformers, lines,
and cables. There could also be one or more locations at which the local power system is
interconnected to a larger grid. These locations are commonly known as “utility-interface”
points. The objective of the short-circuit study is to properly determine the short-circuit currents and voltages at various system locations. In view of the dynamic nature of the shortcircuit current, it is essential to relate any calculated fault currents to a particular instant in
time from the onset of the short circuit. AC decrement analysis serves the purpose of correctly determining the symmetrical rms values of the fault currents, while dc decrement analysis will provide the necessary dc component of the fault current, thus yielding a correct
estimate of the total fault current. It is the total fault current which, in general, must be used
for breaker and switchgear rating and in some cases for protective device coordination. System topology considerations are equally important because the system layout and electrical
proximity of the rotating machinery to the fault location will determine the actual magnitude
of the short-circuit current. It therefore becomes necessary to devise a model for the system
as a whole and analyze it as such in a flexible, accurate, and computationally convenient
manner.
7.3.3 Three-phase vs. symmetrical components representation
It is customary for three-phase electrical systems to be represented on a single-phase basis.
This simplification, successfully employed for power flow and transient stability studies, rests
on the premise that the system is balanced or at least can be assumed to be so for practical
purposes (Anderson [B1], Blackburn [B3] Stevenson [B10], Wagner and Evans [B13]). Modeling the system, however, on a single-phase basis is inadequate for analyzing phenomena
that involve serious system unbalances (Anderson [B1], Arrilaga, Arnold, and Harker [B2]).
Within the context of short-circuit analysis, only the three-phase shunt fault lends itself to
single-phase analysis, because the fault condition is balanced involving all three phases,
assuming a balanced three-phase system. Any other fault condition will introduce unbalances
that require including in the analysis the remaining two phases. There are two alternatives to
address the problem:
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a)
b)
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Three-phase system representation. When the system is represented on a three-phase
basis, we explicitly retain the identity of all three phases. The advantage of threephase representation is that any kind of fault unbalance can be readily analyzed,
including simultaneous faults. Furthermore, the fault condition itself is specified with
somewhat greater flexibility, particularly for arcing faults. The main disadvantages of
the technique are the following:
1) It is not tractable for hand calculations, even for small systems.
2) Supposing that a suitable computer program is used, it can be data-intensive.
Symmetrical components representation. The symmetrical components analysis is a
technique that, instead of requiring analysis of the unbalanced system, allows for the
creation of three subsystems, the positive, the negative, and the zero-sequence systems, properly interconnected at the fault point, depending on the nature of the system unbalance. Once modeled, the fault currents and voltages, anywhere in the
network, are then obtained by properly combining the results of the analysis of the
three-sequence networks (Anderson [B1], Blackburn [B3] Stevenson [B10], Wagner
and Evans [B13]). The distinct advantage of the symmetrical components approach is
that it allows modeling unbalanced fault conditions, while still retaining the conceptual simplicity of the single-phase analysis. Another important advantage of the symmetrical components method is that system equipment impedances can be easily
measured in the symmetrical components reference frame. This simplification is only
true if the system is balanced in all three phases (except at the fault location which
then becomes the interconnection point of the sequence networks), an assumption
that can be entertained without introducing significant modeling errors for most systems. The main disadvantage of the technique is that for complicated fault conditions,
it may introduce more problems than it solves. The symmetrical components technique remains the preferred analytical tool today for fault analysis for both hand and
computer-based calculations.
7.3.4 System impedances and symmetrical components analysis
The symmetrical components theory dictates that for a three-phase system, three sequence
systems need, in general, to be set up for the analysis of an unbalanced fault condition. The
first is the positive sequence system, which is defined by a balanced set of voltages and
currents, equal in magnitude, following the normal phase sequence of a, b, and c. The second
is the negative sequence system, which is similar to the positive sequence system, but is
defined by a balanced set of voltages and currents with a reverse phase sequence of a, c, and
b. Finally, the zero sequence system is a system defined by a set of voltages and currents that
are in phase with each other and not displaced by 120 degrees, as is the case with the other
two systems. The topology of the zero sequence system can be quite different from that of the
positive and negative sequence systems due to the fact that it depends heavily on the power
transformer connections (Anderson [B1], Blackburn [B3] Stevenson [B10]) and system neutral grounding, factors which are not of importance when determining the topology of the
other two sequence networks.
Static system equipment like transformers, lines, cables, busways, and static loads present,
under balanced conditions, the same impedances to the flow of positive and negative
sequence currents. The same components present, in general, different impedances to the
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flow of zero sequence currents. Rotating equipment like synchronous generators, motors,
condensers, and induction motors have different impedances in all three sequence networks.
The positive sequence impedances are the ones normally used for balanced power flow studies. All sequence impedances must be either calculated, measured, provided by the equipment
manufacturers, or estimated. The zero sequence impedance may not exist for some rotating
equipment, depending on the machine grounding.
For a balanced three-phase fault analysis, only the positive sequence system components
impedances Z1 (R1 + jX1) are required. For line-to-line faults, negative sequence impedances
Z2 (R2 + jX2) are also required. For all shunt faults involving ground, i.e., line-to-ground and
double line-to-ground, the zero sequence system impedances Z0 (R0 + jX0) are needed in
addition to the other two. System neutral grounding equipment data like grounding resistors,
reactors, transformers, etc., form an integral part of the zero sequence system impedance
data.
AC decrement considerations dictate that rotating equipment impedances vary from the onset
of the short circuit. This applies only to positive sequence impedances, which vary from subtransient through transient to steady-state values. The negative and zero sequence impedances
for the rotating equipment are considered unchanged. The same holds true for the impedances
of the static system equipment.
7.3.5 Computational approaches
7.3.5.1 Time domain fault analysis
Time-domain fault analysis pertains to techniques that allow for the calculation of the shortcircuit current as a function of time from the moment of the fault inception. For large electric
power systems, with many machines and generators contributing to the fault current, the contributions of many machines will have to be taken into account concurrently. Machine models
have been developed that let predictions of considerable accuracy be made regarding the
behavior of any machine for a fault either at or beyond its terminals. These models are rather
complex because they tend to represent in detail not only the machine itself but also several
nonlinear controllers, such as excitation systems and their related stabilization circuitry, with
nonlinearities. It can therefore be seen that the calculating requirements could be stupendous,
because the problem is reduced to simultaneously solving a large number of differential equations. Despite its inherent power, the use of time-domain fault analysis is not very widespread
and is only used for special studies because it is data-intensive (data requirements can be at
least as demanding as transient stability analysis) and it requires special software.
7.3.5.2 Quasi-steady-state fault analysis
Quasi-steady-state fault analysis pertains to techniques that represent the system at steady
state. Phasors are used to represent system voltages, currents, and impedances at fundamental
frequency. System modeling and the resulting computational techniques are based on the
assumption that the system and its components can be represented by linear models.
Retaining linearity simplifies considerably the necessary calculations (Anderson [B1], Arrilaga, Arnold, and Harker [B2], Blackburn [B3] Stevenson [B10], Wagner and Evans [B13]) as
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demonstrated in Chapter 3. Furthermore, linear algebra theory and the numerical advances in
matrix computations make it possible to implement very elegant computer solutions for
relatively large systems. These techniques have been favored by the various industry standards and will be briefly examined next.
7.4 Fault analysis according to industry standards
Industry standards dictate certain analytical techniques that adhere to specific guidelines,
suited to address the questions of ac and dc decrement in multimachine systems in
compliance with well-established, industry-accepted practices. They are also closely linked
to and harmonize quite well with existing switchgear rating structures. Typical standards are
the North American ANSI and IEEE C37 standards and recommended practices (see 7.4.1),
the international standard, IEC 60909 (1988) and others, such as the German VDE 0102-1972
and the Australian AS 3851-1991 (see 7.8). The analytical and computational framework in
the calculating procedures recommended by these standards remains algebraic and linear, and
the calculations are kept tractable by hand for small systems. The extent of the data base
requirements for computer-based solutions is carefully kept to a necessary maximum for the
results to be acceptably accurate. This type of analysis represents the best compromise
between solution accuracy and simulation simplicity. The great majority of commercialgrade short-circuit analysis programs fall under this category.
In 7.4.1, an outline of ANSI and IEEE standards is presented, while in 7.4.2, the relevant
aspects of IEC 60909 (1988) are described. It is not the intent of these subclauses to fully
explore and describe in detail all pertinent clauses of either standard. Instead, a rather brief
summary is presented in an effort to make any potential user conscious of the salient aspects
of each technique. Because only a brief summary is presented, it is strongly recommended
that the standards be consulted for further clarifications and details.
7.4.1 The North American ANSI and IEEE standards
IEEE standards addressing fault calculations for medium and high voltage are IEEE Std
C37.010-1979, IEEE Std C37.5-1979, IEEE Std 141-1993, IEEE Std 241-1990, and IEEE
Std 242-1986. IEEE standards addressing fault calculations for low-voltage systems (below
1000 V), are the IEEE Std C37.13-1990, IEEE Std 141-1993, IEEE Std 241-1990, and IEEE
Std 242-1986. Three types of short-circuit currents are defined, depending on the time frame
of interest taken from the inception of the fault, as
a)
b)
c)
First cycle currents
Interrupting currents
Time delayed currents
First-cycle currents, also called momentary currents, are the currents at 1/2 cycle after fault
initiation; they relate to the duty circuit breakers face when “closing against” or withstanding
short-circuit currents. That is why these currents are also called “close and latch” currents.
Often these currents contain dc offset, and they are calculated on the premise of no ac decre-
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ment in the contributing sources (i.e., the machine reactances remain subtransient [see Table
7-1]). Since low-voltage breakers operate in the first cycle, their interrupting ratings are compared to these currents.
Table 7-1— Generic impedance types required for short-circuit studies
Electrical system equipment
type
Momentary
1/2 cycle
Interrupting
3–5 cycles
Time delayed
6–30 cycles
Induction motor
X "d, R
X "d, R
Neglect
Synchronous motors
X "d, R
X "d, R
See Note 3
Synchronous generators
X "d, R
X "d, R
X 'd, Xd, R
Synchronous condensers:
electric utility systems
Xs , Rs
Xs , Rs
Xs , Rs
Passive components:
transformers, cables, etc.
X, R
X, R
X, R
where
X "d
X 'd
Xd
X
R
Xs, Rs
is the subtransient reactance. For induction motors, X "d is approximately equal to the
locked rotor reactance.
is the transient reactance
is the synchronous reactance
is the equivalent reactance
is the equivalent modified resistance (see Table 7-2)
is the power company system equivalent reactance and resistance
NOTES
1—See Table 7-2 for exact values.
2—X "d of synchronous machines is the rated voltage (saturated) direct axis subtransient reactance.
3—X 'd of synchronous machines is the rated voltage (saturated) direct axis transient reactance.
4—For calculations of minimum short-circuit current, contribution is neglected. For calculation of
maximum short-circuit current values, use X 'd and R values.
5—For more details on IEEE-related induction motor modeling aspects, see Huening [B4].
Interrupting currents are the short-circuit currents in the time interval from 3 to 5 cycles after
fault initiation. They relate to the currents sensed by the interrupting equipment when isolating a fault. Hence, they are also referred to as “contact-parting” currents. These currents are
asymmetrical; i.e., they contain dc offset, but due consideration is now given to ac decrement
because of the elapsed time from the fault inception. All contributing sources are taken into
account when calculating interrupting currents by virtue of reactances that range from subtransient to transient (see Table 7-1). Interrupting currents in the 3 to 5 cycles interval are
associated with medium- and high-voltage breakers.
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Time delayed currents are the short-circuit currents that exist beyond 6 cycles (and up to
30 cycles) from the fault initiation. They are useful in determining currents sensed by time
delayed relays and in assessing the sensitivity of overcurrent relays. These currents are
assumed to contain no dc offset. Induction and synchronous motor contributions are
neglected, and the contributing generators are assumed to have attained transient or higher
value reactances (see Table 7-1).
7.4.1.1 Accounting for ac and dc decrement
In view of the classification of short-circuit currents in three duty types, different impedances
are used for the rotating equipment for each of these duties. Tables 7-1 and 7-2 portray the
recommended impedances for the system components and for the different types of analysis
and duty currents sought. Once the desired duty type has been selected, the appropriate system impedances may be chosen in accordance with Table 7-2.
Table 7-2—Reactance values for first cycle
and interrupting duty calculations
Duty
calculation
First cycle
(momentary
calculations)
Reactance value
for medium- and
high-voltage
calculations per
IEEE Std
C37.010-1979
and IEEE Std
C37.5-1979
Reactance value
for low-voltage
calculations
(see Note 2)
XS
1.0 X "d
XS
1.0 X "d
Hydrogenerators without amortisseur windings
0.75 X "d
0.75 X 'd
All synchronous motors
1.0 X "d
1.0 X"d
1.0 X"d
1.0 X"d
Above 250 hp at 3600 r/min
1.0 X"d
1.0 X"d
All others, 50 hp and above
1.2 X"d
1.2 X"d
1.67 X"d
(see Note 6)
1.67 X"d
System component
Power company supply
All turbine generators; all
hydrogenerators with amortisseur windings; all condensers
Induction motors
Above 1000 hp
All smaller than 50 hp
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Table 7-2—Reactance values for first cycle
and interrupting duty calculations (Continued)
Duty
calculation
Interrupting
calculations
System component
Power company supply
All turbine generators; all
hydrogenerators with amortisseur windings; all condensers
Reactance value
for medium- and
high-voltage
calculations per
IEEE Std
C37.010-1979
and IEEE Std
C37.5-1979
Reactance value
for low-voltage
calculations
(see Note 2)
XS
1.0 X"d
N/A
Hydrogenerators without amortisseur windings
0.75 X'd
N/A
All synchronous motors
1.5 X"d
N/A
1.5 X"d
N/A
Above 250 hp at 3600 r/min
1.5 X"d
N/A
All others, 50 hp and above
3.0 X"d
N/A
All smaller than 50 hp
Neglect
N/A
Induction motors
Above 1000 hp
NOTES
1—First-cycle duty is the momentary (or close-and-latch) duty for medium-/high-voltage equipment
and is the interrupting duty for low-voltage equipment.
2—Reactance (X) values to be used for low-voltage breaker duty calculations (see IEEE Std C37.131990 and IEEE Std 242-1986).
3—X"d of synchronous-rotating machines is the rated-voltage (saturated) direct-axis subtransient
reactance.
4—X'd of synchronous-rotating machines is the rated-voltage (saturated) direct-axis transient reactance.
5—X"d of induction motors equals 1 divided by per-unit locked-rotor current at rated voltage.
6—For comprehensive multivoltage system calculations, motors less than 50 hp are represented in
medium-/high-voltage, short-circuit calculations (see IEEE Std 141-1993, Chapter 4).
The estimates of 1.2 X"d and 1.67 X"d for induction motor impedances to be used for the first
cycle network are based on locked rotor impedances of 0.20 and 0.50 per unit, respectively,
based on motor rating according to IEEE Std 242-1986. Similarly, the estimate of 3.0 X"d, to
be used for the induction motor impedance for interrupting duty calculations, is based on the
assumption of a locked rotor impedance of 0.28 per unit based on the motor rating, as suggested in IEEE Std 141-1993.
The equivalent Thevenin system impedance at the fault location is then calculated by successive network reduction. Techniques for finding the equivalent short-circuit impedance (reactance) as seen from the fault location are well explained in chapters 3 and 4 of this
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recommended practice, in IEEE Std 141-1993, in IEEE Std 241-1990, and in IEEE Std 2421986. The prefault system voltage, normally assumed to be 1.00 p.u. (rated), divided by the
equivalent short-circuit impedance, will yield the desired symmetrical rms value of the
desired three-phase fault current. The dc component of the fault current is obtained by considering the X/R ratio at the fault point. The X/R ratio is calculated by taking the ratio of the
system reactance (Thevenin equivalent reactance) to the system resistance (Thevenin equivalent resistance) as seen from the fault location. The equivalent reactance must be calculated
from the reactance network (X) which is the impedance network of the system under study
with all resistances absent. Similarly, the equivalent resistance must be calculated from the
resistance network (R), which is the impedance network of the system under study with all
reactances absent.
It should be noted that the separate reactance and resistance network reduction technique will
yield a different X/R ratio (usually higher) than the phasor X/R ratio of the complex fault
impedances.
7.4.1.2 Calculated short-circuit currents and interrupting equipment
The calculating procedures briefly touched upon above are meant to address short-circuit
calculations on Industrial power systems with several voltage levels comprising high-,
medium-, and low-voltage circuits. First cycle currents are useful in calculating the interrupting requirements of low voltage fuses and breakers. Currents resulting from the same simulation are effectively used in calculating the first-cycle requirements for medium- and highvoltage fuses and circuit breakers. The currents resulting from the so-called interrupting network calculations are only used for medium- and high-voltage circuit breakers, which operate
with a certain time delay due to relaying and operating requirements. It must be borne in
mind that since low-voltage fuse and circuit breaker application standards like IEEE Std
C37.13-1990 have adopted the symmetrical rating structure, calculating only the symmetrical
rms fault currents and the X/R ratio may be sufficient, if the calculated X/R ratio is less than
the X/R ratio of the circuit breaker test circuit.
A distinction has to be made between the various rating structures of medium- and highvoltage circuit breakers. Breakers rated with the older rating structure, covered by IEEE Std
C37.5-1979, are assessed on the basis of the total asymmetrical fault current, or total prospective fault MVA, and calculations are normally restricted to minimum parting time for the sake
of safety and simplicity. The more recent rating structure, covered by IEEE Std C37.0101979, assumes breakers to be rated on a symmetrical basis. Depending on service conditions
and the system X/R ratio, the calculated symmetrical short-circuit currents may be sufficient,
because a certain degree of asymmetry is embedded in the breaker rating structure.
When calculation of the total fault current is warranted for medium- and high-voltage breaker
calculations, IEEE Std C37.010-1979 and IEEE Std C37.5-1979 contain tabulated multipliers
that can be applied to the symmetrical rms fault currents in order to obtain asymmetrical rms
currents. For IEEE Std C37.5-1979, these currents are the total asymmetrical fault currents,
whereas IEEE Std C37.010-1979 represents currents that are to be compared with the breaker
interrupting capabilities. In both cases, these multipliers are obtained from curves normalized
against breaker contact parting time. As of 1987, the ANSI C37.06-1987 introduced the peak
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fault current to the preferred ratings as an alternative to the earlier total asymmetrical fault
currents (for first cycle withstand requirements) per ANSI C37.06-1979, in order to better
harmonize with IEC standards.
In summary, it should be stressed that an essential step for the calculation of the total fault
currents in medium- and high-voltage circuit breaker applications is the determination of portions of the fault current coming from “local” and “remote” sources as a means of obtaining a
more reasonable estimate of the breaker interrupting requirements (Huening [B5]). The reason for this distinction is that fault currents from remote sources feature slower, or no, ac current decay as compared to currents coming from local sources. A “remote” contribution, as
defined in IEEE Std C37.010-1979, IEEE Std C37.5-1979, IEEE Std 141-1993, and IEEE Std
242-1986, is the fault current that comes from a generator that
a)
b)
Is located two or more transformations away from the fault, or
Has a per unit X"d that is 1.5 times less than the per unit external reactance on a common MVA basis.
Chapter 4 of IEEE Std 141-1993 provides details on the methods that can be used to determine the appropriate composite adjustment factors that account for local and remote shortcircuit contributions. The ratio of the remote source contributions to the total short-circuit
current is also known as the NACD ratio (Huening [B5]).
7.4.2 The international standard, IEC 60909 (1988)
IEC 60909 (1988) is similar to the German VDE 0102-1972 standard and to the Australian
AS 3851-1991 standard. In what follows, only the very salient aspects are discussed in an
effort to make the potential user conscious of its computational and modeling requirements. It
is strongly recommended that interested readers consult the standard itself for further details.
IEC 60909 (1988) recognizes four duty types that result in four calculated fault currents:
—
—
—
—
The initial short-circuit current I"k
The peak short-circuit current Ip
The breaking short-circuit current Ib
The steady-state fault current Ik
Although, the breaking and steady-state fault currents are conceptually similar to the interrupting and time-delayed currents, respectively, the peak currents are the maximum currents
attained during the first cycle from a fault’s inception and are significantly different from the
first-cycle IEEE currents, which are total asymmetrical rms currents. The initial short-circuit
current is defined as the symmetrical rms current that would flow at the fault point if no
changes are introduced in the network impedances.
The IEC 60909 (1988) provides guidelines for calculating maximum and minimum fault currents. The former are to be used for breaker rating while the latter for protective device coordination. The major governing factors in calculating maximum and minimum fault currents
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are the prefault voltages at the fault point and the fact that minimum fault currents are calculated with minimum connected plant.
The phenomenon of ac decrement is addressed by considering the actual contribution of
every source, depending on the voltage at its terminals during the short circuit. Induction
motor ac decrement is modeled differently than synchronous machinery decrement, because
an extra decrement factor representing the more rapid flux decay in induction motors is
included. AC decrement is only modeled when breaking currents are calculated.
The phenomenon of dc decrement is addressed in IEC 60909 (1988) by applying the principle of superposition for the contributing sources in conjunction with giving due regard to the
topology of the network and the relative locations of the contributing sources with respect to
the fault position. In addition, the standard dictates that different calculating procedures be
used when the contribution converges to a fault point via a meshed or radial path. These considerations apply to the calculation of peak and asymmetrical breaking currents.
Steady-state fault currents are calculated by assuming that the fault currents contains no dc
component and that all induction motor contributions have decayed to zero. Synchronous
motors may also have to be taken into account. Furthermore, provisions are taken not only for
salient and round rotor synchronous machinery but, also for different excitation system
settings.
Prefault system loading conditions are of concern to IEC 60909 (1988) as well. In an attempt
to account for system loads leading to higher prefault voltages, the standard recommends that
prefault system voltages other than 1.00 per unit be used, without requiring a prefault load
flow solution. Furthermore, the standard recommends generator impedance correction factors
that may be applicable to their unit transformers as well.
7.4.3 Differences between the ANSI and IEEE C37 standards and IEC 60909
(1988)
The differences between the two standards are numerous and significant (Rodolakis [B7]).
Despite the conceptual association in the duty types, system modeling and computational
procedures are quite different in the two standards. That is why results calculated using both
standards can be quite dissimilar, with IEC 60909 (1988) having the tendency to yield higher
fault current magnitudes. The essential generic differences between the two standards can be
summarized in the following:
—
—
—
178
AC decrement modeling in IEC 60909 (1988) is fault location-dependent and it quantifies the rotating machinery’s proximity to the fault. The IEEE standard, on the other
hand, recommends universal, system-wide ac decrement modeling.
DC decrement for IEC 60909 (1988) does not always rely on a single X/R ratio. In
general, more than one X/R ratio must taken into account. Furthermore, the notion of
separate X and R networks for obtaining the X/R ratio(s) at the fault point is not applicable to IEC 60909 (1988).
Steady-state fault current calculation in IEC 60909 (1988) takes into account synchronous machinery excitation settings.
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In view of these important differences, computer simulations adhering to the ANSI and IEEE
C37 standards cannot, in general, be used to cover the computational requirements of IEC
60909 (1988) and vice versa.
7.5 Factors affecting the accuracy of short-circuit studies
The accuracy of the calculated fault currents depends primarily on accurate modeling of the
system configuration and the system impedances used for the calculations. Other very important factors include the correct modeling of system rotating load, connected generators, system neutral grounding, and other system components and operating conditions.
7.5.1 System configuration
System configuration consists of the following:
a)
b)
The location of all the potential sources of fault current, i.e., synchronous generators,
synchronous motors, induction motors, and utility connection points, and
How these fault current sources are connected through transformers, lines, cables,
busways, and reactors.
It is conceivable that more than one single-line diagram should be considered for a given system, depending on the system operating modes and on the nature of the study. If the study is
done to assess switchgear adequacy and/or selection, maximum fault currents should be calculated. This entails that fault currents must be calculated under maximum rotating plant and
closed bus-ties (whenever applicable), while any utility interconnections should be assumed
to attain their highest fault levels. If the study is done to assess protection sensitivity requirements, some of these conditions may need to be relaxed. Different system service conditions
may force the study of more than one system topology alternative, particularly in protective
relaying studies.
7.5.2 System Impedances
AC and dc decrement modeling considerations are very important factors in properly selecting the impedances of the rotating equipment for short-circuit studies. It is important to consult manufacturer’s catalogues, data sheets and, if necessary, to perform some calculations to
ascertain reliable impedance values. Typical values can be used in the absence of any other
information, but always with caution and a certain degree of conservatism. Table 7-3 portrays
some typical values for induction motors.
The values used for the impedances should by no means yield lower fault currents than the
ones the system will experience in reality. Underestimating the prospective fault currents can
lead to the undersizing of system equipment and to the selection of circuit breakers with inadequate interrupting capabilities. On the other hand, grossly overestimating the fault currents
may lead to uneconomical design and less sensitive protection settings. The equivalent
impedances representing the power company interconnection points must properly reflect the
anticipated fault MVA level. Any ambiguities concerning the impedances of in-plant equip-
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Table 7-3—Typical values of motor impedances and kVA ratings
to use when exact values are not knowna
Induction motor
Synchronous motor, 0.8 pf
Synchronous motor, 1.0 pf
1 hp = 1 kVA
1 hp = 1 kVA
.1 hp = 0.8 kVA
Motor type
X"d (See Note)
Synchronous motors
2–6 poles
8–14 poles
16 poles or more
0.15
0.20
0.28
Individual large induction motors, usually medium voltage
All others, 50 hp and above
All smaller than 50 hp, usually low voltage
0.67
0.67
0.67
NOTE—Motor impedances are in per unit on motor voltage and kVA rating. X"d for induction
motors is approximately equal to the locked-rotor reactance. For induction motors, the locked-rotor
reactance is the reciprocal value of the locked-rotor current. Reactances and motor base kVA ratings
listed were taken from data and assumptions in IEEE Std 141-1993.
aAs
specified in IEEE Std 141-1993.
ment should be resolved in favor of higher fault currents for the sake of safety in system
design. Impedances of bus ducts, busways, etc., must be accounted for in lower voltage circuits because they effectively limit fault current magnitudes. It is also customary practice to
use the saturated impedance values for synchronous machinery.
Last, but not least, the resistive components of the system impedances should be given proper
regard if operating system temperature is a factor or if significant lengths of cable runs are
present. Although resistance values can usually be omitted for fault current magnitude calculations (E/X calculation), they are important for calculating the system X/R ratio at the fault
point. Generally speaking, the total complex system impedance, Z (R+jX) has to be calculated
at the fault point to yield a more correct estimate of the fault current (E/Z calculation). This is
particularly true for low-voltage systems, where the system resistance is comparable in magnitude to the system reactance and helps limit the fault current.
7.5.3 Neutral grounding
For faults necessitating the inclusion of zero sequence data, i.e., line-to-ground faults, double
line-to-ground shunt faults, and series faults, the flow of fault currents is appreciably affected
by the system grounding conditions. Of particular concern is the presence of multiple
grounding points and the values of system grounding impedances. Grounding impedances
can be used, to various degrees, to limit the value of the ground fault current to a minimum
value, to suppress resulting overvoltages, and to provide “handles” for ground protection.
System grounding can also play an important role in the proper simulation of the system zero
sequence response. More specifically, for solidly, or low-impedance grounded systems, it is
sufficient to include in the study only the occasional current limiting transformer and or gen-
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erator grounding impedances, while disregarding zero sequence line/cable charging shunts.
For high-impedance grounded, floating, and/or resonant-grounded systems, however, the latter will have to be taken into account (per IEC 60909 (1988), since the assumption that
neglecting it yields conservative (higher) fault currents is no longer valid.
7.5.4 Prefault system loads and shunts
It is customary to assume that the system is at steady state before a short circuit occurs. The
simplification of neglecting the prefault load is based on the premise that the magnitude of
the prefault system load current is, usually, much smaller than the fault current. The importance of the prefault load current in the system increases with rated system voltage and certain system loading patterns. That is why it is still justifiable for typical industrial power
system studies to assume a 1.00 per-unit prefault voltage for every bus. For systems in which
prefault loading is a concern, a prefault load flow analysis should precede the fault simulations in order to ascertain a voltage profile for the system that will be consistent with the
existing system loads, shunts, and transformer tap settings. If the actual prefault system condition is modeled, it is important to retain for the fault simulation all the system static loads
(normally neglected when the system is assumed at rest) as well as the capacitive line/cable
shunts.
Standards such as IEC 60909 (1988) and AS 3851-1991 attempt to address this issue by virtue of using elevated prefault voltages and impedance correction factors for the synchronous
generators. The ANSI and IEEE C37 practice, however, is centered around considering the
prefault voltage as being the nominal system voltage with the notable exception being the
assessment of the interrupting requirements of circuit breakers.
7.5.5 Mutual coupling in zero sequence
This phenomenon is of importance when parallel circuits share the same right of way and
their geometrical arrangement is such that current flow in one circuit causes a voltage drop in
the other. A typical example is exposed overhead lines sharing the same support structure. It
should be noted that in reality, mutual coupling exists between phases in the positive
sequence as well. This form of mutual coupling, also known as “interphase coupling,” is not
explicitly modeled in positive sequence because it is restricted within the same circuit of
which only one phase is modeled. Zero sequence coupling, however, is extended between two
(or more) circuits and has to be explicitly modeled in zero sequence (Anderson [B1], Arrilaga, Arnold, and Harker [B2], Blackburn [B3], Stagg and El-Abiad [B9], Wagner and Evans
[B13]). The implications of neglecting or incorrectly modeling this phenomenon leads to
erroneous calculation of ground fault currents and incorrect performance assessment of distance relays. Although relatively infrequent for industrial power system analysis, it should be
borne in mind and treated accordingly.
7.5.6 Phase shifts in delta wye transformer banks
When calculating the distribution of the three-phase fault current throughout a system, it is
often assumed that, going through transformer banks, the phase of the fault current from
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primary to secondary remains the same. This is true only if the transformer is connected wyewye or delta-delta. When a delta-wye transformer is involved, a phase shift is introduced
between the phase quantities of the primary and secondary. The phase shift is present in positive and negative sequence quantities only. Zero sequence quantities are not affected. North
American practice dictates that the positive sequence high side line-to-ground voltage must
lead the positive sequence low side line-to-ground voltage by 30 degrees. Earlier transformer
connections and phase labeling may not comply with that requirement (Wagner and Evans
[B13]). The same may also be true for transformers following overseas phasing standards.
The computational consequence of not accounting for this phase shift for unbalanced faults is
that different current magnitudes are obtained when going through a delta-wye bank because
the sequence currents are manipulated vectorially to obtain phase currents. This can lead to
inaccurate protective device settings which can, in turn, compromise the selectivity of an
overcurrent protection scheme (see also IEEE Std 141-1993 and IEEE Std 242-1986).
7.6 Computer solutions
7.6.1 General
Short-circuit calculations are generally less computationally intensive than other basic power
system studies like power flow or harmonic analysis. In view of the fact that short-circuit calculations are linear systems of small to medium sizes can be computationally tractable by
hand, particularly if the system resistances are neglected to avoid complex arithmetic. Calculations are further simplified for radial systems. Practical industrial systems, however, can
contain several hundred to over one thousand buses, particularly if representation of lowvoltage circuits, smaller rotating loads and protective gear is warranted. Under these conditions, computer solutions are the only practical alternative. It should be noted, however, that
the speed and reliability of computer-based calculations are rapidly rendering hand-calculations a rarity even for small systems.
7.6.2 Computerized network solutions: System matrices
Hand calculations for determining the equivalent system impedance at a fault point rely on
successive and judiciously chosen combinations of the system branches, until the system is
reduced to an equivalent Thevenin impedance. This has to be repeated for every new fault
location. Since this is done by inspecting the network, the intuition of the analyst is essential.
Computers do not have any intuition, that is why different techniques are used. These techniques do not rely on the analyst’s inspection abilities, nor do they assume any system topology. That is why they lend themselves very well to both radial and looped systems and are
capable of accommodating systems of practically any size. The notions of admittance and
impedance matrices are central in realizing any computerized solution scheme.
7.6.2.1 The bus admittance matrix
The bus admittance matrix, also called the Y-matrix, is a square complex matrix (a matrix
whose entries are complex numbers) with as many rows and columns as the system buses
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(Anderson [B1], Arrilaga, Arnold, and Harker [B2], Stevenson [B10], Stagg and El-Abiad
[B9]). The elements of this matrix are either component admittances or sums of component
admittances. The term “component admittance” denotes the inverse of the component complex impedance with a component being a system branch, generator, motor, etc. Once the
system buses have been identified, this matrix can be constructed as follows:
—
—
Assign a diagonal matrix element to every system bus. The value of the matrix diagonal elements is the sum of the admittances of all the power system components connected to that bus.
Assign a nondiagonal element to all the matrix elements that represent a system
branch. For instance, if a branch is connected between buses i and j, the matrix entry
Yij will be nonzero and equal to the negative sum of the admittances of all components directly connected between buses i and j.
Electric power systems are passive and have very few branches compared to all of the possible bus connections, and as a result, typical power system bus admittance matrices are
a)
b)
symmetric (assuming that transformers are not modeled in off-nominal tap positions)
which means that Yij = Yji, and
sparse, i.e., they feature a lot of zero entries.
7.6.2.2 The bus impedance matrix
The bus impedance matrix, also called the Z-matrix, is defined as the inverse of the admittance matrix (Anderson [B1], Arrilaga, Arnold, and Harker [B2], Stevenson [B10], Stagg and
El-Abiad [B9]). This complex matrix is also square and symmetric, i.e., the entry Zij equals
the entry Zji, for passive networks. As the inverse of the sparse Y-matrix, however, this matrix
is a full matrix having no zero entries. It can be proved that the diagonal entries, Zii for bus i,
of this matrix are the equivalent Thevenin impedances used for fault calculations. The entry
Zij, however, does not necessarily represent the value of the impedance of the physical connection between buses i and j. In fact, there is always an impedance Zij despite the fact that
there may not be a branch between buses i and j. The diagonal entries of the Z-matrix are used
in calculating fault currents, while the nondiagonal entries are useful for calculating branch
contributions and system-wide voltage profiles under fault conditions.
7.6.2.3 System topology, matrix sparsity, and solution algorithms
The sparsity of the Y-matrix requires that special techniques be employed for storing the system data, because conceptually straightforward storage techniques may be quite wasteful.
Storing, for instance, the entire Z-matrix is not only impractical but unnecessary because only
a few of its elements may be needed. The development of solution algorithms, therefore, has
been focusing on the efficient retrieval of the necessary Z-matrix entries with the smallest
possible storage and calculation requirements. Modern vintage computer software employs
calculation and system data storage schemes that center around the so-called “sparse vector”
and/or “sparse matrix” solution techniques (Tinney, Brandwajn, Chan [B12]) which render
very rapid and accurate solutions.
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7.6.3 Computer software
7.6.3.1 General
The availability of commercial grade computer software on personal computers has been
steadily increasing in variety and computational power since the early 1980s, although
sophisticated software has existed for more powerful hardware platforms such as mainframes
and minicomputers since the early 1960s (St. Pierre [B11]). The personal computer is now
recognized as a credible computational tool due to the significant advances it has enjoyed in
processor architecture, speed, memory capacity, and in user-friendly operating systems and
environments. Computer programs that addressed short-circuit calculations were among the
first to be developed in all platforms. All programs rely on matrix techniques and require the
analyst to provide accurate system data so that the computer can proceed with the analysis
and produce the results.
7.6.3.2 Selecting software
The great variety of commercially available computer programs for short-circuit calculations
can be attributed to the wide variety of the analytical tasks they perform, the degree of sophistication in user-interface and user-friendliness, and the computer platform for which they are
designed. Because the variety of the available computer software is accompanied by an
equally impressive variation in prices, it is important to acquire software that best corresponds to the bulk of the engineering mandates for which it is purchased. It is questionable,
from an investment point of view, to acquire expensive and very sophisticated software when
the bulk of its analytical features will never be used. On the other hand, it could prove shortsighted to acquire inexpensive software that will rapidly be outgrown by the needs of its user,
compromise the accuracy of the study, or result in a consistent waste of time and resources
due to inherent functional inefficacies. It is also important to assess the degree of user-friendliness of the software versus the computer-literacy of the personnel who will be using it.
Many engineers are reluctant to refamiliarize themselves with bulky user’s guides only to
perform studies with which they are very familiar. It pays to work with software that features
easy data entry, meaningful and helpful diagnostic messages, and comprehensive reports.
Last, it is essential to acquire software that is very well documented, promptly supported, and
regularly updated and upgraded by its vendors.
7.6.3.3 Features of short-circuit analysis software
The previously mentioned general salient principles governing software selection are supplemented by a good number of other features that are particularly applicable to short-circuit
analysis. A very important aspect in short-circuit studies is data preparation, a stage which,
by itself, can be computationally demanding, particularly if the software accepts system data
only on a per-unit basis. It is essential for the program to help the analyst prepare the data for
the study and provide means of identifying and correcting obvious and common mistakes.
Furthermore, whenever international standards are to be used, it is important for the software
to provide sufficient information and results that are transparent enough to allow for more
than one interpretation.
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Table 7-4 contains several features that computer programs may or may not support. These
features have been conceptually categorized as “very desirable,” “desirable,” and “optional.”
“Very desirable” means that the feature is widely encountered and rather indispensable. The
category “desirable” addresses features that will prove of value to more demanding studies.
The category “optional” covers features that may prove to be of value for special studies.
Table 7-4—Analytical features of short-circuit computer programs
Analytical feature
Very
desirable
Systems with more than one voltage level
Yes
Looped and radial system topology
Yes
Ground faults (LG and LLG)
Yes
Series faults (See Note 1)
Desirable
Optional
Yes
Arcing faults (See Note 2)
Yes
Simultaneous faults
Yes
Complex arithmetic
Yes
Explicit negative sequence (See Note 3)
Yes
Interface with power flow (See Note 1)
Yes
Currents in all three phases (See Note 4)
Yes
Currents in all three sequences (See Note 4)
Yes
One-bus-away fault contributions
Yes
Line monitors (See Note 5)
Input data reports
Yes
Yes
Protection coordination interface
Yes
Voltages in nonfaulted buses (See Note 5)
Yes
Summary reports
Yes
Currents in all branches (See Note 5)
Yes
Per-unitization of equipment data
Yes
Rotating equipment impedance adjustment
(IEEE Std C37.010-1979)
Separate X and R reduction for X/R ratios
(IEEE Std C37.010-1979)
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Yes
Mandatory
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Table 7-4—Analytical features of short-circuit computer programs (Continued)
Analytical feature
Very
desirable
Remote and local fault contributions
(IEEE Std C37.010-1979)
Mandatory
First cycle fault currents (IEEE Std C37.010-1979)
Mandatory
Interrupting fault currents (IEEE Std C37.010-1979)
Mandatory
Time-delay fault currents (IEEE Std C37.010-1979)
Desirable
Yes
Symmetrical current multiplying factors
(IEEE Std C37.010-1979)
Yes
Total current multiplying factors
(IEEE Std C37.5-1979 factors)
Yes
Multiplying factors (IEEE Std C37.13-1990)
Yes
X/R—dependent 1/2 cycle multipliers
(IEEE Std C37.010-1979)
Yes
X/R—dependent peak multipliers
(IEEE Std C37.010-1979)
Yes
Transformer phase shifts (See Note 4)
Yes
Mutual coupling in zero sequence (See Note 6)
Methodology in accordance with IEC 60909 (1988)
(See Note 7)
Optional
Yes
Yes
NOTES
1—Series faults are normally modeled when a prefault power flow solution is available and prefault
load can be taken into account.
2—Arcing faults can be of consequence when assessing the sensitivity of ground protection in solidly
grounded systems. Conservative estimates of fault levels, however, will result by assuming bolted
faults.
3—Negative sequence system representation is not warranted by ANSI and IEEE C37 standards,
though it could be of significance when ground faults near large generating stations are calculated, or when simulations compatible with IEC 60909 (1988) require elevated accuracy.
4—It is important to have a good estimate of all three phase and sequence currents, particularly for
protection requirements. In correctly estimating all these currents in magnitude and phase it is
very helpful to take into account phase shifts in transformer banks.
5—When assessing the degree of severity of a fault, it is often of interest to see how nearby system
areas are affected, and if protective devices will be activated as a result of the fault.
6—Mutual coupling in zero sequence normally affects overhead circuitry sharing a common rightof-way. For industrial power systems analysis, zero sequence mutual coupling can be of concern
if such circuits are modeled and the performance of protective devices requiring zero sequence
current compensation is investigated.
7—Support of the IEC 60909 (1988) may require dedicated software.
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7.7 Example
In what follows, a short-circuit study is carried on a typical industrial system in order to illustrate typical steps, calculation requirements, and results. The system is composed of circuits
of several voltage levels, local generation, a utility interconnection, and a variety of rotating
loads. The study is carried out according to the ANSI and IEEE C37 standards (see 7.8).
7.7.1 Determination of the scope and extent of the study
Determination of the scope extent and the desired accuracy of the study is crucial, because
these factors will dictate what types of faults are to be simulated and to what degree system
modeling is to be undertaken. The type and number of fault studies for a given system is
determined by engineering judgment, which is based on the various forms the system layout
may assume during operation or the specific purpose of the study.
The study results may be used for recommending changes to existing plants or for proposing
an initial design for a system in its planning and/or expansion stage. Some important questions for which fault studies may help provide answers are as follows:
a)
b)
c)
d)
e)
f)
g)
Is circuit interrupting equipment adequate for the system interrupting requirements at
all voltage levels? Can the medium- and high-voltage switchgear withstand the
momentary and interrupting duties imposed by the system? Is this switchgear adequate for line to ground faults? If not, should new equipment be purchased or can
some changes to the system be effected to avoid the extra capital expenditure?
Is there any reserve in the interrupting capability of the circuit breakers for accommodating future system expansion? If not, is it necessary to have a safety margin for
future expansion? If so, how can the system be changed to accommodate these concerns?
Is noninterrupting equipment, i.e., reactors, cables, transformers, bus ducts, adequately rated to withstand short-circuit currents until cleared by the interrupting
equipment?
Do load circuit breakers or disconnecting switches have sufficient momentary bracing and/or close-and-latch capabilities?
What will be the effect on the calculated short-circuit currents in the plant system if
there is an increase in the power company’s short-circuit level? Economically, what
can be done to anticipate such an eventuality?
Is special protective equipment or circuitry necessary to provide protective device
selectivity for both maximum and minimum value of short-circuit currents?
During faults, do the voltages on unfaulted buses in the system drop to levels that can
cause motor-starter contactors to drop out or undervoltage relays to operate?
Every study will have to be assessed on its own merits and its results interpreted only for the
purpose the study was conducted. The short-circuit study for the example in question is performed for the purposes of determining the interrupting requirements for low-, medium-, and
high-voltage switchgear. It is not uncommon for these types of studies to consider only threephase fault currents, since, as a rule, they yield the more severe interrupting requirements, as
compared to other shunt faults, and the industrial power systems are often impedance-
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grounded. Line-to-ground fault simulations are necessary for circuit-breaker-adequacy evaluation and/or selection, if the system is such that line-to-ground fault currents may exceed
three-phase fault currents (see 7.2). Only three-phase, bolted, shunt faults will be considered
for the example.
7.7.2 Preparation of the system one-line diagram and collection of data
The one-line diagram of the system is shown in Figure 7-1. We will consider that both inplant generators are connected and that both of the utility service entrance transformers are in
service. The system rotating load, as shown in the one line diagram, represents an operating
condition that is typical for the system operating at or near full capacity. Furthermore, it is
known that the bus ties between buses 3 and 4 (13.8 kV) and between buses 1 and 2 (69 kV)
are open. Cable runs between buses, 9 (FDR E) and 13 (T6 PRI), 28 (T10 SEC) and 38 (480
TIE), 30 (T12 SEC) and 38 (480 TIE), 10 (EMER) and 12 (T5 PRI), are also assumed to be
open. It is this particular system layout that will be studied. In general, however, depending
on the type of study, more than one single-line diagram may have to be considered in practice
(see 7.5.1).
Data necessary for conducting a short-circuit study comprise the following:
a)
b)
c)
d)
Utility interconnection points and associated fault MVA levels (both three-phase and
line-to-ground) in order to determine the equivalent impedance of the utility
In-plant generation data
Rotating load data comprising synchronous motors and induction motors, both standalone and grouped
Static system equipment data, such as transformers, cables, reactors, overhead lines,
busways, bus ducts, etc., switching equipment and, in some cases, static loads (heaters, drives, etc.).
7.7.3 Determination and per-unitization of system impedances
7.7.3.1 Determination of the required system impedances
The choice of system impedances to be used depends on the type of study to be performed
and the actual fault conditions to be simulated. Three-phase fault studies require only positive
sequence impedances, whereas faults involving ground will require zero sequence system
data as well as any neutral grounding data. Negative sequence impedances may also be necessary for line-to-line fault simulations. Impedances of both static and rotating system components are normally known from equipment nameplate data. In the absence of detailed
information, typical values are assumed. Nameplate impedances of rotating equipment are
modified from their rated values in order to account for ac decay, according to the North
American practice (see 7.4.1).
7.7.3.2 Per-unitization of the system impedances
Power system equipment impedances are expressed on a unified per-unit basis because
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a)
b)
Carrying out the calculations in ohms is not practical for systems composed of more
than one voltage level, and
The impedances of the system components are expressed in terms of their rated voltage and power.
When the per-unitization of the system impedances to a common MVA base is done manually, caution is required because this is a common source of errors. It is also one of the most
time-consuming tasks of a short-circuit study. Computer programs, in general, operate internally on per-unitized impedances, and many offer the facility to convert the “raw” system
data to per-unit data in a form ready to be used by the program. If this is the case, it is important to comprehend how the per unitization is carried out. This will greatly facilitate any
future “what if” analysis because of different system layouts or modified system impedances.
In any case, any error in per unitizing the system impedances can seriously compromise the
accuracy of the short-circuit study.
Per-unit impedances are defined as the ratio of the actual ohmic component impedances to a
certain base impedance (see also Chapters 3 and 4). The base impedances are calculated from
a common, arbitrarily chosen, apparent power base and from a base voltage (Anderson [B1],
Stevenson [B10]).
2
Z ohms
V base
Z p.u. = -------------, with Z base = ----------Z base
S base
(7-1)
The power base is usually expressed in MVA and is applicable throughout the system. The
base voltage is expressed in kilovolts (if base power is in MVA) and selected differently for
every system section, following the nominal voltage ratios of the system power transformers.
If single phase power is chosen, line-to-ground voltages should be used. Alternatively, if three
phase power is chosen as base line to line voltages are in order. It is practical to select as base
voltages the rated transformer voltages. All buses in the same network section must share the
same voltage base. Equipment like transformers, generators, motors, etc., have their impedances given in percent (per unit × 100) of their rated voltage and power. It is often necessary
to convert these impedances to new base quantities as follows:
2
Z p.u., new
S new V old
- Z p.u., old
= ---------- ---------S old V 2
new
(7-2)
In what follows, some per-unit impedances used in the example study are calculated.
7.7.3.3 Power company per-unit impedance
Assuming that the three-phase MVA fault level at bus 100-UTIL-69 is 1000 MVA, the perunit impedance of the power company, for a 10 MVA base for the system, is calculated to be
MVA base
10
Z utility = --------------------= ------------ = 0.01 p.u.
1000
MVA fault
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(7-3)
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Figure 7-1—One-line diagram
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Assuming a typical X/R ratio of 22.0, the utility equivalent impedance becomes
2
2
X 2
X
2
2
2
Z = R + -----2- R = R 1.0 + -----2- , from which
R
R
Z utility = 0.000454 + j0.00999 p.u.
This impedance will be applied for both first cycle and interrupting duty calculations because
the power company is considered always to be a “remote” source, thus featuring no ac
decrement.
7.7.3.4 Power cable per-unit impedance
Power cable impedances are normally provided in Ω/1000 ft. Consider, as an example, the
cable connecting buses 3-MILL-1 and 9-FDRE, which is a 250MCM, 3-core, copper conductor, PVC-jacketed cable, and applied at 13.8 kV. The impedance of the cable is Z = 0.0440419
+ j0.0366795 Ω/1000 ft. The cable conduit between the two buses spans 650 ft. The impedance of this cable expressed in per unit of system data will be
10.00
Z cable p.u. = 0.650 ( 0.0440419 + j0.0366795 ) ------------2- = 0.0015032 + j0.0012519 p.u.
13.8
This impedance, as with all impedances of power cables and overhead circuitry, will be
applied as is for both first cycle and interrupting calculations. Impedance correction decrement factors apply only to rotating induction and synchronous loads and in some cases to
synchronous generators (see Table 7-2).
7.7.3.5 Synchronous generator per-unit impedance
The generator connected on bus 4-MILL-2 has a rated power of 12.5 MVA, a rated voltage of
13.8 kV, a subtransient reactance of 12.8%, and an X/R ratio of 35.7. The reactance of the
generator on the common system MVA base of 10 MVA is found to be
2
X p.u., new
MVA new V old 13.8 2
- ------------0.128 = 0.1024 p.u.
= --------------------- ---------MVA old V 2 13.8 2
new
The generator impedance, for the given X/R ratio, therefore becomes
0.1024
Z p.u. = ---------------- + j0.1024 = 0.002868 + j0.1024 p.u.
35.7
For the example at hand, it will be assumed that all generators are turbo generators, thus
ac decrement impedance correction factors will be equal to 1.00 for interrupting duty
calculations.
Copyright © 1998 IEEE. All rights reserved.
191
IEEE
Std 399-1997
CHAPTER 7
7.7.3.6 Synchronous motor per-unit impedance
Consider the synchronous motor connected at bus 8-FDR-L. This motor has a subtransient
reactance of 20.00%, an X/R ratio of 34.00, and is rated 9000 kVA at 13.8 kV. The motor
impedance, expressed on a 10 MVA, 13.8 kV base reference, will then be
2
10.00 13.8
0.20
Z syn p.u. = ---------- + j0.20 ------------- ------------2 = 0.006534 + j0.2222 p.u.
3.40
9.000 13.8
The impedance calculated for the synchronous motor is applicable to both first cycle and
interrupting calculations (see Table 7-2).
7.7.3.7 Induction motor per-unit impedances
7.7.3.7.1 Small induction motors
The motor load at bus 51-AUX partly consists of many small motors rated below 50 hp, totaling 570 hp, connected at 480 V, rotating at 1800 r/min. Assuming a total equivalent locked
rotor reactance of 16.7% and an X/R ratio of 12.00, the per-unit impedance of the group of
motors becomes
2
10.00 0.48
0.167
Z p.u. = ------------- + j0.167 ------------- ------------2 1.67 = 0.40773 + j4.89278 p.u.
12.00
0.570 0.48
Alternative interpretations of IEEE Std C37.010-1979 and IEEE Std C37.13-1990, governing
first-cycle duty calculations, recommend omitting these motors for medium- and high-voltage, while considering them for low-voltage calculations. By applying the factor 1.67, both
high- and how-voltage circuits are modeled in one network and a single computer simulation
will suffice. For medium- and high-voltage interrupting duty calculations, these motors are
neglected (see Table 7-2).
7.7.3.7.2 Medium-sized induction motors
The motor load at bus 51-AUX also comprises a medium-sized induction motor, rotating at
1800 r/min, rated 200 kVA (200 hp with 0.746 power factor) at 480 V, with a locked rotor
reactance of 16.7% and an X/R ratio of 7.00. According to Table 7-2, the per-unit motor
impedances to be used in first cycle (impedance adjustment factor of 1.2) and interrupting
duty (impedance adjustment factor of 3.0) calculations are
2
10.00 0.48
0.167
Z first cycle p.u. = ------------- + j0.167 ------------- ------------2 1.20 = 1.43143 + j10.0200 p.u.
7.00
0.200 0.48
2
10.00 0.48
0.167
Z inter. p.u. = ------------- + j0.167 ------------- ------------2 3.00 = 3.57858 + j25.0500 p.u.
7.00
0.200 0.48
192
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SHORT-CIRCUIT STUDIES
7.7.4 Example system data
The data for the example system components are shown in Tables 7-5 through 7-12. The data
for rotating equipment and transformers are presented in raw form, as would typically be seen
on the nameplates of the various apparatus.
Table 7-5—Example system generator data
Rated
kV
Rated
MVA
X" d
(%)
X/R
ratio
Xo
(%)
Xo/Ro
ratio
GEN-1
13.8
15.625
0.11200
37.4
5.70
37.4
GEN-2
13.8
12.500
12.8
35.7
5.80
35.7
GEN-ID
Table 7-6—Example system utility interconnection data
Connection
point
UTIL-1
3PH-MVA
X/R ratio
L-G MVA level
X/R ratio
1000.00
22.00
765.00
9.70
Table 7-7—Example system overhead line data (impedances in Ω/mi)
Rated kV
Conductor
size
R1
(Ω/mi)
X1
(Ω/mi)
Length
(mi)
L-1
69.00
266.8 MCM
.34940
.744063
1.894
L-2
69.00
266.8 MCM
.34940
.744063
1.894
Line ID
Table 7-8—Example system fixed-tap transformer data
XMR
ID
Rated
MVA
Primary
Secondary
kV
Bus
kV
Bus
Z1
(%)
X1/R1
ratio
Zo
(%)
Xo/Ro
ratio
T-1
15.0
69.00
01
13.80
03
8.00
17.000
7.20
17.000
T-10
1.50
13.80
25
0.48
28
5.75
6.500
5.75
6.500
T-11
1.50
13.80
26
0.48
29
5.75
6.500
5.50
6.500
T-12
1.50
13.80
27
0.48
30
5.75
6.500
5.50
6.500
T-13
2.50
13.80
31
2.40
36
5.75
10.000
50.00
10.000
Copyright © 1998 IEEE. All rights reserved.
193
IEEE
Std 399-1997
CHAPTER 7
Table 7-8—Example system fixed-tap transformer data (Continued)
XMR
ID
Rated
MVA
Primary
Secondary
kV
Bus
kV
Bus
Z1
(%)
X1/R1
ratio
Zo
(%)
Xo/Ro
ratio
T-14
1.00
13.80
32
0.48
37
5.75
5.500
50.00
5.500
T-17
1.25
13.80
05
0.48
49
4.50
6.000
4.50
6.000
T-18
1.50
13.80
50
0.48
51
5.75
5.914
5.75
5.910
T-2
15.0
69.00
02
13.80
04
8.00
17.00
7.40
17.000
T-3
1.725
13.80
05
4.16
39
6.00
8.000
6.00
8.000
T-4
1.50
13.80
06
2.40
11
5.50
6.500
5.50
6.500
T-5
1.50
13.80
12
0.48
17
6.75
6.500
6.75
6.500
T-6
1.50
13.80
13
0.48
18
5.75
6.500
5.75
6.500
T-7
3.75
13.80
06
5.50
19
5.50
12.000
5.50
12.000
T-8
3.75
13.80
15
2.40
20
5.50
12.000
5.50
12.000
T-9
0.75
13.80
16
0.48
21
5.75
5.000
5.50
5.000
Table 7-9—Example system busway data (impedances in Ω/100 ft)
Busway ID
SQD-I-Li
Rated
kV
Size
(A)
R1
(Ω)
X1
(Ω)
Length
(ft)
From
bus
To
bus
0.48
1000
0.0016
0.0010
50.0
28
41
7.7.5 Results
It is not uncommon for computer programs to automatically perform the conversion of the
raw system data to per-unit data ready to be used by the computer package. The results of
such a conversion are illustrated in Figure 7-2, as reported by the computer program. These
data consist of transformer, cable, and line data.
Furthermore, since the study is to follow IEEE Std 141-1993 and IEEE Std 242-1986 and is
carried out for switchgear adequacy verification purposes, two separate studies are required.
Both simulations will generate three-phase fault currents, the first, yielding the first-cycle
fault currents (also known as momentary or close- and latch-currents), and the second, providing the interrupting currents. Strictly speaking, the same studies should be repeated for
line-to-ground faults if system conditions conducive to the generation of line-to-ground fault
currents could exceed the three-phase fault-interrupting requirements (see 7.2).
194
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SHORT-CIRCUIT STUDIES
Table 7-10—Example system cable data (impedances in Ω/1000 ft)
Cable
ID
kV
Length
(ft)
From
bus
To
bus
R1
(Ω)
X1
(Ω)
Ro
(Ω)
Xo
(Ω)
C-E1
13.8
650
03
09
0.04404
0.03668
0.0808
0.07336
C-E2
13.8
1833
09
25
0.04404
0.03668
0.08808
0.07336
C-E3
13.8
75.0
09
13
0.04404
0.03668
0.08808
0.07336
C-E4
13.8
165.0
09
12
0.04404
0.03668
0.08808
0.07336
C-F1
13.8
325.0
03
05
0.04404
0.03668
0.08808
0.07336
C-G1
13.8
680.0
03
26
0.04404
0.03668
0.08808
0.07336
C-H1
13.8
471.0
03
06
0.04404
0.03668
0.08808
0.07336
C-I1
13.8
980.0
04
15
0.04404
0.03668
0.08808
0.07336
C-J2
13.8
619.0
04
27
0.04404
0.03668
0.08808
0.07336
C-J3
13.8
1187.0
16
04
0.04404
0.03668
0.08808
0.07336
C-J4
13.8
200.0
10
13
0.04404
0.03668
0.08808
0.07336
C-J5
13.8
10.0
10
12
0.04404
0.04214
0.08808
0.08429
C-J6
13.8
475.0
10
27
0.04404
0.03668
0.08808
0.07336
C-L1
13.8
510.0
04
08
0.02831
0.03424
0.05661
0.06847
C-M1
13.8
510.0
04
24
0.04404
0.03668
0.08808
0.07336
C-M2
13.8
340.0
24
31
0.04404
0.03668
0.08808
0.07336
C-M3
13.8
485.0
24
32
0.04404
0.03668
0.08808
0.07336
C-T10-1
0.48
50.0
28
38
0.02801
0.02699
0.05602
0.05399
C-T10-2
0.48
20.0
33
28
0.04393
0.02823
0.08786
0.05645
C-T11-1
0.48
66.0
29
38
0.04393
0.02699
0.05602
0.05399
C-T11-2
0.48
20.0
34
29
0.04393
0.02823
0.08786
0.05645
C-T12-1
0.48
50.0
38
30
0.02801
0.02699
0.05602
0.05399
C-T12-2
0.48
20.0
35
30
0.04393
0.02823
0.08786
0.05645
C-T5-1
0.48
20.0
22
17
0.04393
0.02823
0.08786
0.05645
C-T6-1
0.48
20.0
23
18
0.04393
0.02823
0.08786
0.05645
C1A
13.80
2000.0
50
03
0.02314
0.04622
0.02083
0.41595
Copyright © 1998 IEEE. All rights reserved.
195
IEEE
Std 399-1997
CHAPTER 7
Table 7-11—Example system synchronous motor data
Motor ID
Motor
bus #
Rated
kV
Rated
kVA
Rated
hp
r/min
X"d
(%)
X/R
ratio
8
13.8
9000
9000
1800
20.0
34.0
M-FDR-L
Table 7-12—Example system induction motor data
Motor
bus #
Total
hp
Total
kVA
r/min
Rated
kV
XM
(%)
X/R
ratio
Composition
(hp)
M-30
51
200.0
200.0
1800
0.480
16.7
7.00
>50
M-31
51
600.0
570.0
1800
0.480
16.7
12.00
<50
M-T10-1
28
400.0
400.0
1800
0.480
16.7
10.00
<50
M-T10-2
28
500.0
500.0
1800
0.480
16.7
5.00
>50
M-T10-3
33
300.0
287.5
1800
0.480
16.7
12.00
<50
M-T11-1
29
625.0
625.0
1800
0.480
16.7
10.00
>50
M-T11-2
29
465.0
465.0
1800
0.480
16.7
5.00
<50
M-T11-3
34
110.0
110.0
1800
0.480
16.7
7.00
<50
M-T12-1
30
400.0
387.9
1800
0.480
16.7
12.00
>50
M-T12-2
30
500.0
500.0
1800
0.480
16.7
5.00
<50
M-T12-3
35
300.0
287.5
1800
0.480
16.7
12.00
<50
M-T13-1
36
2500.00
2250.00
1800
2.300
16.7
32.85
>50
M-T14-1
37
700.0
678.8
1800
0.480
16.7
12.00
>50
M-T14-2
37
300.0
300.0
1800
0.480
16.7
5.00
<50
M-T17-1
49
1250.00
1250.00
1800
0.460
33.0
10.00
>1000
M-T3-1
39
1750.00
1662.50
1800
4.160
16.7
29.74
>1000
M-T4-1
11
500.0
475.0
1800
2.400
16.7
12.00
>50
M-T5-1
17
850.0
824.2
1800
0.480
16.7
10.00
<50
M-T5-2
17
500.0
500.0
1800
0.480
16.7
5.00
>50
M-T5-3
22
150.0
142.5
1800
0.480
16.7
14.00
<50
M-T6-1
18
850.0
824.2
1800
0.480
16.7
10.00
<50
M-T6-2
18
500.0
500.0
1800
0.480
16.7
5.00
>50
Motor
ID
196
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SHORT-CIRCUIT STUDIES
Table 7-12—Example system induction motor data (Continued)
Motor
bus #
Total
hp
Total
kVA
r/min
Rated
kV
XM
(%)
X/R
ratio
Composition
(hp)
M-T6-3
23
150.0
142.5
1800
0.480
16.7
14.00
<50
M-T7-1
19
1250.00
1125.00
1800
2.400
16.7
26.10
>1000
M-T7-2
19
2500.00
2375.00
1800
2.400
16.7
15.00
>1000
M-T8-1
20
1750.00
1662.50
1800
2.400
16.7
15.00
>1000
M-T8-2
20
2000.00
1800.00
1800
2.400
28.0
26.00
>1000
M-T9-1
21
750.0
727.3
1800
0.480
16.7
12.00
<50
Motor
ID
NOTE—The motor load M-T17-1 at bus 49 (RECT) signifies a dc rectifier load and has been modeled as an induction motor. The reactance was taken to be 33% (0.33 p.u.) considering the rated MVA
of the associated converter transformer, in accordance with Internationally accepted recommendations. The dc rectifier load will be considered only for first-cycle simulation purposes and ignored for
interrupting duty.
Impedances
From
bus #
To
bus #
Crkt #
1
3
2
From
kV
To
kV
Tap
R(p.u.)
X(p.u.)
R(p.u.)
1
.00313
.05324
0.0
69.00
13.800
1.00
4
1
.00313
.05324
0.0
69.00
13.800
1.00
5
39
1
.04314
.34514
0.0
13.80
4.160
1.00
5
49
1
.05918
.35510
0.0
13.80
.480
1.00
6
11
1
.05575
.36240
0.0
13.80
2.400
1.00
6
19
1
.01218
.14616
0.0
13.80
2.400
1.00
12
17
1
.06843
.44477
0.0
13.80
.480
1.00
13
18
1
.05829
.37888
0.0
13.80
.480
1.00
15
20
1
.01218
.14616
0.0
13.80
2.400
1.00
16
21
1
.15036
.75178
0.0
13.80
.480
1.00
25
28
1
.05829
.37888
0.0
13.80
.480
1.00
26
29
1
.05829
.37888
0.0
13.80
.480
1.00
27
30
1
.05829
.37888
0.0
13.80
.480
1.00
Figure 7-2—Positive sequence system branch data (p.u., 10 MVA)
Copyright © 1998 IEEE. All rights reserved.
197
IEEE
Std 399-1997
CHAPTER 7
Impedances
From
bus #
To
bus #
Crkt #
31
36
32
From
kV
To
kV
Tap
R(p.u.)
X(p.u.)
R(p.u.)
1
.02289
.22886
0.0
13.80
2.400
1.00
37
1
.10286
.56573
0.0
13.80
.480
1.00
50
51
1
.06395
.37796
0.0
13.80
.480
1.00
3
5
1
.00075
.00063
0.0
3
6
1
.00109
.00091
0.0
3
9
1
.00150
.00125
0.0
3
26
1
.00157
.00131
0.0
4
8
1
.00076
.00092
0.0
4
15
1
.00227
.00189
0.0
4
24
1
.00118
.00098
0.0
4
16
1
.00274
.00229
0.0
4
27
1
.00143
.00119
0.0
9
12
1
.00038
.00032
0.0
9
25
1
.00424
.00353
0.0
10
13
1
.00046
.00039
0.0
10
27
1
.00110
.00091
0.0
17
22
1
.03813
.02451
0.0
18
23
1
.03813
.02451
0.0
24
31
1
.00079
.00065
0.0
24
32
1
.00112
.00093
0.0
28
33
1
.03813
.02451
0.0
28
41
1
.03429
.02105
0.0
29
34
1
.03813
.02451
0.0
29
38
1
.08024
.07732
0.0
29
38
2
.08024
.07732
0.0
30
35
1
.03813
.02451
0.0
50
3
1
.00243
.00485
0.0
50
3
2
.00243
.00485
0.0
Figure 7-2—Positive sequence system branch data (p.u., 10 MVA) (Continued)
198
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SHORT-CIRCUIT STUDIES
Impedances
From
bus #
To
bus #
Crkt #
1000
1
1000
2
R(p.u.)
X(p.u.)
R(p.u.)
1
.00139
.00296
0.0
1
.00139
.00296
0.0
From
kV
To
kV
Tap
Figure 7-2—Positive sequence system branch data (p.u., 10 MVA) (Continued)
Both studies are needed for medium- and high-voltage breaker calculations (above 1kV)
while only the first one is needed for low-voltage breakers (below 1 kV). The generator
impedances utilized for the first cycle and interrupting duty studies are shown in Figure 7-3
(from Table 7-2). The motor impedances used for momentary and Interrupting calculations
are shown in Figure 7-4 (from Table 7-2). It can be seen that small induction motors (individual motors or groups of motors composed from motors smaller than 50 hp) are neglected for
interrupting duty calculations (Table 7-2).
Generator impedances
Generator
bus #
Bus kV
R'd
X'd
R"d
X"d
4
13.80
0.003
0.102
0.003
0.102
50
13.80
0.002
0.072
0.002
0.072
100
69.00
0.000
0.010
0.000
0.010
Figure 7-3—Generator impedances for momentary and
interrupting duty (p.u., 10 MVA)
Motor
Motor
bus #
Bus kV
11
#
Type
2.40
1
IM
17
0.48
1
17
0.48
18
18
Motor
MVA
Motor impedances
Rmom
Xmom
Rinter
Xinter
0.4750
0.352
4.219
0.879
10.547
IM
0.8242
0.338
3.384
1
IM
0.5000
0.802
4.008
2.004
10.020
0.48
1
IM
0.8242
0.338
3.384
0.48
1
IM
0.5000
0.802
4.008
2.004
10.020
Figure 7-4—Motor impedances for momentary and
interrupting duty (p.u., 10 MVA)
Copyright © 1998 IEEE. All rights reserved.
199
IEEE
Std 399-1997
CHAPTER 7
Motor
Motor
bus #
Bus kV
19
#
Type
2.40
1
IM
19
2.40
1
20
2.40
20
Motor
MVA
Motor impedances
Rmom
Xmom
Rinter
Xinter
1.1250
0.057
1.484
0.085
2.227
IM
2.3750
0.047
0.703
0.070
1.055
1
IM
1.6625
0.067
1.005
0.100
1.507
2.40
1
IM
1.8000
0.060
1.556
0.090
2.333
21
0.48
1
IM
0.7273
0.320
3.835
22
0.48
1
IM
0.1425
1.398
19.571
23
0.48
1
IM
0.1425
1.398
19.571
28
0.48
1
IM
0.5000
0.697
6.972
28
0.48
1
IM
0.4000
0.802
4.008
2.004
10.020
29
0.48
1
IM
0.6250
0.321
3.206
0.802
8.016
29
0.48
1
IM
0.4650
1.199
5.998
30
0.48
1
IM
0.3879
0.431
5.166
1.077
12.916
30
0.48
1
IM
0.5000
1.116
5.578
33
0.48
1
IM
0.2875
0.809
9.701
34
0.48
1
IM
0.1100
3.621
25.354
35
0.48
1
IM
0.2875
0.809
9.701
36
2.40
1
IM
2.2500
0.025
0.818
0.062
2.045
37
0.48
1
IM
0.6788
0.246
2.952
0.615
7.381
37
0.48
1
IM
0.3000
1.859
9.296
39
4.16
1
IM
1.6625
0.034
1.005
0.051
1.507
49
0.48
1
IM
1.2500
0.264
2.640
51
0.48
1
IM
0.2000
1.432
10.020
3.579
25.050
51
0.48
1
IM
0.5700
0.408
4.893
8
13.80
1
SM
9.0000
0.006
0.222
0.010
0.333
Figure 7-4—Motor impedances for momentary and
interrupting duty (p.u., 10 MVA) (Continued)
Typical information pertinent to first-cycle duty calculations is shown in Table 7-4. The symmetrical first cycle fault currents along with the total asymmetrical rms currents using the
standard 1.6 multiplier (assuming an X/R ratio of 25 or less), as well as using the asymmetri-
200
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SHORT-CIRCUIT STUDIES
cal multiplier, calculated from the actual system X/R ratio at the fault point. Similar information is shown for the peak currents. First-cycle asymmetrical fault currents are used to assess
the closing and latching duty of medium- and high-voltage circuit breakers. These currents
can be either total asymmetrical rms currents (ANSI C37.06-1979) or peak currents (ANSI
Std C37.06-1987).
Typical information pertinent to interrupting duty calculations is shown in Figure 7-6. The
same type of information as in Figure 7-5 is essentially displayed, except that in this case the
multipliers applicable to both local and remote contributions are reported with reference to
both breaker rating structures covered in IEEE Std C37.010-1979 and IEEE Std C37.5-1979.
Total asymmetrical interrupting currents are also reported, thus yielding the interrupting
requirements.
Fault at bus
Prefault
#
ID
Zone
Voltage (kV)
Angle (degrees)
19
T7SEC
5
2.40
0.00
Current during fault
Voltage-to-ground during fault
Type
p.u.
Degrees
Amperes
MVA
p.u.
Angle
(degrees)
Module
(kV)
LLL-A
7.67
–85.54
18 449
77
0.00
0.00
0.00
Equivalent impedance (p.u.)
X/R (IEEE Std
C37.010-1979)
Xeq (IEEE Std
C37.010-1979)
ZQQ-1 = 0.0101 + j0.1300
13.69
0.1300
Total asymmetrical first-cycle current
Based on 1.6 multiplier (IEEE Std C37.010-1979)
Based on actual X/R ratio
18 449 × 1.60 = 29 520 A
18 449 × 1.505 = 27 765 A
Peak current
Based on 2.7 multiplier
Based on actual X/R ratio
18 449 × 2.70 = 49 812 A
18 449 × 2.541 = 46 879 A
First ring contributions from:
Bus
#
ID
6
FDR-H 2
Prefault
Zone
Voltage (kV)
1
13.80
0.00
Current during fault
Type
p.u.
Degrees
LLL-A
5.57
–85.11
Angle (degrees)
Voltage-to-ground during fault
Amperes
MVA
p.u.
Angle
(degrees)
Module
(kV)
13 418
56
0.82
–29.87
6.51
Transfer impedance (p.u.)
ZQP-1 = 0.0021 + j0.0236
Figure 7-5—Typical results for first cycle duty calculations
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 7
First ring contribution from:
1 ind. motor(s) each rated 1.13 MVA
Current during fault
Type
p.u.
Degrees
Amperes
MVA
LLL-A
0.67
–87.81
1619
7
First ring contribution from:
1 ind. motor(s) each rated 2.38 MVA
Current during fault
Type
p.u.
Degrees
Amperes
MVA
LLL-A
1.42
–86.19
3414
14
Figure 7-5—Typical results for first cycle duty calculations (Continued)
#
10
Fault at bus
ID
Zone
Voltage (kV)
EMERG
2
13.80
Current during fault
Prefault
Angle (degrees)
0.00
Voltage-to-ground during fault
Angle
Module
(degrees)
(kV)
LLL-A
27.77
–83.05
11 616
278
0.00
0.00
0.00
X/R (IEEE Std
Xeq (IEEE Std
C37.010-1979)
Equivalent impedance (p.u.)
C37.010-1979)
ZQQ-1 = 0.0044 + j0.0358
8.950
0.0360
Local and remote contribution from generating stations
Local/total
Remote/total
49.2%
50.8%
Asymmetrical multipliers for 5 cycle breakers/3 cycles parting time (IEEE Std C37.010-1979)
IEEE Std
IEEE Std
IEEE Std
IEEE Std
C37.010-1979 (local)
C37.010-1979
C37.5-1979 (local)
C37.5-1979 (remote)
(remote)
1.00
1.00
1.00
1.00
Total asymmetrical interrupting fault currents
IEEE Std
IEEE Std
IEEE Std
IEEE Std
C37.010-1979 (weighted)
C37.010-1979
C37.5-1979 (weighted) C37.5-1979 (weighted)
(remote)
11619 A
11619 A
11619 A
11619 A
First ring contributions from:
Bus
Prefault
#
ID
Zone
Voltage (kV)
Angle (degrees)
13
T6PRI
1
13.80
0.00
Type
p.u.
Degrees
Amperes
MVA
p.u.
Figure 7-6—Typical results for interrupting duty calculations
202
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
SHORT-CIRCUIT STUDIES
Type
LLL-A
Current during fault
p.u.
Degrees
Amperes
0.09
–78.76
Type
p.u.
Degrees
LLL-A
27.67
–83.07
#
27
MVA
Voltage-to-ground during fault
p.u.
Angle
Module
(degrees)
(kV)
0.0
00.00
0.00
39
1
Transfer impedance (p.u.)
ZQP-1 = 0.0021 + j0.0236
First ring contribution from:
Bus
Prefault
ID
Zone
Voltage (kV)
Angle (degrees)
T12PRI
1
13.80
0.00
Current during fault
Voltage-to-ground during fault
Amperes
MVA
11577
277
Transfer impedance (p.u.)
ZQP-1 = 0.0033 + j0.0348
p.u.
0.04
Angle
(degrees)
–43.47
Module
(kV)
0.32
Figure 7-6—Typical results for interrupting duty calculations (Continued)
Typical computer-generated results for first-cycle (fault at 3:MILL-1) and interrupting duty
(fault at 5:FDR-F) are shown in Figures 7-7 and 7-8, respectively, in both tabular and graphical form.
7.8 References
This standard shall be used in conjunction with the following publications:
ANSI C37.06-1979, American National Standard for Switchgear—AC High-Voltage Breakers Rated on a Symmetrical Current Basis—Preferred Ratings and Related Interrupting Capabilities.3
ANSI C37.06-1987, American National Standard Preferred Ratings and Related Interrupting
Capabilities for AC High-Voltage Breakers Rated on a Symmetrical Current Basis.
AS 3851-1991, The calculation of short-circuit currents in three-phase a.c. systems.4
IEEE Std 141-1993, IEEE Recommended Practice for Electric Power Distribution for Industrial Plants (IEEE Red Book).
3IEEE
and ANSI C37 publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes
Lane, P.O. Box 1331, Piscataway, NJ 08855-1331, USA.
4AS publications are available from Standards Australia, P.O. Box 1055, Strathfield, NSW 2135, Australia.
Copyright © 1998 IEEE. All rights reserved.
203
SHORT-CIRCUIT STUDIES
204
IEEE
Std 399-1997
Copyright © 1998 IEEE. All rights reserved.
Figure 7-7—Typical computer reports for first-cycle duty calculations
conformable to IEEE Std C37.010-1979
IEEE
Std 399-1997
Copyright © 1998 IEEE. All rights reserved.
CHAPTER 7
205
Figure 7-8—Typical computer reports for interrupting duty calculations
conformable to IEEE Std C37.010-1979
IEEE
Std 399-1997
CHAPTER 7
IEEE Std 241-1990 (Reaff 1997), IEEE Recommended Practice for Electric Power Systems
in Commercial Buildings (IEEE Gray Book).
IEEE Std 242-1986 (Reaff 1991), IEEE Recommended Practice for Protection and Coordination of Industrial and Commercial Power Systems (IEEE Buff Book).
IEEE Std C37.010-1979, IEEE Application Guide for AC High-Voltage Circuit Breakers
Rated on a Symmetrical Current Basis (includes supplement Std C37.010d).
IEEE Std C37.5-1979, IEEE Guide for Calculation of Fault Currents for Application of AC
High-Voltage Circuit Breakers Rated on a Total Current Basis.5
IEEE Std C37.13-1990, IEEE Standard for Low-Voltage AC Power Circuit Breakers Used in
Enclosures.
VDE 0102, Recommendation for the Calculation of Short Circuit Currents, Part 1: Three
Phase systems of voltages above 1 KV, Issued by the Deutsche Elektrotechnische Kommission, Frankfurt, Germany, 1972 (VDE).
IEC 60909 (1988), Short-circuit current calculation in three phase a.c. systems.6
7.9 Bibliography
[B1] Anderson, P., Analysis of Faulted Power Systems, IEEE Power System Engineering
Series, Piscataway, NJ: IEEE Press, 1995.
[B2] Arrilaga J., Arnold, C. P, and Harker, B. J., Computer Modelling of Electrical Power
System, New York: John Wiley & Sons, 1983.
[B3] Blackburn, L. J., Symmetrical Components for Power Systems Engineering, New York:
Marcel Dekker, Inc., 1993.
[B4] Huening, W. C., Calculating short-circuit currents with contributions from induction
motors, IEEE Transactions on Industry Applications, vol. IA-18, pp. 85–92, Mar./Apr. 1982.
[B5] Huening, W. C., Interpretation of new American National Standards for power circuit
breaker applications. IEEE Transactions on Industry and General Applications, vol. IGA-5,
Sep./Oct. 1969.
[B6] NFPA 70-1966, National Electric Code® (NEC®).
5This
Standard has been withdrawn. Copies can be obtained for the IEEE Standards Department at (908) 562-3821.
publications are available from IEC Sales Department, Case Postale 131, 3, rue de Varembé, CH-1211, Genève
20, Switzerland/Suisse. IEC publications are also available in the United States from the Sales Department, American National Standards Institute, 11 West 42nd Street, 13th Floor, New York, NY 10036, USA.
6IEC
206
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SHORT-CIRCUIT STUDIES
IEEE
Std 399-1997
[B7] Rodolakis, A. J., A comparison of North American (ANSI) and European (IEC) Fault
calculation guidelines, IEEE Transactions on Industry Applications, vol. 29, No. 3, pp. 515–
521, May/June 1993.
[B8] Roeper, R., Short Circuit Currents in Three Phase Systems, Siemens Actiengesellschaft,
John Wiley & Sons, 1985.
[B9] Stagg, G. W., and El-Abiad, A. H., Computer Methods in Power System Analysis, New
York: McGraw-Hill, 1968.
[B10] Stevenson, W. D., Elements of Power System Analysis, New York: McGraw-Hill, 1982.
[B11] St. Pierre, C. R., Time Sharing Computer Programs (DATUMS) for Power Data Reduction. Schenectady, NY: General Electric Company, 1973.
[B12] Tinney W., Brandwajn, V., and Chan, S., Sparse Vector Methods, IEEE Transactions on
Power Apparatus and Systems, vol. PAS 104, pp. 295–301, Feb. 1985.
[B13] Wagner, C. F., and Evans, R. D., Symmetrical Components, New York: McGraw-Hill,
1933.
Copyright © 1998 IEEE. All rights reserved.
207
Chapter 8
Stability studies
8.1 Introduction
For years, system stability was a problem almost exclusively to electric utility engineers.
Small independent power producers (IPPs) and co-generation (co-gen) companies were
treated as part of the load and modeled casually. Today, the structure of the utility industry is
going through a revolutionary change under the process of deregulation. A full-scale competition in the generation market is on the horizon. Increasing numbers of industrial and commercial facilities have installed local generation, large synchronous motors, or both. The role
of IPP/co-gen companies and other plants with on-site generation in maintaining system stability is a new area of interest in power system studies (Lee, Chen, and Williams [B20]).1
When a co-generation plant (which, in the context of this chapter, is used in reference to any
facility containing large synchronous machinery) is connected to the transmission grid, it
changes the system configuration as well as the power flow pattern. This may result in stability problems both in the plant and the supplying utility (Flory et al. [B24]; Lee, Chen, and
Williams [B21]). Figure 8-1 and Figure 8-2 are the time-domain simulation results of a system before and after the connection of a co-generation plant. (Lee, Chen, Gim et al. [B20])
The increased magnitude and decreased damping of machine rotor oscillations shown in
these figures indicate that the system dynamic stability performance has deteriorated after the
connection. This requires joint studies between utility and co-gen systems to identify the
source of the problem and develop possible mitigation measures.
8.2 Stability fundamentals
8.2.1 Definition of stability
Fundamentally, stability is a property of a power system containing two or more synchronous
machines. A system is stable, under a specified set of conditions, if, when subjected to one or
more bounded disturbances (less than infinite magnitude), the resulting system response(s)
are bounded. After a disturbance, a stable system could be described by variables that show
continuous oscillations of finite magnitude (ac voltages and currents, for example) or by constants, or both. In practice, engineers familiar with stability studies expect that oscillations of
machine rotors should be damped to an acceptable level within 6 s following a major disturbance. It is important to realize that a system that is stable by definition can still have stability
problems from an operational point of view (oscillations may take too long to decay to zero,
for example).
1The
numbers in brackets preceded by the letter B correspond to those of the bibliography in 8.9
Copyright © 1998 IEEE. All rights reserved.
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Std 399-1997
CHAPTER 8
Figure 8-1—System response—No co-gen plant
8.2.2 Steady-state stability
Although the discussion in the rest of this chapter revolves around stability under transient
and/or dynamic conditions, such as faults, switching operations, etc., there should also be an
awareness that a power system can become unstable under steady-state conditions. The simplest power system to which stability considerations apply consists of a pair of synchronous
machines, one acting as a generator and the other acting as a motor, connected together
through a reactance (see Figure 8-3). (In this model, the reactance is the sum of the transient
reactance of the two machines and the reactance of the connecting circuit. Losses in the
machines and the resistance of the line are neglected for simplicity.)
If the internal voltages of the two machines are EG and EM and the phase angle between them
is θ, it can easily be demonstrated that the real power transmitted from the generator to the
motor is (Westinghouse [B2]; Fitzgerald and Kingsley, Jr. [B3]).
EGEM
P = --------------- sin θ
X
210
(8-1)
Copyright © 1998 IEEE. All rights reserved.
STABILITY STUDIES
IEEE
Std 399-1997
Figure 8-2—Low-frequency oscillation after the
connection of the co-gen plant
Figure 8-3—Simplified two-machine power system
The maximum value of P obviously occurs when θ = 90°. Thus
EGEM
P max = -------------X
Copyright © 1998 IEEE. All rights reserved.
(8-2)
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CHAPTER 8
This is the steady-state stability limit for the simplified two-machine system. Any attempt to
transmit more power than Pmax will cause the two machines to pull out of step (loose
synchronism with each other) for particular values of internal voltages.
This simple example shows that at least three electrical characteristics of a power system
affect stability. They are as follows:
a)
b)
c)
Internal voltage of the generator(s)
Reactance(s) of the machines and transmission system
Internal voltage of the motor(s), if any
The higher the internal voltages and the lower the system and machine reactances, the greater
the power that can be transmitted under steady-state conditions.
8.2.3 Transient and dynamic stability
The preceding look at steady-state stability serves as a background for an examination of the
more complicated problem of transient stability. This is true because the same three electrical
characteristics that determine steady-state stability limits affect transient stability. However, a
system that is stable under steady-state conditions is not necessarily stable when subjected to
a transient disturbance.
Transient stability means the ability of a power system to experience a sudden change in generation, load, or system characteristics without a prolonged loss of synchronism. To see how
a disturbance affects a synchronous machine, consider the steady-state characteristics
described by the steady-state torque equation first (Kimbark [B14]).
2
πP
T = ---------φ SR F R sin δ R
8
(8-3)
where
T
P
φSR
FR
δR
is the mechanical shaft torque
is the number of poles of machine
is the air-gap flux
is the rotor field MMF
is the mechanical angle between rotor and stator field lobes
The air-gap flux φSR stays constant as long as the internal voltage (which is directly related to
field excitation) at the machine does not change and if the effects of saturation of the iron are
neglected. Therefore, if the field excitation remains unchanged, a change in shaft torque T
will cause a corresponding change in rotor angle δR. (This is the angle by which, for a motor,
the peaks of the rotating stator field lead the corresponding peaks of the rotor field. For a
generator, the relation is reversed.) Figure 8-4 graphically illustrates the variation of rotor
angle with shaft torque. With the machine operating as a motor (when rotor angle and torque
are positive), torque increases with rotor angle until δR reaches 90 electrical degrees. Beyond
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STABILITY STUDIES
IEEE
Std 399-1997
90°, torque decreases with increasing rotor angle. As a result, if the required torque output of
a synchronous motor is increased beyond the level corresponding to 90° rotor angle, it will
slip a pole. Unless the load torque is reduced below the 90° level (the pullout torque), the
motor will continue slipping poles indefinitely and is said to have lost synchronism with the
supply system (and become unstable). The problems that can follow from extended operation
in this out-of-step condition will be discussed further in this section.
Figure 8-4—Torque vs. rotor angle relationship for
synchronous machines in steady state
A generator operates similarly. Increasing torque input until the rotor angle exceeds 90°
results in pole slipping and loss of synchronism with the power system, assuming constant
electrical load.
Similar relations apply to the other parameters of the torque equation. For example, air-gap
flux φSR is a function of voltage at the machine. Thus, if the other factors remain constant, a
change in system voltage will cause a change in rotor angle. Likewise, changing the field
excitation will cause a change in rotor angle if constant torque and voltage are maintained.
The preceding discussion refers to rather gradual changes in the conditions affecting the
torque angle, so that approximate steady-state conditions always exist. The coupling between
the stator and rotor fields of a synchronous machine, however, is somewhat elastic. This
means that if an abrupt rather than a gradual change occurs in one or more of the parameters
of the torque equation, the rotor angle will tend to overshoot the final value determined by the
changed conditions. This disturbance can be severe enough to carry the ultimate steady-state
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 8
rotor angle past 90° or the transient swing rotor angle past 180°. Either event results in the
slipping of a pole. If the conditions that caused the original disturbance are not corrected, the
machine will then continue to slip poles; in short, pulling out of step or loosing synchronism
with the power system to which it is connected.
Of course, if the transient overshoot of the rotor angle does not exceed 180°, or if the disturbance causing the rotor swing is promptly removed, the machine may remain in synchronism
with the system. The rotor angle then oscillates in decreasing swings until it settles to its final
value (less than 90°). The oscillations are damped by electrical load and mechanical and electrical losses in the machine and system, especially in the damper windings of the machine.
A change in rotor angle of a machine requires a change in speed of the rotor. For example, if
we assume that the stator field frequency is constant, it is necessary to at least momentarily
slow down the rotor of a synchronous motor to permit the rotor field to fall farther behind the
stator field and thus increases δR. The rate at which rotor speed can change is determined by
the moment of inertia of the rotor plus whatever is mechanically coupled to it (prime mover,
load, reduction gears, etc.).With all other variables equal, this means a machine with high
inertia is less likely to become unstable given a disturbance of brief duration than a low-inertia machine.
Traditionally, transient stability is determined by considering only the inherent mechanical
and electromagnetic characteristics of the synchronous machines and the impedance of the
circuits connecting them. The responses of the excitation or governor systems to the changes
in generator speed or electrical output induced by a system disturbance are neglected. On the
other hand, dynamic stability takes automatic voltage regulator and governor system
responses into account.
The traditional definition of transient stability is closely tied to the ability of a system to
remain in synchronism for a disturbance. Transient stability studies are usually conducted
under the assumptions that excitation and governor-prime mover time constants are much
longer than the duration of the instability-inducing disturbance. This assumption was usually
accurate when power system stability became a relevant problem in the 1920s (about the time
that significant interconnections of systems became more common), because both the generator excitation voltage and the prime mover throttle were controlled either manually or by very
slow feedback mechanisms, and brief short circuits were normally the worst-case disturbances considered.
However, technological advances have rendered the assumption underlying these conventional concepts of transient stability obsolete in most cases. These include the advent of fast
electronic excitation systems and governors, the recognition of the value of stability analysis
for investigating conditions of widely varying severity and duration, and the virtual elimination of computational power as a constraint on system modeling complexity. Most transient
stability studies performed today consider at least the generator excitation system, and are
therefore actually dynamic studies under the conventional conceptual definition.
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Std 399-1997
8.2.4 Two-machine systems
The previous discussion of transient behavior of synchronous machines is based on a single
machine connected to a good approximation of an infinite bus. An example is the typical
industrial situation where a synchronous motor of at most a few thousand horsepower is connected to a utility company system with a capacity of thousands of megawatts. Under these
conditions, we can safely neglect the effect of the machine on the power system.
A system consisting of only two machines of comparable size connected through a transmission link, however, becomes more complicated because the two machines can affect each
other’s performance. The medium through which this occurs is the air-gap flux. This is a
function of machine terminal voltage, which is affected by the characteristics of the transmission system, the amount of power being transmitted, the power factor, etc.
In the steady state, the rotor angles of the two machines are determined by the simultaneous
solution of their respective torque equations. Under a transient disturbance, as in the singlemachine system, the rotor angles move toward values corresponding to the changed system
conditions. Even if these new values are within the steady-state stability limits of the system,
an overshoot can result in loss of synchronism. If the system can recover from the disturbance, both rotors will undergo a damped oscillation and ultimately settle to their new steadystate values.
An important concept here is synchronizing power. The higher the real power transfer capability over the transmission link between the two machines, the more likely they are to remain
in synchronism in the face of a transient disturbance. Synchronous machines separated by a
sufficiently low impedance behave as one composite machine, since they tend to remain in
step with one another regardless of external disturbances.
8.2.5 Multimachine systems
At first glance, it appears that a power system incorporating many synchronous machines
would be extremely complex to analyze. This is true if a detailed, precise analysis is needed;
a sophisticated program is required for a complete stability study of a multimachine system.
However, many of the multimachine systems encountered in industrial practice contain only
synchronous motors that are similar in characteristics, closely coupled electrically, and connected to a high-capacity utility system. Under most type of disturbances, motors will remain
in synchronism with each other, although they can all loose synchronism with the utility.
Thus, the problem most often encountered in industrial systems is similar to a single synchronous motor connected through an impedance to an infinite bus. The simplification should be
apparent. Stability analysis of more complex systems where machines are of comparable
sizes and are separated by substantial impedance will usually require a full-scale computer
stability study.
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 8
8.3 Problems caused by instability
The most immediate hazards of asynchronous operation of a power system are the high transient mechanical torque and currents that usually occur. To prevent these transients from
causing mechanical and thermal damage, synchronous motors and generators are almost universally equipped with pullout protection. For motors of small to moderate sizes, this protection is usually provided by a damper protection of pullout relay that operates on the low
power factor occurring during asynchronous operation. The same function is usually provided for large motors, generators, and synchronous condensers by loss-of-field relaying. In
any case, the pullout relay trips the machine breaker or contactor. Whatever load is being
served by the machine is naturally interrupted. Consequently, the primary disadvantage of a
system that tends to be unstable is the probability of frequent process interruptions.
Out-of step operation also causes large oscillatory flows of real and reactive power over
the circuits connecting the out-of-step machines. Impedance or distance-type relaying that
protects these lines can falsely interpret power surges as a line fault, tripping the line
breakers and breaking up the system. Although this is primarily a utility problem, large
industrial systems or those where local generation operates in parallel with the utility can
be susceptible.
In any of these cases, an industrial system can be separated from the utility system. If the
industrial system does not have sufficient on-site generation, a proper load shedding procedure is necessary to prevent total loss of electrical power. Once separated from the strength of
the utility, the industrial system becomes a rather weakly connected island and is likely to
encounter additional stability problems. With the continuance of problems, protection systems designed to prevent equipment damage will likely operate, thus producing the total
blackout.
8.4 System disturbances that can cause instability
The most common disturbances that produce instability in industrial power systems are (not
necessarily in order of probability):
a)
b)
c)
d)
e)
f)
g)
Short circuits
Loss of a tie circuit to a public utility
Loss of a portion of on-site generation
Starting a motor that is large relative to a system generating capacity
Switching operations
Impact loading on motors
Abrupt decrease in electrical load on generators
The effect of each of these disturbances should be apparent from the previous discussion of
stability fundamentals.
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8.5 Solutions to stability problems
Generally speaking, changing power flow patterns and decreasing the severity or duration of
a transient disturbance will make the power system less likely to become unstable under that
disturbance. In addition, increasing the moment of inertia per rated kVA of the synchronous
machines in the system will raise stability limits by resisting changes in rotor speeds required
to change rotor angles.
8.5.1 System design
System design primarily affects the amount of synchronizing power that can be transferred
between machines. Two machines connected by a low impedance circuit, such as a short
cable or bus run, will probably stay synchronized with each other under all conditions except
a fault on the connecting circuit, a loss of field excitation, or an overload. The greater the
impedance between machines, the less severe a disturbance will be required to drive them out
of step. For some systems, the dynamic stability problems could be resolved by the construction of new connecting circuits (Klein, Rogers, and Kundur [B15]). This means that from the
standpoint of maximum stability, all synchronous machines should be closely connected to a
common bus. Limitations on short-circuit duties, economics, and the requirements of physical plant layout usually combine to render this radical solution impractical.
8.5.2 Design and selection of rotating equipment
Design and selection of rotating equipment and control parameters can be a major contributor
to improving system stability. Most obviously, use of induction instead of synchronous
motors eliminates the potential stability problems associated with the latter. (Under rare circumstances, an induction motor/synchronous generator system can experience instability, in
the sense that undamped rotor oscillations occur in both machines, but the possibility is too
remote to be of serious concern.) However, economic considerations often preclude this
solution.
Where synchronous machines are used, stability can be enhanced by increasing the inertia of
the mechanical system. Since the H constant (stored energy per rated kVA) is proportional to
the square of the speed, fairly small increases in synchronous speed can pay significant dividends in higher inertia. If carried too far, this can become self-defeating because higher speed
machines have smaller diameter rotors. Kinetic energy varies with the square of the rotor
radius, so the increase in H due to a higher speed may be offset by a decrease due to the lower
kinetic energy of a smaller diameter rotor. Of course, specifications of machine size and
speed are dependent on the mechanical nature of the application and these concerns may limit
the specification flexibility with regard to stability issues.
A further possibility is to use synchronous machines with low transient reactances that permit
the maximum flow of synchronizing power. Applicability of this solution is limited mostly by
short-circuit considerations, starting current limitations, and machine design problems.
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8.5.3 Voltage regulator and exciter characteristics
Voltage regulator and exciter characteristics affect stability because, all other things being
equal, higher field excitation requires a smaller rotor angle. Consequently, stability is
enhanced by a properly applied regulator and exciter that respond rapidly to transient effects
and furnish a high degree of field forcing. In this respect, modern solid-state voltage regulators and static exciters can contribute markedly to improved stability. However, a mismatch in
exciter and regulator characteristics can make an existing stability problem even worse
(Demello and Concordia [B1]).
8.5.4 Application of power system stabilizers (PSSs)
The PSS installation has been widely used in the power industry to improve the system damping. The basic function of a PSS is to extend stability limits by modulating generator excitation to provide damping to the oscillation of a synchronous machine rotor. To provide
damping, the PSS must produce a component of electrical torque on the rotor that is in phase
with speed variations. The implementation details differ, depending upon the stabilizer input
signal employed. However, for any input signal, the transfer function of the stabilizer must
compensate for the gain and phase characteristics of the excitation system, the generator, and
the power system, which collectively determine the transfer function from the stabilizer output to the component of electrical torque, which can be modulated via excitation control. To
install the PSS in the power system to solve the dynamic stability problem, one has to determine the installation site and the settings of PSS parameters. This job can be realized through
frequency domain analysis (Fretwell, Lam, and Yu [B5]; Klein, Rogers, Moorty et al. [B16];
Kundur, Klein, Roger et al. [B17]; Kundur, Lee, and Zein El-Din [B18]; Larsen and Swann
[B19]; Martin and Lima [B22]).
8.5.5 System protection
System protection often offers the best prospects for improving the stability of a power system. The most severe disturbance that an industrial power system is likely to experience is a
short circuit. To prevent loss of synchronism, as well as to limit personnel hazards and equipment damage, short circuits should be isolated as rapidly as possible. A system that tends to
be unstable should be equipped with instantaneous overcurrent protection on all of its primary feeders, which are the most exposed section of the primary system. As a general rule,
instantaneous relaying should be used throughout the system wherever selectivity permits.
8.6 System stability analysis
Stability studies, as much or more than any other type or power system study described in this
text, have benefited from the advent of the computer. This is primarily due to the fact that stability analysis requires a tremendous number of iterative calculations and the manipulation of
a large amount of time and frequency-variant data.
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8.6.1 Time- and frequency-domain analysis
Time and frequency-domain (eigenvalue analysis) techniques are, by far, the most common
analytical methods used by power system stability programs. Time-domain analysis utilizes
the angular displacement of the rotors of the machines being studied, often with respect to a
common reference, to determine stability conditions. The differences between these rotor
angles are small for stable systems. The rotor angles of machines in unstable systems drift
apart with time. Thus, time-domain analysis can be used to determine the overall system
response to potentially instability-inducing conditions, but it is limited when one is attempting to identify oscillation modes.
Frequency-domain analysis, on the other hand, can be used to identify each potential oscillation frequency and its corresponding damping factor. Therefore, the powerful frequencydomain techniques are particularly suited for dynamic stability applications whereas timedomain techniques are more useful in transient stability analysis. Fortunately, dynamic stability can also be evaluated by the shapes of the swing curves of synchronous machine rotor
angles as they vary with time. Therefore, time-domain analysis can be used for dynamic stability as well, and it will be the analytical technique on which the remainder of this chapter
will be focused.
8.6.2 How stability programs work
Mathematical methods of stability analysts depend on a repeated solution of the swing equation for each machine:
2
( MVA )H d δ R
P a = ----------------------- ---------2
180 f
dt
(8-4)
where
Pa
MVA
H
f
δR
t
is the accelerating power (input power minus output power) (MW)
is the rated MVA of machine
is the inertia constant of machine (MW·seconds/MVA)
is the system frequency (Hz)
is the rotor angle (degrees)
is the time (seconds)
The program begins with the results of a load flow study to establish initial power and voltage
levels in all machines and interconnecting circuits. The specified disturbance is applied at a
time defined as zero, and the resulting changes in power levels are calculated by a load flow
routine. Using the calculated accelerating power values, the swing equation is solved for a
new value of δR for each machine at an incremental time (the incremental time should be less
than one-tenth of the smallest machine time constant to limit numerical errors) after the disturbance. Voltage and power levels corresponding to the new angular positions of the synchronous machines are then used as base information for next iteration. In this way,
performance of the system is calculated for every interval out to as much as 15 s.
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8.6.3 Simulation of the system
A modern transient stability computer program can simulate virtually any set of power
system components in sufficient detail to give accurate results. Simulation of rotating
machines and related equipment is of special importance in stability studies. The simplest
possible representation for a synchronous motor or generator involves only a constant
internal voltage, a constant transient reactance, and the rotating inertia (H) constant. This
approximation neglects saturation of core iron, voltage regulator action, the influence of
construction of the machine on transient reactance for the direct and quadrature axes, and
most of the characteristics of the prime mover or load. Nevertheless, this classical representation is often accurate enough to give reliable results, especially when the time period being
studied is rather short. (Limiting the study to a short period—say, 1/2 s or less, means that
neither the voltage regulator nor the governor, if any, has time to exert a significant effect.)
The classical representation is generally used for the smaller and less influential machines in
a system, or where the more detailed information required for better simulations is not
available.
As additional data on the machines becomes available, better approximations can be used.
This permits more accurate results that remain reliable for longer time periods. Modern largescale stability programs can simulate all of the following characteristics of a rotating
machine:
a)
b)
c)
d)
e)
f)
Voltage regulator and exciter
Steam system or other prime mover, including governor
Mechanical load
Damper windings
Salient poles
Saturation
Induction motors can also be simulated in detail, together with speed-torque characteristics of
their connected loads. In addition to rotating equipment, the stability program can include in
its simulation practically any other major system component, including transmission lines,
transformers, capacitor banks, and voltage-regulating transformers and dc transmission links
in some cases.
8.6.4 Simulation of disturbances
The versatility of the modern stability study is apparent in the range of system disturbances
that can be represented. The most severe disturbance that can occur on a power system is
usually a three-phase bolted short circuit. Consequently, this type of fault is most often used
to test system stability. Stability programs can simulate a three-phase fault at any location,
with provisions for clearing the fault by opening breakers either after a specified time delay,
or by the action of overcurrent, underfrequency, overpower, or impedance relays. This feature
permits the adequacy of proposed protective relaying to be evaluated from the stability
standpoint.
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Short circuits other than the bolted three-phase fault cause less disturbance to the power system. Although most stability programs cannot directly simulate line-to-line or ground faults,
the effects of these faults on synchronizing power flow can be duplicated by applying a threephase fault with a properly chosen fault impedance. This means the effect of any type of fault
on stability can be studied.
In addition to faults, stability programs can simulate switching of lines and generators. This is
particularly valuable in the load-shedding type of study, which will be covered in a following
section. Finally, the starting of large motors on relatively weak power systems and impact
loading of running machines can be analyzed.
8.6.5 Data requirements for stability studies
The data required to perform a transient stability study and the recommended format for
organizing and presenting the information for most convenient use are covered in detail in the
application guides for particular stability programs. The following is a summary of the
generic classes of data needed. Note that some of the more esoteric information is not essential; omitting it merely limits the accuracy of the results, especially at times exceeding five
times the duration of the disturbance being studied. The more essential items are marked by
an asterisk (*).
a)
b)
c)
System data
1) Impedance (R + jX) of all significant transmission lines, cables, reactors, and
other series components*
2) For all significant transformers and autotransformers
i) kVA rating*
ii) Impedance*
iii) Voltage ratio*
iv) Winding connection*
v) Available taps and tap in use*
vi) For regulators and load tap-changing transformers: regulation range, tap
step size, type of tap changer control*
3) Short-circuit capacity (steady-state basis) of utility supply, if any*
4) kvar of all significant capacitor banks*
5) Description of normal and alternate switching arrangements*
Load data: real and reactive electrical loads on all significant load buses in the
system*
Rotating machine data
1) For major synchronous machines (or groups of identical machines on a common
bus)
i) Mechanical and/or electrical power ratings (kVA, hp, kW, etc.)*
ii) Inertia constant H or inertia Wk2 of rotating machine and connected load or
prime mover*
iii) Speed*
iv) Real and reactive loading, if base-loaded generator*
v) Speed torque curve or other description of load torque, if motor*
vi) Direct-axis subtransient,* transient,* and synchronous reactances*
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vii)
viii)
ix)
x)
xi)
xii)
xiii)
d)
e)
Quadrature-axis subtransient, transient,* and synchronous reactances
Direct-axis and quadrature-axis subtransient and transient* time constants
Saturation information
Potier reactance
Damping data
Excitation system type, time constants, and limits
Governor and steam system or other prime mover type, time constants, and
limits
2) For minor synchronous machines (or groups of machines)
i) Mechanical and/or electrical power ratings*
ii) Inertia*
iii) Speed*
iv) Direct-axis synchronous reactance*
3) For major induction machines or groups of machines
i) Mechanical and/or electrical power ratings*
ii) Inertia*
iii) Speed*
iv) Positive-sequence equivalent circuit data (e.g., R1, X1, XM)*
v) Load speed-torque curve*
vi) Negative-sequence equivalent circuit data (e.g., R2, X2)*
vii) Description of reduced-voltage or other starting arrangements, if used*
4) For small induction machines: detailed dynamic representation not needed,
represent as a static load
Disturbance data
1) General description of disturbance to be studied, including (as applicable) initial
switching status; fault type, location, and duration; switching operations and
timing; manufacturer, type, and setting of protective relays; and clearing time of
associated breakers*
2) Limits on acceptable voltage, current, or power swings*
Study parameters
1) Duration of study*
2) Integrating interval*
3) Output printing interval*
4) Data output required*
8.6.6 Stability program output
Most stability programs give the user a wide choice of results to be printed out. The program
can calculate and print any of the following information as a function of time under time
domain analysis:
a)
b)
c)
d)
e)
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Rotor angles, torques, and speeds of synchronous machines
Real and reactive power flows throughout the system
Voltages and voltage angles at all buses
Bus frequencies
Torques and slips of all induction machines
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The combination of these results selected by the user can be printed out for each printing
interval (also user selected) during the course of the study period.
The value of the study is strongly affected by the selection of the proper printing interval and
the total duration of the simulation. Normally, a printing interval of 0.01 or 0.02 s is used;
longer intervals reduce the solution time slightly, but increase the risk of missing fast swings
of rotor angle. The time required to obtain a solution is proportional to the length or the
period being studied, so this parameter should be closely controlled for the sake of economy.
Avoiding long study periods is especially important if the system and machines have been
represented approximately or incompletely because the errors will accumulate and render the
results meaningless after some point. A time limit of five times the duration of the major disturbance being studied is generally long enough to show whether the system is stable (in the
transient stability sense) or not, while keeping solution time requirements to reasonable
levels.
Frequency domain analysis will calculate the eigenvalues to determine the stability characteristics of the system. For a large utility system, most programs can only provide dominant
eigenvalue(s) of the system. This information is sufficient for most stability studies except
multi-dominant eigenvalues situations.
8.6.7 Interpreting results
The results of a computer stability study are fairly easy to understand once the user learns the
basic principles underlying stability problems. The most direct way to determine from study
results whether a system is stable is to look at a set of swing curves for the machines in the
system. Swing curves are simply plots of rotor angles or machine frequencies (rotor speeds)
versus time; if the curves of all the machines involved are plotted on common axes, we can
easily see whether they diverge (indicating instability) or settle to new steady-state values.
Even if the system is stable, a poor damping situation is not acceptable from the security
operation point of view. As previously mentioned, most utility engineers expect that any
oscillations should be damped to an acceptable value within 6 s. The system responses, as
shown in both Figure 8-1 and Figure 8-5, have good damping factors, and the system returns
to normal within a reasonable time frame. However, the situation depicted in Figure 8-2 is
considered marginal, even though the system is still stable by definition (oscillations are
clearly bounded). In a frequency domain analysis of this same system, one can expect that all
the eigenvalues of the system should lie in the left-half of the s-plane, and most utility engineers consider that the real part of the dominant eigenvalues should be less than –0.2 to –0.3
(time constant is between 3.33 to 5 s) for a normal power system. The time domain responses
for various root locations in the frequency domain are shown in Figure 8-6.
8.7 Stability studies of industrial power systems
The requirement of stability studies depends on the operating conditions of the industrial
power systems. This subclause is intended to summarize the so-called “things to look for”
under different operating conditions and disturbance scenarios.
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Figure 8-5—Oscillation record of the tie line between co-gen and utility
Figure 8-6—The response for various root locations in the frequency domain
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8.7.1 A co-gen plant with excess generation
Consider the situation where a co-gen plant is exporting power to the connected utility company when a severe disturbance occurs. If the tie line(s) between co-gen and utility company
are tripped, the co-gen facility becomes islanded. Because the plant has enough generation to
support its own operation, stability problems within the facility are less likely to happen.
However, the following should be checked to ensure secure operation.
a)
Transient stability problem. Generally speaking, the inertia of the co-gen units are
smaller than the utility generators. They tend to respond faster to a system disturbance. If the fault happens in the vicinity of the plant and is not cleared before the
critical clearing time, the speed of the units inside the plant may increase rapidly and
the units may loose synchronism. Figure 8-7 illustrates this phenomena. A faster circuit breaker can be used to avoid this problem.
Figure 8-7—Out-of-step phenomenon of co-gen unit(s)
under severe disturbance
b)
c)
Potential overfrequency. Because the plant has excess power before islanding occurs,
the frequency within the co-gen plant will rise after the interruption. The exact extent
of the frequency deviation depends on the level of excess power and the response of
the machine governors.
Voltage problems. If the co-gen facility exports reactive power before the disturbance,
the system may experience overvoltage phenomena following the separation. On the
contrary, if the co-gen facility is importing reactive power, then an undervoltage
problem may arise. The ability to overcome the voltage problems depends on the
response of the automatic voltage regulators (AVRs) within the plant.
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d)
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In-plant oscillations. For some co-gen plants, a series reactor is inserted in the line to
limit the fault currents. This may cause the generators to be loosely coupled from the
electrical standpoint, even though are physically located within a plant. During the
disturbance, two generators within the plant may experience oscillations often called
hunting. Figure 8-8 shows a typical hunting oscillation of two generators. One can
see that the output of the generators are basically out-of-phase.
Figure 8-8—Hunting phenomena of two generator units
8.7.2 Co-gen plant that imports power from local utility
A co-gen plant relies on the supply from the local utility under normal operating conditions
and the tie line(s) between the plant and the utility company are interrupted due to a system
fault. Because the plant does not have enough generation to support its own operation, stability problems may happen. In order to protect the system from total blackout, an appropriate
load shedding procedure has to be in place for safe operation. The following problems may
happen under this scenario.
a)
226
Potential underfrequency. Because the co-gen plant has to import power before
islanding, the frequency within the co-gen plant will decline after the interruption.
The exact extent of the frequency deviation depends on the level of the power deficiency and the response of the machine governors. A proper load-shedding procedure
may be needed to maintain continuing operation of critical loads.
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b)
Voltage problems. When the load-shedding procedure is activated, a certain percentage of the real and reactive power are interrupted. The plant may face potential overvoltage problems. The situation may be compounded if the plant is exporting reactive
power before islanding.
c)
In-plant oscillation. Similar to 8.7.1, especially if the co-gen plant uses series reactors to limit the fault currents. A potential in-plant oscillation may occur.
8.7.3 Oscillations between industrial power plant and utility system
The causes of low-frequency oscillation phenomena in a power system are complex. If the
capacity of the industrial power plant is small compared to the connected utility system, the
likelihood of having oscillations between industrial plant and utility system is generally very
small. However, if the size of the industrial power plant is comparable to the capacity of the
local utility, a detailed simulation including both the industrial power plant and the local utility has to be performed to ensure secure operation of the system. The system structure, operation conditions, and excitation systems frequently play important roles in low-frequency
oscillation. It is necessary to distinguish the causes and the reasons before trying to solve the
problem. In time domain analysis, the mode of oscillation cannot be identified exactly, but the
potential oscillation phenomena can be investigated. The following steps can be considered
as standard procedure to identify the possible causes of the low-frequency oscillation.
8.7.3.1 Response of the system before connection
The first step is to identify whether the dynamic stability problem is a pre-existing problem.
If the problem exists before the connection, it is the responsibility of the utility to solve the
problem. However, the plant still needs to provide information for detailed system analysis
including specifics regarding on-site generation.
8.7.3.2 Different line flow between utility and co-gen plant
As a rule of thumb, the stability limit is the upper bound of the transfer capability of the interconnecting line(s) and transformer(s). The system is going to have stability problems if the
transfer limits exceed this value. Therefore, the least-cost step is to investigate the possible
solution(s) for low-frequency oscillation by adjusting the power flow of the interconnecting
equipment.
8.7.3.3 Reduced impedance of the interconnected transmission
A system is less likely to have stability problems if the generators are electrically close
together. Sometimes the stability problem will go away after the construction of a new transmission line, the reconfiguration of the utility supply, or some other system change. Lowering
system impedances will reduce the electrical distance of the generation units and establish a
stronger tie between co-gen and utility systems. The effectiveness of reducing impedances to
increase system stability can be determined easily using modern computer programs.
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8.7.3.4 Reduce fault clearing time
The longer a disturbance is on the system, the larger the frequency deviation and the phase
angle separation can be. Therefore, fault duration may effect the recovery capability of the
system after a severe disturbance. With today’s technology, high-speed relays and breakers
are available to clear the fault within a few cycles. This is an effective method of dealing with
many transient stability problems.
8.7.3.5 System separation
Though this is not the most favorable solution, it is an effective measure to mitigate the
problem if it appeared after the connection of both systems. However, if the utility imports
power from the co-gen plant before the disturbance, system separation means loss of generation capacity. This may compound the problem on the utility side and should therefore be
studied carefully.
8.7.3.6 Installation of a power system stabilizer (PSS)
The PSS installation has been widely used in the power industry to improve the system damping. To install the PSS in the power system to solve the dynamic stability problem, one has to
determine the installation site and the setting of PSS parameters. This job can be realized
through frequency domain analysis. The basic function of a PSS is to extend stability limits
by modulating generator excitation to provide damping to the oscillation of a synchronous
machine rotor. Since site selection and setting of a PSS are very sensitive to the system
parameters, accuracy of the system information is vital in this type of study.
8.8 Summary and conclusions
Power systems are highly nonlinear and the dynamic characteristic of a power system varies
if the system loading, generation schedule, network interconnection, and/or type of system
protection are changed. When a co-gen plant is connected to the utility grid, it changes the
system configuration and power flow pattern in the utility. This may result in some unwanted
system stability problems from low-frequency oscillations. The evaluation of potential problems and solution methods prior to the connection of the co-gen plant becomes a challenging
task for the power engineer.
From the industrial power plant point of view, stability problems appear as over/under voltages and frequencies and may lead to the operation of protection equipment and the initiation
of load-shedding schemes. While there is no single way to design a system that will always
remain stable, the use of low-impedance interconnections (where possible considering fault
duties) and fast-acting control systems are definite options to consider when designing for
maximum stability. It is important to consider problems that could originate in either the
utility or industrial system, or both, and the impact of subsequent disturbances that are
associated with protective device operation and load-shedding initiated by the original event.
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8.9 Bibliography
[B1] Demello, F. P., and Concordia, C., “Concept of synchronous machine stability as
affected by excitation control,” IEEE Transactions on Power Apparatus and System, vol.
PAS-88, no. 4, pp. 316–329, Apr. 1969.
[B2] Electrical Transmission and Distribution Reference Book, Westinghouse Electric Corporation, East Pittsburgh, PA, 1964, Chapter 13.
[B3] Fitzgerald, A. E., and Kingsley Jr., C., Electric Machinery, New York: McGraw-Hill,
1961, Chapter 5.
[B4] Flory, J. E., Key, T. S., Smith, W. M., et al., “The electric utility—industrial user partnership in solving power quality problems,” IEEE Transaction on Power Systems, vol. 5, Aug.
1990.
[B5] Fretwell, A., Lam, D. M., Yu, C. Y., et al., “Design, Installation and Operation of Power
System Stabilizers on the Hong Kong Electric Power System,” IEE International Conference
on Advances in Power System Control, Operation and Management, pp. 534–541, Hong
Kong, Nov. 1991.
[B6] Gim, J. H., “Dynamic characteristic analysis and real time monitoring system for an
industrial power system,” Dissertation, The University of Texas at Arlington, Dec. 1993.
[B7] IEEE Committee Report, “Computer representation of excitation systems,” IEEE Transactions on Power Apparatus and System, vol. PAS-87, no.6, pp. 1460–1464, June 1968.
[B8] IEEE Committee Report, “Current usage and suggested practices in power system
stability simulations for synchronous machines,” IEEE Transactions on Energy Conversion,
vol. EC-1, pp. 77–93, Mar. 1986.
[B9] IEEE Committee Report, “Dynamic models for fossil fueled steam units in power
system studies,” IEEE Transactions on Power Systems, vol. 6, no. 2, pp. 753–761, May 1991.
[B10] IEEE Committee Report, “Dynamic models for steam and hydro turbines in power
system studies,” IEEE Transaction on Power Apparatus and System, vol. 92, no. 6, pp. 1904–
1915, Nov./Dec. 1973.
[B11] IEEE Committee Report, “Excitation system models for power system stability
studies,” IEEE Transactions on Power Apparatus and System, vol. PAS-100, no. 2, pp. 494–
509, Feb. 1981.
[B12] IEEE Std 421.5-1992, IEEE Recommended Practice for Excitation System for Power
Stability Studies.
[B13] IEEE Std 1110-1991, IEEE Guide for Synchronous Generator Modeling Practices in
Stability Analysis.
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[B14] Kimbark, E. W., “Power System Stability,” vol. 1, New York: John Wiley, 1948.
[B15] Klein, M., Rogers, G. J., and Kundur, P., “A fundamental study of inter-area oscillation
in power system,” IEEE Transactions on Power Systems, vol. 6, no. 3, pp. 914–921, Aug.
1991.
[B16] Klein, M., Rogers, G. J., Moorty, S., et al., “Analytical investigation of factors influencing power system stabilizer performance,” IEEE Transactions on Energy Conversion, 92
WM 016-6EC.
[B17] Kundur, P., Klein, M., Roger, G. J., et al., “Application of power system stabilizers for
enhancement of overall system stability” IEEE Transactions on Power Systems, vol. 4, no. 2,
pp. 614–626, May 1989.
[B18] Kundur, P., Lee, D. C., and Zein El-Din, H. M., “Power system stabilizer for thermal
units: analytical techniques and on-site validation,” IEEE Transactions on Power Apparatus
and System, vol. PAS-100, no. 1, pp. 81–95, Jan. 1981.
[B19] Larsen, E. V., and Swann, D. A., “Applying power system stabilizer, Part I: General
concept,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 6, pp.
3017–3024, June 1981.
[B20] Lee, W. J., Chen, M. S., Gim, Y. H., et al., “Dynamic stability analysis of an industrial
power system,” IEEE-IAS Annual Conference, pp. 1477–1482, Oct. 1993.
[B21] Lee, W. J., Chen, M. S., and Williams, L. B., “Load model for stability studies,” IEEE
Transactions on Industry Applications, vol. IA-23, no. 1, Jan./Feb. 1987.
[B22] Martin, N., and Lima, L. T. G., “Determination of suitable locations for power system
stabilizer and static var compensators for damping electromechanical oscillations in large
scale power system,” IEEE PICA, Seattle, WA, pp. 358–365, May 1989.
[B23] Park, R. H., “Two-reaction theory of synchronous machine,” AIEE Transactions, Part I,
vol. 48, pp. 716–730, 1929, Part II, vol. 52, pp. 352–55, 1933.
230
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Chapter 9
Motor-starting studies
9.1 Introduction
This chapter discusses benefits obtained from motor-starting studies and examines various
types of computer-aided studies normally involved in motor-starting studies. Data or information required for these studies as well as the expected results of a motor-starting study
effort are also reviewed.
9.2 Need for motor-starting studies
9.2.1 Problems revealed
Motors on modern industrial systems are becoming increasingly larger. Some are considered
large even in comparison to the total capacity of large industrial power systems. Starting large
motors, especially across-the-line, can cause severe disturbances to the motor and any locally
connected load, and also to buses electrically remote from the point of motor starting. Ideally,
a motor-starting study should be made before a large motor is purchased. A starting voltage
requirement and preferred locked-rotor current should be stated as part of the motor specification. A motor-starting study should be made if the motor horsepower exceeds approximately 30% of the supply transformer(s) base kVA rating, if no generators are present. If
generation is present, and no other sources are involved, a study should be considered whenever the motor horsepower exceeds 10–15% of the generator kVA rating, depending on actual
generator characteristics. The study should also recognize contingent condition(s), i.e., the
loss of a source (if applicable).
It may be necessary to make a study for smaller horsepower sizes depending on the daily
fluctuation of nominal voltage, voltage level, size and length of the motor feeder cable,
amount of load, regulation of the supply voltage, the impedance and tap ratio of the supply
transformer(s), load torque versus motor torque, and the allowable starting time. Finally,
some applications may involve starting large groups of smaller motors of sufficient collective
size to impact system voltage regulation during the starting interval.
A brief discussion of major problems associated with starting large motors, or groups of
motors, and therefore, of significance in power system design and evaluation follows.
9.2.2 Voltage dips
Probably the most widely recognized and studied effect of motor-starting is the voltage
dip experienced throughout an industrial power system as a direct result of starting large
motors. Available accelerating torque drops appreciably at the motor bus as voltage dips to a
lower value, extending the starting interval and affecting, sometimes adversely, overall
motor-starting performance. Acceptable voltage for motor-starting depends on motor and
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load torque characteristics. Requirements for minimum starting voltage can vary over a wide
range, depending on the application. (Voltages can range from 80% or lower to 95% or
higher.)
During motor-starting, the voltage level at the motor terminals should be maintained, as a
minimum, at approximately 80% of rated voltage or above for a standard National Electrical
Manufacturers Association (NEMA) design B motor (as specified in NEMA MG 1-1993)1
having a standard 150% starting torque and with a constant torque load applied. This value
results from examination of speed-torque characteristics of this type motor (150% starting
torque at full voltage) and the desire to successfully accelerate a fully loaded motor
at reduced voltage (that is, torque varies with the square of the voltage
2
T = 0.8 × 150% ≈ 100% ). When other motors are affected, or when lower shaft loadings
are involved, the minimum permissible voltage may be either higher or lower, respectively.
The speed-torque characteristics of the starting motor along with any other affected motors
and all related loads should be examined to specifically determine minimum acceptable
voltage. Assuming reduced voltage permits adequate accelerating torque, it should also be
verified that the longer starting interval required at reduced torque caused by a voltage dip
does not result in the I2t damage limit of the motor being exceeded.
Several other problems may arise on the electrical power system due to the voltage dips
caused by motor-starting. Motors that are running normally on the system, for example, will
slow down in response to the voltage dip occurring when a large motor is started. The running
machines must be able to reaccelerate once the machine being started reaches operating
speed. When the voltage depression caused by the starting motor is severe, the loading on the
running machines may exceed their breakdown torque (at the reduced voltage), and they may
decelerate significantly or even stall before the starting interval is concluded. The decelerating machines all impose heavy current demands that only compound the original distress
caused by the machine that was started. The result is a “dominoing” voltage depression that
can lead to the loss of all load.
In general, if the motors on the system are standard NEMA design B, the speed-torque
characteristics (200% breakdown torque at full voltage) should prevent a stall, provided the
motor terminal voltage does not drop below about 71% of motor nameplate voltage. This is a
valid guideline to follow anytime the shaft load does not exceed 100% rated, since the developed starting torque is again proportional to the terminal voltage squared (V2), and the available torque at 71% voltage would thus be slightly above 100%. If motors other than NEMA
design B motors are used on the system, a similar criterion can be established to evaluate
reacceleration following a motor-starting voltage dip based on the exact speed-torque characteristics of each particular motor.
Other types of loads, such as electronic devices and sensitive control equipment, may be
adversely effected during motor-starting. There is a wide range of variation in the amount of
voltage drop that can be tolerated by static drives and computers. Voltage fluctuations may
also cause objectionable fluctuations in lighting. Tolerable voltage limits should be obtained
from the specific equipment manufacturers.
1Information
232
on references can be found in 9.8.
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By industry standards (see NEMA ICS 1-1993, NEMA ICS 2-1993, NEMA ICS 3-1993,
NEMA ICS 4-1993, NEMA ICS 6-1993), ac control devices are not required to pick-up at
voltages below 85% of rated nameplate voltage, whereas dc control devices must operate
dependably (i.e., pick-up) at voltages above 80% of their rating. Critical control operations
may, therefore, encounter difficulty during motor-starting periods where voltage dips are
excessive. A motor-starting study might be required to determine if this is a problem with
thoughts to using devices rated at 110 V rather than the normal 115 V nominal devices.
Contactors are required to hold-in with line voltage as low as 80% of their rating (see 9.8,
NEMA Standards). The actual dropout voltages of contactors used in industrial applications
commonly range between 60–70% of rated voltage, depending on the manufacturer. Voltages
in this range, therefore, may be appropriate and are sometimes used as the criteria for the
lower limit that contactors can tolerate. Depending on where lighting buses are located, with
respect to large starting motors, this may be a factor requiring a motor-starting study.
Table 9-1 summarizes some critical system voltage levels of interest when performing a
motor-starting study for the purpose of evaluating the effects of voltage dips.
Table 9-1—Summary of representative critical system voltage levels
when starting motors
Voltage drop location or problem
Minimum allowable
voltage (% rated)
At terminals of starting motor
a80%a
All terminals of other motors that must reaccelerate
a71%a
AC contactor pick-up (by standard) (see 9.8, NEMA standards)
DC contactor pick-up (by standard) (see 9.8, NEMA standards)
Contactor hold-in (average of those in use)
Solid-state control devices
Noticeable light flicker
85%
80%
b60–70%b
c90%c
3% change
NOTE—More detailed information is provided in Table 51 of IEEE Std 242-1986.
aTypical
for NEMA design B motors only. Value may be higher (or lower) depending on actual motor and load
characteristics.
bValue may be as high as 80% for certain conditions during prolonged starting intervals.
cMay typically vary by ±5% depending on available tap settings of power supply transformer when provided.
9.2.3 Weak source generation
Smaller power systems are usually served by limited capacity sources, which generally magnify voltage drop problems on motor-starting, especially when large motors are involved.
Small systems can also have on-site generation, which causes an additional voltage drop due
to the relatively higher impedance of the local generators during the (transient) motor-starting
interval. The type of voltage regulator system applied with the generators can dramatically
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influence motor-starting as illustrated in Figure 9-1. A motor-starting study can be useful,
even for analyzing the performance of small systems. Certain digital computer programs can
accurately model generator transient behavior and exciter/regulator response under motorstarting conditions, providing meaningful results and conclusions.
Figure 9-1—Typical generator terminal voltage characteristics
for various exciter/regulator systems
9.2.4 Special torque requirements
Sometimes special loads must be accelerated under carefully controlled conditions without
exceeding specified torque limitations of the equipment. An example of this is starting a
motor connected to a load through gearing. This application requires a special period of lowtorque cushioned acceleration to allow slack in the gears and couplings to be picked up
without damage to the equipment. Certain computer-aided motor-starting studies allow an
instant-by-instant shaft output torque tabulation for comparison to allowable torque limits of
the equipment. This study can be used for selecting a motor or a starting method, or both,
with optimum speed-torque characteristics for the application. The results of a detailed study
are used for sizing the starting resistors for a wound rotor motor, or in analyzing rheostatic
control for a starting wound rotor motor that might be used in a cushioned starting application
involving mechanical gearing or a coupling system that has torque transmitting limitations.
High-inertia loads increase motor-starting time, and heating in the motor due to high currents
drawn during starting can be intolerable. A computer-aided motor-starting study allows
accurate values of motor current and time during acceleration to be calculated. This makes it
possible to determine if thermal limits of standard motors will be exceeded for longer than
normal starting intervals.
Other loads have special starting torque requirements or accelerating time limits that require
special high starting torque (and inrush) motors. Additionally, the starting torque of the load
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or process may not permit low inrush motors in situations where these motors might reduce
the voltage dip caused by starting a motor having standard inrush characteristics. A simple
inspection of the motor and load speed-torque curves is not sufficient to determine whether
such problems exist. This is another area where the motor torque and accelerating time study
can be useful.
9.3 Recommendations
9.3.1 Voltage dips
A motor-starting study can expose and identify the extent of a voltage drop problem. The
voltage at each bus in the system can, for example, be readily determined by a digital
computer study. Equipment locations likely to experience difficulty during motor-starting can
be immediately determined.
In situations where a variety of equipment voltage ratings are available, the correct rating for
the application can be selected. Circuit changes, such as off-nominal tap settings for
distribution transformers and larger than standard conductor-sized cable, can also be readily
evaluated. On a complex power system, this type of detailed analysis is very difficult to
accomplish with time-consuming hand solution methods.
Several methods of minimizing voltage dip on starting motors are based on the fact that
during starting time, a motor draws an inrush current directly proportional to terminal
voltage; therefore, a lower voltage causes the motor to require less current, thereby reducing
the voltage dip.
Autotransformer starters are a very effective means of obtaining a reduced voltage during
starting with standard taps ranging from 50% to 80% of normal rated voltage. A motorstarting study is used to select the proper voltage tap and the lower line current inrush for the
electrical power system during motor start. Other special reduced-voltage starting methods
include resistor or reactor starting, part-winding starting, and wye (Y)-start delta (∆)-run
motors. All are examined by an appropriate motor-starting study, and the best method for the
particular application involved can be selected. In all reduced voltage starting methods,
torque available for accelerating the load is a very critical consideration once bus voltage
levels are judged otherwise acceptable. Only 25% torque is available, for example, with 50%
of rated voltage applied at the motor terminals. Any problems associated with reduced
starting torque imposed by special starting methods are automatically uncovered by a motorstarting study.
Another method of reducing high inrush currents when starting large motors is a capacitor
starting system (see Harbaugh and Ponsting [B6]2). This maintains acceptable voltage levels
throughout the system. With this method, the high inductive component of normal reactive
starting current is offset by the addition, during the starting period only, of capacitors to the
motor bus.
2The
numbers in brackets preceded by the letter B correspond to those of the bibliography in 9.9.
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This differs from the practice of applying capacitors for running motor power factor
correction. A motor-starting study can provide information to allow optimum sizing of the
starting capacitors and determination of the length of time the capacitor must be energized.
The study can also establish whether the capacitor and motor can be switched together, or
because of an excessive voltage drop that might result from the impact of capacitor transient
charging current when added to the motor inrush current, the capacitor must be energized
momentarily ahead of the motor. The switching procedure can appreciably affect the cost of
final installation.
Use of special starters or capacitors to minimize voltage dips can be an expensive method of
maintaining voltage at acceptable levels (see Harbaugh and Ponsting [B6]). Where possible,
off-nominal tap settings for distribution transformers are an effective, economical solution for
voltage dips. By raising no-load voltage in areas of the system experiencing difficulties
during motor-starting, the effect of the voltage dip can often be minimized. In combination
with a load flow study, a motor-starting study can provide information to assist in selecting
proper taps and ensure that light load voltages are not excessively high.
The motor-starting study can be used to prove the effectiveness of several other solutions to
the voltage dip problem as well. With a wound rotor motor, differing values of resistance are
inserted into the motor circuit at various times during the starting interval to reduce maximum
inrush (and accordingly starting torque) to some desired value. Figure 9-2 shows typical
speed-torque characteristic curves for a wound rotor motor. With appropriate switching times
(dependent on motor speed) of resistance values, practically any desired speed-torque
(starting) characteristic can be obtained. A motor-starting study aids in choosing optimum
current and torque values for a wound rotor motor application whether resistances are
switched in steps by timing relays or continuously adjusted values obtained through a liquid
rheostat feedback starting control.
For small loads, voltage stabilizers are sometimes used. These devices provide essentially
instantaneous response to voltage fluctuations by “stabilizing” line voltage variations of as
great as ±15% to within ±1% at the load. The cost and limited loading capability of these
devices, however, have restricted their use mostly to controlling circuit power supply
applications.
Special inrush motors can be purchased for a relatively small price increase over standard
motors. These motors maintain nearly the same speed-torque characteristics as standard
machines, but the inrush current is limited (usually to about 4.6 times full load current
compared with 6 times full load current for a standard motor).
9.3.2 Analyzing starting requirements
A speed-torque and accelerating time study often in conjunction with the previously
discussed voltage dip study permits a means of exploring a variety of possible motor speedtorque characteristics. This type of motor-starting study also confirms that starting times are
within acceptable limits. The accelerating time study assists in establishing the necessary
thermal damage characteristics of motors or verifies that machines with locked-rotor
protection supervised by speed switches will not experience nuisance tripping on starting.
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Figure 9-2—Typical wound rotor motor speed-torque characteristics
The speed-torque/accelerating time motor-starting study is also used to verify that special
motor torque and/or inrush characteristics specified for motors to be applied on the system
will produce the desired results. Mechanical equipment requirements and special ratings
necessary for motor-starting auxiliary equipment are based on information developed from a
motor-starting study.
9.4 Types of studies
From the previous discussion, it is apparent that, depending on the factors of concern in any
specific motor-starting situation, more than one type of motor-starting study can be required.
9.4.1 The voltage drop snapshot
One method of examining the effect of voltage dip during motor-starting is to ensure the
maximum instantaneous drop that occurs leaves bus voltages at acceptable levels throughout
the system. This is done by examining the power system that corresponds to the worst-case
voltage. Through appropriate system modeling, this study can be performed by various
calculating methods using the digital computer. The so-called voltage drop snapshot study is
useful only for finding system voltages. Except for the recognition of generator transient
impedances when appropriate, machine inertias, load characteristics, and other transient
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effects are usually ignored. This type of study, while certainly an approximation, is often
sufficient for many applications.
9.4.2 The detailed voltage profile
This type of study allows a more exact examination of the voltage drop situation. Regulator
response, exciter operation, and, sometimes, governor action are modeled to accurately
represent transient behavior of local generators. This type of study is similar to a simplified
transient stability analysis and can be considered a series of voltage snapshots throughout the
motor-starting interval including the moment of minimum or worst-case voltage.
9.4.3 The speed-torque and acceleration time analysis
Perhaps the most exacting analysis for motor-starting conditions is the detailed speed-torque
analysis. Similar to a transient stability study (some can also be used to accurately investigate
motor-starting), speed-torque analysis provides electrical and accelerating torque calculations
for specified time intervals during the motor-starting period. Motor slip, load and motor
torques, terminal voltage magnitude and angle, and the complex value of motor current
drawn are values to be examined at time zero and at the end of each time interval.
Under certain circumstances, even across-the-line starting, the motor may not be able to
break away from standstill, or it may stall at some speed before acceleration is complete. A
speed-torque analysis, especially when performed using a computer program, and possibly in
combination with one or more previously discussed studies, can predict these problem areas
and allow corrections to be made before difficulties arise. When special starting techniques
are necessary, such as autotransformer reduced voltage starting, speed-torque analysis can
account for the autotransformer magnetizing current and it can determine the optimum time
to switch the transformer out of the circuit. The starting performance of wound rotor motors
is examined through this type of study.
9.4.4 Adaptations
A particular application can require a slight modification of any of the above studies to be of
greatest usefulness. Often, combinations of several types of studies described are required to
adequately evaluate system motor-starting problems.
9.5 Data requirements
9.5.1 Basic information
Since other loads on the system during motor-starting affect the voltage available at the motor
terminals, the information necessary for a load flow or short-circuit study is essentially the
same as that required for a motor-starting study. This information is summarized below.
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Details are available elsewhere in this recommended practice (see Chapters 6 and 7 as well as
the Electrical Transmission and Distribution Reference Book [B4] and Beeman [B2]).
a)
b)
c)
d)
e)
Utility and generator impedances. These values are extremely significant and should be
as accurate as possible. Generally, they are obtained from local utility representatives
and generator manufacturers. When representing the utility impedance, it should be
based on the minimum capacity of the utility system in order to yield the most
pessimistic results insofar as voltage drop problems are concerned. This is in direct
opposition to the approach normally used for a short-circuit analysis discussed in
Chapter 7 of this standard. Where exact generator data cannot be obtained, typical
impedance values are available from the Electrical Transmission and Distribution
Reference Book [B4] and Beeman [B2].
Transformers. Manufacturers’ impedance information should be obtained where
possible, especially for large units (that is, 5000 kVA and larger). Standard
impedances can usually be used with little error for smaller units, and typical X/R
ratios are available in IEEE Std C37.010-1979.
Other components. System elements (such as cables) should be specified as to the
number and size of conductor, conductor material, and whether magnetic duct or
armor is used. All system elements should be supplied with R and X values so an
equivalent system impedance can be calculated.
Load characteristics. System loads should be detailed including type (constant
current, constant impedance, or constant kVA), power factor, and load factor, if any.
Exact inrush (starting) characteristics should also be given for the motor to be started.
Machine and load data. Along with the aforementioned basic information, which is
required for a voltage drop type of motor-starting analysis, several other items are
also required for the detailed speed-torque and accelerating time analysis. These
include the Wk2 of the motor and load (with the Wk2 of the mechanical coupling or
any gearing included), and speed-torque characteristics of both the motor and load.
Typical speed-torque curves are shown in Figure 9-3.
Figure 9-3—Typical motor and load speed-torque characteristics
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For additional accuracy, speed versus current and speed versus power factor characteristics
should be given for as exact a model as possible for the motor during starting. For some
programs, constants for the motor equivalent circuit given in Figure 9-4 can be either required
input information or typical default values. This data must be obtained from the manufacturer
since values are critical. Exciter/regulator data should also be obtained from the manufacturer
for studies involving locally connected generators.
Figure 9-4—Simplified equivalent circuit for a motor on starting
When special motor applications are involved like high starting torque or low starting current,
motor manufacturers may use special “deep bar” or double squirrel cage motor rotor designs.
These designs can be represented either by their torque/speed curves, or by an equivalent
electrical circuit model with two (or more) parallel rotor branches represented. This increases
the complexity of the equivalent circuit and the corresponding mathematical solution beyond
that of the more simplified single rotor model depicted in Figure 9-4.
9.5.2 Simplifying assumptions
Besides using standard impedance values for transformers and cables, it is often necessary to
use typical or assumed values for other variables when making motor-starting voltage drop
calculations. This is particularly true when calculations are for evaluating a preliminary
design and exact motor and load characteristics are unknown. Some common assumptions
used in the absence of more precise data follow:
a)
Horsepower to kVA conversion. A reasonable assumption is 1 hp equals 1 kVA. For
synchronous motors with 0.8 leading, running power factor, and induction motors, it
can easily be seen from the following equation:
kVA(PF)(EFF)
hp = -----------------------------------0.746
(9-1)
The ratio of 0.746 to efficiency times the power factor approaches unity for most
motors given the 1 hp/kVA approximation. Therefore, for synchronous motors operating at 1.0 PF, a reasonable assumption is 1 hp equals 0.8 kVA.
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b)
c)
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Std 399-1997
Inrush current. Usually, a conservative multiplier for motor-starting inrush currents is
obtained by assuming the motor to have a code G characteristic with locked-rotor
current equal to approximately 6 times the full load current with full voltage applied
at motor terminals [see the National Electrical Code® (NEC®) (NFPA 70-1996)]. A
conservative and acceptably accurate method for determining the locked rotor current
to full load running current ratio is to use the reciprocal of the motor's subtransient
reactance when this characteristic is known.
Starting power factor. The power factor of a motor during starting determines the
amount of reactive current that is drawn from the system, and thus, to a large extent,
the maximum voltage drop. Typical data (see Beeman [B2]) suggest the following:
— Motors under 1000 hp, PF = 0.20
— Motors 1000 hp and over, PF = 0.15
The starting power factor can also be determined by knowing the short-circuit X/R
ratio of the machine. Thus:
Starting power factor = cos (arctan X/R)
If a machine has an X ″d equal to 0.17 p.u. impedance on its own machine base, and a shortcircuit X/R ratio of 5.0, then its locked rotor current ratio would be 5.9 and associated starting
power factor would be 20%, or 0.2 p.u.
These power factor values are only rules of thumb for larger, integral horsepower-sized
“standard” design motors. Actual motor power factors may vary dramatically from these
values, especially for small horsepower size machines or any size special-purpose motor. For
example, the starting power factor of a “standard” 5 horsepower motor may be 0.6 or larger,
while the starting power factor of a high starting torque, fractional horsepower motor may be
0.85 or more. Wherever large numbers of small motors or any number of special torque
characteristic motors are connected to a system or circuit, actual power factors should always
be confirmed for purposes of performing accurate motor-starting calculations.
9.6 Solution procedures and examples
Regardless of the type of study required, a basic voltage drop calculation is always involved.
When voltage drop is the only concern, the end product is this calculation when all system
impedances are at maximum value and all voltage sources are at minimum expected level. In
a more complex motor speed-torque analysis and accelerating time study, several voltage
drop calculations are required. These are performed at regular time intervals following the
initial impact of the motor-starting event and take into account variations in system
impedances and voltage sources. Results of each iterative voltage drop calculation are used to
calculate output torque, which is dependent on the voltage at machine terminals and motor
speed. Since the interval of motor-starting usually ranges from a few seconds to 10 or more
seconds, effects of generator voltage regulator and governor action are evident, sometimes
along with transformer tap control depending on control settings. Certain types of motorstarting studies account for generator voltage regulator action, while a transient stability
study is usually required in cases where other transient effects are considered important. A
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summary of fundamental equations used in various types of motor-starting studies follows in
this subclause, along with examples illustrating applications of fundamental equations to actual
problems, including typical computer program outputs shown in Figures 9-10, 9-11, 9-13, and
9-14.
9.6.1 The mathematical relationships
There are basically three ways to solve for bus voltages realized throughout the system on
motor-starting. These are presented in 9.6.1 and then examined in detail by examples in
9.6.2–9.6.5.
a)
Impedance method. This method involves reduction of the system to a simple voltage
divider network (see Manning [B9]) where voltage at any point (bus) in a circuit is
found by taking known voltage (source bus) times the ratio of impedance to the point
in question over total circuit impedance. For the circuit of Figure 9-5,
X1
V = E ------------------X1 + X2
(9-2)
or, more generally,
Z1
V = E -----------------Z1 + Z2
(9-3)
The effect of adding a large capacitor bank at the motor bus is seen by the above
expression for V. The addition of negative vars causes X1 or Z1 to become larger in
both numerator and denominator, so bus voltage V is increased and approaches 1.0
per unit as the limiting improvement.
Figure 9-5—Simplified impedance diagram
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Locked-rotor impedance for three-phase motor is simply
rated voltage L – L
Z LR = E --------------------------------------------- in Ω
( 3LRA )
(9-4)
where
LRA is the locked-rotor current at rated voltage
This value in per unit is equal to the inverse of the inrush multiplier on the motor
rated kVA base:
1
Z LR = E ----------------- in per unit
LRA
------------
FLA
(9-5)
Since a starting motor is accurately represented as a constant impedance, the
impedance method is a very convenient and acceptable means of calculating system
bus voltages during motor-starting. Validity of the impedance method can be seen and
is usually used for working hand calculations. Where other than simple radial
systems are involved, the digital computer greatly aids in obtaining necessary
network reduction. To obtain results with reasonable accuracy, however, various
system impedance elements must be represented as complex quantities rather than as
simple reactances.
b)
Current method. For any bus in the system represented in Figure 9-6 and Figure 9-7,
the basic equations for the current method are as follows:
MVA load
I per unit = --------------------- at 1.0 per unit voltage
MVA base
(9-6)
V drop = I per unit × Z per unit
(9-7)
V drop = V source – V drop
(9-8)
The quantities involved should be expressed in complex form for greatest accuracy,
although reasonable results can be obtained by using magnitudes only for first-order
approximations.
The disadvantage to this method is that, since all loads are not of constant current
type, the current to each load varies as voltage changes. An iterative type solution
procedure is therefore necessary to solve for the ultimate voltage at every bus, and
such tedious computations are readily handled by a digital computer.
c)
Load flow solution method. From the way loads and other system elements are
portrayed in Figure 9-6 and Figure 9-7, it appears that bus voltages and the voltage
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Figure 9-6—Typical single-line diagram
Figure 9-7—Impedance diagram for system in Figure 9-6
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dip could be determined by a conventional load flow program. This is true. By
modeling the starting motor as a constant impedance load, the load flow calculations
yield the bus voltages during starting. The basic equations involved in this process are
repeated here (see Neuenswander [B12] and Stagg and El-Abiad [B13]).
Pk –j Qk
– Y kV k
I k = ----------------V k*
n
V k = V ref +
P i – jQ i
– Y i V i
∑ Z ki -----------------
Vi
(9-9)
(9-10)
i=1
where
Ik
Pk, Qk
Vk
Yk
Vref
n
Zki
is the current in the kth branch of any network
is the real and reactive powers representative of the loads at the kth bus
is the voltage at the kth bus
is the admittance to ground of bus k
is the voltage of the swing or slack bus
is the number of buses in the network
is the system impedance between the kth and ith buses
The load flow solution to the motor-starting problem is very precise for finding bus
voltages at the instant of maximum voltage drop. It is apparent from the expressions
for Ik and Vk that this solution method is ideally suited for the digital computer any
time the system involves more than two or three buses.
9.6.2 Other factors
Unless steady-state conditions exist, all of the above solution methods are valid for one
particular instant and provide the single snapshot of system bus voltages as mentioned earlier.
For steady-state conditions, it is assumed that generator voltage regulators have had time to
increase field excitation sufficiently to maintain the desired generator terminal voltage.
Accordingly, the presence of the internal impedance of any local generators connected to the
system is ignored. During motor-starting, however, the influence of machine transient behavior
becomes important. To model the effect of a close-connected generator on the maximum voltage drop during motor-starting requires inclusion of generator transient reactance in series
with other source reactances. In general, use of the transient reactance as the representation for
the machine results in calculated bus voltages and, accordingly, voltage drops that are reasonably accurate and conservative, even for exceptionally slow-speed regulator systems.
Assuming, for example, that bus 1 in the system shown in Figure 9-7 is at the line terminals
of a 12 MVA generator rather than being an infinite source ahead of a constant impedance
utility system, the transient impedance of the generator would be added to the system. The
resulting impedance diagram is shown in Figure 9-8. A new bus 99 is created. Voltage at this
new bus is frequently referred to as voltage behind the transient reactance. It is actually the
internal machine transient driving voltage (see Chapter 3).
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Figure 9-8—Revised impedance diagram showing
transient reactance of generator
When the steady-state operating voltage is 1.0 per unit, the internal machine transient driving
voltage can be considered the voltage that must be present ahead of the generator transient
reactance with the terminal voltage maintained at 1.0 per unit (within exciter tolerances)
during steady-state conditions while supplying power to the other loads on the system. The
transient driving voltage V is calculated as follows:
V = V terminal + ( jX ′ d )I load
(9-11)
= 1 + ( jX ′ d ) ( I load )
where
V terminal = 1.0 per unit
(9-12)
MVA load
I load = --------------------- per unit
MVA base
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Treatment of a locally connected generator is equally applicable to all three solution methods
described previously. Such an approach cannot give any detail regarding the response of the
generator voltage regulator or changes in machine characteristics with time. For a more
detailed solution that considers time-dependent effects of machine impedance and voltage
regulator action, the appropriate impedance and voltage terms in each expression must be
continuously altered to accurately reflect changes that occur in the circuit. This procedure is
also applicable to any solution methods considered. Figure 9-9 shows a simplified
representation of the machine parameters that must be repeatedly modified to obtain the
correct solution.
Figure 9-9—Simplified representation of generator
exciter/regulator system
Some type of reduced voltage starting is often used to minimize motor inrush current and
thus reduce total voltage drop, when the associated reduction in torque accompanying this
starting method is permissible.
Representation used for the motor in any solution method for calculating voltage drop must
be modified to reflect the lower inrush current. If autotransformer reduced voltage starting is
used, motor inrush will be reduced by the appropriate factor from Table 9-2. If, for example,
normal inrush is 6 times full load current and an 80% tap autotransformer starter is applied,
the actual inrush multiplier used for determining the appropriate motor representation in the
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calculations is (6) (0.64) = 4.2 × full load current. Resistor or reactor starting limits the line
starting current by the same amount as motor terminal voltage is reduced (that is, 65% of
applied bus voltage gives 65% of normal line starting current).
Table 9-2—Autotransformer line starting current
Autotransformer tap
(% line voltage)
Line starting current
(% normal at full voltage)
50
25
65
42
80
64
wye (Y)-start, delta (∆)-run starting delivers 33% of normal starting line current with full
voltage at the motor terminals. The starting current at any other voltage is, correspondingly,
reduced by the same amount. Part winding starting allows 60% of normal starting line current
at full voltage and reduces inrush accordingly at other voltages.
Adjustable speed drives generally provide suitable controls for limiting inrush currents
associated with motor-starting. For this type of apparatus the inrush associated with motorstarting is almost always significantly less than for a motor-starting across the line. For
purposes of motor-starting analysis, the starting current for the drive and motor simply can be
modeled as the maximum imposed by the drive upon the system.
When a detailed motor speed-torque and accelerating time analysis is required, the following
equations found in many texts apply (see Weidmer and Sells [B14]). The equations in general
apply to both induction and synchronous motors, since the latter behave almost exactly as do
induction machines during the starting period.
T ∝ V2
T = I0α
(9-13)
2
Wk
2
I 0 = ----------- ( lb·ft·s )
2g
2
(9-14)
2
ω = ω 0 + 2α ( θ – θ 0 ) ( r/s )
2
(9-15)
∆θ = ω 0 t + 1/2 α t ( r )
(9-16)
T n 2g
2
( r/s )
α = -----------2
Wk
(9-17)
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A simplified approximation for starting time is also available:
2
Wk ( r/min 1 – r/min 2 ) ( 2π )
t ( s ) = ---------------------------------------------------------------60gT n
(9-18)
where
T
V
I0
g
ω
α
t
Tn
θ
Wk2
is the average motor shaft output torque
is the motor terminal voltage
is the moment of inertia
is the acceleration due to gravity
is the angular velocity
is the angular acceleration
is the time in seconds to accelerate
is the net average accelerating torque between rev/min1 and rev/min2
is the electrical angle in degrees
is the inertia
Substituting for g and rearranging yields:
2
Wk ( r/min 1 – r/min 2 )
t ( s ) = ---------------------------------------------------308T n
The basic equation for use with the equivalent circuit of Figure 9-4 is as follows (see
Fitzgerald, Kingsley, and Kusko [B5]; Beeman [B2]; and Peterson [B12]):
2
q 1 V ( r 2 /s )
T = -----------------------------------------------------------------2
2
ω s ( r 1 + r 2 /s ) + ( X 1 + X 2 )
where
T
ωs
(r1 + jX1)
(r2/s + jX2)
q1
V
is the instantaneous torque
is the angular velocity at synchronous speed
is the stator equivalent impedance
is the rotor equivalent impedance
is the number of stator phases (3 for a 3 φ machine)
is the motor terminal voltage
9.6.3 The simple voltage drop determination
To illustrate this type of computer analysis, the system of Figure 9-6 will again be considered.
It is assumed that bus 1 is connected to the terminals of a 12 MVA generator having 15%
transient reactance (1.25 per unit on a 100 MVA base). Prior to starting, when steady-state
load conditions exist, the impedance diagram of Figure 9-7 applies with the motor off-line.
The impedance diagram of Figure 9-8 applies when the 1000 hp motor on bus 4 is started.
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Bus 99 in Figure 9-8 has been assigned a voltage of 1.056 per unit. This value can be confirmed using the expression for the internal machine transient driving voltage V given in 9.6.2
with appropriate substitutions as follows:
V = (1.0 + j0.0) + (j1.25)(0.060114 – j0.042985)
V = (1.0 + 1.0537) + (j0.07514)
V = 1.0564 p.u. voltage ∠4.08°
where values for the current through the X'd element are expressed on a 100 MVA base and
correspond to those that exist at steady state prior to motor-starting with bus 1 operating at
1.0 p.u. voltage. The computer output report of the steady-state load flow results for this case
are shown both graphically and in tabular form in Figures 9-10 (a) and 9-10 (b), respectively.
All system loads are on-line, except the 1000 hp motor. The two non-motor loads are modeled as constant power type loads. The 12 MVA generator must supply steady-state power
equal to 6.011 + j4.299 MVA, as noted in Figure 9-10 (a). The two loads combined require
5.955 + j3.687 MVA. Therefore, the losses in the system, including those through the generator are equal to 0.056 + j0.612 MVA. With these values of power flowing during steady-state,
prior to motor-starting, the swing bus is at the required value of 1.0 p.u. voltage. The voltage
drop at the motor bus, without the motor on line is 4.89%, resulting in an operating voltage
prior to starting of 0.951 p.u. voltage.
Figure 9-10 (a)—Computer-generated graphical output format—steady-state
For convenience, the voltage angle associated with the generator (or swing) bus is assumed to
be zero, which results in the corresponding shift for all other bus voltages. Since the
transformers are assumed to be connected delta-delta, the angular phase shifts indicated are
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Figure 9-10 (b)—Computer-generated output format in tabular form—
steady-state
due to the voltage drops alone. It can be seen from the computer output report, shown both
graphically and in tabular form in Figures 9-11 (a) and 9-11 (b), respectively, that when
subsequent motor-starting calculations are made for this system, the voltage at the motorstarting bus is 0.794 ∠–9.56° per unit.
Although this voltage is very close to the 0.80 p.u. value required to start many motors, it is
well below the 0.85 p.u. criterion established earlier for proper operation of ac control
devices that are connected at most motor-starting buses. Further examination of this problem
with the calculation software would show that when the motor-starting interval is over, and
the motor is operational, the voltage at the motor bus recovers to 0.92 p.u. A second study
could be easily performed to explore the effects of increasing the motor-starting bus voltage,
by adjusting the transformer tap settings.
9.6.4 Time-dependent bus voltages
The load flow solution method for examining effects of motor-starting allows a look at the
voltage on the various system buses at a single point in time. A more exact approach is to
model generator transient impedance characteristics and voltage sources closer to give results
for a number of points in time following the motor-starting event. Although the solution
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Figure 9-11 (a)—Computer-generated graphical output format—motor starting
methods are applicable to multiple generator/motor systems as well, equations can be
developed for a system of the form shown in Figure 9-12 to solve for generator, motor, and
exciter field voltages as a function of time. The digital computer is used to solve several
simultaneous equations that describe the voltage of each bus in a system at time zero and at
the end of successive time intervals.
Figures 9-13 through 9-16 show in detail the type of input information required and the
output obtained from a digital computer voltage drop study. The system shown in Figure 9-12
contains certain assumptions, which include the following:
a)
b)
c)
d)
e)
Circuit losses are negligible—reactances only used in calculations.
Initial load is constant kVA type.
Motor-starting load is constant impedance type.
Motor-starting power factor is in the range of 0 to 0.25.
Mechanical effects, such as governor response, prime mover speed changes, and
inertia constants, are negligible.
Plotted results obtained from the computer compare favorably to those expected from an
examination of Figure 9-1. In the particular computer program used to obtain this report, the
excitation system models available are similar to those described in IEEE Cmte. Paper [B6].
Excitation system models are shown in simplified form in Figure 9-17. Continuously acting
regulators of modern design permit full field forcing for minor voltage variations (as little as
0.5%), and these voltage changes have been modeled linearly for simplicity.
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Figure 9-11 (b)—Computer-generated output format in tabular form—
motor starting
Variations in exciter field voltage (EFV) over each time interval considered are used to
calculate system bus voltages at the end of these same intervals. A single main machine field
circuit time constant is used in the generator representation, and Fromlich’s approximation
(see Kimbark [B8]) for saturation effects is used when the voltage behind the generator
leakage reactance indicates that saturation has been reached. The tabulated output stops just
short of full recovery since a more complex model is necessary to represent overshoot,
oscillation, etc., beyond this point. Of primary concern in this type of study is the maximum
voltage dip and the length of time to voltage recovery as a function of generator behavior and
voltage regulator performance.
9.6.5 The speed-torque and motor-accelerating time analysis
A simplified sample problem is presented for solution by hand. In this way, it is possible to
appreciate how the digital computer aids in solving the more complex problems. The
following information applies to the system shown in Figure 9-18.
a)
b)
Motor hp = 1000 (induction)
Motor r/min = 1800
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Figure 9-12—Simplified system model for generator representation
during motor starting
c)
d)
Motor Wk2 = 270 lb⋅ft2
Load Wk2 = 810 lb⋅ft2
Assuming Figure 9-3 describes the speed-torque characteristic of the motor and the load, it is
possible to find an average value for accelerating torque over the time interval defined by
each speed change. This can be done graphically for hand calculations, and the results are
tabulated in Table 9-3.
Applying the simplified formula for starting time provided earlier,
( 270 + 810 ) ( 450 – 0 )
t 0 – 25 = ---------------------------------------------------- = 0.6981 s
( 308 ) ( 2260.4 )
( 1080 ) ( 900 – 450 )
t 25 – 50 = ---------------------------------------------- = 0.5410 s
( 308 ) ( 2916.7 )
( 1080 ) ( 1350 – 900 )
t 50 – 75 = ------------------------------------------------- = 0.4580 s
( 308 ) ( 3500.0 )
( 1080 ) ( 1710 – 1350 )
t 75 – 95 = ---------------------------------------------------- = 0.6925 s
( 308 ) ( 1822.9 )
and, therefore, the total time to 95% of synchronous speed (or total starting time) is the
sum of the times for each interval, or approximately 2.38 s. It can be seen how a similar
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Figure 9-13—Typical output obtained from input—generator
motor-starting program
Figure 9-14—Typical output—generator motor-starting program
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Figure 9-15—Typical output—plot of generator voltage dip
Figure 9-16—Typical output—plot of motor voltage dip
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MOTOR-STARTING STUDIES
(a)
(b)
(c)
Figure 9-17—Simplified representation of typical regulator/exciter
models for use in computer programs
technique can be applied to the speed-torque starting characteristic of a wound rotor motor
(see Figure 9-2) to determine the required time interval for each step of rotor-starting
resistance. The results of such an investigation can then be used to specify and set timers that
operate resistor switching contactors or program the control of a liquid rheostat.
The current drawn during various starting intervals can be obtained from a speed-current
curve, such as the typical one shown in Figure 9-19. This example has assumed full voltage
available to the motor terminals, which is an inaccurate assumption in most cases. Actual
voltage available can be calculated at each time interval. The accelerating torque will then
change by the square of the calculated voltage. This process can be performed by graphically
plotting a reduced voltage speed-torque curve proportional to the voltage calculated at each
time interval, but this becomes tedious in a hand calculation. Sometimes, in the interest of
simplicity, a torque corresponding to the motor terminal voltage at the instant of the
maximum voltage dip is used throughout the starting interval. More accurate results are
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Figure 9-18—Simplified system model for accelerating time
and speed-torque calculations
Table 9-3—Average values for accelerating torque over time interval
defined by a speed change
Speed
Tmotor
Tload
Tnet
Tnet
0%
100%
30%
—
—
—
—
—
25%
120%
35%
—
—
—
—
—
100%
2916.7 lb⋅ft2
50%
160%
45%
—
—
—
120%
2500.0 lb⋅ft2
75%
190%
65%
—
—
—
—
—
95%
80%
80%
77.5%
2260.4 lb⋅ft2
—
62.5%
—
1822.9 lb⋅ft2
—
possible with digital computer program analysis. A sample output report for the analysis is
shown in Figure 9-20.
As a matter of interest, such a computer analysis using the torque speed and motor
accelerating time method will be applied to the problem that was examined in Figure 9-6.
Previously this problem was examined using the simple voltage drop method, and the results
shown in Figure 9-11 (a) and 9-11 (b). Figure 9-21 shows a computer output plot of motor
speed and motor-starting bus voltage as a function of time. The motor is started one second
into the simulation. Figure 9-21 suggests that the 1000 hp motor in Figure 9-6 would start
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Figure 9-19—Typical motor speed-current characteristic
in about 15 s. The voltage at the motor-starting bus prior to starting the motor is 3.95 kV or
0.95 p.u.; however, the voltage at the motor-starting bus drops to 3.28 kV or 0.79 p.u. at the
instant the motor starts, agreeing with the snapshot results presented in Figure 9-11. Also to
be noted in Figure 9-21 is the steady-state operating voltage of 3.82 kV or 0.92 p.u. at the
motor bus with all three loads in service, as was determined using the calculating procedures
employed in 9.6.3. Figure 9-22 shows a similar computer-generated output plot for a second
study case, where the main transformer taps are used to increase the transformer secondary
bus voltage by 5%, and an autotransformer starter is applied to start the motor. With the +5%
main transformer tap setting and the autotransformer starter modeled, the initial voltage drop
on the motor-starting bus is improved, but not sufficiently to satisfy the 0.85 p.u. voltage
criterion during starting, required for reliable ac controls operation, as mentioned in 9.6.3.
9.7 Summary
Several methods for analyzing motor-starting problems have been presented. Types of
available motor-starting studies range from simple voltage drop calculation to the more
sophisticated motor speed-torque and acceleration time study that approaches a transient
stability analysis in complexity. Each study has an appropriate use, and the selection of the
correct study is as important a step in the solution process as the actual performance of the
study itself. Examples presented here should serve as a guide for determining when to use
each type of motor-starting study, what to expect in the way of results, and how these results
can be beneficially applied. The examples should also prove useful in gathering the required
information for the specific type of study chosen. Experienced consulting engineers and
equipment manufacturers can give valuable advice, information, and direction regarding the
application of motor-starting studies as well.
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Figure 9-20—Typical output—motor speed-torque
and accelerating time program
MOTOR-STARTING STUDIES
IEEE
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Figure 9-21—Computer-generated graphical output of motor speed
and bus voltage vs. time—Motor started across the line
and main transformer taps set on nominal voltage
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Figure 9-22—Computer-generated graphical output of motor speed
and bus voltage vs. time—Main transformer with a tap setting
to produce 5% above normal secondary voltage
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9.8 References
IEEE Std 141-1993, IEEE Recommended Practice for Electric Power Distribution for Industrial Plants (IEEE Red Book).3
IEEE Std 242-1986 (Reaff 1991), IEEE Recommended Practice for Protection and Coordination of Industrial and Commercial Power Systems (IEEE Buff Book).
IEEE Std C37.010-1979 (Reaff 1988), IEEE Application Guide for AC High-Voltage Circuit
Breakers Rated on a Symmetrical Current Basis.
NEMA ICS 1-1993, Requirements for Industrial Control and Systems.4
NEMA ICS 2-1993, Industrial Control Devices, Controllers and Assemblies.
NEMA ICS 3-1993, Industrial Systems.
NEMA ICS 4-1993, Terminal Blocks for Industrial Control and Systems.
NEMA ICS 6-1993, Enclosures for Industrial Control and Systems.
NEMA MG 1-1993, Motors and Generators.
NFPA 70-1996, National Electrical Code® (NEC®).5
9.9 Bibliography
[B1] Alger, P. L., Induction Machines, Their Behavior and Uses, Gordon and Breach Science
Publishers, 1970.
[B2] Beeman, D., Ed., Industrial Power Systems Handbook, New York: McGraw-Hill, 1955.
[B3] Croft, T., Care, C., and Watt, J., American Electrician’s Handbook, New York: McGrawHill, 1970.
[B4] Electrical Transmission and Distribution Reference Book, Westinghouse Electric
Corporation, East Pittsburgh, Pennsylvania, 1964.
3IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
4NEMA publications are available from the National Electrical Manufacturers Association, 1300 N. 17th St., Ste.
1847, Rosslyn, VA 22209, USA.
5The NEC is available from Publications Sales, National Fire Protection Association, 1 Batterymarch Park, P.O. Box
9101, Quincy, MA 02269-9101, USA. It is also available from the Institute of Electrical and Electronics Engineers,
445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331, USA.
Copyright © 1998 IEEE. All rights reserved.
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[B5] Fitzgerald, A., Kingsley, C., and Kusko, A., Electric Machinery, New York: McGrawHill, 1971.
[B6] Harbaugh, J., and Ponsting, L., “How to Design a Capacitor Starting System for Large
Induction and Synchronous Motors,” IEEE IAS 1975 Annual Meeting, Conference Record.
[B7] IEEE Power Generation Committee, “Computer Representation of Excitation Systems,”
Paper 31 TP 67-424, May 1, 1967.
[B8] Kimbark, E., Power System Stability: Synchronous Machines, New York: Dover
Publications, 1956.
[B9] Manning, L., Electrical Circuits, New York: McGraw-Hill, 1966.
[B10] Neuenswander, J., Modern Power Systems, International Text Book Company, 1971.
[B11] Peterson, H., Transients in Power Systems, New York: Dover Publications, 1951.
[B12] Prabhakara, F. S., Smith, Jr., R. L, and Stratford, R. P., Industrial and Commercial
Power Systems Handbook, McGraw-Hill, 1995.
[B13] Stagg, and El-Abiad, Computer Methods in Power System Analysis, New York:
McGraw-Hill, 1971.
[B14] Weidmer, R., and Sells, R., Elementary Classical Physics, 2 vols., Boston, Allyn and
Bacon, Inc., 1965.
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Chapter 10
Harmonic analysis studies
10.1 Introduction
This chapter discusses the basic concepts involved in studies of harmonic analysis of
industrial and commercial power systems. The need for such an analysis, recognition of
potential problems, corrective measures, required data for analysis, and benefits of using a
computer as a tool in a harmonic analysis study are also addressed in this chapter.
Traditionally, the main source of harmonics in power systems has been the static power
converter used as rectifiers for various industrial processes; however, the static power converter is now used in a variety of additional applications such as adjustable speed drives,
switched-mode supplies, frequency changers for induction heating, etc. Semiconductor
devices are being increasingly used as static switches that modulate the voltage applied to
loads. Examples of these are soft starters for motors, static var compensators, light dimmers,
electronic ballasts for arc-discharge lamps, etc. Other examples are devices with nonlinear
voltage-current characteristics such as arc furnaces or saturable electromagnetic devices.
Since nonlinear loads represent an ever-increasing percentage of the total load of an industrial
or commercial power system, harmonic studies have become an important part of overall
system design and operation. Fortunately, the available software for harmonic analysis has
also grown. Guidelines for the acceptance of harmonic distortion are well-defined in IEEE
Std 519-1992.1
By modeling power system impedances as a function of frequency, a study can be made to
determine the effect of the harmonic contributions from nonlinear loads on the voltages and
currents in the power system. Most commercial software for harmonic analysis offer the
following:
a)
b)
c)
Calculation of harmonic bus voltages and branch current flows in the network due to
harmonic sources,
Resonances in the existing or planned system,
Performance indices that calculate the effects of harmonics on voltage or current
waveform distortion, telephone interference, etc.
The software can also help in selecting and locating capacitors or passive filters to optimize
system performance.
Details of system modeling and applicable standards are discussed. The treatment described
here particularly applies to industrial and commercial systems at low and medium voltages,
but the basics are also applicable to other systems and higher voltages. this chapter does not
1Information
on references can be found in 10.9.
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deal with active filters as part of the filter design; however, some reference is made to their
application.
It may be said at the outset that the harmonic filter design is very closely linked to powerfactor (PF) requirements of the system (based on utility tariffs) and both must be considered a
the same time. In the past, many PF compensation studies have been made without regard to
the possible resonances in the system or harmonic absorption by the capacitors.
IEEE Std 519-1992 should be referred to for general information and particularly for harmonic generation from static power converters and other harmonic sources. A considerable
reference list (10.9) and bibliography (10.10) are provided for this chapter, which should be
referred to for details in specific areas.
10.2 Background
As a matter of definition, any load or device that does not draw a sinusoidal current when
excited by a sinusoidal voltage of the same frequency is termed a nonlinear load. The most
common are switching devices, such as solid-state converters, which force the conduction of
currents for only certain periods and, to a lesser extent, saturable impedance devices such as
transformers with nonlinear voltage vs. impedance characteristics. A nonlinear load is considered a source of harmonic currents, where harmonic frequencies are generally considered
to be integer multiples of the system frequency. However, certain nonlinear loads, such as an
arc furnace or a cycloconverter, may also have non-integer harmonic frequencies in addition
to the expected integer harmonics.
Harmonics, by definition, occur in every cycle of the fundamental current and are calculated
as part of the steady-state solution. However, exceptions exist, and harmonics may vary from
cycle to cycle. These are termed time-varying harmonics. Also, harmonics appear in quasisteady-state or transient situations, such as in magnetization inrush current of a transformer.
This chapter does not deal with transient harmonics or time-varying harmonics.
An ideal current source is one which provides a constant current irrespective of the system
impedance seen by the source. In most studies for industrial applications, the nonlinear load
or the harmonic source is considered an ideal current source without a Norton’s impedance
across the source (i.e., Norton impedance is assumed to be infinite). This approximation is
generally reasonable and yields satisfactory results. When the nonlinear device acts like a
voltage source [e.g., a pulse-width-modulated (PWM) inverter], a Norton equivalent current
source model may still be used since most computer programs are based on the current injection method (see 10.5.7).
Since the system is subjected to current injections at multiple frequencies, the network is
solved for voltage and current at each frequency separately. The total voltage or current in an
element is then found either by a root-mean-square sum or arithmetic sum, using the principle of superposition.
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The generated harmonic frequencies are dependent upon the type of nonlinear load. Most
nonlinear loads produce odd harmonics with small even harmonics. However, loads such as
arc furnaces produce the entire spectrum of harmonics: odd, even, and non-integer harmonics
in between (non-integer harmonics are also referred as “interharmonics”). Generally, the
amplitude of the harmonics decreases as the frequency (or the harmonic order) increases.
The effect of harmonic current propagation through the network, including the power source,
produces distortion of the voltage waveform depending upon harmonic voltage drops in
various series elements of the network. Therefore, the voltage distortion at a given bus is
dependent on the equivalent source impedance; the smaller the impedance, the better the voltage quality. Note that the harmonic sources, which are nonlinear loads, are not the sources of
power, but are the cause of additional active and reactive power losses in the system.
10.3 Purpose of harmonic study
With the growing proliferation of nonlinear loads in commercial buildings and industrial
plants, which may be in the range of 30% to 50% of the total load, the effects of harmonics
within the system and their impact on the utility and neighboring loads needs to be examined
before any complaints are made, equipment is damaged, or production is lost.
The following situations may necessitate a harmonic study, which should include recommendations for mitigating the effects of harmonics (see 10.7):
a)
Compliance with IEEE Std 519-1992, which defines the current distortion limits
a user should meet at the point of common coupling (PCC) with the utility. Voltage
distortion limits are also defined as a basis for the system design. The voltage distortion limits are primarily intended for the utility to provide a good sine wave voltage;
however, an individual user is expected to use the voltage limits as a basis for the system design. The chances are that if the current distortion limits are met, the voltage
distortion limits will also be met, except in some unusual circumstances.
b)
A history of harmonic-related problems, such as failure of power-factor compensation capacitors, overheating of cables, transformers, motors, etc., or misoperation of
protective relays or control devices.
Plant expansion where significant nonlinear loads are added or where a significant
amount of capacitance is added.
Design of a new facility or power system, where the load-flow, power factor compensation, and harmonic analyses are considered as one integrated study to determine
how to meet the reactive power demands and harmonic performance limits.
c)
d)
When harmonics appear to be the cause of system problems, it is necessary to determine the
resonant frequencies at the problem sites. With power-factor correction capacitor banks, a
parallel system resonance can occur at or near one of the lower harmonic orders (3, 5, …).
This resonance can be critical if excited by a harmonic current injection at that frequency.
Refer to 10.4.3 for an approximate calculation of harmonic resonant frequency. An estimate
of the resonant frequencies is very useful for an initial evaluation.
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It is not uncommon to encounter a system in which it is more practical to take harmonic measurements as a diagnostic tool rather than to perform a detailed, time-consuming, harmonic
analysis study. In other cases, measurements are used to verify the system model prior to the
performance of a detailed harmonic analysis study. This is especially desirable for arc furnace
installations. In order to ensure that harmonic measurements will produce reliable results,
careful consideration must be given to both test equipment and procedures being applied, see
Schieman and Schmidt [B35] and Shipp [B38]2. The test results may identify the cause of a
harmonic problem so that the need for a detailed harmonic study is either eliminated or the
study is simplified.
10.4 General theory
10.4.1 Harmonic sources
All harmonic sources are referred to as nonlinear loads because they draw non-sinusoidal
currents when a sinusoidal voltage is applied. The non-sinusoidal current may be due to the
inherent characteristic of the load (e.g., arc furnaces), or due to a switching circuit (e.g., a
6-pulse converter that forces conduction of currents for only certain periods). In industrial
and commercial power systems there may be many such harmonic sources distributed
throughout the system.
The harmonic study requires knowledge of the harmonic currents generated by nonlinear
loads. There are three options open to the analytical engineer:
a)
b)
c)
Measure the generated harmonics at each source,
Calculate the generated harmonics by a mathematical analysis where possible, such
as at converters or static var compensators, and
Use typical values based on similar applications or published data.
In practice, all three methods are used and provide reasonable results.
Since the system configuration and load continually change, the harmonics also change and it
would be a formidable task to study all such conditions. Usually, the worst operating condition is determined, and the design is based on the “worst-generated” harmonics. However, It
needs to be recognized that even with the “worst generated” harmonic case, the harmonic
flows within different elements of the network can be different depending upon the number of
transformers or tie breakers in service. This necessitates that for the “worst generated” case,
the “worst operating cases(s)” must be analyzed.
One other difficulty in the analysis arises from the fact that when multiple harmonic sources
are connected to the same bus (or different buses), the phase angles between the harmonics of
the same order are usually not known. This, generally speaking, leads to arithmetic addition
of harmonic magnitudes, which may be reasonable if the harmonic sources are similar and
2Numbers
268
in brackets correspond to those of the bibliography in 10.10.
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have similar operating load points. However, this approach can lead to a more conservative
filter design and distortion calculations, if the sources are different or operate at different load
points. Determination of phase angles of harmonics and vectorial addition can be quite a
complex and expensive approach for general industrial application. This is often resolved by
simplifying assumptions based on experience or by field measurements. More advanced techniques are used in high-voltage dc transmission and other utility applications where accuracy
is important.
Industrial harmonic studies are usually represented on a single-phase basis, i.e., based on the
assumption that the system is balanced and positive sequence analysis applies. A three-phase
study is warranted only if the system or the load is severely unbalanced or a four-wire system
with single-phase loads exist. In such a situation it will be very desirable to determine the harmonics generated in all three phases. If the harmonic generation is assumed to be balanced
and the system is considered unbalanced, a three-phase study may not serve the full purpose.
The cost of a three-phase study could be higher than a single-phase study and should be used
only when such an expense and purpose can be justified.
10.4.2 Effects of harmonics
The effects of harmonics are described here only in the context of the analytical harmonic
system study, details of these effects can be found in referenced literature. IEEE Std
519-1992 (Chapter 6) and Prabhakara, Smith, and Stratford [B30] (Chapter 5) deal with the
subject in detail. The effects of harmonics in a power system are pervasive in that they influence system losses, system operation, and system performance. Unless the harmonics are
controlled to acceptable limits, the power equipment and, even more so, the electronic equipment may be damaged resulting in and costly system outages.
The effects of harmonics are due to both current and voltage, although current-produced
effects are more likely to be seen in day-to-day performance. Voltage effects are more likely
to degrade the insulation and hence shorten the life of the equipment. The following describes
some of the common effects of harmonics:
a)
b)
c)
d)
e)
f)
g)
Increased losses within the equipment and associated cables, lines, etc.,
Pulsating and reduced torque in rotating equipment,
Premature aging due to increased stress in the equipment insulation,
Increased audible noise from rotating and static equipment,
Misoperation of equipment sensitive to waveforms,
Substantial amplification of currents and voltages due to resonances, and
Communication interference due to inductive coupling between power and communication circuits.
Generally, harmonic studies involving harmonic flows and filter design do not involve
detailed analysis of the effects of harmonics if the limits imposed by the user or by a standard
are met. However, in specific cases, analysis of harmonics penetrating into rotating equipment, causing relay misoperation, or interfering with communication circuits may require a
separate study.
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For additional information see the IEEE Task Force Report [B21].
10.4.3 Resonance
Most power system circuit elements are primarily inductive and, therefore, the presence of
shunt capacitors used for power-factor correction or harmonic filtering can cause cyclic
energy transfer between the inductive and capacitive elements at the natural frequency of resonance. At this frequency the inductive and capacitive reactances are equal.
The combination of inductive (L) and capacitive (C) elements as viewed from a bus of interest, generally the bus at which harmonic currents are injected by a nonlinear load (source
bus), can result in either a series resonance (L and C in series) or a parallel resonance (L and
C in parallel). As shown in the following sections, the series resonance results in low impedance and parallel resonance in high impedance. At either series or parallel resonance, the net
impedance is resistive. In harmonic studies, it is essential that the driving-point impedance
(see 10.5.7), as seen from the harmonic source bus (or other bus of interest), be examined to
identify the series and parallel resonance frequencies and resulting impedances.
In practical electrical systems, PF correction capacitors are utilized to offset the power factor
penalty imposed by the utility. This can create an abnormal situation, because the combination of capacitors and inductive elements in the system can result in either series or parallel
resonance or a combination of both depending upon the system configuration. Usually parallel resonance occurs more often because capacitor banks act in parallel with system impedance (inductive); this can be a matter of concern if the resonant frequency happens to be close
to one of the frequencies generated by the harmonic sources in the system.
The result of a series resonance may be the flow of unexpected amounts of harmonic currents
through certain elements. A common manifestation of excessive harmonic current flow is
inadvertent relay operation, burned fuses, and overheating of cables, etc.
The result of a parallel resonance may be the presence of excessive harmonic voltages across
network elements. A common manifestation of excessive harmonic voltages is capacitor or
insulation failure.
10.4.3.1 Series resonance
An example of series resonant circuit is shown in Figure 10-1. Each circuit element is
described in terms of its impedance. The equivalent impedance of the circuit and the current
flow are expressed by Equations (10-1) and (10-2). This circuit is said to be in resonance
when the inductive reactance XL is equal to the capacitive reactance XC . The resonant frequency at which XL = XC is given by Equation (10-3a).
Z = R + j( X L – X C )
(10-1)
V
I = ------------------------------------R + j(X L – X C )
(10-2)
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V
= ---- at resonance ( X L = X C )
R
1
ω 0 = -----------LC
IEEE
Std 399-1997
(10-2a)
(10-3)
1
f 0 = ------------------2π LC
(10-3a)
Given the relatively low values of series resistance usually found in power equipment, the
magnitude of the current in Equation (10-2a) can be large at resonance.
Figure 10-1—Example circuit for series resonance
Figure 10-2 shows the equivalent impedance of the circuit in Figure 10-1 as a function of frequency. The element values are R = 2 Ω , L = 3.98 mH, and C = 36.09 µF. It is clear from
Equation (10-1) that the impedance appears capacitive at low frequencies and becomes
inductive as the frequency increases, and that resonance occurs at 420 Hz (7th harmonic for a
60 Hz system).
A general measure of the shape of the impedance plot of Figure 10-2 is often given in terms
of the quality factor Q. For a series resonant circuit, the Q is defined in Equation (10-4) at any
angular frequency ω.
ωL
Q = ------R
(10-4)
At the resonant frequency, the Q is generally approximated to the ratio of ωo L /RL since
R ≈ RL as capacitors have negligible resistance. As will be demonstrated in 10.6, the parameter Q often plays an important role in filter design because most single-tuned harmonic
filters are simple RLC series-resonant circuits. In general, a higher Q produces a more pronounced “dip” in the plot of Figure 10-2. A lower Q results in a more rounded shape. In most
filter applications, the natural quality factor (with no intentional resistance) is relatively high
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Figure 10-2—Impedance magnitude vs. frequency for series resonant circuit
(>100 at resonant frequency). In special applications it may be necessary to intentionally
reduce the Q.
Typical situations where series resonance can be a problem are shown in the one-line diagrams of Figure 10-3. In Figure 10-3 (a), the utility supply is assumed to contain voltage harmonics. The series resonant path is created from the equivalent series impedance from the
utility supply and the bus transformer and the power factor correction capacitor. In Figure
10-3 (b), the harmonics are generated inside the plant. The series resonant path involves the
two transformer impedances and the PF correction capacitor.
This transformer-capacitor combination could inadvertently act as a filter, and permit the flow
of harmonic current at or near the resonant frequency into the capacitor bank. If unplanned,
these currents can lead to blown fuses, inadvertent relay operation, and loss-of-life for the
capacitor and the transformer.
10.4.3.2 Parallel resonance
There are many forms of parallel resonant circuits. In general, an inductor must be in parallel
with a capacitor to produce parallel resonance. A typical parallel-resonant circuit encountered
in power systems is shown in Figure 10-4. Each element is described by its impedance. This
circuit is said to be in parallel resonance when XL = XC as in the case of series resonance.
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HARMONIC ANALYSIS STUDIES
(a) Utility source containing harmonics
(b) Plant harmonics
Figure 10-3—Potential series resonant situation
The equivalent impedance seen by the current source in Figure 10-4 is given by Equation
(10-5). Note that at a particular frequency, XL = XC and the denominator is reduced to R. This
frequency is the resonant frequency and is given by Equation (10-3). The voltage across the
complete circuit is given by equation (10-6).
– jX C ( R + jX L )
Z = -----------------------------------R + j(X L – X C )
(10-5)
V = I Z
(10-6)
NOTE—Since Z >> XL or XC, V can be very high.
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Figure 10-4—Typical parallel resonant circuit
In most cases, the resistance of power circuits is relatively small. It can be seen from Equation (10-5) that resonances can produce very large equivalent impedances at or near the resonant frequency, since R is generally small. Using the previous values (R = 2 Ω, L = 3.98 mH,
and C = 36.09 µF), a plot of the magnitude of the impedance in Equation (10-5) is shown in
Figure 10-5. The sharpness of Figure 10-5 can be more conveniently calculated by the “current gain factor (rho, ρ)” as the ratio of current in either the inductive branch or the capacitive
branch to the injected current.
Figure 10-5—Impedance magnitude vs. frequency for parallel circuits
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One unique property of the parallel resonant circuit is that when excited from a current source
at this frequency, a high circulating current will flow in the capacitance-inductance loop even
though the source current is small in comparison. The current in the loop circuit is amplified
to a level depending only upon the quality factor Q of the circuit.
Parallel resonance can produce undesirable overvoltages. From Figure 10-4 and Equation
(10-6), a current of 1.0 A at 60 Hz will produce a voltage of approximately 2.6 V across the
capacitor (the net impedance being capacitive, Z = 2.55 @ 35.3°). However, the same 1.0 A
current at 420 Hz (near the resonant frequency) will produce approximately 55 V (the net
impedance being inductive or close to resistive, Z = 55.13 @ –10.8°). This reasoning is often
combined with known current injections for motor drives, rectifiers, etc., to predict potential
harmonic overvoltages in power systems.
Parallel resonance typically involves the following:
a)
b)
The leakage inductance of large transformers and/or the equivalent inductance of the
utility system, and
The power factor correction capacitors. Figure 10-6 shows a possible one-line for
parallel resonance.
Figure 10-6—Possible parallel resonant circuit: Plant harmonics
10.4.3.3 Resonances due to multiple filters
To illustrate the presence of multiple resonances, Figure 10-7 shows a plot of driving-point
impedance as seen from a bus on which three tuned filters (5th, 7th, and 11th), a load, and the
system impedance representing the utility are connected in parallel.
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Figure 10-7—Impedance characteristic of multiple tuned filters
It can be seen that there are as many parallel resonance points as there are filters. The first
parallel resonant frequency (near the 3rd) is due to system and load impedance, the second
resonant frequency is due to the inductive part of the first filter (4.9th) and the capacitive part
of the second filter (7th). Similarly, a third resonant frequency occurs between the 7th and
11th. Note that if the filters are tuned at odd harmonics (5, 7, 11), the parallel resonance are
likely to occur in between, often midway depending upon the filter sizes.
10.5 System modeling
Harmonic analysis is required when a large number of nonlinear loads (typically greater than
25–30% of the total load on a bus or the system) are present or anticipated to be added. Often
PF correction capacitor banks are added without due consideration of resonances, and a study
may be required for corrective action. Frequent failure of power system components may also
justify the undertaking of harmonic studies. The response of the system to harmonics can be
studied by any of the following techniques:
a)
Hand calculations. Manual calculations are restrict to small-size networks since it is
not only very tedious, but quite susceptible to errors as well.
b)
Transient network analyzer (TNA). TNA studies are also restricted to rather small
network sizes because these studies are generally found to be expensive and time
consuming.
c)
Field measurements. Harmonic measurements are often used to determine the
individual harmonic and total harmonic distortion in the power system as part of the
verification of the design, or compliance with a standard, or simply to diagnose a
field problem (see 10.3, last paragraph). The measurements can be used effectively to
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validate and refine system modeling for digital simulations, particularly when noncharacteristic harmonics are present or a parallel resonance is encountered. Great
care should however be taken in using field data for harmonic current injections in
digital simulation, if it significantly differs from the calculated values or generally
acceptable per-unit values. Measurements of interharmonics require special consideration and instrumentation.
It should be recognized that undertaking harmonic measurements in a systematic
fashion can be expensive and time-consuming. Harmonic measurements reflect only
the system conditions that they have been taken at, and do not necessarily represent
the worst condition. Measurements can be in error due to inaccuracies of measuring
instruments or erroneous instrument utilization.
d)
Digital simulation. Digital computer simulation is the most convenient method, and
perhaps a more economical way of analyzing the system. The reason is that the
advent of computer technology has made available quite sophisticated computer programs featuring a large array of system component models to be used in a variety of
cases. Computer simulations are based on system-wide approaches utilizing the ideas
of system impedance and/or admittance matrices, in conjunction with elegant and
powerful numerical calculation techniques.
Chapter 4 of this book deals with system modeling as far as short-circuit, load flow, and transient stability studies are concerned. The short-circuit and load-flow data can also be utilized
for harmonic studies. However, the models require additional information to account for frequency dependence. This is because the behavior of the system equipment must be predicted
for frequencies well above the fundamental. This subclause summarizes system modeling for
harmonic analysis. The models are summarized in Table 10-1. Note that the models and the
constants assumed in the equations are just examples, and there can be many other models or
constants.
10.5.1 Generator model
Generators of relatively modern design produce no significant harmonic voltages; therefore,
they are not harmonic sources and can be represented by an impedance connected to ground.
A reactance derived from either subtransient or negative sequence reactances is often used
(both having similar values). Negative sequence impedance measurements for small units
agree to within 15% as compared with that of the subtransient impedance. In the absence of a
better model, and until more results are reported, a simple series RL circuit representing the
subtransient reactance with X/R ratio (at fundamental frequency) ranging between 15–50 can
be used. However, the generator resistance should be corrected at high frequencies due to
skin effect. The following equation is suggested:
B
R = R dc ( 1.0 + Ah )
(10-7)
where Rdc is the armature dc resistance and h is the harmonic order. Coefficients A and B have
typical values of 0.1 and 1.5, respectively.
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CHAPTER 10
Table 10-1—Power system component models
for harmonic analysis
System
components
Equivalent circuit model
Synchronous
machines
Model parameters
B
R = R dc ( 1+ Ah )
X "d + X "q
X = X " or X 2 = -----------------------2
Transformer
B
R T = R dc ( 1 + Ah )
RT and XT are transformer rated R and X
values
Ignored if not a significant harmonic source
Induction
machines
B
R = R dc ( 1 + Ah )
X "d + X "q
X = X " or X 2 = -----------------------2
NOTE—See 10.5.3 for a more accurate
model.
Load
Static load:
2
V
R = -----P
2
V
X = -----Q
Motor load:
2
V
R = -----P
278
2
V
X = -----Q
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HARMONIC ANALYSIS STUDIES
Table 10-1—Power system component models
for harmonic analysis (Continued)
System
components
Line and
cable
Equivalent circuit model
Short line and cable:
Long line (equivalent Pi):
Model parameters
f
M = 0.001585 -------R dc
f
Rdc
l
R
R
z
y
= frequency (Hz)
= dc resistance (Ω/m)
= length in meters
= Rdc (0.035 M 2 + 0.938), M < 2.4
= Rdc (0.35 M 2 + 0.3), M ≥ 2.4
= r + jxL (Ω/m)
= g + jbC (S/m)
ZC =
z
-- ; γ e =
y
zy
Z
= Z C sinh ( γ e l ) ,
Y
--2
γe l
1
= ------ tanh ----- ---
2
ZC
Shunt
capacitor
10.5.2 Transformer model
A transformer can be modeled as an ideal transformer in series with the nominal leakage
impedance. The leakage reactance varies linearly with frequency, but proper resistance modeling must account for skin effect. A similar expression to the one used for the generator
resistance can be used with similar values of coefficients A and B. Many variants for the
transformer leakage impedance are recommended by CIGRE [B3]. More complex models
suggest considering magnetizing reactance, core loss, and the interturn and interwinding
transformer capacitances. Since the transformer resonance starts to occur at relatively high
frequency, well above the 50th harmonic, capacitances usually are ignored. The magnetizing
branch together with the core losses are also neglected in most cases.
10.5.3 Induction motor model
The standard circuit model of the induction motor, illustrated in Figure 4-17 of Chapter 4,
remains valid in harmonic analysis. Here again, the model consists of a stator impedance, a
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magnetization branch with a core loss resistance, and a slip-dependent rotor impedance. In
harmonic studies, the stator and magnetization branch resistances and inductances are considered independent of frequency. However, considerations are given to the variation of rotor
impedance due to skin effect and the definition of an appropriate harmonic slip at frequencies
other than the fundamental.
The skin effect is an important factor in assessing the rotor impedance of machines with deep
bar rotors or double squirrel cages. At locked rotor, the frequency of the rotor current is high
(slip =1). Due to the skin effect, these rotor constructions provide a high rotor resistance and,
therefore, increased starting torque. At the nominal load, the frequency of the rotor current is
low and the skin effect is negligible. Then the rotor resistance reduces substantially and
allows for a more efficient operation. The same can be said for the rotor inductance, although
the magnitude of the variation is much less pronounced. For harmonic analysis, this variation
of rotor resistance and inductance as function of slip is usually modeled using linear equations. The proportionality coefficients linking the rotor resistance and inductance to the slip
are called cage factors. Expressions for rotor resistance and inductance are as follows:
R r0
R r ( h ) = -------- × ( 1 + CR × S h )
Sh
(10-8)
L r ( h ) = L r0 × ( 1 + CX × S h )
(10-9)
where
R r ( h ), L r ( h )
are the frequency dependent rotor resistance and inductance, respectively,
R r0, L r0
are the dc rotor resistance and inductance, respectively,
CR, CX
are the rotor cage factors for rotor resistance and inductance, respectively
(typical values are CR ≈ 2 and CX ≈ –0.01),
is the slip at the harmonic frequency.
Sh
This rotor model is used in situations where the motor operation forces the rotor currents to
span a wide frequency range. This is the case in harmonic analysis.
Each component 1h of the harmonic current flowing into the motor will each see an impedance whose value is dictated by the above equations at the appropriate slip. A first expression
for that value of the slip (Sh), is obtained from the basic definition of the slip. It states that the
slip is the difference between the stator (harmonic) frequency and the rotor electrical frequency divide by the stator frequency.
h × ω0 – ( 1 – s ) × ω0
ωh – ωr
h+s–1 h–1
- = -------------------- ≅ ------------ = ------------------------------------------------S h = ----------------h
h
h × ω0
ωh
(10-10)
where
h
280
is the harmonic order,
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HARMONIC ANALYSIS STUDIES
ω h, ω r , ω 0
s
IEEE
Std 399-1997
are the harmonic angular frequency, the rotor angular frequency, and the synchronous frequency, respectively,
is the conventional slip at the fundamental frequency.
We note that, at the higher harmonic orders, the harmonic slip approaches 1, and the resistance and inductance terms become constants. In practice, the harmonic slip can be considered 1 for harmonic orders greater than 9.
Balanced harmonic currents of order Nk + 1, Nk + 2, and Nk + 3 (k = 0, 1, 2 for N = 1, 2, 3…)
have been associated with the positive sequence, negative sequence, and zero sequence,
respectively. A negative sequence flux rotates in the opposite direction of the rotor, and the
frequency of the rotor flux is the sum of the rotor and stator frequencies. This is taken into
account in the expression for harmonic slip by replacing the minus sign as follows:
h±1
S h ≅ -----------h
(10-11)
where
“–” is applied to positive sequence harmonics,
“+” is applied to negative sequence harmonics,
Sh = 1 for zero sequence harmonics.
The magnitude of the harmonic slip approaches unity at higher frequencies, and the motor
inductance can be approximated by its locked rotor or subtransient value.
10.5.4 Load model
Various models have been proposed to represent individual loads and aggregate loads in harmonic studies. Specific models are available for individual loads, whether they be passive,
rotating, solid-state, etc. An aggregate load is usually represented as a parallel/series combination of resistances and inductances, estimated from the fundamental frequency load power.
This model can be used to represent an aggregate of passive or motor loads. The model resistance and inductances are considered constant over the frequency range.
10.5.5 Transmission line and cable models
A short line or cable can be represented by a series RL circuit representing the line series
resistance and reactance. The resistance must be corrected to take into account the skin effect
for higher frequencies. For longer lines, modeling of the line shunt capacitance becomes
necessary. Both the lumped parameter model (equivalent pi model for example) and the distributed parameter model are used, but the latter is better suited to represent the line’s
response over a wide frequency range. The distributed line model can be approximated by
cascading several lumped parameter models. Cascading sections of either model to represent
a long line is worthwhile to produce a harmonic voltage profile along the line.
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The variation of line resistance due to skin effect can be evaluated using the following
expression:
2
R = R dc ( 0.35X + 0.938 ), X < 2.4
(10-12a)
R = R dc ( 0.35X + 0.3 ), X ≥ 2.4
(10-12b)
f 0.5
where X = 0.001585 -------
R dc
(10-13)
and f is frequency in Hz, and Rdc is in Ω/mi.
10.5.6 Filter models
Filters, by definition, exhibit small impedances at tuned frequencies. At the fundamental frequency, their impedance is capacitive, thereby supplying reactive power to the electrical network. Many types of filters are applied in power systems for different purposes. Filters most
commonly used for harmonic mitigation are illustrated in Figure 10-8 along with their characteristics. A single-tuned filter is used to suppress a specific harmonic at or near the tuned
frequency. High-pass filters can be of first, second, or third order. The second-order filter is
often used to suppress higher frequencies. A more recent type of high-pass filter, called
C-type filter, is becoming popular due to its smaller losses at the fundamental frequency.
Application of filters is one of the commonly employed solutions to limit the effects of
harmonics. Other remedial measures such as moving the disturbing loads to higher voltage
levels, reinforcing the system, changing capacitor sizes, and adding tuning reactors to capacitor banks are also used. In any case, economics will dictate the most appropriate solution.
Recent studies advocate the utilization of active filtering in an effort to counter the injected
harmonics close to the source, but primarily in low-voltage systems.
10.5.7 Network modeling and computer-based solution techniques
Although several methodologies have been tried successfully, the most common methodology of harmonic analysis is the current injection method. In this method, nonlinear loads are
modeled as ideal harmonic current sources. Each network element is represented by a set of
linear equations corresponding to its previously described circuit. Using Ohm’s law and
Kirchoff’s laws, all the network elements and loads are connected according to the network
topology. Mathematically this operation generates an equation at each bus of the network. At
bus i, connected to a set of buses j, we have the following:
∑ Y ij
j
282
× Vi = I i + ∑ ( Y ij × V j )
(10-14)
j
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HARMONIC ANALYSIS STUDIES
(a) Single-tuned filter
(b) High-pass filter (second order)
(c) Undamped high-pass filter (third order)
Figure 10-8—Filters commonly used for harmonic mitigation
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(d) High-pass C-type filter (third order)
Figure 10-8—Filters commonly used for harmonic mitigation (Continued)
In matrix form, the set of n equations, denoted compactly as YV = I, takes on the form
Y 11 – Y 12 … – Y 1n
V1
– Y 12 Y 22 … – Y 2n
V2
…
– Y 1n – Y 2n … Y nn
…
Vn
I1
=
I2
…
In
(10-15)
The diagonal elements Yii of the matrix are the sums of the admittances connected to the bus.
The off-diagonal element Yij is equal to (–1) times the admittance connected between buses i
and j. The above equation may be expressed in words as follows:
{[The sum of the admittances connected to bus i] multiplied by the voltage Vi} is equal to the
sum of the [(admittances ij taken one by one) multiplied by the corresponding voltages Vj at
the adjacent buses] plus the current Ii injected at bus i.
By solving this system of equations, we obtain the nodal voltages. This computation is performed for each harmonic frequency of interest. From the harmonic voltages we can compute
the harmonic currents in each branch:
I ij = ( V i – V j ) × Y ij
(10-16)
The bulk of the work is in forming the network equations. Excellent linear equation solvers
are readily available.
The inverse of the nodal admittance matrix is called the nodal impedance matrix. This matrix
is rich in quantitative information. The diagonal entry on the ith row is the Thevenin impedance of the network seen from bus i. By computing values of this matrix over a range of
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frequencies, we obtain the frequency response of the network seen from each bus. Exact resonance frequencies can be determined from this computation. The off-diagonal values in the
matrix show the effect of a harmonic current injection on the bus voltages. Consider a single
harmonic source connected at bus i forcing 1.0 A of current into the network. The harmonic
voltage at bus j is simply Zij, the value found in the i th row and j th column of the nodal
impedance matrix. The harmonic voltage at bus j due to numerous sources can be solved by
superposition.
The total harmonic distortion, the rms value, the telephone interference factor, and related
factors (VT and IT) are readily computed from the harmonic voltage or current (Un) and the
fundamental frequency (U1) quantities as follows:
∞
∑ Vn
2
n=2
Total harmonic distortion ( THD ) = 100 ------------------V1
(10-17)
where n is the harmonic order and usually the summation is made up to the 25th or the 50th
harmonic order.
∞
rms value:
U rms =
∑ Vn
2
(10-18)
n=1
∞
VT or IT :
UT =
∑ ( K f × Pf
× Vf )
2
(10-19)
f =0
where U designates either voltage or current.
∞
∑ (K f × P f × V f )
Telephone interference factor:
2
f =0
TIF = ----------------------------------------------------V
(10-20)
where V is the voltage and Kf and Pf are the weighting factors related to hearing sensitivity
(IEEE Std 519-1992 and IEEE Std 597-1983). These useful quantities summarize the harmonic analysis into a few quality-related factors.
In harmonic analysis, two impedance calculations are made to study the system characteristics for series and parallel resonances. These are driving point and transfer impedances. The
driving point impedance is defined as voltage, calculated at a node i, due to current injected at
the same node, in other words:
V
z ii = -----i
Ii
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Since this is the “net” impedance of all circuits seen from that bus, it provides a useful information regarding resonances. By changing the existing circuits (location of capacitors,
cables, etc.) or the design of planned filters, the driving point impedance, and hence resonance, can be changed. The concept of transfer impedance is similar to the driving point
impedance in that it is defined as the voltage measured at one bus due to current injected at
another bus, in other words:
V
z ij = -----i
Ij
(10-22)
where
zij
Vi
Ij
is the transfer impedance to bus i,
is the voltage measured at bus i,
is the current injected at bus j.
The transfer impedance is useful when any location other than the bus where the current is
injected is to be evaluated for harmonic voltages.
10.5.8 Harmonic generation
A harmonic flow study requires the knowledge of harmonics generated by nonlinear devices.
Depending upon the nature of the device and accuracy required, this can be a task by itself.
Very often, typical values are used for most industrial studies. This subject has been dealt
with well in IEEE Std 519-1992 and also in Chapters 3 and 4 of Prabhakara, Smith, and Stratford [B30], therefore a detailed treatment is unnecessary here.
A table is provided here, however, for a 6-pulse and multipulse converter as a matter of
comparison, and to demonstrate how certain characteristic harmonics can be canceled by
phase multiplication. Harmonics generated by a 6-pulse converter for a square wave of 120°
duration are well-known (IEEE Std 519-1992). The magnitude of the harmonic current is
given by Ih = I1 / h, where I1 is the fundament current and the harmonic order h is given by
h = kq ± 1, where k is an integer and q is the number of pulse.
Table 10-2 indicates that for a 12-pulse converter the 5th harmonic and 7th harmonic are
canceled in the ideal case. However, due to system and equipment imbalances (non-ideal
behavior), a perfect cancellation does not occur, and in general the current is assumed to be
10–15% of what would be expected (Prabhakara, Smith, and Stratford [B30], Chapter 11),
e.g., for a 12-pulse converter, the 5th harmonic would be around 2% instead of 20%.
Figure 10-9 (a) and 10-9 (b) show the conceptual arrangement of 12-pulse and 24-pulse converters using 3-phase, 2-winding transformers. In Figure 10-9 (a) the two rectifier transformers are individually phase-shifted 30° with respect to each other. When viewed from bus A
and when they are both equally loaded, they collectively appear to be a 6-phase, 12-pulse
system. Similarly in Figure 10-9 (b), buses C and D appear to have 6-phase, 12-pulse rectification but, due to differing connections of the power transformer, the system becomes a
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Table 10-2—Characteristic ac line harmonic currents in multipulse systems
Rectifier system pulse number
Harmonic
6
12
18
Harmonic
frequency
24
Harmonic current in
percent of fundamental
Theoretical
Typical
5
X
300
20.00
19.20
7
X
420
14.20
13.20
11
X
X
660
9.09
7.30
13
X
X
780
7.69
5.70
17
X
X
1020
5.88
3.50
19
X
X
1140
5.26
2.70
23
X
X
X
1380
4.36
2.00
25
X
X
X
1500
4.00
1.60
29
X
1740
3.45
1.40
31
X
1860
3.23
1.20
35
X
X
2100
2.86
1.10
37
X
X
2220
2.70
1.00
NOTE—The theoretical values are given for a 6-pulse converter with ideal characteristics (i.e.,
square current waves with 120° conduction). The last column gives typical values based on a commutating impedance of 0.12 pu and a firing angle of 30° and infinite dc reactor (IEEE Std 519-1992,
Table 13.1). These values are on the basis of one 6-pulse converter or all converters, assuming that
the harmonics are additive. Since some harmonics will be canceled, but not entirely, a small percentage value may be assumed, as explained earlier in this subclause. Note that if the dc reactor is not
large, some of the harmonics can be greater than typical (or theoretical) and some smaller.
12-phase, 24-pulse system viewed from bus B. This configuration will greatly reduce all
generated harmonics below the 23rd, compared to 6-pulse systems.
10.5.9 Harmonic analysis for industrial and commercial systems
The purpose of a harmonic study was discussed in 10.3. The following summarizes the steps
normally required for a harmonic study in the industrial environment:
a)
Prepare a system one-line diagram. Note that it is important to include capacitor
banks and long lines and cables within the industrial system or the utility system near
the point of common coupling (PCC).
b)
Gather equipment data and ratings (see 10.5.9).
c)
Obtain the locations of nonlinear loads and the generated harmonic currents.
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(a) 12-pulse converter arrangement
(b) 24-pulse converter arrangement
Figure 10-9—Multipulse converter arrangements
d)
288
Obtain from the utility company the relevant data and harmonic requirements at the
PCC. These should include the following:
1)
Minimum and maximum fault levels or preferably system impedances as a function of frequency for different system conditions.
2)
Permissible limits on harmonics including distortion factors and IT factor. The
criteria and limits vary considerably from country to country. Typical values for
different system voltages are given in IEEE Std 519-1992.
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e)
Carry out harmonic analysis for the base system configuration by calculating the
driving point impedance loci at the harmonic source buses as well as at all shunt
capacitor locations.
f)
Compute individual and total harmonic voltage and current distortion factors and IT
values (if required) at the point of common coupling.
g)
Examine the results and, eventually, go back to step a) or step c), depending on
whether the network data or only the parameters of the analysis need to be modified.
h)
Compare the composite (fundamental plus harmonic) loading requirements of shunt
capacitor banks with the maximum rating permitted by the standards. IEEE Std 18-1992
has defined the following operating limits:
—
—
—
—
Continuous operating voltage ≤110% of the rated voltage
rms crest voltage ≤1.2 times the rated rms voltage
kvar ≤135% of the rated kvar
Current ≤180% of the rated rms currents
i)
Relocate the capacitors or change the bank ratings if they are found to exceed their
ratings. Apply a detuning reactor if a resonance condition is found. Go back to step
e). Note that adding a tuning reactor will increase the fundamental voltage on the
capacitor and may also increase harmonic voltage. The capacitor duty must be satisfied as given in h).
j)
Add filters if the harmonic distortion factors and IT values at the PCC exceed the
limit imposed by the utility.
The above steps should be carried out for the base system configuration as well as for system
topologies resulting from likely contingencies. Any future system expansion and utility shortcircuit level changes should also be considered.
10.5.10 Data for analysis
The following data are required for a typical study:
a)
A single-line diagram of the power system to be studied.
b)
The short-circuit capacity and X/R ratio of the utility power supply system. The existing harmonic voltage spectrum of the utility system at the PCC (external to the
system being modeled).
c)
Subtransient reactance and kVA of all rotating machines. If limitations exist, all
machines on a given bus can be lumped together into one composite equivalent
machine.
d)
Reactance and resistance of all lines, cables, bus work, current limiting reactors, and
the rated voltage of the circuit in which the circuit element is located. The units can
be either in per-unit or percent values, or ohmic values, depending on the software
used or preference.
e)
The three-phase connections, percent impedance, and kVA of all power transformers.
f)
The three-phase connections, kvar, and unit kV ratings of all shunt capacitors and
shunt reactors.
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g)
Nameplate ratings, number of phases, pulses, and converter connections, whether
they are diodes or thyristors, and, if thyristors, the maximum phase delay angle, per
unit loading, and loading cycle of each converter unit connected to the system. Actual
manufacturer’s test sheets on each converter transformer are also helpful but not
absolutely mandatory. If this information is not readily available, the kVA rating of
the converter transformer may be used for establishing the harmonic current spectrum being injected into the system.
h)
Specific system configurations.
i)
Maximum expected voltage for the system supplying the nonlinear loads.
j)
For arc furnace installations, secondary lead impedance from the transformer to the
electrodes plus a loading cycle to include arc megawatts, secondary voltages, secondary current furnace transformer taps, and transformer connections.
k)
Utility-imposed harmonic limits at the PCC, otherwise limits, as specified in applicable standards, may be used.
10.6 Example solutions
Several applications of harmonic studies are presented in this subclause. Frequency scans,
capacitor effects, and filter design are demonstrated. Each example uses a variation of the
simple system provided in 10.6.1.
10.6.1 Test system single-line diagram and data
The diagram in Figure 10-10 is used for all examples in 10.6. This system is representative of
an industrial power system and includes multiple voltage levels and power factor correction
capacitors. The data required for basic harmonic studies are provided in Tables 10-3, 10-4,
and 10-5.
The capacitor banks have been sized to provide a power factor of 0.95 lagging at the lowvoltage side of each transformer and may consist of series and parallel units. As previously
mentioned, the modeling of motor load for harmonic studies is not, in most cases, significantly different from static load; therefore, the total load given in Table 10-5 for each bus
includes both types.
10.6.2 Case Study 1: Diode rectifier on 33 kV bus
The impacts of a proposed adjustable-speed motor drive installed at the 33 kV bus are to be
determined. The switches connecting the 33–6.6 kV transformer are open in this case study;
therefore, the 6.6 kV bus is not considered for this case (refer to 10.6.3 for Case Study 2,
where the switch is considered closed). The harmonic source, shown in Figure 10-10 as a
current source, is a standard diode rectifier supplying 25 MW on the dc side. A frequency
scan is performed at the 33 kV bus, and the resultant driving point impedance is shown in
Figure 10-11. Note the distinct impedance peak (indicating a resonant point) near the 8th
harmonic (480 Hz).
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Key:
M—motor load
SL—static load
C—static capacitors
HF—harmonic filters
Figure 10-10—Example system for harmonic studies
Table 10-3—Utility supply data
Parameter
Value
Supply voltage
220 kV
Short-circuit capacity
4000–10 000 MVA
X/R
20.0
Table 10-4—Transformer data
Parameter
T1
T2
Power rating (MVA)
100
30
Voltage rating (kV)
220–33
33–6.6
14
10
10.0
10.0
Impedance (%)
X/R
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Table 10-5—Load and capacitor data
Linear load
33 kV bus
Converter
25 MW
Capacitor
8.4 Mvar
Linear load
6.6 kV bus
25 MVA @ 0.8 lag
15 MVA @ 0.8 lag
Converter
15 MW
Capacitor
5 Mvar
Figure 10-11—Driving point impedance at 33 kV bus
The impacts of this resonance condition are determined using multiple solutions (one solution at each frequency of interest) of the set of nodal equations for the system. The harmonic
content of the diode rectifier current (on the ac side) is given in Table 10-6. Figures 10-12 (a)
and 10-12 (b) and 10-13 (a) and 10-13 (b) show the voltage waveforms and harmonic magnitude spectra (including approximate voltage THDs) at the 33 kV and 220 kV buses, respectively. Note that these are computed waveforms.
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Table 10-6—Harmonic content of diode rectifier current (ac side)
Harmonic number
Frequency (Hz)
Magnitude (A)
Phase (degrees)
1
60
618
0
5
300
124
180
7
420
88
0
11
660
56
180
13
780
47
0
17
1020
36
180
19
1140
33
0
10.6.3 Case Study 2: Effects of including the 6.6 kV bus
This study is identical to that of 10.6.2 except that the switches to the 33–6.6 kV transformer
are closed resulting in the connection of the 6.6 kV bus to the system. A frequency scan is
again conducted at the 33 kV bus, and the resultant driving point impedance is shown in Figure 10-14. Note the presence of two resonance points. This is to be expected because there
are typically the same number of resonance points as number of capacitors.
The impacts of the resonance points on voltage waveforms are shown in Table 10-7 where the
diode rectifier load on the 33 kV bus is given in 10.6.2. Harmonic content and approximate
voltage THD values are given for the 6.6, 33, and 220 kV buses. Relative to the case in
10.6.2, the distortion is worse at the 33 kV bus but not as bad at the 220 kV bus.
10.6.4 Case Study 3: Motor drive on the 6.6 kV bus
The impacts of a proposed motor drive installed on the 6.6 kV bus are to be determined. The
33–6.6 kV transformer switches are closed providing the connection to the supply voltage. A
diode rectifier delivering 15 MW (with the same harmonic current content as given previously for the 33 kV rectifier) on the dc side represents the drive. The rectifier load used in
10.6.2 and 10.6.3 at the 33 kV bus is not present in this study.
The results of a frequency scan conducted at the 6.6 kV bus are shown in Figure 10-15. Note
the multiple resonant frequencies near the 6th (360 Hz) and 12th (720 Hz) harmonics. The
impacts of the resonance points on voltage waveforms are shown in Table 10-8.
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(a) Voltage waveform
(b) Harmonic magnitude spectrum
Figure 10-12—Line-to-neutral voltage waveform at 33 kV bus
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(a) Voltage waveform
126265
(b) Harmonic magnitude spectrum
Figure 10-13—Line-to-neutral voltage waveform at 220 kV bus
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Figure 10-14—Driving point impedance at 33 kV bus
Table 10-7—Harmonic content of bus voltages
including the effects of the 6.6 kV bus
Harmonic
number
Frequency (Hz)
33 kV bus
6.6 kV bus
1
60
126 090
18 108
3559
5
300
1 125
1 114
329
7
420
606
600
278
11
660
945
936
166
13
780
818
810
85
17
1020
303
300
15
19
1140
216
214
8
THD (%)
296
220 kV bus
= 1.4480
= 9.9890
= 13.188000
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HARMONIC ANALYSIS STUDIES
Figure 10-15—Driving point impedance at 6.6 kV bus
Table 10-8—Harmonic content of bus voltages:
Motor drive at 6.6 kV bus
Harmonic
number
Frequency (Hz)
220 kV bus
33 kV bus
6.6 kV bus
1
60
126 296
18 307
3 476
5
300
956
947
461
7
420
807
799
294
11
660
482
477
101
13
780
248
246
100
17
1020
45
45
47
19
1140
25
24
35
= 1.081
= 7.383
THD (%)
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= 16.339000
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10.6.5 Case Study 4: Evaluation of harmonic limits
The analysis of the supply voltage and current assuming that the nonlinear loads at the 33 and
6.6 kV buses are both operating is presented in this case study. This type of study is often
required to demonstrate compliance with harmonics limits imposed by the utility company
serving the facility.
Figures 10-16 and 10-17 show the input current and voltage waveforms at the PCC where harmonic limits are to be satisfied. The frequency contents of the waveforms are given in Table
10-9, including approximate THD values. For compliance evaluation purposes, an average
(typically over a one-year period) maximum demand of 80 MVA is assumed, and this figure is
used to express the existing current harmonics as percentages of the maximum demand current
as required in IEEE Std 519-1992. Based on this assumption, the recommended limits from
IEEE Std 519-1992 are also provided in Table 10-9 and clearly indicate violations of harmonic
limits. For line currents, the concept of total demand distortion (TDD) is used in place of THD
to account for the use of the percentages in terms of average maximum demand instead of in
terms of the fundamental component. All voltage harmonics are expressed in percentages of
the fundamental component and THD is used. Even though the voltage at the PCC is near
compliance, significant voltage distortion is present inside the plant at the 33 and 6.6 kV
buses.
Figure 10-16—Line current at PCC
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Figure 10-17—Line-to-neutral voltage at PCC
Table 10-9—Harmonic limit evaluation at PCC
Harmonic
number
Actual harmonic
current
magnitudes
(percent of
average maximum
demand current)
Limits for
harmonic current
magnitudes
(percent of
average maximum
demand current)
Actual
harmonic
voltage
magnitudes
(kVLN, rms)
Limits for
harmonic
voltage
magnitudes
(percent of
fundamental)
1
N/A
N/A
125.8
N/A
5
15.0
3.00
1.90
1.0
7
5.8
3.00
1.03
1.0
11
1.7
1.50
0.48
1.0
13
1.8
1.50
0.60
1.0
17
0.6
1.15
0.27
1.0
19
0.4
1.15
0.20
1.0
THD = 16.3%
TDD = 3.75%
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THD = 1.85%
THD = 1.5%
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It should be noted that different countries specify different harmonic limits at different points
in the system. The use of methods and limits set forth in IEEE Std 519-1992 are used here for
consistency. If required, other governing limits and methods should be used in place of those
given here.
10.6.6 Representation of the utility system
It is important to consider the effects of variations of the utility supply fault MVA on the frequency response of the industrial system. For this study, the 33–6.6 kV transformer and therefore the entire 6.6 kV bus is disconnected. Figure 10-18 shows a portion of the frequency
scan results for the example system. The utility fault MVA is varied from its minimum
(4000 MVA) to its maximum (10000 MVA) with the system X/R (@ 60 Hz) held constant at
20.0. The plots show the general trend of an increase in resonant frequency as the utility fault
MVA increases. The conclusion is that industrial systems connected to very strong utility
supplies (high fault MVA) are less likely to encounter problematic resonance conditions at
low frequencies.
Figure 10-18—Frequency response at 33 kV bus
as a function of utility fault MVA
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10.6.7 Effects of size of power factor correction capacitors
Power factor correction capacitors are applied in most systems to help achieve lower cost
operation. It is important to consider variations in the frequency response due to the size of
power factor capacitors. For this study, the 33–6.6 kV transformer and the 6.6 kV bus are disconnected. The utility fault MVA is constant at 4000 MVA, X/R = 20.0 (@ 60 Hz). Figure
10-19 shows the variations in the driving point impedance at the 33 kV bus as a function of
capacitor bank size.
Figure 10-19—Variations in frequency response at 33 kV bus
as a function of capacitor bank size
The plots reveal, first, that resonant frequency decreases as capacitor bank size (and therefore
power factor) increases, and second, that peak resonant impedance increases as capacitor
bank size (and therefore power factor) decreases. It should be noted that the tendency of
resistive damping to increase with frequency often lessens the effect of increasing resonant
impedance.
10.6.8 Single-tuned filter application
The distortion in the waveform of Figure 10-13 (a) for the study conditions of 10.6.2 can be
eliminated using one (or more) RLC filters tuned to provide a low impedance path to ground
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at the frequencies of interest. As an example of the effects of filter application, a filter is created using a tuning reactor in series with the power factor correction capacitor bank in the
study of 10.6.2. From the frequency scan results shown in Figure 10-12, it appears that the
resonant point is near the 8th harmonic (480 Hz). The filter could be tuned to eliminate this
resonance, but this action would likely produce a new resonance point at a lower frequency
that coincides more exactly with the rectifier harmonics. For this reason, filter tuning frequencies should be selected based on removing specific harmonic currents (before they can excite
resonant modes) instead of selected the tuning frequency specifically to modify the frequency
response. However, because the application of single-tuned filters inherently produces a new,
lower-frequency resonant point, it is important to apply the filter at the lowest current harmonic frequency. For the rectifier load, the lowest harmonic frequency is 300 Hz (5th harmonic). Ideally, the filter should be tuned to this frequency, but variations in system
parameters (due to utility switching, etc.) are often significant enough to slightly shift the resonant frequency (note Figure 10-18 in 10.6.6). To counter this effect, single-tuned filters are
often constructed using a target frequency that is 3–5% below the frequency of the harmonic
current that is to be removed.
In most applications, a single-tuned filter is created using the existing power factor correction
capacitors. At power frequencies (50–60 Hz), the series RLC combination appears capacitive
and supplies reactive power to the system. At the tuned frequency, the series impedance of the
filter is very low and provides a low impedance path to ground for specific harmonic currents.
For Case Study 1 in 10.6.2, the capacitor bank size is 8.4 Mvar @ 33 kV. Considering a perphase approach, the appropriate tuning reactor can be calculated using standard formulas. A
5.87 Ω reactor (X/R = 14.3 @ 60 Hz) is chosen such that the filter is tuned to 282 Hz (“4.7th”
harmonic). The frequency response of this single-tuned filter using the specified resistance,
inductance, and capacitance is shown in Figure 10-20. Note the very low impedance just
below 300 Hz (5th harmonic).
The driving point impedance and the voltage waveform at the 33 kV bus with the filter in
place are shown in Figures 10-21 (a) and 10-21 (b), respectively. The THD for the voltage
waveform has been reduced from 12.23% for the study case of 10.6.2 to 6.32%. However,
a second single-tuned filter (near the 7th harmonic) would be required at the 33 kV bus if it
was necessary to meet more stringent voltage distortion limits at this location. In many
applications, multiple single-tuned filters (tuned near the frequencies of harmonic currents
generated by the loads) are used in conjunction with a high-pass filter to satisfy realistic
voltage distortion limits.
10.7 Remedial measures
There are several approaches to remedy the harmonic problem in a system. The following is a
general discussion of solutions that are available. An exact solution will depend on whether
the system is an existing one or a new system is planned, whether the system is flexible to
changes, and whether harmonic filters can be added or existing capacitor banks can be
modified.
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Figure 10-20—Frequency response of single-tuned filter
a)
In designing or expanding a system, care should be taken that the total harmonic load is
kept at a relatively low percentage of the total plant load (e.g., 30% would be a good
maximum target). If the measured or calculated distortion levels are high, consideration
should be given to location of harmonic loads, number of buses, size of the transformers, choice of transformer connections, etc., besides the addition of harmonic filters.
b)
Figure 10-22 shows how the harmonic loads may be separated so that the sensitive
loads are not influenced by loads with high harmonics. Note that all heavy loads (in the
order of several MVA) should have their own dedicated transformers, e.g., in the case of
large drives or arc furnaces in a steel mill. If there are several similar loads, their respective transformers feeding them could be connected in delta and wye alternately to provide some cancellation of certain characteristic harmonics. If the secondary buses are
connected together via a tie breaker, precautions should be taken to ensure the harmonic source and sensitive loads are not simultaneously energized during this event.
c)
Multipulsing is a very effective method of reducing harmonics; however, this may not
always be possible because of the high cost of transformers. In several industries, rectifiers are often connected with 6-pulse converters to form a 12-pulse, 24-pulse, or
higher pulse system, see Figure 10-9. Table 10-10 gives the phase shift needed
between bridges to form a multiple system. Note that the harmonic cancellation
between bridges is not complete, and a residual harmonic may still be present (see
10.5.8).
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(a) Driving point impedance
(b) Line-to-neutral voltage
Figure 10-21—33 kV bus impedance and voltage characteristic
with a single-tuned filter
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Figure 10-22—Separation of harmonic loads
Table 10-10—Transformer phase shift for various multipulse system
Pulse number
6
12
18
24
30
36
42
48
Number of units
1
2
3
4
5
6
7
8
Phase shift in degrees
0
30
20
15
12
10
8.57
7.5
Lowest harmonic order
5
11
17
23
29
35
41
47
d)
The most common remedial measure for harmonic mitigation is to provide selected
harmonic filters tuned to appropriate frequencies. Either existing capacitor banks
could be modified to tuned filters by adding tuning reactors, provided the capacitors
are adequately rated, or new filter banks could be added. This is discussed in detail in
10.7.1.
10.7.1 Filter selection
Filter selection can be best described as an art rather than a science because there is no single
solution. Usually, there are many possible solutions, and the system designer must select the
one that best meets the specifications and also make compromises where discretion is
required. In industrial systems, multiple single-tuned filters are the most common solutions.
However, there may be other types of filters as discussed in 10.5.6. The discussion herein is
limited to the procedure of selection rather than the merits and demerits of individual filters.
The Working Group on Harmonic Filters under the Capacitors Subcommittee of the Power
Engineering Society of IEEE is developing a draft entitled “Guide for Application and Specification of Harmonic Filters.” This draft, not yet in circulation, will be designated as IEEE
P1531. It is recommended for futher guidance on filters.
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The procedure for a harmonic study was discussed in 10.5.9. The study often begins with a
preliminary filter design, generally based on past experience, and then refined, as the harmonic performance indices (see 10.5.7 and 10.8) are calculated. In this process of refinement,
several steps are involved that determine the number of filters, effective reactive power compensation, and performance indices. Other considerations also enter, such as filter switching
and protection, loss of one or more filter banks, and space requirements. One rather obvious
design objective is to use as few filters as possible and compare the performance with no filter. Filter location is another consideration. Effective filtering will probably require that the
filters be located near harmonic sources. However, economics may dictate that filters be
located at higher voltage levels (near the PCC or main bus) to meet the demands of all harmonic sources.
Generally, the filters are tuned at one of the dominant odd characteristic harmonics starting
from the lowest order (which in most cases are 5, 7, 11, …). In some cases the lowest order
may be 2 or 3 as in arc furnace applications. Ideally, the filters should be tuned to the exact
harmonic order, i.e., 5, 7, 11, etc. however, the practical considerations may require that it is
safer to tune below the nominal frequency. If it is necessary to offset the parallel resonance
frequency, the filter may be intentionally tuned below or in exceptional cases above the nominal frequency. For example, a 5th harmonic filter may cause a resonance near the 3rd, and it
would be desirable to tune it slightly below or above the 5th to offset the resonance at 3rd.
Another example is one in which the resonant frequency is very close to the 5th (say at 4.7)
for very sharp filters, in which case it will be desirable to tune it below the 5th so that tolerances and temperature deviations, etc., will not let the resonant frequency coincide with the
5th harmonic injection frequency.
The two main components of passive filters, capacitors and reactors, are discussed here. The
nominal fundamental kvar rating of the capacitors determines the effectiveness of harmonic
filtering. Therefore an initial estimate of the capacitor kvar is very important. The larger the
bank size, the easier it is to meet a given harmonic performance criteria. Besides the harmonic requirements, the following additional design factors may need to be considered:
a)
The system power factor (displacement power factor) may be corrected to required or
desirable value (usually above 0.9),
b)
The total kVA demand on the supply transformer may have to be reduced if the transformer is overloaded,
c)
Similarly, the current ratings of buses and cables may have to be reduced.
As a general rule, the capacitor needs to be derated in order to absorb the additional duty
from the harmonics (a derating factor of 15–20% in voltage would be desirable). Note that
because of the derating, the kvar would be reduced by the square of the factor. The loss of
kvars is, however, somewhat compensated by the cancellation of the capacitive reactance by
the inductive reactance of the filter. The effect of this cancellation is to increase the capacitor
voltage (above the bus voltage) by the following factor:
2
h
c = ------------2
h –1
306
pu of the fundamental, where h is the harmonic order
(10-23)
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HARMONIC ANALYSIS STUDIES
For the conventional single-tuned filter this factor is calculated in Table 10-11.
Table 10-11—Fundamental voltage across a single-tuned filter capacitor
Harmonic order
3rd
5th
7th
11th
13th
Per-unit voltage
1.125
1.049
1.021
1.008
1.005
Once the harmonic study is completed and the filter selection has been made, the capacitor
rating with respect to voltage, current, and kvar should be checked. All these three ratings
need to be satisfied independently according to IEEE Std 18-1992 as noted earlier in 10.5.8.
The filter designer could either satisfy these requirements himself if standard units are used,
or request the capacitor supplier to meet these requirements if special units are used.
For industrial applications, either an air-core or iron-core reactor may be used depending
upon the size and cost of the reactor. In general the iron-core reactors are limited to 13.8 kV.
Air-core reactors are available for the complete range of low-voltage, medium-voltage, and
high-voltage applications. The iron-cored reactors should save space and provide the advantage of being enclosed in indoor or outdoor housing along with capacitors and other control
and protection components as required.
The reactors need to be rated for the maximum fundamental current and the “worst” generated harmonics for the “worst” system configuration. Also, the reactor vendor needs to calculate all the losses, fundamental and harmonic, core losses in case of the iron-cored reactors,
and stray losses due to frequency effects so that the hot-spot temperatures are within the
allowable dielectric temperature.
One big unknown factor during the filter design is the Q factor or the ratio of the inductive
reactance to resistance at the tuned frequency. This is usually estimated based on experience.
However, if during the study it is felt that Q is not critical, then the reactor should be specified
to have the “natural Q” (the Q of the reactor that is naturally obtained with no special design
consideration or cost). On the other hand, if the low-Q reactor would help to mitigate amplifications near parallel resonance frequencies then a low-Q reactor should be specified. Manufacturers have a high tolerance on Q (as much as (20%), and this should be recognized. Note
also that the low Q design will produce higher losses.
10.8 Harmonic standards
In the United States, the Industry Application Society (IAS) of IEEE began a standards
development project on harmonics in 1973. The first publication resulting from this project
was IEEE Std 519-1981, entitled “IEEE Guide to Harmonic Control in Electrical Power
Systems.” The IEEE publishes a hierarchy of Standards from the least to the most
prescriptive, which are referred to as Guides, Recommended Practices, and Standards. In
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1986, the Power Engineering Society (PES) joined the IAS to upgrade IEEE Std 519-1981 to
the status of a Recommended Practice, and in 1992, IEEE Std 519-1992, entitled “IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems,”
was published. Since then this document has acquired nearly the status of a Standard. It is
widely used by the utilities and by industrial, commercial, and residential users in North and
South America and around the world. It has become the basis for all new power system
designs and for the interface between the utilities and their customers.
The key prescriptions of IEEE Std 519-1992 are provided in Chapter 10 (Recommended
Practices for Individual Consumers) and in Chapter 11 (Recommended Practices for
Utilities). These two chapters address the harmonic current distortion and harmonic voltage
distortion, respectively, and with maintaining the power quality by both the supplier and the
user. The document also provides limits for notching and IT values for converter applications.
IEEE Std 519-1992 primarily deals with odd harmonics; even harmonics are limited as a
percentage of adjacent odd harmonics. It does not deal with the continuous spectrum of
harmonics or interharmonics that may fall between the odd and even harmonics. Often, interharmonics are treated by engineers in the same manner as even harmonics. However, the
subject of non-integer harmonics is being pursued by two task forces of the Working Group
on Harmonics of the Power Engineering Society (Task Force on Application of the IEEE
Standard 519 and Task Force on Interharmonics). It is expected that the publications of these
task forces will highlight the application issues and interpretations of IEEE Std 519-1992 for
different systems and users.
IEEE Std 519-1992 emphasizes the two following points in applying the harmonic indices:
The point of interface or PCC between the supplier and the user, and the ratio of the system
short-circuit (SC) MVA and the maximum demand load MVA. The PCC is a point mutually
agreeable to the utility and the consumer and, in general, can be considered as the point of
metering or any other point of interface. Within an industrial plant, the PCC is the point
between the nonlinear load and other loads (IEEE Std 519-1992, 10.1). The SC ratio determines the total harmonic current distortion that can be injected into the system and allows
higher limits for higher ratios. For current distortion limits, the fundamental current is calculated from the maximum demand load current, calculated over any 15 or 30 min period and
then averaged over the preceding 12-month period (if the data are available). Note that the
actual fundamental current at any particular time is likely to be less than the maximum
demand fundamental current, so the latter helps to reduce the TDD percentage for any load
less than the maximum demand load.
In conclusion, IEEE Std 519-1992 is a Recommended Practice and not a Standard. The limits
should indeed be a matter of mutual agreement between the supplier and the utility. Further,
strict compliance with IEEE Std 519-1992, within the industrial facility (as compared to the
PCC) can require expenditure that may not be justifiable either technically or economically,
and the effects of harmonics should be evaluated. It is expected that the future revision of
IEEE Std 519-1992 will address the issue of limits for even harmonics and interharmonics
more clearly. Also, it is expected that the future revision will clarify the application and interpretation of the document with several real-life examples.
308
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HARMONIC ANALYSIS STUDIES
IEEE
Std 399-1997
10.9 References
IEEE Std 18-1992, IEEE Standard for Shunt Power Capacitors.3
IEEE Std 444-1973 (Reaff 1992), IEEE Standard Practices and Requirements for Thyristor
Converters for Motor Drives, Part 1.
IEEE Std 519-1992, IEEE Recommended Practices and Requirements for Harmonic Control
in Electrical Power Systems.
IEEE Std 597-1983 (Reaff 1992), IEEE Practices and Requirements for General Purpose
Thyristor DC Drives.
IEEE Std C37.99-1990 (Reaff 1994), IEEE Guide for Protection of Shunt Capacitor Banks.
IEEE Std C57.110-1986 (Reaff 1992), IEEE Recommended Practice for Establishing Transformer Capability When Supplying Nonsinusoidal Load Currents.
10.10 Bibliography
[B1] Adamson, C., and Hingorani, N. G., High Voltage Direct Current Power Transmission.
London: Garraway, Ltd., 1960.
[B2] Arrillaga, J., Bradley, D., and Boger, P. S., Power Systems Harmonics. England: John
Wiley & Sons, Ltd., 1985.
[B3] CIGRE Working Group 36-05 (Disturbing Loads) (1981), Harmonic characteristic
parameters, method of study, estimating of existing values in network, Electra 77: 35–54.
[B4] Czarnecki, L. S., and Tan, O. T., “Evaluation and reduction of harmonic distortion
caused by solid state voltage controllers of induction motors,” IEEE Transactions on Energy
Conversion, Vol. EC-9, No. 3, pp. 528–534, Sept. 1994.
[B5] Day, A. L., and Mahmoud, A. A., “Methods of evaluation of harmonic levels in industrial
plant distribution systems,” IEEE Transactions on Industry Applications, Vol. IA-23, No. 3,
pp. 498–503, May/June 1987.
[B6] Domijan, Jr., A., and Embriz-Santander, E., “A summary and evaluation of recent developments on harmonic mitigation techniques useful to adjustable speed drives,” IEEE Transactions on Energy Conversion, Vol. EC-7, No. 1, pp. 64–71, Mar. 1992.
3IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
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[B7] Domijan, A., and Zaninelli, D., “IEC and IEEE standards on harmonics and
comparisons,” Proceedings of National Science Foundation Conference on Unbundled Power
Quality Services in the Power Industry,” Nov. 17–19, 1996, Key West, Florida.
[B8] Dugan, R. C., McGranghan, F. M., and Beaty, H. W., Electrical Power Systems Quality.
New York: McGraw-Hill, 1996.
[B9] Emanuel, A. E., Orr, J. A., Cyganski, D., and Gulachenski, E. M., “A survey of harmonic
voltages and currents at the customer’s bus,” IEEE Transactions on Power Delivery,
Vol. PWRD-8, No. 1, pp. 411–421, Jan. 1993.
[B10] Etezadi-Amoli, M., and Florence, T., “Voltage and current harmonic content of a utility
system—A summary of 1120 test measurements,” IEEE Transactions on Power Delivery,
Vol. PWRD-5, No. 3, pp. 1552–1557, July 1990.
[B11] Fujita, H., and Akagi, H., “A practical approach to harmonic compensation in power
systems—Series connection of passive and active filters,” IEEE Transactions on Industry
Applications, Vol. IA-27, No. 6, pp. 1020–1025, Nov./Dec. 1991.
[B12] Gluskin, E., “High harmonic currents in fluorescent lamp circuits,” IEEE Transactions
on Industry Applications, Vol. IA-26, No. 2, pp. 347–351, Mar./Apr. 1990.
[B13] Gonzalez, D. A., and McCall, J. C., “Design of filters to reduce harmonic distortion in
industrial power systems,” IEEE Transactions on Industry Applications, Vol. IA, No. 3,
pp. 504–511, May/June 1987.
[B14] Grotzbach, M., and Redman, R., “Analytical predetermination of complex line current
harmonics in controlled AC/DC converter,” IEEE Conference Record, IEEE/Industry
Applications Society Annual Meeting, Orlando, p. 2165, Oct. 1995.
[B15] Hanna, R. A., “Harmonics and technical barriers in adjustable speed drives,” IEEE
Transactions on Industry Applications, Vol. IA-25, No. 5, pp. 894–900, Sept./Oct. 1989.
[B16] Henderson, R. D., and Rose, P. J., “Harmonics: The effects on power quality and
transformers,” IEEE Transactions on Industry Applications, Vol. IA-30, No. 3, pp. 528–532,
May/June 1994.
[B17] Heydt, G.T., Electric Power Quality. Scottsdale, AZ: Stars in a Circle Publications,
1995.
[B18] Heydt, G. T., Kish, D. J., Holcomb, F., and Hill, Y., “A methodology for assessment of
harmonic impact,” IEEE Transactions on Power Delivery, Vol. PWRD-6, No. 4, pp. 1748–
1754, Oct. 1991.
[B19] IEEE Power Engineering Society, IEEE Publication No. 84EH0221-2-PWR, Tutorial
course: “Power system harmonics,” 1984.
310
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Std 399-1997
[B20] IEEE Power Engineering Society, IEEE Publication No. 84TH0115-6 DWR, “Sinewave distortion in power systems and the impact on protective relaying,” 1984.
[B21] IEEE Task Force Report, “Effects of Harmonics on Equipment,” IEEE Transactions on
Power Delivery, Vol. PWRD-8, No. 2, pp. 672–680, Apr. 1993.
[B22] IEEE Task Force on Harmonics Modeling and Simulation, “Modeling and simulation
of the propagation of harmonics in electrical power networks,” Parts 1 & 2, IEEE Transactions on Power Delivery, Vol. PWRD-11, No. 1, pp. 452–474, Jan. 1996.
[B23] Kimbark, E., Direct Current Transmission, New York: John Wiley & Sons, Inc., 1971.
[B24] Lemieux, G., “Power system harmonic resonance—A documented case,” IEEE Transactions on Industry Applications, Vol. IA-26, No. 3, pp. 483–488, May/June 1990.
[B25] Lowenstein, M. Z., “Improving power factor in the presence of harmonics using
low-voltage tuned filters,” IEEE Transactions on Industry Applications, Vol. IA-29, No. 3,
pp. 528–535, May/June 1993.
[B26] Ludbrook, A., “Harmonic filters for notch reduction,” IEEE Transactions on Industry
Applications, Vol. IA-24, No. 5, Sept./Oct. 1988.
[B27] Mahmoud, A. A., and Shultz, R.D., “A method for analyzing harmonic distribution in
ac power systems,” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101,
No. 6, pp. 1815–1824, June 1982.
[B28] Paice, D. A., Power Electronic Converter Harmonics. New York: IEEE Press, 1996.
[B29] Phipps, J. K., Nelson, J. P., and Sen, P. K., “Power quality and harmonic distortion on
distribution systems,” IEEE Transactions on Industry Applications, Vol. IA-30, No. 2,
pp. 476–484, Mar./Apr. 1994.
[B30] Prabhakara, F. S., Smith, R. L., and Stratford, R. P., Industrial and Commercial Power
Systems Handbook, New York: McGraw-Hill, 1995.
[B31] Purkayastha, I., and Savoie, P. J., “Effect of harmonics on power measurement,” IEEE
Transactions on Industry Applications, Vol. IA-26, No. 5, pp. 944–946, Sept./Oct. 1990.
[B32] Rastogi, M., Naik, R., and Mohan, N., “A comparative evaluation of harmonic reduction techniques in three-phase utility interface of power electronic loads,” IEEE Transactions
on Industry Applications, Vol. IA-30, No. 5, pp. 1149–1155, Sept./Oct. 1994.
[B33] Redl, R., Tenti, P., and Van Wyk, J. D., “Power electronics’ polluting effects,” IEEE
Spectrum, pp. 32–39, May 1997.
[B34] Rice, D. E., “Adjustable speed drive and power rectifier harmonics—Their effect on
power system components,” IEEE Transactions on Industry Applications, Jan./Feb.1986.
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CHAPTER 10
[B35] Schieman, R. G., and Schmidt, W. C., “Power line pollution by 3-phase thyristor motor
drives,” IEEE/Industry Application Society Conference, 1976 Annual Meeting.
[B36] Sen, P. K., and Landa, H. A., “Derating of induction motors due to waveform
distortion,” IEEE Transactions on Industry Applications, Vol. IA-26, No. 6, pp. 1102–1107,
Nov./Dec. 1990.
[B37] Sharma, V., Fleming, R. J., and Niekamp, L., “An iterative approach for analysis of
harmonic penetration in the power transmission networks,” IEEE Transactions on Power
Delivery, Vol. PWRD-6, No. 4, pp. 1689–1706, Oct. 1991.
[B38] Shipp, D. D., “Harmonic analysis and suppression for electrical systems supplying
static power converters and other nonlinear loads,” IEEE Transactions on Industry
Applications. Sept./Oct. 1979.
[B39] Shuter, T. C., Volkommer, Jr., H. T., and Kirkpatrick, T. L., “Survey of harmonic levels
on the American electric power distribution system,” IEEE Transactions on Power Delivery,
Vol. PWRD-4, No. 4, pp. 2204–2213, Oct. 1989.
[B40] Subjak, Jr., J., and McQuilkin, J., “Harmonics—Causes, effects, measurements, and
analysis—Update,” IEEE/Industry Application Society Cement Industry Technical
Conference, May 1989.
[B41] Valcarcel, M., and Mayordomo, J. G., “Harmonic power flow for unbalanced systems,”
IEEE Transactions on Power Delivery, Vol. PWRD-8, No. 4, pp. 2052–2059, Oct. 1993.
[B42] Williams, S. M., Brownfield, G. T., and Duffus, J. W., “Harmonic propagation on an
electric distribution system: Field measurements compared with computer simulation,” IEEE
Transactions on Power Delivery, Vol. PWRD-8, No. 2, pp. 547–552, Apr. 1993.
[B43] Yan, Y. H., Chen, C. S., Moo, C. S., and Hsu, C. T., “Harmonic analysis for industrial
customers,” IEEE Transactions on Industrial Applications, Vol. IA-30, No. 2, pp. 462–468,
Mar./Apr. 1994.
312
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Chapter 11
Switching transient studies
11.1 Power system switching transients
11.1.1 Introduction
An electrical transient occurs on a power system each time an abrupt circuit change occurs.
This circuit change is usually the result of a normal switching operation, such as breaker
opening or closing or simply turning a light switch on or off. Bus transfer switching operations along with abnormal conditions, such as inception and clearing of system faults, also
cause transients.
The phenomena involved in power system transients can be classified into two major
categories:
a)
Interaction between magnetic and electrostatic energy stored in the inductance and
capacitance of the circuit, respectively,
b)
Interaction between the mechanical energy stored in rotating machines and electrical
energy stored in the inductance and capacitance of the circuit.
Unlike the first category, which consists solely of electromagnetic transients, the latter deals
with electromechanical transients and will not be treated in this chapter. Electromechanical
transients are considered in Chapters 7 and 8.
Most power system transients are oscillatory in nature and are characterized by their transient
period of oscillation. Despite the fact that these transient periods are usually very short when
compared with the power frequency of 50 Hz or 60 Hz, they are extremely important because
at such times, the circuit components and electrical equipment are subjected to the greatest
stresses resulting from abnormal transient voltages and currents. While overvoltages may
result in flashovers or insulation breakdown, overcurrent may damage power equipment due
to electromagnetic forces and excessive heat generation. Flashovers usually cause temporary
power outages due to tripping of the protective devices, but insulation breakdown usually
leads to permanent equipment damage.
For this reason, a clear understanding of the circuit during transient periods is essential in the
formulation of steps required to minimize and prevent the damaging effects of switching
transients.
11.1.2 Circuit elements
All circuit elements, whether in utility systems, industrial plants, or commercial buildings,
possess resistance, R, inductance, L, and capacitance, C. Ohm’s law defines the voltage
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across a time-invariant linear resistor as the product of the current flowing through the
resistor and its ohmic value. That is,
v (t) = Ri(t)
(11-1)
The other two elements, L and C, are characterized by their ability to store energy. The term
“inductance” refers to the property of an element to store electromagnetic energy in the
magnetic field. This energy storage is accomplished by establishing a magnetic flux within
the ferromagnetic material. For a linear time-invariant inductor, the magnetic flux is defined
as the product of the inductance and the terminal current. Thus,
φ(t) = Li(t)
(11-2)
where φ(t) is the magnetic flux in webers (Wb), L is the inductance in henries (H), and i(t) is
the time-varying current in amperes (A). By Faraday’s law, the voltage at the terminals of the
inductor is the time derivative of the flux, namely,
dφ
v ( t ) = -----dt
(11-3)
Combining this relationship with Equation (11-2) gives the voltage-current relation of a timeinvariant linear inductor as
di
v ( t ) = L ----dt
(11-4)
Finally, the term “capacitance” means the property of an element that stores electrostatic
energy. In a typical capacitance element, energy storage takes place by accumulating charges
between two surfaces that are separated by an insulating material. The stored charge in a
linear capacitor is related to the terminal voltage by
q(t) = Cv(t)
(11-5)
where C is the capacitance in farads (F) when the units of q and v are in coulombs (C) and
volts (V), respectively. Since the electrical current flowing through a particular point in a
circuit is the time derivative of the electrical charge, Equation (11-5) can be differentiated
with respect to time to yield a relationship between the terminal current and the terminal
voltage. Thus,
dq
i ( t ) = -----dt
or
dv
i ( t ) = C -----dt
(11-6)
Under steady-state conditions, the energy stored in the elements swings between the
inductance and capacitance in the circuit at the power frequency. When there is a sudden
change in the circuit, such as a switching event, a redistribution of energy takes place to
accommodate the new condition. This redistribution of energy cannot occur instantaneously
for the following reasons:
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SWITCHING TRANSIENT STUDIES
a)
b)
2
LI - . For a constant inducThe electromagnetic energy stored in an inductor is E = ------2
tance, a change in the magnetic energy requires a change in current. But the change
di
in current in an inductor is opposed by an emf of magnitude v ( t ) = L ----- . For the
dt
current to change instantaneously (dt = 0), an infinite voltage is required. Since this is
unrealizable in practice, the change in energy in an inductor requires a finite time
period.
2
CV and the currentThe electrostatic energy stored in a capacitor is given by E = ---------2
voltage relationship is given by i ( t ) = C d-----v- . For a capacitor, an instantaneous
dt
change in voltage (dt = 0) requires an infinite current, which cannot be achieved in
practice. Therefore, the change in voltage in a capacitor also requires finite time.
These two basic concepts, plus the recognition that the rate of energy produced must be equal
to the sum of the rate of energy dissipated and the rate of energy stored at all times (principle
of energy conservation) are basic to the understanding and analysis of transients in power
systems.
11.1.3 Analytical techniques
The classical method of treating transients consists of setting up and solving the differential
equation or equations, which must satisfy the system conditions at every instant of time. The
equations describing the response of such systems can be formulated as linear time-invariant
differential equations with constant coefficients. The solution of these equations consists of
two parts:
a)
b)
The homogeneous solution, which describes the transient response of the system, and
The particular solution, which describes the steady-state response of the system to the
forcing function or stimulus.
As discussed and described in Chapter 3, the analytical solution of linear differential
equations can also be obtained by the Laplace transform method. This technique does not
require the evaluation of the constants of integration and is a powerful tool for complex
circuits, where the traditional method can be quite difficult.
11.1.4 Transient analysis based on the Laplace transform method
Although they do not represent the types of problems regularly encountered in power
systems, the transient analysis of the simple RL and RC circuits described in Chapter 3 of this
book are useful illustrative examples of how the Laplace transform method can be used for
solving circuit transient problems. Real-life circuits, however, are far more complicated and
often retain many circuit elements in series-parallel combination even after simplification.
These circuits will require several differential or integro-differential equations to describe
transient behavior and must be solved simultaneously to evaluate the response. To do this
efficiently, the Laplace transform method is often used.
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11.1.4.1 LC transients
In this chapter, the more general types of circuits that are described by higher-order
differential equations are discussed. The double-energy transient, or LC circuit, is the first
type of circuit to be considered. In double-energy electric circuits, energy storage takes place
in the magnetic field of inductors and in the electric field of the capacitors. In real circuits, the
interchange of these two forms of energy may, under certain conditions, produce electric
oscillations. The theory of these oscillations is of great importance in electric power systems.
In the circuit shown in Figure 11-1 the circuit elements are represented with Laplace
transform impedances as described in Chapter 3. The response of the circuit to a step input of
voltage due to the closing of the switch at t = 0 will be examined, assuming the capacitor is
initially charged to the potential of Vc (0–) as shown.
1
-----sC
–
V c(0 )
---------------s
Figure 11-1—Double energy network
According to Kirchoff’s voltage law, the sum of the voltages across all the circuit elements
must equal the source voltage at all times. In equation form this is stated as
I (s)
2
V ( s ) = I ( s )s L – sLI ( 0 ) + --------- – V c ( 0 )
C
(11-7)
Since there could be no current flowing in the circuit before the switch closes, the term
LI(0) = 0. Solving for the current I(s) in Equation (11-7),
V (s) + V c(0)
1
I ( s ) = -------------------------------- --------------------2
2
L
( s + ω0 )
316
(11-8)
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SWITCHING TRANSIENT STUDIES
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1
where ω02 is the natural frequency of the circuit, namely ------------ . From the table of inverse
LC
Laplace transforms (see table in Chapter 3), the transient response in the time domain is
V (s) + V c(0)
i ( t ) = -------------------------------- [ sin ( ω 0 t ) ]
Z0
(11-9)
where Z0 is the surge impedance of the circuit defined by
Z0 =
L
---C
(11-10)
Clearly, the transient response indicates a sinusoidal current with a frequency governed by
the circuit parameters L and C only.
Another interesting feature about the time response of the current in the circuit is that the
magnitude of the current is inversely proportional to the surge impedance of the circuit Z0,
which is a function of the circuit parameters of L and C. The importance of this parameter to
the analysis of transient problems will be demonstrated later in the chapter.
In power system analysis, we are often interested in the voltage across the capacitor.
Referring to Figure 11-1, the capacitor voltage is
I (s) V c(0)
V c ( s ) = --------- – ------------sC
s
(11-11)
where Vc(0) is the initial voltage of the capacitor. Solving for I(s) in the above equation
results in
I ( s ) = sCV c ( s ) + CV c ( 0 )
(11-12)
But because the current I(s) is common to both elements L and C, we can substitute Equation
(11-12) into Equation (11-8) to obtain the voltage across the capacitor. After rearranging the
terms,
2
V ( s )ω 0
sV c ( 0 )
V c ( s ) = ------------------------ + --------------------2
2
2
2
s ( s + ω0 ) ( s + ω0 )
(11-13)
From the table of inverse Laplace transforms (see Chapter 3), the transient response is
v c ( t ) = V ( s ) [ 1 – cos ( ω 0 t ) ] + V c ( 0 ) cos ( ω 0 t )
(11-14)
The above equation is plotted in Figure 11-2 for various values of initial capacitor voltage
Vc(0).
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Figure 11-2—Capacitor voltage for various initial voltages
Examination of these curves indicates that, without damping, the capacitor voltage swings as
far above the source voltage V as it starts below. In a real circuit, however, this will not be the
case, since circuit resistance will introduce losses and will damp the oscillations. Treatment
of the effects of resistance in the analysis of circuits is presented next.
11.1.4.2 Damping
Nearly every practical electrical component has resistive losses (I2R losses). To simplify the
calculations and to ensure more conservative results, resistive losses are usually neglected as
a first attempt to a switching transient problem. Once the behavior of the circuit is understood, then system losses can be considered if deemed necessary.
A parallel RLC circuit is depicted in Figure 11-3, in which the circuit elements are represented by their Laplace transform admittances. Many practical transient problems found in
power systems can be reduced to this simple form and still yield acceptable results. With a
constant current source, I(s) and zero initial conditions, the equation describing the current in
the parallel branches is
V (s)
2
I ( s ) = sV ( s )G + s CV ( s ) + ----------L
318
(11-15)
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SWITCHING TRANSIENT STUDIES
Figure 11-3—Parallel RLC circuit
Solving for the voltage results in
1
I (s)
V ( s ) = --------- ----------------------------------
1
C 2
s
s + -------- + -------
RC LC
(11-16)
Equation (11-16) can be written as
1
I (s)
V ( s ) = --------- ------------------------------------C (s + r1)(s + r2)
(11-17)
where r1 and r2 are the roots of the characteristic equation defined as follows:
4
1 2
1
1
r 1 = ----------- + --- -------- – -------
LC
2RC 2 RC
and
4
1 2
1
1
r 2 = ----------- – --- -------- – -------
LC
2RC 2 RC
(11-18)
At the natural resonant frequency ω0, the reactive power in the inductor is equal to the power
in the capacitor but opposite in sign. The source has to supply only the true power PT required
by the resistance in the circuit. The ratio between the magnitude of the reactive power, PR, of
either the inductance or the capacitance at the resonant frequency, and the magnitude of true
power, PT of the circuit, is known as the quality factor of the circuit, or Q. Therefore, for the
parallel circuit,
PR
Q P = -----PT
2
or
V B
Q P = ---------2
V G
or
Copyright © 1998 IEEE. All rights reserved.
R
Q P = ---X
(11-19)
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where B is the susceptance of either the inductor or capacitor, G is the conductance, and V is
the voltage across the element. But, since in a parallel circuit the voltage is common to all
1- in Equation (11-19), the result is
elements, substituting ω0L for --1- and R for --B
G
R
Q P = ---------ω0 L
(11-20)
Furthermore, since the natural frequency of the circuit is
1
ω 0 = -----------LC
(11-21)
then, substituting Equation (11-21) into Equation (11-20) yields
R
Q P = -------L
---C
(11-22)
Rearranging Equation (11-18), we have
1
1
2C
r 1 = ----------- + ----------- 1 – 4R ---L
2RC 2RC
and
1
1
2C
r 2 = ----------- – ----------- 1 – 4 R ---L
2RC 2 RC
(11-23)
Substituting Equation (11-22) into Equation (11-23), the result is
1
1
2
r 1 = ----------- + ----------- 1 – 4Q P
2RC
2RC
and
1
1
2
r 2 = ----------- – ----------- 1 – 4 Q P
2RC 2 RC
(11-24)
Depending on the values of the circuit parameters, the quantity under the radical in Equation (11-24) may be positive, zero, or negative.
For positive values, that is 4QP2 < 1, the roots are real, negative, and unequal. In this case, the
inverse Laplace transform of Equation (11-17) is
t
– -----------
2RC
ω t
–ω t
IRe
v ( t ) = ----------------------- [ e D – e D ]
2
1 – 4Q P
(11-25)
where
2
ωD
320
1 – 4Q P
= ------------------------2RC
(11-26)
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is the damped natural angular frequency. Substituting 2sinh(ωDt) for the exponential
function, the result is
t
– -----------
2RC
2
1 – 4Q P
2IRe
v ( t ) = --------------------------- sinh ----------------------- t
2
2RC
1 – 4Q P
(11-27)
For the case where the quantity 4QP2 = 1, the roots are equal, negative, and real. Therefore,
the solution of Equation (11-25) is
I
v ( t ) = ---- e
C
t
– -----------
2RC
(11-28)
Finally, when the quantity under the radical sign in Equation (11-4) is less than zero, that is,
4QP2 > 1, the roots are unequal and complex. The solution in this case is
t
– -----------
2RC
2
4Q P – 1
2IRe
v ( t ) = --------------------------- sin ----------------------- t
2
2RC
4Q P – 1
(11-29)
Consider the series RLC circuit shown in Figure 11-4.
1
-----sC
Figure 11-4—Series RLC circuit
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With V(s) as a constant source, the equation describing the current in the circuit is
1
V (s)
I ( s ) = ----------- -----------------------------
s
1
R + sL + ------
sC
(11-30)
Rearranging the terms results in
1
V (s)
I ( s ) = ----------- --------------------------------
R
L 2
1
s + s --- + -------
L LC
(11-31)
The expression inside the brackets is similar to the expression for the parallel RLC circuit
shown in Equation (11-16). The only difference is the coefficient of s. Rewriting, as in
Equation (11-17) gives
1
V (s)
I ( s ) = ----------- ------------------------------------L (s + r1)(s + r2)
(11-32)
where r1 and r2 are the roots of the characteristic equation defined as
R
4L
R
r 1 = ------ + ------ 1 – --------2
2L 2L
R C
and
R
4L
R
r 2 = ------ – ------ 1 – --------2
2L 2L
R C
(11-33)
Again, we define the quality factor of the series RLC circuit, Qs, as the ratio of the magnitude
of the reactive power of either the inductor or the capacitor at the resonant frequency to the
magnitude of the true power in the circuit. Stated in equation form, results in
P
Q s = -----RPT
(11-34)
With the reactive power PR = I2X and the true power as PT = I2R, then
2
I X
Q s = -------2
I R
(11-35)
But since, in a series circuit, the current is common to all elements, then
ω0 L
Q s = --------R
322
(11-36)
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1
where ω0L = X. Also, since ω 0 = ------------ , then Equation (11-36) can be written as
LC
L
---C
Q s = -------R
or
Z
Q s = -----0R
(11-37)
Note that the above expression is the ratio of the surge impedance to the resistance in the circuit. This is the reciprocal of the expression developed for the parallel RLC circuit and
described by Equation (11-22). That is,
1
Q s = ------QP
(11-38)
Substituting Equation (11-37) into Equation (11-33) results in
R
R
2
r 1 = ------ + ------ ( 1 – 4Q s )
2L 2L
and
R
R
2
r 2 = ------ – ------ ( 1 – 4 Q s )
2L 2 L
(11-39)
Above expressions have already been solved for the parallel RLC circuit. To obtain the
expression as a function of time, simply substitute Qs for QP and V/R for IR and R/L for 1/RC
in Equation (11-26) and Equation (11-28) and V/L for I/C and R/L for 1/RC in Equation
(11-29). Thus, when the quantity under the radical sign is less than 1, namely, 4Qs2 < 1, the
result is
Rt
– ------
2L
R
2V e
2
i ( t ) = --------------------------- sinh ------ 1 – 4Q s
2L
2
R 1 – 4Q s
(11-40)
For the case where the quantity 4Qs2 = 1, then the roots are equal, negative, and real and the
solution of Equation (11-39) is
V
i ( t ) = ---- e
L
R
– ------t
2L
(11-41)
Finally, when the quantity 4Qs2 > 1, the roots are complex and unequal. Therefore, the
solution is
Rt
– ------
2L
R
2V e
2
i ( t ) = --------------------------- sin ------ 4Q s – 1
2L
2
R 4Q s – 1
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(11-42)
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CHAPTER 11
11.1.5 Normalized damping curves
The response of the parallel and series RLC circuits to a step input of current or voltage,
respectively, can be expressed as a family of normalized damping curves, which can be used
to estimate the response of simple switching transient circuits to a step input of either voltage
or current. To develop a family of normalized damping curves, proceed as follows:
a)
To per-unitize the solutions, we use the undamped response of a parallel LC circuit as
the starting point. Thus, for the voltage,
1
v ( t ) = ---------- sin ( ω 0 t )
ω0 C
(11-43)
and, for the current,
1
i ( t ) = ---------- sin ( ω 0 t )
ω0 L
(11-44)
The maximum voltage or current occurs when the angular displacement ω 0 t = π /2 .
Thus,
1
v ( t ) = ---------ω0 C
b)
1
i ( t ) = ---------ω0 L
and
(11-45)
t
Setting the angular displacement ω 0 t = θ , the quantity ----------- in Equations (11-26)
2RC
θ
through (11-28) can be substituted with the expression ---------- .
2Q P
c)
Finally, dividing Equations (11-26) through (11-28) by the right side of the
expression for v(t) in Equation (11-45) produces a set of normalized curves for the
voltage on the parallel RLC circuit as a function of the dimensionless quantities QP
and the displacement θ.
2
Thus, for 4Q P < 1 ,
θ
– ----------
2Q P
2
θ 1 – 4Q P
2Q P e
f ( Q P ,θ ) = --------------------------- sinh ---------------------------
2
2Q P
1 – 4Q P
(11-46)
2
and, for 4Q P = 1 ,
f ( Q P ,θ ) = θe
324
θ
– ----------
2Q P
,
(11-47)
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SWITCHING TRANSIENT STUDIES
2
and, for 4 ( Q P > 1 ) , the result is
θ
– ----------
2Q P
2
θ 4Q P – 1
2Q P e
f ( Q P ,θ ) = --------------------------- sin ---------------------------
2
2Q P
4Q P – 1
(11-48)
Normalized values of QP, in pu
Equations (11-46) through (11-48) are plotted in Figure 11-5 and Figure 11-6 for various
values of QP . For series RLC circuits, divide the same equations in step c) by the right side of
the expression for i(t), and substitute Qs for QP .
Displacement, θ
Figure 11-5—Normalized damping curves, 0.10 ≤ QP ≤ 0.45
in steps of 0.05, with QP (0.45) = 0.34 pu
11.1.6 Transient example: Capacitor voltage
Very often, in power systems analysis, the form and magnitude of the transient voltage
developed across the capacitor during switching is significant. To develop generalized
expressions for capacitor voltage, start with Equation (11-31), which describes the transient current in the series RLC circuit. The voltage across the capacitor is simply the product of the current and the capacitor impedance, that is,
Vc(s) = I(s)Zc(s)
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(11-49)
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Normalized values of QP, in pu
CHAPTER 11
Displacement, θ
Figure 11-6—Normalized damping curves, 0.50 ≤ QP ≤ 75.0
0.50, 1.0, 2.0, 5.0, 10.0, 15.0, 30.0, and 75.0, with QP (75) = 1.00 p.u.
or
I (s)
V c ( s ) = --------sC
(11-50)
Substituting Equation (11-30) into Equation (11-50), yields the expression for the voltage
across the capacitor in a series RLC circuit. Thus,
1
V (s)
V c ( s ) = ----------- ---------------------------------------LC 2
R
1
s s + s --- + -------
L LC
(11-51)
Equation (11-51) is similar to the Equation (11-30), developed for the current, except that it
has an extra s term in the denominator. The expression can be rewritten as follows:
1
V (s)
V c ( s ) = ----------- --------------------------------------LC s ( s + r 1 ) ( s + r 2 )
(11-52)
where the roots of the equation are the same as those defined by Equation (11-39). Again, the
solution of Equation (11-52) will depend on the values of Qs.
Equation (11-14) shows the voltage across a capacitor due to a step input of voltage when the
resistance in the circuit is zero. For zero initial conditions, that is, no charge in the capacitor,
326
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the last term in Equation (11-14) can be neglected. Then, the maximum voltage occurs when
ω0t = π.
Therefore, the voltage is simply
Vc(t) = 2
(11-53)
Following exactly the same procedures outlined in 11.1.5, Equation (11-51) has three
possible solutions.
2
For the case in which 4 ( Q s < 1 ) ,
2
f ( Q s ,θ ) = 1 – e
θ
– ---------
2 Q s
θ 1 – 4 Qs
sinh --------------------------
2
2 Qs
θ 1 – 4 Qs
------------------------------------------- + cosh --------------------------
2 Qs
2
1 – 4 Qs
(11-54)
2
When 4Q s = 1, the solution is
f ( Q s ,θ ) = 2Q s 1 – e
θ
– ---------
2Q s
θ
1 + -------
2Q s
(11-55)
2
Finally, when 4Q s > 1, the solution takes the form of
2
f ( Q s ,θ ) = 1 – e
θ
– ---------
2 Q s
θ 4 Q s – 1
sin --------------------------
2
2 Qs
θ 4 Q s – 1
---------------------------------------- + cos --------------------------
2 Qs
2
4 Qs – 1
(11-56)
Equations (11-54) through (11-56) are plotted in Figure 11-7 for various values of Qs.
11.1.7 Switching transient examples
In the previous subclauses, some simple circuits were examined that can be used to model
many switching problems in electrical power systems. Very often, practical switching
transient problems can be reduced to either parallel or series RLC circuits for the purpose of
evaluating the response of the network to a particular stimuli on a first-trial basis. To gain
familiarity with the normalized damping curves developed in the previous subclauses, some
typical switching problems in power systems will be examined. Consider, for example, a
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Normalized values of Qs, in pu
CHAPTER 11
Displacement, θ
Figure 11-7—Normalized damping curves, 0.10 ≤ Qs ≤ 100
0.10, 0.30, 0.50, 0.75, 1.0, 1.5, 2.0, 5.0, 10.0, 15.0, 30.0,
and 100.0 with Qs (100) = 1.99
1000 kVA unloaded transformer that, when excited from the 13.8 kV side with its rated
voltage of 13.8 kV, draws a no-load current of 650 mA with a power factor of 10.4%. A test
circuit for the transformer is shown in Figure 11-8. The battery voltage, V, and the resistance,
R, are chosen such that, with the switch closed, the battery delivers 10 mA. For this example,
a shunt capacitance of 2.8 nF per phase is assumed. The goal is to find the voltage across the
capacitance to ground, when the switch is suddenly opened and the flow of current is
interrupted.
From the information provided, the no-load current of the transformer is
INL = 0.067794 – j0.64644 A
The magnetizing reactance XM and inductance LM per phase are, respectively,
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13.8 kV
X M = --------------------------------- = 12.325 kΩ
3 × 0.64644
XM
12.325 k Ω
- = -------------------------- = 32.693 H
L M = -------2πf
377
With a shunt capacitance of CSH = 2.8 nF, the shunt capacitive reactance is
1
1
X SH = ------------------- = ------------------------------- = 947.33 k Ω
377 × 2.8 nF
2πf C SH
Since XSH > XM, the effects of the shunt capacitive reactance at the power frequency are
negligible. The resistance RC is
13.8 kV
R C = ------------------------------------ = 117.524 k Ω
3 × 0.067794
Using delta-wye transformation impedance conversion, the circuit in Figure 11-8 can be
redrawn as shown in Figure 11-9. In the Laplace transform notation, the equation describing
the circuit at t = 0+ is
–
V c (s)
I L ( 0 ) V c (s)
–
- + ------------ + sCV c (s) – CV c ( 0 ) + -----------O = -------------R
s
sL
where
3L
L = ---------M- = 49.035 H
2
2C SH
- = 1.867 nF
C = -----------3
3R
R = ---------c = 176.286 k Ω
2
Assuming that dc steady-state was obtained before the switch opened, the term CVc (0–) = 0
and the initial current in the inductor at t = 0+ is IL (0–) = 10 mA. Solving for Vc(s) and after
rearranging the terms, gives
–
I L ( 0 )
1
V c (s) = – --------------- ----------------------------------
C 2
s
1
s + -------- + -------
RC LC
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CHAPTER 11
Figure 11-8—Test setup of unloaded transformer
Figure 11-9—Equivalent RLC circuit for unloaded transformer
The time-response solution for this expression has already been obtained and, depending on
the values of the circuit parameters, is shown in Equations (11-26), (11-27), and (11-28). The
values for R, L, and C could be inserted into one of those equations to obtain the capacitor
voltage for this problem. But this has also been done through the normalized damping curves
shown in Figure 11-5. Therefore, the answers for this particular problem are obtained as
follows:
a)
The surge impedance of the circuit is
Z0 =
330
L
---- = 162.07 k Ω
C
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SWITCHING TRANSIENT STUDIES
b)
IEEE
Std 399-1997
Without damping, the peak transient voltage would be
–
V peak = I (0 ) Z 0 = – 10 mA × 162.076 k Ω = – 1.621 kV
c)
But since there is damping, the quality factor of the parallel circuit QP is
176.286 k Ω
R
Q P = ------ = ----------------------------- = 1.087
162.076 k Ω
Z0
d)
From the curves shown in Figure 11-5 and with QP ≈ 1.0, the maximum per unit
voltage is 0.57. Therefore, from step b), the maximum voltage developed across the
capacitor is
V max = – 1.621 kV × 0.57 = – 924 V
e)
The maximum peak occurs at approximately θ = 1.2 radians. Since ω0t = θ and
1
ω 0 = ------------ or
LC
ω 0 = 3305 rad/s
Figure 11-10 depicts the actual voltage across the capacitance to ground as calculated by a
computer program.
Another practical case will be examined, which concerns capacitor bank switching as
depicted in Figure 11-11. Capacitor C1 is rated 30 Mvar, three-phase, at 13.8 kV. C2 is
initially uncharged and is rated 10 Mvar, three-phase, also at 13.8 kV. The cable connecting
capacitor C2 to the bus has an inductance L of 35 µH. The following procedure is used to
determine the magnitude of the inrush current and the size of the resistor required to limit this
current to a maximum of 5800 A (peak) during energization.
From the problem statement, the capacitive reactance of the capacitors is
2
2
2
2
13.8
kV
X c1 = --------- = ------------ = 6.348 Ω
30
C1
13.8
kV
X c2 = --------- = ------------ = 19.044 Ω
10
C2
and the capacitance is
1
1
C 1 = ------------------ = --------------------------------------------- = 417.861 µ F
2π × 60 × 6.348 Ω
2πf X c1
C 1 = 139.287 µ F
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Capacitor Voltage, V
CHAPTER 11
Time, s
Figure 11-10—Actual capacitor voltage
Vc (0–)
Figure 11-11—Capacitor bank switching
Assuming worst-case conditions, that is, C1 charged to peak system voltage or
13.8 × 2
kV × 2
V c1 = -------------------- = ------------------------ = 11.268 kV
3
3
and with a surge impedance of
Z0 =
332
L(C1 + C2)
--------------------------= 0.579 Ω
(C1C2)
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SWITCHING TRANSIENT STUDIES
IEEE
Std 399-1997
Then, with no damping, the inrush current would be
V c(0)
11.268 kV
- = ------------------------- = 19.467 kA
i peak = ----------0.579 Ω
Z0
Redrawing the circuit of Figure 11-11 to show the necessary addition of a resistor to limit the
inrush current yields the circuit as shown in Figure 11-12.
Vc (0–)
Figure 11-12—Equivalent circuit for capacitor switching
with pre-insertion resistor
The problem requires that the inrush current should not exceed 5800 A. This represents a perunit value of
I max
5800 A
- = --------------------- = 0.30
I pu = --------19467 A
I peak
Referring to Figure 11-5, since the current problem concerns a series circuit, Qs replaces QP .
With a per-unit value requirement of 0.30 (vertical axis), Figure 11-5 shows that a QP = 0.30
will reduce the current to 5800 (19467 × 0.30) A or less. Now, since
Qs = 0.30
and
Z
Q s = -----0- = 0.30
R
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CHAPTER 11
then
Z
0.579 Ω
R = -----0- = -------------------- = 1.93 Ω
Qs
0.30
Therefore, to limit the inrush current to 5800 A a 1.9 Ω resistor must be placed in series with
capacitor C2 as shown in Figure 11-12.
Current, A
The results of a computer simulation are depicted in Figure 11-13 and Figure 11-14. While
Figure 11-13 shows the current without the pre-insertion resistor, Figure 11-14 reflects the
current with the 1.9 Ω resistor.
Time, s
Figure 11-13—Current in circuit without damping resistor
11.1.8 Transient recovery voltage
Circuit breakers provide the mechanism to interrupt the short-circuit current during a system
fault. When the breaker contacts open, the fault current is not interrupted instantaneously.
Instead, an electric arc forms between the breaker contacts, which is maintained as long as
there is enough current flowing. Since the fault current varies sinusoidally at the power
frequency, the arc will extinguish at the first current zero. However, at the location of the arc,
there are still hot, ionized gases and, if voltages exceeding the dielectric capability of the
contact gap develop across the open contacts of the circuit breaker, it is possible that the arc
will re-ignite. Circuit interruption is a race between the increase of dielectric strength of the
contact gap of the circuit breaker or switch and the recovery voltage. The latter is essentially
a characteristic of the circuit itself.
For inductive circuits, we know that the current lags the voltage by an angle less than ninety
electrical degrees. Thus, when the current is zero, the voltage is at its maximum. This means
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Current, A
SWITCHING TRANSIENT STUDIES
Time, s
Figure 11-14—Current in circuit with damping resistor
that, immediately after interruption of the arc, a rapid buildup of voltage across the breaker
contacts may cause the arc to re-ignite and re-establish the circuit. The rate by which the
voltage across the breaker rises depends on the inductance and capacitance of the circuit.
The simplest form of single-phase circuit that is useful to illustrate this phenomenon is that
shown in Figure 11-15.
Figure 11-15—Simplified diagram to illustrate TRV
In the circuit, L is the inductance of the source and C is the natural capacitance of the circuit
in the vicinity of the circuit breaker. It may include capacitance to ground through bushings,
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CHAPTER 11
current transformers, etc. The voltage source is assumed to vary sinusoidally and, since it is at
its peak at the time the short-circuit current is interrupted, it can be expressed as
v(t) =
2 × V × cos ( ωt )
(11-57)
where ω is the power frequency in radians per second (rad/s). If the switch opens, the flow of
current is interrupted at the first current zero and a voltage known as the transient recovery
voltage (TRV) will appear across the breaker contacts. This voltage is essentially the voltage
across the capacitance. It is zero during the fault but, when the circuit breaker opens to clear
the fault, the voltage across the contacts builds up to approximately twice the peak of the
voltage at the power frequency.
The equivalent circuit of Figure 11-15 may be analyzed by means of the Laplace transform.
The network equation in the s-domain for t = 0+ is
I (s)
–
V (s) = I (s)sL – LI ( 0 ) + --------sC
(11-58)
Solving the current I(s) with I(0–) = 0 (current interruption assumed to occur at zero current,
no current chopping) yields the following:
s
V (s)
I ( s ) = ----------- ---------------L s2 + ω2
0
(11-59)
Substituting sCVc(s) for I(s), the result is
2
ω0
V c ( s ) = V ( s ) ---------------2
2
s + ω0
(11-60)
The Laplace transform of the driving function described by Equation (11-57) is
s
V ( s ) = V max ---------------2
2
s +ω
(11-61)
Combining Equations (11-60) and (11-61), the recovery voltage or the voltage across the
capacitor is
2
sω 0
V c ( s ) = V max -------------------------------------------2
2
2
2
( s + ω ) ( s + ω0 )
336
(11-62)
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SWITCHING TRANSIENT STUDIES
From the table of the inverse Laplace transforms, the transient response is
V max
V c ( t ) = --------------2- [ cos ( ωt ) – cos ( ω 0 t ) ]
ω
1 – -----2ω0
(11-63)
The events before and after the fault are depicted in Figure 11-16 with damping. However,
without damping as described by Equation (11-63), the recovery voltage reaches a maximum
of twice the source voltage (the peak occurs at one half cycle of the natural frequency, after
the switch is opened). This is true when the natural frequency is high as compared with the
fundamental frequency and when losses are insignificant. Losses (damping) will reduce the
maximum value of Vc, as shown in Figure 11-16.
I fault
TRV, V
Time, s
Figure 11-16—Transient recovery voltage
Upon interruption of the fault current by the circuit breaker, the source attempts to charge the
capacitor voltage to the potential of the supply. As a matter of fact, without damping, the
capacitor voltage will overshoot the supply voltage by the same amount as it started below. If
the natural frequency of the circuit is high (L and C very small), the voltage across the
breaker contacts will rise very rapidly. If this rate-of-rise exceeds the dielectric strength of the
medium between the contacts, the breaker will not be able to sustain the voltage and
re-ignition will occur.
11.1.9 Summary
The material covered thus far is by no means an exhaustive discussion of electrical transients
in power systems. The objective of the foregoing material is to provide the reader with the
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basic techniques required to perform simple switching transient calculations. We have seen
that, even for simple series or parallel RLC circuits, the mathematical expressions can be
quite cumbersome and very difficult to solve analytically. It is evident that any slight increase
in circuit complexity will result in expressions very difficult to handle and solve by
conventional methods.
Typical industrial power distribution systems will involve many series and parallel circuit
combinations with very complex relationships. To set down and solve analytically the
equations representing such a system would be a formidable task. This is when solutions by
computer methods are most appropriate. Two of the most common computer methods are
analog and digital. The analog computer makes use of scaled-down components, i.e.,
resistors, inductors, and capacitors, to model a particular system. The digital computer, on the
other hand, utilizes computer programs (software packages) developed especially for the
purpose of transient analysis.
11.2 Switching transient studies
11.2.1 Introduction
Unlike classical power system studies, i.e., short circuit, load flow, etc., switching transient
studies are conducted less frequently in industrial power distribution systems. Capacitor and
harmonic filter bank switching in industrial and utility systems account for most of such
investigations, to assist in the resolution of certain transient behavioral questions in
conjunction with the application or failure of a particular piece of equipment.
Two basic approaches present themselves in the determination and prediction of switching
transient duties in electrical equipment: direct transient measurements (to be discussed later
in this chapter) and computer modeling. The latter can be divided into transient network
analyzer (TNA) and digital computer modeling.
In 11.1, some useful insights regarding the physical aspects prevailing in a circuit during a
transient period were obtained with a minimum of mathematical complications. In fact,
experienced transient analysts use known circuit-response patterns, based on a few basic
fundamentals, to assess the general transient behavior of a particular circuit and to judge the
validity of more complex switching transient results. Indeed, simple configurations consisting
of linear circuit elements can be processed by hand as a first approximation. Beyond these
relatively simple arrangements, the economics and effective determination of electrical power
system transients require the utilization of TNAs or digital computer programs. These two
approaches to the solution of complex switching transients in power systems are the subject
of 11.2. Excerpts from actual switching transient studies are included.
11.2.2 Switching transient study objectives
The basic objectives of switching transient investigations are to identify and quantify
transient duties that may arise in a system as a result of intentional or unintentional switching
events, and to prescribe economical corrective measures whenever deemed necessary. The
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results of a switching transient study can affect the operating procedures as well as the equipment in the system. The following include some specific broad objectives, one or more of
which are included in a given study:
a)
b)
c)
d)
e)
Identify the nature of transient duties that can occur for any realistic switching
operation. This includes determining the magnitude, duration, and frequency of the
oscillations.
Determine if abnormal transient duties are likely to be imposed on equipment by the
inception and/or removal of faults.
Recommend corrective measures to mitigate transient overvoltages and/or
overcurrents. This may include solutions such as resistor pre-insertion, tuning
reactors, appropriate system grounding, and application of surge arresters and surgeprotective capacitors.
Recommend alternative operating procedures, if necessary, to minimize transient
duties.
Document the study results on a case-by-case basis in readily understandable form
for those responsible for design and operation. Such documentation usually includes
reproduction of waveshape displays and interpretation of, at least, the limiting cases.
11.2.3 Control of switching transients
The philosophy of mitigation and control of switching transients revolves around the
following:
a)
b)
c)
d)
e)
f)
Minimizing the number and severity of the switching events
Limitation of the rate of exchange of energy that prevails among system elements
during the transient period
Extraction of energy
Shifting the resonant points to avoid amplification of particular offensive frequencies
Provision of energy reservoirs to contain released or trapped energy within safe limits
of current and voltage
Provision of discharge paths for high-frequency currents resulting from switching
In practice, this is usually accomplished through one or more of the following
methods:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
Temporary insertion of resistance between circuit elements; for example, the
insertion of resistors in circuit breakers
Synchronized closing control for vacuum and SF6 breakers and switches
Inrush control reactors
Damping resistors in filter and surge protective circuits
Tuning reactors
Surge capacitors
Filters
Surge arresters
Necessary switching only, with properly maintained switching devices
Proper switching sequences
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11.2.4 Transient network analyzer (TNA)
11.2.4.1 Introduction
Through the years, a small number of TNAs have been built for the purpose of performing
transient analysis in power systems. A typical TNA is made of scaled-down power system
component models, which are interconnected in such a way as to represent the actual system
under study. The inductive, capacitive, and resistive characteristics of the various power
system components are modeled with inductors, capacitors, and resistors in the analyzer.
These have the same oersted ohmic value as the actual components of the system at the power
frequency. The analyzer generally operates in the range of 10–100 Vrms line-to-neutral, which
represents 1.0 per-unit voltage on the actual system.
The model approach of the TNA finds its virtue in the relative ease with which individual
components can duplicate their actual power system counterparts as compared with the
difficulty of accurately representing combinations of nonlinear interconnected elements in a
digital solution. Furthermore, the switching operation that produces the transients is under the
direct control of the operator, and the circuit can easily be changed to show the effect of any
parameter variation. TNA simulation is also faster than digital simulation especially for larger
systems with many nonlinear elements to model.
11.2.4.2 Modeling techniques
Typical hardware used in a TNA to model the actual system components will be described
now. However, it should be fully recognized that any specific set of components can be
modeled in more than one way, and considerable judgment on the part of the TNA staff is
necessary to select the optimum model for a given situation. Also, it should be recognized
that, while there is a great similarity among the components of the various TNAs in existence
today, there are also unique hardware approaches to any given system. The following is a
general description of some of the hardware models.
a)
b)
c)
d)
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Transmission lines are modeled basically as a four-wire system, with three wires
associated with the phase conductors and the fourth wire encompassing the effects of
shield wire and earth return.
Circuit breakers consist of a number of independent mercury-wetted relay contacts or
solid-state electronic circuitry. The instant of both closing and opening of each
individual switch can be controlled by the operator or the computer system. The
model has the capability of simulating breaker actions like pre-striking, re-striking,
and re-ignition.
Shunt reactors can be totally electronic or analog with variable saturation
characteristics and losses.
Transformers are a critical part of the TNA. This is because many temporary
overvoltages include the interaction of the nonlinear transformer magnetizing branch
with the system inductance and capacitance. Modeling of the nonlinear magnetic
representation of the transformer is very critical to analyzing ferroresonance and
dynamic overvoltages. The model consists of both an array of inductors, configured
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e)
f)
g)
h)
i)
j)
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and adjusted to represent the linear inductances of the transformer, and adjustable
saturable reactors, representing the nonlinear portion of the saturation characteristics.
Arresters of both silicon carbide and metal oxide can be modeled. The models for
both types of arresters can be totally electronic and provide energy dissipation values
to safely size the surge arresters.
Secondary arc, available in some TNA facilities, is a model that can simulate a fault
arc and its action after the system circuit breakers are cleared.
Power sources can be three-phase motor-generator sets or three-phase electronic
frequency converters. The short-circuit impedance of these sources is such that they
appear as an infinite bus on the impedance base of the analyzer.
Synchronous machines can be either totally electronic or analog models, and are used
to study the effects of load rejection or other events that could be strongly affected by
the action of the synchronous machine.
Static var systems include an electronic control circuit, a thyristor-controlled reactor,
and a fixed capacitor with harmonic filters. The control logic circuit monitors the
three-phase voltages and currents and can be set to respond to either the voltage level,
the power factor, or some combination of the two.
Series capacitor protective devices are used in conjunction with series compensated
ac transmission lines. When a fault occurs, the voltage on the series capacitor rises to
a high value unless it is bypassed by protective devices, such as power gap or metaloxide varistors. The TNA can represent both of these devices.
11.2.5 Capacitor bank switching—TNA case study
11.2.5.1 Introduction
The following describes a case study in which a customer planned to install a total of
75 Mvar of switched capacitor banks at a 115 kV substation. The design called for two
separately switched 37.5 Mvar banks to compensate for var loading and voltage drop that
would occur in the system when power was being imported from other sources. Since this
was the customer’s first experience with capacitor bank installation above 34.5 kV, a request
was made for a TNA study to determine the transient overvoltages that could result during
energization of the capacitor banks.
11.2.5.2 Study objectives
The primary objective of this investigation was to determine if any switching surge
overvoltage problems could be experienced when the proposed capacitor banks are added to
the 115 kV substation. The system was modeled in the TNA to determine the switching surge
voltages that can be generated during normal and abnormal switching conditions for the
specific purpose of determining the following:
a)
b)
c)
The influence of the capacitor banks on the existing surge arresters and the
application of protective devices at the buses where the capacitors will be located (see
Figure 11-17)
If pre-insertion resistors are required for the capacitor bank circuit breakers
Current-limiting reactor requirements for both capacitor banks
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d)
If any magnification of the capacitor switching transient voltages at remote system
locations is a possibility
e)
If the system is susceptible to resonance due to added capacitor banks
f)
Traveling wave voltage effects at transformer terminated lines
11.2.5.3 Study results
The system being investigated is depicted in Figure 11-17. The proposed capacitor banks are
connected to the 115 kV bus through the circuit breakers A and B. The entire investigation
consisted of thirty-two different system configurations and switching operations. Due to
space limitations, however, only the results of two of these cases will be presented here,
namely the energization of both capacitor banks. The results of three-phase re-strike, fault
initiation, line energization, etc., which were part of the study, will not be presented.
There are two or more output pages for each case investigated. The first page tabulates the
system voltages recorded for the various system conditions as identified by the headings.
They include both the temporary pre-switching, energizing, and post-switching voltages, as
shown in Figure 11-18 (a), (b), and (c).
The succeeding pages display the statistical distribution curves of the transient voltages and/
or the oscillograms of the voltage, current, and/or waveforms taken during the investigation,
as shown in Figures 11-19, 11-20, and 11-21 for case 1. The results of case 2, that is,
energization of capacitor bank 2, are shown in Figure 11-22 (a), (b), and (c), through
Figure 11-26.
11.2.5.4 Discussion
A maximum transient voltage of 1.38 pu and 1.64 pu was calculated during energization of
each of two 37.5 Mvar banks at the 115 kV bus (66.5 kV line-to-ground), locations 4 and 5.
The 1.64 pu (154 kV peak line-to-neutral) was recorded in case 1, where the first of the two
banks was energized. In case 2, the 1.38 pu (130 kV peak line-to-neutral) transient voltage
was recorded as a result of energizing the second bank. In each of the two cases, the system
was operating under normal conditions and the capacitor switches did not include any closing
resistors or current-limiting reactors. Transient voltages of these magnitudes are generally not
considered to be of sufficient magnitude to cause a 96 kV rated conventional gapped-type
arrester to operate or to cause any undue stress to either a 90 kV or 96 kV rated metal-oxide
type arrester, connected at the line-to-ground system voltage of 66.5 kV.
The switching operations of both capacitor banks did not cause any serious transient
overvoltages at remote locations in the system and no resonant conditions were detected.
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Figure 11-17—System single one-line diagram
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(a) Pre-switching voltages
(b) Energizing voltages
(c) Post-switching voltages
Figure 11-18—System voltages—Case 1
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Figure 11-19—Probability distribution—Case 1
11.2.6 Electromagnetic transients program (EMTP)
11.2.6.1 Introduction
EMTP is a software package that can be used for single-phase and multiphase networks to
calculate either steady-state phasor values or electromagnetic switching transients. The
results can be either printed or plotted.
11.2.6.2 Network and device representation
The program allows for arbitrary connection of the following elements:
a)
b)
c)
d)
e)
Lumped resistance, inductance, and capacitance
Multiphase (π) circuits, when the elements R, L, and C become symmetric matrixes
Transposed and untransposed distributed parameter transmission lines with wave
propagation represented either as distortionless, or as lossy through lumped
resistance approximation
Nonlinear resistance with a single-valued, monotonically increasing characteristics
Nonlinear inductance with single-valued, monotonically increasing characteristics
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Figure 11-20—Voltage oscillations, locations 1 and 4—Case 1
f)
Time-varying resistance
g)
Switches with various switching criteria to simulate circuit breakers, spark gaps,
diodes, and other network connection options
h)
Voltage and current sources representing standard mathematical functions, such as
sinusoidals, surge functions, steps, ramps, etc. In addition, point-by-point sources as
a function of time can be specified by the user.
i)
Single- and three-phase, two- or three-winding transformers
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Figure 11-21—Current oscillograms, location 4—Case 1
11.2.7 Capacitor bank switching—EMTP case study
11.2.7.1 Introduction
As part of a modernization program that included the addition of two paper machine drives to
the existing system, it was determined that a 10 Mvar capacitor bank was required to improve
the plant power factor and the system voltage profile. Further analysis also indicated the need
for a tuning reactor in series with the capacitor bank in order to minimize the effects of
harmonic resonance problems. Because of recent plant outages caused by what appeared to
be normal switching operations, and because the proposed capacitor bank would require
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(a) Pre-switching voltages
(b) Energizing voltages
(c) Post-switching voltages
Figure 11-22—System voltages—Case 2
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Figure 11-23—Probability distribution—Case 2
frequent switching to meet system voltage and power factor requirements, the customer
requested that a switching transient investigation be conducted to determine the voltage and
current waveforms associated with the switching of the proposed capacitor bank.
11.2.7.2 Study objectives
The objectives of the study were to assist the customer in evaluating the effect of filter bank
switching transients and in determining the solution to minimize these effects on the
electrical system and equipment. Specifically, the study addressed the transient voltages and
current waveforms during energization of the filter bank and the effects that these transients
might have on the slip energy recovery drive and on the proposed dc drives for the new paper
machines.
11.2.7.3 Circuit model and cases studied
The study circuit and pertinent system parameters used in the study are depicted in
Figure 11-27.
Table 11-1 describes the cases studied. Various system configurations were investigated to
determine the transient voltage waveforms associated with the energization of the filter bank.
The switching operation for the cases investigated (as listed in Table 11-1) was initiated when
the phase-to-phase voltage (Va–b) at the STPT bus was at its peak (t = 8.4 ms). When resistor
pre-insertion is used, it remains in the circuit for a period of three cycles and then is shorted
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Figure 11-24—Voltage oscillograms, locations 3 and 5—Case 2
out by a second switching operation, as depicted in the single-line diagram shown in
Figure 11-27.
11.2.7.4 Study results and discussion
Selected transient voltage waveforms that were calculated and plotted by the program for
cases 1, 8, and 9 are shown in Figure 11-28 through Figure 11-33.
Tables 11-2 and 11-3 summarize the results of all cases, for the worst peak overvoltages
calculated, in kilovolts and in per units, respectively.
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Figure 11-25—Current oscillograms, locations 4 and 5—Case 2
The following are some observations:
a)
b)
c)
Removal of the 325 kvar capacitor bank on bus L135 (case 2) eliminates the high
frequency oscillations (1000 Hz) experienced in case 1.
The transients are substantially reduced when the 10 Mvar filter bank is divided into
two 5 Mvar banks (cases 4 and 5).
The transient decay is faster when pre-insertion resistors are used (cases 5, 8, and 9).
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Figure 11-26—Current oscillograms, locations 4 and 5—Case 2
expanded time scale
d)
The magnitude of the transient overvoltages is greatly reduced when the 10 Mvar
bank is divided into two 5 Mvar banks and when resistor pre-insertion (5.2 Ω) is used
during energization (cases 8 and 9).
11.2.8 Summary
Complete switching transient study documentation includes not only detailed individual case
study results for transient responses associated with various arrangements and conditions
surveyed, but also analysis, recommendations, and conclusions of the study. The study report
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Table 11-1—Filter energization—Cases studied
Case
Description
1
Energization of 10 Mvar filter bank
2
Energization of 10 Mvar filter bank, with 325 kvar capacitor bank disconnected
3
10 Mvar filter bank divided into two 5 Mvar banks, energization of first
5 Mvar filter bank
4
Energization of second 5 Mvar filter bank
5
Energization of 10 Mvar filter bank with resistor pre-insertion;
(R = 2.6 Ω for 3 cycles)
6
Energization of 10 Mvar filter bank with resistor pre-insertion;
(R = 26 Ω for 3 cycles)
7
Energization of 10 Mvar filter bank with resistor pre-insertion;
(R = 13 Ω for 3 cycles)
8
10 Mvar filter bank divided into two 5 Mvar banks, energization of first 5 Mvar
filter bank, with resistor pre-insertion; (R = 5.2 Ω for 3 cycles)
9
Energization of second 5 Mvar filter bank; with resistor pre-insertion;
(R = 5.2 Ω for 3 cycles)
also includes a complete listing of parameters (R, L, and C) of various system components,
characteristics of protective devices, and a description of any unusual or specialrepresentations used in the study.
11.2.9 Switching transient problem areas
Switching of predominantly reactive equipment represents the greatest potential for creating
excessive transient duties. Principal offending situations are switching capacitor banks with
inadequate or malfunctioning switching devices and energizing and de-energizing
transformers with the same switching deficiencies. Capacitors can store, trap, and suddenly
release relatively large quantities of energy. Similarly, highly inductive equipment possesses
an energy storage capability that can also release large quantities of electromagnetic energy
during a rapid current decay. Since transient voltages and currents arise in conjunction with
energy redistribution following a switching event, the greater the energy storage in associated
system elements, the greater the transient magnitudes become.
Generalized switching transient studies have provided many important criteria to enable
system designers to avoid excessive transients in most common circumstances. The criteria
for proper system grounding to avoid transient overvoltages during a ground fault are a prime
example. There are also several not very common potential transient problem areas that are
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Figure 11-27—System single-line diagram
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Figure 11-28—Voltage oscillograms at STPT and DFBT buses—Case 1
analyzed on an individual basis. The following is a partial list of transient-related problems,
which can and have been analyzed through computer modeling:
a)
b)
c)
d)
e)
Energizing and de-energizing transients in arc furnace installations
Ferroresonance transients
Lightning and switching surge response of motors, generators, transformers,
transmission towers, cables, etc.
Lightning surges in complex station arrangements to determine optimum surge
arrester location
Propagation of switching surge through transformer and rotating machine windings
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Figure 11-29—Voltage oscillograms at DFLT and YFLT buses—Case 1
f)
g)
h)
i)
j)
k)
l)
m)
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Switching of capacitors
Restrike phenomena during line dropping and capacitor de-energization
Neutral instability and reversed phase rotation
Energizing and reclosing transients on lines and cables
Switching surge reduction by means of controlled closing of circuit breaker, resistor
pre-insertion, etc.
Statistical distribution of switching surges
Transient recovery voltage on distribution and transmission systems
Voltage flicker
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Figure 11-30—Voltage oscillograms at STPT and DFBT buses—Case 8
The studies presented in this chapter have been primarily based on closing or opening of
electrical circuits and, therefore, are not generally applicable to transfer switching in
emergency and standby power systems. Here, significant transients often occur when
inductive loads are rapidly transferred between two out-of-phase sources. Transients can also
occur when four-pole transfer switches are both used for line and neutral switching, as may
be necessary for separately derived systems. Typical solutions for such problem areas often
require transfer switch designs that include in-phase monitors and overlapping neutral
conductor switching. For further reading on this subject, see IEEE Std 446-1995.1
1Information
on references can be found in 11.5.
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Figure 11-31—Voltage oscillograms at DFLT and YFLT buses—Case 8
The behavior of transformer and machine windings under transient conditions is also an area
of great concern. Due to the complexities involved, it would be almost impossible to cover
the subject in this chapter. For those interested, Chapter 11 of Greenwood [B5]2 covers the
subject in greater detail. Mazur [B6] and White [B11] also cover transients in transformers
and rotating machines.
2The
358
numbers in brackets correspond to those of the bibliography in 11.6.
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Figure 11-32—Voltage oscillograms at STPT and DFLT buses—Case 9
11.3 Switching transients—field measurements
11.3.1 Introduction
The choice of measuring equipment, auxiliary equipment selection, and techniques of setup
and operation are in the domain of practiced measurement specialists. No attempt will be
made here to delve into such matters in detail, except from the standpoint of conveying the
depth of involvement entailed by switching transient measurements and from the standpoint
of planning a measurement program to secure reliable transient information of sufficient
scope for the intended purpose.
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Figure 11-33—Voltage oscillograms at DFLT and YFLT buses—Case 9
Field measurements seldom, if ever, include fault switching, and often, recommended
corrective measures are not in place to be used in the test program except on a followup basis.
For systems still in the design stage or when fault switching is required, the transient response
is usually obtained with the aid of a TNA or a digital computer program. There are basically
three types of transients to consider in field measurements:
a)
b)
c)
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SWITCHING TRANSIENT STUDIES
Table 11-2—Summary of maximum calculated voltage in kilovolts
Study case
Bus
1
2
3
4
5
6
7
8
9
STPT
26.570
25.710
25.220
22.920
22.300
26.730
25.300
21.450
22.110
L135
5.70
4.79
4.83
4.29
5.02
5.43
4.71
4.40
3.95
L135G
3.70
2.54
2.78
2.07
2.62
2.96
2.74
2.25
2.20
L136
0.95
0.93
0.89
0.81
0.86
N/A
0.87
0.80
0.98
L136G
0.53
0.49
0.47
0.47
0.45
N/A
0.52
0.43
0.44
DFBT
1.62
1.43
1.33
1.14
1.15
N/A
1.25
1.04
1.06
YFBT
1.53
1.40
1.39
1.17
0.96
N/A
1.28
0.98
1.02
DFLT
2.57
2.52
2.37
1.75
1.52
N/A
1.89
1.27
1.32
YFLT
2.70
2.64
2.74
1.73
1.51
N/A
2.00
1.26
1.34
Table 11-3—Summary of maximum calculated voltage in per units
Study case
Bus
1
2
3
4
5
6
7
8
9
STPT
1.36
1.32
1.29
1.17
1.14
1.37
1.30
1.10
1.13
L135
1.75
1.47
1.49
1.32
1.54
1.67
1.45
1.35
1.21
L135G
1.63
1.35
1.48
1.10
1.39
1.58
1.46
1.20
1.17
L136
1.39
1.36
1.31
1.19
1.27
N/A
1.28
1.18
1.44
L136G
1.36
1.25
1.20
1.20
1.14
N/A
1.32
1.10
1.13
DFBT
1.99
1.76
1.63
1.40
1.41
N/A
1.53
1.28
1.31
YFBT
1.88
1.72
1.71
1.43
1.18
N/A
1.57
1.20
1.25
DFLT
3.16
3.10
2.92
2.15
1.87
N/A
2.32
1.56
1.62
YFLT
3.33
3.25
3.37
2.13
1.86
N/A
2.46
1.55
1.65
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The first category includes transients incurred when switching a device on or off. The second
category covers the transients occurring regularly, for example, commutation transients. The
final category refers to transients are those of usually unknown origin, generated by extraneous operations on the system. These may include inception and interruption of faults, lightning strikes, etc. To detect and/or record random transients, it is necessary to monitor the
system continuously.
11.3.2 Signal derivation
The ideal result of a transient measurement, or for that matter, any measurement at all, is to
obtain a perfect replica of the transient voltage and current as a function of time. Quite often,
the transient quantity to be measured is not obtained directly and must be converted, by
means of transducers, to a voltage or current signal that can be safely recorded. However,
measurements in a system cannot be taken without disturbing it to some extent. For example,
if a shunt is used to measure current, in reality, voltage is being measured across the shunt to
which the current gives rise. This voltage is frequently assumed to be proportional to the
current, when, in fact, this is not always true with transient currents. Or, if the voltage to be
measured is too great to be handled safely, appropriate attenuation must be used. In steadystate measurements, such errors are usually insignificant. But in transient measurements, this
is more difficult to do. Therefore, since switching transients involve natural frequencies of a
very wide range (several orders of magnitude), signal sourcing must be by special current
transformers (CTs), non-inductive resistance dividers, non-inductive current shunts, or
compensated capacitor dividers, in order to minimize errors. While conventional CTs and
potential transformers (PTs) can be suitable for harmonic measurements, their frequency
response is usually inadequate for switching transient measurements.
11.3.3 Signal circuits, terminations, and grounding
Due to the very high currents with associated high magnetic flux concentrations, it is
essential that signal circuitry be extremely well shielded and constructed to be as
interference-free as possible. Double-shielded low loss coaxial cable is satisfactory for this
purpose. Additionally, it is essential that signal circuit terminations be made carefully with
high-quality hardware and assure proper impedance match in order to avoid spurious
reflections.
It is desirable that signal circuits and instruments be laboratory-tested as an assembly before
field measurements are undertaken. This testing should include the injection of a known wave
into the input end of the signal circuit and comparison of this waveshape with that of the
receiving instruments. Only after a close agreement between the two waveshapes is achieved
should the assembly be approved for switching transient measurements. These tests also aid
overall calibration.
All the components of the measurement system should be grounded via a continuous
conducting grounding system of lowest practical inductance to minimize internally induced
voltages. The grounding system should be configured to avoid ground loops that can result in
injection of noise. Where signal cables are unusually long, excessive voltages can become
362
Copyright © 1998 IEEE. All rights reserved.
SWITCHING TRANSIENT STUDIES
IEEE
Std 399-1997
induced in their shields. Industrial switching transient measurement systems have not, as yet,
involved such cases.
11.3.4 Equipment for measuring transients
The complement of instruments used depends on the circumstances and purpose of the test
program. Major items comprising the total complement of display and recording
instrumentation for transient measurements are one or more of the following:
a)
b)
c)
One or more oscilloscopes, including a storage-type scope with multichannel
switching capability. When presence of the highest speed transients (that is, those
with front times of less than a microsecond) is suspected, a high speed, single trace
surge test oscilloscope with direct cathode ray tube (CRT) connections is sometimes
used to record such transients with the least possible distortion.
Multichannel magnetic light beam oscillograph with high input impedance
amplifiers.
Peak-holding digital readout memory voltmeter (sometimes called “peakpicker”) that
is manually reset.
The occurrence of most electrical transients is quite unpredictable. To detect and/or record
random disturbances, it is necessary to monitor the circuit on a continuous basis. There are
many instruments available in the market today for this purpose. Most of these instruments
are computer based; that is, the information can be captured digitally and later retrieved for
display or computer manipulation. These instruments vary in sophistication depending on the
type and speed of transient measurements that are of interest.
11.4 Typical circuit parameters for transient studies
11.4.1 Introduction
Tables 11-4 through 11-11 and Figures 11-34 through 11-39 depict typical parameters used in
switching transient analysis. Compared to conventional power system studies, switching transient analysis data requirements are often more detailed and specific. These requirements
remain basically unchanged regardless of the basic analysis tools and aids that are employed,
whether they are digital computer or transient network analyzers.
To determine the transient response of a circuit to a specific form of excitation, it is first
necessary to reduce the network to its simplest form composed of Rs, Ls, and Cs. After
solving the circuit equations for the desired unknown, values must be assigned to the various
circuit elements in order to determine the response of the circuit.
11.4.2 System and equipment data requirements
The following generalized data listed encompass virtually all information areas required in an
industrial power system switching transient study:
Copyright © 1998 IEEE. All rights reserved.
363
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Std 399-1997
CHAPTER 11
Table 11-4—Approximate positive sequence reactance values for standard
25- to 60-cycle, self-cooled, two-winding power transformers
Percent reactance
Rated
high
voltage
Rated
low
voltage
Fully insulated
With reduced
neutral
insulation
Reduced one
insulation
class with
reduced
neutral
insulation
Min.
Max.
Min.
Max.
Min.
Max.
2 400–15 000
440–15 000
4.5
7.0
15 001–25 000
440–15 000
5.5
8.0
25 001–34 500
440–15 000
15 001–25 000
6.0
6.5
8.0
9.0
34 501–46 000
440–15 000
25 001–34 500
6.5
7.0
9.0
10.0
46 001–69 000
400–34 500
34 501–46 000
7.0
8.0
10.5
11.0
69 001–92 000
440–34 500
34 501–69 000
7.5
8.5
10.5
12.5
7.0
8.0
10.0
11.5
92 001–115 000
440–34 500
34 501–69 000
69 001–92 000
8.0
9.0
10.0
12.0
14.0
15.5
7.5
8.5
9.5
10.5
12.5
14.0
7.0
8.0
9.0
10.0
11.5
13.0
115 001–138 000
440–34 500
34 501–69 000
69 001–115 000
8.5
9.5
10.5
13.0
15.0
17.0
8.0
9.0
10.0
12.0
14.0
16.0
7.5
8.5
9.5
10.5
12.0
14.0
138 001–161 000
440–46 000
46 001–92 000
92 001–132 000
9.0
10.5
11.5
14.0
16.0
18.0
8.5
9.5
10.5
13.0
15.0
17.0
8.0
9.0
10.0
12.0
14.0
16.0
161 001–196 000
400–46 000
46 001–92 000
92 001–161 000
10.0
11.5
12.5
15.0
17.0
19.0
9.0
10.5
11.5
14.0
16.0
18.0
8.5
9.5
10.5
13.0
15.0
17.0
196 001–230 000
400–46 000
46 001–92 000
92 001–161 000
11.0
12.5
14.0
16.0
18.0
20.0
10.0
11.5
12.5
15.0
17.0
19.0
9.0
10.5
11.5
14.0
16.0
18.0
364
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IEEE
Std 399-1997
SWITCHING TRANSIENT STUDIES
Table 11-5—Outdoor bushing capacitance to ground
kV
A Rating
Range in pF
kV
A Rating
Range in pF
15.0
600
1200
160–180
190–220
115.0
23.0
400
600
1200
2000
3000
4000
200–450
280
190–450
280–650
370–560
500–620
800
1200
1600
250–450
250–430
250–430
138.0
800
1200
1600
250–450
250–420
250–460
161.0
400
600
1200
2000
3000
200–390
150–220
170–390
240–360
350–620
800
1200
1600
260–440
260–440
260–440
196.0
800
1200
1600
350–550
350–550
350–550
400
600
1200
2000
180–330
150–280
170–330
200–330
330.0
1600
530
345.0
400
600
1200
2000
180–270
250
160–290
210–320
820–2000
BIL:1050
1175
1300
550
500
450
500.0
800–2000
BIL:1425
1550
1675
500
500
520
34.5
46.0
69.0
a)
Single-line diagram of the system showing all circuit elements and connection options
b)
Utility information, for each tie, at the connection point to the tie. This should include
1)
Impedances R, XL, XC, both positive and zero sequence representing minimum
and maximum short-circuit duty conditions
2)
Maximum and minimum voltage limits
3)
Description of reclosing procedures and any contractual limitations, if any
c)
Individual power transformer data, such as rating; connections; no-load tap voltages;
LTC voltages, if any; no-load saturation data; magnetizing current; positive and zero
sequence leakage impedances; and neutral grounding details
d)
Capacitor data for each bank, connections, neutral grounding details, description of
switching device and tuning reactors, if any
e)
Impedances of feeder cables or lines, that is, R, XL, and XC (both positive and zero
sequence)
f)
Information about other power system elements, such as
1)
Surge arrester type, location and rating
Copyright © 1998 IEEE. All rights reserved.
365
IEEE
Std 399-1997
CHAPTER 11
Table 11-6—Synchronous machine constants
Approximate reactances in percentage of machine
kVA rating
Xd
X ′d
X ″d
X2
X0
X eq
Opencircuit
time
constant
T do (s)
9
7–14
11
9–16
3
1–8
Turbine generators,
two-pole
Average
Range
115
95–145
15
12–21
75
60–100
4
3–7
Turbine generators,
four-pole
Average
Range
115
95–145
23
14
16
5
75
20–28 12–17 14–19 1.5–14 60–100
6
4–9
Waterwheel generator,
without amortisseur
windings
Average
Range
100
60–145
35
30
50
20–45 17–40 30–65
7
4–25
65
40–100
5
2–10
Waterwheel generators,
with amortisseur
windings
Average
Range
100
60–145
35
22
22
20–45 13–35 13–35
7
4–25
65
40–100
5
2–10
Synchronous
condensers
Average
180
40
25
25
Range 150–220 30–60 20–35 20–35
8
2–15
70
60–90
8
5–12
Salient-pole motors,
high-speed
Average
Range
80
65–90
25
18
19
15–35 10–25 10–25
5
2–15
50
40–60
2.5
1–4
Salient-pole motors,
low-speed
Average
Range
110
80–150
50
35
35
40–70 25–45 25–45
7
4–27
70
50–100
2.5
1–4
NOTE—With the exception of X ″d for turbine generators and the X eq column, the above figures
represent the approximate average and range of machine constants for both rated voltage and rated
current conditions. The figures given for X ″d for turbine generators represent rated voltage values. The
values given for X eq are representative figures for machines of normal design operating at their fullload ratings.
2)
3)
g)
Grounding resistors or reactors, rating and impedance of buffer reactors
Rating, subtransient and transient reactance of rotating machines, grounding
details, etc.
Operating modes and procedures
The material presented in the following pages is a compendium of parameter values, such as
Rs, Ls, and Cs, for typical power system components that can be used in lieu of actual values.
Most of the tabulated values were obtained from IEEE Std C37.011-1994. (This standard is
in the process of being updated by the TRV Working Group of the IEEE Switchgear
Committee.)
366
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IEEE
Std 399-1997
SWITCHING TRANSIENT STUDIES
Table 11-7—Instrument transformer capacitance (primary winding to ground
and to secondary with its terminals shorted and grounded)
Capacitance in pF
Insulation
class kV
Potential transformers
Current
transformers
Line-to-line
Line-to-neutral
15
260
—
—
25
250–440
270–800
180–260
.34.5
310–440
270–900
160–250
46
350–430
300–970
170–220
69
360–440
340–1300
170–260
115
470–520
480–610
210–320
138
490–550
530–660
—
161
510–580
510–700
310–380
196
—
580–820
330–390
230
600–680
600–810
350–420
345
—
920
—
11.5 References
IEEE Std 446-1995, IEEE Recommended Practice for Emergency and Standby Power
Systems for Industrial and Commercial Applications (IEEE Orange Book).
IEEE Std C37.011-1994, IEEE Application Guide for Transient Recovery Voltage for AC
High-Voltage Circuit Breakers Rated on a Symmetrical Basis.
11.6 Bibliography
[B1] Dommel, H. W., “Electro-Magnetic Transients Program (EMTP) Manual,” Boneville
Power Administration, Portland, 1984.
[B2] Fich, S., Transient Analysis in Electrical Engineering. New York: Prentice-Hall, Inc.,
1971.
[B3] Gill, J. D., “Transfer of motor loads between out-of-phase sources,” IEEE Transactions
on Industry Applications, vol. IA-15, no. 4, pp. 376–381, Jul./Aug. 1979.
Copyright © 1998 IEEE. All rights reserved.
367
IEEE
Std 399-1997
CHAPTER 11
Table 1-8—Generator armature capacitance to ground
Generator size in MVA
(1)
Steam turbine driven:
Conventionally cooled (two-pole 3600 r/min)
15 up to 30
30 up to 50
50 up to 70
70 up to 225
225 up to 275
(four-pole 1800 r/min)
125 up to 225
0.17–0.36
0.22–0.44
0.27–0.52
0.34–0.87
1.49
0.04–1.41
Conductor cooled: gas (two-pole 3600 r/min)
100 up to 300
0.33–0.47
Conductor cooled: liquid (two-pole 3600 r/min)
190 up to 300
300 up to 850
0.27–0.67
0.49–0.68
(four-pole 1800 r/min)
250 up to 300
300 up to 850
Above 850
(2)
Total three-phase
winding capacitance to
ground in µF
0.37–0.38
0.71–0.94
1.47
Hydro driven:
720 to 360 r/min
10 to 30 MVA
0.26–0.53
225 to 85 r/min
25 to 100 MVA
0.90–1.64
NOTE—There is no direct correlation between generator MVA size, size limit, and capacitance limits. For instance, a 50 MVA generator may have an armature capacitance to ground anywhere from
0.27 to 0.52 µF, depending on machine design.
[B4] Goldman, S., Laplace Transform Theory and Electrical Transients. New York: Dover
Publications, 1966.
[B5] Greenwood, A., Electrical Transients in Power Systems, New York: John Wiley & Sons,
Inc., 1971.
[B6] Mazur, A., Kerszenbaum, I., and Frank, J., “Maximum insulation stress under transient
voltages in the HV barrel-type winding of distribution and power transformers,” IEEE
Transactions on Industry Applications, vol. IA-24, no. 3, May/June 1988.
[B7] Miller, R., Algebraic Transient Analysis. San Francisco: Reinhart Press, 1971.
368
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Std 399-1997
SWITCHING TRANSIENT STUDIES
Table 1-9—Phase bus capacitance
Segregated phase
bus
Isolated phase bus
15 kV class
110 kV BIL
(pF/ft)
23 kV class
150 kV BIL
(pF/ft)
15 kV class
110 kV BIL
(pF/ft)
1200
8.9–14.3
8.0–12.4
10.0
2000
2500
10.2–14.3
10.2–14.3
9.0–12.4
9.0–12.4
10.0–10.2
3000
3500
10.2–14.3
10.2–14.3
9.0–12.4
9.0–12.4
10.0–10.2
4000
4500
14.0–14.3
14.0–14.3
12.4–13.5
12.7–13.5
10.0–12.6
5000
5500
14.0–19.0
14.0–19.0
12.7–15.8
12.7–15.8
12.5–14.9
6000
6500
14.0–19.0
14.0–19.0
13.5–15.8
13.5–15.8
15.0–17.1
7000
7500
17.3–22.6
17.3–22.6
14.4–17.6
14.4–17.6
17.1
8000
9000
21.7
21.7
17.6
18.1
—
10 000
11 000
21.7
23.7
18.1
20.5
—
12 000
23.7
20.5
—
A
Rating
Table 1-10—Typical values of inductance between capacitor banks
Rated maximum voltage
(kV)
Inductance per phase of bus
(µH/ft)
Typical inductance between
banks (µH)
15.5 and below
0.214
10–20
38.0
0.238
15-39
48.3
0.256
20–40
72.5
0.256
25–50
121
0.261
35–70
145
0.261
40–80
169
0.268
60–120
Copyright © 1998 IEEE. All rights reserved.
369
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Std 399-1997
CHAPTER 11
Table 1-11—Typical transmission line characteristics 69 kV–230 kV
Nominal voltage (kV)
(line-to-line)
69
115
138
161
230
X1 (Ω/mi)
0.783
0.759
0.771
0.771
0.785
R1 (Ω/mi)
0.340
0.224
0.194
0.137
0.107
0.854
0.792
0.795
0.783
0.792
X1/R1
2.30
3.40
3.98
5.64
7.36
X0
2.37
2.30
2.48
2.22
2.35
R0
1.22
0.755
0.586
0.591
0.576
Z0
2.67
2.42
2.55
2.30
2.42
X0/R0
1.95
3.05
4.23
3.76
4.08
X0/X1
3.03
3.03
03.22
2.88
2.99
X1 (1/B1) (MΩ/mi)
0.184
0.178
0.183
0.174
0.177
B1 (µMho/mi)
5.43
5.63
5.46
5.73
5.63
B0 (Ω/Mi)
3.38
3.09
3.41
3.63
3.33
Surge impedance
Positive sequence (Ω)
397
375
381
370
375
Surge impedance
Zero sequence (Ω)
889
885
865
795
852
Surge impedance
Loading (SIL) (MVA)
0.216
35.3
49.9
70.1
141
Charging current (A/mi)
0.216
0.373
0.435
0.533
0.748
Charging MVA (MVA/mi)
0.0258
0.0774
0.104
0.149
0.298
Line current at SIL (A)
100
177
209
252
354
I2R loss at SIL (kW/mi)
3.43
7.00
8.45
8.60
13.4
Equivalent spacing (ft)
13.8
13.8
15.1
19.8
24.9
Z 1 (Ω/mi)
370
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SWITCHING TRANSIENT STUDIES
IEEE
Std 399-1997
Figure 1-34—Typical values of transformer winding capacitance to ground
Copyright © 1998 IEEE. All rights reserved.
371
IEEE
Std 399-1997
CHAPTER 11
A — Nominal X/R ratio for aluminum conductor reactor = 231K30.461
A'— “High Q” X/R ratio for aluminum conductor reactor = 5.09K30.379
B — Maximum X/R ratio for copper conductor reactor = 3.88K30.398
C — Nominal 60 Hz X/R ratio for tuning reactor = 3.07K30.377
D — Eddy current and stray loss factor for tuning reactor
= (1.16 x 10–3) (K30.353);
Rn = Resistance at fn (any frequency) = R60(1+n2ECF) / (1+ECF),
where R60 = 60 Hz resistance and n = fn/60
Figure 1-35—Typical X/R ratio and resistance of reactors
[B8] Peterson, H. A., Transients in Power Systems. New York: General Electric Company,
1951.
[B9] “Transient Network Analyzer Manual,” General Electric Company, New York, 1978.
[B10] Wadha, C. L., Electrical Power Systems. New Delhi: Wiley Eastner Limited, 1983.
[B11] White, E. L., Surge-Transference Characteristics of Generator/Transformer
Installations, Proceedings of the IEE, vol. 116, p. 575, 1969.
372
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Std 399-1997
Figure 1-36—Typical X/R ratio of generators
Figure 1-37—Typical charging current for cable
Copyright © 1998 IEEE. All rights reserved.
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Std 399-1997
CHAPTER 11
Figure 1-38—Typical X/R ratio of transformers
Figure 1-39—Typical X/R ratio of induction motors
374
Copyright © 1998 IEEE. All rights reserved.
Chapter 12
Reliability studies
12.1 Introduction
An important aspect of power system design involves consideration of service reliability
requirements of loads to be supplied and service reliability provided by any proposed system.
System reliability assessment and evaluation methods based on probability theory allow the
reliability of a proposed system to be assessed quantitatively. Such methods permit
consistent, defensible, and unbiased assessments of system reliability that are not otherwise
defensible, and that are not otherwise possible.
Quantitative reliability evaluation methods permit reliability indexes for any electric power
system computed from knowledge of the reliability performance of the constituent
components of the system. Thus, alternative system designs can be studied to evaluate the
impact on service reliability and cost of changes in component reliability, system
configuration, protection and switching scheme, and system operating policy including
maintenance practice. A detailed treatment of reliability evaluation methods is given in IEEE
Std 493-1997.1
12.2 Definitions
The definitions presented here provide much of the required nomenclature for discussions of
power system reliability.
12.2.1 availability: A term that applies either to the performance of individual components or
to a system. Availability is the long-term average fraction of time that a component or system
is in service satisfactorily performing its intended function. An alternative and equivalent
definition for availability is the steady-state probability that a component or system is in
service.
12.2.2 component: A piece of equipment, a line or circuit, or a section of a line or circuit, or
a group of items that is viewed as an entity for purposes of reliability evaluation.
12.2.3 expected interruption duration: The expected, or average, duration of a single load
interruption event.
12.2.4 exposure time: The time during which a component is performing its intended
function and is subject to failure. Usually expressed in years.
12.2.5 failure: Any trouble with a power system component that causes any of the following
to occur:
1Information
on references can be found in 12.6.
Copyright © 1998 IEEE. All rights reserved.
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IEEE
Std 399-1997
—
—
—
—
CHAPTER 12
Partial or complete shutdown, or below-standard plant operation
Unacceptable performance of user’s equipment
Operation of the electrical protective relaying or emergency operation of the plant
electrical system
De-energization of any electric circuit or equipment
A failure on a public utility supply system can cause the user to have either of the following:
—
—
A power interruption or loss of service
A deviation from normal voltage or frequency outside the normal utility profile
A failure of an in-plant component causes a forced outage of the component, that is, the
component is unable to perform its intended function until repaired or replaced. Syn: forced
outage.
12.2.6 failure rate (forced outage rate): The mean number of failures of a component per
unit of exposure time. Usually, expressed in failures per year.
12.2.7 forced outage: See: failure.
12.2.8 forced outage duration: See: repair time.
12.2.9 forced unavailability: The long-term average fraction of time that a component or
system is out of service due to a forced outage (failure).
12.2.10 interruption: The loss of electric power supply to one or more loads.
12.2.11 interruption frequency: The expected (average) number of power interruptions to a
load per unit time, usually expressed as interruptions per year.
12.2.12 outage: The state of a component or system when it is not available to properly
perform its intended function due to an event directly associated with that component or
system.
12.2.13 repair time: The repair time of a failed component or the duration of a failure is the
clock time from the occurrence of the failure to the time when the component is restored to
service, either by repair of the failed component or by substitution of a spare component for
the failed component. It includes time for diagnosing the trouble, locating the failed
component, waiting for parts, repairing or replacing, testing, and restoring the component to
service. It does not include the time required to restore service to a load by putting alternate
circuits into operation. Syn: forced outage duration.
12.2.14 scheduled outage: An outage that results when a component is deliberately taken out
of service at a selected time, usually for purposes of construction, maintenance, or repair.
376
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RELIABILITY STUDIES
IEEE
Std 399-1997
12.2.15 scheduled outage duration: The period from the initiation of a scheduled outage
until construction, preventive maintenance, or repair work is completed and the affected
component is made available to perform its intended function.
12.2.16 scheduled outage rate: The mean number of scheduled outages of a component per
unit exposure time.
12.2.17 switching time: The period from the time a switching operation is required due to a
component failure until that switching operation is completed. Switching operations include
such operations as throwover to an alternate circuit, opening or closing a sectionalizing
switch or circuit breaker, reclosing a circuit breaker following a tripout from a temporary
fault, etc.
12.2.18 system: A group of components connected or associated in a fixed configuration to
perform a specified function of distributing power.
12.2.19 unavailability: The long-term average fraction of time that a component or system is
out of service due to failures or scheduled outages. An alternative definition is the steadystate probability that a component or system is out of service. Mathematically,
unavailability = (1 – availability).
12.3 System reliability indexes
The basic system reliability indexes that have proven most useful and meaningful in power
distribution system design are as follows:
—
Load interruption frequency
—
Expected duration of load interruption events
These indexes can be readily computed using the methods in IEEE Std 493-1997. The two
basic indexes of interruption frequency and expected interruption duration can be used to
compute other indexes that are also useful:
—
Total expected (average) interruption time per year (or other time period)
—
System availability or unavailability as measured at the load supply point in question
—
Expected, demanded, but unsupplied, energy per year
Note that the disruptive effect of power interruption is often nonlinearly related to the
duration of the interruption. Thus, it is often desirable to compute not only an overall
interruption frequency but also frequencies of interruptions categorized by the appropriate
durations.
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 12
12.4 Data needed for system reliability evaluations
The data needed for quantitative evaluations of system reliability depend to some extent on
the nature of the system being studied and the detail of the study. In general, however, data on
the performance of individual components together with the times required to perform
various switching operations are required.
System component data generally required are summarized as follows:
—
—
—
—
Failure rates (forced outage rates) associated with different modes of component
failure
Expected (average) time to repair or replace failed component
Scheduled (maintenance) outage rate of component
Expected (average) duration of a scheduled outage event
If possible, component data should be based on historical performance of components in the
same environment as those in the proposed system being studied. The reliability surveys
conducted by the Power Systems Reliability Subcommittee provide a source of component
data when such specific data is not available. This data and the data from later surveys are
summarized in Chapter 3 of IEEE Std 493-1997.
Switching time data generally required includes the following:
—
—
—
—
Expected times to open and close a circuit breaker
Expected times to open and close a disconnect or throwover switch
Expected time to replace a fuse link
Expected times to perform such emergency operations as cutting in clear, installing
jumpers, etc.
Switching times should be estimated for the system being studied based on experience,
engineering judgment, and anticipated operating practices.
12.5 Method for system reliability evaluation
The general method for system reliability evaluation that is recommended and presented here
has evolved over a number of years. The method, referred to as the “minimal cut-set method,”
is believed to be particularly well suited to the study and analysis of electric power
distribution systems as found in industrial plants and commercial buildings. The method is
systematic and straightforward and lends itself to either manual or computer computation. An
important feature of the method is that system weak points can be readily identified, both
numerically and non-numerically, thereby focusing design attention on those sections of the
system that contribute most to service unreliability.
The procedure for system reliability evaluation is outlined as follows:
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RELIABILITY STUDIES
a)
b)
c)
d)
IEEE
Std 399-1997
Assess the service reliability requirements of the loads and processes that are to be
supplied and determine appropriate service interruption definition or definitions.
Perform a failure modes and effects analysis (FMEA), which consists of identifying
and listing those component failures and combinations of component failures that
result in service interruptions and that constitute minimal cut-sets of the system.
Compute interruption frequency contribution, expected interruption duration, and the
probability of each of the minimal cut-sets of the system.
Combine the results of step c) to produce system reliability indexes.
These steps will be discussed in more detail in the sections that follow.
12.5.1 Service interruption definition
The first step in any electric power system reliability study should be a careful assessment of
the power supply quality (e.g., sags, surges, harmonics, etc.) and continuity required by the
loads that are to be served. This assessment should be summarized and expressed in a service
interruption definition that can be used in the succeeding steps of the reliability evaluation
procedure. The interruption definition specifies, in general, the reduced voltage level (voltage
dip) together with the minimum duration of such reduced voltage period that results in
substantial degradation or complete loss of function of the load or process being served.
Frequently, reliability studies are conducted on a continuity basis, in which case, interruption
definitions reduce to a minimum duration specification with voltage assumed to be zero
during the interruption.
12.5.2 Failure modes and effects analysis (FMEA)
The FMEA for power distribution systems amounts to the determination and listing of those
component outage events or combinations of component outages that result in an interruption
of service at the load point being studied according to the interruption definition that has been
adopted. This analysis must be made in consideration of the different types and modes of
outages that components may exhibit and the reaction of the system’s protection scheme to
these events. Component outages are categorized as follows:
—
—
—
Forced outages or failures
Scheduled or maintenance outages
Overload outages
Forced outages or failures are either permanent forced outages or transient forced outages.
Permanent forced outages require repair or replacement of the failed component before it can
be restored to service; transient forced outages imply no permanent damage to the
component, thus permitting its restoration to service by a simple reclosing or refusing
operation. Additionally, component failures can be categorized by physical mode or type of
failure. This type of failure categorization is important for circuit breakers and other
switching devices where the following failure modes are possible:
—
—
Faulted, must be cleared by backup devices
Fails to trip when required
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IEEE
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—
Trips falsely
—
Fails to reclose when required
CHAPTER 12
Each will produce a varying impact on system performance.
The primary result of the FMEA as far as quantitative reliability evaluation is concerned is
the list of minimal cut-sets it produces. A minimal cut-set is defined to be a set of components
which, if removed from the system, results in loss of continuity to the load point being
investigated and does not contain as a subset any set of components that is itself a cut-set of
the system. In the present context, the components in a cut-set are just those components
whose overlapping outage results in an interruption according to the interruption definition
adopted.
An important nonquantitative benefit of FMEA is the thorough and systematic thought
process and investigation it requires. Often weak points in system design will be identified
before any quantitative reliability indexes are computed. Thus, the FMEA is a useful
reliability design tool even in the absence of the data needed for quantitative evaluation.
12.5.3 Computation of quantitative reliability indexes
Computation of reliability indexes can proceed once the minimal cut-sets of the system have
been found. The first step is to compute the frequency, expected duration, and expected
down-time per year of each minimal cut-set. Note that expected down-time per year is the
product of the frequency expressed in terms of events per year and the expected duration. If
the expected duration is expressed in years, the expected down-time will have the units of
years per year and can be regarded as the relative proportion of time or probability the system
is down due to the minimal cut-set in question. More commonly, expected duration is
expressed in hours and the expected down-time has the units of hours per year.
Approximate expressions for frequency and expected duration of the most commonly
considered interruption events associated with first-, second-, and third-order minimal cutsets are given in Table 12-1. Note that expressions for the calculation of interruption
frequencies and durations are for forced outages (failures) only. A detailed treatment of
expressions for the calculation of interruption frequency and duration considering forced
outages as well as maintenance outages, switching after faults to restore service, and
incomplete redundancy of parallel facilities is given in IEEE Std 493-1997.
12.6 References
This chapter shall be used in conjunction with the following publication:
IEEE Std 493-1997, IEEE Recommended Practice for the Design of Reliable Industrial and
Commercial Power Systems (IEEE Gold Book).
380
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RELIABILITY STUDIES
Table 2-1—Frequency and expected duration expressions for
interruptions associated with forced outages only
First order
minimum cut-set
Second-order
minimal cut-set
fcs = λi
rcs = ri
fcs = λiλj(ri + rj)
rcs = rirj(ri + rj)
Third-order
minimal cut-set
fcs = λiλjλk(rirj + rirk + rjrk)
rcs = rirjrk / (rirj + rirk + rjrk)
Symbols:
fcs = frequency of cut-set event
rcs = expected duration of cut-set event
λi = forced outage rate of ith component
ri
= expected repair or replacement time of ith component
NOTES
1—The time units of r and λ in expressions for fcs must be the same. Once frequencies and expected
durations have been computed for each minimal cut-set, system reliability indexes at the load point
in question are given by the following:
fs
= interruption frequency
= ∑fcsi
minimum cut-sets
rs
= expected interruption duration
= ∑fcsi rcsi / fs
minimum cut-sets
fsrs = total interruption time per time period
2—These are approximate formulas and should only be used when every λiri, λjrj, λkrk (dimensionless) is less than 0.01.
Copyright © 1998 IEEE. All rights reserved.
381
Chapter 13
Cable ampacity studies
13.1 Introduction
The cables that network a power system together form the backbone of the system. It is only
logical, therefore, that any complete analysis of a power system should include an analysis of
its cable ampacities. This analysis is complicated since the ampacity of a conductor varies
with the actual conditions of use. Ampacity is defined as “the current in amperes a conductor
can carry continuously under the conditions of use (conditions of the surrounding medium in
which the cables are installed) without exceeding its temperature rating.” Therefore, a cable
ampacity study is the calculation of the temperature rise of the conductors in a cable system
under steady-state conditions. The purpose of this chapter is to acquaint the reader with the
use of computer software systems in the solution of cable ampacity problems with emphasis
on underground installations.
The ampacity of a conductor depends on a number of factors. Prominent among these factors
and of much concern to the designers of electrical distribution systems are the following:
a)
b)
c)
d)
Ambient temperature
Thermal characteristics of the surrounding medium
Heat generated by the conductor due to its own losses
Heat generated by adjacent conductors
To account for the various items that affect ampacities of cables, the 1975 edition of the
National Electrical Code® (NEC®) (see NFPA 70-19961) accepted, for the first time, the
Neher-McGrath method (Neher and McGrath [B10]2) of determining the ampacities of
conductors. Since then, the NEC has added new ampacity tables to account for some limited
conditions of use. As an alternative to the ampacity tables, Section 310-15 (b) of NFPA
70-1996 permits, under engineering supervision, the use of an ampacity equation for determining ampacities. A discussion of this evolution and the origin of NEC Tables 310-16
through 310-19 is provided in Knutson and Miles [B1]. This equation is based on the NeherMcGrath method, which is the basis for the calculating procedures discussed in this chapter.
In subsequent paragraphs, various items that affect cable ampacities are discussed and
quantified with the help of ampacity adjustment factor tables and actual computer runs. The
computer program from which the ampacity adjustment factors were generated is based on
the Neher-McGrath method of calculation and has been corroborated by a second,
independently developed computer program of like kind. Under some specific and limited
conditions, the ampacity adjustment tables were compared and verified with the NEC
ampacity tables, including Appendix B of the NEC. Note that the tables provided here
generally cover broader conditions of use with greater resolution than the NEC tables.
1Information
2The
on references can be found in 13.7.
numbers in brackets correspond to those of the bibliography in 13.8.
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CHAPTER 13
Since the ampacity adjustment tables have been developed for some specific conditions, they
cannot be applied for all cases. In general, these tables can be used to size the cables in the
initial stages of a design and to closely approximate ampacities. These preliminary cable
sizes can then be used as the basis for a more rigorous computer analysis to determine actual
conductor temperatures and to finalize the design.
13.2 Heat flow analysis
When designing a power distribution system, the cable ampacity is of primary concern. Once
the size and location of electrical loads are determined, an adequate distribution system must
be designed. The total number of required circuits, their sizes, and the method of routing are
significant elements in the design problem. But in addition, accurate cable sizing becomes
especially critical to ensure that the cables are adequate to carry the required load without
being subjected to temperatures that exceed their temperature ratings.
As an electrical current flows through a cable, it generates heat. The type of cable and how it
is connected and installed determines how many components of heat generation are present,
e.g., I 2R losses, sheath losses, etc. The heat flows from these sources through a series of
thermal resistances to the surrounding environment. The operating temperature that the cable
ultimately reaches is directly related to the amount of heat generated and the net effective
value of the thermal resistance through which it flows.
A detailed discussion of all the heat transfer complexities involved is beyond the scope of this
subclause. However, the heat transfer process will be covered briefly in order to establish a
basic background from which the discussion to follow can proceed.
The calculation of the temperature rise of cable systems involves the application of a series of
thermal equivalents of Ohm’s and Kirchoff’s laws to a relatively simple thermal circuit, as is
illustrated in Figure 13-1. This circuit includes a number of parallel paths with heat entering
at several points. Under conditions of equilibrium, the conductor temperature will be
determined by the temperature differential created across a series of thermal resistances as
the heat flows to the ambient temperature T ' c = T ' a + ∆T ).
To understand the basic calculation procedure used in cable ampacity programs, consider the
fundamental equation for the ampacity of a cable in an underground duct.
T ' c – ( T ' a + ∆T d + ∆T int )
I = ------------------------------------------------------------( R ac ) ( R'ca )
1/2
kA
(13-1)
This equation follows the Neher-McGrath method where
T 'c
T 'a
∆T d
∆T int
384
is the allowable conductor temperature (°C),
is the soil ambient temperature (°C),
is the temperature rise of conductor due to dielectric heating (°C),
is the temperature rise of conductor due to interference heating from cables in
other ducts (°C). (Note that since the temperature rise, due to another conductor,
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CABLE AMPACITY STUDIES
IEEE
Std 399-1997
Figure 13-1—A generalized model for heat flow from heat sources in a cable
system to ambient temperature through a series of thermal resistances
Copyright © 1998 IEEE. All rights reserved.
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IEEE
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R ac
R′ca
CHAPTER 13
depends on the current through it, simultaneous solutions of ampacity equations
are required.)
is the electrical alternating current resistance of conductor including skin,
proximity, and temperature effects (µΩ/ft),
is the effective total thermal resistance from conductor to ambient soil adjusted
to include effects of load factor, shield/sheath losses, metallic conduit losses,
and the effect of multiple conductors in the same duct (thermal-Ω/ft, °C-ft/W).
Note that all effects that produce a conductor temperature rise except the conductor loss
2
( I R ac ) have been treated as adjustments to the basic thermal system. Fundamentally, the
difference between two temperatures (e.g., T ' c – T ' a ) divided by a separating thermal
resistance equals the heat flow in watts (or W/ft of conductor). The similarity of the procedure
used in the cable ampacity program to that used with the traditional approach is apparent if
both sides of the ampacity equation are squared and then multiplied by R ac . The result is as
follows:
T ' c – ( T 'a + ∆T d + ∆T int )
2
I R ac = ------------------------------------------------------------- W/ft
( R'ca )
(13-2)
Although it is not necessary to understand these heat transfer concepts in order to use cable
ampacity programs, such knowledge may be helpful for understanding how physical
parameters affect ampacity. Observation of the ampacity equation shows how lower
ampacities are inherent with the following:
—
—
—
—
—
—
—
Lower conductor operating temperatures
Higher soil ambient temperatures
Smaller conductors (higher R ac )
Higher thermal resistivities of earth, concrete, insulation, duct, etc. (higher R'ca )
Deeper burial depths (higher R'ca )
Closer cable spacing (higher ∆T int )
Cables located in inner, rather than outer, ducts (higher ∆T int )
Other factors that decrease ampacity and whose relationship to the ampacity equation is not
readily apparent include the following:
—
—
—
—
Higher load factor (higher R'ca )
Higher voltage (higher ∆T d )
Higher insulation SIC and power factor (higher ∆T d )
Lower shield/sheath electrical resistance (higher R'ca )
13.3 Application of computer program
The calculations used in cable ampacity programs are normally based on the Neher-McGrath
method. In computing cable ampacities in duct banks, only power cables need to be
considered, since control cables, carrying very little current, contribute very little to the
overall temperature rise. Cable ampacity programs deal only with the temperature-limited,
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CABLE AMPACITY STUDIES
IEEE
Std 399-1997
current-carrying capacity of cables. Voltage drop, future load growth, and short-circuit
capability are also important factors that should be considered when selecting cables.
The calculation of cable ampacity in underground installations is a very complicated
procedure requiring the evaluation of a multitude of subtle effects. In order to make the
calculations possible for a wide variety of cases, certain assumptions are made. Most of the
assumptions are developed by Neher and McGrath in [B10] and are widely accepted. Some
programs may make other assumptions that should be understood.
The basic steps in applying cable ampacity programs follow. It is important to follow
methodical procedures in order to obtain good results with minimum effort.
a)
b)
c)
d)
e)
f)
The first step in designing an underground cable installation is to establish which
circuits are to be routed through the duct bank. Consideration should be given to
present circuits as well as to circuits that may be added in the future. Only power
cables need to be included as current-carrying conductors in analysis; but space
allowances must be made for spare ducts or for control and instrumentation circuits.
The duct bank should be designed with consideration given to the circuits contained,
the space available for the bank, cable separation criteria, and factors that affect
ampacity. For example, cables buried deeply or surrounded by other power cables
often have greatly reduced ampacity. It should be decided if ducts will be directly
buried or encased in concrete. The size(s) and type(s) of duct to be used should be
determined. Finally, a sketch of the duct bank should be prepared with burial depth
and spacing of ducts clearly shown. Physical data on the duct installation should be
compiled, including thermal resistivity of the soil, ambient temperature of the soil,
and thermal resistivity of the concrete. Note that soil thermal resistivity and
temperature at some locations (e.g., desert) may be much greater than the typical
values often used.
Complete data on all power cables used in the installation must be assembled. Some
data may be taken from standard tables; but certain data should be based on
manufacturer’s specifications. Conductor size, conductor material, operating voltage,
type of shield or sheath, temperature rating, insulation type, and jacket type are
especially important.
An initial cable placement layout should be designed, based on anticipated loads and
load factors. Circuits with high currents and load factors (ratio of average to peak
load over a given load cycle) should be placed in outside ducts near the top of the
bank to eliminate the need for larger conductors due to unnecessarily reduced
ampacity. Frequently, a good compromise between best use of duct space and highest
ampacity is achieved by installing each three-phase circuit in a separate duct.
However, nonshielded single- conductor cables may have a higher ampacity with
each phase conductor in a separate nonmetallic duct. If the load factor cannot be
evaluated readily, a conservative value of 1.0 may be entered, which implies that the
circuit always operates at peak load.
The manual method presented in this chapter can be used to initially size the cables
based on the ambient temperature, soil thermal resistivity, and grouping of the cables.
Once the initial design is established and all necessary data have been collected, the
user should enter the program data interactively or prepare an input data file for a
Copyright © 1998 IEEE. All rights reserved.
387
IEEE
Std 399-1997
g)
h)
CHAPTER 13
batch program. Normally, the data should be prepared for standard ampacity
calculation, using the worst-case conditions. If actual load currents are known, these
may be entered to find the temperatures of cables within each duct. Temperature
calculations are especially useful if some circuits are lightly loaded, while others
carry heavy loads that push ampacity limits. If the lightly loaded circuits were to
operate at rated temperature, as the ampacity calculation assumes, the load capability
of the heavily loaded circuits would be reduced. Temperature calculations may also
be used as a rough indicator of the reserve capacity of each duct.
After a program is run, the user should carefully analyze the results to verify that
design currents are less than ampacities (if an ampacity calculation is performed), or
that actual temperatures are less than rated temperatures (if a temperature calculation
is performed). If the initial design is shown to be inadequate, various corrective
measures should be considered. These include increasing conductor sizes, modifying
cable locations, and changing the physical design of the bank. The effects of various
parameters may be analyzed by repeating these steps until a satisfactory overall
design is achieved.
The results of such an analysis should be documented and permanently archived for
use in properly controlling and/or analyzing future changes in duct bank usage (i.e.,
installation of cables in spare ducts).
13.4 Ampacity adjustment factors
The ampacity values stated (specified) by the cable manufacturer and/or other authoritative
sources, such as the NEC and IEEE Std 835-1994, are usually based on some very specific
conditions relative to the cable’s immediate surrounding environment. The following are
examples of some specific conditions:
—
—
—
—
Installation under an isolated condition
Installation of groups of three or six circuits
Soil thermal resistivity (RHO) of 90 °C–cm/W
Ambient temperature of 20 °C or 40 °C
In practice, the surrounding medium or environment in which cables are to be installed rarely
matches those conditions under which the stated ampacities apply. The differences can be
thought of as an intermediate medium (requiring adjustment factors for conditions of use)
inserted between the base conditions (an environment at which the base ampacity is specified
by the manufacturer or other authoritative sources) and the actual conditions of use. This
process is presented pictorially in Figure 13-2. It illustrates that the nature of the practical
problem is to adjust the specified (base) ampacities of the cables by an adjustment factor to
account for the effects of the various intermediate elements or conditions of use.
A simple manual method of determining cable ampacities is presented here to illustrate the
concept of cable derating and to present the different factors that have a direct effect on the
operating temperatures of the conductors.
388
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CABLE AMPACITY STUDIES
IEEE
Std 399-1997
Figure 13-2—Simplified illustration of the heat transfer model used
to determine the cable ampacity (3–1/C cables shown)
This method is based on the concept of an adjustment (derating) factor applied against a base
ampacity to provide the allowable cable ampacity.
I ' = FI
(13-3)
where
I'
is the allowable ampacity under the actual installation conditions,
F
is the overall cable ampacity adjustment factor,
I
is the base ampacity, i.e., the ampacity specified by the manufacturers or other
authoritative sources, such as the ICEA. For example, the ampacity of a cable that is
installed in an underground conduit under isolated conditions with an ambient
temperature of 20 °C and soil thermal resistivity RHO of 90 °C-cm/W.
The overall cable adjustment factor is a correction factor that takes into account the
differences in the cable’s actual installation and operating conditions from the base
conditions. This factor establishes the maximum load capability that results in an actual cable
Copyright © 1998 IEEE. All rights reserved.
389
IEEE
Std 399-1997
CHAPTER 13
life equal to or greater than that expected when operated at the base ampacity under the
specified conditions.
The overall ampacity adjustment factor is composed of several components as indicated in
Equation (13-4).
F = Ft F th Fg
(13-4)
where
Ft
is the adjustment factor to account for the differences in the ambient and conductor
temperatures from the base case,
F th
is the adjustment factor to account for the difference in the soil thermal resistivity,
from the RHO of 90 °C–cm/W at which the base ampacities are specified,
Fg
is the adjustment factor to account for cable grouping.
To obtain the values of the adjustment factors F th and Fg , an elaborate computer program
was developed based on the Neher-McGrath method and was used to calculate the conductor
temperatures for various arrangements. The program takes into account each adjustment
factor in Equation (13-4) which together account for the more significant effects indicated in
Figure 13-1 for underground installations. Thousands of computer runs were made to
determine the adjustment factor tables. These tables were then verified by utilizing the NEC,
IEEE Std 835-1994, and the Underground Systems Reference Book [B18]. Knutson and Miles
[B1], Shokooh and Knutson 1988 [B14], and Shokooh and Knutson 1983 [B15] report the
results of similar efforts for ampacity adjustment factors based on the Neher-McGrath
method.
The various adjustment factors in Equation (13-4) are largely, but not completely,
independent from each other. Although the computer program can simulate any complex
configuration, for the sake of clarity, the ampacity adjustment tables reported here are based
on the following simplifying assumptions:
a)
Cables for some voltage ratings and sizes are combined for the Fth tables. For some
applications where RHO is considerably high (more than 180 °C–cm/W) and a mixed
group of cables are installed, the interdependencies of the adjustment factors for
different cable sizes may not be negligible and up to a 4% error in the overall
conductor temperatures may be expected.
b)
The effect of the temperature rise due to the insulation dielectric losses is neglected
for the temperature adjustment factor, Ft. This temperature rise for rubber and polyethylene insulated cables rated 15 kV and below (sizes 1000 kcmil and below) is less
than 2 °C. However, this effect can be included in Ft by adding the temperature rise
due to the dielectric losses to the ambient temperatures T a and T a' .
c)
The often negligible effects of any applicable sheath, shield, and metallic conduit
losses depicted in Figure 13-1 are ignored.
390
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
CABLE AMPACITY STUDIES
In the final design case where accuracy and precision are required, the previously mentioned
assumptions cannot be disregarded, and the ampacities obtained from the manual method can
be used as an initial approximation for computer simulation of the actual design conditions.
13.4.1 Ft (ambient and conductor temperature adjustment factor)
This adjustment factor is used to determine the cable ampacity when the operating ambient
temperature and/or the maximum allowable conductor temperature differ from the original
temperatures at which the cable base ampacity is specified. The expression for calculating the
effect of changes in the conductor and ambient temperatures on the base ampacity is given by
Ft in Equations (13-5) and (13-6) for copper and aluminum conductors, respectively.
234.5 + T c
T c' – T a'
F t = -------------------- × --------------------------T c – T a 234.5 + T c'
1/2
228.1 + T c
T c' – T a'
F t = -------------------- × --------------------------T c – T a 228.1 + T c'
1/2
(copper)
(13-5)
(aluminum)
(13-6)
where
Tc
is the conductor rated temperature in °C at which the base ampacity is specified,
T c'
is the maximum allowable conductor operating temperature in °C,
Ta
is the ambient temperature in °C at which the base ampacity is specified,
T a'
is the actual (maximum) soil ambient temperature in °C.
The maximum operating ambient temperature is usually difficult to obtain and has to be
estimated based on historical meteorological data. For application in underground cables, T a'
is the maximum soil temperature at the depth of installation at peak summertime. In general,
seasonal temperature variations of the soil follow a roughly sinusoidal cycle with soil temperature peaking during the summer months. The effect of seasonal variation in soil temperature
decreases with depth until the depths of 20–30 ft are reached, at which the soil temperature
remains fairly constant.
Certain characteristics of the soil (texture, density, and moisture content) and soil pavement
(asphalt, cement, etc.) have a noticeable effect on the soil temperature profile. For maximum
accuracy, it is important to obtain Ta via a field test rather than using an approximate value
based on the maximum atmospheric temperature. For cable installation in air, T a is the
maximum air temperature at peak summertime. Special attention should be given for cable
applications in the shade or under direct sunlight.
Adjustment factors for typical copper conductor temperatures ( T c = 90 °C and 75 °C) and
ambient temperatures ( T a = 20 °C for underground installation and 40 °C for above-ground
installation) at which the base ampacities are specified, are calculated from Equation (13-5)
and tabulated in Tables 13-1 through 13-4.
Copyright © 1998 IEEE. All rights reserved.
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IEEE
Std 399-1997
CHAPTER 13
Table 13-1—Ft : Adjustment factor for various copper conductors and
ambient temperatures when T c = 75 ° C and T a = 40 ° C
T 'a in ° C
T'c in ° C
30
35
40
45
50
55
60
0.95
0.87
0.77
0.67
0.55
0.39
75
1.13
1.07
1.00
0.93
0.85
0.76
90
1.28
1.22
1.17
1.11
1.04
0.98
110
1.43
1.34
1.34
1.29
1.24
1.19
Table 13-2—Ft : Adjustment factor for various copper conductors and
ambient temperatures when T c = 90 ° C and T a = 40 ° C
T ' a in ° C
T' c in ° C
30
35
40
45
50
55
75
0.97
0.92
0.86
0.79
0.72
0.65
85
1.06
1.01
0.96
0.90
0.84
0.78
90
1.10
1.05
1.00
0.95
0.89
0.84
110
1.23
1.19
1.15
1.11
1.06
1.02
130
1.33
1.30
1.27
1.23
1.19
1.16
Table 13-3—Ft : Adjustment factor for various copper conductors and
ambient temperatures when T c = 75 ° C and T a = 20 ° C
T ' a in ° C
T' c in ° C
392
10
15
20
25
30
35
60
0.98
0.93
0.87
0.82
0.76
0.69
75
1.09
1.04
1.00
0.95
0.90
0.85
90
1.18
1.14
1.10
1.06
1.02
0.98
110
1.29
1.25
1.21
1.18
1.14
1.11
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
CABLE AMPACITY STUDIES
Table 13-4—Ft : Adjustment factor for various copper conductors and
ambient temperatures when T c = 90 ° C and T a = 20 ° C
T ' a in ° C
T' c in ° C
10
15
20
25
30
35
75
0.99
0.95
0.91
0.87
0.82
0.77
85
1.04
1.02
0.97
0.93
0.89
0.85
90
1.07
1.04
1.00
0.96
0.93
0.89
110
1.16
1.13
1.10
1.06
1.02
0.98
130
1.24
1.21
1.18
1.16
1.13
1.10
13.4.2 F th (thermal resistivity adjustment factor)
Soil thermal resistivity (RHO) indicates the resistance to heat dissipation of the soil in °C–
cm/W. Tables 13-5 through 13-7 indicate the adjustment factors required when the actual soil
thermal resistivity is different from the RHO of 90 °C–cm/W at which the base ampacities
are specified. These tables are calculated based on an assumption that the soil has a uniform
and constant thermal resistivity.
Table 13-5—Fth: Thermal resistivity adjustment factor for 0–1000 V cables in
duct banks with base ampacity given at an RHO of 90 ° C-cm/W
RHO ( °C-cm/W )
Number
of
CKT
60
90
120
140
160
180
200
250
#12–#1
1
3
6
9+
1.03
1.06
1.09
1.11
1.0
1.0
1.0
1.0
0.97
0.95
0.93
0.92
0.96
0.92
0.89
0.87
0.94
0.89
0.85
0.83
0.93
0.87
0.82
0.79
0.92
0.85
0.79
0.76
0.90
0.82
0.75
0.71
1/0–4/0
1
3
6
9+
1.04
1.07
1.10
1.12
1.0
1.0
1.0
1.0
0.97
0.94
0.92
0.91
0.95
0.90
0.87
0.85
0.93
0.87
0.84
0.81
0.91
0.85
0.81
0.78
0.89
0.83
0.78
0.75
0.86
0.80
0.74
0.70
250–1000
1
3
6
9+
1.05
1.08
1.11
1.13
1.0
1.0
1.0
1.0
0.96
0.93
0.91
0.90
0.94
0.89
0.86
0.84
0.92
0.86
0.83
0.80
0.90
0.83
0.80
0.77
0.88
0.81
0.77
0.74
0.85
0.77
0.72
0.69
Cable
Size
Copyright © 1998 IEEE. All rights reserved.
393
IEEE
Std 399-1997
CHAPTER 13
Table 13-6—Fth: Thermal resistivity adjustment factor for 1001–35 000 V cables
in duct banks with base ampacity given at an RHO of 90 ° C-cm/W
RHO ( °C-cm/W )
Number
of
CKT
60
90
120
140
160
180
200
250
#12–#1
1
3
6
9+
1.03
1.07
1.09
1.10
1.0
1.0
1.0
1.0
0.97
0.94
0.92
0.91
0.95
0.90
0.87
0.85
0.93
0.87
0.84
0.81
0.91
0.84
0.80
0.77
0.90
0.81
0.77
0.74
0.88
0.77
0.72
0.69
1/0–4/0
1
3
6
9+
1.04
1.08
1.10
1.11
1.0
1.0
1.0
1.0
0.96
0.93
0.91
0.90
0.94
0.89
0.86
0.84
0.92
0.86
0.82
0.80
0.90
0.83
0.79
0.76
0.88
0.80
0.77
0.73
0.85
0.75
0.71
0.68
250–1000
1
3
6
9+
1.05
1.09
1.11
1.12
1.0
1.0
1.0
1.0
0.95
0.92
0.91
0.90
0.92
0.88
0.85
0.84
0.90
0.85
0.81
0.79
0.88
0.82
0.78
0.75
0.86
0.79
0.75
0.72
0.84
0.74
0.70
0.67
Cable
Size
Table 13-7—Fth: Thermal resistivity adjustment factor for cables directly
buried with base ampacity given at an RHO of 90 ° C-cm/W
RHO ( °C-cm/W )
Number
of
CKT
60
90
120
140
160
180
200
250
#12–#1
1
2
3+
1.10
1.13
1.14
1.0
1.0
1.0
0.91
0.90
0.89
0.86
0.85
0.84
0.82
0.81
0.79
0.79
0.77
0.75
0.77
0.74
0.72
0.74
0.70
0.67
1/0–4/0
1
2
3+
1.13
1.14
1.15
1.0
1.0
1.0
0.91
0.90
0.89
0.86
0.85
0.84
0.81
0.80
0.78
0.78
0.76
0.74
0.75
0.73
0.71
0.71
0.69
0.67
250–1000
1
2
3+
1.14
1.15
1.16
1.0
1.0
1.0
0.90
0.89
0.88
0.85
0.84
0.83
0.81
0.80
0.78
0.78
0.76
0.74
0.75
0.73
0.71
0.71
0.69
0.67
Cable
Size
394
Copyright © 1998 IEEE. All rights reserved.
CABLE AMPACITY STUDIES
IEEE
Std 399-1997
Typical values of thermal resistivity for various materials are as follows (see the NEC).
Material type
Solid paper insulation
Varnished cambric
Polyvinyl chloride (PVC)
Paper
Neoprene
Rubber, jute, textiles
Fiber duct
Polyethylene (PE)
Transite duct
Somastic
Concrete
Average soil
Very dry soil (rocky or sandy)
Damp soil (coastal areas, high water table)
EPR
Crosslinked polyethylene
(°C-cm/W)
700
600
650
550
519
500
480
450
200
100
55–85
90
120
60
400
370
The thermal resistivity of the soil depends on a number of factors, such as soil texture,
moisture content, density, and structural arrangement of the soil grains. In general, higher density or moisture content of the soil results in a better heat dissipating ability and lower thermal
resistivity. There is a tremendous variation in the soil thermal resistivities ranging from a RHO
of less than 40 to more than 300 °C–cm/W. Based on these facts, it is apparent that direct testing of the soil is essential. Furthermore, it is important that this test be conducted after a prolonged dry spell at a peak summer temperature when the soil moisture content is minimal. The
result of such a field test usually indicates a wide range of soil thermal resistance for a given
depth over a test site. For the purpose of cable ampacity deratings, the maximum value of the
thermal resistivities for a given cable route should be used.
The effect of soil dryout, which is caused by the continuous loading of the cables, can be
taken into account by considering a RHO higher than the actual value obtained from the soil
test. Use of dense sandy soil as backfill can lower the effective overall thermal resistivity and
can offset the soil dryout effect. Dryout curves of RHO versus moisture content can be
obtained to help select an appropriate value.
In cases where the soil thermal resistivity is very high and corrective backfill with low
thermal resistivity is used, Tables 13-5 through 13-7 are inaccurate and may not produce
cable ampacity values that are acceptable even on an approximate basis.
13.4.3 F g (grouping adjustment factor)
Grouped cables will operate at a higher temperature than isolated cables. The increase in the
operating temperature is due to the presence of the other cables in the group, which act as
heat sources. Therefore, the amount of interference temperature rise from other cables in the
group depends on the separation of the cables and the surrounding media.
Copyright © 1998 IEEE. All rights reserved.
395
IEEE
Std 399-1997
CHAPTER 13
In this subclause, adjustment factors for cables installed with maintained separation in
underground duct banks and for directly buried cables are given in Tables 13-8 through
13-11. For cable separations other than those considered in these tables, one can use one’s
own judgment for estimating the value of Fg or use a computer program directly without an
initial approximation for the grouping effect. In general, increasing the horizontal and vertical
spacing between the cables would decrease the temperature interference between them and,
therefore, increase the value of Fg.
Table 13-8—Fg : Grouping adjustment factor for 0–5000 V 3/C, or triplexed
cables in duct banks (no spare ducts, nonmetallic conduits of 5 in
with center-to-center spacing of 7.5 in)
Number of columns
No.
of
rows
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
#8
1
2
3
4
1.00
.930
.870
.820
.942
.840
.772
.710
.885
.772
.694
.629
.835
.723
.632
.571
.795
.687
.596
.536
.768
.660
.569
.509
.745
.638
.548
.490
.727
.620
.532
.472
.710
.604
.519
.458
.698
.592
.508
.446
.688
.582
.498
.436
.679
.572
.490
.428
.671
.564
.482
.420
.664
.557
.476
.412
.658
.550
.470
.405
#6
1
2
3
4
1.00
.920
.860
.810
.930
.813
.747
.700
.874
.747
.679
.620
.826
.700
.625
.565
.790
.665
,588
.531
.760
.638
.560
.503
.737
.615
.540
.484
.718
.598
.525
.467
.702
.583
.510
.452
.690
.572
.498
.440
.680
.561
.490
.431
.671
.552
.481
.422
.663
.544
.473
.415
.656
.537
.467
.408
.650
.530
.460
.400
#4
1
2
3
4
1.00
.920
.850
.805
.925
.809
.742
.690
.871
.742
.668
.610
.817
.693
.615
.560
.781
.659
.578
.524
.750
.632
.551
.497
.726
.610
.531
.477
.707
.593
.514
.460
.691
.579
.500
.447
.678
.567
.489
.435
.668
.555
.480
.425
.659
.547
.471
.418
.651
.539
.464
.410
.646
.530
.458
.401
.640
.525
.450
.395
#2
1
2
3
4
1.00
.920
.840
.800
.918
.800
.723
.685
.858
.723
.657
.608
.808
.680
.608
.553
.770
.648
.568
.518
.741
.623
.540
.490
.720
.602
.520
.471
.701
.586
.504
.453
.688
.572
.490
.440
.677
.560
.479
.429
.667
.549
.470
.420
.658
.540
.461
.411
.650
.530
.454
.402
.641
.522
.447
.395
.635
.514
.440
.390
#1
1
2
3
4
1.00
.920
.830
.740
.918
.795
.702
.634
.849
.702
.618
.551
.799
.650
.562
.497
.753
.613
.525
.465
.721
.583
.500
.440
.699
.563
.480
.421
.682
.546
.464
.405
.669
.530
.450
.392
.659
.520
.440
.383
.650
.510
.430
.374
.643
.502
.421
.366
.639
.494
.413
.359
.632
.488
.406
.352
.630
.482
.400
.348
1/0
1
2
3
4
1.00
.915
.817
.735
.910
.790
.700
.629
.842
.700
.610
.546
.791
.642
.554
.492
.745
.604
.520
.460
.716
.575
.494
.435
.694
.555
.474
.417
.678
.537
.457
.402
.665
.523
.444
.391
.655
.511
.432
.381
.646
.503
.424
.371
.639
.494
.415
.363
.635
.486
.408
.355
.628
.480
.400
.349
.626
.475
.394
.343
2/0
1
2
3
4
1.00
.915
.817
.735
.910
.790
.700
.629
.842.
700
.610
.546
.791
.642
.554
.492
.745
.604
.520
.460
.716
.575
.494
.435
.694
.555
.474
.417
.678
.537
.457
.402
.665
.523
.444
.391
.655
.511
.432
.381
.646
.503
.424
.371
.639
.494
.415
.363
,635
.486
.408
.355
.628
.480
.400
.349
.626
.475
.394
.343
3/0
1
2
3
4
1.00
.915
.817
.735
.910
.790
.700
.629
.842
.700
.610
.546
.791
.642
.554
.492
.745
.604
.520
.460
.716
.575
.494
.435
.694
.555
.474
.417
.678
.537
.457
.402
.665
.523
.444
.391
.655
.511
.432
.381
.646
.503
.424
.371
.639
.494
.415
.363
.635
.486
.408
.355
.628
.480
.400
.349
.626
.475
.394
.343
4/0
1
2
3
4
1.00
.910
.810
.730
.908
.770
.684
.624
.830
.684
.602
.541
.780
.635
.548
.487
.737
.599
.515
.456
.709
.570
.489
.431
.690
.550
.469
.414
.673
.532
.452
.399
.660
.518
.440
.388
.650
.506
.429
.378
.642
.498
.420
.368
.635
.489
.411
.360
.628
.481
.403
.352
.623
.475
.397
.346
.619
.470
.391
.341
250
1
2
3
4
1.00
.890
.780
.694
.905
.770
.675
.588
.830
.675
.579
.512
.777
.609
.518
.460
.725
.570
.480
.422
.692
.542
.454
.397
.668
.519
.434
.379
.646
.500
.420
.364
.628
.485
.408
.352
.615
.474
.398
.345
.603
.466
.390
.338
.597
.458
.383
.331
.590
.450
.378
.327
.583
.445
.373
.323
.580
.440
.370
.320
Cable
size
396
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
CABLE AMPACITY STUDIES
Table 13-8—Fg : Grouping adjustment factor for 0–5000 V 3/C, or triplexed
cables in duct banks (no spare ducts, nonmetallic conduits of 5 in
with center-to-center spacing of 7.5 in) (Continued)
Number of columns
No.
of
rows
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
350
1
2
3
4
1.00
.887
.775
.690
.905
.749
.664
.587
.830
.664
.575
.511
.770
.609
.515
.457
.720
.570
.479
.421
.688
.540
.453
.395
.661
.518
.433
.377
.640
.499
.419
.362
.622
.484
.406
.351
.608
.474
.397
.343
.597
.465
.389
.336
.590
.458
.382
.330
.583
.450
.377
.325
.578
.445
.372
.321
.573
.440
.369
.318
500
1
2
3
4
1.00
.882
.770
.685
.897
.745
.656
.585
.815
.656
.570
510
.762
.608
.514
.454
.708
.569
.478
.420
.678
.539
.452
.393
.652
.516
.432
.374
.630
.498
.417
.360
.613
.483
.404
.349
.599
.473
.395
.340
.588
.463
.388
.333
.581
.457
.381
.328
.575
.450
375
.323
.570
.444
.370
.319
.565
.439
.367
.315
750
1
2
3
4
1.00
.870
.760
.680
.890
.725
.641
.579
.802
.641
.560
.501
.747
.591
.507
.448
.700
.552
.470
.413
.670
.522
.445
.389
.640
.500
.425
.371
.622
.484
.410
.357
.605
.469
.398
.346
.590
.457
.389
.337
.580
.448
.380
.330
.572
.440
.374
.323
.566
.434
.369
.318
.560
.430
.363
.314
.555
.425
.360
.310
1000
1
2
3
4
1.00
.858
.748
.676
.885
.716
.632
.574
.795
.632
.551
.497
.740
.582
.499
.444
.695
.544
.464
.409
.665
.513
.439
.385
.639
.493
.419
.367
.618
.474
.403
.353
.600
.460
.392
.342
.585
.448
.383
.333
.574
.439
.375
.326
.567
.431
.369
.319
.561
.425
.363
.315
.555
.420
.358
.311
.551
.415
.355
.308
Cable
size
Table 13-9—Fg : Grouping adjustment factor for 5001–35 000 V 3/C, or triplexed
cables in duct banks (no spare ducts, nonmetallic conduits of 5 in
with center-to-center spacing of 7.5 in)
Number of columns
No.
of
rows
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
#6
1
2
3
4
1.00
.920
.840
.770
.920
.800
.714
.642
.854
.714
.625
560
.803
.660
.569
.506
.758
.620
.530
.469
.726
.590
.501
.441
.699
.570
.484
.422
.678
.552
.470
.406
.660
.540
.459
.394
.646
.530
.450
.385
.635
.521
.442
.378
.628
.515
.436
.371
.620
.509
.429
.367
.615
.503
.423
.362
.610
.500
.420
.358
#4
1
2
3
4
1.00
.920
.835
.760
.920
.795
.709
.630
.852
.714
.615
.548
.800
.660
.561
.498
.755
.620
.521
.460
.722
.590
.493
.430
.695
.570
.474
.410
.673
.552
.459
.395
.655
.540
.488
.382
.642
.530
.439
.374
.630
.521
.430
.367
.623
.515
.424
.361
.615
.434
.420
.356
.610
.430
.416
.352
.605
.425
.412
.350
#2
1
2
3
4
1.00
.920
.820
.746
.910
.782
.689
.622
.836
.689
.600
.539
.784
.639
.544
.484
.748
.599
.505
.445
.714
.570
.479
.415
.688
.548
.460
.396
.665
.531
.445
.382
.649
.518
.433
.370
.635
.508
.424
361
.625
.500
.417
.353
.616
.494
.410
.348
.609
.489
.405
.342
.602
.484
.400
.338
.598
.480
.395
.334
#1
1
2
3
4
1.00
.920
.816
.785
.905
.771
.681
.605
.827
.681
.588
.524
.777
.629
.532
.471
.731
.590
.497
.435
.697
.560
.469
.410
.670
.538
.448
.390
.645
.519
.432
.376
.626
.502
.418
.364
.610
.491
.407
.353
.598
.480
.397
.347
.588
.471
.389
.340
.579
.462
.382
.333
.571
.455
.376
.328
.565
.450
.370
.323
1/0
1
2
3
4
1.00
.912
.811
.730
.904
.765
.671
.604
.825
.671
.581
.518
.775
.619
.525
.464
.729
.580
.488
.431
.695
.549
.460
.406
.668
.527
.440
.385
.643
.509
.423
.372
.624
.494
.409
.359
.609
.481
.398
.349
.597
.471
.387
.341
.587
.462
.379
.335
.578
.453
.372
.329
.570
.446
.365
.324
.564
.440
.359
.320
Cable
size
Copyright © 1998 IEEE. All rights reserved.
397
IEEE
Std 399-1997
CHAPTER 13
Table 13-9—Fg : Grouping adjustment factor for 5001–35 000 V 3/C, or triplexed
cables in duct banks (no spare ducts, nonmetallic conduits of 5 in
with center-to-center spacing of 7.5 in) (Continued)
Number of columns
No.
of
rows
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2/0
1
2
3
4
1.00
.903
.800
.722
.904
.761
.667
.597
.823
.667
.578.
511
.773
.612
.520
.460
.728
.573
.482
.425
.694
.542
.454.
400
.668
.520
.433
.380
.643
.500
.418
.365
.624
.488
.402
.353
.609
.475
.391
.343
.580
.463
.382
.335
.597
.455
.374
.329
.587
.448
.367
.322
.578
.441
.360
.317
.570
.434
.353
.312
3/0
1
2
3
4
1.00
.898
.802
.720
.898
.752
.664
.593
.814
.664
.572
.508
.765
.609
.514
.456
.722
.570
.479
.421
.690
.539
.451
.396
.661
.451
.430
.377
.637
.498
.414
.362
.618
.483
.399
.350
.602
.471
.388
.340
.590
.461
.379
.332
.580
.451
.371
.327
.571
.443
.364
.320
.563
.437
.357
.314
.556
.429
.350
.310
4/0
1
2
3
4
1.00
.896
.795
.711
.894
.743
.656
.584
.811
.656
.564
.502
.762
.603
.513
.450
.717
.565
.474
.417
.682
.536
.447
.392
.653
.513
.427
.374
.631
.496
.411
.359
.612
.480
.397
.348
.597
.468
.386
.338
.585
.459
.377
.329
.574
.449
.369
.324
.566
.441
.362
.317
.558
.434
.355
.311
.550
.427
.349
.307
250
1
2
3
4
1.00
.885
.785
.701
.892
.741
.654
.580
.811
.654
.559
.500
.762
.594
.498
.448
.715
.552
.459
.414
.679
.523
.429
.385
.645
.500
.408
.365
.620
.482
.388
.348
.600
.469
.373
.332
.583
.457
.361
.321
.572
.447
.351
.311
.564
.438
.342
.302
.557
.430
.335
.295
.552
.422
.328
.288
.550
.416
.321
.281
350
1
2
3
4
1.00
.872
.772
.681
.890
.733
.641
.572
.807
.641
.550
.491
.754
.580
.492
.440
.700
.538
.451
.402
.661
.510
.420
.375
.634
.488
.396
.354
.609
.470
.377
.337
.589
.455
.362
.322
.572
.443
.350
.311
.561
.432
.340
.300
.552
.423
.331
.292
.548
.415
.323
.285
.542
.408
.316
.278
.540
.400
.310
.271
500
1
2
3
4
1.00
.862
.765
.676
.885
.728
.634
.574
.801
.634
.542
.497
.745
.572
.483
.444
.692
.531
.446
.409
.650
.502
.415
.385
.620
.480
.391
.367
.593
.462
.373
.353
.573
.447
.358
.342
.559
.435
.346
.333
.548
.425
.335
.326
.539
.415
.327
.319
.533
.407
.319
.315
.529
.400
.311
.311
.526
.391
.305
.308
750
1
2
3
4
1.00
.850
.755
.671
.879
.710
.622
.560
.790
.622
.530
.480
.780
.560
.479
.430
.682
.520
.441
.392
.647
.490
.410
.366
.615
.469
.387
.345
.589
.450
.368
.328
.570
.436
.352
.314
.556
.424
.341
.302
.545
.412
.331
.292
.536
.402
.322
.284
.530
.394
.314
.277
.524
.388
.307
.270
.520
.381
.300
.263
1000
1
2
3
4
1.00
.844
.745
.663
.873
.705
.614
.552
.786
.614
.523
.473
.730
.554
.472
.422
.680
.514
.434
.385
.642
.485
.403
.359
.609
.463
.381
.338
.582
.445
.363
.321
.562
.430
.348
.307
.548
.418
.337
.295
.537
.406
.327
.285
.528
.397
.318
.278
.521
.390
.309
.270
.516
.383
.301
.263
.512
.376
.294
.256
Cable
size
Table 13-10—Fg : Grouping adjustment factor for directly buried 3/C,
or triplexed cables (7.5 in horizontal and 10 in
center-to-center vertical spacing)
Number
of layers
398
Number of horizontal cables
1
2
3
4
6
9
12
1
1.0
0.82
0.70
0.63
0.56
0.51
0.49
2
0.81
0.62
0.53
0.48
0.41
—
—
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
CABLE AMPACITY STUDIES
Table 13-11—Fg: Grouping adjustment factor for directly buried 1/C,
or triplexed cables (7.5 in horizontal and 10 in
center-to-center vertical spacing)
Number of
layers
Number of horizontal cables
3
6
9
12
1
1.0
0.79
0.71
0.68
2
0.73
0.58
—
—
Based on the computer studies for duct bank installations, it was found that the size and
voltage rating of the cables make a noticeable difference in the value of Fg. Therefore, the
adjustment factors for cable groupings are tabulated as functions of cable sizes and voltage
ratings. For applications where a mixed group of cables are installed in a duct bank, the value
of Fg will be different for each cable size. In this case, it is recommended that cable
ampacities be determined as the location of the cables is progressively changed from the
worst (hottest) conduit locations and the best (coolest) conduit locations to establish the most
economical arrangement.
Note that no grouping adjustment factor is given for cables installed in air or in conduits in
air. Refer to the NEC and IEEE Std 835-1994 for the allowable ampacities of cable installed
in conduits in air.
13.5 Example
To illustrate the use of the method described in this chapter, a 3 × 5 duct bank system (3 rows,
5 columns) is considered. The duct bank contains 350 kcmil and 500 kcmil (15 kV, 3/C)
copper cables. Ducts are a diameter of 5 in (trade size) of PVC, and are separated by 7.5 in
(center-to-center spacing), as shown in Figure 13-3. The soil thermal resistivity (RHO) is
120 °C-cm/W, and the maximum soil ambient temperature is 30 °C.
The objective of this example is to determine the maximum ampacities of the cables under
the specified conditions of use, i.e., to limit the conductor temperature of the hottest location
to 75 °C (an NEC requirement for wet locations). To achieve this, the base ampacities of the
cables are found first. These ampacities are then derated using the adjustment factors. The
computer program is then used to verify the derated ampacities by calculating the actual
conductor temperatures.
The depth of the duct bank is set at 30 in for this example. For average values of soil thermal
resistivity, the depth can be varied by approximately ±10% without drastically affecting the
resulting ampacities. However, larger variations in the bank depth, or larger soil thermal
resistivities, may significantly affect ampacities.
Copyright © 1998 IEEE. All rights reserved.
399
IEEE
Std 399-1997
CHAPTER 13
Figure 13-3—3 × 5 duct bank arrangement
13.5.1 Base ampacities
From the NEC ampacity tables, the base ampacities of 15 kV three-conductor cables under an
isolated condition and based on a conductor temperature of 90 °C, ambient soil temperature
of 20 °C, and thermal resistivity (RHO) of 90 °C-cm/W are as follows:
I = 375 A (350 kcmil)
I = 450 A (500 kcmil)
13.5.2 Manual method
The required ampacity adjustment factors for the ambient and conductor temperatures,
thermal resistivity, and grouping are as follows:
Ft
= 0.82 for adjustment in the ambient temperature from 20–30 °C and conductor
temperature from 90–75 °C (see Table 13-4).
Fth = 0.90 for adjustment in the thermal resistivity from a RHO of 90–120 °C–cm/W (see
Table 13-6).
400
Copyright © 1998 IEEE. All rights reserved.
CABLE AMPACITY STUDIES
IEEE
Std 399-1997
Fg = 0.479 for grouping adjustment of 15 kV, 3/C 350 kcmil cables installed in a 3 × 5 duct
bank (see Table 13-8).
Fg = 0.478 for grouping adjustment of 15 kV, 3/C 500 kcmil cables installed in a 3 × 5 duct
bank (see Table 13-8).
The overall cable adjustment factors are:
F = 0.82 × 0.90 × 0.479 = 0.354 (350 kcmil cables)
F = 0.82 × 0.90 × 0.478 = 0.353 (500 kcmil cables)
The maximum allowable ampacity of each cable size is the multiplication product of the
cable base ampacity by the overall adjustment factor. This ampacity adjustment would limit
the temperature of the hottest conductor to 75 °C when all of the cables in the duct bank are
loaded at 100% of their derated ampacities.
I' = 375 × 0.354 = 133 A (350 kcmil cables)
I' = 450 × 0.353 = 159 A (500 kcmil cables)
13.5.3 Computer method
As the last step, a computer program is run to simulate the actual conductor temperature
using the ampacities determined by the manual method. The computer program used here is
the same program that was used to generate the ampacity adjustment factors. The output
report of the program is shown in Figure 13-4, where (a) indicates all input parameters and
(b) indicates conduit locations and conductor temperatures.
The objective for this design was to find the cable ampacities that would limit the conductor
temperature to 75 °C. The results of the computer study indicate that the hottest conductor is
located in the middle row (2) and middle column (3) with a temperature of 74.3 °C. The
ampacities obtained from the manual method for this simplified example case exactly agree
with the ampacities obtained by the computer calculations. In more general cases, however,
where the assumptions listed in 13.4 do not apply, computer calculation would be necessary
to establish final ampacities
Copyright © 1998 IEEE. All rights reserved.
401
IEEE
Std 399-1997
Project:
Location:
Contract:
Engineer:
CHAPTER 13
Example
Irvine, California
1234567
F. S.
Page:
Date:
Study:
1
09-01-1989
SC-100
Cable ampacity derating example—3 × 5 duct bank application
O.D.
(in)
Insul.
thermal
R
(Ω/ft)
Dielectric
losses
(W/ft)
Yc
Ys
Cable
size
No. of
cond.
Volt
(kV)
Type
DC
resistance
(µΩ/ft)
500
3
15
CU
21.60
2.590
1.430
0.056
0.018
0.000
350
3
15
CU
30.80
2.290
1.564
0.048
0.009
0.000
Installation
Conduit
type
No.
of
rows
No.
of
cols.
Ref .
depth
(in)
Height
(in)
Width
(in)
Soil
Fill
Ambient
temp.
°C
Duct
bank
PVC
3
5
30.0
27.0
42.0
120.0
90.0
30.0
RHO
Cable
Conduit (in)
Row
Col.
Horiz.
dist.
(in)
Vert.
dist.
(in)
Load
current
(A)
No.
C/C
Size
kV
Type
Size
Thickness
1
1
6.00
6.00
159.0
1
3
500
15
CU
5.040
0.260
2
1
6.00
13.50
159.0
1
3
500
15
CU
5.040
0.260
3
1
6.00
21.00
159.0
1
3
500
15
CU
5.040
0.260
1
2
13.50
6.00
159.0
1
3
500
15
CU
5.040
0.260
2
2
13.50
13.50
159.0
1
3
500
15
CU
5.040
0.260
3
2
13.50
21.00
159.0
1
3
500
15
CU
5.040
0.260
1
3
21.00
6.00
133.0
1
3
350
15
CU
5.040
0.260
2
3
21.00
13.50
133.0
1
3
350
15
CU
5.040
0.260
3
3
21.00
21.00
133.0
1
3
350
15
CU
5.040
0.260
1
4
28.50
6.0
133.0
1
3
350
15
CU
5.040
0.260
2
4
28.50
13.50
133.0
1
3
350
15
CU
5.040
0.260
3
4
28.50
21.00
133.0
1
3
350
15
CU
5.040
0.260
1
5
37.00
6.00
133.0
1
3
350
15
CU
5.040
0.260
2
5
37.00
13.50
133.0
1
3
350
15
CU
5.040
0.260
3
5
37.00
21.00
133.0
1
3
350
15
CU
5.040
0.260
(a) Input parameters
Figure 13-4—Computer program output report
for cable ampacity derating
402
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
CABLE AMPACITY STUDIES
Project:
Location:
Contract:
Engineer:
Example
Irvine, California
1234567
F. S.
Page:
Date:
Study:
2
09-01-1989
SC-100
Cable ampacity derating example—3 × 5 duct bank application
Columns
1
2
3
4
5
Row 1
Cable:
Amp:
Temp:
500.0
159.0
66.8
500.0
159.0
69.7
350.0
133.0
70.9
350.0
133.0
69.9
350.0
133.0
66.6
Row 2
Cable:
Amp:
Temp:
500.0
159.0
69.7
500.0
159.0
73.0
350.0
133.0
74.3
350.0
133.0
73.1
350.0
133.0
69.3
Row 3
Cable:
Amp:
Temp:
500.0
159.0
69.3
500.0
159.0
72.3
350.0
133.0
73.5
350.0
133.0
72.4
350.0
133.0
69.0
(b) Conduit locations and conductor temperatures
Figure 13-4—Computer program output report
for cable ampacity derating (Continued)
13.6 Conclusion
Analytical derating of cable ampacity is a complex and tedious process. A manual method
was developed in this chapter that uses adjustment factors to simplify cable derating for some
very specific conditions of use and produce close approximations to actual ampacities. The
results from the manual method can then be entered as the initial ampacities for input into a
cable ampacity computer program. The speed of the computer allows the program to use a
more complex model, which considers factors specific to a particular installation and can
iteratively adjust the conductor resistances as a function of temperature. The following is a
list of factors that are specific for the cable system:
—
—
—
—
—
—
—
—
—
—
Conduit type
Conduit wall thickness
Conduit inside diameter
Asymmetrical spacing of cables or conduits
Conductor load currents and load cycles
Height, width, and depth of duct bank
Thermal resistivity of backfill and/or duct bank
Thermal resistance of cable insulation
Dielectric losses of cable insulation
AC/DC ratio of conductor resistance
The results from the computer program should be compared with the initial ampacities found
by the manual process to determine whether corrective measures, i.e., changes in cable sizes,
Copyright © 1998 IEEE. All rights reserved.
403
IEEE
Std 399-1997
CHAPTER 13
duct rearrangement, etc., are required. Many computer programs alternatively calculate cable
temperatures for a given ampere loading or cable ampacities at a given temperature. Some
recently developed computer programs perform the entire process to size the cables
automatically. To find an optimal design, the cable ampacity computer program simulates
many different cable arrangements and loading conditions, including future load expansion
requirements. This optimization is important in the initial stages of cable system design since
changes to cable systems are costly, especially for underground installations. Additionally,
the downtime required to correct a faulty cable design may be very long.
13.7 References
IEEE Std 835-1994, Standard Power Cable Ampacity Tables.3
NFPA 70-1996, National Electrical Code® (NEC®).4
13.8 Bibliography
[B1] Knutson, H. M., and Miles, B. B., “Cable derating parameters and their effects,” IEEE
paper no. PCIC-77-5, 1977.
[B2] National Electrical Code Committee Report, 1980 Annual Meeting.
[B3] National Electrical Code Committee Report, 1986 Annual Meeting.
[B4] National Electrical Code Committee Report, 1989 Annual Meeting.
[B5] National Electrical Code Technical Committee Documentation, 1980 Annual Meeting.
[B6] National Electrical Code Technical Committee Documentation, 1983 Annual Meeting.
[B7] National Electrical Code Technical Committee Documentation, 1986 Annual Meeting.
[B8] National Electrical Code Technical Committee Documentation, 1989 Annual Meeting.
[B9] National Electrical Code Technical Committee Report, 1983 Annual Meeting.
[B10] Neher, J. H., and McGrath, M. H., “The calculations of the temperature rise and load
capability of cable systems,” AIEE Transactions on Power Applications Systems, vol. 76,
pt. III, pp. 752–772, Oct. 1957.
3IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
4The NEC is available from Publications Sales, National Fire Protection Association, 1 Batterymarch Park, P.O. Box
9101, Quincy, MA 02269-9101, USA. It is also available from the Institute of Electrical and Electronics Engineers,
445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331, USA.
404
Copyright © 1998 IEEE. All rights reserved.
CABLE AMPACITY STUDIES
IEEE
Std 399-1997
[B11] NEMA Subcommittee Final Report, Determination of Maximum Permissible Current
Carrying Capacity of Code Insulated Wires and Cable for Building Purposes, Part IV.5
[B12] Rosch, S. J., “The current-carrying capacity of rubber insulated conductors,” paper
presented at AIEE meeting, New York City, Jan. 27, 1938.
[B13] Schurig, O. R., and Frick, G. W., “Heating and current-carrying capacity of bare
conductors for outdoor service,” General Electric Review, vol. 33, Schenectady, NY, p. 141,
1930.
[B14] Shokooh, F., and Knutson, H. M., “Ampacity derating of underground cables,” IEEE
paper no. CH2581-7/88, presented at the I&CPS conference in Baltimore, MD, May 1988.
[B15] Shokooh, F., and Knutson, H. M., “A simple approach to cable ampacity rating,” IEEE
paper no. PCIC-83-16, presented at the IEEE/PCIC conference in Denver, CO, Sept. 1983.
[B16] Simmons, D. M., “Calculation of the Electrical Problems of Underground Cables,” The
Electric Journal, East Pittsburgh, PA, May–Nov. 1932.
[B17] Study NBSIR 78-1477, Department of Energy, Washington, DC.
[B18] Underground Systems Reference Book, EEI publication no. 55-16, Edison Electric
Institute, New York City, NY, 1957.
[B19] Zipse, D. W., “Ampacity Tables—Demystifying the Myths,” presented at the IEEE/
PCIC conference, Sept. 1988.
5NEMA
publications are available from the National Electrical Manufacturers Association, 1300 N. 17th St., Ste.
1847, Rosslyn, VA 22209, USA.
Copyright © 1998 IEEE. All rights reserved.
405
Chapter 14
Ground mat studies
14.1 Introduction
A ground mat study has one primary purpose: to determine if a ground mat design will limit
the neutral-to-ground voltages normally present during ground faults to values that the
average person can tolerate. Equipment protection or system operation is rarely an objective
of a ground mat study. Historically, only utilities and unusually large industrial plants have
been concerned with this type of study. However, the trend of power systems toward everincreasing short-circuit capability has made safe ground mat design a criterion for all sizes of
substations. This chapter will briefly review the theoretical background behind ground mat
studies and discuss its application in the design of a ground mat by computer program.
14.2 Justification for ground mat studies
Virtually every exposed metallic object in an industrial facility is connected to ground, either
deliberately or by accident. Under normal operating conditions, these conductors will be at
the same potential as the surrounding earth. However, during ground faults, the absolute
potential of the grounding system will rise (often to thousands of volts) along with any
structural steel tied to the grounding system. Because any metal is a relatively good
conductor, the steelwork everywhere will be at essentially the same voltage for most
industrial installations. Most soils are poor conductors, however, and the flow of fault current
through the earth will create definite and sometimes deadly potential gradients. Ground mat
studies calculate the voltage difference between the grounding grid and points at the earth’s
surface and evaluate the shock hazard involved. Moreover, a computerized ground mat
analysis of the type described herein allows the designer to specifically identify unsafe areas
within a proposed mat and to optimize the mat design while verifying that the design is safe
throughout the area in question.
14.3 Modeling the human body
To properly understand the analytical techniques involved in a ground mat study, it is
necessary to understand the electrical characteristics of the most important part of the circuit:
the human body. A normal healthy person can feel a current of about 1 mA. (Tests have long
ago established that electric shock effects are the result of current and not voltage.) Currents
of approximately 10–25 mA can cause lack of muscular control. In most men, 100 mA will
cause ventricular fibrillation. Higher currents can stop the heart completely or cause severe
electrical burns.
For practical reasons, most ground mat studies use the threshold of ventricular fibrillation,
rather than muscular paralysis or other physiological factors, as their design criterion.
Ventricular fibrillation is a condition in which the heart beats in an abnormal and ineffective
Copyright © 1998 IEEE. All rights reserved.
407
IEEE
Std 399-1997
CHAPTER 14
manner, with fatal results. Accordingly, most ground mats are designed to limit body currents
to values below this threshold. Tests on animals with body and heart weights comparable to
those of a human have determined that 99.5% of all healthy humans can tolerate a current
through the heart region defined by
0.116
I b = ------------T
(14-1)
where
Ib
T
is the maximum body current in amperes, and
is the duration of current in seconds,
without going into ventricular fibrillation. This equation applies to both men and women with
0.116 used as the constant of proportionality, but is valid only for 60 Hz currents. In practice,
most fault currents have a dc offset. This dc component is represented by a correction factor
described in 14.4.2.
Tests indicate that the heart requires about 5 min to return to normal after experiencing a
severe electrical shock. This implies that two or more closely spaced shocks (such as those
that would occur in systems with automatic reclosing) would tend to have a cumulative
effect. Present industry practice considers two closely spaced shocks (T1 and T2) to be
equivalent to a single shock (T3) whose duration is the sum of the intervals of the individual
shocks (T1 + T2 = T3).
Although there are many possible ways that a person may be shocked, industry practice is to
evaluate shock hazards for two common, standard conditions. Figures 14-1 and 14-2 show
these situations and their equivalent resistance diagrams. Figure 14-1 shows a touch contact
with current flowing from the operator’s hand to his feet. Figure 14-2 shows a step contact
where current flows from one foot to the other. In each case, the body current Ib is driven by
the potential difference between points A and B. Exposure to touch potential normally poses
a greater danger than exposure to step potential. The step potentials are usually smaller in
magnitude, the corresponding body resistance greater, and the permissible body current
higher than for touch contacts. (The fibrillation current is the same for both types of contact.
In the case of step potentials, however, not all current flowing from one leg to the other will
pass through the heart region.) The worst possible touch potential (called “mesh potential”)
occurs at or near the center of a grid mesh. Accordingly, industry practice has made the mesh
potential the standard criterion for determining safe ground mat design. In most cases,
controlling mesh potentials will bring step potentials well within safe limits. Step potentials
can, however, reach dangerous levels at points immediately outside the grid.
Since the body of an individual who is exposed to an electrical shock forms a shunt branch in
an electrical circuit, the resistance of this branch must be determined to calculate the
corresponding body current. Generally, the hand and foot contact resistances are considered
to be negligible. However, the resistance of the soil directly underneath the foot is usually
significant. Treating the foot as a circular plate electrode gives an approximate resistance of
3 ρs, where ρs is the soil resistivity. The body itself has a total measured resistance of about
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GROUND MAT STUDIES
Figure 14-1—Touch potential
Figure 14-2—Step potential
2300 Ω hand to hand or 1100 Ω hand to foot. In the interest of simplicity and conservatism,
IEEE Std 80-19861 recommends the use of 1000 ohms as a reasonable approximation for
body resistance in both models. This yields a total branch resistance
R = 1000 Ω + 6 ρ s
(14-2)
for foot-to-foot resistance, and
R = 1000 Ω + 1.5 ρ s
1Information
(14-3)
on references can be found in 14.10.
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for hand-to-foot resistance where ρs is the surface resistivity in ohm-meters ( Ω ⋅ m ) and R is
expressed in ohms (Ω). If the station surface has been dressed with crushed rock or some
other high resistivity material, the resistivity of the surface layer material should be used in
Equations (14-2) and (14-3).
Because potential is easier to calculate and measure than current, the fibrillation threshold
given by Equation (14-1) is normally expressed in terms of voltage. Combining Equations
(14-1), (14-2), and (14-3) gives the maximum tolerable step and touch potentials:
( 1000 Ω + 6 ρ s ) ( 0.116 )
E step-tolerable = ---------------------------------------------------------T
(14-4)
( 1000 Ω + 1.5 ρ s ) ( 0.116 )
E touch-tolerable = -------------------------------------------------------------T
(14-5)
Because these voltages are dependent on surface resistivity, most industrial facilities have
several different values for each tolerable voltage to match the various surface materials
found in the plant.
Although in each of the cases discussed, body resistance shunts a part of the ground
resistance, its actual effect on voltage and current distribution in the overall system is
negligible. This becomes obvious when the normal magnitude of the ground fault current (as
much as several thousand amperes) is compared to the desired body current (usually no more
than several hundred milliamperes).
14.4 Traditional analysis of the ground mat
The voltage rise of any point within the grid depends upon three basic factors: groundresistivity, available fault current, and grid geometry.
14.4.1 Ground resistivity
Most ground mat studies assume that the ground grid is buried in homogeneous soil. This is a
good model for most soils and simplifies the calculations considerably. Also, many
nonhomogeneous soils can be modeled by two-layer techniques. Although reasonably
straightforward, these methods involve quite a bit of calculation, making computation by
hand difficult. Normally, the two-layer model is necessary only for locations where bedrock
and other natural soil layers with different resistivities are close enough to the surface and/or
grid to severely affect the distribution of current.
Of far more serious concern are soils that experience drastic and unpredictable changes in
resistivity at various points on the surface. These situations present the following problems:
a)
b)
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Difficulty of modeling the soil in calculations
Physical difficulties in finding the area boundaries in the field and measuring each
area’s local resistivity
Copyright © 1998 IEEE. All rights reserved.
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Std 399-1997
GROUND MAT STUDIES
At present, these cases are normally handled by the inclusion of a safety margin in the value
used for soil resistivity.
IEEE Std 80-1986 and Sunde [B19]2 contain descriptions of simple methods of measuring
soil resistivity for homogeneous and two-layer soils. Because soil resistivity varies with
moisture content and, to a lesser degree, with temperature, ideally these measurements should
be made over a period of time under different weather conditions. If, for some reason, an
actual measurement of resistivity is impractical, Table 14-1 gives approximate values of
resistivity for different soil types. These values are only approximations and should be
replaced by measured data whenever possible.
Table 14-1—Representative values of soil resistivities
Type of ground
Resistivity ( Ω ⋅ m )
Wet organic soil
10
Moist soil
102
Dry soil
103
Bedrock
104
14.4.2 Fault current—magnitude and duration
Since shock hazard is a function of both time and current, a strictly rigorous ground mat
analysis would require checking every possible combination of time and current. In practice,
the worst shock hazard normally occurs at the maximum fault current. Determination of
ground fault current and clearing time normally requires a separate system study. The
techniques and problems of making fault studies are covered in numerous sources (including
this book). Therefore, this section will only cover aspects peculiar to ground grid studies.
After the system impedance and grid resistance have been determined, the maximum ground
fault current (assuming a bolted fault) is given as follows:
3V
I = --------------------------------------------------------------------------------------------------3R g + ( R 1 + R 2 + R 0 ) + j ( X "1 + X 2 + X 0 )
(14-6)
where
I
V
Rg
2Numbers
is the maximum fault current in amperes [note that this is not the same as the
current in Ib in Equation (14-1)],
is the phase-to-neutral voltage in volts,
is the grid resistance to earth in ohms,
in brackets correspond to those of the bibliography in 14.11.
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R1
R2
R0
X "1
X2
X0
CHAPTER 14
is the positive sequence system resistance in ohms,
is the negative sequence system resistance in ohms,
is the zero sequence system resistance in ohms,
is the positive sequence subtransient system reactance in ohms,
is the negative sequence system reactance in ohms,
is the zero sequence system reactance in ohms.
This current will, in general, be a sinusoidal wave with a dc offset. Since dc current can also
cause fibrillation, the current value I must be multiplied by a correction factor called the
decrement factor to account for this effect. Table 14-2 gives approximate values for this
factor. For more accurate results, the exact value for the decrement factor D is given by the
following equation:
D =
1
--T
–2 ω T
---------------
1 X
X⁄R
T + ---- ⋅ ---- 1 – e
ω R
(14-7)
where
T
ω
X
R
is the duration of fault in seconds,
is the system frequency in radians per second,
is the total system reactance in ohms,
is the total system resistance in ohms.
Table 14-2—Decrement factor for use in calculating
electrical shock effect of asymmetrical ac currents
Shock and fault duration
Decrement factor
Seconds
Cycles (60 Hz)
0.008
1/2
1.65
0.100
6
1.25
0.250
15
1.10
0.5 or more
30 or more
1.00
The current value calculated in Equation (14-6) must be multiplied by this factor to find the
effective fault current. Note that the time T in Equation (14-7) is the same as that used in
Equations (14-1), (14-4), and (14-5).
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A common mistake in calculating ground mat current is to ignore alternate current paths. In
most systems, only a portion of the ground fault current will return to the source through the
earth. Because of the time and expense involved in running a full scale short-circuit study to
accurately account for the division of fault current, the worst-case situation based upon the
full short-circuit capability of the fault source is generally used.
To determine the fault duration, it is necessary to analyze the relaying scheme to find the
interrupting time for the current calculated by Equation (14-6). The choice of the clearing
time of either the primary protective devices or the backup protection for the fault duration
depends upon the individual system. Designers must choose between the two on the basis of
the estimated reliability of the primary protection and the desired safety margin. Choice of
backup device clearing time is more conservative, but it will result in a more costly ground
mat installation. Substitution of this time in Equations (14-4) and (14-5) will fix the
maximum allowable step and touch potentials at the appropriate values.
14.4.3 Fault current—the role of grid resistance
In most power systems, the grid resistance is a significant part of the total ground fault
impedance. Accurate calculation of ground fault currents requires an accurate and dependable
value for the grid resistance. Equation (14-8) (taken from IEEE Std 80-1986) gives a quick
and simple formula for the calculation of resistance when a minimum of design work has
been completed.
ρ ρ
R = ----- + --4r L
(14-8)
where
R
ρ
L
r
is the grid resistance to ground in ohms,
is the soil resistivity in ohm-meters,
is the total length of grid conductors in meters,
is the radius of a circle with area equal to that of the grid in meters.
The first term gives the resistance of a circular plate with the same area as the grid. The
second term compensates for the grid’s departure from the idealized plate model. The more
the length of the grid conductors increases, the smaller this term becomes. This equation is
surprisingly accurate and is ideal for the initial stages of a study where only the most basic
data about the ground mat is available.
By inspecting Equation (14-8), it also becomes evident that adding grid conductors to a mat
to reduce its resistance eventually becomes ineffective. As the conductors are crowded
together, their mutual interference increases to the point where new conductors tend only to
redistribute fault current around the grid, rather than lower its resistance.
Any computer program that can calculate the grid voltage rise can also calculate the grid
resistance (with greater accuracy than the method described immediately above). The grid
resistance is simply the total grid voltage rise (relative to a “remote” ground reference)
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divided by the total fault current. In many cases, such programs perform this calculation
automatically. This method can be applied to any grid configuration with any number of
conductor elements. However, because the more advanced of these programs calculate grid
voltages by solving hundreds of simultaneous equations, the same procedure is usually not
practically achievable with hand calculations.
Since grid resistance is viewed as a measure of the grid’s ability to disperse ground fault
current, many designers are tempted to use resistance as an indicator of relative safety of a
ground mesh. In general, however, there is no direct correlation between grid resistance and
safety. At high fault currents, dangerous potentials exist within low resistance grids. The only
occasion where a low grid resistance can guarantee safety is when the maximum potential
rise of the entire grid (that is, grid potential) is less than the allowable touch potentials. In
these cases, the ground mat is inherently safe.
14.4.4 Grid geometry
The physical layout of the grid conductors plays a major role in ground mat analysis. The
step and touch potentials depend upon grid burial depth, length and diameter of conductors,
spacing between each conductor, distribution of current throughout the grid, location of the
grid with respect to a different resistivity soil layer, and proximity of the fault electrode and
the system grounding electrodes to the grid conductors, along with many other factors of
lesser importance. A perfectly rigorous analysis of all these variables would require both
simultaneous linear and complex differential equations to exactly describe the distribution of
current throughout the grid.
Historically, IEEE Std 80-1986 provides the only practical method for computing the effects
of the grid geometry upon the step and touch potentials [Equations (14-9) and (14-10)].
I
E mesh = K m K i ρ --L
I
E step = K s K i ρ --L
(14-9)
(14-10)
where
Emesh is the worst-case touch potential at the surface above any individual grid area (i.e.,
“mesh”) within the mat,
Estep is the worst-case step potential anywhere above the mat,
ρ
is the soil resistivity in ohm-meters,
I
is the maximum total fault current in amperes (adjusted for the decrement factor)
L
is the total length of grid conductors in meters,
Km
is the mesh coefficient,
Ks
is the step coefficient,
Ki
is the irregularity factor.
Coefficients Km and Ks are calculated by two reasonably simple equations based upon the
number of grid elements, their spacing and diameters, and the burial depth of the grid.
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Because these equations do not take into account the many other factors that influence grid
voltages, they are not meant to rigorously model a grid design, but are instead intended to
make hand calculation of touch and step potentials feasible. Equations (14-9) and (14-10)
incorporate an irregularity factor Ki to compensate for the inaccuracies introduced by these
simplifying assumptions. Except for applications involving very simple grid configurations,
proper selection of a value for Ki is dependent upon the experience and judgment of the
designer. Km and Ks can only be calculated for regular grid designs and must be estimated for
irregular grid geometries. Most often, a high value picked for all these factors in the interest
of conservatism usually results in an overdesigned mat. Conversely, there is no way to determine if the selected values are too low, resulting in an unsafe ground mat design.
The values of Emesh and Estep calculated by Equations (14-9) and (14-10) must be compared
to the tolerable touch and step potentials, Etouch-tolerable and Estep-tolerable as determined from
Equations (14-4) and (14-5) in order to establish whether or not the design is safe. If, in fact,
one of the tolerable voltage limits is exceeded, it is sometimes possible, by inspection of the
grid, to determine mesh locations where additional cross-conductors should be added in order
to achieve a safe design. The more general approach, however, is to uniformly increase the
number of grid conductors.
Although this traditional hand calculation method for determining step and mesh potentials
was considered acceptable in the past, modern ground mat studies normally use one of the
new generation of computer programs. There are two types of computer programs available
for ground mat studies. One type performs the aforementioned traditional hand calculations
for empirically determining step and mesh potentials, but does it faster and more efficiently
than possible by hand. The other type of program calculates the step and touch potentials for
each individual grid (i.e., “mesh”) within the overall ground mat. The results, therefore, allow
a more detailed analysis of ground mat design effectiveness, pinpointing any mesh locations
where shock hazards may exist. The discussion that follows will concentrate on the latter of
the two program types.
14.5 Advanced grid modeling
The key to an accurate ground grid analysis is the individual modeling of each single grid element, rather than the en masse treatment used in IEEE Std 80-1986. For example, Figure 14-3
shows a single grid element located at depth h below the earth’s surface in a homogenous
medium. The element runs from point (x1, y1, z1) to (x1, y2, z1) and is radiating current to the
surrounding earth at the linear current density σl (the current per unit length). By integrating
σl over the length of the grid element, the current flux ξ can be found at any desired point (a)
as follows:
ξ =
y2
∫y 1
σ dy
-----1- -----2- r
4π R
Copyright © 1998 IEEE. All rights reserved.
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where
ξ
σl
is the current per unit area at any point,
is the current flowing to ground per unit length of conductor (current density),
R
=
r
( i – x 1 )i + ( j – y 1 ) j + ( k – z 1 )k
= ------------------------------------------------------------------------------ .
2
2
2
( i – x1 ) + ( j – y1 ) + ( k – z1 )
2
2
2
( i – x1 ) + ( j – y1 ) + ( k – z1 ) ,
NOTE—For the purposes of illustration, Equation (14-11) shows a special-case expression that is only
valid for lines running parallel to the y axis. The more general form is derived in the same manner, but is
much more difficult to follow.
Figure 14-3—Physical model used in calculating voltage
at point (a) due to a single conductor
Once ξ has been determined, the E field at the same point can be expressed as follows
(assuming a homogeneous soil):
E = ρξ
(14-12)
where ρ is the soil resistivity.
From this, the voltage at point (a) can be obtained by performing the following integration:
a
V al = – ∫ E ⋅ dl
∞
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(14-13)
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GROUND MAT STUDIES
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Std 399-1997
or
ρσ 1
2
2
2
V a1 = --------- ln [ j a – y 2 + ( i a – x 1 ) + ( j a – y 2 ) + ( k a – z 1 ) ]
4π
(14-14)
ρσ 1
2
2
2
+ --------- ln [ j a – y 1 + ( i a – x 1 ) + ( j a – y 1 ) + ( k a – z 1 ) ]
4π
where Val is the absolute potential at any point (a) due to line l. This process must be repeated
for every element in the grid.
NOTE—This process is complicated somewhat by the presence of the “current density” factor σl in the
equations. Although Equations (14-1) and (14-2) treat σl as a constant, in actuality it varies
continuously along the length of each grid element, as well as from element to element. In practice, the
variation of σl along the length of an element has little effect upon the calculated voltages, especially
when calculating mesh potentials. The variation between elements is very significant, however, and
must be obtained by solving a set of simultaneous equations. These can be written by using Equation
(14-14) to calculate the voltage at known points (the surface of each grid element, for example). When
the variation of current density along an element is important, it can be approximated by modeling the
element as several segments, each with its own value of σl .
Finally, the individual contribution of each grid element can be summed to determine the
total voltage at point (a).
An extension of this same basic approach involving multiple images of each conductor is
used to perform calculations for multilayer (typically two layer) soils (Dawalibi and
Mukhedkar [B7] and Joy, Meliopoulos, and Webb [B13]). The number and complexity of the
equations that must be solved is greater, but they can be readily managed using the computer
solution methods.
The advantages of this analytical method are immediately apparent. This technique automatically accounts for the finite length of each element, a particularly important consideration
when finding the potential at points near the end of an element. It can handle grid designs
with large degrees of asymmetry with no sacrifice in accuracy. In cases involving multilayer
soils, the effect of the resulting redistribution of currents within such soil systems (Dawalibi
and Mukhedkar [B7]) on touch and step potentials is accurately quantified. Furthermore,
since point a can be located anywhere and any number of points can be examined, detailed
analysis of the grid design is possible.
The grid layout also determines which points should be checked for touch and step potentials.
Touch potentials are normally calculated at the mesh centers, at control stations (where
operators may be present), at the entrances to the facility, and at the corners of the grid. Step
potentials are rarely a problem inside the grid. However, they may be a danger in the areas
just outside the grid, such as the exterior of a perimeter switchyard adjacent to the fence. The
worst step potentials usually occur along a diagonal line at the corners of a grid (see
Figure 14-4). Shock hazard voltages can be accurately determined at all such critical
locations using these calculating procedures. In addition, the absolute (earth) surface potentials and the ground potential rise (EPR) of the grid is an automatic by-product.
Copyright © 1998 IEEE. All rights reserved.
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Figure 14-4—Typical ground mat layout showing possible locations
of critical step and touch potentials near grid corners
14.6 Benchmark problems
Every analytical procedure (including computer programs) needs to be checked against a
benchmark problem. Figure 14-5 shows six different grid layouts where the mesh potentials
were measured on small scale models. Measured results from these configurations are the
best available benchmark information for ground mat analysis programs. The calculated
voltages (expressed as a percentage of the grid voltage) are shown in the center of each mesh.
For easy reference, the actual measured voltages are also shown in parentheses. For this
analysis, a program calculated the absolute voltage at the center of each mesh. In an actual
ground mat study, the calculated point voltages would be subtracted from the calculated grid
potential to determine the touch potential hazard. Figure 14-5 (especially grids E and F)
illustrates the accuracy of the program. Errors are typically on the order of 5%, and never
worse than 10%.
The most impressive feature of this type of program, however, is its ability to calculate voltages at any point of interest within or around the mat’s geometric boundaries. By repeated
use of the program throughout the design process, a ground mat layout can be fine-tuned to
achieve the desired protection without the need to overdesign any section of the mat.
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Figure 14-5—Experimental grids showing various (mesh) arrangements
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14.7 Input/output techniques
The increased use of personal computers in the workplace has led to higher expectations
for all software. Ground mat analysis programs are no exception. Since the grid layout is so
important to this class of programs, they are especially well suited to graphical input/output
methods. First generation programs typically required the user to input the end coordinates
of each grid element, and simply printed the calculated voltages at user-designated coordinates. Figure 14-6 shows a typical output report from an early ground mat analysis
computer program (in this particular case, grid E of Figure 14-5). Although the basic calculations are correct, interpretation is difficult and checking the data entry is laborious and
time-consuming.
Most modern ground-mat analysis programs support some form of graphical output, either
two-dimensional or three-dimensional, as illustrated by Figures 14-7, 14-8, and 14-9. Many
programs allow the user to draw the grid design and then calculate the endpoints internally.
With proper preparation, others are capable of reading the grid design directly from CADD
drawing files. Some programs can select the points to calculate automatically or plot
equipotential lines. Ground mat programs can also do material take-offs as well as material
and labor cost estimates. The latest programs not only calculate raw voltages, they compute
the mesh and touch potentials and compare them to the limits (also automatically calculated).
New programs can also manage multiple surface materials and soil layers. They can also
store intermediate calculations for later use. Advanced programs can edit their input data,
ignoring unimportant detail. Figure 14-11 shows a graphical output from a typical ground
mat analysis program of recent vintage. The conductor arrangement shown affords a ground
mat design that has a 5% minimum safety margin.
14.8 Sample problem
Figures 14-10 through 14-13 demonstrate the use of a ground mat analysis program on a
typical ground mat design in a homogeneous soil. Figure 14-10 shows the grid layout and all
pertinent data along with the mesh design resulting from application of the traditional hand
calculating procedures described in 14.4.4. Figure 14-11 gives the results of a computer
analysis of the grid clearly indicating that grid corners and certain perimeter meshes are
unsafe using these procedures. Figures 14-12 and 14-13 show the results of modified grid
designs. Note that in Figure 14-13 the amount of additional grid conductor required to safely
control mesh potentials in the grid has been minimized and that the use of the computer
program has permitted the location of this conductor to be optimally determined (that is, the
conductor has been added only where required). This is the great advantage of a computer
ground mat analysis.
14.9 Conclusion
The adaptation of classical analytical techniques and calculating procedures to the digital
computer has made ground mat analysis much more precise, reliable, and useful. Many
ground mat analysis programs can all but eliminate unnecessary grid overdesign and, if
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Figure 14-6—Sample output report—
Computer calculated ground grid potentials
properly applied, all ground mat analysis programs can detect unsafe conditions that might
otherwise go undiscovered until made apparent by serious mishap.
A ground mat study requires, as a minimum, the following data:
—
—
—
—
Soil resistivity, both at the level of the grid and (if appropriate) at any other soil layer
Resistivity of any special soil surface dressing material
Estimated duration of a ground fault
System frequency
Copyright © 1998 IEEE. All rights reserved.
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Figure 14-7—Sample computer generated graphical output
report for a typical ground mat analysis
—
—
—
—
System X/R ratio
Maximum symmetrical ground fault current, both future and present
Grid layout showing the precise location of every conductor
Coordinates where the potential rise must be calculated
Consideration of all this information will lead to a reliable, accurate, and useful study.
Two basic types of ground mat computer programs have been discussed. When considering
the purchase or use of such programs, it first must be determined what level of analytical
accuracy is required and, accordingly, which type of calculating method is desired—either
the traditional empirical method discussed in 14.4 of this chapter or the detailed mesh-bymesh method described in 14.5 through 14.8. Care should be taken to select a program that is
appropriate for the application chosen.
Although ground mat analysis programs provide an invaluable design tool, they are by no
means infallible. If at all possible, a followup investigation should be made of each grid after
it has been installed. This should include a measurement of grid resistance at the very least,
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Figure 14-8—Sample 2-dimensional program graphical output showing
equipotential plots of touch voltage at earth’s surface above mat
and preferably the measurement of the ac mesh potential at several locations within the grid.
If these measured values differ appreciably from the calculated ones, the results of the grid
study should be rechecked and supplemental rods or buried conductors provided as required
to establish safe conditions (see IEEE Std 80-1986).
14.10 References
IEEE Std 80-1986 (Reaff 1991), IEEE Guide for Safety in AC Substation Grounding.
Copyright © 1998 IEEE. All rights reserved.
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Figure 14-9—Sample 3-dimensional program graphical output showing
equipotential plots of touch voltage at earth’s surface above mat
14.11 Bibliography
[B1] Dalziel, C. F., “Dangerous electric currents,” AIEE Transactions, vol. 65, pp. 579–585;
1123–1124.
[B2] Dalziel, C. F., “Threshold 60-cycle fibrillating currents,” AIEE Transactions, vol. 79,
pp. 667–673, 1960.
[B3] Dalziel, C. F., “A study of the hazards of impulse currents,” AIEE Transactions, vol. 72,
pp. 1032–1043, 1953.
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Std 399-1997
Figure 14-10—Typical ground mat design showing all pertinent
soil and system data
[B4] Dalziel, C. F., and Lee, W. R., “Re-evaluation of lethal electric currents,” IEEE
Transactions on Industry General Applications, vol. IGA-4, pp. 467–476, Sept./Oct. 1968.
[B5] Dawalibi, F., and Mukhedkar, D., “Multi-step analysis of interconnected grounding
electrodes,” IEEE Transactions on Power Applications Systems, vol. PAS-95, pp. 113–119,
Jan./Feb. 1976.
[B6] Dawalibi, F., and Mukhedkar, D., “Optimum design of a substation grounding in a twolayer earth structure,” IEEE Transactions on Power Applications Systems, vol. PAS-94, no. 2,
pp. 252–272, Mar./Apr. 1975.
[B7] Dawalibi, F., and Mukhedkar, D., “Parametric analysis of grounding grids,” IEEE PES
Winter Meeting, Paper No. F79243-7, 1979.
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CHAPTER 14
Figure 14-11—Typical ground mat design showing meshes with
hazardous potentials as identified by computer analysis
[B8] Ferris, L. P., King, B. G., Spence, P. W., and Williams, H. B., “Effect of electrical shock
on the heart,” AIEE Transactions, vol. 55, pp. 498–515 and 1263, May 1936.
[B9] Geddes, L. A., and Baker, L. E., “Response to passage of electric current through the
body,” Journal of the Association for the Advancement of Medical Instruction, vol. 2, pp. 13–
18, Feb. 1971.
[B10] Gross, E. T. B., Chitnis, B. V., and Stratton, L. J., “Grounding grids for high-voltage
stations,” AIEE Transactions, vol. 72, pp. 799–810, 1953.
[B11] Gross, E. T. B., and Hollitch, R. F., “Grounding grids for high-voltage stations—III,
resistance of rectangular grids,” AIEE Transactions, vol. 75, pt. III, pp. 926–935, 1953.
[B12] Gross, E. T. B., and Wise, R. B., “Grounding grids for high-voltage stations—II,” AIEE
Transactions, vol. 74, pt. III, pp. 801–809, 1955.
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Figure 14-12—Typical ground mat design, first refinement
showing meshes with hazardous touch potentials
[B13] Joy, E. B., Meliopoulos, A. P., Webb, R. P., “Touch and step calculation for substation
systems,” IEEE PES Winter Meeting, Paper No. A79052-2, 1979.
[B14] Koch, W., “Grounding methods for high-voltage stations with grounded neutrals,”
Electrotechnische Zeit, vol. 71, no. 4, pp. 89–91, 1950.
[B15] Laurent, P., “General fundamentals of electrical grounding techniques,” Bulletin de la
Societé Francaise des Electriciens, vol. I, series 7, pp. 368–402, Jul. 1951.
[B16] Niemann, J., “Changeover from high-tension grounding installation to operation with a
grounded star point,” Electrotechnische Zeit, vol. 73, no. 10, pp. 333–337, May 15, 1952.
[B17] Rudenberg, R., “Grounding principals and practice I—fundamental considerations on
ground currents,” Electrical Engineering, vol. 64, pp. 1–13, Jan. 1945.
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Figure 14-13—Typical ground mat design, final refinement
with no hazardous touch potentials
[B18] Schwartz, S. J., “Analytical expression for resistance of grounding system,” AIEE
Transactions, vol. 73, pp. 1011–1016, 1954.
[B19] Sunde, E. D., “Earth Conduction Effects in Transmission Systems,” New York: Van
Nostrand, 1949.
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Chapter 15
Coordination studies
15.1 Introduction
An important step in the design of any power distribution system is the time-current
coordination of all overcurrent protective devices required for the protection of the system
and the connected equipment. When a short circuit or an abnormal power flow occurs for a
sustained period of time, the protective devices should react to isolate the problem with
minimum disruption to the balance of the system. This is the goal of a well-coordinated
electrical power system.
The reader should be aware that this chapter addresses only one aspect of system protection:
overcurrent protection. For most large, medium-voltage systems, overcurrent protection acts
only as backup for primary protection and, as such, this chapter is not a complete study of
system protection. The operation of protective devices can be estimated by graphic
representation of the time-current characteristics curves (TCCs) of these devices. By plotting
these characteristics on a common graph, the relationship of the characteristics among the
devices is immediately apparent. Any potential trouble spots, such as overlapping of curves
or unnecessarily long time intervals between devices, are revealed. By indicating on the
current scale the maximum and minimum value of short-circuit currents (three-phase and
line-to-ground) that can occur at various points in the circuit, the operation of circuit
protective devices can be estimated for various fault conditions. An accompanying single-line
diagram can indicate the components and define their location in the circuit.
The time-honored method of plotting these curves, as illustrated in Figure 15-1, is to
superimpose a transparent sheet of graph paper over the manufacturer’s published curves on
an illuminated drafting table. When the time scales have been carefully matched and the
current scales adjusted, the curve is traced onto the graph paper using a French curve or
flexible spline. The process is then repeated for the remaining protective devices. Damage
curves for equipment, such as motors, transformers, and cables, are also plotted to assess the
level of protection provided for the equipment and to provide a graphical representation of
the protection achieved. This process can be both tedious, time consuming, and prone to
error.
Since the selection of device parameters follows well-defined rules, for the most part,
computer programs are available for the task of producing these curves. Some programs will
select the settings necessary to achieve a well-coordinated system as well as plotting curves
and related data.
The application of the computer and computer software to time-current coordination studies
is a viable alternative to the manual approach. With the availability of dependable hardware
for digitizing, plotting, computing, and communicating with a computer, time-current
coordination studies using these tools are a practical alternative that is now possible. Device
settings and ratings may be calculated and tabulated. The TCC drawings may be displayed on
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Figure 15-1—Manual method of producing time-current curves
by using a light table
graphic monitors, and the output reports may be routed to graphical printers and plotters. The
popular K&E form 48-5258 may be used as a plot background, when desired.
15.2 Basics of coordination
Whether the coordination is done manually or by computer, it is necessary for the engineer to
“describe” the system. The information needed to perform a coordination study is a singleline diagram showing the following:
—
—
—
—
—
—
—
—
—
—
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Protective device manufacture and type
Protective device ratings
Trip settings and available range
Short-circuit current at each system bus (three-phase and line-to-ground)
Full load current of all loads
Voltage level at each bus
Transformer kVA, impedance, and connections (delta-wye, etc.)
Current transformer (CT) and potential transformer (PT) ratios
Cable size, conductor material, and insulation
All sources and ties
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Figure 15-2 shows a section of the single-line diagram presented in Chapter 1 (Figure 1-1)
with protective devices and other information added using the normal conventions employed
for coordination studies. Figure 15-3, which is a copy of Figure 5-26 in IEEE Std 141-1993
[B1],1 shows a coordination study with a manually drawn plot. Figure 15-4 uses the same
single-line diagram and shows the study done by a computer program.
Information is needed regarding the time-current characteristics of the devices in the circuit.
Traditionally, this information is in the form of manufacturer’s TCC curves on 11 in by 17 in,
4-1/2 by 5 cycle log-log paper. When coordination is performed on a computer, this
information is stored in a device data file or “library.”
The coordination can start at the farthest downstream device but coordination must exist with
the utility’s last protective device and the plant’s main protective device. It is in this area, at
the very least, where engineering judgment is required in order to achieve the best
coordination. The downstream device is sketched with a characteristic and setting that allows
full-load current to flow and prevents tripping for transient and normal overload conditions.
The device immediately upstream is next, with its characteristics and settings selected to
satisfy the specified current requirements and to coordinate with the downstream device. This
procedure is followed for each device either by use of a light table or a grid on transparent
paper and shifting and sketching or by giving the information to the computer and allowing it
to show the coordination. When a transformer is encountered, the impedance, connection,
and rating are needed to properly select settings and ratings of upstream devices. A typical
procedure for organizing data before beginning a coordination study by computer is as
follows:
a)
b)
c)
d)
e)
f)
g)
Note motor horsepower, full load current, acceleration time, and locked rotor current.
For each protective device: note short circuit current, full load current, and voltage
level at each device. List device manufacturer and type, and program file name for
device.
For each low-voltage breaker, indicate long time, short time, instantaneous. Note
settings if existing device.
For each fuse, note rating.
For each relay, note tap range, CT ratio, tap and time dial, if known, and whether
relay has instantaneous.
For each transformer, note kVA, fan cooled rating, impedance, and transformer
connection.
For cable damage curves: note cable size, conductor material, and cable insulation.
The engineer, when working on a coordination study, either manually or by computer, will
encounter situations in which there are devices of a specific size or settings that cannot be
adjusted, or when other constraints make it impossible to obtain perfect coordination. In a
situation such as this, the design engineer must make a compromise judgment, based on his
or her training and experience. The ability to try various settings to determine the best
coordination must be a feature of any computer program.
1The
numbers in brackets correspond to those of the bibliography in 15.9.
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Figure 15-2—Single-line diagram showing notations
relative to coordination
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*As defined in ANSI C57.12.00-1973, which has been superseded by IEEE C57.12.00-1993.
Figure 15-3—Manually produced time-current curve
Once the phase overcurrent coordination study is made, a ground-fault current coordination
study should be performed using separate plots because of different fault current levels. The
results of the ground fault coordination study should be compared with the phase overcurrent
protection to verify the coordination.
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Figure 15-4—Computer-produced time-current curve
plotted on a printer
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15.3 Computer programs for coordination
Several types of computer programs for the coordination of circuit protective devices are
available to the design engineer. For some time to come, this area is expected to be a rapidly
changing technology. For purposes of illustration, two types of programs will be introduced.
One type of program is designed to select circuit breaker trip settings, overcurrent relay tap
and time dial settings, and fuse ratings and to plot the TCC curves on the standard 4-1/2 by
5 cycle format. The second type stores the data, makes it accessible to the user, but does not
automatically perform the coordination. It produces the plot based on the settings chosen by
the engineer as would be done manually.
15.3.1 Coordination programs
Coordination programs perform setting and rating selections, unless these items are input by
the user. In such programs, the time-current characteristic data for various manufacturers’
fuses, relays, and circuit breakers must be stored. Ratings and settings are selected by the
program to satisfy the stated input conditions. The engineer must accept the responsibility of
reviewing the selections and making the final determinations in any coordination study.
Such programs may include the necessary logic to calculate and plot transformer inrush and
withstand as well as cable damage and motor-starting curves. When there is a motor in the
system, the program will select the setting of the first device to ensure that it will not trip on
motor starting and will provide the commands to plot the curve. From that point on, the
program will perform the coordination based upon the type of device used and the parameters
built into the program for clearance with downstream devices using accepted coordination
criteria for separation between curves.
Programs that perform the coordination are usually structured so that any device parameter
may be specified by the user. If the settings are input, the program may not perform the usual
checks for that device, since the logic of the program may be based on the assumption that
the user is aware of what he or she is doing. With this method, the engineer may add one
device at a time and view it before continuing, or may input the entire set of data before
viewing it on a graphics screen.
With this type of program, information on the type of device selected may be available for
use in developing and plotting the single-line diagram.
15.3.2 TCC plotting programs
The second type of coordination program is the type that provides a library for drawing
coordination curves and other features usually shown on a TCC curve. The same type of plot
can be obtained as in the first coordination type by having the user enter the settings and
ratings and construct the curves using the library files. All decisions associated with the
selection of device settings are performed by the engineer. These programs may include the
logic for calculating and plotting transformer inrush and withstand, cable damage, motorstarting curves, and one-line diagrams.
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15.4 Common structure for computer programs
The primary task of the protective device coordination computer program is to permit the
engineer to produce coordination studies that are similar to the studies that would be created
using manual techniques.
A well-structured program will contain features to model various types of protective devices
and equipment damage characteristics and to store these characteristics in a device library.
The program should be able to call these devices from the library and to accurately reproduce
the manufacturer’s curves on the graphic output device. In addition, the program should
generate documentation of the studies including output reports indicating the device settings
and single-line diagrams indicating the elements of the power system described on the TCC
drawing.
15.4.1 Project data base files
A data base is a method by which the program stores the information contained in the study.
Ideally, the data base will be structured to hold all of the information on all of the devices
located in the power system. Coordination studies usually consist of a number of different
TCC drawings. A single device, say a main breaker, may be shown on more than one
drawing. Keeping the characteristics of the devices in specified groups will eliminate
duplicate data entries for displaying the same device on a second or third drawing. This data
base structure will eliminate the possibility of plotting the same device with different settings
on different drawings.
15.4.2 Interactive data entry
The selection of the devices is normally an interactive procedure whereby the program
prompts the user for the designation and description of the devices to be used. It is common
for the program to display information on all the devices in the library when requested by the
user.
Regardless of whether the program is the type that performs the coordination, the type that
gives the engineer the choice of entering his or her own settings, or the type that does both,
rapid feedback of the information to the engineer via the computer monitor is important.
Viewing the screen should provide enough accurate information so that a paper copy is not
necessary until the coordination process is complete.
The program should provide the engineer the capability of modifying data, by changing
plotting voltage, current setting, or any device parameter, and immediately viewing results.
Features to zoom into areas of the display and directly measure time intervals between device
operating characteristics should be available.
Figure 15-5 shows an engineer zooming a plot on the screen to determine which area needs to
be examined in more detail.
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Figure 15-5—Engineer shown viewing a CRT monitor
while using the computer method
15.4.3 User-defined libraries
The program should provide for a user-defined device library. Protective devices, when not
available in the program library, are modeled by the engineer using some type of digitizing
procedure, mathematical modeling, or a combination of the two. Since many output devices
cannot draw curves, curves are simulated by the software as very short straight lines.
For some types of protective devices, mathematical modeling is preferred. For example, the
solid-state trip device for a low-voltage circuit breaker may have literally thousands of possible settings. It is not practical to store each of these as individual digitized curves in the data
base. Mathematical modeling can accomplish this task with a minimum of data.
15.4.4 Single-line diagram generator
To assist in documentation for the study, the program should include some means of
generating a single-line diagram. One method of accomplishing this is to have the software
automatically generate the single-line diagram as the devices are entered. With other
programs, the single-line diagram may be developed using computer-aided design and
drafting (CADD) software. The CADD software may be included with the coordination
software. Some programs produce graphic commands that may be used with popular CADD
systems, so that the single-line diagram may be added or enhanced with the software residing
on the computer.
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15.4.5 Graphics monitor
Most software has provisions to view the coordination study on a graphics monitor before
requesting a plot on the printer or plotter. Some method should be provided with the screen
plot so that the engineer can determine if the results of the coordination are satisfactory and
can accurately establish the clearance between devices. An opportunity to modify, add, or
delete devices should be available. Figures 15-6, 15-7 and 15-8 show the input screen for
three programs.
Figure 15-6—Example of screen plot
15.4.6 Plotter/printer graphical interface
In today’s computer environment, the engineer is able to select from a wide range of graphic
display plotters and printers. Programs are normally written to use plot drivers so that the
software will be able to support a wide range of hardware. Plot drivers are programs that
translate the data from the internal computer representation to a form usable by the plotting
hardware.
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Figure 15-7—Example of screen plot using the latest PC technology
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Figure 15-8—Example of how the light table of Figure 15-1 has been replaced by the computer screen “light table”
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Software is used to provide the output format of the data that the graphics software sends to
the plot driver. The use of this method permits a single program to support most types of
hardware without modification of the basic software package. Such programs permit the
engineer to upgrade hardware without fear that the software will become obsolete.
15.4.7 Graphical output reports
The computer program should be capable of producing the TCC drawings in a wide range of
formats on a wide range of graphical output devices. As a minimum, the software should be
able to produce the drawings on a 4 1/2 by 5 cycle K&E form 48-5258 pre-printed log-log
paper. Figure 15-9 shows such output.
Most computer programs support graphical output to both printers and plotters. Features are
provided for the engineer to specify if the software is required to generate a background grid.
Pen colors and fill patterns may also be an option available to the user. Figure 15-4 is a plot
produced by a printer.
15.4.8 Device setting report generator
The software should contain provisions for reporting the device settings and ratings of all
devices. These reports may be sent to a printer when desired.
15.5 How to make use of coordination software
There are various ways that an engineer can make use of existing coordination software,
depending on the type of equipment that is used for computing and plotting, the frequency of
program use, and the money invested in the program. With the increase in the number of
personal computers (PCs; see 15.5.1), use of the programs on a main frame computer has
become obsolete. Coordination software is typically used with a PC, either in-house or
through a consultant or manufacturer. A description of equipment needs, cost, and advantages
or disadvantages follows in 15.5.1 and 15.5.2.
15.5.1 Personal computers (PCs)
Engineering departments have purchased PCs, which are single-user, stand-alone computers
that sit on (or under) the desk of the engineer. This type of computer provides engineering
departments better control over and flexibility in their computing chores. Time-current
coordination programs are available for these computers. Display graphics is an important
feature common to this type of computer and, with low-cost graphic printers and desktop
plotters, results of the studies may be quickly generated.
Early PCs suffered from a lack of random access memory (RAM) and a limited amount of
permanent data storage capacity. There have been very rapid technical enhancements in
hardware and computer operating systems so that engineers have very powerful PCs with
large memory and high speed readily available.
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Figure 15-9—Example of plot on K&E 48-5258 form
PCs offer all firms, large and small, the opportunity to use advanced software. PCs offer
engineers the opportunity to examine many design alternatives in no more time than a manual
analysis would have taken.
An additional use of the PC is as a storage device for coordination curves. Storing the 11 in
by 17 in forms is often done haphazardly and, when one is needed, can be hard to find. When
the curves are created on the computer, the data can be copied to a floppy, if hard disk space is
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short. This disk can then be stored and, when a copy of the curve is needed, it can be printed
again. Some companies are using the computer to “replot” old curves and store the
information to be available when needed
15.5.2 Consulting service
Some manufacturers will provide coordination studies (usually for a fee) for customers using
their equipment. Often, their device libraries are limited, particularly on existing devices of
another manufacturer.
Consulting firms who have purchased or written a coordination program are another source
for having the studies done. Consulting engineers develop experience with these programs.
The time-current curves can often be produced more cost-effectively by a consultant familiar
with the program in cases in which the company would need to add and/or train manpower.
There is a learning curve with any new program and, if the knowledge will not be used on a
continuing basis, it is often better to let someone else do it. Also, consultant’s experience in
performing coordination studies can provide important “know-how” and expertise that may
be required, especially when complex power systems and circuit arrangements are involved.
15.6 Verifying the results
Whether the coordination is performed by computer or manually, the engineer needs to have
confidence in the results. Checking the computer plot may be difficult since often the
engineer does not have a copy of the manufacturer’s curve. When it is available, it is often
plotted to a different scale, making it difficult to compare. Ideally, the program should have a
method for plotting out the entire family of fuse or relay curves at a scale and in a manner that
permits comparison to the manufacturer’s curves. For example, for any fuse, there would be a
plot of all the minimum melt curves and a plot of all the total clearing curves. Software
vendors may be willing to supply these plots along with copies of the manufacturer’s curves
to the engineer for verification. Plots could also be made for low-voltage circuit breakers for
specific settings for direct comparison with the manufacturer’s curves.
15.7 Equipment needs
A computer with hard drive and monitor, a plotter and/or a printer are required to run most
programs.
Before purchasing equipment, the supplier of the software should be contacted for
information. The required hardware configuration may vary with the software being
purchased. Particular attention should be given to the minimum conventional memory
requirements of these programs and compatibility to specific LAN environments.
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15.8 Conclusion
The increased popularity of the computer and its availability in most engineering facilities
has resulted in liberating the design engineer from the tedious task of manually drawing
coordination curves, thereby allowing him or her to be free to design. With the use of
simplified input of interactive software, even those engineers who have an inherent fear of
computers can become confident. The engineer is still needed to make those critical judgment
decisions and to establish the criteria that should not be left to a computer. With state-of-theart coordination software, the engineer is no longer required to struggle with the mundane
tasks of manually drawing curves and tabulating results.
15.9 Bibliography
Additional information may be found in the following sources:
[B1] IEEE Std 141-1993, IEEE Recommended Practice for Electric Power Distribution for
Industrial Plants (IEEE Red Book).
[B2] IEEE Std 242-1986 (Reaff 1991) IEEE Recommended Practice for Protection and Coordination of Industrial and Commercial Power Systems (IEEE Buff Book).
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Chapter 16
DC auxiliary power system analysis
16.1 Introduction
Attention concerning the reliable operation of dc electrical power systems (e.g., emergency
standby power supplies used for generating stations, data processing facilities, long-distance
telephone companies, etc.) has steadily increased in recent years. This has increased the role
of the dc systems engineer by not only requiring in-depth dc power systems analysis, but
keeping that analysis up to date. The recent introduction of computer techniques to dc power
systems analysis allows more rapid and rigorous analysis of complex problems in
comparison with earlier manual techniques.
This chapter discusses two primary aspects of performing dc power systems analysis (load
flow and short-circuit) including system modeling, developing the appropriate one-line
diagram for the power system of interest; including source bus, branches and loads. A one
line diagram from a representative system is used for illustrating the examples.
16.2 Purpose of the recommended practice
Several standards, guidelines, and technical papers exist that provide guidance to dc power
systems analysis. These documents are available through various standards organizations
(i.e., IEEE, UL, NEMA, etc.) and manufacturers. Many of these sources treat the subject with
the assumption that hand calculations are being performed. It is the intent of this chapter to
consolidate this information and provide guidance for performing dc power systems analyses.
16.3 Application of dc power system analysis
As with ac industrial and commercial power systems, the planning, design, installation, and
operation of dc electrical power systems require engineering studies to evaluate existing and
proposed system performance, reliability, safety, and economics. For new systems, studies
can ensure proper sizing of equipment such as batteries, chargers, distribution equipment, etc.
For existing systems, studies can aid in improving system reliability by identifying
weaknesses within the system and providing the recommended design for system
improvement.
DC systems are becoming increasingly complex, making studies more difficult and timeconsuming. The recent introduction of software specifically designed for analysis of dc
power systems greatly simplifies computational tasks that have traditionally been done by
hand.
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The elements of dc power systems analysis that are essential to ensuring an adequate dc
system include the following:
a)
System modeling. Sources, branches, and loads
b)
Load flow/voltage drop. During battery discharge and motor starting
c)
Short-circuit. Peak, time constant, rate of rise, and steady-state
d)
Battery sizing. See IEEE Std 485-1997 [B8]1
e)
Charger sizing. See IEEE Std 946-1992 [B9]
16.4 Analytical procedures
With the development of the digital computer and advanced programming techniques, dc
power system problems of the most complex types can be rigorously analyzed. An engineer
involved in the creation or modification of dc power systems analysis programs must
thoroughly understand the application of the basic analytical solution methods used. It is
equally important for those who assemble and prepare data for input to a dc power systems
analysis program, as well as those who interpret and apply the results generated by such a
program to understand the basic analytical solution methods.
Some of the analytical procedures utilized on dc systems are very similar to the procedures
presented in Chapters 3 and 4 for use with ac systems. In fact, for steady-state conditions on
dc systems it is possible to obtain reasonable approximations by carefully manipulating
resistance or reactance values in load flow or short-circuit software designed for analysis of
ac power systems.
The remainder of this chapter will cover the analytical procedures specific to dc systems
which are not covered in Chapters 3 and 4.
16.5 System modeling
The normal dc system analysis process begins with preparation of a system one-line diagram
similar to Figure 16-1. The actual values for the load and branch information to be modeled
are added as the diagram is developed. The completed one-line diagram with all of the cable
resistances and source/load models included is often referred to as an impedance diagram. A
separate impedance diagram showing the connected impedances, sources, and loads is
sometimes prepared for each system being analyzed.
For steady-state load flow/voltage drop studies the inductances are ignored and a resistance
diagram results. For transient load flow/voltage drop studies the inductance values cannot be
ignored.
1The
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numbers in brackets correspond to those of the bibliography in 16.9.
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Figure 16-1—Sample dc one-line diagram
Load flow/voltage drop and short-circuit studies are usually performed with different
conductor temperatures assumed. Therefore, the resistance portion of the one-line diagrams
for these two types of studies are typically not identical.
When developing the one-line diagram, each component should be represented as the
equivalent model necessary for the study. Components commonly modeled are sources,
branches, and loads. Sources include batteries, chargers, power converters (rectifiers),
generators, and motors (for short-circuit studies). Branches include wire, cables, bus duct,
protective devices, and combination starter units (contactors and thermal overload devices).
Loads include motors, relay coils, solenoid valves, lamps, inverters, and dc/dc power
supplies. The basic considerations for modeling the three general component classes are
presented herein.
Most loads can be adequately modeled as having a constant resistance, constant power, or
constant current characteristic. Correctly categorizing the loads is absolutely essential if load
flow/voltage drop studies are to provide useful results.
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Constant resistance loads draw a current that is directly proportional to the load terminal
voltage (load current falls as terminal voltage falls). Heaters, relays, solenoid valves, some
motors, and most lamps fall into the category of constant resistance loads.
Constant power loads draw a current that is inversely proportional to the load terminal
voltage (load current rises as terminal voltage falls). Inverters, dc/dc power supplies, and
many motors fall into this category.
Constant current loads draw essentially the same current for a wide range of input voltages.
Some dc power supplies (i.e., shunt regulated supplies) draw a constant current. Some motors
(most notably, valve actuator motors) can be most accurately modeled as constant current
loads rather than constant power.
If a load’s characteristic is unknown, it is usually conservative to model it as constant power
for load flow/voltage drop studies. This is because the current rises as load terminal voltage
falls, which tends to amplify the effects of voltage drop, producing a lower (usually worse)
terminal voltage at the load.
As stated above, steady-state load flow calculations do not require inductance values. Shortcircuit calculations require inductance values for all elements only if the short-circuit time
constant and rate of rise are to be calculated in addition to the peak and steady-state shortcircuit current values. The various circuit elements will be discussed in the following order:
sources, loads, and branches. Load flow system modeling will be discussed first, followed by
short-circuit system modeling in each clause.
After the sources, loads, and branches have been modeled, the data from the models and the
one line are typically entered into a computer program to perform the required analyses.
16.5.1 Battery characteristics
16.5.1.1 Voltage during discharge
Battery voltage begins to decline as soon as discharge begins. There is an initial drop in voltage due to the ohmic resistance of the battery and chemical action within the battery. This
abrupt drop in voltage is usually followed by a slight rise in voltage (the coup de fouet effect)
as the diffusion process allows fresh electrolyte to come in contact with the plates. Thereafter,
the voltage resumes its decline. The voltage drop and the rate of decline are proportional to
the current density. The higher the current, the greater the initial voltage drop and rate of
decline.
After the initial effects have disappeared, increasing or decreasing the discharge current will
cause the battery voltage to react accordingly, based on the ohmic resistance of the battery at
the time of the load change. The voltage will then continue to decline at a rate determined by
the new discharge current, state of charge (the amount of energy previously removed), cell
temperature, and age. For load reductions, the battery voltage will rise immediately due to
ohmic effects, then more slowly for up to several minutes due to chemical action in the
battery.
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Battery manufacturers typically provide either tabular or graphical data to describe cell
discharge characteristics for a given discharge rate. This data is based on new cells having
nominal specific gravity electrolyte at 77 °F (25 °C).
16.5.1.2 Short-circuit characteristics
The peak short-circuit current that a battery can deliver is limited by its internal resistance
(including intercell connectors). The greater the battery’s resistance, the lower the peak shortcircuit current is. In addition, it takes a finite time for the battery to reach its peak short-circuit
current value. The rise time is dependent on the ratio of the battery’s inductance to its
reactance (i.e., time constant). The battery’s time constant increases in direct proportion to its
inductance (with resistance held constant).
Figure 16-2 shows the equivalent circuit used for calculating battery short-circuit current.
where
RB is the battery internal resistance,
LC is the battery circuit inductance,
RC is the battery connector resistance (sum of all internal cable and connector resistances).
Source: [B6]. Reprinted with the permission of General Electric Company.
Figure 16-2—Battery equivalent circuit
The internal resistance of a cell is calculated from the slope of the initial volts line on the
characteristic curve for the cell. Figure 16-3 shows a discharge characteristic curve for a
typical 2500 Ah, 35 plate (total) cell.
In accordance with IEEE Std 946-1992 [B9]:
R
R cell = ------p
Np
(16-1)
where
Rcell
Rp
Np
is the total internal cell resistance, in ohms (Ω),
is the resistance per positive plate, in ohms (Ω),
is the number of positive plates.
Copyright © 1998 IEEE. All rights reserved.
449
CHAPTER 16
Figure 16-3—Discharge curve for a typical lead acid cell
IEEE
Std 399-1997
450
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DC AUXILIARY POWER SYSTEM ANALYSIS
IEEE
Std 399-1997
and
V1 – V2
R p = ------------------ Ω /positive plate
I2 – I1
(16-2)
where
I1 and I2 are the amperes (A)/positive plate
for any two voltage and current points along the line.
Example 1:
From Figure 16-3 (initial volts curve):
V1 = 1.95 V
I1 = 20 A/positive plate
V2 = 1.50 V
I2 = 300 A/positive plate
Per equation (16-2):
1.95 – 1.50
R p = --------------------------300 – 20
Rp = 0.00161 Ω/positive plate
Given:
35 total plates (17 positive plates/cell)
60 cells @ 2.00 Vpc
Then
–3
R
161 × 10
–6
R cell = ------p = ------------------------- = 94.7 × 10 Ω
17
Np
RB
= Rcell × number of cells
(16-3)
= 94.7 × 10–6 × 60 = 5.68 × 10–3 Ω
The next step is to calculate the battery cable resistance (Rc). Intercell connector resistance is
often included in the manufacturer’s discharge curves and initial cell voltage lines. If it is not
included, the connector resistance must be added to the cable resistance. Intercell connector
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 16
resistance should be available from the manufacturer. Battery cable resistance can be
calculated as follows:
Rcable = battery cable resistance per unit length (e.g., Ω/1000 ft)
total length of battery connector (in ft) × R cable
R c = ---------------------------------------------------------------------------------------------------------------1000 ft
(16-4)
Example 2:
As shown in Figure 16-4, there are four segments of battery cable to be included in this
portion of the calculation. The 100 ft (30.5 m) cable to the 125 Vdc bus (as shown in
Figure 16-1) is addressed separately. The battery cable length is
1)
Total length of battery connector (lengthcc):
lengthcc = 15 ft + 6 ft + 3.25 ft + 1 ft = 25.25 ft (7.7 m)
2)
(16-5)
Connector cable is 3-350 kcmil copper, which has a resistance of 10.2 × 10–3 Ω/
1000 ft (33.65 × 10–3 Ω/km) at 25 °C (see Example 4 in 16.5.4.1).
Source: [B6]. Reprinted with the permission of General Electric Company.
Figure 16-4—Typical 60-cell battery installation—
arranged in three rows
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IEEE
Std 399-1997
DC AUXILIARY POWER SYSTEM ANALYSIS
Therefore,
25.25 ft
–3
–6
R c = ------------------- × 10.2 × 10 Ω /1000 ft = 257.6 × 10 Ω
1000
The inductance of the battery circuit (Lc) is equal to the inductance of the battery connector
(Lbc) plus the inductance of the battery string (Lbs).
Lc = Lbc + Lbs
(16-6)
The inductance of the cell connectors (Lbc) can also be determined by analysis from data
similar to that shown in Figure 16-4 using the following formula (General Electric [B6]):
d
–9
L bc = 30.5 × 10 2 ln --- + 0.5 H/ft
r
(16-7)
where
d
r
is the distance between conductor centers = 25 in and 12 in (63.5 cm and 30.5 cm),
is the radius of the conductor = 0.590 in (15.0 mm).
NOTE—effective radius =
number of conductors × area of one conductor
----------------------------------------------------------------------------------------------------------------
π
2
=
3 × 0.364 in
-----------------------------
π
= 0.590 in (15.0 mm)
The 3-350 kcmil battery connector cable is in parallel with a 15 ft (4.57 m) section of the
battery string. To simplify the calculation, the 15 ft (4.57 m) section of the battery string is
considered to be equivalent to the 3-350 kcmil connector cable.
Then
25
–9
L bc = 30.5 × 10 2 ln ------------- + 0.5 H/ft
0.590
= 0.244 µ H/ft
12
–9
L bc = 30.5 × 10 2 ln ------------- + 0.5 H/ft
0.590
= 0.199 µ H/ft
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CHAPTER 16
and
180
–9
L bc = 30.5 × 10 2 ln ------------- + 0.5 H/ft
0.590
= 0.364 µ H/ft
Therefore,
Lbc = (0.244 µH/ft × 15 ft) + (0.199 µH/ft × 3.25 ft) + (0.364 µH/ft × 1 ft) = 4.67 µH
Battery cell inductance values obtained from testing on lead acid cells ranging from 10–
1600 Ah are presented in Willihnganz and Rohner [B11]. These results indicate that battery
cell inductances are on the order of 0.1 µH/cell. Specifically, the inductance ranges are shown
in Figure 16-5 Willihnganz and Rohner [B11]. Note that the battery inductance levels off
at ≈ 0.145 µH/cell for batteries larger than 1300 Ah; therefore, 0.145 µH/cell can be used for
large installations (i.e., above 1300 Ah).
Figure 16-5—Inductance vs. battery size for lead-acid cells [B11]
If 0.145 µH/cell is used for a 2500 Ah, 60 cell battery, then
Lbs = 0.145 µH/cell × 60 cells = 8.7 µH,
454
(16-8)
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Std 399-1997
Then, using Equation (16-6), the inductance of the battery circuit (Lc) is as follows:
Lc = Lbc + Lbs
= 4.67 µH + 8.7 µH
= 13.4 µH
which gives a battery equivalent circuit as shown in Figure 16-6.
Source: [B6]. Reprinted with the permission of General Electric Company.
Figure 16-6—Battery equivalent circuit with component values
As an alternative, the inductance of the battery string can be calculated if the time constant
and resistance are known (e.g., from actual field test data). For example, if short-circuit tests
show a time constant of 1.07 ms, and the battery resistance (including intercell connectors) is
0.0127 Ω , the inductance can be calculated as follows:
Lbs = tRb
(16-9)
Therefore,
Lbs = 1.07 × 10–3 s × 12.7 × 10–3 Ω = 13.6 µH
16.5.2 Battery charger
16.5.2.1 Voltage regulation
One of the main goals of load flow/voltage drop studies is to determine whether sources are
applied above their output current ratings. Battery chargers are generally considered to be
“stiff” sources in that their output voltage is well-regulated from no load to full load. Battery
chargers are normally modeled as a constant voltage source in series with a small resistance
for load flow/voltage drop studies, assuming that they are applied within their ratings.
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Std 399-1997
CHAPTER 16
Chargers typically have an integral current-limiting circuit designed to limit steady-state
output current. This circuit reduces their output voltage to limit output current to a safe value
for their internal components. The current limiting circuit normally has an adjustable setpoint
that will not exceed ≈ 150% of the nominal output current rating. When the set point is
reached, the charger begins to behave more like a constant current source than a constant
voltage source.
16.5.2.2 Short-circuit characteristics
Most standard battery chargers are supplied with a current-limiting feature that limits the
steady-state fault current the charger can deliver. Generally, it is conservatively assumed that
the charger will deliver no more than 150% of its rated output for an extended period of time.
IEEE Std 946-1992 [B9] states that for a typical dc system, the short-circuit current from the
charger has already peaked and decayed before the short-circuit from the battery reaches its
peak.
The magnitude and duration of the transient short-circuit current are dependent on the
charger design (including rectifier type and control circuit response time) and the X/R ratio of
the ac supply, as well as the inductance and resistance of the fault circuit. The transient
typically peaks very quickly, at 5–20 times the charger current rating, and then decays. In
some charger designs the transient is as short as 1/2 cycle (8 ms), and in others it has been
shown to last as long as 100 ms. Therefore, in conjunction with the battery time constant, the
maximum coincident short-circuit current is conservatively calculated as the sum of the peak
short-circuit from the battery and the peak value from the charger. Steady-state short-circuit
current is conservatively calculated as the sum of the peak short-circuit current from the
battery and the current limit value from the charger.
16.5.3 Motors
16.5.3.1 Voltage/power characteristics
DC motors are available in a wide variety of designs for many different applications. All of
the various designs behave somewhat differently as terminal voltage changes. This behavior
is dependent upon both the motor design and the load characteristic as speed varies. Some
motors exhibit a constant current characteristic for a fairly wide range of terminal voltage
conditions. Others exhibit a constant power characteristic. If the behavior of a motor for
varying terminal voltage is unknown, it is usually conservative to assume that it has a
constant power characteristic when performing load flow/voltage drop studies.
When dc motors are started, there can be a significant inrush current depending upon motor
design and starting technique. As with ac motors, the duration of this inrush current is
dependent on both motor design and load speed/torque characteristics. Motor starting can be
evaluated by using static techniques, such as the voltage drop snapshot described in 9.4.1 of
Chapter 9, or by using dynamic techniques. Dynamic studies can be performed to evaluate
system conditions on motor starting when more rigorous analysis is required. Dynamic
analyses are not normally required for dc auxiliary power systems. They are more often
456
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DC AUXILIARY POWER SYSTEM ANALYSIS
IEEE
Std 399-1997
performed for larger dc systems such as those found in mining operations, where large
motors constitute a higher proportion of the dc system load.
16.5.3.2 Short-circuit characteristics
The short-circuit model and approximate curve of short-circuit vs. time of a dc motor can be
represented as shown in Figure 16-7.
Source: [B5]. Reprinted with the permission of General Electric Company.
Figure 16-7—Typical short-circuit characteristic of dc motor
The approximate short-circuit characteristic of a dc motor, Figure 16-7, can be drawn by
determining two factors:
a)
b)
Initial rate of rise of current
Peak short-circuit current
The section of the curve between points a and b in Figure 16-7 (initial rate of rise of the shortcircuit current) is determined by the motor circuit time constant. Point b is the point at which
the current magnitude is equal to two thirds of its maximum value. Point c is the peak shortcircuit current. Section b-c-d of the curve is drawn to represent an exponential type of curve
with section b-c being an exponential approaching I a ' and section c-d being an exponential
approaching the reduced steady-state fault current level (General Electric [B5]).
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 16
The initial rate of rise of the fault current is as follows [B5]:
V
di a /dt = ------1
La '
(16-10)
where
V1
is the rated machine voltage, in V,
La '
is the machine armature circuit unsaturated inductance, in H.
19.1 C x V 1
L a ' = ------------------------- H
PN 1 I a
(16-11)
where
P
N1
Ia
Cx
is the number of poles,
is the base speed, in r/min,
is the rated machine current, in A,
is 0.4 for dc motors without pole face windings,
is 0.1 for dc motors with pole face windings.
The peak short-circuit current is as follows:
I
I a ' = ------aRd '
(16-12)
where
Rd '
is the transient effective armature resistance of machine (per unit on the machine
base)
Example 3:
Given the following motor parameters:
4-pole motor
Constant speed (1900 r/min) (N1)
Shunt
3 hp
115 V (V1)
Noncompensated (no pole face winding)
22 A rated current (Ia)
R d ' = 0.175 p.u.
458
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DC AUXILIARY POWER SYSTEM ANALYSIS
IEEE
Std 399-1997
The peak short-circuit current is as follows:
I
22
I a ' = ------a- = ------------- = 126 A
0.175
Rd '
The initial rate of rise is as follows:
V
di
-------a = -----1La
dt
(16-13)
where
19.1 C x V 1
L a ' = ------------------------PN 1 I a
19.1 × 0.4 × 115
= --------------------------------------- = 5.3 mH
4 × 1900 × 22
Therefore,
di
115
-------a = ---------------- = 21 700 A/s
0.0053
dt
The time constant of the motor is as follows:
La '
T m = -----Rd '
Converting to ohms,
115 V
R d ' = 0.175 p.u. ---------------
22 A
= 914.8 × 10
–3
Ω
Therefore,
La '
T m = -----Rd '
–3
5.3 × 10 H
= -----------------------------------–3
914.8 × 10 Ω
= 5.8 ms
Copyright © 1998 IEEE. All rights reserved.
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CHAPTER 16
16.5.4 Branches
Load flow/voltage drop studies are normally done for steady-state conditions; therefore,
branches are modeled as purely resistive elements. DC resistance values for most common
bus and cable sizes can be found easily in many codes and handbooks. Care must be taken to
ensure that resistance values used are dc resistance values. Many standard sources and
manufacturers’ data give ac resistance, which includes skin and proximity effects. The
resistance tables in most codes and handbooks also include formulas or multipliers for
correcting the table values to match the temperatures to be used for the study in question. The
resistivity of metal conductors rises with temperature. The effect of this additional resistance
can be quite significant for circuits with long runs and/or small conductor sizes. Therefore, it
is common practice to assume that a cable is at its rated conductor temperature (typically,
75 °C or 90 °C) when performing load flow/voltage drop studies. An example of a widely
accepted method of correcting cable resistances for temperature is included in the following
discussion of branch models for short-circuit studies. Cable resistances must include the total
resistance of the circuit. If two conductor cables are used for a circuit, the total circuit length
is twice the cable length. Circuit length can be difficult to determine for control circuits in
which one leg is longer than the other.
For load flow/voltage drop studies, the resistance for each branch must include the resistance
of overload heaters in motor starters where applicable.
Reversing motors are a special case. Usually four wires are required between the motor and
the starter. This effectively quadruples the conductor length if the starter is in a motor control
center (MCC) rather than local to the motor. Some compound wound motors used in reversing applications utilize a fifth wire for the shunt field. In most studies, this wire is conservatively ignored. The current in the fifth wire is added to the armature current. This results in a
larger calculated voltage drop and consequently a lower calculated motor terminal voltage
than would actually be found in the field.
16.5.4.1 Cable/bus
Typical branches modeled in dc power systems analysis connecting equipment are cables and
buses. These resistances are usually taken at 25 °C for conservatism in short-circuit studies,
since cable resistance decreases as cable temperature decreases.
To correct cable resistance from any initial temperature to any final temperature, the
following formula is used (the National Electrical Code® [ΝΕC®] [NFPA 70-1996] [B10]):
R2 = R1 [ 1 + α ( T 2 – T 1 ) ]
(16-14)
where
R2
R1
α
T1
T2
460
is the resistance at desired temperature (T2), Ω,
is the resistance at initial temperature (T1), Ω,
is 3.23 × 10–3 for copper conductors, 3.30 × 10–3 for aluminum conductors, Ω/°C,
is the initial temperature, °C,
is the desired temperature, °C.
Copyright © 1998 IEEE. All rights reserved.
DC AUXILIARY POWER SYSTEM ANALYSIS
IEEE
Std 399-1997
Example 4:
Given:
3-350 kcmil copper conductors in parallel at 75 °C, R = 0.0367 Ω/1000 ft (120.4 × 10–3 Ω/
km) per conductor (NFPA 70-1996 [B10]).
–3
36.7 × 10 Ω/1000 ft
–3
–3
R = ----------------------------------------------------- = 12.2 × 10 Ω /1000 ft ( 40.14 × 10 Ω /km )
3
Correcting to 25 °C:
R2 = 12.2 × 10–3(1 + 3.23 × 10–3(25 – 75))
= 10.2 × 10–3 Ω/1000 ft (33.65 × 10–3 Ω/km)
16.5.4.2 Motor control circuits
The resistance of thermal overload heaters in motor-starters is often a significant portion of a
branch circuit’s resistance. For small motors it is often greater than the resistance of the
circuit conductors. The resistance of fuses, circuit breakers, starter contacts, and switches is
usually negligible and is not included in most branch models.
For conservatism, the effects of combination motor-starter unit and thermal overload heater
resistance on short-circuit current are normally ignored; but can be accounted for if removing
conservatism is desirable.
16.6 Load flow/voltage drop studies
When the resistance diagram has been completed, a load flow/voltage drop analysis can be
performed. As with ac systems, an iterative approach is required unless the system loads are
all constant resistance (a substation with only switchgear and lamps for example) or all
constant current. If all loads on the system are of one of these two types, a direct solution can
be obtained using voltage divider circuits. The iterative approach gives the same results as the
direct approach, and software that will quickly and accurately perform the necessary
iterations is available. Since the iterative approach usually converges regardless of the load
mix and system configuration, it is normally used for all but single branch circuit
calculations.
Load flow calculations on a dc system utilize the same basic network solution techniques as
an ac system. The base equation is essentially the same as Equation (6-2):
[I] = [Y] [V]
For steady-state dc systems inductance is ignored and the equation becomes the following:
[I] = [G] [V]
Copyright © 1998 IEEE. All rights reserved.
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Std 399-1997
CHAPTER 16
where
[I]
is the vector of total currents flowing into the network nodes,
[G]
is the network conductance matrix,
[V]
is the vector of voltage at the network nodes.
The conductance [G] matrix is built in the same manner as the [Y] matrix for ac systems. A
number of solution algorithms (Gauss-Seidel, Newton-Raphson, and others) are available to
solve the resulting networks. These techniques are described in Chapters 5 and 6, as well as
in many other texts and papers. Therefore, they will not be discussed here.
If the system is being supplied by a charger or other stiff source, the load flow needs to be run
only once to determine bus voltages and currents.
If the system is supplied by a battery, neither the power nor the voltage at the swing bus can
be held constant as it can with other sources. It is prudent to calculate load flow and voltage
drop for a minimum of three conditions on each battery system. These conditions are the
battery equalizing voltage (if the battery is equalized on line), the battery float voltage, and
the battery end of discharge voltage. These three cases usually cover the highest, normal, and
lowest voltages seen on the dc system. If the battery experiences large load additions during
its discharge cycle, the load flow/voltage drop and battery terminal voltage should also be
determined for these points in the profile to verify that the voltage available at the loads does
not drop below their minimum requirement. While most relays can withstand a considerable
voltage dip before they drop out, electronic equipment is often more sensitive to voltage dips.
It should also be noted that, depending on system configuration, the lowest voltage seen at a
panel or load may not occur when the battery voltage is lowest.
Table 16-1 shows the program output for several iterations of a load flow/voltage drop
calculation, which illustrates the iterative technique previously described for dc systems
being supplied by a battery (refer to Figure 16-1 for the corresponding system). Terminal
voltages from a load flow/voltage drop analysis of all nodes in Figure 16-1 are shown with
the short-circuit results in Table 16-2.
Table 16-1
Load flow/voltage drop iteration results at the battery
462
Vbatt
Ibatt
I/positive plate
Vpc
125.0
885.7
52.1
2.08
115.3
955.4
56.2
1.92
114.2
963.9
56.7
1.90
113.4
969.0
57.0
1.89
Copyright © 1998 IEEE. All rights reserved.
DC AUXILIARY POWER SYSTEM ANALYSIS
IEEE
Std 399-1997
16.6.1 Load profile
As noted above, a load flow/voltage drop analysis is essentially an analysis of a single
“snapshot” in time. To accurately determine system voltages during a discharge event, it
is essential to know how much energy has been removed from the battery (the initial state of
charge) as well as the current being drawn from it for the instant in time being considered. This
is typically done by developing a load profile showing time vs. current on an x - y plot for the
duration of the anticipated discharge event. Developing this profile can be fairly simple for a
battery feeding a relatively constant load or extremely complex for a system involving a large
number of relays, controls, and instruments. Additional discussion regarding the development
of battery duty cycles can be found in IEEE Std 485-1997 [B8] and IEEE Std 946-1992 [B9].
16.6.2 Battery terminal voltage
Batteries are not well-regulated sources because their output voltage at any given time is
dependent upon both the load current being supplied and the total energy removed from the
battery prior to the time in question. Since load flow/voltage drop studies are steady-state
problems, battery voltage is generally assumed or calculated for the specific time in question.
Battery manufacturers typically provide discharge characteristic curves (Figure 16-3) or
tables for use in determining battery capacity and terminal voltage.
The energy removed from a battery is measured in ampere-hours (Ah) removed, and can be
calculated fairly easily for any load profile. Ampere-hours removed is calculated by adding
the product of current drawn and the duration for each period in the load profile.
Given the ampere-hours removed, the number of positive plates in a cell, and the current
flowing through the cell, the cell voltage can be determined from the manufacturer’s data
(usually graphs or tables) by interpolating as necessary. Multiplying the cell voltage by the
number of cells in the string gives battery terminal voltage.
Battery terminal voltage for a given load current and energy removed can be determined from
the battery curves. Energy removed is calculated by determining the ampere-hours removed
from the battery and dividing it by the number of positive plates per cell. The amperes per
positive plate is simply the current flowing in the battery string divided by the number of
positive plates in each cell. These numbers are plotted on the battery curves and the cell
voltage is read by interpolating as necessary between cell voltage curves. The cell voltage is
multiplied by the number of cells in the string to obtain battery voltage. For example, using
the battery curve for a 35-plate cell in Figure 16-3, if the battery has had 1480 Ah removed,
and the battery load is 1020 A, the cell voltage would be found by plotting the following:
Ah
1480 Ah
------------------------------------------ = 87 -------------------------------positive plate
17 positive plates
Copyright © 1998 IEEE. All rights reserved.
(16-15)
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IEEE
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CHAPTER 16
vs.
A
1020 A
------------------------------------------ = 60 -------------------------------positive plate
17 positive plates
This yields a cell voltage of 1.75 Vpc, or 105 Vdc for the terminal voltage on a 60-cell battery.
Some cell manufacturer’s curves have derating factors for discharge rate or time,
temperature, and different cells on each curve. These must be carefully accounted for in
determining cell voltages. A great deal of care must be taken in reading the cell voltages. Any
error in reading the cell voltage from the graphs or tables will be multiplied by the number of
cells and significantly affect the calculated battery terminal voltage.
16.7 Short-circuit studies
Resistance and inductance diagrams are generally created to aid in calculation of short-circuit
values. The resistance diagram is used to calculate the maximum current for a short-circuit at
any point in the system. The inductance diagram is used to calculate the initial rate of rise and
time constant of the total short-circuit current. The resistances and inductances are combined
in parallel or series until one equivalent system resistance and inductance, respectively, is
determined to represent the system from the point of short-circuit back to the voltage source.
The maximum short-circuit current is calculated based on the equivalent resistance (Req)
from the resistance diagram as follows (General Electric [B4]):
E
I T = ------R eq
(16-16)
where
E
Req
IT
is the system voltage, in V,
is the equivalent system resistance, in Ω,
is the maximum short-circuit current, in A.
The initial rate of rise is calculated based on the equivalent inductance (Leq) from the
inductance diagram as follows:
di
E
------t = rate of rise of total current = ------- (A/s)
L eq
dt
(16-17)
The time constant is calculated based on the equivalent inductance (Leq) divided by the
equivalent resistance (Req) from the inductance and resistance diagrams as follows:
L eq
T = ------- (s)
R eq
464
(16-18)
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
DC AUXILIARY POWER SYSTEM ANALYSIS
Example 5:
From Figure 16-6, the time constant of the 2500 Ah battery is as follows:
L eq
13 µ H
T = ------= --------------------------------------------------------------------------------- = 2.2 ms
–3
–6
R eq
( 5.68 × 10 Ω + 214.2 × 10 Ω )
This is consistent with the 2 ms time constant determined by recent testing on a two cell
1870 Ah battery (ATI Test No. 0792-1 [B1]).
The separate resistance and inductance diagrams for Figure 16-1 are not drawn here. The
cable resistance values were taken from Chapter 9, Table 8 of NFPA 70-1996 [B10] and
adjusted to 25 °C before short-circuit calculations were made. For simplicity in this example,
cable inductances were calculated from ac reactance values in steel conduit taken from
Chapter 9, Table 9 of NFPA 70-1996 [B10], which may not be representative of the dc
inductances for a particular dc system. In addition, the short-circuit calculation was run with
both the battery and battery charger connected to the main bus. Table 16-2 shows the shortcircuit and time-constant results for each node in Figure 16-1.
Table 16-2
Short circuit and load flow/voltage drop results
Node
Fault current (A)
Terminal voltage (V)
Time constant (ms)
Battery
21 843
113.4
1.630
Charger 1
13 158
110.9
1.230
Charger 2
—
110.9
—
16 444
110.9
1.450
499
107.8
0.152
DC Ckts
6 636
103.8
1.870
Emergency lighting
6 115
108.1
1.890
Inverter
5 225
107.1
1.440
MCC-1
4 623
108.2
0.311
MOV-1
1 095
101.9
0.0765
MOV-2
847
108.2
0.0720
MOV-3
378
108.2
0.0534
3 821
107.8
0.0914
Load center
Annunciators
PUMP-1
Copyright © 1998 IEEE. All rights reserved.
465
IEEE
Std 399-1997
CHAPTER 16
16.8 International guidance on dc short-circuit calculations
Papers have been presented (Berizzi et al. 1995 [B2] and Berizzi et al. 1994 [B3]) discussing
the short-circuit calculation methods for dc systems contained in IEC Project #73.6.1 [B7]. It
is not within the scope of this chapter to present the methods contained in the IEC draft standard. However, these papers and IEC Project #73.6.1 are mentioned here to provide additional information sources on the subject of dc short-circuit calculations. It should be noted
that one of the papers [B2] shows substantial agreement between some of the models used in
IEC Project #73.6.1 and detailed simulations performed using a PC-based version of a
well-accepted transient analysis program.
16.9 Bibliography
Additional information may be found in the following sources:
[B1] ATI Test No. 0792-1 “Test Report Stationary Battery Short-Circuit Test,” Albér Technologies, Inc., Boca Raton, FL, July 9 & 10, 1992.
[B2] Alberto Berizzi, et al. “IEC draft standard evaluation and dynamic simulation of shortcircuit currents in dc systems,” Presented at the IEEE Industrial Applications Society Fall
Conference, Oct. 1995.
[B3] Alberto Berizzi, et al. “Short-circuit current calculations for dc systems,” presented at
the IEEE Industrial Applications Society Fall Conference, Oct. 1994.
[B4] General Electric, D-C System Short-Circuit Current Calculations,” GE Industrial Power
Systems Data Book, .178.
[B5] General Electric, “Short-Circuit Characteristics of D-C Motors and Generators,” GE
Industrial Power Systems Data Book, .171.
[B6] General Electric, “Short-Circuit Characteristics of Lead-Acid Storage Batteries,” GE
Industrial Power Systems Data Book, Project .173.
[B7] IEC Project #73.6.1 Draft Standard, “Calculation of short-circuit currents in dc auxiliary
installations in power plants and substations,” Version V4, TC73, May 1994.
[B8] IEEE Std 485-1997, IEEE Recommended Practice for Sizing Lead Acid Batteries for
Stationary Applications.2
2IEEE publications are available from the Institute of Electrical and Electronics Engineers, 445 Hoes Lane, P.O. Box
1331, Piscataway, NJ 08855-1331, USA.
466
Copyright © 1998 IEEE. All rights reserved.
DC AUXILIARY POWER SYSTEM ANALYSIS
IEEE
Std 399-1997
[B9] IEEE Std 946-1992, IEEE Recommended Practice for the Design of DC Auxiliary
Power Systems for Generating Stations.
[B10] NFPA 70-1996, National Electrical Code®(NEC®). Table 430-147, and Chapter 9,
Tables 8 and 9.3
[B11] Willihnganz, E., and Rohner, P., “Battery impedance: farads, milliohms, microhenrys,”
presented at the AIEE Summer and Pacific General Meeting and Air Transportation
Conference, Seattle WA, June, 1959.
3The
NEC is available from Publications Sales, National Fire Protection Association, 1 Batterymarch Park, P.O. Box
9101, Quincy, MA 02269-9101, USA. It is also available from the Institute of Electrical and Electronics Engineers,
445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331, USA.
Copyright © 1998 IEEE. All rights reserved.
467
INDEX
A
Acceleration factors, 116
Acceleration time analysis (motors), 236–
239, 248, 253–262
AC decrement, 168, 174–176, 178–179
Active network elements, 46–49
Active power, 47
Admittance, 45
Air-core reactors, 307
Algebra. See Matrix algebra; Simultaneous
algebraic equations, solution of
Aluminum conductors, 60–61, 391. See also
Conductors
Ambient and conductor temperature (Ft),
391–393
American National Standards Institute
(ANSI) standards (short-circuit analysis), 172–179, 203
American Standard Code for Information
Interchange (ASCII), definition of, 122
Ampacity, cable. See Cable ampacity studies
Analog computers (ac network analyzer), 1,
103, 140
Analytical procedures
errors, common causes of, 13
Fourier representation, 21–22
Laplace expansion, 112
Laplace transform, 22–29, 120, 315–323,
336–337
linearity, 14–15
Norton equivalent circuit, 19
overview, 13–14
per-unit method, 38–40, 71–73
phasor representation, 20–21
single-phase equivalent circuit, 29–31
sinusoidal forcing function, 19–20
superposition, 15–16
symmetrical component analysis, 32–37,
170
Thevenin equivalent circuit, 16–19, 175–
176
Copyright © 1998 IEEE. All rights reserved.
ANSI (American National Standards Institute) standards (short-circuit analysis),
166, 172–179, 203
Arc furnace installations
switching transient analysis, 355
Armature capacitance to ground, 368
Arresters, surge, 339, 341
ASCII (American Standard Code for Information Interchange), definition of, 122
Autotransformers, 247–248, 259
data requirement for stability studies, 221
starters, 235
Availability, definition of, 375
B
Batch computer programs, 127
Battery characteristics, 448–455
battery terminal voltage, 463–464
short-circuit characteristics, 449–455
voltage during discharge, 448–449
Battery charger characteristics, 455–456
short-circuit characteristics, 456
voltage regulation, 455–456
Binary, definition of, 122
Bit, definition of, 122
Branch
data (load flow studies), 139–140
definition of, 54
Branch element modeling, 55–71
cables, 59–60. See also Cable
capacitors, 64. See also Capacitors
constants. See Conductors, constants
lines, 55–59. See also Lines
reactors, 64. See also Reactors
transformers. See Transformers, models
Breaking short-circuit current, 177
Bug, definition of, 122
Bus
dc power systems studies, 460–461
definition of, 54, 107, 122
load flow studies, 138–139, 143
models. See Bus element models
swing, 138, 143
Bus admittance matrix, 182–183
Bus element models, 80–98
469
IEEE
Std 399-1997
electric furnaces, 98
induction motors. See Induction motors,
models
lighting and electric heating, 97–98
loads in general, 80–81
shunt capacitors, 98
shunt reactors, 98
synchronous machines. See Synchronous
machine models
Bushing capacitance to ground, 365
Bus impedance matrix, 183
Byte, definition of, 122
C
Cable
ampacity. See Cable ampacity studies
charging current, 373
dc power systems studies, 460–461
energizing and reclosing transients, 356
grouped, 395–400
harmonic analysis models, 279, 281–282
lightning and switching surge response,
355
models, 55–60, 281–282
overhead lines, compared to, 60
short-circuit studies, 173
Cable ampacity studies, 383–404
ampacity adjustment factors, 388–398
ambient and conductor temperature
(Ft), 391–393
grouping (Fg), 395–398
thermal resistivity (Fth), 393–395
computer program application, 386–388
example, 399–403
heat flow analysis, 383–386
overview, 10–11, 383–384, 403–404
CADD (computer-aided design and drafting) software, 437
Capacitance, 313–314
generator armature, 368
phase bus, 369
transformer winding to ground, 371
Capacitive susceptance, 45
Capacitor banks. See also Capacitors
data requirement for stability studies, 221
470
INDEX
inductance between, 369
switching examples
electromagnetic transients program
(EMTP), 347–361
transient network analyzer (TNA),
341–345
Capacitors
banks. See Capacitor banks
de-energization, restrike phenomena
during, 356
harmonic filters, 306–307
modeling, 64
power factor correction, 270, 275–276,
302
shunt, 98, 279
starting systems, 235–236
surge, 339
switching transient analysis, 325–237,
356
Central processing unit (CPU), definition of,
122
Circuit breakers, 378, 431, 435
rating structure, 176–177
transient network analyzer (TNA), 340
“Close and latch” currents, 172
Co-generation (co-gen) companies, 209
co-gen plant that imports power from
local utility, 226–227
co-gen plant with excess generation, 225–
226
oscillations between industrial plant and
utility, 224, 227–228
Column matrix, 104
Column vector, 104
Commercial power systems. See Power
systems analysis
Complex power, 47
Component, definition of, 375
Computer-aided design and drafting
(CADD) software, 437
Computer solutions and systems
cable ampacity studies, 386–388
computer systems, 122–129
hardware, 124–126
software, 126–129, 162
terminology, 122–124
consulting services, 443
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
INDEX
ground mat studies, 420–428
harmonic analysis, 277, 282, 284–286
history of
digital computers, 1–2, 103, 140
early analog computers, 1, 103, 140
load flow studies, 133, 140–141, 162
motor-starting studies, 250–253, 255–
257, 260–262
numerical solution techniques, 104–121
differential equations, 120–121
matrix algebra fundamentals, 104–107
power system network matrices, 107–
110
simultaneous algebraic equations. See
Simultaneous algebraic equations,
solution of
overview, 7–10, 12–13, 103–104
protective coordination studies, 435–443
common structure, 436–441
coordination programs, 435, 441–443
TCC plotting programs, 435, 441
short-circuit studies, 182–186
software, 184–186
system matrices, 182–183
stability studies, 218–223
data requirements, 221–222
disturbances, simulation of, 220–221
interpreting results, 223–224
program functioning, 219
program output, 222–223
system simulation, 220
time- and frequency-domain analysis,
171, 219, 223–224
Condensers, synchronous
constants, 88
loss-of-field relaying, 216
modeling, 88
short-circuit studies, 173
Conductance, 45
Conductors
aluminum, 60–61, 391
constants, 60–64
inductive reactance, 62–63
resistance, 60–62
shunt capacitive reactance, 63–64
copper, 60–61, 391–393, 461
effective resistance, 60
Copyright © 1998 IEEE. All rights reserved.
galvanized steel, 61
temperature adjustment factor (Ft), 391–
393
Constant admittance model (lighting and
electric heating loads), 97–98
Constant impedance model (induction
motors), 87
Constant load model (induction motors), 83,
85
Coordination studies. See Protective coordination studies
Copper conductors, 60–61, 391–393, 461.
See also Conductors
Copy protection, definition of, 122
CPU (central processing unit), definition of,
122
Cramer’s rule, 112
C-type filter, 282, 284
Current method of bus voltage determination for motor-starting, 243–244
Current transformers (CTs), 362, 430
Cursor, definition of, 123
D
Damping, 318
curves, 324–325, 328
resistors, 334–335, 339
DC decrement, 168–169, 174–176, 178–179
DC motors, 456–459
motor control circuits, 461
short-circuit characteristics, 457–459
voltage and power characteristics, 456–
457
DC power system studies, 445–466
analytical procedures, 446
application of dc power system analysis,
445–446
load flow or voltage drop studies, 461–
464
battery terminal voltage, 463–464
load profile, 463
overview, 12, 445
short-circuit studies, 464–466
international guidance on, 466
system modeling, 446–461
471
IEEE
Std 399-1997
battery characteristics. See Battery
characteristics
battery chargers, 455–456
branches, 460–461
motors. See DC motors
Deep bar motor rotor design, 240, 280
Default, definition of, 123
Determinants, 111
Diagonal elements, 104
Differential equations, 120–121
Digital computers, 1–2, 7, 43, 103, 140. See
also Computer solutions and systems
Dimension, matrix, definition of, 104
Disconnected buses, 138
Disconnect switches, 378
Disk (disc), definition of, 123
Distribution and transmission systems,
transient recovery voltage, 356
Disturbance data requirement for stability
studies, 222
Double-energy transients, 316
Double line-to-ground faults, 167
Double squirrel cage motor rotor design,
240, 280
Duct banks, 396–400
Dynamic stability, 212–214. See also Stability studies
E
Eigenvalue analysis, 219
Electric furnace modeling, 98
Electric heating modeling, 97–98
Electromagnetic transients program (EMTP),
345–361
capacitor bank switching case study, 347–
361
network and device representation, 345–
347
Element, matrix, definition of, 104
Element, network
active, 46–49
passive, 44–46
Emergency power systems. See DC power
system studies
472
INDEX
EMTP. See Electromagnetic transients
program
Engineering workstations, 125
Euler’s method, 121
Even harmonics, 308
Exciter field voltage (EFV), 253
Exciter systems, 234, 247, 257
models, 95–97
stability and, 218
Expected interruption duration, definition
of, 375
Expert systems, 12
definition of, 123
Exposure time, definition of, 375
F
Failure, 379
definition of, 375–376
frequency and expected duration expressions, 381
rate, 378
definition of, 376
Failure modes and effects analysis (FMEA),
379
Faraday’s law, 314
Fault analysis. See also Short-circuit studies
industry standards, 172–179
quasi-steady-state, 171–172
time domain, 171
Fault clearing time, 228
Fault current and grid resistance (ground mat
studies), 413–414
Fault current magnitude and duration
(ground mat studies), 411–413
Ferroresonance transients, 10, 355
Field measurements
harmonics, 276–277
switching transients, 359–363
equipment for measuring transients,
363
overview, 359–360, 362
signal circuits, terminations, and grounding, 362–363
signal derivation, 362
Filters, 339
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
INDEX
harmonic analysis models, 282–284
resonance due to multiple filters, 275–276
selection, 305–307
single-tuned, 282–283, 302–304, 307
First cycle currents, 172–173, 176
Fluorescent lighting, 98
FMEA (failure modes and effects analysis),
379
Forced outage, 379
definition of, 376
duration, definition of, 376
frequency and expected duration expressions, 381
rate, 378
definition of, 376
Forced unavailability, definition of, 376
Fourier representation, 21–22
Fourth-order Runge-Kutta method, 121
Frequency-domain analysis, 219, 223–224
Fromlich’s approximation, 253
Ft (ambient and conductor temperature),
391–393
Fuses, 378, 431, 435
G
Galvanized steel conductors, 61
Gauss-Seidel method, 112, 115–118, 141–
146, 148, 462
Generators
armature capacitance to ground, 368
buses, 138, 143
constants, 366, 368
data
load flow studies, 139
motor-starting studies, 239
harmonic analysis models, 277–278
hunting phenomena, 226
lightning and switching surge response,
355
modeling, 87–88, 277–278
motor starting solution procedure and
example, 245–256
pullout protection, 216
steady-state torque equation, 212–213
transient reactance, 245–246
Copyright © 1998 IEEE. All rights reserved.
turbine, constants for, 366, 368
waterwheel, constants for, 366, 368
X/R ratio, 373
Governor models, 97
Graphical user interface (GUI), definition
of, 123
Grids (ground mat studies)
geometry, 414–415
modeling, 415–418
Ground-fault current coordination study,
433
Ground mat studies, 407–423
advanced grid modeling, 415–418
benchmark problems, 418–419
input/output techniques, 420–424, 426
justification, 407
modeling the human body, 407–410
overview, 11, 407, 420–423
sample problem, 420, 425–428
traditional analysis, 410–415
fault current and grid resistance, 413–
414
fault current magnitude and duration,
411–413
grid geometry, 414–415
ground resistivity, 410–411
Grouped cables, 395–400
GUI (graphical user interface), definition of,
123
H
Hardware, 124–126. See also Computer
solutions and systems
Harmonic analysis, 265–308
examples, 290–302
general theory, 268–276
effects of harmonics, 269–270
harmonic sources, 268–269
resonance. See Resonance
overview, 9–10, 265–267
purpose, 267–268
remedial measures, 302–307
filter selection, 305–307
standards, harmonic, 307–308
system modeling, 276–290
473
IEEE
Std 399-1997
data for analysis, 289–290
filter models, 282–284
generator model, 277–278
harmonic generation, 286–287
induction motor model, 278–281
for industrial and commercial systems,
287–289
load model, 278, 281
network modeling and computer-based
solution techniques, 282, 284–286
shunt capacitor, 279
transformer model, 278–279
transmission line and cable models,
279, 281–282
H constant (stability studies), 93–94
Heat flow analysis (cable), 383–386
High-pass filters, 282–284
Horsepower to kVA conversion (motorstarting studies), 240
Human body, modeling, 407–410
Hunting phenomena (generators), 226
I
IAS (Industry Application Society), 307
IEC 60909 standard, 177–179
IEEE (Institute of Electrical and Electronics
Engineers), 1
Industry Application Society (IAS), 307
Power Engineering Society (PES), 305,
308
standards
cable ampacity studies, 388, 404
coordination studies, 431, 433, 444
dc power system studies, 445–446, 449,
463
ground mat studies, 413–415, 423
harmonics, 266–267, 269, 286, 288,
305, 307–309
load flow studies, 149, 161, 163
motor-starting studies, 233, 239, 252,
263
reliability evaluation, 377–378
short-circuit studies, 166, 172–179,
203, 206
stability studies, 95–96
474
INDEX
switching transient studies, 357, 366–
367
Impedance, 45, 227, 239
data requirement for stability studies, 221
dc decrement and, 168–169
diagram, 53–54
induction motors, 180
method of bus voltage determination for
motor-starting, 242–243
Independent power producers (IPPs), 209
Inductance, 55, 313–314
between capacitor banks, 369
Induction motors
impedance values, 180
models
constant impedance model, 87
constant load model, 83, 85
harmonic analysis, 278–281
overview, 81–84
short-circuit studies, 85–87, 173
reactance values, 174–175
X/R ratio, 374
Inductive reactance, 62–63
Inductive susceptance, 45
Industrial power systems. See Power systems
analysis
Industry Application Society (IAS), 307
Infinite utility systems, 54
Initial short-circuit current, 177
Inrush control reactors, 339
Inrush current, 241, 247–248
Inrush motors, 236
Instability. See also Stability studies
problems caused by, 216
system disturbances that produce, 216
Institute of Electrical and Electronics
Engineers. See IEEE
Instrument transformer capacitance, 367
Interactive computer programs, 127. See
also Computer solutions and systems
Interharmonics, 308
Interrupting currents, 172–173
Interrupting equipment, 176–177
Interruption, definition of, 376
Interruption frequency, definition of, 376
IPPs (independent power producers), 209
Iron-core reactors, 307
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
INDEX
Isolated systems, 54
Iterative solution algorithms, 115–120, 141–
142
Gauss-Seidel technique, 112, 115–118,
141–146, 148
Newton-Raphson technique, 118–120,
141, 146–149, 462
K
K, definition of, 123
Kirchoff’s law, 11, 108–109, 133, 282, 316,
384
Kvar for capacitor banks, 221, 306
L
Laplace expansion, 112
Laplace transform, 22–29, 120, 315–323,
336–337
LC transients, 316
Lighting modeling, 97–98
Lightning, switching surge response to, 355
Linearity, 14–15
Line dropping, restrike phenomena during,
356
Lines
definition of, 55
energizing and reclosing transients, 356
models, 55–59
long lines, 57–58
medium lines, 58–59
short lines, 58–59
Line-to-line faults, 167
Load flow studies, 133–163
bus voltage determination for motor starting, 243–245
dc power systems, 446–448, 461–464. See
also DC power system studies
constant current loads, 447–448
constant power loads, 447–448
constant resistance loads, 447–448
example, 151–161
input data, 137–140
branch data, 139–140
Copyright © 1998 IEEE. All rights reserved.
bus data, 138–139
generator data, 139
system data, 137
transformer data, 140
load buses, 138, 143, 221
load flow analysis, 149–151
overview, 7–8, 133–134, 162–163
programs, 162
solution methods, 140–149
comparison, 149
Gauss-Seidel iterative technique, 112,
115–118, 141–146, 148, 462
Newton-Raphson iterative technique,
118–120, 141, 146–149, 462
problem formulation, 141
system representation, 134–137
Loads, nonlinear, 265–268, 276
proliferation of, 265, 267
Long line models, 57–58
Loops, 107
Loss-of-field relaying, 216
M
M, definition of, 123
Mainframes, 126
Maintenance outages, 378–379
Matrix algebra
fundamentals, 104–107
power systems, 107–110
Matrix sparsity, 183
Mat studies. See Ground mat studies
Medium line models, 58–59
Mercury vapor lighting, 98
Mesh, 107
ground mat studies, 414–415, 417–419,
426–427
Microprocessor, definition of, 123
Minicomputers, 126
Modeling
human body, 407–410
power system. See System modeling
Modem, definition of, 123
Momentary currents, 172
Monitor, definition of, 123
475
IEEE
Std 399-1997
Motors
acceleration time analysis, 236–239, 248,
253–262
dc. See DC motors
impedances, 180
induction. See Induction motors
lightning and switching surge response,
355
NEMA design specifications, 232–233
pullout protection, 216
salient-pole, reactance constants, 366
speed-torque analysis, 221–222, 236–239,
248, 253–262
starting. See Motor-starting studies
synchronous. See Synchronous machine
models
trend to become increasingly larger, 231
Motor-starting studies, 231–262
data requirements, 238–241
basic information, 238–240
simplifying assumptions, 240–241
dc motors, 456–457
need for, 231–235
problem revelation, 231
special torque requirements, 234–235
voltage dips, 231–233
weak source generation, 233–234
overview, 8–9, 231, 259
recommendations, 235–237
starting requirements, analyzing, 236–
237
voltage dips, 235–236
solution procedures and examples, 241–
262
mathematical relationships, 242–249
simple voltage drop method, 249–251
speed-torque and motor-accelerating
time analysis, 236–239, 248, 253–
262
time-dependent bus voltages, 251–253
types of studies, 237–238
adaptations, 238
detailed voltage profile, 238
speed-torque and acceleration time
analysis, 236–239, 248, 253–262
voltage drop snapshot, 237–238
Multipulsing, 287–288, 303
476
INDEX
Mutual coupling in zero sequence
short-circuit studies, 181
N
National Electrical Code (NEC), 166, 383,
388, 460
National Electrical Manufacturers Association (NEMA), 445
motor design specifications, 232–233
Neher-McGrath conductor ampacity method, 383–384, 386–387
Networks
harmonic analysis models, 282, 284–286
reduction, 110
Neutral grounding (short-circuit studies),
180–181
Newton-Raphson iterative technique, 118–
120, 141, 146–149, 462
Nodes, 54, 107
Non-integer harmonics, 308
Nonlinear loads, 176, 265–268
proliferation of, 265, 267
Norton equivalent circuit, 19
Norton impedance, 266
Numerical solution techniques, 104–121
differential equations, 120–121
matrix algebra fundamentals, 104–107
power system network matrices, 107–110
simultaneous algebraic equations. See
Simultaneous algebraic equations,
solution of
O
Odd harmonics, 308
Off-diagonal elements, 104
Ohm’s law, 11, 283, 313–314, 384
Operating system, definition of, 123
Oscillographs, 363
Oscilloscopes, 363
Outage, definition of, 376
Outdoor bushing capacitance to ground, 365
Out-of-step operation, 216
Copyright © 1998 IEEE. All rights reserved.
INDEX
Overcurrent protective devices. See Protective coordination studies
Overfrequency, 225
Overhead lines, 55
cables, compared to, 60
P
Parallel resonance, 272–275
Partitioning, matrix, 106–107, 110
Passive network elements, 44–46
PCC (point of common coupling), 287–290,
308
PCs (personal computers), 125, 441–443.
See also Computer solutions and
systems
“Peakpickers,” 363
Peak short-circuit current, 177
Personal computers (PCs), 125, 441–443.
See also Computer solutions and
systems
Per-unit method, 38–40, 71–73
PES (Power Engineering Society), 305, 308
PF. See Power factor
Phase bus capacitance, 369
Phase overcurrent coordination study, 433
Phase-shifting transformer models, 70–71
Phasors, 20–21, 171
Pixel (pel), definition of, 124
Point of common coupling (PCC), 287–290,
308
“Post-mortem” analysis, 166–167
Potential transformers (PTs), 362
Potier reactance, 222
Power company. See Utilities
Power Engineering Society (PES), 305, 308
Power factor (PF), 47, 306
correction capacitors, 270, 275–276, 302
harmonic filter design and, 266
motor during starting, 241
Power quantities, defining expressions for,
47
Power system data development, 71–79
per-unit representations, 71–79
applications example, 73–79
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
Power systems analysis
application to industrial and commercial
power systems, 2, 7
cable ampacity studies. See Cable ampacity studies
coordination studies. See Protective
coordination studies
DC auxiliary power system analysis. See
DC power system studies
digital computers, 7. See also Computer
solutions and systems
ground mat studies. See Ground mat
studies
harmonic analysis studies. See Harmonic
analysis
history of, 1–2
load flow studies. See Load flow studies
most common system studies, 3–5
motor-starting studies. See Motor-starting
studies
overview, 1–5
preparing for, 2–3
recommended practice, purposes of, 2–5
reliability studies. See Reliability studies
short-circuit studies. See Short-circuit
studies
stability studies. See Stability studies
switching transient studies. See Switching
transient studies
transient network analyzer (TNA), 7, 10,
276, 338, 340–345
Power system stabilizer (PSS), 218, 228
Power transformers (PTs), 430
reactance, 364
Prime mover models, 97
Protective coordination studies, 429–444
basics, 430–434
computer programs, 435–443
common structure, 436–441
coordination programs, 435, 441–443
TCC plotting programs, 435, 441
equipment needs, 443
overview, 11–12, 429–430, 444
verifying the results, 443
PSS (power system stabilizer), 218, 228
PTs (potential transformers), 362
Pullout protection (generators), 216
477
IEEE
Std 399-1997
Q
Q factor, 307
Quasi-steady-state fault analysis, 171–172
R
RAM (random access memory), definition
of, 124
Reactance
power transformers, 364
synchronous machines, 366
Reactive power, 47
Reactors, 306, 307
air-core, 307
modeling, 64
shunt, 98, 340
tuning, 339
X/R ratio and resistance, 372
Recommended practice
most common system studies, 3–5
purposes of, 2–5
Rectifiers, 265
Regulators, 221, 234, 247, 257
Relays, 431, 435
Reliability studies, 375–381
data needed, 377–378
definitions, 375–377
method, 378–381
failure modes and effects analysis
(FMEA), 379
service interruption definition, 379
overview, 10, 375
system reliability indexes, 377
Repair time, definition of, 376
Resistance, 55, 60–62, 313–314
Resistivity, soil, 388, 393–395, 410–411,
413–414, 421
Resonance, 270–276
due to multiple filters, 275–276
parallel, 272–275
series, 270–273
Restrike phenomena (switching transient
analysis), 356
Reversed phase rotation (switching transient
analysis), 356
478
INDEX
RLC circuits, 318–323, 338
ROM (read-only memory), definition of,
124
Rotating machines
data requirement for stability studies,
221–222
design and selection of, 217
Rotation, equations for, 49
Rotor angle, 212–215
Row matrix, 104
Row vector, 104
Runge-Kutta method, 121
S
Salient-pole motor constants, 366
SCADA (Supervisory Control And Data
Acquisition) systems, 133
Scale modeling, 43
Scheduled outage, 378–379
definition of, 376
duration, definition of, 376
rate, definition of, 377
Secondary arc (transient network analyzer),
341
Series capacitors, 64. See also Capacitors
Series faults, 167–168
Series inductance, 55
Series resistance, 55
Series resonance, 270–273
Servers, 125–126
Service interruption method 379
Shock hazards. See Ground mat studies
Short-circuit studies, 85–87, 165–205. See
also Fault analysis
accuracy, factors affecting, 179–182
mutual coupling in zero sequence, 181
neutral grounding, 180–181
phase shifts in delta wye transformer
banks, 181–182
prefault system loads and shunts, 181
system configuration, 179
system impedances, 179–180
batteries, 449–455
battery chargers, 456
computational approaches, 171–172
Copyright © 1998 IEEE. All rights reserved.
INDEX
quasi-steady-state fault analysis, 171–
172
time domain fault analysis, 171
computer solutions, 182–186
software, 184–186
system matrices, 182–183
dc power systems, 447, 464–466
international guidance on, 466
motors, 457–459
example, 187–203
extent and requirements, 166–168
fault analysis according to industry
standards, 172–179
international standard IEC 60909 (1988),
177–179
North American ANSI and IEEE standards, 172–179
ac and dc decrement, 174–176, 178–
179
calculating short-circuit currents and
interrupting equipment, 176–177
overview, 8, 12, 165–166
synchronous machines, 89–93
system modeling, 168–171
ac decrement and rotating machinery,
168
dc decrement, 168–169
system configuration, 179
system impedances, 168–171, 179–180
system loads, 181
three-phase versus symmetrical components, 169–170
Short line models, 58–59
Shunt reactors, 98
transient network analyzer (TNA), 340
Shunts
capacitance, 55
capacitive reactance, 63–64
capacitor modeling, 98, 279
conductance, 55–57
faults, 167
reactor modeling, 98, 340
short-circuit studies, 181
Simultaneous algebraic equations, solution
of, 110–120
direct methods, 111–115
iterative methods, 115–120, 141–142
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
Gauss-Seidel technique, 112, 115–118,
141–146, 148, 462
Newton-Raphson technique, 118–120,
141, 146–149, 462
overview, 110–111
Single-line diagram, 3–4
Single line-to-ground faults, 167
Single-phase equivalent circuit, 29–31
Single-tuned filters, 282–283, 302–304, 307.
See also Filters
Sinusoidal forcing function, 19–20
Skin effect, 280
Software, 126–129, 162. See also Computer
solutions and systems
Soil
resistivity, 410–411, 413–414, 421
temperature, 384–385, 391
thermal resistivity, 388, 393–395
Speed-torque analysis, 221–222, 236–239,
248, 253–262
Square matrix, 104
Stability studies
fundamentals, 209–215
definition of stability, 209
multimachine systems, 215
steady-state stability, 210–212
transient and dynamic stability, 212–
214
two-machine systems, 215
industrial power systems, 223, 225–228
co-gen plant that imports power from
local utility, 226–227
co-gen plant with excess generation,
225–226
oscillations between industrial power
plant and utility system, 227–228
instability
problems caused by, 216
system disturbances that produce, 216
most complex type of study, 8
overview, 8, 209, 228
solutions to stability problems, 217–218
design and selection of rotating equipment, 217
power system stabilizer application,
218
system design, 217
479
IEEE
Std 399-1997
system protection, 218
voltage regulator and exciter characteristics, 218
synchronous machines, 92–97
classical, 93
exciters, 95–97, 218. See also Exciter
systems
governors, 97
H constant, 93–94
prime movers, 97
variations, 94–95
system stability analysis, 218–223
data requirements, 221–222
disturbances, simulation of, 220–221
interpreting results, 223–224
program functioning, 219
program output, 222–223
system simulation, 220
time- and frequency-domain analysis,
219, 223–224
Standby power systems. See DC power system studies
Starting, motor. See Motor-starting studies
Static power converters as harmonics
source, 265
Static var systems (transient network analyzer), 341
Steady-state fault current, 177–178
Steady-state stability, 210–212. See also
Stability studies
definition of, 8
Steady-state torque equation, 212–213, 215
Step potential, 408–409
Stiff utility systems, 54
Super-minicomputers, 126
Superposition, 15–16
Supervisory Control And Data Acquisition
(SCADA) systems, 133
Surge arresters, 339, 341
Surge capacitors, 339
Swing buses, 138, 143
Switching time, definition of, 377
Switching transient studies, 313–374
capacitor bank switching examples
electromagnetic transients program
(EMTP), 347–361
480
INDEX
transient network analyzer (TNA),
341–345
circuit parameters, typical, 363–374
cable charging current, 373
generator armature capacitance to
ground, 368
inductance between capacitor banks,
369
instrument transformer capacitance,
367
outdoor bushing capacitance to ground,
365
phase bus capacitance, 369
power transformer reactance, 364
synchronous machine constants, 366
transformer winding capacitance to
ground, 371
transmission line characteristics, 370
X/R ratio and resistance of reactors, 372
X/R ratio of generators, 373
X/R ratio of induction motors, 374
X/R ratio of transformers, 374
control of switching transients, 339
electromagnetic
transients
program
(EMTP), 345–361
capacitor bank switching case study,
347–361
network and device representation,
345–347
field measurements, 359–363
equipment for measuring transients,
363
overview, 359–360, 362
signal circuits, terminations, and
grounding, 362–363
signal derivation, 362
general description, 313–338
analytical techniques, 315
circuit elements, 313–315
examples, 325–334
normalized damping curves, 324–325
overview, 313, 337–338
transient analysis based on the Laplace
transform method, 24–29, 315–
323
transient recovery voltage (TRV), 334–
337
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
INDEX
objectives, 338–339
overview, 10, 338, 352–353
problem areas, 353, 355–358
transient network analyzer (TNA), 340–
345
capacitor bank switching case study,
341–345
Symmetrical component analysis, 32–37,
170
Symmetric matrix, 104
Synchronous condensers
constants, 366
loss-of-field relaying, 216
modeling, 88
short-circuit studies, 173
Synchronous machine models, 87–97
constants, 366
generators, 87–88, 173
short-circuit models, 89–93, 167, 173
stability models, 92–97
classical, 93
data requirement, 221–222
exciters, 95–97, 218. See also Exciter
systems
governors, 97
H constant, 93–94
prime movers, 97
variations, 94–95
synchronous condensers, 88
synchronous motors, 88–89
impedance values, 180
modeling, 88–89
pullout protection, 216
short-circuit studies, 173
torque versus rotor angle, 213
transient network analyzer (TNA), 341
System, definition of, 377
System modeling
branch element modeling, 55–71
cables, 59–60. See also Cable
capacitors, 64. See also Capacitors
constants. See Conductors, constants
lines, 55–59. See also Lines
reactors, 64. See also Reactors
transformers. See Transformers, models
bus elements, 80–98
electric furnaces, 98
Copyright © 1998 IEEE. All rights reserved.
induction motors. See Induction motors,
models
lighting and electric heating, 97–98
loads in general, 80–81
shunt capacitors, 98
shunt reactors, 98
synchronous machines. See Synchronous machine models
digital computers, 43. See also Computer
solutions and systems
extent of model, 54
isolated systems, 54
utility supplied systems, 54
impedance, 53–54, 168–171, 179–180
overview, 43
power network solution, 49–53
power system data development, 71–79
applications example, 73–79
per-unit representations, 71–79
reactors. See Reactors
review of basics, 44–49
active elements, 46–49
passive elements, 44–46
scale modeling, 43
short-circuit studies, 168–171. See also
Short-circuit studies
ac decrement and rotating machinery,
168
dc decrement, 168–169
system configuration, 179
system impedances, 168–171, 179–180
system loads, 181
three-phase versus symmetrical components, 169–170
System protection and power system stability,
218
System reliability indexes, 377. See also
Reliability studies
System separation (industrial power plant
and utility), 228
T
TCCs. See Time-current characteristics
curves
Telephone interference, 9
481
IEEE
Std 399-1997
telephone interference factor (TIF), 285
Terminal, definition of, 124
Thermal overload heaters, 461
Thermal resistivity (Fth), 393–395
Thevenin equivalent circuit, 16–19, 175–
176
Three-circuit transformer models, 67–70
Three-phase faults, 167, 181
analysis, 171
simulation, 220
Throwover switches, 378
TIF (telephone interference factor), 285
Time-current characteristics curves (TCCs),
429–430, 433–436, 441
plotting programs
435, 441
Time delayed currents, 172–174
Time-dependent bus voltages (motor starting), 251–253
Time-domain analysis, 171, 219
Time-sharing services, 124
TNA (transient network analyzer), 7, 10,
276, 338, 340–345
capacitor bank switching case study, 341–
345
Torque
equation, 212–213, 215
motor starting, 234–235, 238
speed analysis, 257–260
Touch potential, 408–409
Transformers
current, 362, 430
data
coordination studies, 431
load flow studies, 140
motor-starting studies, 239
stability studies, 221
instrument, capacitance of, 367
lightning and switching surge response,
355
line-to-ground short-circuit currents and,
167
models, 65–71
harmonic analysis, 278–279
phase-shifting transformers, 70–71
three-circuit transformers, 67–70
transformer taps, 67
482
INDEX
two-circuit transformers, 65–66
phase shifts in delta wye transformer
banks, 181–182
potential, 362
reactance, 364
short-circuit studies, 173
taps, 67
transient network analyzer (TNA), 340–
341
winding capacitance to ground, 371
X/R ratio, 374
Transient network analyzer (TNA), 7, 10,
276, 338, 340–345
capacitor bank switching case study, 341–
345
Transient recovery voltage (TRV), 334–337
distribution and transmission systems,
356
Transients, 216–217
studies. See Switching transient studies
Transient stability, 212–214. See also Stability studies
definition of, 8
Translation, equations for, 48–49
Transmission lines
characteristics, 370
harmonic analysis models, 279, 281–282
transient network analyzer (TNA), 340
Transmission towers
lightning and switching surge response,
355
TRV (transient recovery voltage), 334–337
distribution and transmission systems,
356
Tuning reactors, 339
Turbine generator constants, 366, 368
Two-circuit transformer models, 65–66
U
Unavailability, definition of, 377
Undamped high-pass filter, 283
Underfrequency, 226
Utilities, 54
co-gen plant that imports power from
local utility, 226–227
Copyright © 1998 IEEE. All rights reserved.
IEEE
Std 399-1997
INDEX
data requirements for motor-starting
studies, 239
infinite, 54
oscillation between industrial plant and
utility, 224, 227–228
point of common coupling (PCC), 287–
290, 308
power factor, 270
reactance values, 174–175
separating industrial system from, 216
short-circuit capacity data for stability
studies, 221
stiff, 54
V
Ventricular fibrillation, 407–408
Voltage dips (motor starting), 231–233,
235–236
Voltage drop studies
dc systems, 461–464
motor starting, 237–238, 249–251
Voltage flicker
switching transient analysis, 356
Voltage profile (motor starting), 238
Voltage regulators and stability, 218
Copyright © 1998 IEEE. All rights reserved.
Voltage stabilizers, 236
Voltmeters, 363
W
Waterwheel generators
constants, 366, 368
Watthour meter error, 9
Winding capacitance to ground, 371
Word, definition of, 124
Workstation, engineering, 125
X
X/R ratio
generators, 373
induction motors, 374
resistance of reactors and, 372
transformers, 374
Y
Y-matrix, 182–183
483
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