PowerandEnergySeries51
Short-circuit
Currents
J.Schlabbach
To my wife Bettina and my children Marina and Tobias
Contents
List of figures
xiii
List of tables
xxiii
Foreword
xxvii
1
Introduction
1.1
Objectives
1.2
Importance of short-circuit currents
1.3
Maximal and minimal short-circuit currents
1.4
Norms and standards
2
Theoretical background
2.1
General
2.2
Complex calculations, vectors and phasor diagrams
2.3
System of symmetrical components
2.3.1
Transformation matrix
2.3.2
Interpretation of the system of symmetrical
components
2.3.3
Transformation of impedances
2.3.4
Measurement of impedances of the symmetrical
components
2.4
Equivalent circuit diagram for short-circuits
2.5
Series and parallel connection
2.6
Definitions and terms
2.7
Ohm-system, p.u.-system and %/MVA-system
2.7.1
General
2.7.2
Correction factor using %/MVA- or p.u.-system
2.8
Examples
2.8.1
Vector diagram and system of symmetrical
components
1
1
1
3
4
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viii
Contents
2.8.2
2.8.3
2.8.4
2.8.5
Calculation of impedances of a three-winding
transformer in %/MVA
Conversion of impedances (; %/MVA; p.u.)
Impedances in %/MVA-system based on
measurement
Representation of a line in the RYB-system and in the
system of symmetrical components
3
Calculation of impedance of electrical equipment
3.1
General
3.2
Equipment in a.c. systems
3.2.1
General
3.2.2
Impedance calculation
3.3
Equipment in d.c. systems
3.3.1
General
3.3.2
Impedance calculation
3.4
Examples for calculation
3.4.1
a.c. equipment
3.4.2
d.c. equipment
4
Calculation of short-circuit current in a.c. three-phase
HV-systems
4.1
Types of short-circuits
4.2
Methods of calculation
4.3
Calculation of parameters of short-circuit currents
4.3.1
General
4.3.2
Calculation of short-circuit current parameters
according to IEC 60909-0
4.4
Influence of motors
4.5
Minimal short-circuit currents
4.6
Examples
4.6.1
Three-phase near-to-generator short-circuit
4.6.2
Line-to-earth (single-phase) short-circuit
4.6.3
Calculation of peak short-circuit current
4.6.4
Short-circuit currents in a meshed 110-kV-system
4.6.5
Influence of impedance correction factors on
short-circuit currents
4.6.6
Short-circuit currents in a.c. auxiliary supply of a
power station
5
Influence of neutral earthing on single-phase short-circuit
currents
5.1
General
5.2
Power system with low-impedance earthing
5.3
Power system having earthing with current limitation
5.4
Power system with isolated neutral
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46
50
50
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Contents
5.5
5.6
5.7
Power system with resonance earthing (Petersen-coil)
5.5.1
General
5.5.2
Calculation of displacement voltage
5.5.3
Tuning of the Petersen-coil
Handling of neutrals on HV-side and LV-side of
transformers
Examples
5.7.1
Increase of displacement voltage for systems with
resonance earthing
5.7.2
Limitation of single-phase short-circuit current by
earthing through impedance
5.7.3
Design of an earthing resistor connected to an artificial
neutral
5.7.4
Resonance earthing in a 20-kV-system
5.7.5
Calculation of capacitive earth-fault current and
residual current
5.7.6
Voltages at neutral of a unit transformer
ix
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112
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116
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123
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126
6
Calculation of short-circuit currents in low-voltage systems
6.1
General
6.2
Types of faults
6.3
Method of calculation
6.4
Calculation of short-circuit parameters
6.4.1
Impedances
6.4.2
Symmetrical short-circuit breaking current Ib
6.4.3
Steady-state short-circuit current Ik
6.5
Minimal short-circuit currents
6.6
Examples
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135
7
Double earth-fault and short-circuit currents through earth
7.1
General
7.2
Short-circuit currents during double earth-faults
7.2.1
Impedances and initial symmetrical short-circuit
current Ik
7.2.2
Power system configurations
7.2.3
Peak short-circuit current ip
7.2.4
Symmetrical short-circuit breaking current Ib and
steady-state short-circuit current Ik
7.3
Short-circuit currents through earth
7.3.1
Introduction
7.3.2
Short-circuit inside a switchyard
7.3.3
Short-circuit at overhead-line tower
7.4
Examples
7.4.1
Double earth-fault in a 20-kV-system
7.4.2
Single-phase short-circuit in a 110-kV-system
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148
x
8
9
10
Contents
Factors for the calculation of short-circuit currents
8.1
General
8.2
Correction using %/MVA- or p.u.-system
8.3
Impedance correction factors
8.4
Factor κ for peak short-circuit current
8.5
Factor μ for symmetrical short-circuit breaking current
8.6
Factor λ for steady-state short-circuit current
8.7
Factor q for short-circuit breaking current of asynchronous
motors
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160
Calculation of short-circuit currents in d.c. auxiliary installations
9.1
General
9.2
Short-circuit currents from capacitors
9.3
Short-circuit currents from batteries
9.4
Short-circuit currents from rectifiers
9.5
Short-circuit currents from d.c. motors with independent
excitation
9.6
Total short-circuit current
9.7
Example
9.7.1
Calculation of the impedances of cables and busbar
conductors
9.7.2
Calculation of the short-circuit currents of the
individual equipment
9.7.3
Calculation of the correction factors and corrected
parameters
9.7.4
Calculation of partial short-circuit currents
9.7.5
Calculation of total short-circuit current
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Effects of short-circuit currents
10.1 General
10.2 a.c. systems
10.2.1 Thermal effects and thermal short-circuit strength
10.2.2 Mechanical short-circuit strength of rigid
conductors
10.3 d.c. auxiliary installations
10.3.1 Substitute rectangular function
10.3.2 Mechanical short-circuit strength of rigid
conductors
10.3.3 Thermal short-circuit strength
10.4 Calculation examples (a.c. system)
10.4.1 Calculation of thermal effects
10.4.2 Electromagnetic effect
10.5 Calculation examples (d.c. system)
10.5.1 Thermal effect
10.5.2 Electromagnetic effect
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Contents
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13
xi
Limitation of short-circuit currents
11.1 General
11.2 Measures
11.2.1 Measures in power systems
11.2.2 Measures in installations and switchgear
arrangement
11.2.3 Measures concerning equipment
11.3 Structures of power systems
11.3.1 General
11.3.2 Radial system
11.3.3 Ring-main system
11.3.4 Meshed systems
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Special problems related to short-circuit currents
12.1 Interference of pipelines
12.1.1 Introduction
12.1.2 Calculation of impedances for inductive
interference
12.1.3 Calculation of induced voltage
12.1.4 Characteristic impedance of the pipeline
12.1.5 Voltage pipeline-to-earth
12.2 Considerations on earthing
12.2.1 General
12.2.2 Resistance of human body
12.2.3 Soil conditions
12.2.4 Relevant currents through earth
12.2.5 Earthing impedance
12.3 Examples
12.3.1 Interference of pipeline from 400-kV-line
12.3.2 Calculation of earthing resistances
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Data of equipment
13.1 Three-phase a.c. equipment
13.1.1 System feeders
13.1.2 Transformers
13.1.3 Generators
13.1.4 Overhead lines
13.1.5 Cables
13.1.6 Reactors and resistors
13.1.7 Asynchronous motors
13.2 d.c. equipment
13.2.1 Conductors
13.2.2 Capacitors
13.2.3 Batteries
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xii
Contents
Symbols, superscripts and subscripts
287
References
293
Index
299
List of figures
Figure 1.1
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Importance of short-circuit currents and definition of tasks as
per IEC 60781, IEC 60865, IEC 60909 and IEC 61660
Vector diagram and time course of a.c. voltage
Definition of vectors for current, voltage and power in
three-phase a.c. systems. (a) Power system diagram and
(b) electrical diagram for symmetrical conditions
(positive-sequence component)
Vector diagram of current, voltage and power of a three-phase
a.c. system represented by the positive-sequence component.
(a) Consumer vector system and (b) generator vector
system
Differentially small section of homogeneous three-phase
a.c. line
Vector diagram of voltages in RYB-system and in the
zero-sequence component, positive- and negative-sequence
components are NIL
Vector diagram of voltages in RYB-system and
positive-sequence component, zero- and negative-sequence
components are NIL
Vector diagram of voltages in RYB-system and
negative-sequence component, zero- and positive-sequence
components are NIL
Measurement of impedance in the system of symmetrical
components. (a) Positive-sequence component (identical with
negative-sequence component) and (b) zero-sequence
component
Measuring of zero-sequence impedance of a two-winding
transformer (YNd). Diagram indicates winding arrangement
of the transformer: (a) measuring at star-connected winding
and (b) measuring at delta-connected winding
Measurement of positive-sequence impedance of a
three-winding transformer (YNyn + d). Diagram indicates
winding arrangement of the transformer
2
12
14
15
16
18
19
19
21
22
22
xiv
List of figures
Figure 2.11
Figure 2.12
Figure 2.13
Figure 2.14
Figure 2.15
Figure 2.16
Figure 2.17
Figure 2.18
Figure 2.19
Figure 2.20
Figure 2.21
Figure 4.1
Figure 4.2
Figure 4.3
Measurement of zero-sequence impedance of a three-winding
transformer (YNyn + d). Diagram indicates winding
arrangement of the transformer
General scheme for the calculation of short-circuit currents in
three-phase a.c. systems using the system of symmetrical
components
Equivalent circuit diagram of a single-phase short-circuit in
RYB-system
Equivalent circuit diagram in the system of symmetrical
components for a single-phase short-circuit
Equations for impedance analysis in power systems
Equivalent circuit diagram of a power system with different
voltage levels
Graphical construction of voltages in the system of
symmetrical components: (a) vector diagram RYB, (b) vector
diagram of voltage in the zero-sequence component, (c) vector
diagram of voltage in the positive-sequence component and
(d) vector diagram of voltage in the negative-sequence
component
Simplified equivalent circuit diagram in RYB-components
Equivalent circuit diagram in the system of symmetrical
components
Equivalent circuit diagram of an overhead line of infinitesimal
length with earth return in RYB-system
Equivalent circuit diagram of an overhead line of infinitesimal
length with earth return in 012-system. (a) Positive-sequence
component, (b) negative-sequence component and
(c) zero-sequence component
Types of short-circuits and short-circuit currents.
(a) Three-phase short-circuit, (b) double-phase short-circuit
without earth/ground connection, (c) double-phase
short-circuit with earth/ground connection and
(d) line-to-earth (line-to-ground) short-circuit
Time-course of short-circuit currents. (a) Near-to-generator
short-circuit (according to Figure 12 of IEC 60909:1988),
(b) far-from-generator short-circuit (according to Figure 1 of
IEC 60909:1988). Ik – initial (symmetrical) short-circuit
current, ip – peak short-circuit current, Ik – steady-state
short-circuit current and A – initial value of the aperiodic
component idc
Example for short-circuit current calculation with an
equivalent voltage source at s.-c. location. (a) Three-phase
a.c. system with three-phase short-circuit, (b) equivalent
circuit diagram in 012-system (positive-sequence system),
(c) equivalent circuit diagram in 012-system with equivalent
voltage source
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34
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41
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43
43
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70
List of figures
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
Figure 4.18
Figure 4.19
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Estimate of maximal initial short-circuit current for different
types of short-circuit and different impedance ratios Z1 /Z0
and Z2 /Z1 . Phase angle of Z 0 , Z 1 and Z 2 assumed to be
identical. Parameter r: ratio of asymmetrical short-circuit
current to three-phase short-circuit current
Equivalent circuit diagram for the calculation of short-circuit
currents inside power plant
Equivalent circuit diagram for single-fed three-phase
short-circuit
Factor κ for the calculation of peak short-circuit current
Equivalent circuit diagram for three-phase short-circuit
fed from non-meshed sources
Equivalent circuit diagram of a three-phase short-circuit in
a meshed system
Factor μ for calculation of symmetrical short-circuit breaking
current
Factors λmax and λmin for turbine generators (Figure 17 of DIN
EN 60909.0 (VDE 0102)). (a) Series one and (b) series two
Factors λmax and λmin for salient-pole generators (Figure 18 of
DIN EN 60909.0 (VDE 0102) 1988). (a) Series one and (b)
series two
Factor q for the calculation of symmetrical short-circuit
breaking current
Equivalent circuit diagram of a 220-kV-system with
short-circuit location
Equivalent circuit diagram of a 110-kV-system with
220-kV-feeder
Equivalent circuit diagram of a 10-kV system, f = 50 Hz
A 110-kV system with short-circuit location
System with different voltage levels with short-circuit
location
High-voltage system configuration for the auxiliary supply of
a power station
Equivalent circuit diagram of a single-phase short-circuit
(system with low-impedance earthing). (a) Diagram in
RYB-system, (b) equivalent circuit diagram in the system of
symmetrical components
Ratio of single-phase to three-phase short-circuit current
depending on Z1 /Z0 and (γ1 − γ0 )
Earth-fault factors in relation to Z 1 /Z 0 and (γ1 − γ0 ).
(a) Earth-fault factor δY and (b) earth-fault factor δB
Earth-fault factor δ depending on X0 /X1 for different ratios
R0 /X0 and R1 /X1 = 0.01
/I depending on X /X
Earth-fault factor δ and ratio Ik1
0
1
k3
xv
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xvi
List of figures
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15
Figure 5.16
Figure 5.17
Figure 5.18
Figure 5.19
Figure 5.20
Figure 5.21
Power system with isolated neutral with single-phase
earth-fault. (a) Equivalent circuit diagram in RYB-system and
(b) equivalent circuit diagram in the system of symmetrical
components
Limit for self-extinguishing of capacitive currents in air
according to VDE 0228 part 2
Vector diagram of voltages, power system with isolated
neutral. (a) Prior to fault and (b) during earth-fault
Time course of phase-to-earth voltages, displacement voltage
and earth-fault current. System with isolated neutral,
earth-fault in phase R
System with resonance earthing, earth-fault in phase R.
(a) Equivalent diagram in RYB-system and (b) equivalent
diagram in the system of symmetrical components
Current limits according to VDE 0228 part 2:12.87 of
ohmic currents IRes and capacitive currents ICE
Equivalent circuit diagram of a power system with
asymmetrical phase-to-earth capacitances. (a) Equivalent
circuit diagram in the RYB-system and (b) equivalent circuit
diagram in the system of symmetrical components
Polar plot of the displacement voltage in a power system with
resonance earthing
Voltages and residual current in the case of an earth-fault;
displacement voltage without earth-fault
Current–voltage
√ characteristic of a Petersen-coil;
Ur = 20 kV/ 3; Ir = 640 A. (a) Minimal adjustment (50 A)
and (b) maximal adjustment (640 A)
Displacement voltage in non-faulted operation and residual
current under earth-fault conditions; non-linear characteristic
of the Petersen-coil
Transformation of voltage in the zero-sequence component of
transformers in the case of single-phase faults. (a) Equivalent
circuit diagram in RYB-system and (b) equivalent circuit
diagram in the system of symmetrical components
Alternate earthing of transformer neutrals by Petersen-coils.
(a) Two parallel transformers and (b) earthing at artificial
neutral with reactor XD2
Fault current in the MV-system in the case of a short-circuit in
the HV-system
Resonance curve (displacement voltage) for different detuning
factors in a 20-kV-system for different conditions
Voltages in a 20-kV-system with resonance earthing for
different tuning factors. (a) Phase-to-earth voltages and
(b) displacement voltage (resonance curve)
105
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107
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112
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115
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118
119
121
121
122
123
List of figures xvii
Figure 5.22
Figure 5.23
Figure 5.24
Figure 6.1
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Figure 8.6
Figure 8.7
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Equivalent circuit diagram of a 20-kV-system with resonance
earthing
Connection of a power station to a 220-kV-system with
short-circuit location
Equivalent diagram in the zero-sequence component for fault
location F
Equivalent circuit diagram of a LV-installation
Equivalent circuit diagram with short-circuit inside
switchyard B
Equivalent circuit diagram with short-circuit at overhead-line
tower
Equivalent circuit diagram of a 20-kV-system
Equivalent circuit diagram of a 110-kV-system with
short-circuit location
Equivalent circuit diagram of a power system with different
voltage levels
Equivalent circuit diagram for the calculation of impedance
correction factor using %/MVA- or p.u.-system
Generator directly connected to the power system.
(a) Equivalent system diagram and (b) equivalent circuit
diagram in the positive-sequence component
Determination of the short-circuit current by superposition
Equivalent circuit diagram of a power system with three-phase
short-circuit. (a) Circuit diagram, (b) simplified diagram of
a single-fed three-phase short-circuit and (c) time course of
voltage with voltage angle ϕU
Characteristic saturation curve method for determination of
Potier’s reactance
Calculated and measured values of factor q for the calculation
of short-circuit breaking current of asynchronous motors;
values of q as per Figure 4.13 (According to Figure 20 of
IEC 60909-1:1991.)
Equivalent circuit diagrams of equipment in d.c. auxiliary
installations; typical time course of short-circuit current
(according to Figure 1 of DIN EN 61660-1 (VDE 0102
Teil 10)). (a) Capacitor, (b) battery, (c) rectifier in three-phase
a.c. bridge connection and (d) d.c. motor with independent
excitation
Standard approximation function of the short-circuit current
(according to Figure 2 of IEC 61660-1:1997)
Factor κC for the calculation of peak short-circuit current of
capacitors (according to Figure 12 of IEC 61660-1:1997)
Time-to-peak tpC for the calculation of short-circuit
currents of capacitors (according to Figure 13 of
IEC 61660-1:1997)
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xviii List of figures
Figure 9.5
Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9
Figure 9.10
Figure 9.11
Figure 9.12
Figure 9.13
Figure 9.14
Figure 9.15
Figure 9.16
Figure 9.17
Figure 9.18
Figure 9.19
Figure 9.20
Factor k1C for the calculation of rise-time constant (according
to Figure 14 of IEC 61660-1:1997)
Factor k2C for the calculation of decay-time constant
(according to Figure 15 of IEC 61660-1:1997)
Rise-time constant τ1B and time to peak tpB of short-circuit
currents of batteries (according to Figure 10 of
IEC 61660-1:1997)
Factor λD for the calculation of quasi steady-state short-circuit
current of rectifiers (according to Figure 7 of
IEC 61660-1:1997)
Factor κD for the calculation of peak short-circuit currents of
rectifiers. Factor: R ∗ = (RN /XN )(1 + 2RDBr /3RN )
(according to Figure 8 of IEC 61660-1:1997)
Factor κM for the calculation of peak short-circuit current of
d.c. motors with independent excitation (according to
Figure 17 of IEC 61660-1:1997)
Time to peak of short-circuit currents for d.c. motors with
independent excitation and τMec < 10 ∗ τF (according to
Figure 19 of IEC 61660-1:1997)
Factor k1M in the case of d.c. motors with independent
excitation and τMec ≥ 10 ∗ τF (according to Figure 18 of
IEC 61660-1:1997)
Factor k2M in the case of d.c. motors with independent
excitation and τMec < 10 ∗ τF (according to Figure 19 of
IEC 61660-1:1997)
Factor k3M in the case of d.c. motors with independent
excitation and τMec < 10 ∗ τF (according to Figure 20 of
IEC 61660-1:1997)
Factor k4M in the case of d.c. motors with independent
excitation and τMec < 10 ∗ τF (according to Figure 21 of
IEC 61660-1:1997)
Equivalent circuit diagram of a d.c. auxiliary installation
Typical time curves of total short-circuit current in
d.c. auxiliary installations, e.g., (a) with dominating part of
motors, (b) with dominating part of rectifiers, (c) with
dominating part of batteries and (d) in the case of low rectifier
load (according to Figure 22 of DIN EN 61660-1 (VDE 0102
Teil 10))
Equivalent circuit diagram of the d.c. auxiliary installation
(220 V), e.g., of a power station
Partial short-circuit currents and total short-circuit current,
d.c. auxiliary system as per Figure 9.18
Total short-circuit current, obtained by superposition of the
partial short-circuit currents and approximated short-circuit
current, d.c. auxiliary system as per Figure 9.18
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181
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194
List of figures
Figure 10.1
Figure 10.2
Figure 10.3
Figure 10.4
Figure 10.5
Figure 10.6
Figure 10.7
Figure 10.8
Figure 10.9
Figure 10.10
Figure 10.11
Figure 10.12
Figure 10.13
Figure 10.14
Figure 10.15
Figure 10.16
Figure 10.17
Figure 11.1
Figure 11.2
Figure 11.3
Factor n for the calculation of thermal short-time current
(heat dissipation of a.c. component) (according to Figure 22 of
IEC 60909-0:2001)
Factor m for the calculation of thermal short-time current
(heat dissipation of d.c. component) (according to Figure 21 of
IEC 60909-0:2001)
Rated short-time current density of conductors. δ0 is the
temperature at beginning of short-circuit and δ1 is the
temperature at end of short-circuit. (a) ____: Copper;
- - - -: unalloyed steel and steel cables and (b) Al, aluminium
alloy, ACSR
Maximal permissible thermal short-circuit current for
impregnated paper-insulated cables Un up to 10 kV
Arrangement of parallel conductors
Correction factor k12 for the calculation of effective distance
(according to Figure 1 of IEC 61660-2:1997)
Factors Vσ and Vσ s for the calculation of bending stress
(according to Figure 4 of IEC 60865-1:1993)
Factors Vr and Vrs for the calculation of bending stress
(according to Figure 5 of IEC 60865-1:1993)
Factor VF for the calculation of bending stress (according to
Figure 4 of IEC 60865-1:1993)
Calculation of mechanical natural frequency (Factor c).
Arrangement of distance elements and calculation equation
(according to Figure 3 of IEC 60865-1:1993)
Standard approximation function (a) and substitute rectangular
function (b) (according to Figure 4 of IEC 60660-2:1997). Not
to scale
Factors Vσ and Vσ s for the calculation of bending stress on
conductors (according to Figure 9 of IEC 61660-2:1997)
Factor VF for the calculation of forces on supports (according
to Figure 9 of IEC 61660-2:1997)
Equivalent circuit diagram, data of equipment, resistance
at 20◦ C
Equivalent circuit diagram of a power system with
wind power plant
Arrangement of busbar conductor (data, see text)
Standardised rectangular function and approximated total
short-circuit current
Selection of suitable voltage level for the connection of power
stations
Schematic diagram of a 400/132-kV-system for urban load;
values of short-circuit currents in case of operation as two
subsystems
Schematic diagram of a 132-kV-system with power station
xix
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229
xx
List of figures
Figure 11.4
Figure 11.5
Figure 11.6
Figure 11.7
Figure 11.8
Figure 11.9
Figure 11.10
Figure 11.11
Figure 11.12
Figure 11.13
Figure 11.14
Figure 11.15
Figure 11.16
Figure 11.17
Figure 11.18
Equivalent circuit diagram of a 30-kV-system with feeding
132-kV-system: (a) Operation with transformers in parallel and
(b) limitation of short-circuit current. Result of three-phase
= 3.2 GVA; S = 40 MVA;
short-circuit current: SkQ
rT
ukrT = 12%; trT = 110/32; OHTL 95Al; ltot = 56 km
Equivalent circuit diagram of a 380-kV-system and results of
three-phase short-circuit current calculation: (a) Radial fed
= 8 GVA; OHTL
system and (b) ring fed system. SkQ
ACSR/AW 4 × 282/46; li = 120 km
Schematic diagram of a 110-kV-substation fed from the
220-kV-system: (a) Operation with buscoupler closed and
(b) operation with buscoupler open. Result of three-phase
short-circuit current calculation
Equivalent circuit diagram of a 6-kV-industrial system. Results
of three-phase short-circuit current calculation: (a) Busbar
sectionaliser closed and (b) Busbar sectionaliser open
Equivalent circuit diagram of switchgear with single
busbar
Time course of short-circuit current in installations with and
without Ip-limiter
Cutaway view of an Ip-limiter support: (1) insulating tube,
(2) explosive loading, (3) main conductor, (4) fuse element
and (5) transducer
Equivalent circuit diagram of a 10-kV-system with incoming
feeder. Results of three-phase short-circuit current calculation:
(a) impedance voltage 13% and (b) impedance
voltage 17.5%
Equivalent circuit diagram of a 10-kV-system with
short-circuit limiting reactors. Results of three-phase
short-circuit current calculation
Equivalent circuit diagram of 11.5-kV-system fed from the
132-kV-system
Equivalent circuit diagram of a power station with
132-kV-busbar. Results of three-phase short-circuit current
calculation: SrG = 150 MVA; xd = 12–17.8%
General structure of a radial system with one incoming
feeder
General structures of ring-main systems: (a) Simple structure
with one feeding busbar and (b) structure with two feeding
busbars (feeding from opposite sides)
Principal structure of a high voltage system with different
voltage levels
Principal structure of meshed low voltage system:
(a) Single-fed meshed system and (b) meshed system with
overlapping feeding
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
List of figures
Outline and distances of a high-voltage transmission-line
tower. B: counterpoise; P: pipeline; L: conductor nearest to
pipeline. E: first earth conductor (nearest to pipeline), also
named E1; E2: second earth conductor
Figure 12.2 Oblique exposure and crossing of pipeline and overhead line.
(a) Plot plan and (b) elevation plan (detail from crossing
location)
Figure 12.3 Impedance of the human body (hand-to-hand) depending on
the touch voltage
Figure 12.4 Permissible touch voltage depending on the time of
exposure
Figure 12.5 Plot plan of the exposure length pipeline and transmission
line
Figure 12.6 Elevation plan of the overhead transmission tower and the
pipeline
Figure 12.7 Specific electric field strength (a) and specific induced
voltage (b) of the pipeline between towers 2 and 11
Figure 12.8 Voltage pipeline-to-earth along the exposure length
(0–6400 m)
Figure 13.1 Principal structure of a power supply system and typical values
of initial short-circuit power of public supply system
Figure 13.2 Typical values for the impedance voltage of two-winding
transformers
Figure 13.3 Typical values for the ohmic losses, no-load losses and
no-load current of two-winding transformers
Figure 13.4 Tower outline of high-voltage transmission lines.
(a) Single-circuit line and (b) double-circuit line
Figure 13.5 Capacitances MV-cables (Un < 20 kV)
Figure 13.6 Capacitances C1 (a) and capacitive loading current Ic (b) of
HV-cables
Figure 13.7 Reactance (positive-sequence system) of three-phase cables
(Un ≤ 110 kV)
Figure 13.8 Arrangement of a short-circuit limiting
reactor
√
Figure 13.9 Adjustable Petersen-coil 21 kV/ 3; 4 MVAr;
Ir = 70.1–330 A; adjustable in 64 steps, 4.13 A each
Figure 13.10 Earthing resistor made from CrNi-alloy steel fabric for indoor
installation 3810 , 5 A for 10 s, 170 kV BIL, IP 00
Figure 13.11 Earthing resistor made from meandering wire for outdoor
installation 16 , 400 A for 10 s, 75 kV BIL, IP 20
xxi
Figure 12.1
250
256
258
259
263
264
265
266
268
269
269
271
276
277
277
278
279
280
281
List of tables
Table 1.1
Table 1.2
Table 1.3
Table 2.1
Table 2.2
Table 2.3
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Table 3.5
Table 3.6
Table 3.7
Table 3.8
Table 3.9
International documents and norms for short-circuit current
calculation
Selection of norms as referred in standards for short-circuit
current calculation and as mentioned in this book
Cross-reference list of standards on short-circuit current
calculation
Equivalent circuit diagrams and equations to represent
short-circuits (single fault location) in three-phase a.c. systems
in the RYB-system and in the 012-system
Definitions of quantities in physical, relative and semirelative
units
Conversion of quantities between %/MVA-system, -system
and p.u.-system (100 MVA-base)
Impedance of system feeder, equivalent circuit diagram,
calculation equations and remarks
Impedance of two-winding transformer, equivalent circuit
diagram, calculation equations and remarks
Impedance of three-winding transformer, equivalent circuit
diagram, calculation equations and remarks
Equivalent circuit diagram of two- and three-winding
transformers in the positive- and zero-sequence component
Impedance of synchronous generator, equivalent circuit
diagram, calculation equations and remarks
Impedance of power-station unit, equivalent circuit diagram,
calculation equations and remarks
Impedance of overhead line (single-circuit), equivalent circuit
diagram, calculation equations and remarks
Impedance of short-circuit limiting reactor, equivalent circuit
diagram, calculation equations and remarks
Impedance of asynchronous motor, equivalent circuit diagram,
calculation equations and remarks
5
6
9
28
33
35
46
47
48
50
51
52
53
54
55
xxiv
List of tables
Table 3.10
Table 3.11
Table 3.12
Table 3.13
Table 3.14
Table 3.15
Table 3.16
Table 3.17
Table 3.18
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 5.1
Table 5.2
Table 5.3
Table 5.4
Table 6.1
Table 6.2
Table 6.3
Table 7.1
Impedance of static converter fed drive, equivalent circuit
diagram, calculation equations and remarks
Impedance of system load, equivalent circuit diagram,
calculation equations and remarks
Impedance of a conductor, equivalent circuit diagram,
calculation equations and remarks
Impedance of capacitor, equivalent circuit diagram,
calculation equations and remarks
Impedance of battery, equivalent circuit diagram, calculation
equations and remarks
Impedance of rectifier, equivalent circuit diagram, calculation
equations and remarks
Impedance of d.c. motor with independent excitation,
equivalent circuit diagram, calculation equations and
remarks
Results of calculation of impedance in three-phase a.c.
equipment
Results of calculation of impedance of equipment in d.c.
installations (without common branch as per IEC 61660-1)
Voltage factor c according to IEC 60909-0
Equations for the calculation of initial symmetrical
short-circuit currents
Equivalent frequency for the calculation of decaying
component
Calculation of short-circuit currents of asynchronous
motors
Characteristics of different types of neutral handling in power
systems
Voltages in the zero-sequence component U0 transferred
through 110/10-kV-transformer in the case of single-phase
fault in the 110-kV-system according to Figure 5.17; Un :
nominal system voltage
Capacitive asymmetry CE for different parameters in a
20-kV-system
Characteristics of a 20-kV-system with respect to resonance
earthing
Voltage factor c according to IEC 60909-0. (Voltage factors as
per IEC 60781 are of different values. The standard is under
review)
Example for the calculation of maximal short-circuit currents
in LV-system
Example for the calculation of minimal short-circuit currents
in LV-system
System configurations and equations for the calculation of
short-circuit currents in the case of double earth-faults
56
57
58
59
60
61
62
63
65
71
73
81
84
99
120
122
126
132
136
137
141
List of tables
Table 8.1
Table 8.2
Table 8.3
Table 9.1
Table 10.1
Table 10.2
Table 10.3
Table 10.4
Table 10.5
Table 10.6
Table 11.1
Table 11.2
Table 12.1
Table 12.2
Table 12.3
Table 12.4
Table 12.5
Table 12.6
Table 13.1
Table 13.2
Table 13.3
Table 13.4
Table 13.5
Table 13.6
Table 13.7
Impedances of equipment and short-circuit current as per
Figure 8.1
Impedances of equipment and short-circuit current using
correction factor
Assumed values of uf max for the calculation of the factor λ
Resistances Rij and equivalent resistances Rresj for the
calculation of correction factors; U : Voltage at short-circuit
location prior to the short-circuit
Maximal permissible conductor temperature and rated
short-time current density; 1) – Normal operating condition;
2) – Short-circuit condition
Data of materials for screening, armouring and sheaths of
cables
Factors α, β and γ for different arrangement of supports
(according to Table 3 of IEC 60865-1:1993)
Factors for the calculation of mechanical natural frequency.
Swing is at right angle to the area of sub-conductor
Results of calculation of thermal equivalent currents
Results of short-circuit current calculation
Selection of recommended voltage as per IEC 60038:1987
Result of loadflow and short-circuit analysis as per
Figure 11.11
Interference between power system, communication circuits
and pipelines
Resistivity of soil ρ for different types of soil conditions
Resistance of pipeline coatings
Currents through earth for the design of earthing
installations
Reduction factor for typical power system installations;
distance of earth conductor to phase conductor D ≈ 20 m;
ρ = 100 m
Resistance of earthing installations REI for different types and
arrangement
Data of transformers
Typical data of synchronous generators
Typical values of impedance of the positive-sequence
component of MV-overhead lines
Typical values of impedances of the positive- and
zero-sequence component of HV-overhead lines
(ρE = 100 m)
Characteristic parameters of resistor elements
Data of asynchronous motors
Typical values of MKP-capacitors; self-healing dry insulation;
different make of capacitor can and fuse
xxv
153
154
162
179
196
201
204
209
217
218
227
237
246
248
254
260
261
262
270
271
274
274
280
282
283
xxvi
List of tables
Table 13.8
Table 13.9
Typical values of MKP-capacitors; resin insulation;
round can
Resistance of loaded batteries (data from several
manufacturers)
284
284
Foreword
Short-circuit currents are the dominating parameters for the design of equipment
and installations, for the operation of power systems and for the analysis of outages
and faults. Besides the knowledge about design of equipment in power systems,
in auxiliary installations and about system operation constraints, the calculation of
short-circuit currents is a central task for power system engineers. The book describes
the individual equipment in power systems with respect of the parameters needed for
short-circuit current calculation as well as methods for analysing the different types
of short-circuits in power systems using the system of symmetrical components.
Besides detailed explanation of the calculation methods for short-circuit currents and
their thermal and electromagnetic effects on equipment and installations, short-time
interference problems and measures for the limitation of short-circuit currents are
explained. Detailed calculation procedures for the parameters and typical data of
equipment are given in a separate chapter for easy reference. All aspects of the book
are explained with examples based on engineering studies carried out by the author.
The preparation of the book was finalised in December 2004 and reflects the actual
status of the technique, norms and standards. Carrying out short-circuit studies always
requires the application of the latest editions of standards, norms and technical recommendations, which can be obtained from the IEC-secretariat or from the national
standard organisation. All comments in this book are given in good faith, based on
the comprehensive technical experience of the author.
The author wishes to thank very much his former colleague Dipl.-Ing. Heiner
Rofalski for revising the text and improving the book. Most of the drawings were
prepared by my students Stefan Drees and Elmar Vogel who spent much effort to
obtain clear and understandable presentation. My thanks go also to the staff of IEEpublishers, especially to Ms. Sarah Cramer who encouraged me to write this book.
Comments are highly appreciated.
Bielefeld, December 2004
juergen.schlabbach@fh-bielefeld.de
xxviii
Foreword
Professor Dr.-Ing. Jürgen Schlabbach, born 1952, member IEEE and VDI,
studied power system engineering at the Technical University of Darmstadt/Germany,
from where he got his Ph.D. in 1982. Until 1992 he was working in a consulting
engineering company, responsible for planning and design of public and industrial
supply systems. Since 1992 he has been professor for ‘Power system engineering
and utilisation of renewable energy’ at the University of Applied Sciences in Bielefeld/Germany. His main working areas are planning of power systems, analysis of
faults, power quality, interference problems and connection of renewable energy
sources to power systems. He is also a consulting engineer in the mentioned fields.
More information can be found on the author’s web-page http://www.
fh-bielefeld.de/fb2/labor-ev.
Chapter 1
Introduction
1.1 Objectives
This book deals with the calculation of short-circuit currents in two- and three-phase
a.c. systems as well as in d.c. systems, installed as auxiliary installations in power
plants and substations. It is not the objective of this book to repeat definitions and
rules of norms and standards, but to explain the procedure for calculating short-circuit
currents and their effects on installations and equipment. In some cases repetition
of equations, tables and diagrams from norms and standards however are deemed
necessary for easy understanding. It should be emphasised in this respect that the
presentation within this book is mainly concentrated on installations and equipment
in high voltage systems, i.e., voltage levels up to 500 kV. Special considerations have
to be taken in the case of long transmission lines and in power systems with nominal
voltages above 500 kV. The calculation of short-circuit currents and of their effects
are based on the procedures and rules defined in the IEC documents 61660, 60909,
60865 and 60781 as outlined in Table 1.1.
1.2 Importance of short-circuit currents
Electrical power systems have to be planned, projected, constructed, commissioned
and operated in such a way to enable a safe, reliable and economic supply of the
load. The knowledge of the loading of the equipment at the time of commissioning
and as foreseeable in the future is necessary for the design and determination
of the rating of the individual equipment and of the power system as a whole.
Faults, i.e., short-circuits in the power system cannot be avoided despite careful
planning and design, good maintenance and thorough operation of the system. This
implies influences from outside the system, such as short-circuits following lightning strokes into phase-conductors of overhead lines and damages of cables due to
earth construction works as well as internal faults, e.g., due to ageing of insulation
2
Short-circuit currents
materials. Short-circuit currents therefore have an important influence on the design
and operation of equipment and power systems.
Switchgear and fuses have to switch-off short-circuit currents in short time and
in a safe way; switches and breakers have to be designed to allow even switch-on to
an existing short-circuit followed by the normal switch-off operation. Short-circuit
currents flowing through earth can induce impermissible voltages in neighbouring
metallic pipelines, communication and power circuits. Unsymmetrical short-circuits
cause displacement of the voltage neutral-to-earth and are one of the dominating criteria for the design of neutral handling. Short-circuits stimulate mechanical
oscillations of generator units which will lead to oscillations of active and reactive power as well, thus causing problems of stability of the power transfer which
can finally result in system black-out. Furthermore, equipment and installations
must withstand the expected thermal and electromagnetic (mechanical) effects of
short-circuit currents.
In Figure 1.1 the typical time course of a short-circuit current is shown, which
can be measured at high-voltage installations in the vicinity of power stations with
synchronous generators, characterised by decaying a.c. and d.c. components of the
•
•
Electromagnetic effects
IEC 60865-1
Switch-on capability
IEC 60265-1
•
•
•
Peak short-circuit current
(Maximal instantaneous value)
Thermal effects
IEC 60865-1
Protection measures against
touch voltages in LV-installations
DIN VDE 0100-470
Protection in HV-systems
r.m.s. value
•
Breaking capability
IEC 60265-1
IEC 60265-1 etc.
Short-circuit breaking current
(r.m.s. value at switching instant tmin)
24
19
•
kA
14
•
9
4
•
–1
•
tmin
–6
•
–11
–16
Total time duration
•
Figure 1.1
Further aspects
Touch and step voltages
DIN VDE 0141
Interference
DIN VDE 0228
Surge arresters
IEC 60099-4
Overvoltages
IEC 60071-1
IEC 60071-2
Neutral point earthing
DIN VDE 0141
Insulation co-ordination
IEC 60071-1
IEC 60071-2
Surge arresters
IEC 60099-4
Importance of short-circuit currents and definition of tasks as per
IEC 60781, IEC 60865, IEC 60909 and IEC 61660
Introduction
3
current. It is assumed that the short-circuit is switched-off approximately 14 periods
after its initiation, which seems a rather long time, but was chosen for reason of a better
visibility in the figure. Attention shall be put on four parameters of the short-circuit
current.
• The total time duration of the short-circuit current consists of the operating time
of the protection devices and the total breaking time of the switchgear.
• The peak short-circuit current, which is the maximal instantaneous value of the
short-circuit current, occurs approximately a quarter period after the initiation of
the short-circuit. As electromagnetic forces are proportional to the instantaneous
value of the current, the peak short-circuit current is necessary to know in order
to calculate the forces on conductors and construction parts affected by the shortcircuit current.
• The r.m.s.-value of the short-circuit current is decaying in this example due to the
decaying a.c. component. Currents through conductors will heat the conductor
due to ohmic losses. The r.m.s. value of the short-circuit current, combined with
the total time duration, is a measure for the thermal effects of the short-circuit.
• The short-circuit breaking current is the r.m.s.-value of the short-circuit current
at switching instant, i.e., at time of operating the circuit-breaker. While opening
the contacts of the circuit-breaker, the arc inside the breaker will heat up the
installation, which depends obviously on the breaking time as well.
1.3 Maximal and minimal short-circuit currents
Depending on the task of engineering studies, the maximal or minimal short-circuit
current has to be calculated. The maximal short-circuit current is the main design
criteria for the rating of equipment to withstand the effects of short-circuit currents,
i.e., thermal and electromagnetic effects. The minimal short-circuit current is needed
for the design of protection systems and the minimal setting of protection relays.
The short-circuit current itself depends on various parameters, such as voltage level,
actual operating voltage, impedance of the system between any generation unit and
the short-circuit location, impedance at the short-circuit location itself, the number
of generation units in the system, the temperature of the equipment influencing the
resistances and other parameters. The determination of the maximal and the minimal
short-circuit current therefore is not as simple as might be seen at this stage [36]. It
requires detailed knowledge of the system operation, i.e., which cables, overheadlines, transformers, generators, machines and reactors are in operation and which are
switched-off. The assessment of the results of any calculation of short-circuit currents
must take into account these restrictions in order to ensure that the results are on the
safe side, i.e., that the safety margin of the calculated maximal short-circuit current is
large enough without resulting in an uneconomic high rating of the equipment. The
same applies to the minimal short-circuit current for which the safety margin must
be assessed in such a way as to distinguish between the highest operating current and
any short-circuit current, which has to be switched-off.
4
Short-circuit currents
1.4 Norms and standards
Technical standards are harmonised on international basis. The international
organisation to coordinate the works and strategies is the ISO (International Standards
Organisation), whereas IEC (International Electrotechnical Commission) is responsible for the electrotechnical standardisation. The national standard organisations
such as CENELEC in Europe, BSI in the United Kingdom, DKE in Germany, ANSI
in the United States, JSI in Japan as well as national electrotechnical organisations
such as IEE, VDE, IEEE, JES etc. are working in the working groups of IEC to
include their sight and knowledge on technical items in the international standards
and documents. On national basis, standards are adopted to the widest extent to
the internationally agreed standards and documents. In some cases additions to the
international standards are included in the national standards, however their status is
‘for information only’.
The application of norms and standards has to be based on the latest issues, which
can be obtained in Germany from Beuth-Verlag GmbH, Burggrafenstr. 6, D-10787
Berlin or from VDE-Verlag GmbH, Bismarckstr. 33, D-10625 Berlin. English versions are available from British Standards Institution, London/UK, in the United
States from American National Standards Institute or any national standard organisation. Standards can also be searched and ordered through the web on the following
URLs (appearance in alphabetical order):
American National Standards Institute
British Standards Institute
Deutsches Institut für Normung
International Electrotechnical Commission
VDE-Verlag
http://www.nssn.org/help.html
http://www.bsonline.techindex.co.uk
http://din.de
http://www.iec.ch
http://vde-verlag.de
The structure of standards and norms dealing with short-circuit current calculation
as published in IEC or EN-norms are outlined in Table 1.1. The listing should not
be understood as a complete catalogue of standards but represents an overview only.
Some of the mentioned standards are actually in draft status; others include corrections, additions and appendices. For details reference should be made to the
IEC-homepage or the homepage of the national standards committee. The official
actual standards catalogue is the only relevant document for any technical application. IEC-documents and national standards refer to other norms and standards.
A short overview of these references is outlined in Table 1.2.
With respect to the calculation of short-circuit currents and their effects, the
standards are harmonised in most of the countries. The procedures and methods
described are identical to those defined in the mentioned IEC-documents. Table 1.3
shows a cross-reference list between IEC, EN and BS.
The classification numbers of the different standards differ in some cases from
those of the IEC-documents or the EN-norms, however in most of the cases, classification numbers are similar, e.g.: Australian standard AS 3865, Swedish standard
SS-EN 60865-1 and British standard BS EN 60865-1 are identical to IEC 60865-1
Introduction
5
(Short-circuit currents – calculation of effects; Part 1: Definitions and calculation
methods). The standards’ catalogue of American National Standards Institute, to be
accessed through the home-page of National Standards Systems Network, directly
indicates the IEC- resp. EN-documents under the heading ‘short-circuit currents’. It is
therefore fully sufficient to apply the IEC-documents for calculation of short-circuit
currents and the analysis of their effects.
Table 1.1
International documents and norms (with related VDE-classification) for
short-circuit current calculation
IEC (year)
EN (year)
DIN; VDE (year)
Title, contents
61660-1
(1997)
61660-1
(1997)
VDE 0102
Part 10
(1998–06)
Short-circuit currents in d.c. auxiliary
installations in power plants and
substations
Part 1: Calculation of short-circuit currents
61660-2
(1997)
61660-2
(1997)
VDE 0103
Part 10
(1998–05)
Short-circuit currents in d.c. auxiliary
installations in power plants and
substations
Part 2: Calculation of effects
61660-3
(2000)
61660-3
Appendix 1
VDE 0102
Part 10
(2002–11)
Short-circuit currents in d.c. auxiliary
installations in power plants and
substations
Examples of calculation of short-circuit
current and effects
60781
(1989)
HD 581 S1
(1991)
Appendix 2
VDE 0102
(1992–09)
Application guide for calculation
of short-circuit currents in
low-voltage radial systems
60865-1
(1993)
60865-1
(1993)
VDE 0103
(1994–11)
Short-circuit currents – Calculation
of effects
Part 1: Definitions and calculation methods
—
—
Appendix 1
VDE 103
(1996–06)
Short-circuit currents – Calculation of
effects
Part 1: Definitions and calculation methods
Examples for calculation
60909-0
(2001)
60909-0
(2001)
VDE 0102
(2002–07)
Short-circuit current calculation
in a.c. systems
6
Short-circuit currents
Table 1.1
Continued
IEC (year)
EN (year)
DIN; VDE (year)
Title, contents
60909-1
(1991)
—
Appendix 3
VDE 0102
(1997–05)
Short-circuit current calculation
in three-phase a.c. systems
Part 1: Factors for the calculation of
short-circuit currents in three-phase
a.c. systems according to IEC 909
60909-2
(1992)
—
Appendix 4
VDE 0102
(2003–02)
Electrical equipment
Data for short-circuit current calculations
in accordance with IEC 909 (1998)
60909-3
(1995)
—
VDE 0102
Part 3
(1997–06)
Short-circuit current calculation in
three-phase a.c. systems
Part 3: Currents during two separate
simultaneous single-phase line-to-earth
short-circuits and partial short-circuit
currents flowing through earth
60909-4
(2000)
—
Appendix 1
VDE 0102
(1992–09)
Examples for the calculation
of short-circuit currents
Table 1.2
Selection of norms as referred in standards for short-circuit current
calculation and as mentioned in this book
IEC (year)
EN (year)
DIN; VDE (year)
Title, contents
60038 (mod)
(2002)
HD472 S1
(1989)
DIN IEC 60038
(2002)
IEC standard voltages
60050(131)
(1978)
DIN IEC 60050-131
(1983)
International Electrotechnical
Vocabulary (IEV)
Chapter 131: Electric and
magnetic circuits
60050(151)
(2001)
DIN 40200
(1981–12)
International Electrotechnical
Vocabulary (IEV)
Chapter 151: Electric and magnetic
devices
Some parts of DIN 40200 are identical
to IEC 60050
Introduction
Table 1.2
Continued
IEC (year)
EN (year)
DIN; VDE (year)
Title, contents
60050(195)
(1998)
International Electrotechnical
Vocabulary (IEV)
Chapter 195: Earthing and protection
against electric shock
60050(441)
(1998)
International Electrotechnical
Vocabulary (IEV)
Chapter 441: Switchgear,
controlgear and fuses
60071-1
(1993)
60071-1
(1995)
VDE 0111 Part 1
(1996–07)
Insulation coordination
Definitions, principles and rules
60071-2
(1996)
60071-2
(1997)
VDE 0111 Part 2
(1997–09)
Insulation coordination
Application guide
TS 60479-1
(1994)
Effects of currents on
human being and livestock
Part 1: General aspects
TS 60479-2
(1982)
Effects of currents passing
through the human body
Part 2: Special aspects
60986
(2000)
Short-circuit temperature limits
of electric cables with rated
voltages from 6 kV up to 30 kV
60949
(1988)
Calculation of thermally permissible
short-circuit currents, taking into
account non-adiabatic heating effects
60896-1
(1987)
60896-1
(1991)
61071-1 (mod)
(1991)
61071-1
(1996)
60265, 60282,
60298, 60420,
60517, 60644,
60694, etc.
60099, 61643
60949
(1988)
7
60099-1
(1994)
Stationary lead-acid batteries – General
requirements and methods of
test – Part 1: Vented types
VDE 0560 Part 120
(1997–08)
Capacitors for power electronics
VDE 0670
Switchgear, circuit-breakers, fuses, etc.
Various documents of IEC,
parts of VDE 0670
VDE 0675 Part 1
(2000–08)
Surge arresters
Various documents of IEC,
parts of VDE 0670
Calculation of thermally permissible
short-circuit currents, taking into
account non-adiabatic heating effects
8
Short-circuit currents
Table 1.2
Continued
IEC (year)
EN (year)
DIN; VDE (year)
60986
(2000)
Title, contents
Guide to the short-circuit temperature
limits of electrical cables with a rated
voltage from 1.8/3 (3.6) kV to
18/30 (36) kV
EN 50160
(1999)
VDE 0141
(2000–01)
Earthing of special systems for
electrical energy with nominal
voltages above 1 kV
VDE 0228 Part 1
(1987–12)
Proceedings in the case of interference
on telecommunication installations
by electrical power
installations – General
VDE 0228 Part 2
(1987–12)
Proceedings in the case of interference
on telecommunication installations
by electrical power installations
interference by three-phase
installations
VDE 0228 Part 3
(1988–09)
Proceedings in the case of interference
on telecommunication installations
by electrical power installations
interference by alternating current
traction systems
VDE 0228 Part 4
(1987–12)
Proceedings in the case of interference
on telecommunication installations
by electrical power installations
interference by direct current systems
VDE 0226 Part 1000
(1995–06)
Current carrying capacity
General, conversion factors
DIN 13321
(1978–04)
Electric power engineering;
components in three-phase networks
Concepts, quantities and their letter
symbols
DIN 40110-1
(1994)
Quantities used in alternating
current theory; two-line circuits
DIN 40110-2
(2002)
Quantities used in alternating
current theory; three-line circuits
DIN EN 50160
(2000–03)
Voltage characteristics of electricity
supplied by public distribution
systems
Introduction
Table 1.3
9
Cross-reference list of standards on short-circuit current calculation
IEC
(year)
EN
(year)
BS EN
(year)
Remarks (title see Table 1.1)
61660-1
(1997)
61660-1
(1997)
61660-1
(1997)
Short-circuit currents in d.c. auxiliary
installations in power plants and substations
Part 1: Calculation of short-circuit currents
61660-2
(1997)
61660-2
(1997)
61660-2
(1997)
Short-circuit currents in d.c. auxiliary
installations in power plants and substations
Part 2: Calculation of effects
61660-3
(2000)
61660-1
98/202382 DC
Short-circuit currents in d.c. auxiliary
installations in power plants and substations
Examples of calculation of short-circuit
current and effects
60781
(1989)
HD 581 S1
(1991)
7638
(1993)
Application guide for calculation of
short-circuit currents in low-voltage
radial systems
60865-1
(1993)
60865-1
(1993)
60865-1
(1994)
Short-circuit currents – Calculation
of effects
Part 1: Definitions and calculation methods
—
—
PD 6875-2
(1995)
Short-circuit currents – Calculation of effects
Part 1: Definitions and calculation methods;
Examples for calculation
60909-0
(2001)
60909-0
(2001)
60909-0
(2001)
Short-circuit current calculation in
a.c. systems
60909-1
(1991)
—
PD IEC TR
60909-1
(2002)
Short-circuit current calculation in
three-phase a.c. systems
Part 1: Factors for the calculation of
short-circuit currents in three-phase
a.c. systems according to IEC 909
60909-2
(1992)
—
PD 7639-2
(1994)
60909-3
(1995)
prEN 60909-3
95/203556 DC
60909-4
(2000)
—
—
Electrical equipment
Data for short-circuit current calculations
in accordance with IEC 909 (1988)
Short-circuit current calculation in three-phase
a.c. systems
Part 3: Currents during two separate
simultaneous single-phase line-to-earth
short-circuits and partial short-circuit
currents flowing through earth
Examples for the calculation of short-circuit
currents
Chapter 2
Theoretical background
2.1 General
A detailed deduction of the mathematical procedure is not given within the context
of this book, but only the final equations are quoted. For further reading, reference is
made to [1], [13]. In general, equipment in power systems is represented by equivalent
circuits, which are designed for the individual tasks of power system analysis. For the
calculation of no-load current and the no-load reactive power of a transformer, the
no-load equivalent circuit is sufficient. Regarding the calculation of short-circuits,
voltage drops and load characteristic a different equivalent circuit is required. The
individual components of the equivalent circuits are resistance, inductive and capacitive reactance (reactor and capacitor), voltage source and ideal transformer. Voltage
and currents of the individual components and of the equivalent circuit are linked by
Ohm’s law.
2.2 Complex calculations, vectors and phasor diagrams
When dealing with two- and three-phase a.c. systems, it should be noted that currents
and voltages are generally not in phase. The phase position depends on the amount of
inductance, capacitance and resistances of the impedance. The time course, e.g., of a
current or voltage in accordance with
√
u(t) = 2 ∗ U ∗ sin(ωt + ϕU )
(2.1a)
√
i(t) = 2 ∗ I ∗ sin(ωt + ϕI )
(2.1b)
can in this case be shown as a line diagram as per Figure 2.1. In the case of sinusoidal variables, these can be shown in the complex numerical level by rotating
pointers, which rotate in the mathematically positive sense (counterclockwise) with
12
Short-circuit currents
p
2
v
p
u (t)
i (t)
3p
2
0
^
i
wI
p
2
wU
u^
u
3p
2
Figure 2.1
u (t ); i (t )
u^
^i
i
p
2p
vt
wU
wI
Vector diagram and time course of a.c. voltage
angular velocity ω as follows:
√
U = 2 ∗ U ∗ e(j ωt+ϕU )
√
I = 2 ∗ I ∗ e(j ωt+ϕI )
(2.2a)
(2.2b)
where U and I are the r.m.s-values of voltage and current, ω is the angular frequency
and ϕU and ϕI are the phase angle of voltage and current. The time course in this
case is obtained as a projection on the real axis, see Figure 2.1.
The generic term for an impedance Z is given as impedance or apparent resistance
Z = R + jX
(2.3a)
The generic term for an admittance Y is admittance or apparent admittance
Y = G + jB
(2.3b)
where R is the active resistance, X the reactance, G the active conductance and
B the susceptance. The terms for the designation of resistances and admittances as
per the above are stipulated in DIN 40110 (VDE 0110).
The reactance depends on the particular angular frequency ω under consideration
and can be calculated for capacitances C or inductances L from
1
ωC
(2.4a)
XL = ωL
(2.4b)
XC =
For sinusoidal variables, the current i(t) through a capacitor, or the voltage u(t) at
an inductance, can be calculated by the first derivative of the voltage, respectively
Theoretical background
13
current, as follows:
i(t) = C ∗
du(t)
dt
(2.5a)
u(t) = L ∗
di(t)
dt
(2.5b)
The derivation for sinusoidal currents and voltages at a reactance establishes that
the current achieves its maximum value a quarter period after the voltage. When
considering the process in the complex level, the pointer of the voltage precedes the
pointer of the current by 90◦ . This corresponds to a multiplication by +j .
For a capacitance on the other hand, the voltage does not reach its maximum
value until a quarter period after the current; the voltage pointer lags behind the
current by 90◦ , which corresponds to a multiplication by −j .
This enables the relationships between current and voltage for inductances and
capacitances to be shown in a complex notation
U = j ωL ∗ I
I=
1
∗U
j ωC
(2.6a)
(2.6b)
The individual explanations of the quantities are given in the text above.
Vectors are used to describe electrical processes. They are therefore used in d.c.,
a.c. and three-phase systems. Vector systems can, by definition, be chosen as required,
but must not be changed during an analysis or calculation. It should also be noted that
the appropriate choice of the vector system is of substantial assistance in describing
and calculating special tasks. The need for vector systems is clear if one considers
the Kirchhoff’s laws, for which the positive direction of currents and voltages must
be specified. In this way, the positive directions of the active and reactive powers are
then also stipulated.
For comparison and transfer reasons, the vector system for the three-phase network (RYB components) is also to be used for other component systems (e.g., system
of symmetrical components), which describe the three-phase network.
If vectors are drawn as shown in Figure 2.2, the active and reactive powers
generated by a generator in overexcited operation mode are positive. This vector
system is designated as a generator vector system. Accordingly, the active and reactive
power consumed by the load are positive when choosing the consumer vector system.
When describing electrical systems voltage vectors are drawn from the phase
conductor (named L1, L2, L3 or also R, Y, B) to earth (E). In other component
systems, for instance in the system of symmetrical components (Section 2.3), the
direction of the voltage vector is drawn from the conductor towards the particular
reference conductor. On the other hand, vectors in phasor diagrams are shown in
the opposite direction. The vector of a voltage conductor to earth is therefore shown
in the phasor diagram from earth potential to conductor potential.
14
Short-circuit currents
+P
+Q
(a)
IRQ
Three-phase
a.c. system
E
R(L3)
IYQ
Y(L2)
IBQ
UYQ
URQ
IRQ + IYQ + IBQ = 0; URQ + UYQ + UBQ = 0;
Figure 2.2
Positive
sequence
system
B(L1)
UBQ
+P
+Q
(b)
01
IRQ
UYQ= U1Q
UYQ = UQ / 3
Definition of vectors for current, voltage and power in three-phase
a.c. systems. (a) Power system diagram and (b) electrical diagram for
symmetrical conditions (positive-sequence component)
Based on the definition of the vector system, the correlation of voltage and current
of an electrical system can be shown in phasor diagrams. Where steady-state or quasisteady-state operation is shown, r.m.s. value phasors are generally used. Figure 2.3
shows the phasor diagram of an ohmic-inductive load in the generator and in the
consumer vector system.
2.3 System of symmetrical components
2.3.1 Transformation matrix
The relationships between voltages and currents of a three-phase system can be represented by a matrix equation, e.g., with the aid of the impedance or admittance matrix.
The equivalent circuits created by electrical equipment, such as lines, cables, transformers and machines, in this case have couplings in the three-phase system which
are of an inductive, capacitive and galvanic type. This can be explained by using
any short element of an overhead line in accordance with Figure 2.4 as an example,
see also [1], [7].
The correlation of currents I and voltages U of the RYB system is as follows:
⎡
UR
⎤
⎡
Z RR
⎢ ⎥ ⎢
⎣U Y ⎦ = ⎣Z YR
UB
Z BR
Z RY
Z YY
Z BY
⎤ ⎡
⎤
IR
⎥ ⎢ ⎥
Z YB ⎦ ∗ ⎣I Y ⎦
Z RB
Z BB
(2.7)
IB
where Z RR , Z YY , Z BB are the self-impedances of each phase; Z RY , Z RB the coupling
(mutual) impedances between phase R and Y, respectively, B; Z YR , Z YB the coupling
(mutual) impedances between phase Y and R, respectively, B; and Z BR , Z BY the
coupling (mutual) impedances between phase B and R, respectively, Y.
All the values of this impedance matrix can generally be different. Because of the
cyclic-symmetrical construction of three-phase systems only the self-impedance and
Theoretical background
(a)
Q
Positive
sequence
system
I1
U
–Q
3
R
X
U
–R
U
–x
15
P1, Q1
01
+Re
U
–Q =U
– R+ U
–X
3
U
–R
jQ1
–SI
–I1
P
–1
w
U
– x = jXI
–1
–UR = RI
–1
+ Im
Q
(b)
Positive
sequence
system
I1
U
–Q
3
R
X
U
–R
U
–X
P1, Q1
01
+Re
U
–R
U
–Q =U
– R + –UX
3
a
U
–X
w –jQ
1
+ Im
RI1 = – UR
–
–
–I1
Figure 2.3
–P1
= –U
–Ux = jXI
–1 – X
–S1
Vector diagram of current, voltage and power of a three-phase
a.c. system represented by the positive-sequence component. (a) Consumer vector system and ( b) generator vector system
two coupling impedances are to be considered. A cyclic-symmetrical matrix is thus
obtained.
⎤ ⎡ ⎤
⎡ ⎤ ⎡
Z A Z M1 Z M2
IR
UR
⎥ ⎢ ⎥
⎢ ⎥ ⎢
Z
Z
U
Z
I
=
∗
(2.8)
⎣ Y ⎦ ⎣ M2
A
M1 ⎦ ⎣ Y ⎦
UB
Z M1 Z M2 Z A
IB
16
Short-circuit currents
Self coefficients
Coupling coefficients
L⬘Ldl
R⬘dl L⬘dl
B
L⬘Edl
C⬘Ldl
C⬘Edl
G⬘Ldl
G⬘Edl
Y
R
E
R⬘Edl L⬘Edl
Longitudinal
Figure 2.4
Transversal
Differentially small section of homogeneous three-phase a.c. line
where Z A is the self-impedances of each phase and Z M1 , Z M2 the coupling (mutual)
impedances between the phases.
The multiplicity of couplings between the individual components of three-phase
systems complicates the application of the solution methods, particularly when calculating extended networks. For this reason, a mathematical transformation is sought
which transfers the RYB-components to a different system. The following conditions
should apply for the transformation:
• The transformed voltages should still depend only on one transformed current.
• For symmetrical operation only one component should be unequal to zero.
• The linear relationship between current and voltage should be retained, i.e., the
transformation should be linear.
• For symmetrical operation, the current and voltage of the reference component
should be retained (reference component invariant).
The desired transformation should in this case enable the three systems to be decoupled
in such a way that the three components, named 0, 1 and 2, are decoupled from each
other in the following manner:
⎡ ⎤ ⎡
⎤ ⎡ ⎤
U0
0
I0
Z0 0
⎢ ⎥ ⎢
⎥ ⎢ ⎥
(2.9)
⎣U 1 ⎦ = ⎣ 0 Z 1 0 ⎦ ∗ ⎣ I 1 ⎦
U2
0
0
Z2
I2
These requirements are fulfilled by the transformation to the system of symmetrical
components [29], which is realised for voltages and currents by the transformation
matrix T according to Equations (2.10). It should be noted that the factor 13 is part of
the transformation and therefore belongs to the matrix T.
U 012 = T ∗ U RYB
(2.10a)
I 012 = T ∗ I RYB
(2.10b)
Theoretical background
⎡ ⎤
⎡
1 1
U0
⎢ ⎥ 1⎢
⎣U 1 ⎦ = ⎣1 a
3
U2
1 a2
1
⎤ ⎡
UR
⎤
⎥ ⎢ ⎥
a 2 ⎦ ∗ ⎣U Y ⎦
1
(2.10c)
UB
a
⎡ ⎤
⎡
1 1
I0
⎢ ⎥ 1⎢
⎣I 1 ⎦ = ⎣1 a
3
I2
1 a2
17
⎤ ⎡
IR
⎤
⎥ ⎢ ⎥
a 2 ⎦ ∗ ⎣I Y ⎦
IB
a
(2.10d)
The voltage vector of the 012-system is linearly linked to the voltage vector of
the RYB-system (the same applies for the currents). The system of symmetrical
components is defined according to DIN 13321.
The reverse transformation of the 012-system to the RYB-system is achieved by
the matrix T−1 in accordance with
U RYB = T −1 ∗ U 012
(2.11a)
I RYB = T −1 ∗ I 012
(2.11b)
⎡
1
⎢ ⎥ ⎢
⎣U Y ⎦ = ⎣1
UB
1
⎡
⎡
UR
IR
⎤
⎤
⎡
1
1
a2
1
⎤
U0
⎥ ⎢ ⎥
a ⎦ ∗ ⎣U 1 ⎦
a
a2
1
1
⎢ ⎥ ⎢
⎣I Y ⎦ = ⎣1 a 2
IB
1 a
⎤ ⎡
(2.11c)
U2
⎤ ⎡ ⎤
I0
⎥ ⎢ ⎥
a ⎦ ∗ ⎣I 1 ⎦
I2
a2
(2.11d)
The following applies for both transformation matrices T and T−1 :
T ∗ T−1 = E
(2.12)
with the identity matrix E. The complex rotational phasors a and a 2 have the following
meanings:
√
◦
a = ej 120 = − 21 + j 21 3
(2.13a)
√
◦
a 2 = ej 240 = − 12 − j 21 3
(2.13b)
1 + a + a2 = 0
(2.13c)
Multiplication of a vector with either a or a 2 will only change the phase shift of the
vector by 120◦ or 240◦ but will not change the length (amount) of it.
18
Short-circuit currents
2.3.2 Interpretation of the system of symmetrical components
If only one zero-sequence component exists, the following applies:
⎤ ⎡ ⎤ ⎡ ⎤
⎡ ⎤ ⎡
1 1 1
U0
U0
UR
⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢
2
1
a
U
a
U
0
=
∗
=
⎦ ⎣ ⎦ ⎣ 0⎦
⎣ Y⎦ ⎣
UB
U0
0
1 a a2
(2.14)
No phase displacement exists between the three a.c. voltages of the RYB-conductors.
The zero-sequence component is thus a two-phase a.c. system. Figure 2.5 shows the
phasor (vector) diagram of the voltages of the RYB-system and the voltage of the
zero-sequence component.
+ Re
–UR; –UY; –UB; –U0
+Im
Figure 2.5
e jvt
Vector diagram of voltages in RYB-system and in the zero-sequence
component, positive- and negative-sequence components are NIL
Where only a positive-sequence exists, the following applies:
⎡ ⎤ ⎡
⎤
⎤ ⎡ ⎤ ⎡
1 1 1
U1
0
UR
⎥
⎥ ⎢ ⎥ ⎢
⎢ ⎥ ⎢
⎣U Y ⎦ = ⎣1 a 2 a ⎦ ∗ ⎣U 1 ⎦ = ⎣a 2 U 1 ⎦
UB
0
a U1
1 a a2
(2.15)
A three-phase system with a positive rotating phase sequence R, Y, B is represented
by the positive-sequence component only. Figure 2.6 shows the phasor (vector) diagram of the voltages of the RYB-system and the voltage of the positive-sequence
component.
Where only a negative-sequence component exists, the following applies:
⎤
⎤ ⎡ ⎤ ⎡
⎡ ⎤ ⎡
1 1 1
U2
0
UR
⎣U Y ⎦ = ⎣1 a 2 a ⎦ ∗ ⎣ 0 ⎦ = ⎣ a U 2 ⎦
(2.16)
U2
UB
a2 U 2
1 a a2
A three-phase system with a positive counter-rotating phase sequence R, B, Y is represented by the negative-sequence component only. Figure 2.7 shows the phasor (vector)
diagram of the voltages of the RYB-system and the voltage of the negative-sequence
component.
Theoretical background
19
+ Re
–UR = –U1
+ Im
e jvt
UY = a2 U1
UB = a U1
Figure 2.6
Vector diagram of voltages in RYB-system and positive-sequence component, zero- and negative-sequence components are NIL
+ Re
UR = U2
+ Im
e jvt
UB = a2 U2
UY = a U2
Figure 2.7
Vector diagram of voltages in RYB-system and negative-sequence
component, zero- and positive-sequence components are NIL
2.3.3 Transformation of impedances
For the transformation of the impedance matrix, Equations (2.17) apply in accordance with the laws of matrix multiplication, taking account of Equations (2.10)
and (2.12)
T URYB = T ZRYB T−1
U012 = Z012
T IRYB
(2.17a)
I012
(2.17b)
As T URYB is equal to U012 and T IRYB is equal to I012 the impedance matrix Z012
in the system of symmetrical components is obtained by multiplying the impedance
matrix ZRYB in the RYB-system with the matrix T from left and with T−1 from right.
Based on Equation (2.8) the impedances as per Equations (2.18) for the conversion
20
Short-circuit currents
of the impedances of the three-phase system to the 012-system are obtained.
Z 0 = Z A + Z M1 + Z M2
(2.18a)
Z 1 = Z A + a 2 Z M1 + a Z M2
(2.18b)
Z 2 = Z A + a Z M1 + a 2 Z M2
(2.18c)
The impedance values of the positive-sequence and negative-sequence components
are generally equal. This applies to all non-rotating equipment. The impedance
of the zero-sequence component mainly has a different value from the impedance
of the positive-sequence component. If mutual coupling is absent, as perhaps with
three single-pole transformers connected together to form a three-phase transformer,
the impedance of the zero-sequence component is equal to the impedance of the
positive- or (negative-) sequence component.
2.3.4 Measurement of impedances of the symmetrical components
Any equipment can be represented by an equivalent circuit diagram using the system
of symmetrical components, which can be determined. To obtain the parameters of the
equivalent circuit diagram either short-circuit measurement or no-load measurement
has to be carried out in accordance with Figure 2.8 with a voltage system representing
the individual component of the system of symmetrical components, i.e.,
• Three-phase voltage system with positive rotating phase sequence RYB to
measure the positive-sequence component.
• Three-phase voltage system with positive counter-rotating phase sequence RBY
to measure the negative-sequence component.
• a.c. voltage system without any phase displacement of the voltage in the three
phases of the equipment to measure the zero-sequence component.
It should be noted in this respect that the type of measurement, i.e., short-circuit measurement or no-load measurement, depends on the type of analysis to be carried out.
For short-circuit current calculation or voltage drop estimate, the parameters obtained
by short-circuit measurement are needed. Impedance will be calculated based on the
measured voltage and current as outlined in Figure 2.8.
Special attention must be taken in case of transformers, as the impedance in
the zero-sequence component depends on the type of winding arrangement (star or
delta connection) and on the handling of the neutral of the transformer. Figure 2.9
indicates the measurement of the zero-sequence impedance in the case of a twowinding transformer with Yd-arrangement. The Y-connected winding is fed from a
single-phase a.c. system; the zero-sequence impedance is present in case the neutral
of the transformer is earthed (Figure 2.9(a)). Due to the delta-connection of the
second winding a current in the delta winding is present in this case. If the neutral
of the transformer remains isolated, the current in the Y-winding is Zero and thus
the impedance is infinite. If the zero-sequence impedance of the delta-connected
Theoretical background
(a)
B
Y
GS
3~
21
1)
2)
R
IR
(b)
B
Y
G
1~
R
IR
Figure 2.8
UR
1)
2)
UR
Measurement of impedance in the system of symmetrical components.
(a) Positive-sequence component (identical with negative-sequence
component) and (b) zero-sequence component
winding shall be measured, the current in the delta winding is Zero in any case and the
zero-sequence impedance of the delta-connected winding is infinite (Figure 2.9(b)).
To measure the impedance of three-winding transformers in the positive- and
zero-sequence component three different measurements have to be carried out. For
the positive-sequence component the measurement is outlined in Figure 2.10 and
further explained as below:
1. Feeding with three-phase a.c. system at winding 1 (HV-side)
Short-circuit at winding 2 (MV-side)
Measurement of uk12 , respectively, ukHVMV
2. Feeding with three-phase a.c. system at winding 1 (HV-side)
Short-circuit at winding 3 (LV-side)
Measurement of uk13 , respectively, ukHVLV
3. Feeding with three-phase a.c. system at winding 2 (MV-side)
Short-circuit at winding 3 (LV-side)
Measurement of uk23 , respectively, ukMVLV
The impedances (reactances) as per the equivalent circuit diagram (see
Table 3.3) are
ukHVLV
ukMVLV
ukHVMV
(2.19a)
XTHV = 0.5 ∗
+
−
SrTHVMV
SrTHVLV
SrTMVLV
ukMVLV
ukHVMV
ukHVLV
XTMV = 0.5 ∗
+
−
(2.19b)
SrTMVLV
SrTHVMV
SrTHVLV
ukHVLV
ukMVLV
ukHVMV
(2.19c)
XTLV = 0.5 ∗
+
−
SrTHVLV
SrTMVLV
SrTHVMV
22
Short-circuit currents
(a)
ü I
0
3
1U I0
3I0
2U
1V I0
2V
1W I0
G
1
U0
2W
0
ü:1
(b)
1U
2U
1V
3I0 = 0
2V
1W
2W
Figure 2.9
G
1
Measuring of zero-sequence impedance of a two-winding transformer
( YNd). Diagram indicates winding arrangement of the transformer:
(a) measuring at star-connected winding and (b) measuring at
delta-connected winding
1
(a)
3
2
1
(b)
G
3~
3
2
I11
U11
(c)
1
3
2
(d)
1
3
2
G
G
3~
Figure 2.10
I11
I12
U11
U12
3~
Measurement of positive-sequence impedance of a three-winding transformer ( YNyn + d). Diagram indicates winding arrangement of the
transformer
Theoretical background
(a)
1
OS
3
2
US
MS
(b)
1
3
23
2
G
1~ U01
3I 01
(c)
1
G
1~
3I01
Figure 2.11
U01
3
2
(d)
1
3
2
U02
G
1~
3I02
Measurement of zero-sequence impedance of a three-winding transformer ( YNyn + d). Diagram indicates winding arrangement of the
transformer; for explanations see text
The measurement in the zero-sequence component is carried out in a similar
way as outlined in Figure 2.11.
4. Feeding with single-phase a.c. system at winding 1 (HV-side)
Short-circuit at winding 2 (MV-side)
Open-circuit at winding 3 (LV-side)
Measurement of uk012 , respectively, uk0HVMV
5. Feeding with single-phase a.c. system at winding 1 (HV-side)
Short-circuit at winding 3 (LV-side)
Open-circuit at winding 2 (MV-side)
Measurement of uk013 , respectively, uk0HVLV
6. Feeding with single-phase a.c. system at winding 2 (MV-side)
Short-circuit at winding 3 (LV-side)
Open-circuit at winding 1 (HV-side)
Measurement of uk023 , respectively, uk0MVLV
The impedances (reactances) as per the equivalent circuit diagram (see
Table 3.3) are
u0kHVLV
u0kMVLV
u0kHVMV
X0THV = 0.5 ∗
(2.20a)
+
−
SrTHVMV
SrTHVLV
SrTMVLV
u0kMVLV
u0kHVMV
u0kHVLV
X0TMV = 0.5 ∗
(2.20b)
+
−
SrTMVLV
SrTHVMV
SrTHVLV
u0kHVLV
u0kMVLV
u0kHVMV
X0TLV = 0.5 ∗
(2.20c)
+
−
SrTHVLV
SrTMVLV
SrTHVMV
24
Short-circuit currents
Any impedance in the neutral of the transformer has no effect on the impedance in
the positive-sequence component, as the three phase current are summing up to zero
at neutral point, and no current flows through the neutral impedance. In the zerosequence component the neutral impedance will appear with three-times its value in
the RYB-system, as the current through the neutral is three-times the phase current
during the measurement of zero-sequence impedance.
2.4 Equivalent circuit diagram for short-circuits
The system of symmetrical components can be used for the analysis of symmetrical
and asymmetrical operation of power systems. Faults in general and short-circuit
currents in particular are the most severe operating conditions in power systems.
Each of the different faults, e.g., single-phase-to-ground, three-phase, etc., can be
represented by an equivalent circuit diagram in the RYB-system and by this in the
012-system (system of symmetrical components) as well. The calculation of shortcircuits in the system of symmetrical components is generally carried out as per
Figure 2.12.
1. Draw the equivalent circuit diagram in RYB-components (RYB-system).
2. Draw the short-circuit location at the connection of the RYB-system, the shortcircuit should be assumed symmetrical to phase R.
3. Definition of fault equations in RYB-components, equations should be given
preference indicating which voltages and/or currents are Zero or are equal to
each other.
4. Transformation of fault conditions with the matrices T and T−1 into the
012-system (system of symmetrical components). Rearrange the transformed
fault equations in such a way that voltages and/or currents are Zero or are equal
to each other.
5. Draw the equivalent circuit diagrams in the system of symmetrical components.
6. Draw connection lines between the three components to realise the fault
conditions.
7. Calculation of currents and voltages in the system of symmetrical components.
8. Transformation of current and voltages into the RYB-system using transformation
matrix T−1 .
The eight steps as defined above are explained in case of a single-phase short-circuit
in a three-phase a.c. system. Figure 2.13 indicates the equivalent circuit diagram in
the RYB-system (item 1 above) as well as the type of short-circuit at the short-circuit
location (item 2 above).
Any fault in the three-phase a.c. system has to be described by three independent conditions for the voltages of currents of combinations of both. In case of the
single-phase short-circuit, the fault equations in the RYB-system (item 3 above) are
U R = 0;
I Y = 0;
IB = 0
(2.21)
Theoretical background
Equivalent circuit
diagram
RYB
(1) + (2)
25
Currents and
voltages
RYB
(8)
T –1
Fault equations
RYB
Currents and
voltages
012
(3)
(7)
T –1
T
Fault equations
012
Equivalent circuit
diagram
012
(5) + (6)
(4)
Figure 2.12
General scheme for the calculation of short-circuit currents in threephase a.c. systems using the system of symmetrical components. For
explanations see text
Z0; Z1; Z2
B
Y
R
IR
UY; UB
Figure 2.13
Equivalent circuit diagram of a single-phase short-circuit in
RYB-system
26
Short-circuit currents
The transformation into the system of symmetrical components (item 4 above) is
carried out using the transformation matrices by Equations (2.10) and (2.11). The
fault equations for the voltages in the system of symmetrical components are
UR = 0 = U0 + U1 + U2
(2.22a)
and for the currents
I0 = I1
(2.22b)
I0 = I2
(2.22c)
The fault conditions as per Equations (2.22) can only be realised by a series connection of the positive-, negative- and zero-sequence component. The equivalent
circuit diagram (item 5 above) in the system of symmetrical components is outlined
in Figure 2.14 as well as the connection of the individual components to realise the
fault conditions (item 6 above).
The positive-, negative- and zero-sequence component are represented by the
impedances Z 1 ; Z 2 ; Z 0 . The currents and voltages of the system of symmetrical
I1
Z1
U1
E1
01
I2
Z2
U2
02
I0
Z0
U0
00
Figure 2.14
Equivalent circuit diagram in the system of symmetrical components
for a single-phase short-circuit
Theoretical background
27
components (item 7 above) are then calculated as
I0 = I1 = I2 =
U 0 = −Z 0 ∗
E1
Z0 + Z1 + Z2
E1
Z0 + Z1 + Z2
U 1 = (Z 0 + Z 2 ) ∗
U 2 = −Z 2 ∗
E1
Z0 + Z1 + Z2
E1
Z0 + Z1 + Z2
(2.23a)
(2.23b)
(2.23c)
(2.23d)
The currents and voltages of the RYB-components (item 8) are calculated using the
transformation matrix, Equation (2.11) and the voltages as per Equations (2.24) are
obtained
U Y = E1 ∗
Z 0 ∗ (a 2 − 1) + Z 2 ∗ (a 2 − a)
Z0 + Z1 + Z2
(2.24a)
U B = E1 ∗
Z 0 ∗ (a − 1) + Z 2 ∗ (a − a 2 )
Z0 + Z1 + Z2
(2.24b)
IR =
3 ∗ E1
Z0 + Z1 + Z2
(2.24c)
E1
3
∗
Z1 2 + k
(2.25b)
In the case of Z 1 = Z 2 and |Z 0 /Z 1 | = k the voltages and the current can be
expressed by
√
√
k2 + k + 1
|U Y | = |U B | = 3E1
(2.25a)
2+k
Ik1 =
All other faults in three-phase a.c. systems can be analysed in the same manner by
the system of symmetrical components. Table 2.1 represents the equivalent circuit
diagrams of all short-circuits which can occur in power systems (single fault location
only) and the equations to describe the fault both in the RYB-system and in the system
of symmetrical components.
2.5 Series and parallel connection
Power systems include numerous equipment, such as lines, transformers, reactors
and generators which are connected in series and/or in parallel to other equipment
according to their location in the system’s topology. The related total impedance at
the short-circuit location has to be obtained by mathematical procedures, including
Table 2.1
Equivalent circuit diagrams and equations to represent short-circuits (single fault location) in three-phase a.c. systems in
the RYB-system and in the 012-system
RYB-system
012-system
Equivalent
circuit
diagram
Fault condition
Voltages
Three-phase
B
Y
R
I
E
U
Fault condition
Currents
Voltages
Currents
Without earth connection
UR = UY
IR + IY + IB = 0
Without earth connection
U1 = 0
I0 = 0
UY = UB
U2 = 0
With earth connection
UR = 0
With earth connection
U1 = 0
UY = 0
U2 = 0
UB = 0
U0 = 0
Equivalent
circuit
diagram
I1
“1”
U1
01
I2
“2”
U2
02
I0
“0”
00
U0
Double-phase
Without earth connection
UY = UB
IR = 0
B
Without earth connection
U1 = U2
I 1 = −I 2
I0 = 0
I Y = −I B
Y
R
I
U
E
With earth connection
IR = 0
UY = 0
UB = 0
I1
With earth connection
U1 = U2
“1”
U1
01
I1 + I2 + I0 = 0
I2
U1 = U0
U2
“2”
02
I0
“0”
U0
00
Line-to-ground
UR = 0
B
Y
IY = 0
IB = 0
U1 + U2 + U0 = 0
I1 = I2
I1 = U0
I1
R
“1”
I
E
U1
01
U
I2
U2
“2”
02
I0
“0”
00
U0
30
Short-circuit currents
star-delta- and delta-star-transformation. The equations to calculate total impedance
within a given system topology are outlined in Figure 2.15.
2.6 Definitions and terms
A clear usage of terms defined in standards and norms is essential in all areas of
technique. The knowledge of IEC-documents, national standards and norms therefore
Diagram
Impedance–admittance
Current–voltages
2
Z = Z1 + Z2
I = I 1 = I2
2
Y =
Y 1Y 2
Y1 + Y2
U = U1 + U2
2
Z=
Z1 Z2
Z1 + Z2
I = I 1 + I2
2
Y = Y1 + Y2
Serial connection
1
I1
I2
Z1
Z2
1
Z = 1/Y
Parallel connection
I1 Z = 1/Y1
1
I
1
I2 Z = 1/Y
2
2
I
Z = 1/Y
U = U1 = U2
Star-delta-transformation
1
I1
Z1 = 1/Y1
0
Z3 = 1/Y3
3
Z 12 =
Z1 Z2 + Z1 Z3 + Z2 Z3
Z3
Z 13 =
Z1 Z2 + Z1 Z3 + Z2 Z3
Z2
Z 23 =
Z1 Z2 + Z1 Z3 + Z2 Z3
Z1
Y 12 =
Y 1Y 2
Y1 + Y2 + Y3
I Z − I 2 Z2
I 12 = 1 1
Z 12
Z2 = 1/Y2
I2 2
I3
1
I13
I12
I 1 + I 2 + I3 = 0
Z13= 1/Y13
Z12= 1/Y12
Y 13 =
Y 1Y 3
Y1 + Y2 + Y3
I Z − I 3 Z3
I 13 = 1 1
Z 13
I31
I21
Y 23 =
Y 2Y 3
Y1 + Y2 + Y3
I Z − I 3 Z3
I 23 = 2 2
Z 23
3
I32
I23
Z23 = 1/Y23
Impedance: Z = R + jX
Figure 2.15
2
Equations for impedance analysis in power systems [30,31]
Theoretical background
Diagram
Impedance–admittance
31
Current–voltages
Delta-star-transformation
1
I13
I12
Z13
Z12
I31
3
I21
I32
I23
Z23
2
Z1 =
Z 12 Z 13
Z 12 + Z 23 + Z 13
I 1 = I 12 + I 13
Z2 =
Z 12 Z 23
Z 12 + Z 23 + Z 13
I 2 = I 21 + I 23
Z3 =
Z 13 Z 23
Z 12 + Z 23 + Z 13
I 3 = I 31 + I 32
1
I1
Z1
Z3
3
0
Z2
I2 2
I3
Admittance: Y = G + jB
Figure 2.15
Continued
is absolutely necessary. Some definitions of terms as related to short-circuit currents
are based on the German standard DIN 40110 and IEC 60050 as stated below. Further
reference is made to the IEC-documents as per Tables 1.1 and 1.2.
Nominal value
Example
A suitable rounded value of a physical quantity to define or
identify an element, a group of elements or an installation.
The nominal value 110 kV defines a voltage level for an
electrical power system. Actual voltages in the system are
different from the nominal voltage 110 kV.
Limiting value
Example
A defined minimal or maximal value of a physical quantity.
The minimal value of a current setting has to be defined
in order to guarantee the suitable operation of a protection
or control device.
Rated value
The value of a physical unit for operating conditions as
defined for the element, group of elements or installation
by the manufacturer.
The rated apparent power of a transformer should not be
exceeded at the defined operating conditions, in order to
protect the transformer from damage by overheating.
Example
32
Short-circuit currents
Rated data
Example
Summary of rated data and operating conditions.
The definition of the rated current of a cable is not sufficient,
as the thermal constraints are fixed besides others by ambient
temperature, thermal resistance of the soil, duration of current
loading and pre-load conditions.
2.7 Ohm-system, p.u.-system and %/MVA-system
2.7.1 General
It is necessary to calculate the values of the equipment of electrical supply systems
in order, for instance, to examine the behaviour of the supply system during normal
operation and in the state of disturbed operation. In this connection, equipment such
as generators, transformers, lines, motors and capacitors are of interest. Simulation
of consumers is only necessary in special cases. It may also be possible to determine
the equipment data from name plate rating or tabulated data. Various systems of units
are available for calculation.
2.7.1.1 Physical quantities
To describe the steady-state conditions of equipment and of the system, four basic
unit quantities are required, i.e., voltage U , current I , impedance Z and power S
with the units Volt, Ampere, Ohm and Watt. Other units have to be converted into the
ISO-standard unit system [32]. The units are linked to each other by Ohm’s law and
the power equation.
U
I
(2.26a)
S = U ∗ I∗
(2.26b)
Z=
where U is the voltage across the impedance Z and, I the current through the
impedance (I ∗ is the conjugate-complex value of I ).
In the case of a three-phase system the apparent power S is calculated as per
Equation (2.26c) with the voltage U being the phase-to-phase voltage, e.g., the rated
voltage of an equipment.
√
S = 3 ∗ U ∗ I∗
(2.26c)
If physical quantities are taken to be measurable properties of physical objects, occurrences and states from which useful sums and differences can be formed, the following
then applies:
Physical quantity = numerical value × unit
Theoretical background
33
2.7.1.2 Relative quantities
On the contrary, the unit of a relative quantity is One or 1 p.u. by definition, i.e.,
Relative quantity = quantity/reference quantity
Because the four quantities voltage, current, impedance and power required for system
calculations are linked to each other, two reference quantities only are required to
specify a relative system of units. Voltage and power are usually chosen for this
purpose. This system is called the per-unit system. As reference voltage either the
phase-to-phase or the phase-to-earth voltage can be chosen. If the power of 100 MVA
is selected as reference quantity, the system is called the p.u.-system on 100 MVAbase. Table 2.2 gives the definitions in the p.u.-system. It should be observed that
the phase-to-phase voltage is chosen as reference voltage. In case the phase-to-earth
voltage is selected as reference voltage, the current i has to be calculated as per [28]
as indicated by (∗1) in Table 2.2.
2.7.1.3 Semirelative quantities
In the semirelative system of units only one quantity is freely chosen as the reference
quantity. If the voltage is chosen, the %/MVA system is obtained, which is outstandingly suitable for network calculations because the values of the equipment
Table 2.2
Definitions of quantities in physical, relative and semirelative units
Ohm-system
Physical units
%/MVA-system
Semirelative units
p.u.-system
Relative units
No reference
quantity
One reference quantity
Two reference quantities
Voltage U
u=
Current I
i = I ∗ UB = {I } ∗ {UB }
{U }
U
∗ 100%
=
UB
{UB }
∗ MVA
Impedance Z
U
{U }
∗1
=
UB
{UB }
i = I ∗ UB /SB = {I } ∗ {UB }/{SB } ∗ 1
(∗1) remark see text
i = I ∗ UB /SB = {I }
√
∗{ 3UB }/{SB } ∗ 1
z = Z/UB2 = {Z} ∗ 100/{UB2 }
z = Z ∗ SB /UB2 = {Z} ∗ {SB }/UB2 ∗ 1
s = S = {S} ∗ 100% ∗ MVA
s = S/SB = {S}/{SB } ∗ 1
∗ %/MVA
Power S
applies also
to P and Q
u =
34
Short-circuit currents
can be calculated very easily. Table 2.2 gives the definitions for the %/MVA-system.
The reference voltage UB in the %/MVA-system should be equal to the rated voltage of
equipment Ur or to the nominal system voltage Un , i.e., it should be a phase-to-phase
voltage.
The impedances or reactance of electrical equipment are determined from the
data of the respective rating plate (name plate) or from geometrical dimensions. The
reactance, resistance or impedance should generally be calculated based on the nominal apparent power or on the nominal voltage of the system in which the equipment
is used. Conversion between the different unit systems is made using the data in
Table 2.3.
2.7.2 Correction factor using %/MVA- or p.u.-system
In case the nominal voltage of the system is unequal to the rated voltages of the equipment connected to this system (most often the case with transformers) a correction
factor for the impedances must be applied [33]. The derivation of the correction factor
is explained based on Figure 2.16.
Q
ZB
Figure 2.16
Ti
T2
T1
Un
Equivalent circuit diagram of a power system with different voltage
levels
The impedance correction factor KB for equipment B is calculated by
2
UrT1E UrT2E UrT3E
UrB 2
∗
∗
∗ ··· ∗
KB =
UrT1A UrT2A UrT3A
Un
(2.27)
where UrT.. are the rated voltages of the transformers at side A or E, UrB the rated
voltage of the equipment B and, Un the nominal system voltage at short-circuit
location.
The impedance correction factor while using the %/MVA- or the p.u.-system must
be applied for any equipment except power station units for which special correction
factors (see Table 3.6) are valid.
2.8 Examples
2.8.1 Vector diagram and system of symmetrical components
The voltages U R , U Y and U B are measured in a power system with nominal voltage
Un = 10 kV.
0
U R = 6.64 kV ∗ ej 0 ;
0
U Y = 6.64 kV ∗ ej 250 ;
U B = 6.64 kV ∗ ej 110
0
Table 2.3
Conversion of quantities between %/MVA-system, -system and p.u.-system (100 MVA-base)
%/MVA-system
→ -system
-system
→ %/MVA-system
-system
→ p.u.-system (100 MVA)
%/MVA-system
→ p.u.-system (100 MVA)
U
u
1
UB
=
∗
∗
kV
% 100 kV
u
U
1
=
∗ 100 ∗
%
kV
UB /kV
u
U
1
=
∗
p.u.
kV UB /kV
u
u
1
=
∗
p.u.
% 100
i
1
I
=
∗
kA
MVA UB /kV
i
I
UB
=
∗
MVA
kA kV
i
I
UB
1
=
∗
∗
p.u.
kA kV SB /MVA
i
i
=
p.u.
MVA
Z
100
z
=
∗
%/MVA
(UB /kV)2
Z
SB
1
z
=
∗
∗
p.u.
MVA (UB /kV)2
z
1
z
=
∗
p.u.
%/MVA SB /MVA
s
S
=
∗ 100
%MVA
MVA
s
S
1
=
∗
p.u.
MVA SB /MVA
s
s
0.01
=
∗
p.u.
%MVA SB /MVA
z
1
Z
=
∗
∗
%/MVA 100
s
1
S
=
∗
MVA
%MVA 100
UB 2
kV
36
Short-circuit currents
The voltages in the system of symmetrical components U 0 , U 1 and U 2 have to be
calculated by using the transformation equations. The individual voltages in units
of kV are obtained by using Equation (2.10). The voltage in the zero-sequence
component is
U0 =
1
3
∗ (U R + U Y + U B )
U0 =
1
3
∗ (6.64 ∗ ej 0 + 6.64 ∗ ej 250 + 6.64 ∗ ej 110 )
0
0
0
U 0 = 13 [(6.64 + 6.64 ∗ cos 2500 + 6.64 ∗ cos 1100 )
+ j (6.64 ∗ sin 00 + 6.64 ∗ sin 2500 + 6.64 ∗ sin 1100 )]
U 0 = 13 [(6.64 − 6.64 ∗ 0.342 − 6.64 ∗ 0.342)
+ j (6.64 ∗ 0 − 6.64 ∗ 0.94 + 6.64 ∗ 0.94)]
U 0 = (0.699 + j 0.0)kV
The voltage in the positive-sequence component is
U1 =
1
3
∗ (U R + a ∗ U Y + a 2 ∗ U B )
U1 =
1
3
∗ (6.64 ∗ ej 0 + ej 120 ∗ 6.64 ∗ ej 250 + ej 240 ∗ 6.64 ∗ ej 110 )
U1 =
1
3
∗ (6.64 ∗ ej 0 + 6.64 ∗ ej 370 + 6.64 ∗ ej 350 )
0
0
0
0
0
0
0
0
U 1 = 13 [(6.64 + 6.64 ∗ cos 3700 + 6.64 ∗ cos 3500 )
+ j (6.64 ∗ sin 00 + 6.64 ∗ sin 3700 + 6.64 ∗ sin 3500 )]
U 1 = 13 [(6.64 + 6.64 ∗ 0.985 + 6.64 ∗ 0.985)
+ j (6.64 ∗ 0 + 6.64 ∗ 0.174 − 6.64 ∗ 0.174)]
U 1 = (6.57 + j 0.0) kV
The voltage in the negative-sequence component is
U2 =
1
3
∗ (U R + a 2 ∗ U Y + a ∗ U B )
U2 =
1
3
∗ (6.64 ∗ ej 0 + ej 240 ∗ 6.64 ∗ ej 250 + ej 120 ∗ 6.64 ∗ ej 110 )
0
0
0
0
0
Theoretical background
U2 =
1
3
0
0
37
0
∗ (6.64 ∗ ej 0 + 6.64 ∗ ej 490 + 6.64 ∗ ej 230 )
U 2 = 13 [(6.64 + 6.64 ∗ cos 4900 + 6.64 ∗ cos 2300 )
+ j (6.64 ∗ sin 00 + 6.64 ∗ sin 4900 + 6.64 ∗ sin 2300 )]
U 2 = 13 [(6.64 − 6.64 ∗ 0.643 − 6.64 ∗ 0.643)
+ j (6.64 ∗ 0 + 6.64 ∗ 0.766 − 6.64 ∗ 0.766)]
U 2 = (−0.633 + j 0.0) kV
The voltages in the system of symmetrical components can be obtained by graphical
construction as well as outlined in Figure 2.17. The voltages are assumed to have the
same value and phase displacement as mentioned above.
2.8.2 Calculation of impedances of a three-winding transformer
in %/MVA
The impedances of a three-winding transformer are calculated in units of %/MVA
by using the Equations (2.19). The data as mentioned below are taken from the
name-plate:
Rated voltages
Rated apparent power
Rated impedance voltage
UrHV /UrMV /UrTLV = 110 kV/30 kV/10 kV
SrHV = 30 MVA; SrMV = 20 MVA; SrLV = 10 MVA
ukrHVMV = 10%; ukrHVLV = 4.5%;
ukrMVLV = 10.2%
It should be noted that the rated apparent power Sr as per above indicates the
apparent power of the individual winding. The maximal permissible apparent
power to be transferred between the windings involved for measurement of the
impedance voltage had to be taken for the calculation of impedances, i.e., SrHVMV =
MAX{SrHV ; SrMV } = 20 MVA. The impedance of the high-voltage winding is
calculated by
ukHVLV
ukMVLV
ukHVMV
+
−
XTHV = 0.5 ∗
SrTHVMV
SrTHVLV
SrTMVLV
XTHV = 0.5 ∗
4.5%
10.2%
10%
+
−
20 MVA 10 MVA 10 MVA
XTHV = −0.035
%
MVA
38
Short-circuit currents
The impedance of the medium-voltage winding is calculated by
ukHVMV
ukHVLV
ukMVLV
+
−
XTMV = 0.5 ∗
SrTMVLV
SrTHVMV
SrTHVLV
XTMV = 0.5 ∗
XTMV = 0.535
10.2%
10%
4.5%
+
−
10 MVA 20 MVA 10 MVA
%
MVA
+ Re
(a)
UR
+ Im
UB
UY
+ Re
(b)
UY
UR
UB
3 · U0
+ Im
UB
Figure 2.17
UY
Graphical construction of voltages in the system of symmetrical components: (a) vector diagram RYB, (b) vector diagram of voltage in
the zero-sequence component, (c) vector diagram of voltage in the
positive-sequence component and (d) vector diagram of voltage in the
negative-sequence component
Theoretical background
+ Re
(c)
1
3
a2 ·UB
1
3
a ·UY
1
3
+ Im
UR
U1
·UR
UB
UY
+ Re
(d)
a2 ·UY
UR
a ·UB
+ Im
UB
Figure 2.17
3 · U2
UY
Continued
The impedance of the low-voltage winding is calculated by
ukHVLV
ukMVLV
ukHVMV
XTLV = 0.5 ∗
+
−
SrTHVLV
SrTMVLV
SrTHVMV
XTLV = 0.5 ∗
XTLV = 0.485
10.2%
10%
4.5%
+
−
10 MVA 10 MVA 20 MVA
%
MVA
39
40
Short-circuit currents
It should be noted that the negative value of XTHV shall not be interpreted to have a
physical meaning. The model (see Table 3.4) describes the performance of the threewinding transformer at the connection point HV, MV and LV only. The impedance
between any two connection points appears physically correct with a positive value.
2.8.3 Conversion of impedances (; %/MVA; p.u.)
Impedance values shall be converted from one unit system into the other unit
systems (%/MVA-system and p.u.-system on 100 MVA-base) by using equations
as per Tables 2.2 and 2.3.
•
Overhead line Z = (6.2 + j 53.4) at Un = 380 kV
The impedance in the %/MVA-system is calculated by
z = (6.2 + j 53.4) ∗
%
100%
= (0.0043 + 0.037)
(380 kV)2
MVA
The impedance in the p.u.-system on 100 MVA-base is calculated by
z = (6.2 + j 53.4) ∗
100 MVA
= (0.0043 + 0.037) p.u.
(380 kV)2
• Apparent power of a transformer S = (1.22 + j 300) MVA
The apparent power in the %/MVA-system is calculated by
s = (1.22 + j 300) MVA ∗ 100% = (122 + j 30000)%/MVA
The apparent power in the p.u.-system on 100 MVA-base is calculated by
s =
(1.22 + j 300) MVA
= (0.0122 + j 3) p.u.
100 MVA
• Voltage drop U = 12.5 kV in a 132 kV-system
The voltage drop in the %/MVA-system is calculated by
100%
= 9.47%
132 kV
u = 12.5 kV ∗
The voltage drop in the p.u.-system on 100 MVA-base is calculated by
u = 12.5 kV ∗
1
= 0.0947 p.u.
132 kV
• Rated current of a 320 MVA-transformer on 115 kV-side Ir = 1.607 kA
The current in the %/MVA-system is calculated by
ir = 1.607 kA ∗ 115 kV = 184.81 MVA
The current in the p.u.-system on 100 MVA-base is calculated by
ir = 1.607 kA ∗
115 kV
= 0.185 p.u.
100 MVA
Theoretical background
41
2.8.4 Impedances in %/MVA-system based on measurement
A simplified example for the application of the system of symmetrical components in
representing electrical equipment is outlined in Figure 2.18 by using a short element
of an overhead line in accordance to Figure 2.4. The impedance matrix (RYB-system)
is thus obtained
⎤
⎡
jXM
R + jX
jXM
⎥
⎢
R + jX
jXM ⎦
ZRYB = ⎣ jXM
(2.28)
jXM
jXM
R + jX
where R is the resistance of the line, X the reactance of the line and XM the mutual
reactance between the individual phases.
IRYB
R
jX
R
jX
R
jX
jXM
B
Y
R
URYB
Figure 2.18
Simplified equivalent circuit diagram in RYB-components
Voltages at the individual components R, Y and B are obtained from the current
and the impedance by Ohm’s law.
The voltage of phase R (component R) is given as follows:
U R = (R + jX) ∗ I R + jXM ∗ I Y + jXM ∗ I B
(2.29)
and depends on the currents of all three phases (components) I R , I Y and I B . Using
the system of symmetrical components, see Equation (2.18), the impedance matrix
is simplified in such a way that the three components are decoupled from each other
in the following manner:
Z012
⎤
0
0
R + j (X + 2XM )
⎦
0
R + j (X − XM )
0
=⎣
0
0
R + j (X − XM )
⎡
(2.30)
The equivalent circuit diagram in the system of symmetrical components is outlined
in Figure 2.19.
42
Short-circuit currents
I1
R
j(X – XM)
U1
01
I2
R
j(X – XM)
U2
02
I0
R
j(X + 2XM)
U0
00
Figure 2.19
Equivalent circuit diagram in the system of symmetrical components
The voltage U 1 of the component 1 (positive-sequence component) is given as
below:
U 1 = (R + j (X − XM )) ∗ I 1
(2.31a)
and depends only on the current I 1 of the same component. Impedance of the components 1 and 2 (positive- and negative-sequence components) is identical. Voltage
of the negative-sequence system therefore is given by
U 2 = (R + j (X − XM )) ∗ I 2
(2.31b)
The impedance of the component 0 (zero-sequence component) is different from the
others and the voltage in the zero-sequence component is given by
U 0 = (R + j (X + 2XM )) ∗ I 0
(2.31c)
2.8.5 Representation of a line in the RYB-system and in the system of
symmetrical components
Overhead lines are represented in the RYB-system by lumped elements, indicating
the inductive and capacitive coupling as well as the impedance of earth return path.
The equivalent circuit diagram of an infinitesimal small section of a line is outlined
in Figure 2.20.
The representation in the system of symmetrical components (012-system) can
either be done by transformation of the individual elements using the transformation
Theoretical background
X
R
~
XM
CL
CE
B
EB
Y
~
~
EY
R
ER
Figure 2.20
43
XE
RE
E
Equivalent circuit diagram of an overhead line of infinitesimal length
with earth return in RYB-system
(a)
E1
R
X
–XM
3CL
CE
C1 = 3CL + CE
01
(b)
R
X
–XM
3CL
CE
02
(c)
R
2XM
X
C0 = CE
3XE
3RE
00
Figure 2.21
Equivalent circuit diagram of an overhead line of infinitesimal length
with earth return in 012-system. (a) Positive-sequence component,
(b) negative-sequence component and (c) zero-sequence component
matrix as per Equations (2.18) or by measurement according to the procedure as
explained in Section 2.3.4. The line is represented in the system of symmetrical
components as outlined in Figure 2.21. The reactance XE and resistance RE of the
earth return path appear with its threefold value in the zero-sequence component, as the
current through the neutral is three-times the phase current during the measurement of
zero-sequence impedance. The capacitances between the phases CL are represented
with three times its capacitance in the positive- and negative-sequence component,
as they appear in the RYB-system as delta-connected and have to be transferred to
44
Short-circuit currents
a star-connection for representation in the 012-system. The capacitance line-to-earth
CE is represented in the three components with the same value.
Please observe that the mutual reactance XM appears with different values in
the positive- (negative-) and zero-sequence component as outlined in Section 2.3.3.
The negative value of the reactance in the positive-, respectively, negative-sequence
system has no physical meaning, as the total of the mutual reactance XM and the
self-reactance X still remains positive and therefore physically correct.
The equivalent voltage source E 1 , which is the internal voltage of a generator, is
only present in the positive-sequence component as the symmetrical operation of the
system is represented by this component.
Chapter 3
Calculation of impedance of
electrical equipment
3.1 General
In general, equipment in power systems are represented by equivalent circuits, which
are designed for the individual tasks of power system analysis, e.g., for the calculation
of no-load current and the no-load reactive power of a transformer, the no-load equivalent circuit is sufficient. Regarding the calculation of short-circuits, voltage drops
and load characteristic a different equivalent circuit is required. The individual components of the equivalent circuits are resistance, inductive and capacitive reactance
(reactor and capacitor), voltage source and ideal transformer. Voltage and currents of
the individual components and of the equivalent circuit are interlaced by Ohm’s law,
which is valid for the three-phase system (RYB-system) as well as for the system
of symmetrical components (012-system). A detailed deduction of the mathematical
methods and equations is not given within the context of this section of the book, but
only the final equations are quoted. For further reading, reference is made to [1,13].
3.2 Equipment in a.c. systems
3.2.1 General
Impedances of equipment are calculated based on name plate data, from
manufacturer’s data or from geometrical arrangement. For the calculation of
impedances of generators, power plants, step-up and step-down transformers, correction factors are necessary. The calculation equations as per Tables 3.1–3.11 are given
in the Ohm-system only. For conversion to %/MVA-system, respectively, p.u.-system
Tables 2.2 and 2.3 can be used. If not marked by index ‘1’, e.g., Z 1Q , in a different
way, impedances are given for the positive-sequence component. The impedance in
the zero-sequence system is marked with index ‘0’, e.g., Z 0Q .
3.2.2 Impedance calculation
Table 3.1
Impedance of system feeder, equivalent circuit diagram, calculation equations and remarks
Figure
Symbol
RQ
Impedance
Remarks
Positive-sequence system
2
c ∗ UnQ
Z1Q =
SkQ
c
Ik1Q
Ik3Q
SkQ
UnQ
c ∗ UnQ
Z1Q = √
3 ∗ Ik3Q
XQ
X1Q =
1 + (R1Q /X1Q )2
Zero-sequence component
E
01
Equivalent circuit diagram
Z1Q
Ik3Q
Z0Q = 3 ∗ Z1Q ∗ − 2 ∗ Z1Q
Ik1Q
Voltage factor according to Table 4.1
Initial short-circuit current (single-phase short-circuit)
Initial short-circuit current (three-phase short-circuit)
Initial short-circuit power (three-phase short-circuit)
Nominal system voltage at connection point
Assumptions for resistance:
• RQ ≈ 0 if UnQ ≥ 35 kV; system with overhead transmission
lines
• RQ = 0.1XQ respectively XQ = 0.995ZQ
• The impedance in the zero-sequence component normally is
given as a ratio of the impedance in the positive-sequence
component. If the single-phase short-circuit current from the
system feeder is known, the zero-sequence impedance can be
calculated.
• Reference Item 3.2 of IEC 60909
Table 3.2
Impedance of two-winding transformer, equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
HV
LV
ZT =
RT =
Symbol
RT =
HV
RT
XT
LV⬘
01
Equivalent circuit diagram
Remarks
2
ukr ∗ UrT
∗ KT
100% ∗ SrT
2
uRr ∗ UrT
∗ KT
100% ∗ SrT
PkrT
2
3 ∗ IrT
∗ KT
XT = ZT2 − RT2
Correction factor KT :
UnQ
KT =
Ub max
cmax
∗
1 + xT (Ib max T /IrT ) sin ϕbT
Approximation:
KT = 0.95 ∗
cmax
1 + 0.6xT
ukr
UrT
SrT
Impedance voltage
Rated voltage (HV- or LV-side)
Rated apparent power
cmax
Ib max T
IrT
PkrT
uRr
Ub max
UnQ
ϕbT
Voltage factor according to Table 4.1 at LV-side
Maximal current prior to short-circuit
Rated current (HV- or LV-side)
Ohmique losses
Ohmique part of impedance voltage
Maximal voltage prior to short-circuit
Nominal system voltage
Phase angle of current prior to short-circuit
– Transformer with high rated power: XT ≈ ZT
– Correction factor for positive-, negative- and zero-sequence
component [35]
– Impedances as per this table are related to HV-side
– The impedance in the zero-sequence component can either be
given as a ratio of the impedance in the positive-sequence component (see Chapter 13) or can be calculated from the impedance
voltage and the losses in the zero-sequence component.
– The impedance in the zero-sequence component depends on the
handling of transformer neutral, see Table 3.6
– Reference Item 3.3.1 of IEC 60909
Table 3.3
Impedance of three-winding transformer, equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
LV
|Z THVMV | = ukrHVMV ∗
MV
HV
|Z THVLV | = ukrHVLV ∗
Symbol
|Z TLVMV | = ukrLVMV ∗
∗K
SrtHVMV
2
UrTHV
SrtHVLV
∗K
2
UrTHV
SrtHVMV
∗K
ZTMV = 0.5 ∗ (Z TLVMV + Z THVMV
−Z THVLV ) ∗ KTMV
ZTLV = 0.5 ∗ (Z THVLV + Z TLVMV
−Z THVMV ) ∗ KTLV
RTLV
XTLV
HV
MV⬘
XTHV
2
UrTHV
ZTHV = 0.5 ∗ (Z THVMV + Z THVLV
−Z TLVMV ) ∗ KTHV
LV⬘
RTHV
Remarks
XTMV
RTMV
01
Equivalent circuit diagram
2
uRr ∗ UrT
∗ KT
100% ∗ SrT
P
RT = krT2 ∗ KT∗∗
3 ∗ IrT
RT =
XT =
ZT2 − RT2
Voltage factor according to
Table 4.1
IrT
Rated current
PkrT Ohmique losses
SrT
Rated apparent power
UrT Rated voltage
uRr
Ohmique part of impedance voltage
ukr
Impedance voltage
– HV, MV, LV related to voltage levels
– Calculation of RT and XT similar to
ZT for HV, MV and LV
– Transformers with high rating: XT ≈
ZT
– Correction factor for positive, negative and zero-sequence component
[35]
– Impedances as per this table are
related to HV-side
– Correction factor K not defined herewith
– Correction factor KT∗∗ equal KTHV ;
KTMV ; KTLV
cmax
Correction factor KT :
cmax
KTHVMV = 0.95
1 + 0.6xTHVMV
cmax
KTHVLV = 0.95
1 + 0.6xTHVLV
cmax
KTMVLV = 0.95
1 + 0.6xTMVLV
– The impedance in the zero-sequence
component can either be given
as a ratio of the impedance in the
positive-sequence component (see
Chapter 13) or can be calculated from
the impedance voltage and the losses
in the zero-sequence component.
– The impedance in the zero-sequence
component depends on the handling
of transformer neutral, see Table 3.4
– Reference Item 3.3.2 of IEC 60909
50
Short-circuit currents
Table 3.4
Equivalent circuit diagram of two- and three-winding transformers in the
positive- and zero-sequence component
Type of transformer Equivalent diagram Equivalent diagram in system of symmetrical components
(any vector group)
in RYB-system
Positive-sequence component
Zero-sequence component
1
2
1
2
1
X1
1
X1
2
1
X01
1
X01
2
YNy
01
ZS
YNd
1
YNy + d
1
1
X1
2
ZS
2
3ZS
01
ZS
ZNy; ZNd
2
00
2
1
2
X01
3ZS
01
2
1 X1
ZS
3
X3
3
X2 2
1 3ZS
X01
3
2
1 X1
ZS1 ZS2
3
X3
3
1
ES
2
1 X1
2
00
3
X2 2
1 3ZS1 X01
X02
2
X03 3ZS2(U1/U2)2
00
01
3
X3
X02
X03
01
YNyn + d
00
or
3
1
00
3
X2 2
XL
01
1
ES
X01
X02
2
X03 3XL(U1/U2)2
00
3.3 Equipment in d.c. systems
3.3.1
General
For the calculation of short-circuit currents in d.c. systems, the parameters of equipment contributing to the short-circuit current, i.e., capacitor, battery, rectifier and
d.c. motor need to be known besides the parameter of conductors. The calculation
equations as per Tables 3.12 to 3.16 are given in the Ohm-system only. For conversion
to %/MVA system, p.u. system, respectively, Tables 2.2 and 2.3 can be used.
Tables 3.12 to 3.16 mention the term ‘common branch’. The common branch in
d.c. systems is the branch (conductor) leading parts of the short-circuit current from
several different sources (capacitor, battery, rectifier and d.c. motor) according to
IEC 61660-1.
Table 3.5
Impedance of synchronous generator, equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
GS
3~
X1G =
2
xd ∗ UrG
100% ∗ SrG
Remarks
∗ KG
Salient pole generators:
X2G = 0.5 ∗ (Xd + Xq ) ∗ KG
Symbol
RGf
Zero-sequence component
X0G ≈ (0.4–0.8)Xd ∗ KG
XG
Correction factor KG :
UnQ
cmax
KG =
∗
UrG (1 + pG ) 1 + xd ∗ sin ϕrG
EG
01
Equivalent circuit diagram
cmax
pG
SrG
UrG
xd
ϕrG
Voltage factor according to Table 4.1
Voltage control range: UrG = const ⇒ pG = 0
Rated apparent power
Rated voltage
Subtransient reactance
Phase angle between UrG and IrG
Fictitious resistance of stator RGf (for peak s.-c. current)
HV: RGf = 0.05 ∗ XG : SrG ≥ 100 MVA
RGf = 0.07 ∗ XG : SrG < 100 MVA
LV: RGf = 0.15 ∗ XG
– Correction factor for positive-, negative- and zero-sequence
component
– Synchronous motors identical to synchronous generators
– Calculation of decaying d.c. component with real resistance of
stator
– Impedance in the zero-sequence component depends on the type
of winding of the generator
– Reference Item 3.6 of IEC 60909
Table 3.6
Impedance of power-station unit, equivalent circuit diagram, calculation equations and remarks
Figure
G
Impedance
Remarks
2 +Z
Z KW = (Z G ∗ trT
THV ) ∗ KKWi
UnG Nominal system voltage
T
Correction factor KKWi with tap-changer:
GS
3~
KKWs =
trT
Symbol
(RGf + jXG)t 2rT
RT
(UrG (1 + pG ))2
U2
cmax
∗
∗ rTLV
− x | ∗ sin ϕ
2
1
+
|x
UrTHV
T
rG
d
Correction factor KKWi without tap-changer:
UnQ
KKWo =
UrG (1 + pG )
cmax
UrTLV
∗ (1 ± pT ) ∗
∗
UrTHV
1 + xd sin ϕrG
XT
EGitrT
01
Equivalent circuit diagram
2
UnQ
Generator:
pG Control range of voltage:
UrG = const ⇒ pG = 0
UrG Rated voltage
xd Subtransient reactance
ZG Impedance according to
Table 3.5
ϕrG Phase angle between UrG
and IrG
Unit transformer:
cmax Voltage factor according to
Table 4.1
pT
Permanent setting of winding
trT
Rated transformation ratio
UrTLV Rated voltage LV-side
UrTHV Rated voltage HV-side
xT
Reactance (equal impedance
voltage)
ZT
Impedance according to
Table 3.2 (two-winding transformer)
• Correction factor for positive-,
negative- and zero-sequence
component [37]
• Reference Item 3.7 of IEC 60909
Table 3.7
Impedance of overhead line (single-circuit), equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
=
R1L
Symbol
R⬘1L·
=
RLδ
X⬘1L·
01
Equivalent circuit diagram
Remarks
n
R1L
Geometric mean distance between
conductors dRY ; dYB ; dBR
Distances of conductors R, Y, B
Number of conductors per phase
Resistance per length (pos. seq.)
r
rt
qn
X1L
α
δ
ρ
ρE
μ0
μr
Radius of conductor
Radius of conductor arrangement per phase
Nominal cross-section of conductor
Reactance per length (pos. seq.)
Temperature coefficient α = 0.004 K −1
Equivalent depth of earth conductor
Resistivity in mm2 /m
Specific earth resistance
Permeability of vacuum
Relative permeability
D
ρ
n ∗ qn
(1 + α ∗ (δ − 20◦ C)) ∗ RL20
n
Single-circuitoverhead line:
= ω μ0 ln D + μr
X1L
2π
rB
4n
Zero-sequence component:
R
= 1L + 3ω μ0
R0L
n
8
δ
μr
μ0
3 ln 3
+
X0L = ω
2
2π
4n
rB ∗ D
with
D=
rB =
√
3
n
Reactance:
dRY ∗ dYB ∗ dBR
(n−1)
n ∗ r ∗ rt
1.85
δ= √
μ0 ∗ ω/ρE
–
–
–
Calculation from geometrical arrangement,
see Section 13.1.4
Specific earth resistance ρE = 30 m (swamp
soil)–3000 m (stone)
Reference Item 3.4 of IEC 60909
Note: Impedances of other arrangements of overhead lines needed for special technical problems are dealt with in Section 12.1 and Section 13.1.4. Impedances of
cables can be calculated from geometrical data only in a very time consuming manner. It is recommended to use manufacturer’s data. Tables and diagrams can be
found in [1,2,8,9].
Table 3.8
Impedance of short-circuit limiting reactor, equivalent circuit diagram, calculation equations
and remarks
Figure
Impedance
ZD =
ukr
Un
∗√
100%
3 ∗ IrD
√
SrD = 3 ∗ Un ∗ IrD
Symbol
ZD ≈ XD
Remarks
IrD
rated current
SrD
Un
ukr
Rated apparent power
Nominal system voltage
Rated voltage drop (impedance
voltage)
–
01
Equivalent circuit diagram
–
–
–
Impedances in positive-, negativeand zero-sequence component
identical in case of symmetrical
construction
RD ≈ 0
Impedance in the zero-sequence
component equal to the impedance
in the positive-sequence component
in case three single-phase reactors
are used
Reference Item 3.5 of IEC 60909
Table 3.9
Impedance of asynchronous motor, equivalent circuit diagram, calculation equations and
remarks
Figure
Impedance
ZM =
M
3~
XM =
Symbol
RM
SrM =
XM
Remarks
2
IrM UrM
∗
IanM SrM
ZM
1 + (RM /XM )2
PrM
ηrM ∗ cos ϕrM
IanM Locked rotor current
IrM
Rated current
PrM Rated active power
SrM Rated apparent power
ϕrM Phase angle at rated power
ηrM Rated power factor
MV:
RM = 0.1 ∗ XM with PrMp ≥ 1 MW
RM = 0.15 ∗ XM with PrMp < 1 MW
PrMp Rated active power per pole pair
LV:
RM = 0.42 ∗ XM including connection cable
E
01
Equivalent circuit diagram
–
–
Asynchronous motors are normally operated
with isolated neutrals, zero-sequence
impedance therefore can be neglected
Reference Item 3.8 of IEC 60909
Table 3.10
Impedance of static converter fed drive, equivalent circuit diagram, calculation equations and
remarks
Figure
Impedance
ZM =
2
IrM UrM
∗
IanM SrM
RM
= 0.1
XM
Symbol
RM
XM
Remarks
IanM
IrM
SrM
Locked rotor current
Rated current
Rated apparent power
UrM
Rated voltage
–
–
–
–
E
01
Equivalent circuit diagram
IanM /IrM = 3
Only for rectifiers, able to transfer energy for
deceleration during the duration of
short-circuit
Static converters for photovoltaic generators
or fuel cells contribute to short-circuit
currents only with their rated current
Reference Item 3.9 of IEC 60909
Table 3.11
Impedance of system load, equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
ZL
01
XL
01
RL
01
XC
01
Symbol, Equivalent circuit diagram
ZL =
2
UrL
SrL
RL =
2
UrL
PrL
U2
ωLL = rL
QrL
U2
1
= rL
ωCL
QrL
Remarks
PrL
QrL
SrL
UrL
Rated active power
Rated reactive power
Rated apparent power
Rated voltage
In case UrL is unknown, the nominal system voltage
Un is to be used
• Reference Item 3.10 of IEC 60909
•
3.3.2 Impedance calculation
Table 3.12
Impedance of a conductor, equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
RL =
ρ
qn
= (1 + α ∗ (δ − 20◦ C)) ∗ R
RLδ
L20
Symbol
Loop by single cables
R⬘L ·
L⬘ ·
R⬘J
L =
a
μ0
∗ 0.25 + ln
π
r
Loop by busbars
L =
Equivalent circuit diagram
a
μ0
∗ 1.5 + ln
π
d +b
Resistance of bolted joint:
Conductor
qn
d
Conductor joint
14 ∗ ρ ∗ d
RJ =
qn
Remarks
a
b
d
RJ
RL
RL20
RLδ
r
qn
α
δ
ρ
μ0
Distance of conductors
Height of conductor (busbar)
Thickness of conductor
Resistance of bolted joint
Specific resistance
Resistance at 20◦ C
Resistance at temperature δ
Radius of conductor
Cross-section of conductor
Temperature coefficient (for Al, Cu)
α = 0.004 K −1
Temperature in ◦ C
Resistivity in mm2 /m
Permeability
RJ for calculation of minimal s.-c. currents
only
– Calculation of L for simple arrangement
only
– Reference Item 2.3 of IEC 61660-1
–
Table 3.13
Impedance of capacitor, equivalent circuit diagram, calculation equations and remarks
Figure
RCBr ; LCBr
RCL
Remarks
RCBr = RC + RCL + RY
LCBr = LCL + LY
Conductor for connection:
RCL ; LCL Resistance;
inductance
Approximation:
C ≈ 1.2 ∗ Ca.c.
Symbol
RC
Impedance
LCL
RY
LY
Common branch:
RY ; LY Resistance;
inductance
Capacitor:
C
d.c.-capacitance
Ca.c. a.c.-capacitance at
100 Hz
RC
EC
Equivalent circuit diagram
–
Equivalent series d.c.
resistance. If unknown:
maximal a.c. resistance
Reference Item 2.6 of
IEC 61660-1
Table 3.14
Impedance of battery, equivalent circuit diagram, calculation equations and remarks
Figure
RBBr; LBBr
LB
RBL
LBL
EB
Equivalent circuit diagram
Remarks
RBBr = 0.9 ∗ RB + RBL + RY
LBBr = LB + LBL + LY
Conductor for connection:
RBL ; LBL Resistance;
inductance
Approximation:
RBun = 1.7 ∗ RB
LBZ = 0.2 µH
EBge = 1.05 ∗ UnB
EBun = 0.9 ∗ UnB
EBge = 1.05 ∗ UnB
Symbol
RB
Impedance
RY
LY
Common branch:
RY ; LY Resistance;
inductance
Battery:
EBge Voltage of charged
battery
EBun Voltage of discharged
battery
LBZ
Inductance of one cell
including connection
RB ; LB Resistance;
inductance of a charged
battery
RBun Resistance of
discharged battery
UnB
Nominal voltage of
battery
Lead-acid battery:
UnB = 2.0 V per cell
– Reference Item 2.5 of
IEC 61660-1
Table 3.15
Impedance of rectifier, equivalent circuit diagram, calculation equations and remarks
Figure
Impedance
Remarks
ac-side
RN = RQ + RP + RT + RD
Indices for ac-side:
D Commutating
reactor
N Total
Q System feeder
P Connection at
secondary side of
transformer
– ZQmin (RQ and XQ )
for calculation of max.
short-circuit current
– ZQmax (RQ and XQ )
for calculation of min.
short-circuit current
XN = XQ + XP + XT + XD
Symbol
RDBr ; LDBr
R Q ; XQ
RS
RQ ; XQ
Q
RP ; XP
RT ; XT
LS
RL
LL
RY
LY
RD ; XD
Equivalent circuit diagram
dc-side
RDBr = RS + RL + RY
LDBr = LS + LL + LY
Indices for dc-side:
L Power supply cable
S DC saturated
smoothing reactor
Coupling branch:
LS
saturated
inductance
RY ; LY Resistance;
inductance
–
Reference Item 2.4
of IEC 61660-1
Table 3.16
Impedance of d.c. motor with independent excitation, equivalent circuit diagram, calculation equations and remarks
Figure
M
Motor
Symbol
RMBr; LMBr; M
F
RF
LF
J
M
RM
n
EF
Equivalent circuit diagram
EM
LM
RML
LML
RY
LY
Impedance
Remarks
RMBr = RM + RML + RY
LMBr = LM + LML + LY
Conductor for
connection:
RML
Resistance
LML
Inductance
Time constants:
LMZw
τM =
Common branch:
RMZw
Resistance
R
2π ∗ n0 ∗ J ∗ RMBr ∗ IrM Y
LY
inductance
τMec =
Mr ∗ UrM
Motor:
LF
Rated current
IrM
τF =
RF
J
Moment of
inertia of whole
rotating part
Mr
Rated torque
n0
No-load speed
RM ; LM Resistance;
inductance of
armature circuit
including brushes
UrM
Rated voltage
τF
Field circuit
time constant
τM
Time constant
of armature circuit
up to s.-c. location
τMec
Mechanical
time constant
–
Reference Item 2.7
of IEC 61660-1
63
Calculation of impedance of electrical equipment
3.4 Examples for calculation
3.4.1 a.c. equipment
The impedance (resistance and reactance) of equipment in a.c. three-phase power
systems has to be calculated based on the data as below. Results are summarised in
Table 3.17.
Power system feeder Q:
Two-winding
transformer:
Three-winding
transformer:
= 3000 MVA; U
SkQ
nQ = 110 kV
SrT = 70 MVA; UrTHV /UrTLV = 115 kV/10.5 kV;
ukr = 12%; uRr = 0.5%
Synchronous machine:
SrG = 70 MVA; UrG = 10.5 kV; cos ϕrG = 0.9;
xd = 17%; pG = ±10%
SrG = 70 MVA; UrG = 10.5 kV; xd = 17%;
pG = ±10%
SrT = 70 MVA; UrTHV /UrTLV = 115 kV/10.5 kV;
ukr = 12%;
URr = 0.5%; without tap-changer
Power plant consisting
of synchronous machine
and two-winding
transformer:
Overhead transmission line:
s.-c. limiting reactor:
Asynchronous motor:
Rectifier:
System load:
Table 3.17
Equipment
System feeder
Two-winding
transformer
UrT = 110 kV/30 kV/10 kV;
SrT = 30 MVA/20 MVA/10 MVA
ukrHVMV = 10%; ukrHVLV = 4.5%; ukrMVLV = 10.2%
uRrHVMV = 0.5%; uRrHVLV = 0.6%; uRrMVLV = 0.65%
Al/St 240/40; r = 10.9 mm; Line length 10 km
Flat arrangement, distance between phase wires 4 m
ukr = 6%; IrD = 630 A; Un = 10 kV
PrM = 1.2 MW; UrM = 6 kV; cos ϕrM = 0.84;
ηrM = 0.93; Ian /IrM = 5.6; 2 pairs of poles
SrM = 4 MVA; UrM = 6.2 kV
SrL = 6 MVA; cos ϕL = 0.87; Un = 10 kV
Results of calculation of impedance in three-phase a.c. equipment
Z ()
4.437
R ()
0.441
X ()
4.414
22.76
0.945
22.74
22.19
0.921
22.17
Remark
Tab.
RQ /XQ not defined
XQ ≈ 0.995 × ZQ
3.1
Without correction factor
Impedance related to 110 kV
KT = 0.975
Impedance related to 110 kV
3.2
64
Short-circuit currents
Table 3.17
Continued
Equipment
Z ()
R ()
X ()
Remark
Three-winding
transformer
−3.24
62.89
58.65
1.31
1.67
6.08
−2.97
62.87
58.33
Impedance related to 110 kV
3.3
including correction factors
KT = 0.986–1.018–0.985
Values from top: HV ∗ MV ∗ LV
Synchronous
machine
0.268
0.238
0.019
0.017
0.267 Without correction factor
0.237 KG = 0.887; UnQ = 10 kV
Tables
3.5
KKWo = 0.891; UnQ = 110 kV
3.6
4.01
D = 5.04 m
3.7
0
0.55
XD ≈ ZD
3.8
4.14
0.41
4.12
SrM = 1.53 MVA
3.9
Rectifier
28.83
2.88
28.69
RM /XM = 0.1
3.10
System load
16.67
—
3.11
Power plant
65.14
3.22
54.77
Overhead
transmission line
4.19
1.23
s.-c. limiting
reactor
0.55
Asynchronous
machine
14.5
8.22
3.4.2 d.c. equipment
The impedance (resistance and reactance) of equipment in d.c. systems has to be
calculated based on the data as below. Results are summarised in Table 3.18.
Conductor Busbar arrangement, copper (120 × 10): qn = 1200 mm2 ;
with joint: Distance a = 50 mm; Length of line loop 30 m
Capacitor:
MKP dry-type, self-healing; C = 9000 µF; RC = 0.5 m
Connected to short-circuit location with conductor as above,
l = 20 m
Two bolted joints
Battery:
Sealed lead-acid-type; 108 cells, each:
150 Ah; UnB = 2.0 V; RB = 0.83 m; LB = 0.21 mH
Connected to short-circuit location with conductor as above,
l = 15 m
Two bolted joints
Calculation of impedance of electrical equipment
Rectifier:
d.c. motor
(independent
excitation):
Table 3.18
65
= 40 MVA; R /X = 0.25
AC-system: UnQ = 600 V; SkQ
Q
Q
Transformer: trT = 600 V/240 V; SrT = 400 kVA; ukrT = 3.5%;
PkrT = 4.2 kW
Rectifier: IrD = 1.2 kA; commutating reactor: LS = 0.31 µH;
RS = 0.91 m
Connected to short-circuit location with conductor as above,
l = 10 m
UrM = 225 V; PrM = 110 kW; IrM = 500 A; RM = 0.043 ;
LM = 0.41 mH
RF = 9.85 M; LF = 9.97 H
Connected to short-circuit location with conductor as above,
l = 10 m
Results of calculation of impedance of equipment in d.c. installations
(without common branch as per IEC 61660-1)
Equipment
R (m)
L
Others
Remarks
Tab.
Conductor
926
0.653 µH
—
3.12
—
—
Loop length
60 m
Resistance of
bolted joint
—
218 nH
—
218 nH
9000 µF
—
—
9000 µF
2.16
Capacitor
Battery
0.5
309
4.32
313.82
RB = 89.6
LB = 21.6 µH EBge = 226.8 V
RBun = 152.4
EBun = 194.4 V
231.5
4.32
325.42
388.22
Rectifier
0.367
1.512
1.879
163.2 nH
—
21.76 µH
—
—
—
4.667 µH
4.813 µH
9.48 µH
ZQ = 1.51 m
ZT = 5.04 m
—
3.13
Conductor
Two joints
Total
Voltage of
discharged
battery
e.g., 1.8 V/cell
Conductor
Two joints
Total
3.14
System feeder
Transformer
Total
a.c. system
3.15
66
Short-circuit currents
Table 3.18
Equipment
Continued
R (m)
L
Others
Remarks
0.91
0.31 µH
—
154.3
155.21
0.11 µH
0.42 µH
—
—
Commutating
reactor
Conductor
Total
d.c. system
0.41 mH
0.11 µH
0.411 mH
—
d.c. motor
43.3
with
154.3
independent 197.6
excitation
Motor
Conductor
τM = 2.08 ms Total
τF = 1.01 s
Tab.
3.16
Chapter 4
Calculation of short-circuit current in a.c.
three-phase HV-systems
4.1 Types of short-circuits
In three-phase a.c. systems it has to be distinguished between different types of
short-circuits (s.-c.), as outlined in Figure 4.1.
Short-circuit currents can be carried out with different methods and in different
details, depending on the available data and the technical needs. IEC 60909-0 calculates characteristic parameters of the short-circuit current, which are necessary for
the design of power system equipment. The course of time of short-circuit currents
is outlined in Figure 4.2. Generally, one has to distinguish between near-to-generator
and far-from-generator short-circuits. A near-to-generator short-circuit exists if the
contribution to the short-circuit current of one synchronous generator is greater than
twice its rated current, or if the contribution to the short-circuit current of synchronous
or asynchronous motors is greater than 5 per cent of the short-circuit current without
motors.
The analysis of the short-circuit current in the case of near-to-generator shortcircuit as per Figure 4.2(a) indicates two components, besides the decaying d.c.
component, a subtransient and a transient decaying a.c. component. The first or
subtransient component is determined by the impedance between stator and damping winding, called subtransient reactance Xd . The subtransient component decays
with the subtransient time constant T which is normally in the range of some
periods of the system frequency, i.e., T < 70 ms. The transient component is
determined by the reactance between the stator and exciter winding, called transient
reactance Xd . The transient component decays with the transient time constant T ,
which can last up to 2.2 s for large generators. Finally, the short-circuit current is
determined by the saturated synchronous reactance Xd .
In the case of a far-from-generator short-circuit as per Figure 4.2(b), the a.c.
component remains constant throughout the total time duration of the short-circuit, as
the influence of the changing reactance of generators can be neglected. The decaying
68
Short-circuit currents
(a)
B
(b)
Y
Y
R
R
I ⬙k2
I ⬙k3
B
(c)
B
(d)
B
Y
Y
R
R
I ⬙k2E
I ⬙kE2E
Figure 4.1
I ⬙k1
Types of short-circuits and short-circuit currents. (a) Three-phase
short-circuit, (b) double-phase short-circuit without earth/ground connection, (c) double-phase short-circuit with earth/ground connection
and (d) line-to-earth (line-to-ground) short-circuit
d.c. component is due to the ohmic-reactive short-circuit impedance and the instant
of initiation of the short-circuit.
4.2 Methods of calculation
Short-circuit current calculation according to IEC 60909-0 is carried out based on
the method of ‘equivalent voltage source at the short-circuit location’, which will be
explained with the equivalent circuit diagram of a power system outlined in Figure 4.3.
The method is based on the presuppositions as below:
•
•
•
•
Symmetrical short-circuits are represented by the positive-sequence component; unsymmetrical (unbalanced) short-circuits are represented by connection
of positive-, negative- and zero-sequence component as per Table 2.1.
Capacitances and parallel admittances of non-rotating load of the positive(and negative-) sequence component are neglected. Capacitances and parallel
admittances of the zero-sequence component shall be neglected, except in systems
with isolated neutral or with resonance earthing (systems with Petersen coil) as
they have an influence on fault currents in power.
Impedance of the arc at the short-circuit location is neglected.
The type of short-circuit and the system topology remain unchanged during the
duration of short-circuit.
Calculation of short-circuit current
69
2 2I k⬙
ip
(a)
A
2 2Ik
(b)
A
2 2I ⬙k
ip
2 2Ik
Figure 4.2
Time-course of short-circuit currents. (a) Near-to-generator shortcircuit (according to Figure 12 of IEC 60909:1988), (b) far-fromgenerator short-circuit (according to Figure 1 of IEC 60909:1988). Ik
– initial (symmetrical) short-circuit current, ip – peak short-circuit current, Ik – steady-state short-circuit current and A – initial value of the
aperiodic component idc
• The tap-changers of all transformers are assumed to be in main-position (middle
position).
• All internal voltages of system feeders, generators √
and motors are short-circuited
and an equivalent voltage source with value cUn / 3 is introduced at the shortcircuit location. The voltage factor c shall be selected in accordance with Table 4.1.
The voltage factor c takes account of the differences between the voltage at the
short-circuit location and the internal voltage of system feeders, motors and generators
70
Short-circuit currents
(a)
Q
T
F
L
k3
S kQ
⬙
(b)
RQt
XQt
XT
RT
RL
XL
F
F
E1
ZB
CL
2
CL
2
I ⬙k
ZB
01
(c)
RQt
XQt
XT
RT
Zk
RL
XL
F
cUn
3
I ⬙k
01
Figure 4.3
Example for short-circuit current calculation with an equivalent voltage source at s.-c. location. (a) Three-phase a.c. system with three-phase
short-circuit, (b) equivalent circuit diagram in 012-system (positivesequence system), (c) equivalent circuit diagram in 012-system with
equivalent voltage source
due to voltage variations (time and place), operating of transformer tap-changer, etc.
Assuming the voltage factor as per Table 4.1 will result in short-circuit currents on
the safe side, that are higher than in the real power system, however, avoids an
uneconomic high safety margin.
4.3 Calculation of parameters of short-circuit currents
4.3.1
General
The calculation of the impedances of power system equipment was explained in
Chapter 3. It should be noted that the impedances shall be related to the voltage level
of the short-circuit location and that all equipment belonging to the same voltage level
shall have the same nominal voltage. The impedances, therefore, have to be calculated
Calculation of short-circuit current
Table 4.1
71
Voltage factor c according to IEC 60909-0
Nominal system voltage Un
LV: 100 V up to 1000 V (inclusive)
(IEC 60038, Table I)
Voltage tolerance +6%
Voltage tolerance +10%
MV: >1 kV up to 35 kV (inclusive)
(IEC 60038, Table III)
HV: >35 kV (IEC 60038, Table IV)
Voltage factor c for calculation of
Maximal
s.-c. current
cmax
Minimal
s.-c. current
cmin
1.05
1.10
1.10
0.95
0.95
1.00
1.10
1.00
Remark: cmax Un shall not exceed the highest voltage of equipment Um as per IEC 60071.
in relation to the rated apparent power Sr of the equipment itself, respectively to the
nominal system voltage Un . In case the %/MVA-system or the p.u.-system is used,
attention must be given to deviations of rated votages of the equipment from nominal
system voltages, see Section 2.7.2.
The presentation within the following sections is closely related to IEC 60909-0.
The IEC-document includes items for the calculation of impedances and for the
calculation of the short-circuit parameters for balanced and unbalanced short-circuits,
both near-to-generator and far-from-generator short-circuits. Calculation of shortcircuit currents during two separate single-phase to earth short-circuits and the partial
short-circuit currents flowing through earth are dealt with in IEC 60909-3, which is
currently under review. These items are explained in Chapter 7.
Depending on the task, the maximal or minimal short-circuit current has to be
calculated. The maximal short-circuit current is the main design criteria for the rating
of equipment to withstand the effects of short-circuit currents. For the calculation of
maximal short-circuit current, the items shall be considered as below:
•
•
•
•
•
•
For the equivalent voltage source at the short-circuit location the voltage factor
cmax as per Table 4.1 shall be used. National standards can define voltage factors
different from those in Table 4.1.
Short-circuit impedance of system feeders shall be minimal (ZQmin ), so that the
contribution to the short-circuit current is maximal.
The contribution of motors has to be assessed and eventually be taken into account,
see Section 4.4.
Resistance of lines is to be calculated for a temperature of 20◦ C.
Operation of power plants and system feeders shall be in such a way that the
contribution to short-circuit currents will be maximal.
System topology leading to the maximal short-circuit currents shall be taken into
account.
72
Short-circuit currents
The minimal short-circuit current is needed for the design of protection systems and
the minimal setting of protection relays; details of the presuppositions for calculation
are dealt with in Section 4.5.
4.3.2 Calculation of short-circuit current parameters according to
IEC 60909-0
4.3.2.1 Initial symmetrical short-circuit current Ik
The initial symmetrical short-circuit current Ik is calculated for balanced and unbalanced short-circuits based on the equivalent voltage source at the short-circuit location
and the short-circuit impedance seen from the short-circuit location, which has to be
determined with the system of symmetrical components. The results obtained for
the short-circuit currents (and the voltages of the non-faulted phases, if required)
in the 012-system have to be transferred back into the RYB-system. The results for
the different types of short-circuits are outlined in Table 4.2.
Quantities as per Table 4.2:
c
Un
Z1; Z2; Z0
Voltage factor according to Table 4.1
Nominal system voltage
Short-circuit impedance in the positive-, negative- and zerosequence component
As can be obtained from Table 4.2, the value of the initial short-circuit current
depends on the impedances in the positive-, negative- and zero-sequence component. Based on the impedance ratios Z0 /Z1 and Z2 /Z1 , it can be estimated which
type of short-circuit will cause the maximal initial short-circuit current. Figure 4.4
outlines the initial short-circuit currents for different types of short-circuits related to
the short-circuit current of a three-phase short-circuit in variation of the impedance
ratios mentioned above. As the phase angle of the impedances are different in the
positive-, negative- and zero-sequence component, the diagram shall only be used
for a preliminary estimate.
Figure 4.4 can be used as explained below. In the case of a far-from-generator
short-circuit, Z1 is equal to Z2 (Z2 /Z1 = 1). The maximal short-circuit current will
occur in the case of a three-phase short-circuit if Z1 /Z0 ≤ 1. For ratios Z1 /Z0 > 1,
the single-phase short-circuit will result in the highest short-circuit currents. In the
case of near-to-generator short-circuits the ratio of negative- to positive-sequence
impedance Z2 /Z1 mainly determines which type of short-circuit will cause the maximal short-circuit current. If Z1 /Z0 > 1 the maximal short-circuit current will always
occur in the case of a single-phase short-circuit.
4.3.2.2 Short-circuit currents inside power plant
When calculating short-circuit currents inside power plants, the short-circuit location itself and the installation of the unit transformer will result in a different
Calculation of short-circuit current
Table 4.2
73
Equations for the calculation of initial symmetrical short-circuit currents
Type of
short-circuit
Three-phase
Double-phase
short-circuit
without earth
connection
Double-phase
short-circuit with
earth connection
General
Equation
Section
Remarks
IEC 60909-0
= √cUn
Ik3
3|Z 1 |
=
Ik2
cUn
|2Z 1 |
√
− 3cUn Z 2
IkE2E
=
Z1 Z2 + Z1 Z0 + Z2 Z0
−j cUn (Z 0 − a Z 2 )
Ik2EY
=
Z Z +Z Z +Z Z
1 2
1 0
2 0
4.2.1
4.3.1
4.6.1–4.6.3
Short-circuited
phases R, Y
and B
4.2.2
4.3.2
4.6.4
Short-circuited
phases Y and B
4.2.3
4.3.3
4.6.4
Current flowing
through earth
Current of
phase Y
j cU (Z − a 2 Z )
n
0
2
Ik2EB
=
Z1 Z2 + Z1 Z0 + Z2 Z0
Far-from-generator
(Z 1 = Z 2 )
IkE2E
=
Line-to-earth
single-phase
short-circuit
General
Far-from-generator
(Z 1 = Z 2 )
√
3cUn
|Z 1 + 2Z 0 |
Ik2EY
=
cUn |Z 0 /Z 1 − a|
|Z 1 + 2Z 0 |
=
Ik2EB
cUn |Z 0 /Z 1 − a 2 |
|Z 1 + 2Z 0 |
√
3cUn
=
Ik1
|Z 1 + Z 2 + Z 0 |
√
3cUn
=
Ik1
|2Z 1 + Z 0 |
Current of
phase B
Short-circuited
phases Y and B
4.2.3
4.3.3
4.6.4
Current flowing
through earth
Current of
phase Y
Current of
phase B
Short-circuited
phases Y and B
4.2.4
4.3.4
4.6.4
Short-circuited
phase R
74
Short-circuit currents
1.4
kE2E
r = 1.2
1.2
k1
1.0
2.5
2.0
0.8
r
1.5
Z
Z1/ Z0
1.3
1.2
0.6
k2E
0.4
k3
k2
0.2
1.5
1.3
1.2 r
1.0
0
0
Figure 4.4
0.2
0.4
Z2 / Z1
0.6
0.8
1.0
Estimate of maximal initial short-circuit current for different types of
short-circuit and different impedance ratios Z1 /Z0 and Z2 /Z1 . Phase
angle of Z 0 , Z 1 and Z 2 assumed to be identical. Parameter r: ratio of
asymmetrical short-circuit current to three-phase short-circuit current
approach, respectively considerations, concerning the impedances. According to
Figure 4.5 different locations have to be considered, that is,
• Short-circuit between generator and unit transformer (F1)
• Short-circuit at HV-side of auxiliary transformer (F2)
• Short-circuit at MV-side of auxiliary transformer (F3)
Furthermore, the arrangement of the unit transformer, that is,
• Equipped with tap-changer
• Without tap-changer
has an important influence on the short-circuit currents.
The short-circuit current for location F1 between generator and unit transformer,
if the unit transformer is equipped with tap-changer, is calculated by
+ IkT
=
Ik = IkG
1
c ∗ UrG
1
∗
+
√
2 )∗Z
|KGs ∗ Z G | |Z TLV + (1/trT
3
Q min |
(4.1)
Calculation of short-circuit current
G
GS
3~
75
T
I ⬙kG
F1
I ⬙kT
trT
UnQ
I ⬙kEB
S ⬙kQ
F2
Auxiliary transformer
ürE
I ⬙kEBt
F3
I ⬙kM
M
3~
Figure 4.5
Equivalent circuit diagram for the calculation of short-circuit currents
inside power plant
with the impedance correction factor KGs
KGs =
cmax
1 + xd ∗ sin ϕrG
(4.2)
When the unit transformer is not equipped with tap-changer, the short-circuit current
is given as
+ IkT
=
Ik = IkG
1
c ∗ UrG
1
∗
+
√
2 )∗Z
|KGo ∗ Z G | |Z TLV + (1/trT
3
Q min |
(4.3)
with the impedance correction factor KGo
KGo =
1
cmax
1 + pG 1 + xd ∗ sin ϕrG
(4.4)
The short-circuits at the HV-side of the auxiliary transformer at location F2 are
given by
=
IkEB
c ∗ UrG
∗
√
3
1
1
+
2 )∗Z
|KGs ∗ Z G | |KTs ∗ Z TLV + (1/trT
Q min |
(4.5)
76
Short-circuit currents
with impedance correction factor for the generator
cmax
1 + xd ∗ sin ϕrG
(4.6a)
and the unit transformer
cmax
KTs =
1 + xT ∗ sin ϕrG
(4.6b)
KGs =
If the unit transformer is installed without tap-changer the impedance correction
factors are given for the generator
KGo =
1
cmax
∗
1 + pG 1 + xd ∗ sin ϕrG
(4.7a)
and for the unit transformer
KTo =
1
cmax
∗
1 + pG 1 + xT ∗ sin ϕrG
(4.7b)
which shall be used instead of correction factors KGs and KTs as per Equations (4.6),
respectively. The impedance Zrsl
Zrsl =
1
1
+
2 )∗Z
|KGs ∗ Z G | |KTs ∗ Z TLV + (1/trT
Q min |
(4.8)
including the correction factors is called the coupling impedance.
Quantities of Equations (4.1)–(4.8) are
ZG
Z TLV
Z Q min
trT
UrG
c; cmax
xT
xd
pG
sin ϕrG
Impedance of the generator
Impedance of the unit transformer at LV-side (generator voltage)
Impedance of the system feeder for maximal short-circuit current
Transformation ratio of the unit transformer (trT ≥ 1)
Rated voltage of generator
Voltage factor as per Table 4.1
Reactance of the transformer in p.u. (impedance voltage)
Subtransient reactance of the generator in p.u.
Voltage control range of generator in p.u.
Power factor of generator at rated operating conditions
Other quantities are explained previously.
Short-circuits at MV-side of the auxiliary transformer at location F3 or at the
auxiliary busbar are a superposition of the partial short-circuit current IkEBt
of the
auxiliary transformer related to the voltage level of the short-circuit location and of
the contribution of the motors in the auxiliary system to the short-circuit current. In
the case of a unit transformer without tap-changer the impedance correction factors
KGo and KTo have to be considered accordingly.
77
Calculation of short-circuit current
4.3.2.3 Peak short-circuit current ip
Depending on the feeding source of the short-circuit different considerations have to
be taken to calculate the peak short-circuit current.
Figure 4.6 indicates an equivalent circuit diagram with single-fed short-circuit.
The short-circuit impedance is represented by a series connection of the individual
impedances.
The peak short-circuit current, which is a peak value, can be calculated for
the different types of short-circuits based on the initial short-circuit current (r.m.s.
value) by
√
(4.9a)
ip3 = κ ∗ 2Ik3
√
ip2 = κ ∗ 2Ik2
(4.9b)
√
ip1 = κ ∗ 2Ik1
(4.9c)
System feeder Q
Transformer T
Q
F
A
Line L
k3
S ⬙kQ
trT:1
UnQ
RQt
XQt
Q
RT
XT
A
RL
XL
F
~
cUn
3
01
Figure 4.6
Equivalent circuit diagram for single-fed three-phase short-circuit
The peak short-circuit current ip2E in the case of a double-phase short-circuit with
earth connection is always smaller than either of a three-phase or single-phase shortcircuit and need not be calculated separately. The factor κ can be obtained from
Figure 4.7 or calculated by
κ = 1.02 + 0.98 ∗ e−3(R/X)
(4.10)
; I ; I are the initial symmetrical short-circuit currents for three-phase,
where Ik3
k2 k1
double-phase and line-to-earth short-circuit and R; X are the resistance and reactance
of the short-circuit impedance.
Figure 4.8 indicates an equivalent circuit diagram in the case of a short-circuit fed
from non-meshed sources. The peak short-circuit current is calculated by superposing
the contributions of different sources.
78
Short-circuit currents
2
1.9
1.8
1.7
1.6
k
1.5
1.4
1.3
1.2
1.1
1
0
Figure 4.7
0.2
0.4
0.6
0.8
1
R/X
1.2
1.4
1.6
1.8
2
Factor κ for the calculation of peak short-circuit current
Q1
T1
B
I ⬙kT1
ipT1
S ⬙kQ1
I ⬙k ; ip
Q2
T2
S ⬙kQ2
Figure 4.8
I ⬙kT2
ipT2
Equivalent circuit diagram for three-phase short-circuit fed from
non-meshed sources
The peak short-circuit currents ip3T1 and ip3T2 of each branch, fed through the
transformers T1 and T2, are calculated separately as well as the factors κT1 and κT2 .
√
(4.11a)
ip3T1 = κ1 ∗ 2 ∗ Ik3T1
√
(4.11b)
ip3T2 = κ2 ∗ 2 ∗ Ik3T2
Calculation of short-circuit current
79
The total peak short-circuit current ip3 is given by
(4.11c)
ip3 = ip3T1 + ip3T2
Particular considerations are to be taken in the case of short-circuits in meshed networks according to Figure 4.9. The peak short-circuit current at the short-circuit
location cannot be calculated by superposition as the R/X-ratios of the individual branches feeding the short-circuit are different and the direction of the branch
short-circuit currents through the system is defined by the Ohmic law.
Q
F
Figure 4.9
Equivalent circuit diagram of a three-phase short-circuit in a meshed
system
In principle, the peak short-circuit current in a meshed system will be calculated by
√
ip3 = κ ∗ 2Ik3
(4.9a)
as given for three-phase short-circuits, and for other types of short-circuits accordingly. The factor κ, however, will be determined with different methods as
below.
Uniform (smallest) ratio R/X. The factor κ is calculated based on the smallest
ratio R/X of all branches of the network. Only those branches need to be taken
into account which contribute to the short-circuit current in the power system corresponding to the short-circuit location, respectively those branches connected through
transformers to the short-circuit location. The results are always on the safe side,
however the accuracy is low.
Ratio R/X at short-circuit location. Based on the ratio R/X of the total system
impedance at the short-circuit location, the factor κ is calculated taking account of
a safety factor of 1.15 to allow for deviations due to the different ratios R/X in the
different branches.
κb = 1.15 ∗ κ
(4.12)
The factor 1.15 ∗ κ should not exceed the value of 1.8 in LV-systems and shall not
exceed 2.0 in HV-systems. The safety factor 1.15 is neglected when R/X ≤ 0.3.
80
Short-circuit currents
Equivalent frequency fc . The factor κ is found from Figure 4.7 or can be calculated
based on the ratio R/X:
Rc f c
R
∗
=
X
Xc f
(4.13)
Rc and Xc are the equivalent effective resistance and reactance at the short-circuit
location at equivalent frequency fc which is
fc = 20 Hz (nominal power system frequency 50 Hz)
fc = 24 Hz (nominal power system frequency 60 Hz)
The calculation of the equivalent impedance at equivalent frequency fc is to be
carried out additionally to the calculation of impedance at nominal power system
frequency [34].
IEC 60909-1 mentions accuracy limits for the different methods of calculating
short-circuit currents in meshed systems.
Results obtained by the method of uniform (smallest) ratio R/X are always on
the safe side, if all branches contributing to the short-circuit current are taken into
account. Errors can reach in rare cases up to 100 per cent. If only those branches
are considered, which contribute up to 80 per cent to the short-circuit current and
if the ratios R/X are in a wide range, the results can even be on the unsafe side. The
method therefore shall be applied only if the ratios R/X are in a small bandwidth and
if R/X < 0.3.
The method ratio R/X at short-circuit location (safety factor 1.15) will lead to
results on the safe and unsafe side. Applying the method to ratios 0.005 ≤ R/X ≤ 1.0
the error will be in the range +10 to −5 per cent.
The method of equivalent frequency has an accuracy of ±5 per cent, if the ratio
R/X of each branch is in the range of 0.005 ≤ R/X ≤ 5.0.
4.3.2.4 Decaying (aperiodic) component idc
The maximal decaying aperiodic component idc is calculated by
√
idc = 2 ∗ Ik ∗ e−2πf t∗(R/X)
(4.14)
where Ik is the initial symmetrical short-circuit current, f is the power system frequency, t is the time parameter and R, X are the resistance and reactance of the
short-circuit impedance.
The resistance of the short-circuit impedance shall be calculated with the real stator
resistance RG of generators instead of the fictitious resistance RGf , see Table 3.5. The
ratio R/X shall be calculated in meshed systems with an equivalent frequency fc
which depends on the duration t of the short-circuit as outlined in Table 4.3.
4.3.2.5 Symmetrical short-circuit breaking current Ib
In the case of near-to-generator short-circuits, the a.c. component of the short-circuit
current is decaying to a steady-state value, see Figure 4.2(a). The short-circuit current
is interrupted by the switchgear at the instant of minimal time delay tmin of the
Calculation of short-circuit current
Table 4.3
81
Equivalent frequency for the calculation of decaying component
Factor
f ∗t
<1
<2.5
<5
<12.5
Ratio of equivalent frequency
to power system frequency
Example: f = 50 Hz
fc /f
0.27
0.15
0.092
0.055
t
fc
<0.02 s
13.5 Hz
<0.05 s
7.5 Hz
<0.1 s
4.6 Hz
<0.25 s
2.75 Hz
Example: f = 60 Hz
t
fc
<0.02 s
16.2 Hz
<0.05 s
9.0 Hz
<0.1 s
5.52 Hz
<0.25 s
3.3 Hz
protection. The calculation of the symmetrical short-circuit breaking current Ib is
based on the initial short-circuit current and on the factor μ,
Ib = μ ∗ Ik
(4.15)
The factor μ can be taken from Figure 4.10 or calculated by
μ = 0.84 + 0.26 ∗ e−0.26(IkG /IrG )
for tmin = 0.02 s
(4.16a)
μ = 0.71 + 0.51 ∗ e
for tmin = 0.05 s
(4.16b)
for tmin = 0.10 s
(4.16c)
for tmin ≥ 0.25 s
(4.16d)
μ = 0.62 + 0.72 ∗ e
μ = 0.56 + 0.94 ∗ e
/I )
−0.30(IkG
rG
/I )
−0.32(IkG
rG
/I )
−0.38(IkG
rG
1.1
1
Minimum time delay
0.9
0.02 s
m 0.8
0.05 s
0.7
0.1 s
> 0.25 s
0.6
0.5
0
2
4
6
8
10
12
0 /IrM
I 0kG /IrG or I kM
Figure 4.10
Factor μ for calculation of symmetrical short-circuit breaking current
82
Short-circuit currents
is the initial symmetrical short-circuit current of the generator, I
where IkG
rG is
the rated current of the generator and tmin is the minimal time delay of the protection, switchgear and auxiliaries, that is, minimal time for switching the short-circuit
current off.
The factor μ is valid for high-voltage synchronous generators, excited by rotating machines of rectifiers. If the excitation system is not known the factor shall be
set to μ = 1.
In the case of far-from-generator short-circuits, the symmetrical short-circuit
breaking current Ib is equal to the initial short-circuit current Ik as the a.c. component
is not decaying.
4.3.2.6 Steady-state short-circuit current Ik
The steady-state short-circuit current Ik in the case of near-to-generator short-circuits
depends on various factors such as saturation effects, power factor of generators,
change of system topology due to operation of switching, etc. and can therefore only
be determined with a certain inaccuracy. The method as per IEC 60909 determines
lower and upper limits only when one synchronous machine is feeding the shortcircuit. The calculation is based on the generator-rated current assuming a factor λ
which depends on the ratio of initial symmetrical short-circuit current to rated current
of the generator and on the saturated synchronous reactance.
Maximal excitation of the synchronous machine leads to the maximal symmetrical
short-circuit breaking current
Ik max = λmax ∗ IrG
(4.17a)
The factor λmax is valid for turbine generators according to Figure 4.11 and for
salient-pole generators as per Figure 4.12.
The minimal symmetrical short-circuit breaking current is calculated for constant
no-load excitation of the generator with the factor λmin .
Ik min = λmin ∗ IrG
(4.17b)
The values for λmin are included in Figures 4.11 and 4.12. Reference is made to the
remarks in IEC 60909-0 on the factors.
Quantities as used above are
IrG
IkG
xdsat
Rated current of the generator
Initial synchronous short-circuit current of the generator
Saturated synchronous reactance of the generator, equal to the reciprocal
of the short-circuit ratio of the generator
In the case of far-from-generator short-circuits, the symmetrical short-circuit
breaking current Ib is equal to the initial short-circuit current Ik as the a.c. component
is not decaying.
83
Calculation of short-circuit current
(a) 2.8
(b) 2.8
2.6
2.6
xdsat
2.4
max
2.2
2.0
1.8
1.6
1.2
1.4
1.6
1.8
2.0
2.2
2.2
2.0
1.8
1.6
1.2
1.0
1.0
1.4
0.8
0.8
min
0.6
0.4
0.2
0.2
0
0
1 2 3 4 5 6 7 8
Three-phase short circuit I 0kG/IrG
Figure 4.11
min
0.6
0.4
0
1 2 3 4 5 6 7 8
Three-phase short circuit I kG
0 /IrG
Factors λmax and λmin for turbine generators (Figure 17 of DIN EN
60909-0 (VDE 0102)). (a) Series one and (b) series two
xdsat
0.6
(b) 5.5
(a) 5.5
5.0
5.0
xdsat
4.5
max
4.0
0.6
3.5
3.0
2.5
1.0
1.2
1.7
2.0
3.0
1.5
min
1.0
0.5
0.8
4.0
0.8
2.0
max
4.5
3.5
0
1.2
1.4
1.6
1.8
2.0
2.2
max
2.4
1.4
1.2
0
xdsat
1.0
1.2
1.7
2.0
2.5
2.0
1.5
min
1.0
0.5
0
1 2 3 4 5 6 7 8
Three-phase short circuit I 0kG/IrG
Figure 4.12
0
0
1 2 3 4 5 6 7 8
Three-phase short circuit I 0kG /IrG
Factors λmax and λmin for salient-pole generators (Figure 18 of DIN
EN 60909-0 (VDE 0102) 1988). (a) Series one and (b) series two
84
Short-circuit currents
4.4 Influence of motors
Asynchronous motors and synchronous motors have to be taken into account in
MV-systems and in auxiliary supply systems of power plants and industrial networks for the calculation of maximal short-circuit currents. They contribute to the
initial symmetrical short-circuit current, to the peak short-circuit current, to the symmetrical short-circuit breaking current and in case of unbalanced short-circuits to
the steady-state short-circuit current as well, see Table 4.4. Synchronous motors are
modelled like generators and asynchronous generators are treated as asynchronous
motors. Motors of any kind, which are not in operation at the same time, e.g., due
to the process or due to any interlocking, can be neglected for the calculation of
short-circuit currents. Motors fed by static-rectifiers need to be considered in the case
of three-phase short-circuits only, if they are able to transfer energy for deceleration
for the duration of the short-circuit, as they contribute to the initial symmetrical and
to the peak short-circuit current.
Table 4.4
Calculation of short-circuit currents of asynchronous motors
Parameter
Initial
symmetrical
short-circuit
current
Peak
short-circuit
current
Type of short-circuit
Three-phase
Double-phase
Ik3M
= √cUn
Ik2M
= 23 Ik3M
√
ip3M = κM 2Ik3M
ip2M = 23 ip3M
3ZM
√
√
Line-to-earth
√
c 3Un
Ik1M
= Z +Z
1M
2M +Z0M
in systems with earthed
neutral only
√
ip1M = κM 2Ik1M
MV-motors:
κM = 1.65 (RM /XM = 0.15) for active power per pole-pair <1 MW
κM = 1.75 (RM /XM = 0.10) for active power per pole-pair ≥1 MW
LV-motors including connection cables
κM = 1.30 (RM /XM = 0.42)
Symmetrical
short-circuit
breaking
current
Ib3M = μqIk3M
Steady-state
short-circuit
current
Ik3M = 0
√
Ib2M ≈ 23 Ik3M
Ib1M ≈ Ik1M
μ as per Figure 4.10, q as per Figure 4.13
√
Ik2M ≈ 23 Ik3M
Quantities used in the equations are explained in the text.
Ik1M ≈ Ik1M
Calculation of short-circuit current
85
Asynchronous motors in public supply systems are considered when
• the sum of the rated currents is greater than 1 per cent of the initial symmetrical
short-circuit current without motors;
• the contribution to the initial symmetrical short-circuit current without motors
is greater or equal to 5 per cent of the initial symmetrical short-circuit current
without motors.
Medium- and low-voltage asynchronous motors connected through two-winding
transformers to the short-circuit are considered if
PrM
0.8
(4.18)
>
√
SrT
|((c ∗ 100 SrT )/( 3UnQ /Ik )) − 0.3|
where UnQ is the nominal system voltage, SrT is the sum of rated apparent power
of all transformers, directly connected to motors feeding the short-circuit,
Ik is the
initial symmetrical short-circuit current without motors and
PrM is the sum of
rated active power of all low- and medium-voltage motors.
In order to calculate the branch short-circuit current of asynchronous motors,
Table 4.4 can be used. The factor q (three-phase short-circuit) depending on the
minimal time delay of the protection tmin can be obtained from Figure 4.13 or by
q = 1.03 + 0.12 ∗ ln(m),
q = 0.79 + 0.12 ∗ ln(m),
q = 0.57 + 0.12 ∗ ln(m),
q = 0.26 + 0.10 ∗ ln(m),
tmin = 0.02 s
(4.19a)
tmin = 0.05 s
(4.19b)
tmin ≥ 0.25 s
(4.19d)
tmin = 0.10 s
(4.19c)
where tmin is the minimal time delay of the protection, switchgear and auxiliaries,
i.e., minimal time for switching the short-circuit current off and m is the active power
of the motor per pole-pair in p.u. based on 1-MW-base. The factor q should not be
greater than 1.
4.5 Minimal short-circuit currents
In order to calculate the minimal short-circuit current, the voltage factor cmin according to Table 4.1 for the equivalent voltage source at the short-circuit location has to
be considered. Furthermore,
• System topology, generator dispatch and short-circuit power of feeding networks
have to be defined in such a way that the minimal short-circuit current is expected.
This normally applies for low-load conditions.
• Motors are to be neglected.
• Resistances of overhead lines and cables shall be calculated with the maximal
permissible temperature at the end of the short-circuit, e.g., 80◦ C in low-voltage
systems.
86
Short-circuit currents
1
Minimum time delay
0.9
0.8
0.02 s
0.7
0.6
0.05 s
q 0.5
0.1 s
0.4
0.3
> 0.25 s
0.2
0.1
0
0.01
Figure 4.13
0.1
1
10
m (active power per pair of poles)
MW
100
Factor q for the calculation of symmetrical short-circuit breaking
current
These assumptions have to be taken in the case of balanced and unbalanced shortcircuits, except where other presuppositions are mentioned.
4.6 Examples
Examples for the calculation of short-circuit currents are included in IEC 60909-4
besides those given below. Reference is made to the relevant chapters where the
individual quantities are explained.
4.6.1 Three-phase near-to-generator short-circuit
Figure 4.14 outlines the equivalent circuit diagram of a 220-kV system. For threephase short-circuit at busbar B (maximal s.-c. currents) the branch short-circuit
currents of the generators and the system feeder as well as the contribution of the
generators to the symmetrical short-circuit breaking current and to the steady-state
short-circuit current (tmin = 0.1 s; xdsat = 140%; Turbine generator (Series 1)) are
to be calculated.
Data of equipment taken from nameplates:
SrG1 = 120 MVA; UrG1 = 10.5 kV; cos ϕrG1 = 0.8; xdG1
= 18%
SrG2 = 80 MVA; UrG2 = 10.5 kV; cos ϕrG2 = 0.85; xdG2 = 16%
SrT1 = 120 MVA; UrT1HV /UrT1LV = 220 kV/10.5 kV; ukrT1 = 14%
SrT2 = 80 MVA; UrT2HV /UrT2LV = 220 kV/10.5 kV; ukrT2 = 12%
SrT3 = 200 MVA; UrT3HV /UrT3LV = 400 kV/220 kV; ukrT3 = 12%
87
Calculation of short-circuit current
B
A
Q
Figure 4.14
G1
GS
3~
L1
L2
UnQ = 380 kV
T1
T2
G2
GS
3~
Un = 220 kV
Equivalent circuit diagram of a 220-kV-system with short-circuit
location
= 5 GVA; U
SkQ
nQ = 380 kV
XL1 = XL2 = 0.4 /km; l = 50 km
The impedances of equipment (positive-sequence component) including correction
factors are
XQK = 9.632
XT3 = 28.31
XL1 = XL2 = 20.04
XKW1K = 138.67
XKW2K = 182.52
The short-circuit impedance at the short-circuit location is XkB = 29.82
The initial symmetrical short-circuit currents are:
= 4.69 kA,
at s.-c. location
Ik3
Ik3Q = 1.69 kA, branch s.-c. current of system feeder
The contribution of the generators to the short-circuit currents is outlined in the table
below.
Parameter
Generator 1
Generator 2
Section
Ik3
μ
Ib
λmax /λmin
Ik max /Ik min
21.11 kA
0.879
18.55 kA
1.75/0.46
11.55 kA/3.04 kA
16.04 kA
0.844
13.54 kA
1.82/0.47
8.01 kA/2.07 kA
4.3.2.1
4.3.2.5
4.6.2 Line-to-earth (single-phase) short-circuit
The initial short-circuit current for a single-phase short-circuit at location F according
to Figure 4.15 shall be calculated.
88
Short-circuit currents
Q
T1
A
B
1
Un = 110 kV
T2
UnQ = 220 kV
Figure 4.15
C
L2
L3
Equivalent circuit diagram of a 110-kV-system with 220-kV-feeder
The data of equipment are:
= 5 GVA; X /X = 4
SkQ
0
1
SrT = 100 MVA; ukr = 12%; trT = 220 kV/115 kV; X0 /X1 = 3
XL = 0.13 /km; lL = 10 km; X0 /X1 = 3.5
The impedances in the positive- (negative-) and zero-sequence component including
correction factors are:
X1QK = 2.904 ; X0QK = 11.62
X1T1K = X1T2K = 15.49 ; X0T1K = X0T2K = 46.34
X1L1 = X1L2 = X1L3 = 1.295 ; X0L1 = X0L2 = X0L3 = 4.538
The impedances at the short-circuit location are X1k = 11.507 ; X0k = 49.368
= 2.895 kA.
and the initial short-circuit current is Ik1
4.6.3 Calculation of peak short-circuit current
The peak short-circuit current for a three-phase short-circuit at location F according
to Figure 4.16 shall be calculated with the different methods, i.e., ‘superposition
method’, ‘ratio R/X at s.-c. location’ and ‘equivalent frequency’. The accuracy of
the results is to be assessed.
The data of equipment are
= 1000 MVA; U
SkQ
nQ = 110 kV
SrT = 10 MVA; ukrT = 10%; PkT = 70 kW; trT = 125 kV/12 kV
= 10%
SrG = 20 MVA; UrG = 10.5 kV; cos ϕrG = 0.8; xdG
XL = 0.09 /km; RL = 0.123 /km; lL = 5 km
The impedances of equipment including correction factors are
XGK = 0.5448 ; RGfK = 0.5448
XL = 0.45 ; RL = 0.615
Calculation of short-circuit current
89
B
GS
3~
A
Q
UnQ = 110 kV
Figure 4.16
Un = 10 kV
Equivalent circuit diagram of a 10-kV system, f = 50 Hz
XQ = 0.122 ; RQ = 0.0122
XT = 1.416 ; RT = 0.0994
The results of short-circuit calculation are outlined in the table below:
Method
Impedance
Ik3
κ
ip
Impedance at
s.-c. location
Zk = (0.244 + 0.649)
9.16 kA
—
—
Superposition
of feeders A
and B
= 4.118 kA 1.837 i
ZkA = (0.1116 + 1.5381) Ik3A
pA = 10.7 kA
= 5.337 kA 1.157 i
ZkB = (0.653 + 0.9948) Ik3B
pB = 8.73 kA
ipges = 19.43 kA
Ratio R/X at
s.-c. location
Equivalent
frequency
Zk = (0.244 + 0.649)
9.16 kA
1.337 19.92 kA
ZkAc = (0.1116 + 0.6112) 9.16 kA
ZkBc = (0.653 + 0.3979)
Zkc = (0.2509 + 0.4036)
1.485 19.24 kA
The results obtained with the superposition method are the correct results.
4.6.4 Short-circuit currents in a meshed 110-kV-system
The three-phase short-circuit current for the short-circuit location at busbar E in
a meshed 110-kV-system shall be calculated. The system diagram is outlined in
Figure 4.17.
90
Short-circuit currents
GS
3~
G1
GS
3~
G2
A
B
T2
T1
UnQ = 380 kV
Q
T3
D
C
L4
L3
L1
Un = 110 kV
Un = 220 kV
L2
T4
E
Figure 4.17
A 110-kV system with short-circuit location
The rated data of equipment (positive-sequence component) are given below:
System Q:
G1:
G2:
T1:
T2:
T3:
T4:
Lines:
= 4000 MVA
UnQ = 380 kV; SkQ
= 18%; cos ϕ
SrG1 = 200 MVA; UrG1 = 10.5 kV; xd1
rG1 = 0.85;
pG1 = 10%
= 15%; cos ϕ
SrG2 = 120 MVA; UrG2 = 10.5 kV; xd2
rG2 = 0.8;
pG2 = 10%
SrT1 = 200 MVA; ukrT1 = 16%;
UrT1HV /UrT1LV = 110 kV/10.5 kV; pT1 = 12%
SrT2 = 100 MVA; ukrT2 = 14%;
UrT2HV /UrT2LV = 110 kV/10.5 kV; pT2 = 10%
SrT3 = 300 MVA; ukrT3 = 16%;
UrT3HV /UrT3LV = 400 kV/220 kV; pT3 = 15%
SrT4 = 300 MVA; ukrT4 = 16%;
UrT4HV /UrT4LV = 220 kV/110 kV; pT4 = 12%
= 0.4 /km; l
XL1
L1 = 40 km; XL2 = 0.36 /km; lL2 = 30 km
XL3 = 0.4 /km; lL1 = 20 km; XL4 = 0.42 /km;
lL4 = 100 km
91
Calculation of short-circuit current
The impedances of the equipment in %/MVA calculated according to the equations
as per Section 3.2.2 are outlined in the table below:
No.
Equipment
Ik3
x
Correction Corrected Sk3
(%/MVA) as per
impedance (MVA) (kA)
Tables 3.2 (%/MVA)
and 3.6
1
2
1+2
3
4
3+4
5
6
7
Y5; 6; 7
Generator 1
Transformer 1
Power station 1
Generator 2
Transformer 2
Power station 2
Line 1
Line 2
Line 3
L1L2
L1L3
L2L3
System Q
Transformer 3
Line 4
Transformer 4
0.09
0.08
0.17
0.125
0.14
0.265
0.1322
0.0893
0.0661
0.8996
0.153
0.8576
0.227
0.041
0.0304
0.0205
8
9
10
11
8 + 9 + 10 + 11
12
Total impedance
at E
0.02736
0.0533
0.0868
0.05
1.004
0.0535
1.004
0.0502
0.2212
0.096
1146
6.01
4.6.5 Influence of impedance correction factors on short-circuit currents
The three-phase short-circuit current for the short-circuit location in the 110-kVsystem in a single-fed system of different voltage levels as outlined in Figure 4.18
shall be calculated.
Q
T1
C
B
T2
A
L
Un = 220 kV
UnQ = 380 kV
Figure 4.18
UnB = 110 kV
System with different voltage levels with short-circuit location
92
Short-circuit currents
The rated data of equipment (positive-sequence component) are given below:
System Q:
T1:
T2:
Line:
= 5000 MVA
UnQ = 380 kV; SkQ
SrT1 = 200 MVA;
ukrT1 = 16%;
220 kV; pT1 = 12%
SrT2 = 200 MVA;
ukrT2 = 16%;
120 kV; pT2 = 10%
XL = 0.4 /km; lL = 100 km
UrT1HV /UrT1LV = 400 kV/
UrT2HV /UrT2LV = 220 kV/
The impedances of the equipment in %/MVA and the short-circuit current calculated
according to the equations as per Section 3.2.2 are outlined in the table below:
No.
Equipment
x
(%/MVA)
Sk3
(MVA)
Ik3
(kA)
1
2
3
4
5
System Q
Transformer 1
Line
Transformer 2
Total impedance
at 110 kV
0.0219
0.08
0.0826
0.08
0.2645
415.9
2.18
As indicated by the rated data of equipment and as can be seen in Figure 4.18, the
rated voltage of the transformers differ from the nominal voltages of the 380-kV- and
the 110-kV-systems. Correction factors for those equipment connected to the shortcircuit location through transformers must be taken into account. The results of the
calculation with impedance correction factor as per Figure 2.16 and Equation (2.27)
are given below.
No.
Equipment
x
(%/MVA)
Correction
factor
as per
Figure 2.16
Corrected
impedance
(%/MVA)
1
2
3
4
5
System Q
Transformer 1
Line
Transformer 2
Total impedance
at 110 kV
0.0219
0.08
0.0826
0.08
1.074
1.19
1.19
1.19
0.0235
0.0952
0.0983
0.0952
0.3122
Sk3
(MVA)
Ik3
(kA)
352.3
1.85
Calculation of short-circuit current
93
The calculation of transformer impedances furthermore requires impedance
correction factor as outlined in Table 3.3. The results of the analysis, taking account
of both correction factors (Figure 2.16 and Table 3.3), are outlined in the table below:
No. Equipment
x
Correction Correction Corrected Sk3
Ik3
(%/MVA) factor
factor
impedance (MVA) (kA)
as per
as per
(%/MVA)
Figure 2.16 Table 3.3
1
2
3
4
5
0.0219
0.08
0.0826
0.08
0.2645
System Q
Transformer 1
Line
Transformer 2
Total
impedance
at 110 kV
1.074
1.19
1.19
1.19
1.004
1.004
0.0235
0.0956
0.0983
0.0956
0.313
351.4
1.84
Calculating the short-circuit current in the p.u.-system is similar to the calculation
in the %/MVA-system. Correction factors as per Table 3.3 for the transformers and
Figure 2.16 due to differences between the rated voltages of equipment and the nominal system voltages have to be taken into account. The reference voltage is equal
to the nominal voltage at the short-circuit location UB = 110 kV and the reference
power is 100 MVA. The results are outlined in the table below:
No. Equipment
x
(p.u.)
Correction
factor
as per
Figure 2.16
1
2
3
4
5
0.0219
0.08
0.0826
0.08
0.2645
1.074
1.19
1.19
1.19
System Q
Transformer 1
Line
Transformer 2
Total
impedance
at 110 kV
ik3
Correction Corrected sk3
impedance (p.u.) (p.u.)
factor
(p.u.)
as per
Table 3.3
1.004
1.004
0.0235
0.0956
0.0983
0.0956
0.313
3.514 2.024
If the short-circuit current is calculated using the -system only the correction factor
as per Table 3.3 for the transformers has to be taken into account. The results as per
94
Short-circuit currents
calculation in the -system are outlined in the table below:
No.
Equipment
X
()
1
2
3
4
5
System Q
Transformer 1
Line
Transformer 2
Total
impedance
at 110 kV
2.844
11.519
11.894
11.519
37.78
Correction
as per
Table 3.3
1.004
1.004
Corrected
impedance
()
2.844
11.565
11.894
11.565
37.87
Sk3
(MVA)
Ik3
(kA)
351.4
1.84
The greatest influence on the short-circuit current is given by the correction factors
due to the difference of the rated voltage of the transformers and the nominal voltages
of the 380-kV- and the 110-kV-systems as can be seen clearly from the results. The
correction factor of the transformers as per Table 3.3, however, have only a negligible
effect on the system under investigation. As the individual correction factors depend
on the rated data of the equipment the influence on the short-circuit current may be
different in other system configurations. Thus, it should be noted that the correction
factors should be taken into account in general.
4.6.6 Short-circuit currents in a.c. auxiliary supply of a
power station
Figure 4.19 indicates the high-voltage system configuration of the a.c. power supply
of a power station. Auxiliary supply is connected to the 6-kV-busbar E. During start-up
of the power station, i.e., prior to synchronization of the generator, the power supply is
taken from the start-up supply through transformer T5 either from the 30-kV-system
connected to busbar B or from the 110-kV-system (busbar A) or 220-kV-system
(busbar Q). After synchronization the transformers T2 and T5 are both in operation
for the auxiliary supply, transformer T5 is switched-off finally and the auxiliaries are
supplied through transformer T2 only.
The rated data of equipment are given below:
System Q
System A
System B
= 10,000 MVA
UnQ = 220 kV; SkQ
UnA = 110 kV; SkA = 3000 MVA
= 300 MVA
UnB = 30 kV; SkB
Calculation of short-circuit current
UnQ = 220 kV
Q
95
UnA = 110 kV
A
T3
×
T1
T4
B
UnB = 30 kV
T5
×
GS
3~
T2
G
UrG = 11.5 kV
Figure 4.19
G
T1
T2
T3
T4
T5
UnE = 6 kV
E
High-voltage system configuration for the auxiliary supply of a
power station
SrG = 300 MVA; UrG = 11.5 kV; xd = 18%; cos ϕrG = 0.85
SrT1 = 300 MVA; ukrT1 = 14%; UrT1HV /UrT1LV = 220 kV/11.5 kV
SrT2 = 25 MVA; ukrT2 = 8%; UrT2HV /UrT2LV = 11.5 kV/6 kV
SrT3 = 150 MVA; ukrT3 = 12%; UrT3HV /UrT3LV = 220 kV/110 kV
SrT4 = 40 MVA; ukrT4 = 10%; UrT4HV /UrT4LV = 110 kV/30 kV
SrT5 = 25 MVA; ukrT5 = 8%; UrT5HV /UrT5LV = 11.5 kV/6 kV
Data on the voltage control of the generator and the tap-changers of the transformers are not known. The short-circuit currents have to be calculated for threephase short-circuit at the auxiliary busbar E for the three operating conditions as
mentioned.
96
Short-circuit currents
The impedances of the equipment are given in the table below.
No.
Equipment
x
(%/MVA)
1
2
3
4
5
6
7
8
9
System Q
System A
System B
Generator
Transformer 1
Transformer 2
Transformer 3
Transformer 4
Transformer 5
0.0109
0.0365
0.3652
0.06
0.0467
0.32
0.8
0.25
0.32
Correction
factor as per
Table 3.2
and Table 3.3
Corrected
impedance
(%/MVA)
1.005
0.964
0.997
0.975
0.986
0.997
0.0603
0.045
0.31
0.7799
0.246
0.31
For start-up operation of the power station the total impedance is xkS =
= 22.58 kA.
0.469%/MVA, resulting in a three-phase short-circuit current of Ik3S
For the intermediate operation state (both transformers T2 and T5 are in operation) the total impedance is xkI = 0.197%/MVA, the three-phase short-circuit
= 53.82 kA. For normal operation of the power station the total
current is Ik3I
impedance is xkN = 0.339%/MVA, resulting in a three-phase short-circuit current
= 31.23 kA.
of Ik3N
The highest short-circuit current appears in case the auxiliaries are supplied
through the transformers T2 and T5. This condition is only present for a short-time
while switching from one supply to the other. It is therefore not recommended to
take this condition for the design rating of the switchgear and equipment, but to
take the highest short-circuit current occurring under other operating conditions
= 31.23 kA).
(Ik3
Calculation can also be done using p.u.-system, which gives identical numerical
values for the impedances. The short-circuit currents in p.u.-system are calculated
with reference voltage UB = 6 kV (nominal system voltage at short-circuit location).
Results of calculation in %/MVA-system and p.u.-system are outlined in the table
below.
Operating condition
x
(%/MVA)
Ik3
(kA)
x
(p.u.)
ik3
(p.u.)
Start-up operation
Transformers T2 and T5 are in operation
Normal operation of the power station
0.469
0.197
0.339
22.58
53.82
31.23
0.469
0.197
0.339
1.35
3.23
1.87
Chapter 5
Influence of neutral earthing on single-phase
short-circuit currents
5.1 General
The theoretical approach to calculate short-circuit (s.-c.) currents with symmetrical
components in general and especially in the case of single-phase short-circuit was
explained in detail in Chapter 2. Current and voltages in case of short-circuits with
earth connection (e.g., single-phase short-circuits) depend on the positive- and zerosequence impedances Z1 and Z0 . If the ratio of zero-sequence to positive-sequence
impedance is k = Z0 /Z1 the voltages in the non-faulted phases (see Equation (2.25a))
and the single-phase short-circuit current (see Equation (2.25b)) are
|U Y | = |U B | = E1 ∗
Ik1
=
3
E1
∗
Z1 2 + k
√
3∗
√
k2 + k + 1
2+k
(5.1a)
(5.1b)
√
If the voltage E1 is set to E1 = Un / 3, similar to the equivalent voltage at
short-circuit location then
√
k2 + k + 1
(5.1c)
|U Y | = |U B | = Un ∗
2+k
3
Un
(5.1d)
∗
Ik1
=√
3 ∗ Z1 2 + k
The impedances in the positive-sequence (and negative-sequence) system are
determined only by the network topology. The single-phase short-circuit current and
the voltages of the non-faulted phases can be changed only by changing the ratio
of positive-sequence to zero-sequence impedance, i.e., by changing the handling of
transformer neutrals.
98
Short-circuit currents
The type of neutral earthing determines the impedance Z0 of the zero-sequence
component and has a dominating influence on the short-circuit current through earth,
in case of single-phase short-circuits and I
i.e., Ik1
kE2E in case of two-phase shortcircuit with earth connection. In order to change the zero-sequence impedance of
the system, it is possible to earth any number of neutrals, i.e., none, a few or all
transformer neutrals, leading to the highest zero-sequence impedance (no neutral
earthed), respectively the lowest zero-sequence impedance (all neutrals earthed).
The system is characterised less by the number of neutrals to be earthed, than by the
value of the single-phase short-circuit current and by the voltages in the non-faulted
phases.
The different types of neutral handling in power systems (high-voltage systems
only) are outlined in Table 5.1.
Quantities as per Table 5.1
Un
U0max
ω
CE
Z0; Z1
δ0
v
5.2
Nominal system voltage
Maximal voltage in the zero-sequence system, i.e., at neutral of
transformer
Angular velocity of the power system
Line-to-earth capacitance of the power system
Zero-sequence, respectively positive-sequence, impedance of the
system
Damping of the power system (see Section 5.5)
Ratio indicating capacitance to reactance (see Section 5.5)
Power system with low-impedance earthing
Low-impedance earthing is applied in medium-voltage and high-voltage systems
worldwide with nominal voltages above 10 kV. Power systems having nominal
voltages Un ≥ 132 kV are generally operated with low-impedance earthing. In order
to realise a power system with low-impedance earthing, it is not necessary that the
neutrals of all transformers are earthed, but to fulfil the criteria, that the voltages of
the non-faulted phases remain below 140 per cent of the nominal system voltage in
the case of a single-phase short-circuit. The disadvantage while earthing all neutrals
is seen in an increased single-phase short-circuit current, sometimes exceeding the
three-phase short-circuit current. The neutral of unit transformers in power stations
shall not be earthed at all, as the single-phase short-circuit current will then depend
on the generation dispatch. As the contribution of one unit transformer is in the range
of up to 8 kA, the influence on the single-phase short-circuit currents is significant.
Based on Figure 5.1 and assuming a far-from-generator short-circuit with positivesequence impedance equal to negative-sequence impedance Z 1 = Z 2 , the singlephase short-circuit current is calculated by
√
c ∗ 3 ∗ Un
(5.1e)
I k1 =
2 ∗ Z1 + Z0
Table 5.1
Characteristics of different types of neutral handling in power systems
Single-phase fault
current (short-circuit
current)
Increase of voltages at
non-faulted phases
Isolated
neutral
Low-impedance
earthing
Earthing
with current
limitation
Resonance
earthing
Capacitive earth-faults
current
Single-phase (earth-fault)
short-circuit current
Single-phase (earth-fault)
short-circuit current
Residual earth-fault
current
√
I CE ≈ j 3ωCE Un
√
I k1 = c 3Un /(2Z 1 + Z 0 )
√
I k1 = c 3Un /(2Z 1 + Z 0 )
√
I Rest ≈ j 3Un ωCE (δ0 + j v)
Present
U0max /Un ≈ 0.6
√
≈ 3
Generally high
No increase
U0max /Un < 0.3–0.45
No increase
U0max /Un ≈ 0.45–0.6
√
1.38– 3
>4
Present
U0max /Un ≈ 0.6
√
√
≈ 3–1.1 ∗ 3
→Infinite
Extinguishing of fault arc
Self-extinguishing
(see Figure 5.11)
Not self-extinguishing
Self-extinguishing in
rare cases
Self-extinguishing
(see Figure 5.11)
Repetition of faults
Double earth-fault
Reignition of
earth-fault
None
None
Double earth-fault
Voltage at earthing
electrode UE
UE ≤ 125 V
UE > 125 V permitted
UE > 125 V permitted
UE ≤ 125 V
Touching voltage UB
UB ≤ 65 V
See VDE 0141
See VDE 0141
UB ≤ 65 V
Earth-fault factor δ
Ratio of impedances
Z0 /Z1
<1.38
2–4
100 Short-circuit currents
(a)
Z0; Z1; Z2
B
Y
R
IR
UY; UB
I1
(b)
Z1
E1 = E 0
U1
01
I2
Z2
U2
02
I0
Z0
U0
00
Figure 5.1
Equivalent circuit diagram of a single-phase short-circuit (system with
low-impedance earthing). (a) Diagram in RYB-system, (b) equivalent
circuit diagram in the system of symmetrical components
with voltage factor c according to Table 4.1. If the single-phase short-circuit current
is related to the three-phase short-circuit current
c ∗ Un
I k3 = √
3 ∗ Z1
(5.2)
I k1
3 ∗ Z1
=
I k3
2 ∗ Z1 + Z0
(5.3)
it follows that
The relation of single-phase to three-phase short-circuit current depending on the
ratio of Z1 /Z0 with the difference of phase angles (γ1 − γ0 ) of the impedances
as parameter is outlined in Figure 5.2. The phase angles γ1 and γ0 are defined
by the arcustangens-function γ1 = arctan(X1 /R1 ) in the positive-sequence system
respectively γ0 = arctan(X0 /R0 ) in the zero-sequence system.
Influence of neutral earthing 101
3.5
3
2.5
I 0k1/I k3
0
0°
2
Degree
30°
1.5
60°
90°
1
150°
120°
0.5
0
0
0.5
1
1.5
2
2.5
3
Z1/Z0
Figure 5.2
Ratio of single-phase to three-phase short-circuit current depending on
Z1 /Z0 and (γ1 − γ0 )
The voltages (power-frequency voltage) of the non-faulted phases Y and B,
as calculated in Chapter 2 in detail,
U Y = E1 ∗
Z 0 ∗ (a 2 − 1) + Z 2 ∗ (a 2 − a)
Z0 + Z1 + Z2
(5.4a)
U B = E1 ∗
Z 0 ∗ (a − 1) + Z 2 ∗ (a − a 2 )
Z0 + Z1 + Z2
(5.4b)
can be simplified if Z 1 = Z 2 is assumed and by taking account of the meaning of a
and a 2 as below:
√
√
3
U Y = −0.5 3 ∗ E 1 ∗
(5.5a)
1 + (2Z 1 /Z 0 ) + j
√
√
3
(5.5b)
U B = −0.5 3 ∗ E 1 ∗
1 + (2Z 1 /Z 0 ) − j
Relating the voltages to the voltage E1 the earth-fault factors of the phases Y and B,
δY and δB are obtained.
3
δY = −0.5 ∗
(5.6a)
1 + (2Z 1 /Z 0 ) + j
3
(5.6b)
δB = −0.5 ∗
1 + (2Z 1 /Z 0 ) − j
102 Short-circuit currents
which are different from each other, depending on the impedances and the phase
angle.
The effect of the earthing can be described by the earth-fault factor δ according
to VDE 0141/07.89 and is defined to be the maximum of the earth-fault factors δY
and δB
δ = MAX{δY ; δB } =
ULE max
√
U/ 3
(5.7)
where ULEmax is the highest value of the power-frequency voltage phase-to-earth of
the non-faulted phases in the case of a short-circuit with earth connection and U is
the voltage between phases prior to fault.
Power systems having an earth-fault factor δ < 1.4 are defined as systems with
low-impedance earthing. It should be noted that the single-phase short-circuit currents
shall be below the permissible limits, which are defined by the breaking capability
of circuit-breakers, the short-circuit withstand capability of switchgear, installations
and equipment and by other criteria such as earthing voltage, induced voltages, etc.
Figure 5.3 indicates the earth-fault factors δY and δB in dependence of the ratio
Z 1 /Z 0 and the difference of impedance angles (γ1 − γ0 ). An impedance angle above
90◦ is only possible in the case of a capacitive impedance of the zero-sequence
component but not in systems with low-impedance earthing.
Figure 5.4 presents the earth-fault factor δ in relation to X0 /X1 with the parameter
R0 /X0 , whereas the impedance angle in the positive-sequence component remains
constant. The earth-fault factor δ remains below 1.4 if X0 /X1 ≤ 5 can be achieved
and if R0 /X0 is kept below 0.2 (alternatively X0 /X1 ≤ 4 and R0 /X0 < 0.3).
An impedance ratio X0 /X1 = 2–4 can easily be achieved in power systems as
the relation of zero-sequence to positive-sequence impedances of equipment is
X0 /X1 ≈ 4
X0 /X1 ≈ 3
X0 /X1 ≈ 0.3
5.3
Parallel double-circuit overhead lines
Single-circuit overhead lines and HV-transformers Yy(d)
Unit transformers Yd in power stations (normally not to be
earthed)
Power system having earthing with current limitation
Earthing with current limitation can be seen in some cases as a special case of the
low-impedance earthing, provided the earth-fault factor is below 1.4. Earthing with
current limitation is applied in urban power systems having rated voltage Un ≤ 20 kV.
Some applications are known in systems with nominal voltage up to 132 kV.
The criterion for the design of the earthing conditions is the value of the singlephase short-circuit current, which can be limited to some kA (1 kA or 2 kA) in mediumvoltage systems or to some 10 kA in high-voltage systems (e.g., below the three-phase
short-circuit current). To realise the scheme of earthing with current limitation, the
neutrals of some or all transformers are earthed through reactances or resistances
Influence of neutral earthing 103
(a)
2.5
2
120°
1.5
Degree
0°
30°
150°
dY
1
0.5
60°
90°
0
0
(b)
0.5
1
1.5
Z1/Z0
2
2.5
3
2
2.5
3
4
3.5
3
30°
2.5
dB
Degree
0°
2
60°
1.5
120°
1
90°
150°
0.5
0
0
Figure 5.3
0.5
1
1.5
Z1/Z0
Earth-fault factors in relation to Z 1 /Z 0 and (γ1 − γ0 ). (a) Earth-fault
factor δY and (b) earth-fault factor δB
to such an amount that the condition for the single-phase short-circuit is fulfilled.
As a disadvantage it should be noted that the earth-fault factor δ might exceed the
value of 1.4, which seems to be acceptable in medium-voltage systems with nominal
voltages Un = 10–20 kV. In high-voltage systems with Un = 110–132 kV the
advantages and disadvantages have to be analysed in more detail.
In order to estimate the required value of the earthing impedance, the zero /I
sequence impedance is considered based on Figure 5.5, indicating the ratio of Ik1
k3
as well as the earth-fault factor δ in relation to X0 /X1 . As an example, a mediumvoltage system with Un = 10 kV having an initial three-phase short-circuit power
= 100–250 MVA (I = 5.8–14.4 kA) is regarded. In this case for the limitation
Sk3
k3
104 Short-circuit currents
1.8
0.01
0.02
1.6
0.5
1.4
1.0
2.0
3.0
d 1.2
R0 /X0
1
0.8
0.6
0
Figure 5.4
1
2
3
4
5
6
X0/X1
7
8
9
10
11
12
Earth-fault factor δ depending on X0 /X1 for different ratios R0 /X0 and
R1 /X1 = 0.01
2.5
d
2.0
3
0.8 3
I ⬙k3
d;
I ⬙k1
1.5
1.0
Low-impedance
earthing
d
0.5
I ⬙k1/I ⬙k3
0
–12
Figure 5.5
–10
–8
–6
–4
–2
0
X0/X1
2
4
6
8
10
/I depending on X /X
Earth-fault factor δ and ratio Ik1
0
1
k3
12
Influence of neutral earthing 105
= 2 kA the ratio X /X = 6.7–19.6
of the single-phase short-circuit current to Ik1
0
1
is required. The earth-fault factor in this case will be δ = 1.44–1.61. By this, the
system is no longer a system with low-impedance earthing.
5.4 Power system with isolated neutral
The operation of power systems with isolated neutrals is applicable to systems
with nominal voltages up to 60 kV, however the main application is seen in power
station auxiliary installations and industrial power systems with voltages up to 10 kV.
In public supply systems, isolated neutrals are not very common.
The analysis of a single-phase earth-fault is based on Figure 5.6.
(a)
~
B (L3)
~
Y (L2)
~
R (L1)
U
E = n
3
GE
CE
(b)
~
E1
01/02
GE
CE U0
00
Figure 5.6
Power system with isolated neutral with single-phase earth-fault.
(a) Equivalent circuit diagram in RYB-system and (b) equivalent circuit
diagram in the system of symmetrical components
Contrary to power systems with low-impedance earthing or earthing with
current limitation the capacitances phase-to-earth capacitances in the zero-sequence
component cannot be neglected in power systems with isolated neutral as can be
seen from Figure 5.6. To determine the respective parameters of the equipment,
no-load measurements are necessary. The single-phase earth-fault current, in general,
106 Short-circuit currents
is calculated by
I1
√
c ∗ 3 ∗ Un
=
|2 ∗ Z 1 + Z 0 |
(5.8)
where Un is the nominal system voltage, c is the voltage factor as per Table 4.1 and
Z 1 ; Z 0 are the positive- and zero-sequence impedances, respectively.
The zero-sequence impedance Z 0 is determined by the capacitance phase-toearth CE , and is significantly higher than the positive-sequence impedance Z 1 . The
single-phase earth-fault current is determined through the capacitive component by
√
I R = I CE = j ω ∗ CE ∗ 3 ∗ Un
(5.9)
and is called capacitive earth-fault current I CE . As the capacitive earth-fault current
is significantly lower than a typical short-circuit current, in most of the cases even
lower than the normal operating current, the single-phase fault in a system with
isolated neutral is called earth-fault instead of short-circuit. The earth-fault current
increases with increasing capacitance phase-to-earth and by this with increasing line
length as can be seen from Equation (5.9). Small capacitive currents in the case of
faults through air can be extinguished by themselves if they remain below some 10 A
depending on the voltage level. Figure 5.7 indicates the limits for self-extinguishing
of capacitive currents ICE according to VDE 0228 part 2/12.87.
A
140
120
100
ICE
80
60
ICE
40
20
0
Figure 5.7
3
10
20
Un
30
60
110 kV
Limit for self-extinguishing of capacitive currents in air according to
VDE 0228 part 2
The voltages (phase-to-earth) of the non-faulted phases in the case of an earthfault are increasing to the amount of the phase-to-phase voltage, as can be seen
from Figure 5.8. Prior to fault the voltage potential of earth (E) and neutral (N)
are identical, the phase-to-earth voltages are symmetrical as well as the line-to-line
voltages. During the earth-fault, the voltage of the faulted phase (R) is identical to
Influence of neutral earthing 107
(a)
R;E
(b)
UR
UB
E,N
UB
UY
UY
N
B
Figure 5.8
UNE
Y
Vector diagram of voltages, power system with isolated neutral. (a) Prior
to fault and (b) during earth-fault
the voltage of the earth (E). The voltage potential of the neutral (N) is given, by
definition, as the mean value of the three phases R, Y and B which is not changed by
the earth-fault. A voltage displacement U NE between neutral and earth equal to the
line-to-earth voltage is originating from the earth-fault. The voltage displacement is
equal to the voltage U 0 of the zero-sequence component. As the impedance of the
zero-sequence component is significantly higher than the impedances of the positiveand negative-sequences system, the displacement voltage is identical with the voltage
at the transformer neutral. The voltages of the non-faulted phases are increased, but
the three voltages phase-to-phase remain symmetrical as outlined in Figure 5.8(b).
The capacitive earth-fault current and the recovery voltage at the fault location
have a phase displacement of nearly 90◦ . At the instant of the maximum of the recovery
voltage or shortly after it, a reignition of the fault arc is possible and probable. The
time courses of the phase-to-earth voltages uR , uY and uB and of the displacement
voltage uNE as well as the earth-fault current iCE are outlined in Figure 5.9 indicating
the time prior, during and after the occurrence of the earth-fault.
The earth-fault occurs at time instant t1 , phase R having the maximal voltage. The
phase-to-earth voltage of the non-faulted phases Y and B are increasing to the value
of the phase-to-phase voltage. The displacement voltage uNE increases from a very
low value, ideally zero, to the phase-to-earth voltage. The transient frequency can be
calculated by
f ≈
2∗π ∗
√
1
3 ∗ L 1 ∗ C0
(5.10)
where L1 is the inductance of the positive-sequence system and C0 the capacitance
of the zero-sequence system.
The earth-fault arc is extinguished at time t2 approximately 10 ms after ignition
of the earth-fault; the current iCE has its zero-crossing, whereas the displacement
voltage has nearly reached its peak value. The three phase-to-earth voltages uR , uY
and uB are symmetrical to each other, however with a displacement determined by the
displacement voltage at the time of arc extinguishing, i.e., the displacement voltage
is equal to the peak value of the phase-to-earth voltage. Approximately 10 ms after
the extinguishing
√ of the arc the phase-to-earth voltage of phase R reaches the new
peak value 2 ∗ 2 ∗ Un .
108 Short-circuit currents
^
u/U ^
2.0
1.0
uR
3
1.5
uNE
uY
uR
uNE
uY
0.5
uB
uR
0
uB
–0.5
–1.0
Voltages
uY
uB
–1.5
t1
t3
Earth-fault
current
iCE
Figure 5.9
t2
Time courses of phase-to-earth voltages, displacement voltage and
earth-fault current. System with isolated neutral, earth-fault in phase R
This voltage may cause a reignition of the earth-fault due to the very high-voltage
stress. This reignition takes place at time instant t3 with the phase-to-earth voltage
of phase R having its peak value. The voltages of the non-faulted phases again
are increasing,
this time starting from a higher value and reaching the peak value
√ √
nearly to 3 ∗ 2 ∗ Un .
Besides the power-frequency overvoltage in the case of an earth-fault, the transient
overvoltage with frequency according to Equation (5.10) has to be considered. The
overvoltage factor kLE , taking account of both types of overvoltages, is given by the
maximal peak voltage related to the peak value of phase-to-earth voltage
uü
(5.11)
kLE = √
√
2 ∗ U/ 3
where uü is the maximal peak voltage during the earth fault and U the phase-to-earth
voltage (power-frequency).
In theory, the overvoltage factor after multiple reignition of the earth-fault can
reach kLE = 3.5. Due to the system damping, the overvoltage factor will be below
kLE < 3 in most of the cases.
5.5
Power system with resonance earthing (Petersen-coil)
5.5.1 General
Power systems with resonance earthing are widely in operation in Central European
countries. The German power system statistic [3] indicates that 87 per cent of the
Influence of neutral earthing 109
MV-systems having nominal voltages Un = 10–30 kV and nearly 80 per cent of
110-kV-systems are operated with resonance earthing (Criteria: Total line lengths).
Some MV-systems are operated with a combined scheme of resonance earthing
under normal operating conditions and low-impedance earthing in case of earthfault. Resonance earthing, therefore, is the dominating type of system earthing in
Germany for power systems with voltage 10 kV up to 110 kV. In other countries such
as India, South Africa and China, power systems with resonance earthing have gained
an increasing importance during the last decades, however are still not so common
as systems with low-voltage earthing.
Resonance earthing is realised by earthing of one or several neutrals of transformers through reactances (Petersen-coils), normally adjustable, which will be set in
resonance to the phase-to-earth capacitances of the system. The principal arrangement
of a power system with resonance earthing is outlined in Figure 5.10.
The impedances of transformers and lines of the positive-sequence component can
be neglected compared with those of the zero-sequence component due to the order
of magnitude of the impedances. The admittance of the zero-sequence component is
(a)
~
B (L3)
~
Y (L2)
~
R (L1)
E =
LD
Un
3
GE
CE
U0
310
(b)
~
E1
01/02
3RD
3LD
GE
CE
U0
00
Figure 5.10
System with resonance earthing, earth-fault in phase R. (a) Equivalent
diagram in RYB-system and (b) equivalent diagram in the system of
symmetrical components
110 Short-circuit currents
given by
Y 0 = j ω ∗ CE +
1
+ GE
3 ∗ RD + j 3 ∗ X D
(5.12)
where CE is the phase-to-earth capacitance of the system, ω is the angular frequency
of the system, RD is the resistance of the Petersen-coil, XD is the reactance of the
Petersen-coil XD = ωL and GE is the admittance representing the phase-to-earth line
losses.
After some conversions it follows that
1
+ GE (5.13a)
Y 0 = j ω ∗ CE ∗ 1 −
3 ∗ ω2 ∗ LD ∗ CE ∗ (1 − j (RD /XD ))
The impedance of the Petersen-coil appears with its threefold value in the zerosequence component [1]. It is assumed that RD ≪ XD and that the losses of the
Petersen-coil are summed up with the phase-to-earth losses and are represented as
admittance GE of the line. The admittance in the zero-sequence component is then
1
(5.13b)
+ GE
Y 0 = j ω ∗ CE ∗ 1 −
2
3 ∗ ω ∗ LD ∗ CE
The maximal impedance is obtained if the imaginary part as per Equation (5.13b) is
equal to zero; the current from the Petersen-coil ID is equal to the capacitive current
ICE of the system. As indicated in Figure 5.10, the capacitance phase-to-earth CE ,
the reactance 3LD and the ohmic losses R0 = 1/GE are forming a parallel resonance
circuit with the resonance frequency
1
ω= √
3 ∗ L D ∗ CE
(5.14)
The resonance frequency in the case of resonance earthing shall be the nominal
frequency of f = 50 Hz or f = 60 Hz, respectively. Defining the detuning factor v
v=
ID − ICE
1
=1−
ICE
3 ∗ ω 2 ∗ L D ∗ CE
(5.15a)
GE
ω ∗ CE
(5.15b)
and the damping d
d=
the admittance of the zero-sequence component is given by
Y 0 = ω ∗ CE ∗ (j v + d)
(5.16)
The admittance will be minimal and the impedance will be maximal in the case of
resonance tuning (v = 0). The earth-fault current I Res , in general, is obtained by
√
(5.17a)
I Res ≈ 3 ∗ Un ∗ ω ∗ CE ∗ (j v + d)
In case of resonance tuning (v = 0) the earth-fault current is a pure ohmic current
√
(5.17b)
I Res ≈ 3 ∗ Un ∗ ω ∗ CE ∗ d
Influence of neutral earthing
111
The phase-to-earth voltages of the non-faulted phases increase to the value of the
phase-to-phase voltage in the case of a single-phase earth-fault, which is furthermore
increased due to asymmetrical system voltages resulting in a higher displacement
voltage between neutral and earth. In order to avoid the high voltages in the case of
exact resonance tuning a small detuning of 8–12 per cent is chosen in practice.
The task of resonance earthing is to reduce the earth-fault current at the fault
location to the minimum or nearly to the minimum by adjusting the Petersen-coil to
resonance or nearly to resonance with the phase-to-earth capacitances. The ohmic
part of the residual current IRes cannot be compensated by this. If the residual current
is small enough, a self-extinguishing of the arc at the fault location is possible.
VDE 0228 part 2:12.87 defines the limits for self-extinguishing of residual currents
IRes (and capacitive earth-fault currents ICE ) for different voltage levels as outlined in
Figure 5.11. It can be seen from Figure 5.11 that the limit for ohmic currents, e.g., in
30-kV-systems, is twice the limit for capacitive currents.
The Petersen-coil can only be tuned for one frequency (nominal frequency) in
resonance; harmonics present in the system voltage are increasing the residual current
at the fault location.
As the phase-to-earth capacitances are changing during system operation, e.g.,
due to switching of lines, the Petersen-coil has to be changed also to keep system
operation with resonance tuning. Reliable criteria have to be established to tune the
Petersen-coil in resonance with the phase-to-earth capacitances.
A
140
120
100
IRes
IRes ; ICE
80
60
ICE
40
20
0
3
10
20
30
60
110 kV
Un
Figure 5.11
Current limits according to VDE 0228 part 2:12.87 of ohmic currents
IRes and capacitive currents ICE
112 Short-circuit currents
5.5.2 Calculation of displacement voltage
In real power systems, the phase-to-earth capacitances are unequal, e.g., in the case of
a transmission line due to different clearance of the phase-wires above ground or the
case of cables due to manufacturing tolerances. Under normal operating conditions,
a displacement voltage between transformer neutral and earth U NE can be measured.
As mentioned in previous sections, this voltage is equal to the voltage U 0 in the zerosequence component. The calculation of the displacement voltage can be carried
out in the RYB-system (Figure 5.12(a)) as well as with the system of symmetrical
components (Figure 5.12(b)).
Based on Figure 5.12(a) the displacement voltage is calculated as
j ω ∗ (CRE + a 2 ∗ CYE + a ∗ CBE )
Un
U NE = √ ∗
3 j ω ∗ (CRE + CYE + CBE ) − j (1/(ω ∗ LD )) + 3 ∗ GE
(a)
~
B (L3)
~
Y (L2)
~
R (L1)
E=
(5.18)
Un
3
CBE
CYE
GE
CRE
ΔC
U0 = UNE
LD
310
(b)
~
1+2
E1
01/02
1
3
3LD
GE
CE
U0
ΔCE
0
00
Figure 5.12
Equivalent circuit diagram of a power system with asymmetrical
phase-to-earth capacitances. (a) Equivalent circuit diagram in the
RYB-system and (b) equivalent circuit diagram in the system of
symmetrical components
Influence of neutral earthing 113
where Un is the nominal system voltage, ω is the angular frequency of the system, CRE ;
CYE ; CBE are the line-to-earth capacitances as per Figure 5.12(a), LD is the inductance
of the Petersen-coil and GE is the admittance representing the phase-to-earth line
losses.
If the phase-to-earth capacitances are different and if the asymmetry is assumed
to be placed in phases R and Y, the capacitances are
CRE = CE +
CRE
(5.19a)
CYE = CE +
CYE
(5.19b)
(5.19c)
CBE = CE
where CRE ; CYE are the asymmetry of the line-to-earth capacitances.
The displacement voltage is given by
Un
U NE = √ ∗
3 (3 ∗ CE +
CRE + a 2 ∗ CYE
CRE + CYE ) − j (1/(ω ∗ LD )) + 3 ∗ GE
(5.20)
Defining the asymmetry factor k
k=
=
CRE + a 2 ∗ CYE + a ∗ CBE
CRE + CYE + CBE
CRE + a 2 ∗ CYE
3 ∗ CE + CRE + CYE
(5.21a)
the system damping d
3 ∗ GE
ω ∗ (CRE + CYE + CBE )
3 ∗ GE
=
ω ∗ (3 ∗ CE + CRE + CYE )
d=
(5.21b)
and the detuning factor v
1/(ω ∗ LD ) − ω ∗ (CRE + CYE + CBE )
ω ∗ (CRE + CYE + CBE )
1/(ω ∗ LD ) − ω ∗ (3 ∗ CE + CRE + CYE )
=
ω ∗ (3 ∗ CRE + CRE + CYE )
v=
(5.21c)
the displacement voltage U NE is calculated by
k
Un
U NE = √ ∗
3 v + jd
(5.22)
Assuming the asymmetrical capacitance CE concentrated in phase R ( CE ≫
CRE and CE ≫ CYE ) the displacement voltage U NE , equal to the voltage
in the zero-sequence component U 0 , is calculated with the system of symmetrical
114 Short-circuit currents
components based on Figure 5.12(b)
j ω ∗ CE
Un
1
U0 = √ ∗
∗
2 ∗ L ∗ C )) − j (G (ω ∗ C ))
j
ω
∗
3
∗
C
1
−
(1/(3
∗
ω
3
E
D
E
E
E
(5.23)
The asymmetry factor k, the system damping d and the detuning factor v can be
calculated based on these assumptions:
CE
3 ∗ CE
GE
d=
ω ∗ CE
(5.24a)
k=
v =1−
3 ∗ ω2
(5.24b)
1
∗ LD ∗ C E
(5.24c)
The displacement voltage U NE , equal to the voltage in the zero-sequence component
U 0 , is calculated by
k
Un
U NE = U 0 = √ ∗
3 v + jd
(5.25)
The polar plot of the displacement voltage U NE as per Equations (5.22) and (5.25)
and outlined in Figure 5.13 indicates a circular plot through the zero point. The
phase angle of the diameter location at v = 0 is determined by the phase angle of the
capacitive asymmetry. The diameter of the polar plot is defined as per Equation (5.25)
as the ratio of capacitive asymmetry k and damping d.
The capacitive asymmetry is comparatively high in power systems with overhead
transmission lines, resulting in a sufficient high-displacement voltage. Cable systems
have a comparative small asymmetry, resulting for most of the cable systems in
+ Re
UR
+ Im
y=0
UY
UB
Figure 5.13
Polar plot of the displacement voltage in a power system with resonance
earthing
Influence of neutral earthing 115
an insufficient low-displacement voltage and problems while tuning the Petersencoil into resonance. Capacitors between two phases or between one phase and earth
will increase the displacement voltage to the required value.
5.5.3 Tuning of the Petersen-coil
ULE
The Petersen-coil can be constructed as a plunger-coil (tuning-coil) with continuous
adjustment of the reactance, which can be tuned into resonance by successive operation. The displacement voltage measured at the Petersen-coil is maximal in the case
of resonance tuning; the value depends on the capacitive asymmetry and on the losses
of the reactor. The earth-fault current will be minimal in this case and the power frequency component of the capacitive earth-fault current is compensated by the reactive
current of the Petersen-coil. Figure 5.14 indicates the displacement voltage and the
residual current for different tuning of the reactor.
The displacement voltage shall be limited to UNE < 10 kV. It is obvious that the
residual current is increased as can be seen from Figure 5.14. Residual currents
above 130 A in 110-kV-systems, respectively 60 A in 10-kV-systems, are not
self-extinguishing; both parameters define the tuning limits of the Petersen-coil as
indicated in Figure 5.14. Tuning of the Petersen-coil can be done in such a way
that the resonance circuit is either capacitive (undertuning; v < 0), resulting in an
Umax
ULE
Unsymmetry
UR
UY
100%
Umin
ULE
U0; IRes
UB
IRes
IRes< 130 A
U0 < 10 kV
U0
–30%
–15%
0
Max. voltage Umax
Min. voltage Umin
15%
Permissible
tuning range
V
30%
Max. displacement voltage Umin
Max. voltage unbalance ku
Max. residual earth-fault current IRes
Figure 5.14
Voltages and residual current in the case of an earth-fault; displacement voltage without earth-fault
116 Short-circuit currents
ohmic-capacitive residual current or inductive (overtuning; v > 0), resulting in an
ohmic-inductive residual current at the earth-fault location. A small overtuning (overcompensation) up to v = 10% is often recommended as the displacement voltage
will not increase in the case of switching of lines, because the capacitances will be
reduced by this and the resonance circuit will be detuned without any further adjustment. The limits for the displacement voltage and the residual current as indicated in
Figure 5.14 have to be guaranteed even under outage conditions.
Figure 5.14 also indicates the phase-to-earth voltages for different tuning factors
(system parameters are: Un = 110 kV; ICE = 520 A; d = 3%; k = 1.2%) which
also limit the range of detuning√
of the Petersen-coil. Assuming a minimal permissible
voltage of Umin = 0.9 ∗ (Un / 3) according to IEC 60038,
√ a maximal permissible
voltage according to IEC 60071-1 of Umax = 123 kV/ 3 and a permissible asymmetry of the three voltages according to DIN EN 50160 of p = 2% it can be seen
that the permissible tuning range of the Petersen-coil is v = 12–22%.
All considerations carried out so far are based on a linear current-voltagecharacteristic of the Petersen-coil.
√ Figure 5.15 indicates the non-linear characteristic
of a Petersen-coil (Ur = 20 kV/ 3; Ir = 640 A) for minimal and maximal adjustment.
Due to the non-linear characteristic, the minimum of the residual current is not
achieved at the maximal displacement voltage (adjustment criteria of the Petersencoil). The difference is typically in the range of 3–15% of the rated current as outlined
in Figure 5.16.
5.6 Handling of neutrals on HV-side and LV-side of transformers
Special attention must be placed while selecting the type of neutral handling on
HV-side and LV-side of transformers. The neutral earthing on one side of the transformer has an influence on the system performance on the other side, in case of
earth-faults or single-phase short-circuits as the voltages in the zero-sequence component are transferred from one side of the transformer to the other. The neutral
earthing of a 110/10-kV-transformer (vector group Yyd) according to Figure 5.17 is
taken as an example. It is assumed [15] that XC0 /R0 = 0.1–0.05, first value is applied
for systems with overhead lines, second value with cables.
The impedances Z E1 and Z E2 as per Figure 5.17 representing the earthing are
different depending on the type of neutral earthing. In case of a single-phase fault in
the high-voltage system (110 kV), the voltage U0 in the zero-sequence component
is transferred to the medium-voltage system (10 kV) with the same amount. Similar
consideration indicates that the voltage in the zero-sequence component is transferred
to the HV-side in case of a single-phase fault in the LV-system. In both cases, a fault
current is measured in the system, having no fault. Table 5.2 indicates the results of
a fault-analysis [16] with the voltages transferred through the transformer in case of
faults.
The 110/10-kV-transformer can be operated with low-impedance earthing on both
sides if a third winding (compensation winding, vector group d) is available, as can be
seen from Table 5.2. If the transformer is not equipped with compensation winding,
Influence of neutral earthing 117
(a) 70
60
Current in A
50
40
30
20
10
0
0
2000
4000
6000
8000
Voltage in V
10,000
12,000
14,000
0
2000
4000
6000
8000
Voltage in V
10,000
12,000
14,000
(b) 800
700
Current in A
600
500
400
300
200
100
0
Figure 5.15
√
Current-voltage characteristic of a Petersen-coil; Ur = 20 kV/ 3;
Ir = 640 A. (a) Minimal adjustment (50 A) and (b) maximal adjustment
(640 A)
the voltages in the zero-sequence component may reach values up to 70 per cent of
the phase-to-earth voltage.
Low-impedance earthing on the 110-kV-side and resonance earthing on the
10-kV-side should be avoided due to high voltages in the zero-sequence component,
which furthermore depend on the tuning of the Petersen-coil. The maximal voltage
in this case is not reached for resonance tuning but depends on the ratio XC0 /R0 . The
strategy to limit the displacement voltage under normal operation conditions as per
Section 5.5.2 may result in an increased displacement voltage in the 10-kV-system
with resonance earthing in the case of an earth-fault in the 110-kV-system with
low-impedance earthing.
118 Short-circuit currents
IRest
UEN
ΔI
IRest = f (IEN)
UEN = f (IEN)
Vl
IEN = IDr
VU
Figure 5.16
Displacement voltage in non-faulted operation and residual current under earth-fault conditions; non-linear characteristic of the
Petersen-coil
Resonance earthing in the 110-kV-system can be combined with all types of neutral earthing in the 10-kV-system if the transformer is equipped with a compensation
winding. The connection of Petersen-coils to both neutrals (110- and 10-kV) has
to be investigated for special cases and is not generally recommended. The voltage
transfer by stray capacitances in the case of isolated neutral in the 10-kV-system can
be reduced by installing capacitances in the 10-kV-system. If the earthing of both
neutrals of transformers by Petersen-coils cannot be avoided in the same substation,
the earthing should be alternate in the case of two parallel transformers as indicated
in Figure 5.18(a). If only one transformer is installed, the connection of one Petersencoil XD1 can be carried out directly to the transformer, the second one XD2 should be
connected at an artificial neutral as per Figure 5.18(b).
If the feeding system (e.g., 110 kV) is operated with low-impedance earthing and
the medium-voltage system (e.g., 20 kV) is earthed through Petersen-coils or by fault
limiting impedance, fault currents will occur in the medium-voltage system in the case
of a single-phase short-circuit in the high-voltage system, as outlined in Figure 5.19,
the value of which depends on the impedance of the earthing in the medium-voltage
system. In some cases, this current may exceed the rated current of the transformer,
thus causing operation of power system protection on MV-side [5,16].
Influence of neutral earthing 119
(a)
110-kV-system
Transformer
10-kV-system
~
B
Y
~
R
~
E
~
~
ZE1
ZE2
k1
X1Q
(b)
~
X11T
C1Q
U1F
X12T
C1
X13T
01
X1Q
X11T
C1Q
U2F
X12T
C1
X13T
02
X0Q
X01T + 3ZE1
~
C0Q
U0F
X12T + 3ZE2
~
X03T
R0
C0
U0
00
Figure 5.17
Transformation of voltage in the zero-sequence component of transformers in the case of single-phase faults. (a) Equivalent circuit
diagram in RYB-system and (b) equivalent circuit diagram in the system
of symmetrical components
5.7 Examples
5.7.1 Increase of displacement voltage for systems with resonance earthing
The capacitive asymmetry k of cable systems normally is below k < 0.1%; the
system damping is in the range of d ≈ 2–4% resulting in a displacement voltage
120 Short-circuit currents
Table 5.2
Voltages in the zero-sequence component U0 transferred through
110/10-kV-transformer in the case of single-phase fault in the 110-kVsystem according to Figure 5.17; Un : nominal system voltage
10-kV-system
110-kV-system and compensation winding
of the transformer
Limitation
Ik1
ZE2
Low-resistance earthing
ZE2 = 0
Current limitation
ZE2 inductive
Current limitation
ZE2 ohmic
Resonance earthing
Isolated neutral
2000 A
500 A
2000 A
500 A
Low-impedance√earthing
u0 = U0 /(Un / 3)
Resonance
earthing
With
compensation
winding
Without
compensation
winding
With
compensation
winding
0.2
0.6
0.03
0.2
0.6
0.03
0.25
0.7
0.04
0.2
0.6
0.03
0.2
0.6
0.03
<7
>10
<0.3
Voltage transfer through stray capacitances
which is too low for the operation of the resonance controller. For a given capacitive
asymmetry of k = 0.1% and
√ system damping d = 2% the displacement voltage
detuning shall
will be UNE = 0.005 ∗ Un / 3 for resonance tuning (v = 0%); if the √
be v = 5% the displacement voltage will be UNE = 0.002 ∗ Un / 3 only [17].
Resonance controller normally
operates sufficiently if the displacement voltage is
√
above UNE > 0.03∗Un / 3. The displacement voltage, therefore, had to be increased
by installing an additional capacitor in one phase [18]. The required capacitance CE
for different parameters is given in Table 5.3 for a 20-kV-system earthed through
Petersen-coil.
Documentation of system data normally do not indicate exact values of the lineto-earth capacitances CE in each phase, furthermore the system damping d, the exact
length of the cables and the non-linear characteristic of the Petersen-coil are also
unknown or only to an insufficient extent and are not suitable to determine the displacement voltage UNE without measurement. It should be noted that the system
damping changes with the system load as well as in the case of a power system with
a significant number of overhead lines and with external climatic conditions. It can
be deducted from this, that:
•
•
•
•
the asymmetrical capacitance (absolute value)
the angle of the asymmetrical capacitance (with respect to the three phases)
the value of the displacement voltage, and
the resonance curve for different detuning factors
Influence of neutral earthing 121
(a)
XD1
110 kV
10 kV
XD2
(b)
110 kV
10 kV
XD1
Figure 5.18
XD2
Alternate earthing of transformer neutrals by Petersen-coils. (a) Two
parallel transformers and (b) earthing at artificial neutral with
reactor XD2
T1
MV-system
e.g., 20 kV
Feeding system
e.g., 110 kV
k1
T2
Z0; Z1; Z2
Currents in the positive-,
negative- and zero-sequence
system
Figure 5.19
Fault current in the MV-system in the case of a short-circuit in the
HV-system
122 Short-circuit currents
Table 5.3
Capacitive asymmetry
CE for different
parameters in a 20-kV-system
UNE = U0
(V)
Damping d
(%)
Detuning v
(%)
Asymmetry k
(%)
CE
(nF)
346.4
2
4
6
4
6
0.134
0.19
0.17
0.216
177
251
225
285
4
6
4
6
0.224
0.316
0.283
0.361
296
417
374
477
4
577.4
2
4
0.9
0.8
05.08.98; 11:42 till 11:57 Uhr
02.07.98; 22:10 till 22:21 Uhr
12.07.98; 15:51 till 16:04 Uhr
+ Calculation
Displacement voltage in V
0.7
0.6
0.5
0.4
+
+++
+
+
+
0.3
0.2
0.1
+
++
+
+
+ +
+
+
+
+
+
+ +
0
300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600
Reactor current in A
Figure 5.20
+
+ +
Resonance curve (displacement voltage) for different detuning factors
in a 20-kV-system for different conditions
cannot be determined from system studies, but require measurement for different load
conditions.
The resonance curves of a 20-kV-system with total cable length of 176 km as
measured for different load conditions are outlined in Figure 5.20. Furthermore, the
resonance curve calculated from documented system data is also given in Figure 5.20.
Voltages√refer to the secondary side of a voltage transformer in the neutral (ratio
20 kV/ 3 : 100 V). The maximal value of each of the resonance curves differ by
Line-to-earth voltages in kV
Influence of neutral earthing 123
12.2
U phase R
12.1
12.0
U phase Y
11.9
U phase B
11.8
11.7
11.6
11.5
11.4
11.3
11.2
11.1
11.0
10.9
10.8
250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625
Reactor current in A
Figure 5.21
Voltages in a 20-kV-system with resonance earthing for different tuning
factors. (a) Phase-to-earth voltages and (b) displacement voltage
(resonance curve)
more than 60 per cent for different load conditions and is higher than the calculated
value. The resonance tuning (v = 0) of calculated and measured resonance curves
differs also by 6 per cent as can be seen from Figure 5.20.
A capacitance of CE = 173.3 nF was installed in phase B in order to increase the
displacement voltage up to 3–5 per cent (350–580 V) of the nominal system voltage
for resonance tuning. The total phase-to-earth capacitance of the 20-kV-system was
determined to be CE = 44 μF. The results of measurement of the phase-to-earth
voltages and the displacement voltage for different detuning
√ of the Petersen-coil are
outlined in Figure 5.21 (voltage transformer ratio 20 kV/ 3 : 100 V).
It can be seen from Figure 5.21 that the maximal displacement voltage is in the
required range between 3 and 5 per cent of the phase-to-earth voltage. For resonance
tuning, the maximal phase-to-earth voltage appears in phase R with UR ≈ 12.1 kV
and the minimal voltage in phase Y with UY ≈ 10.9 kV, which is only 94 per cent of
the nominal phase-to-earth voltage. Exact resonance tuning shall be avoided in this
system due to the low voltage in one phase and the resulting high asymmetry factor.
Other measures to increase the displacement voltage such as installation of a
reactor in one phase to earth and voltage additions are not considered in detail
here [19].
5.7.2 Limitation of single-phase short-circuit current by earthing through
impedance
An urban 11.5-kV-system is fed from a 132-kV-system; both systems are pure cable
systems. The substations are equipped with four transformers each, having rated
values of Sr = 40 MVA, ukr1 = 14% and ukr0 = 16%. Both systems are earthed
124 Short-circuit currents
by low impedance in such a way that all transformer neutrals are solidly connected
to earth. The short-circuit current level in the 132-kV-system is high, whereas the
single-phase short-circuit currents is higher than the three-phase short-circuit currents
≈ 29.3 kA; I ≈ 37.3 kA). The maximal permissible short-circuit current in
(Ik3
k1
the 132-kV-system is Ikmax
= 25 kA. The
= 40 kA; in the 11.5-kV-system Ikmax
132/11.5-kV-transformers cannot be operated in parallel due to the high short-circuit
currents. In the case of low-impedance earthing the neutrals of each transformer, the
= 15.04 kA in case of a
single-phase short-circuit current of one transformer is Ik1
short-circuit at the 11.5-kV-busbar. In the case of two transformers in parallel, the
= 29.27 kA, which is above the maximal
short-circuit current is increased to Ik1
permissible short-circuit current of the 11.5-kV-system.
The limitation of single-phase short-circuit currents to 25 kA in case two transformers shall be operated in parallel is possible by earthing of the 11.5-kV-transformer
neutrals through a resistor having RE = 0.31 or through a reactor having
XE = 0.1 (see also Figure 11.13).
5.7.3 Design of an earthing resistor connected to an artificial neutral
The short-circuit limitation in a 20-kV-system having initial short-circuit power
= 700 MVA and an impedance ratio Z /Z = 4 is explained below. The singleSkQ
0
1
= 12.1 kA. The
phase short-circuit current in case of low-impedance earthing is Ik1
= 3I )
single-phase short-circuit current shall be limited to Ik1max = 1.5 kA (Ik1
0
realised by earthing through a resistor to be connected to an artificial neutral. As the
current through the artificial neutral is only one-third of the single-phase short-circuit
current, the rated power shall be
√
SrS min = 0.33 ∗ 3 ∗ Ik1
max ∗ Un = 17.55 MVA
A transformer with vector group Zz is selected for the artificial neutral with
SrS = 20 MVA; PCu = 1.3 kW and uk0 = 45% resulting in a required zero-sequence
impedance of Z0S = 9 (R0S = 3.75 ; X0S = 8.18 ). The required rating of
the earthing resistor resulting from the calculation is RE = 4.21 . The value of the
initial short-circuit power of the system has only a marginal effect on the rating of
the earthing.
A typical design rating of the resistor is given below:
Rn = 4.21 at 20◦ C (Tolerance ±5 per cent or ±10 per cent); CrNi-alloyed steel
Ir = 1.5 kA, rated short-time duration t = 5 s or 10 s
Ur = 13.3 kV
Isolation according to IEC 60071-1, Table 5.2: Um = 17.5 kV IP 00
5.7.4
Resonance earthing in a 20-kV-system
Figure 5.22 shows a 20-kV-system fed from two sides by one transformer in each of
the substations S/S A and S/S B. The system is split at S/S C into two subsystems A
and B during summer months. During winter months, a part of system A is connected
to system B, but both systems are still operated separately.
Influence of neutral earthing 125
S/S C
300 A
Figure 5.22
S/S B
System B
110 kV
190 A
System A
60–590 A
S/S A
50–500 A
110 kV
6.4 MW
Equivalent circuit diagram of a 20-kV-system with resonance earthing
The 20-kV-system A is a pure cable-system having a total system length of 490 km
of the XLPE-type. The capacitive earth-fault current is calculated to be Ires = 453 A.
The transformer in substation A has a rating of 12.5 MVA; current carrying capacity of the neutral is 361 A. The system shall be operated with resonance earthing,
i.e., a reactor has to be installed in the neutral having a rated current of more than
450 A to compensate the capacitive earth-fault current. It is obvious that the reactor cannot be connected to the neutral of the transformer in substation S/S A as the
required current for resonance earthing exceeds the permissible current of the transformer neutral. An additional fixed reactor with rated current 300 A is installed in
substation S/S C and connected at an artificial neutral of a Zz-transformer.
System B is a pure cable-system as well, having a total system length of 386 km
of different type (XLPE and mass-impregnated cables); the earth-fault current is
calculated to be Ires = 632 A. The feeding transformer in substation B has a rated
power Sr = 25 MVA and a current carrying capacity of the neutral of 721 A. Resonance earthing is done with a fixed reactor (190 A) in parallel with a tuning-reactor
(60–590 A) capable of compensating the total capacitive earth-fault current of
system B.
Operation in wintertime differs from the described scheme as additional generation in a combined-cycle plant is in operation with a power of 6.4 MW. In order to
avoid back-feeding into the 110-kV-system additional load from system A is supplied
from system B. The capacitive earth-fault current of system A is reduced and the current in system B is increased accordingly (see Table 5.4 for details). As the earthing
reactor in S/S C is connected in wintertime as well to system B, the compensation
scheme with different reactors at different substations are capable of realising the
resonance earthing for both systems under different operating conditions.
5.7.5 Calculation of capacitive earth-fault current and residual current
A 10-kV-system with isolated neutral (overhead lines, system length 170 km,
CB = 9.5 nF/km; CE /CL = 5) shall be extended by cables. The existing earthing
126 Short-circuit currents
Table 5.4
Condition
Characteristics of a 20-kV-system with respect to resonance earthing
System A
System B
Ires (A)
Available reactors
Ires
Available reactors
Summertime
453
30–300 A
S/S A
300 A
S/S C
721 A
60–590 A
S/S B
190 A
S/S B
—
Wintertime
261
30–300 A
S/S A
—
913 A
60–590 A
S/S B
190 A
S/S B
300 A
S/S C
concept shall be checked with respect to its suitability. The phase-to-earth capacity is calculated as CE = 1009 μF and the earth-fault current is given according
to Equation (5.9) to be IE = 6.05 A which is self-extinguishing as can be seen in
Figure 5.7. The capacitive earth-fault current is increased by 1.42 A/km if cables of
the NAKBA-type will be installed (CE cable /CE OHL = 40) reaching the permissible
limit for self-extinguishing ICE = 35 A for a cable length of 23 km.
If the system earthing shall be changed to resonance earthing, the Petersen-coil
shall have a reactance for resonance tuning (v = 0) of XD = 165 , the inductance
will be LD = 0.525 H for 50 Hz. The resistance of the reactor is assumed to be RD =
6 k (parallel equivalent diagram). The residual current is given by IRes = 2.9 A.
The ratio of residual current to capacitive current is IRes /ICE = 8.3 per cent.
5.7.6 Voltages at neutral of a unit transformer
A power station with rated power 400 MVA is connected to a 220-kV-system.
The neutral of the unit transformer can be earthed either through an impedance
(reactor), directly without impedance or kept isolated without earthing as outlined in
Figure 5.23. The rating of the reactor shall be determined in such a way to guarantee a
ratio X0 /X1 = 2 in case of a short-circuit at location F. The single-phase short-circuit
G
A
T1
F
GS
3~
Un = 220kV
Us
S2
S1
Xs
T2
Q
UnQ = 380 kV
Figure 5.23
Connection of a power station to a 220-kV-system with short-circuit
location
Influence of neutral earthing 127
current and the voltage at the transformer neutral shall be calculated for the different
operating conditions.
The rated data of the equipment are given below:
= 15,000 MVA; X /X
System Q UnQ = 380 kV; SkQ
0Q
1Q = 3
System A
G
T1
T2
= 5,000 MVA; X /X
UnQ = 220 kV; SkQ
0A
1A = 3
= 15%;
SrG = 400 MVA; UrG = 21 kV; xd1
cos ϕrG1 = 0.8; pG1 = 10%
SrT1 = 400 MVA; ukrT1 = 14%;
UrT1HV /UrT1LV = 220 kV/21 kV; pT1 = 12%
SrHVT2 = 660 MVA; SrMVT2 = 660 MVA;
SrLVT2 = 198 MVA
ukrHVMVT2 = 10.2%; ukrHVLVT2 = 13.5%;
ukrMVLVT2 = 10.5%
UrHVT2 = 380 kV; UrMVT2 = 220 kV; UrLVT2 = 30 kV;
pT2 = 10%
Line L
X0T2 /X1T2 = 1 (three single-phase transformers)
= 0.3 /km; X = 1.0 /km; l = 10 km
X1L
L
0L
The impedances of the equipment calculated in the -system are given below.
No.
Equipment
X1
()
X0
()
1
2
3
4
3+4
System Q
System A
Generator G
Transformer 1
Power station
10.595
3.532
18.15
16.94
35.09
42.38
10.596
5
Line
3.0
16.94
16.94
+3XS
10.0
Correction X1K
as per
()
Table 3.3
X0K
()
10.595 42.38
3.532 10.596
1.093
38.35
3.0
From
right
18.681
From
left
16.94+
3Xs
10.0
Ik1
Ik3
(kA) (kA)
128 Short-circuit currents
No. Equipment
X1
()
X1
()
6
7.422
7.422
Transformer
2 HV
MV
LV
Correction X1K
as per
()
Table 3.2
HV-MV
0.985
0.3436 0.3436 MV-LV
0.983
25.59 25.59 HV-LV
0.967
Total
impedance
at 220 kV
(X0 /X1 = 2)
Ik3
(kA)
X0K
()
7.023
7.023
0.399
0.399
Ik1
(kA)
25.586 25.586
6.945
13.89 !! 18.29 15.09
X0A
X0T1
XL
X0MVT2
Us
S2
X0HVT2
X0HVT2
S1
X0LVT2
3Xs
Figure 5.24
Equivalent diagram in the zero-sequence component for fault
location F
It should be noted that the impedance of the earthing reactor appears threefold
(X0S = 3XS ) in the zero-sequence component. As the rating of the reactor shall be
determined to guarantee the ratio X0 /X1 = 2 for short-circuit at location F the single = 15.09 kA. By this the impedance of the reactor
phase short-circuit current is Ik1
will be XS = 12.4 .
In order to calculate the voltage U S at the transformer neutral while earthed with
the reactor, the zero-sequence current through the reactor I 0S needs to be calculated
as outlined in Figure 5.24 indicating the equivalent diagram in the zero-sequence
component.
The zero-sequence current through the reactor is determined by the ratio of the
impedances
I 0S =
I
13.89 15.09 kA
X0k
X0k
∗
= 1.29 kA
∗ I0 =
∗ k1 =
X0left
X0left
3
54.16
3
Influence of neutral earthing 129
The voltage across the reactor USe at the transformer neutral is calculated by
U Se = I 0S ∗ 3 ∗ XS = 1.29 kA ∗ 3 ∗ 12.41 = 48.03 kV
In this case, the transformer neutral is kept isolated (S1 and S2 open); the zerosequence current through the reactor will be Zero. The voltage is determined by the
zero-sequence current determined by the remaining equipment (see Figure 5.24). The
zero-sequence impedance is X0 = 18.681 . The single-phase short-circuit current
without earthing is calculated by
√
√
1.1 ∗ 3 ∗ 220 kV
1.1 ∗ 3 ∗ Un
=
= 12.87 kA
I k1 =
X0 + 2 ∗ X 1
18.681 + 2 ∗ 6.945
The voltage across the reactor U Si is calculated by
I k1
12.87 kA
∗ X0 =
∗ 18.681 = 80.14 kV
3
3
The insulation of the transformer neutral must be designed for the maximal voltage,
i.e., 80.14 kV.
U Si =
Chapter 6
Calculation of short-circuit currents in
low-voltage systems
6.1 General
IEC 60781 presents an application guide for the calculation of short-circuit currents
in low-voltage radial systems. The methods described there are identical to those
as per IEC 60909-0 and as outlined in Chapter 3. The short-circuits are treated as
far-from-generator short-circuits. This assumption is valid in the future as well, even
with an increasing number of distributed generation units in low-voltage systems.
New generation will be connected to the system in the case of
• Photovoltaic installations by rectifier (six-pulse bridge or PWM-rectifier).
• Small wind turbines, low-rated combined heat and power units and small
hydro power plants by asynchronous generators (rotor fed).
Installations with synchronous generators are comparatively rare.
The following chapters describe the approach and special conditions for the
calculation of short-circuit currents in low-voltage systems.
6.2 Types of faults
Depending on the type of protection against electrical shock in low-voltage systems
all types of short-circuit, i.e., three-phase, double-phase with and without earth connection and single-phase-to-earth short-circuits can occur. The maximal short-circuit
current depends on the impedances of the positive- and zero-sequence component.
Reference is made to Figure 4.4. The ratio Z2 /Z1 can be set to 1, as low-voltage
systems for the most part have no generation by synchronous generators. The threephase short-circuit will lead in those cases to the maximal short-circuit current. Special
attention must be given to the currents flowing through earth, as the phase angles of
the impedances in the positive- and zero-sequence components differ a lot from each
other in low-voltage systems.
132 Short-circuit currents
6.3 Method of calculation
The method of the equivalent voltage source at the short-circuit location is applied
for the calculation of short-circuit currents in low-voltage systems.
•
•
•
•
•
•
Symmetrical short-circuits are represented by the positive-sequence component, asymmetrical (unbalanced) short-circuits are represented by connection of
positive-, negative- and zero-sequence components (see also Table 2.1).
Capacitances and parallel admittances of non-rotating load of the positivesequence component are neglected. Capacitances and parallel admittances of
the zero-sequence component have only an influence on fault currents in power
systems with isolated neutral or with resonance earthing.
Impedance of the arc at the short-circuit location is neglected.
The type of short-circuit and the system topology remain unchanged during the
duration of the short-circuit.
The tap-changer of any transformer is assumed to be in main-position.
All internal√voltages are short-circuited and an equivalent voltage source with
value cUn / 3 is introduced at the short-circuit location. The voltage factor c
shall be selected in accordance with Table 6.1.
6.4 Calculation of short-circuit parameters
6.4.1 Impedances
Calculation of impedances of equipment and the analysis of the short-circuit
impedance at short-circuit location was explained in Sections 2.5 and 3.2. It should
be noted that the impedances of the equipment must be related to the voltage level of
the short-circuit location.
Table 6.1
Voltage factor c according to IEC 60909-0. (Voltage factors
as per IEC 60781 are of different values. The standard is
under review)
Nominal system voltage Un
LV: 100 V up to 1000 V (inclusive)
(IEC 60038, Table 1)
Voltage tolerance +6%
Voltage tolerance +10%
Voltage factor c for calculation of
Maximal
short-circuit
current, cmax
Minimal
short-circuit
current, cmin
1.05
1.10
0.95
0.95
Remark: cmax Un shall not exceed the highest voltage of equipment Um according to
IEC 60071
Calculation of short-circuit currents 133
When data for generation in LV-systems are not known, approximation as below
shall be used for the branch short-circuit currents [12]:
Synchronous generator
Asynchronous generator
Generator with rectifier
Branch short-circuit current equal to eight times rated
current
Branch short-circuit current equal to six times rated
current
Branch short-circuit current equal to rated current
This approximation does not include the effect of conductors between the generation
unit and the short-circuit location.
6.4.1.1 Initial symmetrical short-circuit current Ik
The initial symmetrical short-circuit current Ik is calculated for balanced and unbalanced short-circuits based on the equivalent voltage source at the short-circuit
location. The short-circuit impedance seen from the short-circuit location has to be
determined with the system of symmetrical components. The results obtained for
the short-circuit currents (and the voltages of the non-faulted phases, if required)
in the 012-system have to be transferred back into the RYB-system. The results for
the different types of short-circuits are outlined in Table 4.2.
6.4.1.2 Peak short-circuit current ip
Depending on the feeding source of the short-circuit, different considerations have
to be taken to calculate the peak short-circuit current. Short-circuits in low-voltage
systems normally are single-fed short-circuits. The short-circuit impedance is represented by a series connection of the individual impedances. The peak short-circuit
current can be calculated for the different types of short-circuit by
√
ip3 = κ ∗ 2Ik3
(6.1a)
√
ip2 = κ ∗ 2Ik2
(6.1b)
√
ip1 = κ ∗ 2Ik1
(6.1c)
The factor κ
κ = 1.02 + 0.98 ∗ e−3R/X
(6.2)
obtained from the ratio R/X for three-phase short-circuit shall be taken for all types
of short-circuits as well. All other assumptions as per Section 4.3.2.3 are valid in
low-voltage systems as well.
6.4.2 Symmetrical short-circuit breaking current Ib
Short-circuits in low-voltage systems normally are far-from-generator short-circuits.
The symmetrical short-circuit breaking current is identical to the initial symmetrical
short-circuit current.
134 Short-circuit currents
6.4.3 Steady-state short-circuit current Ik
Short-circuits in low-voltage systems normally are far-from-generator shortcircuits. The steady-state short-circuit current is identical to the initial symmetrical
short-circuit current.
6.4.3.1 Influence of motors
Asynchronous motors contribute to the initial symmetrical short-circuit current, to
the peak short-circuit current, to the symmetrical short-circuit breaking current and in
the case of unbalanced short-circuits to the steady-state short-circuit current as well.
Synchronous motors are modelled like generators and asynchronous generators are
treated as asynchronous motors. Motors of any kind, which are not in operation at
the same time, e.g., due to the process or due to any interlocking, can be neglected
for the calculation of short-circuit current to such an extent that only those motors
are to be taken into account which lead to the highest contribution of the short-circuit
current under realistic operating conditions. Motors fed by static-rectifiers need to be
considered in the case of three-phase short-circuits only if they are able to transfer
energy for deceleration at the time of short-circuit, as they contribute to the initial
symmetrical and to the peak short-circuit current.
Asynchronous motors in public supply systems are considered when
•
•
the sum of the rated currents is greater than 1 per cent of the initial symmetrical
short-circuit current without motors;
the contribution to the initial symmetrical short-circuit current is greater or equal
to 5 per cent without motors.
Medium-and low-voltage motors connected through two-winding transformers to the
short-circuit are considered if
0.8
PrM
(6.3)
>
√
SrT
|(c100 SrT /( 3UnQ /Ik )) − 0.3|
The influence of asynchronous motors in low-voltage systems can be neglected if
IrM ≤ 0.01 ∗ Ik
(6.4)
SrT is the
where UnQ is the nominal system voltage at short-circuit location Q,
sum of rated apparent power of all transformers, directly connected to motors feeding
the short-circuit, Ik is the initial symmetrical short-circuitcurrent without motors,
PrM is the sum of rated active power of all motors and IrM is the sum of rated
currents of all motors.
6.5 Minimal short-circuit currents
In order to calculate the minimal short-circuit current the voltage factor cmin according
to Table 6.1 for the equivalent voltage source at the short-circuit location has to be
Calculation of short-circuit currents 135
considered. Furthermore,
• System topology, generator dispatch and short-circuit power of feeding networks
have to be defined in such a way that the minimal short-circuit current is expected.
This normally applies for low-load conditions.
• Motors are to be neglected.
• Resistances of overhead lines and cables shall be calculated with the maximal
permissible temperature at the end of the short-circuit, e.g., 80◦ C in low-voltage
systems.
These assumptions have to be made in the case of balanced and unbalanced shortcircuits, except when other presuppositions are mentioned.
6.6 Examples
Examples for the calculation of short-circuit currents in LV-systems are included in
IEC 60781. The calculation is carried out with form-sheets, which are used for the
calculation of impedances of equipment as well as for short-circuit current calculation
itself. The voltage factor cmax = 1.0 as given in the examples shall be cmax = 1.05
or cmax = 1.1 according to IEC 60909-0. Reference is made to Table 6.1.
Short-circuit current calculation can easily be carried out with spreadsheet analysis
using, e.g., EXCEL [14]. Figure 6.1 outlines an example for a low-voltage installation. The initial symmetrical and the peak short-circuit current both for minimal and
maximal conditions shall be calculated for short-circuit either at location A or at
location B. The data of the equipment are given below:
System feeder
Transformer
Cable
Motor
Un = 10 kV; SkQmax
= 190 MVA
= 240 MVA; SkQmin
SrT = 630 kVA; ukrT = 6%; urRT = 1.1%; trT = 10/0.4 kV;
R0 /R1 = 1; X0 /X1 = 1
= 77.4 m/km; X = 78 m/km;
Each 4 × 240 mm2 ; RK
K
R0 /R1 = 4; X0 /X1 = 4.1; l = 35 m
PrM = 50 kW; UrM = 0.41 kV; cos ϕrM = 0.84; ηrM = 0.94;
IanM /IrM = 6
A
B
35 m
UnQ = 10 kV
Q
Un = 0.4 kV ± 6%
Figure 6.1
Equivalent circuit diagram of a LV-installation
M
3~
Table 6.2
Example for the calculation of maximal short-circuit currents in LV-system
Data Input
Low voltage
Nominal voltage in kV
cmax/cmin
System feeder
Pos. seq. system
S ⬙kQmax in MVA
S ⬙kQmin in MVA
Transformer
SrT in kVA
HHV-side
LV-side
Pos. seq. system
u1kT in %
u1rRT in %
Zero seq. system
u0kT in %
u0rRT in %
Cable
Length in m
Cables in parallel
Max. temp. °C
Pos. seq. system
X1⬘ in mΩ/km
R1⬘ in mΩ/km
Zero seq. system
X0⬘ in mΩ/km
R0⬘ in mΩ/km
0.4
1.1
R
mΩ
240
190
0.0421
0.0333
X
mΩ
0.4203
0.3328
Z1
mΩ
Z0
mΩ
0.4225
0.3345
s.-c. at location A
630
10
0.4
I ⬙k3min
kA
15.2381
6
1.1
2.7937
6
1.1
2.7937
15.493
15.2381
15.97
I ⬙k2min
kA
ip3
kA
Rk/Xk
0.182
13.83
35.86
ip2
kA
31.83
I ⬙kE2E
kA
16.25
ipE2E
kA
36.49
I ⬙k1max
kA
16.11
ip1
kA
36.18
15.493
s.-c. at location B
35
3
80
I ⬙k3min
kA
0.91
78
77.4
0.903
319.8
309.6
3.612
1.282
3.731
14.78
ip3
kA
Rk/Xk
0.226
31.72
I ⬙k2min
kA
12.8
ip2
kA
27.47
I ⬙kE2E
kA
13.02
i pE2E
kA
27.95
I ⬙k1max
kA
13.84
ip1
kA
29.7
5.193
Branch s.-c. current of motor
Motor
Active power in kW
Ur in V
cos
h
Ia/Ir
Zero seq. Imp. in mΩ
Rated current
Assessment of motors
Assessment of motors
50
410
0.84
0.94
6
99999999
171.34
407.93
442.46
I ⬙k3max
kA
0.58
0.42
1.07
I ⬙k2max
kA
0.51
ip2
kA
0.93
Branch s.-c. current of motor to be neglected
99999999
Ampere
1% of s.-c. current in Ampere
5% of s.-c. current in Ampere
ip3
kA
Rk/Xk
89.17
147.8
739
I ⬙kE2E
kA
0.01
ipE2E
kA
0.02
I ⬙k1max
kA
0.01
ip1
kA
0.02
Table 6.3
Example for the calculation of minimal short-circuit currents in LV-system
Input data
Low voltage
Nominal voltage in kV
cmax/cmin
System feeder
Pos. seq. system
S ⬙kQmax in MVA2
S ⬙kQmin in MVA1
Transformer
SrT in kVA
HV-side
LV-side
Pos. seq. system
u1kT in %
u1rRT in %
Zero seq. system
u0kT in %
u0rRT in %
Cable
Length in m
Cables in parallel
Max. temp. °C
Pos. seq. system
X1⬘ in mΩ/km
R1⬘ in mΩ/km
Zero seq. system
X0⬘ in mΩ/km
R0⬘ in mΩ/km
0.4
0.95
R
mΩ
X
mΩ
Z1
mΩ
240
0.0421
0.4203
0.4225
190
0.0333
0.3328
0.3345
Z0
mΩ
s.-c. at location A
630
10
0.4
6
1.1
2.7937
6
1.1
2.7937
15.2381
15.493
15.2381
I ⬙k3min
kA
Rk/Xk
ip3
kA
I ⬙k2min
kA
ip2
kA
I ⬙kE2E
kA
ipE2E
kA
I kmin
⬙
kA
ip1
kA
13.78
0.182
30.87
11.84
26.81
14.04
31.53
13.81
31.24
I ⬙k3min
kA
Rk/Xk
ip3
kA
I ⬙k2min
kA
ip2
kA
I ⬙kE2E
kA
ipE2E
kA
I ⬙k1min
kA
ip1
kA
12.64
0.239
26.79
10.95
23.21
10.97
23.25
11.75
24.9
15.493
s.-c. at location B
35
3
80
78
77.4
1.1198
319.8
309.6
4.4789
0.91
3.731
1.443
5.83
138 Short-circuit currents
The calculation is carried out with EXCEL; the spreadsheet is shown in Tables 6.2
and 6.3. Data for the calculation of minimal short-circuit currents are automatically
transferred; results are highlighted in the tinted boxes. All fields, except the input
fields, are blocked against unintentional modification.
Chapter 7
Double earth-fault and short-circuit currents
through earth
7.1 General
IEC 60909-3 describes methods and procedures for the calculation of currents during
two separate simultaneous single-phase line-to-earth short-circuits (s.-c.) at different
locations of the system, which are called ‘double earth-fault’ in the context of
this section. The double earth-fault is not identical to a double-phase short-circuit,
where two phases have a short-circuit at the same location. Furthermore, the branch
short-circuit currents flowing through earth are dealt with.
7.2 Short-circuit currents during double earth-faults
7.2.1 Impedances and initial symmetrical short-circuit current Ik
in the case of a double earth-fault,
In order to calculate the short-circuit currents IkEE
the mutual impedance in the positive- and zero-sequence component between the
two short-circuit locations is needed. As the mutual impedance comparatively difficult
to determine, the double earth-fault can only be analysed in a simple manner for special
system configurations. The following cases are considered:
•
•
•
•
Both short-circuit locations are on the same line.
The short-circuit locations are on different lines.
Single-fed line.
Double-fed line.
In the case of a single-fed (radial) line, both short-circuit locations are on the same
line and the double earth-fault is identical to a double-phase short-circuit without earth
connection, as seen from the feeding point of the line. In the case of a double-fed
140 Short-circuit currents
single-circuit line, the voltage line-to-earth is changing significantly between the
two short-circuit locations, whereas the line-to-line voltages are remaining almost
unchanged. Details are outlined in [3].
In general, the short-circuit current in the case of a double earth-fault with shortcircuit locations A and B is calculated according to [4] by
IkEE
=
3 ∗ c ∗ Un
|Z 1A + Z 2A + Z 1B + Z 2B + M 1 + M 2 + Z 0 |
(7.1)
where Z 1A ; Z 1B are the short-circuit impedances in the positive-sequence system
at location A and B, respectively, Z 2A ; Z 2B the short-circuit impedances in the
negative-sequence system at location A and B, respectively, M 1 ; M 2 the mutual
impedances in the positive-sequence system between the short-circuit locations A
and B, respectively, and Z 0 the short-circuit impedance in the zero-sequence system
between locations A and B.
The impedance M 1 and M 2 , representing the mutual impedances of the positiveand negative-sequence components between the two short-circuit locations, can be
measured by short-circuiting all voltages in the system and feeding the voltage U 1A
(positive-sequence component), respectively the voltage U 2A (negative-sequence
component), at short-circuit location A. The mutual impedances M 1 and M 2 are
calculated using the voltage U 1B at short-circuit location B and the current I 1A at
short-circuit location A by
M1 =
U 1B
I 1A
(7.2a)
M2 =
U 2B
I 2A
(7.2b)
In case feeding shall be carried out at location B, the mutual impedances given are
obtained by the voltage U 1A at short-circuit location A and the current I 1B at shortcircuit location B as below:
M1 =
U 1A
I 1B
(7.3a)
M2 =
U 2A
I 2B
(7.3b)
7.2.2 Power system configurations
In the case of far-from-generator short-circuits the impedances in the positive- and
negative-sequence components Z 1 = Z 2 and M 1 = M 2 are equal. Table 7.1 outlines
simple systems configurations in accordance with IEC 60909-3.
Table 7.1
System configurations and equations for the calculation of short-circuit currents in the case of double earth-faults
Single-fed radial line
Both short-circuit locations on the same line (circuit)
d
=
IkEE
f
3 ∗ c ∗ Un
|6 ∗ Z 1d + 2 ∗ Z 1f + Z 0f |
d: Impedance to short-circuit location A
f: Impedance between short-circuit locations A and B
B
A
Two single-fed radial lines
Short-circuit locations on different lines (circuits)
d
g
=
IkEE
d: Impedance of system feeder including busbar connection
g: Impedance between busbar and short-circuit location A
h: Impedance between busbar and short-circuit location B
A
B
h
3 ∗ c ∗ Un
|6 ∗ Z 1d + 2 ∗ Z 1g + Z 1h + Z 0g + Z 0h |
Table 7.1 Continued
Double-fed line
Both short-circuit locations on the same line (circuit)
d
System feeder
left
f
A
=
IkEE
e
B
System feeder
right
3 ∗ c ∗ Un
|(6 ∗ Z 1d ∗ Z 1e + 2 ∗ Z 1f (Z 1d + Z 1e ))/(Z 1d + Z 1f + Z 1e ) + Z 0f |
d: Impedance from left till short-circuit location A
e: Impedance from right till short-circuit location B
f: Impedance between short-circuit locations A and B
Earthing impedances in the switchyard and at the overhead tower can be neglected.
The voltage factor c shall be selected in accordance to Table 4.1.
Indices: 0, 1 and 2: zero-, positive- and negative-sequence components. d, e, f, g, h: length of the line as indicated in the drawing of the first column.
Double earth-fault and short-circuit currents 143
7.2.3 Peak short-circuit current ip
The calculation of peak short-circuit current is carried out in the same manner as
described in Section 4.3.2.3. The peak short-circuit current is calculated by
√
ipEE = κ ∗ 2 ∗ IkEE
(7.4)
The factor κ shall be the maximum of the factors obtained for three-phase short-circuit
at location A or B.
κ = MAX{κA ; κB }
(7.5)
For explanations and calculation method of quantities reference is made to Chapter 4.
7.2.4 Symmetrical short-circuit breaking current Ib and steady-state
short-circuit current Ik
In the case of far-from-generator short-circuits, the symmetrical short-circuit breaking
current and the steady-state short-circuit current are identical to the initial symmetrical
short-circuit current. In other cases, reference is made to Chapter 4.
7.3 Short-circuit currents through earth
7.3.1 Introduction
Branch short-circuit currents can flow through earth in the case of unbalanced shortcircuits with earth connection, i.e., line-to-earth (single-phase) short-circuit and
double-phase short-circuit with earth connection. Single-phase short-circuits are the
dominating fault type in power systems with earthed neutrals and are leading to the
flowmaximal branch short-circuit currents flowing through earth. The current Ik1
ing through earth is equal to three-times the current in the zero-sequence component
flowing towards the short-circuit location. Type, number and arrangement of earth
conductors of overhead lines, the installation of counterpoise, shielding, armouring
and sheaths of cables, and their connection to the earthing grid of the switchyard
determine the part of the short-circuit current that will flow through the earthing
installations. This part is described by means of the reduction factor pE , sometimes
represented as r. The reduction factor is defined as the ratio of branch short-circuit
current I Etot flowing through earth to the total short-circuit current 3I 0 , which can
be described by the ratio of impedances
pE =
I Etot
Z
= 1 − LE
3I 0
ZE
(7.6)
where Z E is the impedance of earth conductor with earth return, Z LE is the mutual
impedance between earth and line conductor with common earth return, I Etot is
the branch short-circuit current through earth and I 0 is the total zero-sequence
short-circuit current.
144 Short-circuit currents
The loop-impedance Z E of the earth conductor and earth return is given by
μ0
μ0
δ
μr
ZE = R +
∗ω+j
∗ ω ∗ ln
+
(7.7)
8
2π
r
4
The mutual impedance Z LE of the loop earth conductor and conductor with earth
return is given by
μ0
μ0
δ
Z LE =
(7.8)
∗ω+j
∗ ω ∗ ln
8
2π
dLE
where r is the radius of earth conductor, μ0 is the absolute permeability, μr is the
relative permeability, ω is the angular frequency, R is the resistance
of earth wire per
√
unit length, δ is the depth of the earth return path δ = 1.85/ ω ∗ μ0 /ρE , ρE is the
resistivity of soil depending on soil conditions and dLE is the distance between the
earth conductor and phase conductor. Equations for the calculation of the impedances
Z E and Z LE are also given in [1,6].
7.3.2 Short-circuit inside a switchyard
Figure 7.1 outlines the equivalent circuit diagram of a power system (HV) with shortcircuit inside the switchyard B. All quantities are defined according to Figure 7.1.
The single-phase short-circuit current is calculated using
I k1 = 3 ∗ I 0A + 3 ∗ I 0B + 3 ∗ I 0C
(7.9)
The total current through the earth grid at location B is
(7.10)
I ZB = pA ∗ 3 ∗ I 0A + p C ∗ 3 ∗ I 0C
(I–pA)3I0A
(I–pC)3I0C
3I0A
3I0B
3I0B
3I0C
I 0K1
A
ZEA
IZA
B
UEB
pA3I0A
Figure 7.1
Earthwire
C
ZEB
IZB
ZEC
IZC
pC3I0C
Earth
Equivalent circuit diagram with short-circuit inside switchyard B
Double earth-fault and short-circuit currents 145
The potential at the earth grid is then
(7.11)
U EB = Z EB ∗ I ZB
with the earthing impedance of the switchyard
Z EB =
1
1/RE + 1/Z P
(7.12)
where R E is the resistance of the earth grid and Z P is the driving point impedance
(impedance of earth conductor with earth return and earthing impedance of overhead
towers, respectively, input impedance of shielding, sheaths and armouring of cables).
The considerations mentioned above are only valid when the short-circuit location
(switchyard B) is far away from other switchyards (A and C in Figure 7.1). The
current through earth is lower when the distance to the short-circuit is lower than the
far-from-station distance dF defined as
dF = 3 ∗
RTo ∗
dTo
Re{ Z E }
(7.13)
where RTo is the footing resistance of tower, dTo is the distance between two towers
and Z E is the impedance of earth conductor with earth return.
7.3.3 Short-circuit at overhead-line tower
Figure 7.2 outlines the equivalent circuit diagram of a power system with short-circuit
outside the switchyard (distance d > dF , see Equation (7.13)). The single-phase
short-circuit current is calculated by
I k1 = 3 ∗ I 0A + 3 ∗ I 0B + 3 ∗ I 0C
(7.14)
The branch current through the earthing of the tower at short-circuit location is
I ZF = pC ∗ (3 ∗ I 0A + 3 ∗ I 0C ) + p C ∗ 3 ∗ I 0C = pC ∗ I k1
(7.15)
and the potential at the earth grid
U EF = Z EF ∗ I ZF
(7.16)
with the earthing impedance of the installation
Z EF =
1
1/RTo + 2/Z P
(7.17)
where RTo is the footing resistance of tower and Z P is the driving point impedance
(impedance of earth conductor with earth return and earthing impedance of overheadline towers, respectively, input impedance of shielding, sheaths and armouring
of cables).
146 Short-circuit currents
(I–pA)3I0A
(I–pC)3I0A + 3I0B
3I0A
(I–pC)3I0C
Earthwire
3I0C
3IA+3I0B
I 0K1
3I0B
A
~
~
ZEB
IZB
pA3I0A
UEF
IZF
pC(3I0A+3I0B)
C
ZEC
IZC
~
ZEA
IZA
Figure 7.2
F
B
pC3I0C
Earth
Equivalent circuit diagram with short-circuit at overhead-line tower
If the short-circuit occurs in a short distance from the switchyard, the branch
short-circuit current through the earth conductor and back to the switchyard can
be comparatively high. The branch short-circuit current through earth is reduced
accordingly. The branch short-circuit current through the earthing grid at switchyard B
in the case of a short-circuit at location F is
I ZB = pC ∗ (3 ∗ I 0A + 3 ∗ I 0C ) − p A ∗ 3 ∗ I 0A
(7.18)
The branch short-circuit current through the earthing grid can be higher or lower for
the short-circuit location inside the switchyard or at any overhead-line tower outside
depending on the actual earthing conditions.
IEC 60909-3 presents the method to calculate the reduction factor on overhead
lines. Further reference is made to [2,3,5,6].
7.4 Examples
Examples for the calculation of short-circuit currents in the case of double earthfault and for the calculation of branch short-circuit currents flowing through earth
are included in IEC 60909-3 Annexes A and B. Configurations with nominal system
voltage U n = 132 kV similar to those as per Figures 7.1 and 7.2 are presented.
7.4.1 Double earth-fault in a 20-kV-system
Figure 7.3 represents the equivalent circuit diagram of a 20-kV-system with overhead
lines to calculate the short-circuit current in the case of a double earth-fault. Data of
Double earth-fault and short-circuit currents 147
A: Phase R
8 km
B: Phase Y
12 km
Figure 7.3
Equivalent circuit diagram of a 20-kV-system
equipment are
System feeder
Transformer
Overhead lines
= 1 GVA; U = 110 kV
SkQ
n
SrT = 40 MVA; ukrT = 14%; trT = 110 kV/20 kV
ACSR 95/15; R1 = 0.384 /km; X1 = 0.35 /km;
R0 = 1.35 /km; X0 = 0.6 /km.
The impedances of equipment are calculated in accordance with Section 7.3.
Equipment
Positive-sequence
component
Zero-sequence
component
System feeder
(0.044 + j 0.438) —
Transformer
(0.0 + j 1.35)
—
Line to short-circuit (3.07 + j 2.8)
(10.08 + j 3.07)
location A
Line to short-circuit (4.61 + j 4.2)
(16.2 + j 4.61)
location B
Total impedance
(42.624 + j 32.408)
(Table 7.1)
The short-circuit current in the case of a double earth-fault is calculated as
= 1.23 kA. When both short-circuits occur at a tower, a part of the shortIkEE
circuit current flows through the tower and the footing resistance. Depending on
the resistivity of the earth, the footing resistance and the surge impedance of the
148 Short-circuit currents
line, this part is approximately 10–25 per cent smaller than the total short-circuit
current.
7.4.2 Single-phase short-circuit in a 110-kV-system
Figure 7.4 indicates the equivalent circuit diagram of a 110-kV-system with shortcircuit location F. The single-phase short-circuit current, the branch short-circuit
currents flowing through the earth and the potential of the earth grid shall be calculated.
The data of equipment are
Earth wire
A
45 km
B
Un = 110 kV
ZEB
Figure 7.4
Feeder A
Feeder B
Feeder C
80 km
C
UEB
Equivalent circuit diagram of a 110-kV-system with short-circuit
location
SkQA
= 1.5 GVA; SrT = 250 MVA; u1kT = 16%; u0kT = 20%;
u0rR = 0.2%
= 2.0 GVA; SrT = 350 MVA; u1kT = 14%; u0kT = 18%;
SkQB
u0rR = 0.2%
= 1.1 GVA; SrT = 200 MVA; u1kT = 15%; u0kT =
SkQC
19%; u0rR = 0.2%
Line ACSR 2 × 240/40; earth wire 240/40; reduction factor
p ≈ 0.6
Z 1 = (0.059 + j 0.302) /km; Z 0 = (0.27 + j 1.51) /km
Double earth-fault and short-circuit currents 149
The impedances of equipment are calculated in accordance to Section 7.3 and
Section 3.2.
Equipment
Positive-sequence
component
Zero-sequence
component
Feeder A including
transformer
Feeder B including
transformer
Feeder C including
transformer
Line A–B
Line B–C
Impedance at short-circuit
location
(0.0 + j 8.87)
(0.1 + j 9.68)
(0.0 + j 6.66)
(0.06 + j 6.22)
(0.0 + j 12.1)
(0.12 + j 11.5)
(2.655 + j 13.59) (12.15 + j 21.6)
(4.72 + j 24.16)
(67.95 + j 120.8)
(0.178 + j 4.512) (0.407 + j 5.11)
The single-phase short-circuit current is calculated as I k1 = (0.798 − j 14.785) kA
and |I k1 | = 14.8 kA, respectively. The current in the zero-sequence component is
I 0 = 13 I k1 = (0.266 − j 4.928) kA and the branch short-circuit currents of the three
feeders A, B and C are
I 0A = (0.179 − j 0.732) kA
I 0B = (0.502 − j 4.036) kA
I 0C = (0.057 − j 0.16) kA
The total current through the earth grid is I ZB = (0.425−j 1.426) kA. The impedance
of the earth grid depends on the resistivity of the soil, the footing resistance of the tower
and the surge impedance of the line. Assuming (in accordance with IEC 60909-3) the
impedance of the earthing grid to be Z EB = (0.68 + j 0.49) the potential of the
earth grid is U EB = (0.987 − j 0.761) kV and |U EB | = 1.246 kV, respectively.
Chapter 8
Factors for the calculation of
short-circuit currents
8.1 General
Several factors for the calculation of short-circuit (s.-c.) currents have been introduced
in previous sections, the origin of which will be explained within this section.
• Voltage factor cmax and cmin for different voltage levels as per Table 4.1.
• Correction factor using the %/MVA- or the p.u.-system as mentioned in Chapter 2.
• Impedance correction factors for synchronous machines, power station units and
transformers as per Tables 3.2, 3.3, 3.5 and 3.6.
• Factors for the calculation of different parameters of the short-circuit current based
on the initial short-circuit current as per Chapter 4.
The factors are necessary as the method of the equivalent voltage source at the
short-circuit location is used for the calculation of short-circuit currents which is based
on some simplifications such as neglecting the load current prior to fault, assuming
the tap-changer of transformers in middle-position, calculating the impedance of
equipment based on the name-plate data or on data for rated operating conditions
and neglecting voltage control gear for generators and transformers. The main task
of short-circuit analysis is to determine the maximal short-circuit current which is
one of the main criteria for the rating of equipment in electrical power systems.
It is obvious that the parameters of the short-circuit current as calculated with the
equivalent voltage source at the short-circuit location will differ from those currents,
which may be measured during short-circuit tests or may be calculated with transient
network analysing programmes. In order to obtain results on the safe side without
uneconomic safety margin the correction factors will be applied. Detailed deductions
of the various correction factors are given in IEC 60909-1:1991-10.
152 Short-circuit currents
8.2 Correction using %/MVA- or p.u.-system
The need to use special correction factors for the impedances using the %/MVAsystem, applies also to the p.u.-system. The calculation of short-circuit currents can
be carried out using the %/MVA- or the p.u.-system as outlined in Section 2.7. The
rated voltage of equipment Ur is chosen as reference voltage UB for the calculation
of the impedance of transformers, generators, etc. For system feeders and lines the
rated voltage is not defined, therefore the nominal system voltage Un is taken as
reference voltage. The rated voltages of transformers in most cases are unequal to
the nominal voltage of the power system, connected to the transformer. Figure 8.1
indicates a 110/10-kV-system as an example.
Q1
T2
T1
F
L
Q2
Un = 220 kV
Figure 8.1
Equivalent circuit diagram of a power system with different voltage
levels
Rated data of equipment are given below:
= 1 GVA; U
UnQ1 = 110 kV; SkQ1
nQ2 = 220 kV; SkQ2 = 3 GVA
SrT2 = 300 MVA; ukrT2 = 15%; trT2 = 225 kV/115 kV
SrT1 = 250 MVA; ukrT1 = 17%; trT1 = 220 kV/115 kV
RL = 0.03 /km; XL = 0.12 /km; l = 100 km
The impedances of the equipment using the Ohm-system related to the short-circuit
location (Column 2), the %/MVA-system (Column 3) and the p.u.-system (Column 4)
are summarised in Table 8.1. The values in the %/MVA-system are converted to
the Ohm-system by using the equations as per Table 2.3 (Column 5). The result
(Column 6) indicates a difference of 6.3 per cent for the total short-circuit impedance.
The differences of the impedances result from the fact that the calculation in the
Ohm-system takes account of the real transformation ratio of the transformers tr =
UrTHV /UrTMV , whereas the calculation in the %/MVA- and in the p.u.-system assumes
a voltage of 100 per cent for all different voltage levels. The final conversion to the
Ohm-system is done using the voltage at the short-circuit location UnF , thus denying
the differences between rated voltages of the transformers and nominal voltages of the
connected power systems. These differences obviously will result in differences of
the short-circuit currents as outlined in Table 8.1. A correction of the impedances
is necessary in case the rated voltages of transformers and the nominal system voltages
differ from each other.
Factors for the calculation of short-circuit currents 153
Table 8.1
Impedances of equipment and short-circuit current as per Figure 8.1
1
2
3
Equipment
Impedance Impedance Impedance Impedance
Difference
related to
(%/MVA) (p.u.)
(%/MVA) → () (%)
s.-c.
(p.u.) → ()
location
()
System Q1
13.929
Transformer T2
6.917
System Q2
4.849
Line L
3.279
Transformer T1
8.993
Total s.-c.
16.206
impedance
Short-circuit
4.31 kA
current
Q
ZB
4
5
0.11
0.05
0.037
0.0248
0.068
0.126
0.11
0.05
0.037
0.0248
0.068
0.126
4.60 kA
10.12 p.u.
Ti
6
13.31
6.05
4.48
3.0
8.228
15.246
4.7
14.3
8.2
9.3
9.3
6.3
—
6.3
T2
T1
Un
Figure 8.2
Equivalent circuit diagram for the calculation of impedance correction
factor using %/MVA- or p.u.-system
The impedance correction factors (see Figure 8.2 for explanation) are obtained
starting from the short-circuit location F indicated in Figure 8.1 by multiplying the
ratios of the rated voltages of all transformers T1 –Ti between the short-circuit location F and the equipment under consideration B, then going back with the ratio of the
rated voltage of the equipment UrB and the nominal voltage Un at the short-circuit
location. For system feeders and lines the nominal system voltage at the very location
UnB has to be taken instead of the rated voltage, which is not defined for feeders and
lines. The impedance correction factor KB is calculated by
KB =
UrT1E UrT2E UrT3E
∗
∗
∗ ···
UrT1A UrT2A UrT3A
2
∗
UrB
UnF
2
(8.1)
The impedance correction factor using the %/MVA- or the p.u.-system must be applied
for any equipment [33] except power station units for which special correction factors
are valid.
154 Short-circuit currents
Table 8.2
Impedances of equipment and short-circuit current using correction
factor
1
2
3
Equipment
Impedance
Impedance Correction factor Impedance Impedance
(%/MVA) (p.u.)
related to s.-c. (%/MVA) KB as per
location ()
Equation (8.1)
using KB
using KB
System Q1
13.929
Transformer T2
6.917
System Q2
4.849
Line L
3.279
Transformer T1
8.993
Total s.-c.
16.206
impedance
Short-circuit
4.31 kA
current
4
5
6
0.11
0.05
0.037
0.0248
0.068
0.126
1.046
1.093
1.143
1.093
1.093
—
0.115
0.057
0.04
0.027
0.0743
0.134
0.115
0.057
0.04
0.027
0.0743
0.134
—
—
4.31 A
9.48 p.u.
Applying the impedance correction factors as per Equation (8.1) the impedances
calculated with the %/MVA- and the p.u.-system are identical to those obtained by
using the Ohm-system as outlined in Table 8.2.
8.3 Impedance correction factors
Within this book the deduction of the impedance correction factor KG for synchronous
machines (generators) is given. The factor is valid for generators connected directly
without unit transformers to the power system which is normally the fact in mediumvoltage and low-voltage systems [37]. Assuming an overexcited turbine generator as
per Figure 8.3 with voltage control at the terminal connection to UG = (1 ± pG )∗UrG ,
the control range normally is set to (1 + pG ) = 1.05. Prior to fault the generator
generates the apparent power S G = PG + j QG to be fed into the system.
(a)
X ⬙d
(b)
GS
3~
UrG
Figure 8.3
~
UG
Un
E⬙
RG
IG
UrG
3
01
Generator directly connected to the power system. (a) Equivalent system
diagram and ( b) equivalent circuit diagram in the positive-sequence
component
Factors for the calculation of short-circuit currents 155
ZG
~
~
ZG
IG
UG
3
E0
ZG
I 0kUb
~
UG
3
~
~
~
E0
I 0kG
UG
3
~
UG
3
~
UG
= UrG (1⫾PG)
3
Figure 8.4
Determination of the short-circuit current by superposition
In the case of a short-circuit as indicated F in Figure 8.3(a) the short-circuit current
can be calculated by superposition of the generator current I G prior
√ to fault and the
short-circuit current I kUb based on the voltage Ub = UG = UrG / 3 prior to fault as
outlined in Figure 8.4.
The generator current I G prior to fault is given by
IG
√
E − (U rG / 3)
=
ZG
(8.2)
and the fault current I kUb can be calculated by
I kUb
√
U rG / 3
=
ZG
(8.3)
The short-circuit current of the generator I kG is obtained by superposition of the two
currents
I kG
=
I kUb
+ IG
√
√
U rG / 3 E − (U rG / 3)
=
+
ZG
ZG
(8.4)
where E is the subtransient voltage of the generator, U rG is the rated generator
voltage and Z G is the generator impedance.
If the method of the equivalent voltage source with the voltage E is used the
short-circuit current I kG is calculated by
I kG =
E
ZG
(8.5a)
which is different from the calculation
√ as per Equation (8.4). If the voltage at the
short-circuit location E = c ∗ Un / 3 with the voltage factor c as per Table 4.1 and
156 Short-circuit currents
an impedance correction factor K G are introduced, the short-circuit current I kG is
found to be
c ∗ Un
c ∗ Un
I kG = √
=√
(8.5b)
3 ∗ ZG ∗ K G
3 ∗ (RG + j Xd ) ∗ K G
where K G is the impedance correction factor (to be determined), Un is the nominal
system voltage, Z G is the generator impedance, RG is the stator resistance of the
generator and Xd is the subtransient reactance of the generator.
Equations (8.5a) and (8.5b) are set to be equal. The unknown internal subtransient
voltage of the generator E can be found in accordance with Figure 8.3(b).
U
E = √rG + IG ∗ (cos ϕG − j sin ϕG ) ∗ (RG + j Xd )
3
(8.6)
where U rG is the rated voltage of the generator and ϕG is the phase angle of the
generator current (power factor: cos ϕ) and other quantities are explained above.
The impedance correction factor K G is then calculated by
√
3 ∗ IG
Un
KG =
∗c∗ 1+
∗ (RG ∗ cos ϕG + Xd ∗ sin ϕG )
UrG
UrG
√
−1
3 ∗ IG
(8.7)
∗ (Xd ∗ cos ϕG − RG ∗ sin ϕG )
+j
UrG
The resistance RG normally can be neglected against the subtransient reactance Xd
of the generator; the correction factor then results in
KG ≈
Un
c
∗
UrG 1 + (IG /IrG ) ∗ xd ∗ sin ϕG
(8.8)
2 )∗S .
whereas the subtransient reactance is introduced as a p.u.-value xd = (Xd /UrG
rG
The correction factor is maximal when the maximal voltage factor cmax and rated
operating conditions with IG = IrG and ϕG = ϕrG are applied. The impedance
correction factor KG is given by
KG ≈
cmax
Un
∗
UrG 1 + xd ∗ sin ϕG
(8.9)
For other equipment such as power station units and transformers impedance correction factors can be deducted in a similar way [35] as explained for the correction
factor of the generator. Details can be found in IEC 60909-1.
8.4 Factor κ for peak short-circuit current
The peak short-circuit current is the maximal instant value of the short-circuit current
which occurs normally within the first few milliseconds after the occurrence of the
short-circuit. The system configuration with a single-fed three-phase short-circuit is
dealt with in Figure 8.5.
Factors for the calculation of short-circuit currents 157
(a)
Q
T
Un
Uk; SrT; UrT
S ⬙kQ
t=0
(b)
Rtot
ik(t)
Xtot
~
U
u (t) =c 2 n .sin (v)
3
(c)
u
u (t)
vt
wu
Figure 8.5
Equivalent circuit diagram of a power system with three-phase shortcircuit. (a) Circuit diagram, ( b) simplified diagram of a single-fed threephase short-circuit and (c) time course of voltage with voltage angle ϕU
The time course of the short-circuit current ik (t) is calculated from the differential
equation
L∗
√
dik (t)
c ∗ Un
+ R ∗ ik (t) = 2 ∗ √ ∗ sin(ωt + ϕU )
dt
3
(8.10)
The solution of the differential equation is given by
ik (t) =
√
1
c ∗ Un
2∗ √ ∗ √
∗ (sin(ωt + ϕU − γ ) − e−t/T
2
3
R + X2
∗ sin(ϕU − γ ))
(8.11)
where Un is the nominal system voltage, ϕU is the angle of voltage related to zero
crossing as per Figure 8.5, c is the voltage factor according to Table 4.1, T is the time
constant: T = L/R, X is the reactance of the short-circuit impedance: X = ωL, R
is the resistance of the short-circuit impedance, ω is the angular velocity and γ the
angle of the short-circuit impedance: γ = arctan(X/R).
158 Short-circuit currents
The initial short-circuit current Ik is equal to the first part (periodic term) of
Equation (8.11), the second term is the aperiodic and decaying d.c.-component of the
current. If the time course of the short-circuit current as per Equation (8.11) is related
to the peak value of the initial short-circuit current
Ik =
√
1
c ∗ Un
2∗ √ ∗ √
2
3
R + X2
(8.12)
the peak factor κ is obtained
κ(t) = sin(ωt + ϕU − γ ) − e−t/T ∗ sin(ϕU − γ )
(8.13a)
The maximum of the peak factor κ determines the maximum of the short-circuit
current (peak short-circuit current ip ) to be calculated by partial differentiation of
Equation (8.13a) with respect to ϕU and t. The maximum of the peak factor always
occurs for short-circuits at ϕU = 0 and t ≤ 10 ms (50 Hz), respectively t ≤ 8.33 ms
(60 Hz), whatever the ratio R/X might be
κ(t) = sin(ωt − γ ) + e−(R/X)∗ωt ∗ sin γ
(8.13b)
A sufficient approximation of the peak factor κ is given by
κ = 1.02 + 0.98 ∗ e−3(R/X)
(8.14)
Special attention for the calculation of peak short-circuit current must be given in
the case of short-circuits in meshed systems or in systems having parallel lines with
R/X-ratios different from each other [34]. A detailed analysis of these conditions is
given in IEC 60909-1 and is mentioned in Chapter 4. The peak factor κ according to
Equation (8.14) is outlined in Figure 4.7.
8.5 Factor µ for symmetrical short-circuit breaking current
The short-circuit current in the case of a near-to-generator short-circuit decays significantly during the first periods after initiation of the short-circuit due to the change
of the rotor flux in the generator. This behaviour cannot be calculated exactly as eddy
currents in the forged rotor of turbine generators, non-linearities of the iron and saturation effects especially in the stator tooth are difficult to be calculated. Furthermore, the
decay of the short-circuit current and by this the breaking current depend on different
generator and system parameters such as time constants of the generator itself, location of short-circuit in the system, operational condition prior to the fault, operation
of excitation and voltage control device, tap-changer position of transformers, etc.
which cause unpredictable deviations of the calculated results from those obtained
from measurements. Detailed calculations with digital programmes are therefore only
applicable in special cases if high safety requirements are to be met.
Factors for the calculation of short-circuit currents 159
The time decay of the a.c. part of the short-circuit current is calculated by
Ikac (t) = (Ik − Ik ) ∗ e−t/TN + (Ik − Ik ) ∗ e−t/TN + Ik
(8.15)
Equation (8.15) is composed of the initial short-circuit current
Ik =
Xd
E
+ XN
(8.16a)
the transient short-circuit current
Ik =
E
Xd + XN
(8.16b)
and the steady-state short-circuit current
Ik =
E
Xd + X N
(8.16c)
where E; E ; E are the steady-state, transient and subtransient voltages, Xd ; Xd ;
Xd is the steady-state, transient and subtransient reactance of the generator and XN
is the reactance between the generator and the short-circuit location, e.g., including
the reactance of the unit transformer.
The individual components of the short-circuit currents as per Equations (8.15)
and (8.16) are declining with different time constants, i.e., the subtransient time con
stant TN which can be set approximately equal to the subtransient time constant TdN
in the direct axis. Typical values of the time constants are TN ≈ 0.03–0.04 s and
TN ≈ 1.0–1.5 s and are calculated by
TN = T ∗
Xd Xd + XN
∗
Xd Xd + XN
(8.17a)
TN = T ∗
Xd Xd + XN
∗
Xd Xd + XN
(8.17b)
The units T and T are the subtransient and transient time constants of the generator. Regarding the time course of the decaying a.c. part of the current as per
Equation (8.15), the approximation e−t/TN ≈ 1 can be assumed if the time range
(minimal time delay of circuit-breakers) t = tmin = 0.02–0.25 s is considered:
Ikac (t) ≈
Ik
X + XN
∗ 1 − d
Xd + XN
∗e
−tmin /TN
X + XN
+ d
Xd + XN
= Ik ∗ μ
(8.18)
with tmin being the minimal time for breaking the short-circuit current. The second part
of Equation (8.18) is taken as the factor μ for the calculation of the breaking current.
160 Short-circuit currents
As the exponent tmin /TN may be greater or smaller than 1, the infinite progression
√
for e−t/TN is applied. For no-load conditions, where E ≈ E ≈ E ≈ UrG / 3, the
factor μ is obtained by
X − Xd
X − Xd
tmin
+ e−tmin /TN − 1
∗
μ ≈ 1 − d
√ ∗ Ik ∗ tmin + d
Xd + XN
TN
T0 ∗ (UrG / 3)
(8.19)
with the no-load subtransient time constant
T0 = T ∗
Xd
Xd
(8.17c)
When if tmin ≪ TN the last part of Equation (8.19) can be neglected. The factor μ
can be presented depending on the minimal time delay of the circuit-breaker tmin and
the ratio Ik /IrG as outlined in Figure 4.10.
8.6 Factor λ for steady-state short-circuit current
The steady-state short-circuit currents of generators are determined by the method
of excitation, the maximal possible excitation voltage, the type of voltage control
and strongly by the saturation effects. As salient-pole and turbine generators differ
significantly with respect to their reactances and are mostly equipped with different
types of excitation, the steady-state short-circuit currents of both generators will differ
even if all other conditions are equal. The calculation is carried out with the factor λ
based on the rated current IrG of the generator which is determined separately for
minimal and maximal current.
Ik max = λmax ∗ IrG
(8.20a)
Ik min = λmin ∗ IrG
(8.20b)
The factor λ is found from the characteristic curve method as per Figure 8.6 defining
Potier’s reactance XP .
Ik =
E0 (If )
XP + XN
(8.21)
where XP is the Potier’s reactance as per Figure 8.6, E0 is the no-load voltage, If is the
excitation current and XN is the reactance between the generator and the short-circuit
location, e.g., including the reactance of the unit transformer.
The value of Potier’s reactance is between the transient reactance Xd (pole
saturation only) and the stator leakage reactance Xσ (teeth saturation only).
The method to determine Potier’s reactance requires detailed knowledge of the saturation within the machine and is not practicable for the determination of the factor λ.
Factors for the calculation of short-circuit currents 161
E0
UL
UrG / 3 UrG/ 3
XP
E0 (If)
UL
E0 (If)
XP/IrG
UrG/ 3
1
Ifkr/IfOd
UL (If)
1
Figure 8.6
If /IfOd
Characteristic saturation curve method for determination of Potier’s
reactance
IEC 60909-1 recommends a simplified method. Potier’s reactance and the source voltage E0 which is a function of the field current If are reduced due to the saturation as
can be seen from Figure 8.6. Both effects compensate each other to a certain extent
and are ignored therefore. The current Ik is calculated by
Ik =
uf max ∗ Er
Xdsat + XN
(8.22)
where uf max is the highest possible excitation voltage (p.u.-value), Er is the internal
steady-state voltage of the generator at rated operating conditions, Xdsat is the saturated value of the synchronous reactance (equal to the reciprocal of the short-circuit
ratio) and XN is the reactances between the generator and the short-circuit location,
e.g., including the reactance of the unit transformer.
Furthermore, the subtransient internal voltage is given as
E = IG ∗ (Xd + XN )
(8.23)
By this the factor λ is determined to be
λ=
Ik
uf max ∗ Er
=
)
IrG
(Xdsat − Xd ) ∗ IrG + E ∗ (IrG /IkG
(8.24)
162 Short-circuit currents
Table 8.3
Assumed values of uf max for the calculation of the factor λ
uf max = Uf max /Ufr
Type of synchronous machine
Salient-pole generator
Turbine generator
Series 1
1.3
1.6
Series 2
1.6
2.0
The voltages E and Er can be determined if RG ≪ Xd by
UrG
E ≈ √ ∗ (1 + xd ∗ sin ϕrG )
3
(8.25)
Ur
Er ≈ √ ∗
3
(8.26)
2 +2∗x
1 + xdsat
dsat ∗ sin ϕrG
√
√
where xd = Xd ∗ (( 3 ∗ IrG )/UrG ) and xdsat = Xdsat ∗ (( 3 ∗ IrG )/UrG ).
√
The rated impedance is ZrG = UrG /( 3 ∗ IrG ). The values for λmax and λmin as
per Figures 4.11 and 4.12 are calculated by Equations (8.23)–(8.25), cos ϕrG = 0.85
and Xd = 0.2 ∗ ZrG . The highest possible excitation voltage (p.u.-value) uf max is
assumed for the calculation in accordance with the values as per Table 8.3.
8.7 Factor q for short-circuit breaking current of asynchronous
motors
Asynchronous motors are contributing to the short-circuit current as outlined in
Chapter 4. As the short-circuit current of asynchronous motors decays faster as
compared with the short-circuit current of synchronous machines, the short-circuit
using
breaking current is based on the initial short-circuit current of the motor IkM
the factor μ which is identical to the factor for the calculation of breaking current of
synchronous generators and an additional factor q
IbM = q ∗ μ ∗ IkM
(8.27)
The factor q as per Figure 4.13 is derived from the results of transient calculations and measurements using 28 motors with different rating PrM = 11–160 kW
in the low-voltage range and up to PrM = 160 kW–10 MW in the mediumvoltage range. A detailed list of the rated data of the asynchronous motors is
included in Table 2 of IEC 60909-1:1991 (similar to those given in Tables 13.6
Factors for the calculation of short-circuit currents 163
and 13.7). The minimum time delay of the circuit-breakers was assumed in four steps
tmin = 0.02–0.05–0.1–≥0.25 s. The results are outlined in Figure 8.7.
1
0.02 s
0.05 s
0.1 s
0.9
0.8
0.7
>0.25 s
0.6
q 0.5
Minimal time delay
0.02 s
0.05 s
0.1 s
0.4
0.3
0.2
> 0.25 s
0.1
0
0.01
Figure 8.7
0.1
1
m (active power per pair of poles)
10
MW
100
Calculated and measured values of factor q for the calculation of shortcircuit breaking current of asynchronous motors; values of q as per
Figure 4.13 (According to Figure 20 of IEC 60909-1:1991.)
As can be seen from Figure 8.7 the values of the factor q (approximation) as per
Figure 4.13 are mean values of the calculated and measured ones with the 50 per centfrequency deviation between the exact values and the approximated values in the range
of q < 5%.
Chapter 9
Calculation of short-circuit currents in
d.c. auxiliary installations
9.1 General
The calculation of short-circuit currents in d.c. auxiliary installations, e.g., in power
plants and substations is dealt with in IEC 61660-1. Contrary to the approach for the
calculation of short-circuit currents in a.c. three-phase systems, the determination of
the exact time course of the short-circuit current is needed besides the calculation of
defined parameters [42]. The equipment as below contribute to short-circuit currents
in d.c. installations:
• Smoothing capacitors
• Stationary batteries (normally of the lead-acid type)
• Rectifiers (IEC 61660-1 deals only with rectifiers in three-phase a.c. bridge
connection for 50 Hz, parameters for 60 Hz are actually under consideration)
• d.c. motors with independent excitation.
The branch short-circuit currents from the equipment mentioned above are characterised by different time course, depending on the ohmic, inductive and capacitive
parts, the d.c. voltage source and other parameters. Figure 9.1 presents the equivalent
circuit diagrams of the equipment and the typical time course of the short-circuit
current.
d.c. installations in auxiliary supply systems include several pieces of equipment;
the total short-circuit current at the short-circuit location is the superposition of the
individual branch short-circuit currents from the different equipment. In principle the
short-circuit current can be defined by a time function i1 (t), describing the time span
tp from short-circuit initiation till the maximal short-circuit current ip (peak shortcircuit current) and a time function i2 (t), describing the decaying time course to the
166 Short-circuit currents
(a)
iC
ipC
tpC
t
(b) iB
ipB
IkB
tpB
t
(c) iD
ipD
IkD
~
–
IkD
tpD
t
(d) iM
ipM
IkM
IkM
tpM
M
t
Motor without additional inertia mass
Motor with additional inertia mass
Figure 9.1
Equivalent circuit diagrams of equipment in d.c. auxiliary installations;
typical time course of short-circuit current (according to Figure 1 of DIN
EN 61660-1 (VDE 0102 Teil 10)). (a) Capacitor, (b) battery, (c) rectifier in three-phase a.c. bridge connection and (d) d.c. motor with
independent excitation
Calculation in d.c. auxiliary installations 167
1.2
1
1
ip
i1(t)
0.8
i/ip
i2(t)
tP
0.6
Ik
0.4
t2
0.2
Tk
0
0
2
4
6
8
10
12
14
16
18
t
Figure 9.2
Standard approximation function of the short-circuit current (according
to Figure 2 of IEC 61660-1:1997)
quasi steady-state short-circuit current Ik as outlined in Figure 9.2. Time functions
can be calculated according to [50] by
1 − e−t/τ1
for 0 ≤ t ≤ tp
1 − e−tp /τ1
Ik
Ik
i2 (t) = ip
1−
∗ e−(t−tp )/τ2 +
ip
ip
(9.1a)
i1 (t) = ip ∗
for tp ≤ t ≤ Tk
(9.1b)
If no distinct maximum of the short-circuit current is present, the peak shortcircuit current ip and the quasi steady-state short-circuit current Ik are equal. This
time course as well is described by the standard approximation function according to
Figure 9.2.
The approach to calculate the parameter ip and Ik , the time constants τ1 and τ2
and the time to peak tp is explained below:
•
•
The parameter of the short-circuit current will be calculated for each branch
separately, i.e., for each individual equipment contributing to the short-circuit
current.
In case of several sources, the short-circuit parameters will be calculated by
superposition of the branch short-circuit currents by:
– Calculation of the branch short-circuit currents taking account of the common branch, i.e., that branch in the installation carrying branch short-circuit
currents from different sources.
– Correction of the branch short-circuit currents by a factor σ , which depends
on the different resistances.
168 Short-circuit currents
– Calculation of the time course of the branch short-circuit currents with the
corrected impedances.
– Superposition of the calculated time functions of the branch short-circuit
currents to determine the total short-circuit current.
– The thermal and electromagnetic effects of short-circuit currents are calculated
using the standard approximation function.
The calculation of impedances of the sources in d.c. auxiliary installations feeding
the short-circuit current was dealt with in Section 3.3. The calculation of the shortcircuit parameters is carried out as mentioned above and explained below. For the
calculation of maximal short-circuit currents in d.c. auxiliary installations the items
as below had to be taken into account:
•
•
•
•
•
•
•
Short-circuit impedance of system shall be minimal (ZQmin ), so that the
contribution to the short-circuit current is maximal.
Resistance of lines is to be calculated for a temperature of 20◦ C.
System topology leading to the maximal short-circuit currents shall be taken into
account.
Joint resistance of busbars shall be neglected.
Control circuits to limit the contribution of rectifiers are disconnected, i.e.,
limitation is not active.
Batteries are fully charged.
Systems coupled by diodes are regarded as directly connected.
For the calculation of minimal short-circuit currents in d.c. auxiliary installations
the items as stated below shall be considered:
•
•
•
•
•
•
•
Short-circuit impedance of system shall be maximal (ZQmax ), so that the
contribution to the short-circuit current is minimal.
System topology leading to the minimal short-circuit currents shall be taken into
account.
Resistance of lines shall be calculated for maximal permissible operating
temperature.
Joint resistance of busbar has to be taken into account.
Contribution of rectifiers to the short-circuit current is limited to the rated value
of the current limiter.
Voltage of batteries shall be defined equal to the minimal discharge voltage as per
manufacturer’s information.
Systems coupled by diodes are regarded as disconnected.
The current limiting characteristic of fuses and switchgear in d.c. auxiliary installations has to be taken into account for the calculation of both maximal and minimal
short-circuit current.
Calculation in d.c. auxiliary installations 169
9.2 Short-circuit currents from capacitors
The quasi steady-state short-circuit current of a capacitor is zero
IkC = 0
(9.2)
The peak short-circuit current ipC is calculated using
ipC = κC ∗
EC
RCBr
(9.3)
where EC is the capacitor voltage at the instant of short-circuit and RCBr is the
resistance of capacitor including connection and common branch (see Section 3.3.1
and Table 3.13).
The factor κC depends on the eigen-frequency ω0
1
ω0 = √
LCBr ∗ C
(9.4a)
2 ∗ LCBr
1
=
δ
RCBr
(9.4b)
and on the decay coefficient δ
where LCBr is the inductance of the capacitor connection including common branch
(see Table 3.13) and C is the capacitance.
The reasonable range of values for κC is outlined in Figure 9.3; equations for
the calculation of κC are included in the Annex of IEC 61660-1:1997 and are not
repeated here.
1
0.9
103
0.8
0.7
kC 0.6
. 0.5
104
0.4
v0
0.3
105
0.2
0.1
0
0.1
1
10
ms
100
1/d
.
Figure 9.3
Factor κC for the calculation of peak short-circuit current of capacitors
(according to Figure 12 of IEC 61660-1:1997)
170 Short-circuit currents
10
ms
tpC
103
1
.
v0
104
0.1
105
0.01
0.1
10
1
ms
100
.
1/d
Figure 9.4
Time-to-peak tpC for the calculation of short-circuit currents of
capacitors (according to Figure 13 of IEC 61660-1:1997)
The time-to-peak tpC also depends on the eigen-frequency ω0 and the decay
coefficient δ. The reasonable range of values for tpC is outlined in Figure 9.4; equations
for the calculation of tpC are included in the Annex of IEC 61660-1:1997.
The rise-time constant τ1C
τ1C = k1C ∗ tpC
(9.5a)
and the decay-time constant τ2C
τ2C = k2C ∗ RCBr ∗ C
(9.5b)
depend upon the factors k1C and k2C , i.e., upon eigen-frequency and decay component. Reasonable ranges of values are outlined in Figures 9.5 and 9.6. Quantities of
Equations (9.5) are explained above.
Calculation equation for the factors k1C and k2C are not included in IEC
61660-1:1997.
9.3 Short-circuit currents from batteries
As rise-time constants of the short-circuit current of batteries are always below 100 ms,
the quasi steady-state short-circuit current Ik is calculated for the time instant of 1 s
after initiation of the short-circuit.
0.95 ∗ EB
IkB =
(9.6)
RBBr + 0.1 ∗ RB
where EB is the open-circuit voltage of the battery, RBBr is the resistance of the
battery including connection and common branch (see Section 3.3.1 and Table 3.14)
and RB is the resistance of the charged battery.
Calculation in d.c. auxiliary installations 171
0.8
0.7
0.6
.
k1C
105
0.5
104
0.4
v0
0.3
103
0.2
0.1
0
0.1
1
10
ms
100
.
1/d
Figure 9.5
Factor k1C for the calculation of rise-time constant (according to
Figure 14 of IEC 61660-1:1997)
10,000
k2C
.
1000
105
100
104
v0
10
103
1
0.1
1
10
.
Figure 9.6
ms
100
1/d
Factor k2C for the calculation of decay-time constant (according to
Figure 15 of IEC 61660-1:1997)
The peak short-circuit current ipB is calculated using the battery voltage EB by
ipB =
EB
RBBr
(9.7)
Reasonable ranges of the values for the rise-time constant τ1B and the time-to-peak
tpB , both depending on the decay component δ
1
2
=
δ
RBBr /LBBr + 1/TB
(9.8)
172 Short-circuit currents
100
ms
tpB
tpB
10
1B
.
1B
1
0.1
0.1
1
.
Figure 9.7
1/d
10
ms
100
Rise-time constant τ1B and time to peak tpB of short-circuit currents of
batteries (according to Figure 10 of IEC 61660-1:1997)
where LBBr is the reactance of the battery including connection and common branch
(see Table 3.14) and TB is the time constant of the battery assumed to be TB = 30 ms
(outlined in Figure 9.7). The decay-time constant is set to τ2B = 100 ms.
More details of the calculation of short-circuit currents fed from batteries are
included in [56].
9.4 Short-circuit currents from rectifiers
The quasi steady-state short-circuit current IkD of a rectifier in three-phase a.c. bridge
connection is
√
3∗ 2
UrTLV
c ∗ Un
∗
(9.9)
∗√
IkD = λD ∗
π
3 ∗ ZN UrTHV
where Un is the nominal system voltage on a.c. side of rectifier, ZN is the network
impedance a.c. side, UrTLV is the rated voltage at low voltage side of transformer (a.c.
side) and UrTHV is the rated voltage at high voltage side of transformer (a.c. side).
The factor λD depends on the ratio RN /XN of the a.c. side of the rectifier as well
as on the ratio of the resistances RDBr /RN (ratio of resistance d.c. side to resistance
a.c. side). A reasonable range of values of the factor λD is outlined in Figure 9.8.
Equations for the calculation of λD are included in IEC 61660-1:1997.
The peak short-circuit current is calculated using
ipD = κD ∗ IkD
(9.10)
whereas the factor κD depends on the ratio of the inductivities LDBr /LN (ratio of
inductance d.c. side to inductance a.c. side) and on the resistances and reactances
Calculation in d.c. auxiliary installations 173
1.1
1
0.01
0.9
0.8
1.5
0.7
RDZw/RN
D 0.6
5.0
.
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
.
Figure 9.8
0.7
0.8
RN/XN
0.9
1
1.1
1.2
Factor λD for the calculation of quasi steady-state short-circuit current
of rectifiers (according to Figure 7 of IEC 61660-1:1997)
2
1.9
1.8
kD
1.7
1.6
.
1.5
1.4
RDBr /RN
1.3
0.01
1.2
1.0
1.1
1
5.0
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
.
Figure 9.9
0.8
0.9
1
1.1
1.2
1.3
1.4
R*
Factor κD for the calculation of peak short-circuit currents of rectifiers.
Factor: R ∗ = (RN /XN )(1 + 2RDBr /3RN ) (according to Figure 8 of
IEC 61660-1:1997)
RN , RBBr and XN . The reasonable range of values of the factor κD is outlined in
Figure 9.9. Equations for the calculation are included in IEC 61660-1:1997.
In the case of κD < 1.0 no distinct maximum of the short-circuit current is present,
thus the peak short-circuit current can be neglected as the quasi steady-state shortcircuit current will be the maximal value of the current. IEC 61660-1 determines for
174 Short-circuit currents
κD < 1.05 the time to peak tpD equal to the duration of the short-circuit Tk . For all
other cases (κD ≥ 1.05) the time to peak is calculated using
tpD = (3 ∗ κD + 6) ms
LDBr
tpD = (3 ∗ κD + 6) + 4
−1
LN
ms
for
LDBr
≤1
LN
(9.11a)
for
LDBr
>1
LN
(9.11b)
The rise-time constant τ1D for rectifiers fed from 50 Hz (60-Hz-values are actually
under consideration) is
LDBr
(9.12a)
ms for κD ≥ 1.05
τ1D = 2 + (κD − 0.9) ∗ 2.5 + 9 ∗
LN
RN
2 LDBr
τ1D = 0.7 + 7 −
∗ 1+ ∗
XN
3
LN
LDBr
(9.12b)
∗ 0.1 + 0.2 ∗
ms
for κD < 1.05
LN
A suitable approximation, giving results on the safe side is
τ1D =
tpD
3
(9.12c)
The decay-time constant τ2D for 50 Hz is calculated using
τ2D =
2
ms
RN /XN ∗ (0.6 + 0.9 ∗ (RDBr /RN ))
(9.13)
Quantities as per Equations (9.11)–(9.13) are: XN is the inductance of system N
(a.c. side) XN = ωLN , RN is the resistance of system N (a.c. side), RDBr is the
resistance of d.c. side including smoothing reactor, connection and common branch
(see Section 3.3.1 and Table 3.15), LDBr is the inductance of d.c. side including
smoothing reactor (saturated value), connection and common branch (see Table 3.15)
and κD the factor as per Figure 9.9.
More details of the calculation of short-circuit currents fed from rectifiers are
included in [54].
9.5 Short-circuit currents from d.c. motors with
independent excitation
Branch short-circuit currents from d.c. motors with independent excitation are only
considered if the total sum of the rated currents of all d.c. motors is greater than
1 per cent of the branch short-circuit currents of one rectifier.
The parameters of the motor, i.e., rated voltage UrM , rated current IrM , saturated
inductance of the field circuit at short-circuit LF and the unsaturated inductance
Calculation in d.c. auxiliary installations 175
of the field circuit at no-load LOF , determine the quasi steady-state short-circuit
current IkM which is calculated using
LF
UrM − IrM ∗ RM
IkM =
(9.14)
∗
LOF
RMBr
where LF is the field inductance, LOF is field inductance at no-load, UrM is the
rated voltage, IrM is the rated current of the motor, RM is the resistance of the motor
and RMBr the resistance of the motor including connection and common branch (see
Section 3.3.1 and Table 3.16).
Equation (9.14) is valid when the speed of the motor remains constant during the
duration of the short-circuit. When the speed of the motor decreases to zero (n → 0)
the quasi steady-state short-circuit current is IkM = 0.
The mechanical time constant τMec , the time constants of the field circuit τF
and the armature circuit τM determine the peak short-circuit current ipM , which is
calculated using
ipM = κM ∗
UrM − IrM ∗ RM
RMBr
(9.15)
The factor κM is equal to One in the case of nominal speed and in all cases where
τMec ≥ 10∗ tF , else (τMec < 10 ∗ τF ). κM depends on the eigen-frequency ω0 ,
ω0 =
RM ∗ IrM
1
∗ 1−
τMec ∗ τM
UrM
(9.16a)
where τMec is the mechanical time constant (see Table 3.16) and τM the time constant
of armature circuit up to short-circuit location (see Table 3.16); and on the decay
coefficient δ.
1
= 2 ∗ τM
δ
(9.16b)
The reasonable range of the factor κM is outlined in Figure 9.10.
The time to peak tpM , the rise-time and the decay-time constants τ1M and τ2M
depend on the value of the mechanical time constant τMec . IEC 61660-1:1997 defines
four factors k1M , k2M , k3M and k4M which are outlined in Figures 9.12 to 9.15.
Calculation equations are not included in IEC 61000-1:1997.
When the motor speed remains constant or in all cases when τMec ≥ 10 ∗ τF the
time to peak tpM is calculated using
tpM = k1M ∗ τM
(9.17)
the rise-time constant τ1M
τ1M = k2M ∗ τM
(9.18)
176 Short-circuit currents
1.1
1
1
0.9
M
0.8
20
0
0.7
0.6
0.5
.
100
0.4
0.3
0.2
0.1
0
1
ms
10
.
Figure 9.10
100
1/d
Factor κM for the calculation of peak short-circuit current of d.c.
motors with independent excitation (according to Figure 17 of IEC
61660-1:1997)
1000
ms
tpM
100
1
10
.
0
100
10
1
1
Figure 9.11
10
.
1/
ms
100
Time to peak of short-circuit currents for d.c. motors with independent excitation and τMec < 10 ∗ τF (according to Figure 19 of IEC
61660-1:1997)
and the decay-time constant τ2M
τ2M = τF
τ2M = k4M ∗
n = nn = const.
LOF
∗ τMec
LF
n→0
(9.19a)
(9.19b)
For all other cases, i.e., τMec < 10 ∗ τF the reasonable range of the time to peak tpM
is outlined in Figure 9.11.
Calculation in d.c. auxiliary installations 177
12
10
k1m
0.9
.
LF /LOF
0.6
8
0.2
6
4
1
10
Figure 9.12
F/M
.
100
1000
Factor k1M in the case of d.c. motors with independent excitation and
τMec ≥ 10 ∗ τF (according to Figure 18 of IEC 61660-1:1997)
1
0.9
k2M
0.8
0.7
LF/LOF
.
0.6
0.2
0.5
0.4
0.4
0.9
0,.
0.2
0.1
0
1
Figure 9.13
10
F/M
.
100
1000
Factor k2M in the case of d.c. motors with independent excitation and
τMec < 10 ∗ τF (according to Figure 18 of IEC 61660-1:1997)
The rise-time constant and the decay-time constant τ1M and τ2M are
calculated using
τ1M = k3M ∗ τM
τ2M = k4M ∗ τMec
The factors k1M , k2M , k3M and k4M are outlined in Figures 9.12 to 9.15.
(9.20a)
(9.20b)
178 Short-circuit currents
1.1
1
1
0.9
10
0.8
k3M 0.7
0
.
0.6
100
0.5
0.4
0.3
0.2
0.1
0
1
Figure 9.14
10
1/
.
100
ms
Factor k3M in the case of d.c. motors with independent excitation and
τMec < 10 ∗ τF (according to Figure 20 of IEC 61660-1:1997)
10
9
k4M 8
7
.
6
5
4
100
0
3
50
2
1–10
1
0
1
Figure 9.15
10
1/
ms
100
Factor k4M in the case of d.c. motors with independent excitation and
τMec < 10 ∗ τF (according to Figure 21 of IEC 61660-1:1997)
More details of the calculation of short-circuit currents fed from motors with
independent excitation are included in [52].
9.6 Total short-circuit current
The calculation of the total short-circuit current is carried out taking into account
the common branches of the d.c. installation. A common branch is a branch,
Calculation in d.c. auxiliary installations 179
e.g., conductor circuit, which carries branch short-circuit currents from different
sources.
When no common branch exists in the d.c. installation, the total short-circuit
current is calculated by adding the branch (partial) short-circuit currents. Otherwise,
the partial short-circuit currents of the different sources (Index j) are to be corrected
(Index kor) with factors σj , which reflect the different resistances of the sources
and the common branches RY . The peak short-circuit current and quasi steady-state
short-circuit current are calculated by
(9.21a)
ipkorj = σj ∗ ipj
(9.21b)
Ikkorj = σj ∗ Ikj
with the correction factor σj for each source j
σj =
Rresj ∗ (Rij + RY )
Rresj ∗ Rij + Rij ∗ RY + Rresj ∗ RY
(9.22)
The resistance up to the common branch of a source is named Rij and the equivalent
resistance of the other sources up to the common branch which contributes to the
short-circuit current is named Rresj . Resistances of capacitors are neglected and the
resistance of motors shall only be taken into account if the motor contributes to
the quasi steady-state short-circuit current.
The calculation equations of the resistances Rij and equivalent resistances Rresj
are outlined in Table 9.1. It is assumed, that all four sources are contributing to the
short-circuit current through one common branch, as outlined in Figure 9.16.
Table 9.1
Resistances Rij and equivalent resistances Rresj for the calculation of correction factors; U: Voltage at short-circuit location
prior to the short-circuit [42,50,57]
Source j
Resistance Rij
Equivalent resistance Rresj
Capacitor
RiC = RC + RCL
RresC =
Battery
RiB = RB + RBL
Rectifier
RiD =
d.c. motor with
independent
excitation
RiM = RM + RML
U
− RY
IkD
1
1/RiD + 1/RiB + 1/RiM
1
RresB =
1/RiD + 1/RiM
1
RresD =
1/RiB + 1/RiM
1
RresM =
1/RiD + 1/RiB
Remarks: Index L – Connection of equipment; index Y – Common branch (see Section 3.3.1);
and indices C; B; D; M – Capacitor; battery; rectifier; motor
180 Short-circuit currents
M
Motor
Load
Line
(Coupling branch)
Rectifier
Battery
Capacitor
Figure 9.16
Equivalent circuit diagram of a d.c. auxiliary installation
Rise-time and decay-time constants τ1M and τ2M of the partial short-circuit
currents are not corrected.
The total short-circuit current is calculated by superposition of the corrected partial
short-circuit currents of the different sources.
i1 (t) =
m
i2 (t) =
m
j =1
j =1
ipkorj ∗
ipkorj
1 − e−t/τ1j
1 − e−tpj /τ1j
1−
Ikkorj
ipkorj
for 0 ≤ t ≤ tpj
(9.23a)
∗ e−(t−tpj )/τ2j +
Ikkorj
ipkorj
for tpj ≤ t ≤ Tk
(9.23b)
The calculation of the thermal and electromagnetic (mechanical) effects of shortcircuit currents as per IEC 61660-2 is based on the standard approximation function.
Typical time-curves of total short-circuit currents are outlined in Figure 9.17.
The peak short-circuit current ip , the quasi steady-state short-circuit current Ik
and the decay-time constant τ2 are determined graphically from the time curve of
the total short-circuit current. The rise-time constant is calculated in accordance with
Calculation in d.c. auxiliary installations 181
(a)
i
ip
0.9 (ip – Ik)
31
22
Ik
tp
(b)
t
i
1 = min
Ik
t
(c)
i
ip
0.9 (ip – Ik)
22
31
Ik
t
tp
(d)
i
ip
1 = min
0.9 (ip – Ik)
2t2
Ik
tp
t
Total short-circuit current
Standard approximation function
Figure 9.17
Typical time curves of total short-circuit current in d.c. auxiliary installations, e.g., (a) with dominating part of motors, (b) with dominating
part of rectifiers, (c) with dominating part of batteries and (d) in the
case of low rectifier load (according to Figure 22 of DIN EN 61660-1
(VDE 0102 Teil 10))
182 Short-circuit currents
Figure 9.17(a) and (c), respectively by
τ1 =
tp
3
(9.24a)
and for time curves as per Figure 9.17(b) or (d)
τ1 = MIN{τ1C ; τ1B ; τ1D ; τ1M }
(9.24b)
The rise-time constant τ1C of capacitors is neglected in the case where
ipC ≤ 0.5 ∗ ip
(9.25)
The decay-time constant τ2 is equal to 50 per cent of that time span in which the
short-circuit current i2 (t) is reduced to 0.9 ∗ (ip –Ik ), i.e., the short-circuit current has
the value of (Ik + 0.1 ∗ ip ). Reference is made to Figure 9.17.
i2 (tp + 2τ2 ) = 0.9 ∗ (ip − Ik )
(9.26a)
i2 (tp + 2τ2 ) = Ik + 0.1 ∗ ip
(9.26b)
and
9.7 Example
The short-circuit currents at location F at the main busbar of the auxiliary supply
installation of a power station are to be calculated. The 220-V-installation as outlined
in Figure 9.18 include a battery of 1100 Ah, feeding from an a.c. LV-system through
rectifier with smoothing capacitor and a d.c. motor. The data and parameter of the
equipment are given below:
Q
T
D
C
Cable
Rectifier
Capacitor
Q
Battery
Figure 9.18
F
L1
L4
L3
M
M
L2
Cable
coupling branch
Cable
Motor
Load
Busbar
Equivalent circuit diagram of the d.c. auxiliary installation (220 V),
e.g., of a power station
Calculation in d.c. auxiliary installations 183
Equipment
Quantity
Parameter
Remarks
Battery B
Nominal voltage
Capacity for
10 h-discharge
Resistance per cell
Inductance per cell
Number of cells
UnB = 225 V
K10 = 1100 Ah
Table 3.14
Power system
AC/DCtransformer
Nominal voltage
Initial short-circuit
power
Resistance
UnQ = 660 V
= 33.3 MVA
SkQ
Rated voltages
UrTHV /UrTLV =
660 V/240 V
SrT = 380 kVA
ukT = 3.2%
PCu = 4 kW;
uRT = 0.5%
Table 3.2
Rated voltage
Rated current
Resistance
Reactance
a.c. capacitance
Resistance
UrD = 220 V
IrD = 1000 A
RDBr = 0.87 m
XDBr = 29 µH
Cac = 70 mF
RC = 8 m
Table 3.15
Rated voltage
Rated power
Rated current
No-load speed
Inductance of
armature circuit
Resistance of
armature
Inductance of
field circuit
Resistance of
field circuit
Moment of inertia
Motor speed
Tk > 1 s
UrM = 220 V
PrM = 95 kW
IrM = 432 A
n0 = 25 s−1
LM = 0.4 mH
Rated power
Impedance voltage
Losses; resistance
voltage
Rectifier D
Smoothing
reactor
Smoothing
capacitor C
d.c. motor M
RBZ = 0.12 m
LBZ = 0.2 µH
n = 109
Table 3.1
RQ /XQ = 0.28
RM = 41.9 m
LF = 9.9 H
RF = 9.9
J = 6.5 kg m2
n→0
Table 3.15
Table 3.13
Table 3.16
184 Short-circuit currents
Equipment
Quantity
Parameter
Remarks
Cables
L1; L2; L3
Cross section
Number of cables
Specific resistance
Radius of conductor
Length of cables
Table 3.12
Conductor
bars L4
Cross section
Number of bars
Width and height
Specific resistance
Distance between
sub-conductors
Distance of bars
Length of bar
qn = 300 mm2
3 in parallel (triangle)
ρ = 0.0173 mm2 /m
r = 10.5 mm
L1 = 2 m; L2 = 6 m;
L3 = 20 m
qn = 400 mm2
2 in parallel
d × b = 10 mm × 40 mm
ρ = 0.0173 mm2 /m
aS = 10 mm
Table 3.12
am = 75 mm
L4 = 14 m
The calculation of the short-circuit current is as follows:
1 Calculation of the impedances of cables and busbar conductors (Section 9.7.1).
2 Calculation of the short-circuit currents of the individual equipment
(Section 9.7.2).
3 Calculation of the correction factors (Section 9.7.3).
4 Calculation of partial short-circuit currents (Section 9.7.4).
5 Calculation of total short-circuit current (Section 9.7.5).
9.7.1 Calculation of the impedances of cables and busbar conductors
Three cables are laid in triangle arrangement. Specific resistance and inductance are
calculated according to Table 3.12.
0.0173 mm2 /m
ρ
=2
= 0.0384 m/m
3qn
3 ∗ 300 mm2
a
μ0
1
L =
∗ ln
+
π
rB
4n
R = 2
with rB =
n
n ∗ rT ∗ r n−1 =
3
3 ∗ 17.3 mm ∗ (10.5 mm)2 = 17.9 mm
1
75 mm
4π ∗ 10−7 H/m
L =
+
ln
π
17.9 mm 4 ∗ 3
= 0.607 µH/m
Calculation in d.c. auxiliary installations 185
Two rectangular busbars are laid in parallel. Specific resistance and inductance are
calculated according to Table 3.12.
ρ
0.0173 mm2 /m
= 0.0433 m/m
=2∗
2qn
2 ∗ 400 mm2
μ0
a
∗ ln
L =
π
0.223 ∗ (2d + dS + b)
4π ∗ 10−7 H/m
75 mm
=
∗ ln
= 0.628 µH/m
π
0.223 ∗ (30 + 40) mm
R = 2
The resistances and inductances of the individual connections by cables and bars
as per Figure 9.18 are summarised below:
Connection
L1
Length(m)
2
Resistance R(m) 0.0768
Inductance L(µH) 1.212
L2
6
0.2304
3.636
L3
L4
20
14
0.768 0.606
12.12 8.792
9.7.2 Calculation of the short-circuit currents of the individual equipment
9.7.2.1 Short-circuit current from capacitor
Total impedance of the capacitor according to Table 3.13 with common branch (cable
L2) and capacitor connection (cable L1)
RCBr = RC + RL1 + RL2 = (8 + 0.0708 + 0.2304) m = 8.301 m
LCBr = LL1 + LL2 = (1.212 + 3.636) µH = 4.848 µH
Peak short-circuit current as per Equation (9.3)
ipC = κC ∗
EC
RCBr
Factor κC depends on the eigen-frequency ω0
1
1
= 1.717 ∗ 103 /s
ω0 = √
=√
4.848 µH ∗ 70 mF
LCBr ∗ C
and on the decay coefficient δ
1
2 ∗ LCBr
2 ∗ 4.848 µH
=
= 1.168 ms
=
δ
RCBr
8.301 m
→ κC = 0.65 as per Figure 9.3
ipC = κC ∗
EC
225 V
= 17.62 kA
= 0.65 ∗
RCBr
8.301 m
Time-to-peak tpC = 1.1 ms as per Figure 9.4.
186 Short-circuit currents
Rise-time constant and decay-time constant as per Equation (9.5) are
τ1C = k1C ∗ tpC
τ2C = k2C ∗ RCBr ∗ C
With factors k1C and k2C depending on eigen-frequency and decay component k1C ≈
0.38 as per Figure 9.5 and k2C ≈ 1.4 as per Figure 9.6
τ1C = 0.38 ∗ 11 ms = 0.42 ms
τ2C = 1.4 ∗ 8.301 m ∗ 70 mF = 0.813 ms
9.7.2.2 Short-circuit current from battery
Total impedance of the battery as per Table 3.14 with common branch (cable L2) and
battery connection (cable L4). The battery consists of 109 cells.
RBBr = 0.9 ∗ RB + RL4 + RL2
= (0.9 ∗ 109 ∗ 0.12 + 0.606 + 0.2304) m = 12.608 m
LBBr = LB + LL4 + LL2 = (109 ∗ 0.2 + 8.792 + 3.636) µH = 34.228 µH
Quasi steady-state short-circuit current as per Equation (9.6)
IkB =
0.95 ∗ 1.05 ∗ 225 V
0.95 ∗ EB
= 16.13 kA
=
RBBr + 0.1 ∗ RB
(12.608 + 0.1 ∗ 109 ∗ 0.12) m
Peak short-circuit current according to Equation (9.7) is
ipB =
1.05 ∗ 225 V
EB
=
= 18.74 kA
RBBr
12.608 m
The decay component δ is
1
2
2
=
=
= 4.98 ms
δ
RBBr /LBBr + 1/TB
12.608 m/34.228 µH + 1/30 ms
Time-to-peak and rise-time constant as per Figure 9.7 are tpB = 13.7 ms and τ1B =
2.6 ms. The decay-time constant is τ2B = 100 ms per definition.
9.7.2.3 Short-circuit current from rectifier
Impedance of system feeder Q related to 220 V as per Table 3.1
ZQ =
2
c ∗ UnQ
SkQ
=
1.0 ∗ (220 V)2
= 1.73 m
33.3 MVA
Calculation in d.c. auxiliary installations 187
Resistance and reactance due to the ratio of RQ /XQ
XQ = 0.963 ∗ ZQ = 1.666 m
RQ =
2 − X 2 = 0.466 m
ZQ
Q
Impedance of transformer according to Table 3.2
ZT =
2
ukT UrTLV
3.2%
(240 V)2
∗
=
∗
= 4.85 m
SrT 100%
380 kVA
100%
RT =
2
uRT UrTLV
1.05%
(240 V)2
∗
=
∗
= 1.592 m
SrT 100%
380 kVA
100%
XT =
ZT2 − RT2 = 4.581 m
Total impedance a.c. side according to Table 3.15
RN = RQ + RT = (0.466 + 1.592) m = 2.058 m
XN = XQ + XT = (1.666 + 4.581) m = 6.247 m
and inductance
LN = XN /ω = (6.247/100 ∗ π) µH = 4.581 µH
Total impedance d.c. side according to Table 3.15 with common branch (cable L2)
RDBr = RS + RL1 + RL2 = (0.87 + 0.0768 + 0.2304) m = 1.177 m
LDBr = LS + LL1 + LL2 = (29 + 1.212 + 3.636) µH = 33.85 µH
Quasi steady-state short-circuit current of the rectifier as per Equation (9.9)
√
UrTLV
3∗ 2
c ∗ Un
∗
∗√
IkD = λD ∗
π
3 ∗ ZN UrTHV
Factor λD depending on RN /XN and RDBr /RN
RN /XN = 0.33 and RDBr /RN = 0.572 → λD = 0.92 as per Figure 9.8
IkD
√
240 V
1.0 ∗ 240 V
3∗ 2
∗
∗√
= 26.44 kA
= 0.92 ∗
π
3 ∗ 6.58 m 660 V
Peak short-circuit current as per Equation (9.10)
ipD = κD ∗ IkD
Factor κD depends on LDBr /LN and RN , RBBr and XN
2 RDBr
RN
LDBr /LN = 1.702 and
1+ ∗
= 0.455 → κD = 1.12
XN
3
RN
188 Short-circuit currents
as per Figure 9.9
ipD = 1.12 ∗ 26.44 kA = 29.61 kA
Time to peak according to Equation (9.11b) as κD = 1.12 ≥ 1.05 and LDBr /LN =
1.702 > 1
LDBr
tpD = (3 ∗ κD + 6) + 4
−1
LN
33.85
= (3 ∗ 1.12 + 6) + 4
−1
ms = 12.17 ms
19.89
Rise-time constant τ1D as per Equation (9.12a)
LDBr
τ1D = 2 + (κD − 0.9) ∗ 2.5 + 9 ∗
LN
33.85
= 2 + (1.12 − 0.9) ∗ 2.5 + 9 ∗
19.89
ms = 5.92 ms
Decay-time constant τ2D as per Equation (9.13)
2
RN /XN ∗ (0.6 + 0.9 ∗ RDBr /RN )
2
=
ms = 5.44 ms
2.058/6.247 ∗ (0.6 + 0.9 ∗ 1.177/2.058)
τ2D =
9.7.2.4 Short-circuit currents from d.c. motor
Total impedance of the motor as per Table 3.16 with connection cable L3, common
branch is neglected in this case
RMBr = RM + RL3 = (41.9 + 0.768) m
LMBr = LM + LL3 = (400 + 12.12) µH = 412.12 µH
Quasi steady-state short-circuit current (n → 0)
IkM = 0
Peak short-circuit current according to Equation (9.15)
ipM = κM ∗
UrM − IrM ∗ RM
RMBr
With factor κM depending on the eigen-frequency ω0 and on the decay coefficient δ
as per Equation (9.16)
1
RM ∗ IrM
ω0 =
∗ 1−
τMec ∗ τM
UrM
1
= 2 ∗ τM
δ
Calculation in d.c. auxiliary installations 189
Mechanical time constant as per Table 3.16
τMec =
2π ∗ n0 J ∗ RMZw ∗ IrM
2π ∗ n0 ∗ J ∗ RMZw ∗ IrM
=
Mr ∗ UrM
(PrM /2π n0 ) ∗ UrM
τMec =
2π ∗ 25/s ∗ 6.5 kgm2 ∗ 42.67 m ∗ 432 A
= 134.4 ms
(100 kW/2π ∗ 25/s) ∗ 220 V
Time constant of armature circuit as per Table 3.16
τM =
412.12 µH
LMBr
= 9.66 ms
=
RMBr
42.67 m
Time constant of the field circuit
τF =
9.9 H
LF
=1s
=
RF
9.9
Giving the eigen-frequency
ω0 =
1
41.9 m ∗ 432 A
∗ 1−
134.4 ms ∗ 9.66 ms
220 V
= 37.61/s
and the decay coefficient 1/δ = 2 ∗ 9.66 ms = 19.32 ms.
As τMec = 134.4 ms < 10∗ tF = 100 s → κM = 0.81 as per Figure 9.10
ipM = 0.81 ∗
220 V − 432 A ∗ 41.9 m
= 3.83 kA
42.67 m
Time to peak as τMec < 10 ∗ τF
tpM = 28 ms
according to Figure 9.11
Rise-time and decay-time as per Equation (9.20) constants depend on the mechanical
time constant as per Equations (9.18) to (9.19)
τ1M = k3M ∗ τM
τ2M = k4M ∗ τMec
with factors k3M and k4M
k3M = 0.78 according to Figure 9.14
k4M = 1.05 according to Figure 9.15
τ1M = 0.78 ∗ 9.66 ms = 7.53 ms
τ2M = 1.05 ∗ 134.4 ms = 141.1 ms
190 Short-circuit currents
9.7.2.5 Summary of results
The result of the short-circuit currents of the individual equipment are summarised
below
Symbol Equipment
ip (kA) Ik (kA) tp (ms) τ1 (ms)
τ2 (ms)
C
B
D
M
17.62
18.74
29.61
3.83
0.98
100
5.44
141.1
Capacitor
Battery
Rectifier
Motor
0.0
16.13
26.44
0.0
1.1
13.7
12.17
28
0.42
2.6
5.92
7.53
9.7.3 Calculation of the correction factors and corrected parameters
Correction factors σ as per Table 9.1 are only to be calculated for rectifier, capacitor
and battery, as the motor feeds the short-circuit directly.
Source j
Resistance Rij
Capacitor
Battery
Rectifier
Motor
Equivalent resistance Rresj
1
(1/RiD ) + (1/RiB ) + (1/RiM )
1
RresC =
RiC = RC
m
(1/8.29) + (1/13.686)
RiC = 8 m
RresC = 5.16 m
1
RresB =
RiB = RB + RL4
(1/RiD ) + (1/RiM )
RiB = (109 ∗ 0.12 + 0.606) m RresB = RiD
RiB = 13.686 m
RresB = 8.29 m
U
1
− RL2
RresD =
RiD =
IkD
(1/RiB) + (1/RiM )
225 V
− 0.2304 m RresD = RiB
RiD =
26.44 kA
RiD = 8.29 m
RresD = 13.686 m
RiM = RM + RL3
As the motor is feeding
RiM = (41.9 + 0.768) m
the short-circuit directly, RiM is
RiM = 42.668 m
neglected
RiC = RC + RCL
RresC =
Calculation of correction factors σ as per Equation (9.22)
• Capacitor
RresC ∗ (RiC + RL2 )
RresC ∗ RiC + RiC ∗ RL2 + RresC ∗ RL2
5.16 ∗ (8 + 0.2304)
= 0.958
σC =
5.16 ∗ 8 + 8 ∗ 0.2304 + 5.16 ∗ 0.2304
σC =
Calculation in d.c. auxiliary installations 191
•
•
Battery
σB =
RresB ∗ (RiB + RL2 )
RresC ∗ RiB + RiB ∗ RL2 + RresB ∗ RL2
σB =
8.29 ∗ (13.686 + 0.2304)
= 0.973
8.29 ∗ 13.686 + 13.686 ∗ 0.2304 + 8.29 ∗ 0.2304
Rectifier
σD =
RresD ∗ (RiD + RL2 )
RresD ∗ RiD + RiD ∗ RL2 + RresD ∗ RL2
σD =
13.686 ∗ (8.29 + 0.2304)
= 0.984
13.686 ∗ 8.29 + 8.29 ∗ 0.2304 + 13.686 ∗ 0.2304
• Motor
Correction factor of the motor is set to 1, as the motor is feeding the short-circuit
directly.
• Correction of parameters
Correction of peak short-circuit current and quasi steady-state short-circuit current
is carried out for the individual results based on Equation (9.21)
ipkorj = σj ∗ ipj
Ikkorj = σj ∗ Ikj
Symbol Equipment σ
C
B
D
M
Capacitor
Battery
Rectifier
Motor
ip
Ik
ipkor Ikkor tp
(kA) (kA) (kA) (kA) (ms)
τ1
τ2
(ms) (ms)
0.958 17.62 0.0 16.88 0.0
1.1 0.42
0.973 18.74 16.13 18.23 15.69 13.7 2.6
0.984 29.61 26.44 29.14 26.02 12.17 5.92
1
3.83 0.0
3.83 0.0 28
7.53
9.7.4 Calculation of partial short-circuit currents
The partial short-circuit currents are calculated based on Equation (9.1)
1 − e−t/τ1
1 − e−tp /τ1
Ik
Ik
i2 (t) = ip
∗ e−(t−tp )/τ2 +
1−
ip
ip
i1 (t) = ip ∗
for 0 ≤ t ≤ tp
for tp ≤ t ≤ Tk
0.98
100
5.44
141.1
192 Short-circuit currents
•
Capacitor
1 − e−t/0.42
1 − e−1.1/0.42
i1C (t) = 16.88 kA ∗
•
i1C (t) = 18.21 kA ∗ (1 − e−t/0.42 )
for 0 ≤ t ≤ 1.1 ms
i2C (t) = 16.88 kA ∗ (1 − e−(t−1.1)/0.98 )
for 1.1 ms ≤ t ≤ Tk
Battery
1 − e−t/2.6
1 − e−13.7/2.6
i1B (t) = 18.23 kA ∗
i1B (t) = 18.32 kA ∗ (1 − e−t/2.6 )
i2B (t) = 16.88 kA ∗
1−
15.69
18.23
for 0 ≤ t ≤ 13.7 ms
∗ e−(t−13.7)/100 +
15.69
18.23
i2B (t) = 16.88 kA ∗ (0.139 ∗ e−(t−13.7)/100 + 0.861)
for 13.7 ms ≤ t ≤ Tk
•
Rectifier
i1D (t) = 29.14 kA ∗
1 − e−t/5.92
1 − e−12.17/5.92
i1D (t) = 33.42 kA ∗ (1 − e−t/5.2 )
26.02
i2D (t) = 29.14 kA ∗
1−
29.14
for 0 ≤ t ≤ 12.17 ms
∗ e−(t−12.17)/5.44 +
26.02
29.14
i2D (t) = 29.14 kA ∗ (0.107 ∗ e−(t−12.17)/5.44 + 0.893)
for 12.17 ms ≤ t ≤ Tk
•
Motor
i1M (t) = 3.83 kA ∗
1 − e−t/7.53
1 − e−28/7.53
i1M (t) = 3.83 kA ∗ (1 − e−t/7.53 )
for 0 ≤ t ≤ 28 ms
i2M (t) = 3.83 kA ∗ e−(t−28)/141.1
for 28 ms ≤ t ≤ Tk
The corrected partial short-circuit currents of the different equipment (sources) are
outlined in Figure 9.19.
Calculation in d.c. auxiliary installations 193
60
0.9(ip–Ik)
Total
ip = 50.5 kA
50
Ik = 46.6 kA
2t2 = 34.5 ms
Current in A
40
30
Rectifier
20
Battery
10
0
Capacitor tp = 12.1 ms
0
Figure 9.19
10
Motor
20
30
Time in ms
40
50
60
Partial short-circuit currents and total short-circuit current, d.c.
auxiliary system as per Figure 9.18
9.7.5 Calculation of total short-circuit current
The total short-circuit current is calculated by superposition of the corrected partial
short-circuit currents of the different sources as per Equation (9.23)
i1 (t) =
m
ipkorj ∗
i2 (t) =
m
ipkorj
j =1
j =1
1 − e−t/τ1j
1 − e−tpj /τ1 j
1−
Ikkorj
ipkorj
for 0 ≤ t ≤ tpj
∗ e−(t−tpj )/τ2j +
Ikkorj
ipkorj
for tpj ≤ t ≤ TK
The total short-circuit current obtained by superposition is outlined in Figure 9.19.
The peak short-circuit current
ip = 50.5 kA
the quasi steady-state short-circuit current
Ik = 46.6 kA
the time-to-peak
tp = 12.1 ms
and the decay-time constant
τ2 = 17.3 ms
194 Short-circuit currents
60
Iappr
50
Isup
Current in A
40
Approximated total short-circuit current Iappr
30
Total short-circuit current by superposition Isup
20
10
0
0
10
Figure 9.20
20
30
Time in ms
40
50
60
Total short-circuit current, obtained by superposition of the partial
short-circuit currents and approximated short-circuit current, d.c.
auxiliary system as per Figure 9.18
are obtained from Figure 9.19 as indicated. The rise-time constant can be estimated
according to Equation (9.24) either by
τ1 =
tp
12.1 ms
=
= 4.03 ms
3
3
(time course as per Figure 9.17(a) and (c))
or
τ1 = MIN{τ1C ; τ1B ; τ1D ; τ1M }
(time course as per Figure 9.17(b) and (d))
τ1 = MIN{5.92; 2.6; 0.42; 7.53} ms = 0.42 ms
Comparison of the two different values revealed that the approximated time course
of the total short-circuit current fits best to the superposition of the partial short-circuit
currents, see Figure 9.20, if the rise-time constant is τ1 = 4.03 ms, which also is in
line with the typical time course of the short-circuit current as per Figure 9.17(b)
and (d).
Chapter 10
Effects of short-circuit currents
10.1
General
Calculation methods for the thermal and electromagnetic effects of short-circuit
currents are dealt with within IEC 61660-2, which is applicable to short-circuit currents in d.c. auxiliary installations in power plants and substations and IEC 60865-1,
related to three-phase a.c. systems.
10.2
10.2.1
a.c. systems
Thermal effects and thermal short-circuit strength
The thermal withstand capability (thermal short-circuit strength) of equipment is
determined by the maximal permissible conductor temperature prior to the shortcircuit, the duration of the short-circuit and the short-circuit current itself. The
maximal permissible temperature of conductors under normal operating conditions
and in the case of short-circuits, e.g., as per DIN VDE 0276, is summarised in
Table 10.1. Figures are given for a short-circuit duration of Tk = 5 s. It is assumed that
no heat transfer is taking place during the short-circuit duration (adiabatic heating).
Skin- and proximity-effects are neglected, the specific caloric heat of the conductor
and insulation is constant and the relation resistance-to-temperature is linear. Special
consideration is to be taken for conductors in a.c. installations with cross-section
above 600 mm2 , as the skin-effect has to be taken into account. Additional requirements according to IEC 60986:1989 and IEC 60949:1988 for cables and isolated
conductors are to be met.
10.2.1.1 Conductors and equipment
The analysis is based on the calculation of the thermal equivalent short-time current Ith
Tk 2
Q
0 ik (t) dt
Ith =
=
(10.1)
R ∗ Tk
R ∗ Tk
196 Short-circuit currents
Table 10.1
Maximal permissible conductor temperature and rated short-time
current density; 1) – Normal operating condition; 2) – Short-circuit
condition
Type of cable
Maximum Temperature at beginning of short-circuit (◦ C)
permissible
temperature 90 80 70 65 60 50 40 30 20
(◦ C)
Rated short-time current density in A/mm2 ; tkr = 1 s
1) 2)
Copper conductor
Cables with
soft-soldering
XPE-cables
PVC-cables (mm2 )
≤300
>300
Mass-impregnated
cables (kV)
0.6/1
3.6/6
6/10
12/20
18/30
Radial-screen
cable 12/20
— 160
100 108 115 119 122 129 136 143 150
90 250
143 149 154 157 159 165 170 176 181
70 160
70 140
—
—
—
—
115 119 122 129 136 143 150
103 107 111 118 126 133 140
80
80
70
65
60
65
—
—
—
—
—
—
149
113
—
—
—
—
154
120
120
—
—
—
250
170
170
170
150
170
157
124
124
124
—
124
159
127
127
127
117
127
165
134
134
134
124
134
170
141
141
141
131
141
176
147
147
147
138
147
181
154
154
154
145
154
Aluminium conductor
XPE-cables
PVC-cables (mm2 )
≤300
>300
Mass-impregnated
cables (kV)
0.6/1
3.6/6
6/10
12/20
18/30
Radial-screen
cable 12/20
90 250
94
98 102 104 105 109 113 116 120
70 160
70 140
—
—
—
—
80
80
70
65
60
65
—
—
—
—
—
—
98
75
—
—
—
—
250
170
170
170
150
170
76
68
78
71
81
73
85
78
90
83
95
88
99
93
102 104 105 109 113 116 120
80 82 84 89 93 97 102
80 82 84 89 93 97 102
—
82 84 89 93 97 102
— —
77 82 87 91 96
—
82 84 89 93 97 102
where Q is the thermal heat produced by the short-circuit current, R the resistance
of the equipment, Tk the short-circuit duration and ik (t) the time course of shortcircuit current which produces the same thermal heat Q within the conductor as the
short-circuit current ik (t) during the short-circuit duration Tk . The thermal equivalent
Effects of short-circuit currents 197
short-time current is calculated from the initial short-circuit current Ik by using
√
Ith = Ik ∗ m + n
(10.2)
The factors m and n represent the heat dissipation of the d.c. component and the
a.c. component of the short-circuit current [38]. Suitable ranges of values for m and
n are outlined in Figures 10.1 and 10.2. If an interrupting short-circuit is present or if
multiple short-circuits (number n) occur with different duration Tki and current Ithi
the resulting thermal equivalent rated short-time current is
n
1
2 ∗T
Ith =
(10.3)
Ithi
ki
Tk
i=1
where
Tk =
n
Tki
i=1
IEC 60909-0 includes calculation equations for the factors m and n.
1.1
1
.
1
0.9
.
2.5
0.8
0.7
n
0.6
I ⬙k / Ik
10
0.5
0.4
0.3
0.2
0.1
0
0.01
Figure 10.1
0.1
1
Tk
10
Factor n for the calculation of thermal short-time current (heat dissipation of a.c. component) (according to Figure 22 of IEC 60909-0:2001)
According to IEC 60865-1 separate considerations have to be taken concerning the
thermal strength of equipment, i.e., transformers, transducers, etc., and conductors,
i.e., busbars, cables, etc.
Equipment have a suitable thermal short-circuit strength if the rated short-time
short-circuit current Ithr (as per manufacturer’s data) for the short-circuit duration
Tk < Tkr (e.g., Tkr = 1 s) is above the thermal equivalent short-circuit current Ith .
Ith ≤ Ithr
for Tk ≤ Tkr
(10.4a)
198 Short-circuit currents
2
1.8
1.6
1.4
1.2
m
1
1.95
0.8
0.6
1.6
0.4
0.2
k
1.1
0
0.1
1
Figure 10.2
10
100
Tk
1000
Factor m for the calculation of thermal short-time current (heat dissipation of d.c. component) (according to Figure 21 of IEC 60909-0:2001)
In case the short-circuit duration Tk is longer than the rated short-circuit duration Tkr ,
the thermal short-circuit strength is fulfilled if
Tkr
Ith ≤ Ithr ∗
for Tk ≥ Tkr
(10.4b)
Tk
The maximal permissible values for conductor material can be obtained from
Figure 10.3.
In the case of bare conductors the thermal short-time current density Jth is calculated on the basis of the thermal equivalent short-time current and the conductor
cross-section qn .
Jth =
Ith
qn
(10.5)
where qn is the nominal cross-section of conductor and Ith is the equivalent
short-circuit current.
In the case of overhead line conductors of the Al/St-type, only the cross-section
of the aluminium part is considered. Conductors have sufficient thermal short-circuit
strength if
Tkr
(10.6)
Jth ≤ Jthr ∗
Tk
Values for the rated short-time current density are included in Table 10.1 and
Figure 10.3. In IEC 60865 equations for the calculation of rated short-time current
density are included.
Effects of short-circuit currents 199
A/mm2
(a) 200
180
d1
160
300°C
140
250°C
120
200°C
180°C
100
Jthr
80
160°C
60
140°C
40
300°C 250°C 200°C
100°C
120°C
20
0
A/mm2
(b) 140
120
d1
100
300°C
250°C
80
200°C
180°C
160°C
140°C
Jthr 60
40
100°C
20
0
20
40
60
80
100
120°C
120 130 °C
d0
Figure 10.3
Rated short-time current density of conductors. δ0 is the temperature
at beginning of short-circuit and δ1 is the temperature at end of shortcircuit [1]. (a) ——: Copper; - - - -: unalloyed steel and steel cables
and (b) Al, aluminium alloy, ACSR
Regarding non-insulated conductors, e.g., bare conductors and busbars, the
thermal equivalent short-time current density is allowed to exceed the rated short-time
current density in the case Tk < Tkr .
Manufacturer’s cable lists usually include data on the maximal permissible
thermal short-circuit currents Ithz . An example is outlined in Figure 10.4. The rated
short-time current is given for a short-circuit duration of Tk = 1 s.
10.2.1.2 Cable screening, armouring and sheath
Sheaths, screening and armouring of cables carry parts of the short-circuit current in
the case of asymmetrical short-circuits. Depending on the type of the short-circuit
and the method of cable-laying, this current can be in the range of the short-circuit
current itself, e.g., when the cables are laid in air or on wall-racks. In case cables are
laid in earth the part of the short-circuit current through the sheaths, armouring and
200 Short-circuit currents
kA
100
90
80
70
60
50
40
30
25
20
mm2
15
300
240
Ithz
300
185
240
150
185
120
150
95
10
9
8
7
6
5
120 q
70
95
4
3
50
70
2.5
2
1.5
1
0.1
Figure 10.4
35
50
Cu
Al
0.2
0.3
0.4 0.5 0.6 0.8
tk
1
1.5
2
3
4
5s
25
35
25
16
Maximal permissible thermal short-circuit current for impregnated
paper-insulated cables Un up to 10 kV
Source: KABELRHEYDT
screening is lower than the short-circuit current, as one part is flowing through earth
as well. Due to the comparatively high specific resistance of lead (used for sheaths)
and steel (used for armouring), the short-circuit current preferably flows through
the screening made from copper or aluminium. Data of the different materials are
outlined in Table 10.2. Due to different production processes and degree of purity of
the material, data obtained from other tables can be slightly different.
The maximal permissible temperature δ0 or screening and sheaths of cables are
to be observed in the case of short-circuits. It is assumed for the analysis that the heat
Effects of short-circuit currents 201
Table 10.2
Data of materials for screening, armouring and sheaths
of cables
Material
Specific
caloric heat
(J/K ∗ mm3 )
Specific
resistance
( ∗ mm2 /m)
Temperature
coefficient of
resistance (K−1 )
Copper
Aluminium
Lead
Steel
3.48 ∗ 10−3
2.39 ∗ 10−3
1.45 ∗ 10−3
3.56 ∗ 10−3
17.28 ∗ 10−6
28.6 ∗ 10−6
214 ∗ 10−6
143 ∗ 10−6
3.8 ∗ 10−3
4.0 ∗ 10−3
4.35 ∗ 10−3
4.95 ∗ 10−3
production within the cable during the short-circuit duration is an adiabatic process.
The heat is dissipated to the surroundings only after the short-circuit is switched-off.
The maximal permissible short-circuit current density Jthz , respectively the maximal
permissible short-circuit current Ithz , for the given cross-section qn is calculated by
δ1 + β
1
Qc ∗ (β + 20◦ C)
∗ ln
(10.7a)
∗√
Jthz =
ρ20
δ0 + β
Tkr
Ithz =
δ1 + β
Qc ∗ (β + 20◦ C)
∗ ln
ρ20
δ0 + β
qn
∗√
Tkr
(10.7b)
where Qc is the specific caloric heat, α0 is the thermal coefficient of resistance,
β is the parameter: β = 1/α0 − 20◦ C, δ1 is the maximal permissible temperature
at end of short-circuit, δ0 is the maximal permissible temperature at beginning of
short-circuit, ρ20 is the specific resistance at 20◦ C, qn the nominal cross-section of
screening or sheath and Tkr is the rated short-circuit duration.
The calculation for short-circuit duration different from the rated short-circuit
duration is performed using
Tkr
for Tk ≥ Tkr
(10.4b)
Ith ≤ Ithr ∗
Tk
10.2.2
Mechanical short-circuit strength of rigid conductors
10.2.2.1 General
Currents in conductors induce electromagnetic forces into other conductors. The
arrangement of parallel conductors, such as busbars and conductors of overhead
lines, is of special interest as the electromagnetic forces will be maximal as compared
with transversal arrangements. Three-phase and double-phase short-circuits without
earth connection normally cause the highest forces. The currents inducing the electromagnetic forces are a function of time; therefore the forces are also a function of
202 Short-circuit currents
time. Electromagnetic forces lead to stresses in rigid conductors, to forces (bending,
compression and tensile stress) on support structures and to tensile forces in slack
conductors. Within this section, only stresses on rigid conductors are explained.
10.2.2.2 Electromagnetic forces
Figure 10.5 shows the arrangement of parallel conductors as can be found in busbar
arrangements. In the case of a double-phase short-circuit without earth connection
the forces on the conductors Y and B are
μ0 l
∗ ∗ i Y ∗ iB
(10.8)
Fk2 =
2π a
where μ0 is the permeability, a is the spacing of conductors, l is the length of
conductors and iY ; iB are the peak values of short-circuit current in phases Y and B.
= 2r
B
a
Y
a
R
l
Figure 10.5
Arrangement of parallel conductors
Assuming that the distance between the support structures is large as compared
with the spacing between the conductors [13], which on the other hand is assumed
to be large as compared with the conductor radius r (l/a > 10; a/r > 10), typical
in high-voltage installations, the Skin- and Proximity-effects can be neglected. The
amplitude of the force Fk2 is acting always in the same direction.
In the case of a three-phase short-circuit, the central conductor is exposed to the
maximal force, as the magnetic fields caused by the outer conductors R and B are
superimposed. The force Fk3 on the central conductor acting in the opposite direction
is
μ0 l
(10.9)
∗ ∗ iY ∗ (iR − iB )
Fk3 =
2π a
Due to the decaying d.c. component, the maximal forces act immediately after the
initiation of the short-circuit. With the peak short-circuit current ip2 , respectively
ip3 , being the maximal value of the short-circuit current the maximal forces are
l
μ0
∗
∗ i2
(10.10)
2π am p2
√
l
μ0
3 2
∗
∗ ip3
∗
(10.11)
Fk3 max =
2π am
2
In the case the conductors are arranged at the edges of an isosceles triangle, the
calculation of the forces is identical to that mentioned above.
Fk2 max =
Effects of short-circuit currents 203
The assumptions made above for the spacing and lengths (l/a > 10; a/r > 10),
are normally not fulfilled in low-voltage and medium-voltage installations. The
Skin- and Proximity-effects cannot be neglected and the influence on the electromagnetic forces is considered by the effective distance am using a correction
factor k12
a
(10.12)
am =
k12
A suitable range of values of the correction factor k12 for conductors with rectangular
cross-section is given in Figure 10.6. The electromagnetic force is increased in the
case of flat conductor arrangement and reduced in the case of standing arrangement
as compared with circular cross-section.
1.3
1.2
< 0.1
1.1
k12
1
1.0
0.9
k1ss 0.8
.
5.0
0.7
0.6
b/d
0.5
20.0
0.4
0.3
0.2
1
10
Figure 10.6
a1s /d
100
Correction factor k12 for the calculation of effective distance
(according to Figure 1 of IEC 61660-2:1997)
If the main conductor is composed of several sub-conductors (number n), each
sub-conductor carries only the nth part of total current and the maximal force Fs max
on the sub-conductors is
ip 2
μ0 ls
(10.13)
∗
∗
Fs max =
2π as
n
where ls is the length of sub-conductor, n is the number of sub-conductors, as is the
spacing between sub-conductors and ip is the peak short-circuit current.
The three-phase short-circuit in three-phase systems, respectively the two-phase
short-circuit in two-phase systems, will cause the maximal force. The effective
distance of the sub-conductors as in the case of circular cross-section is
n
1
a1i
=
as
i=2
(10.14a)
204 Short-circuit currents
and for rectangular cross-section
n
k1i
1
=
as
a1i
(10.14b)
i=2
whereas the factor k1i (named k1ss ) can be taken from Figure 10.6.
10.2.2.3 Calculation of stresses in rigid conductors
The forces on the support structures of conductors and the stresses in the conductors
themselves depend on the type of mechanical fixing and the elasticity. The mechanical
system composed of conductor, fixing and supporting structure has a mechanical
natural frequency, which can be actuated by the frequency of the current (50 Hz or
60 Hz), thus increasing the mechanical forces. As axial forces in rigid conductors can
be neglected, the bending stress σm for the main conductor, respectively σs for the
sub-conductors, are calculated using
Fm ∗ l
8∗Z
Fs ∗ ls
σs = Vσ s ∗ Vrs ∗
8 ∗ Zs
(10.15a)
σm = Vσ ∗ Vr ∗ β ∗
(10.15b)
where l; ls are the lengths of conductor, respectively sub-conductor, Fm ; Fs are the
forces on main conductor respectively on sub-conductor, V is the factor as explained
below and Zs is the section moduli depending on the shape of conductor. Whereas Fm
respectively Fs is the force Fm3 in the case of three-phase short-circuits in three-phase
systems. In two-phase systems the force Fm2 in the case of two-phase short-circuits
has to be used. Typical ranges of values of the factors Vσ , Vr (respectively Vσ s , Vrs )
are shown in Figures 10.7 and 10.8. The factor β takes account of type and number
of supports and can be obtained from Table 10.3.
Table 10.3
Factors α, β and γ for different arrangement of supports
(according to Table 3 of IEC 60865-1:1993)
Type of bar and fixing of support
Single-span bar
Multiple-span bar
with equidistant
simple supports
Factor α
Factor β
Factor γ
Support A
Support B
Both simple
A: fixed
B: simple
Both fixed
0.5
0.625
0.5
0.375
1
0.73
1.57
2.45
0.5
0.5
0.5
3.56
2 spans
A:fixed
B: simple
3 and more bars
A: fixed
B: simple
0.375
1.25
0.73
2.45
0.4
1.1
0.73
3.56
Effects of short-circuit currents 205
1.2
1
V
0.8
1.25
.
V s
. 0.6
Three-phase and
double-phase
short-circuits
1.0
>1.6
0.4
0.2
0
0.01
Figure 10.7
0.1
1
fc /f
10
Factors Vσ and Vσ s for the calculation of bending stress (according to
Figure 4 of IEC 60865-1:1993)
1.9
1.8
1.7
Vr
1.6
1.5
.
Vrs
1.4
1.3
1.2
1.1
1
0.01
Figure 10.8
0.1
1
fc /f
10
Factors Vr and Vrs for the calculation of bending stress (according to
Figure 5 of IEC 60865-1:1993)
If the busbar consists of multiple bars supported in unequal distances, the maximal
supporting distance is regarded. If the supporting distance is less than 20 per cent of
the distance of neighbouring bars, the bars have to be coupled by joints. Joints between
two supports are permitted, if the distance between the supporting points is less than
70 per cent of the supporting distances of neighbouring bars.
The values of section moduli Z, respectively Zs , for typical arrangements
of rectangular cross-sections with stiffening elements are outlined in Table 5 of
IEC 60865-1:1993. Values for Z are between 0.867 d 2 b and 3.48 d 2 b, with b being
the height of the conductor and d the thickness of the conductor, respectively the
206 Short-circuit currents
stiffening element. Furthermore, the factor of plasticity q has to be considered.
Values of q are given for typical arrangements in Table 4 of IEC EN 608651:1993. Values for q are between 1.19 and 1.83 for rectangular cross-section and for
U-, H- and I-shape profiles. In the case of circular ring section type conductors, the
factor of plasticity depends on the diameter and the thickness of the wall. Reference
is made to IEC 61865-1:1993.
The conductor has sufficient electromagnetic strength with respect to bending
stress, if the bending stress value σm is below the product of factor of plasticity and
the stress corresponding to the yield point.
σm ≤ q ∗ Rp0.2
(10.16)
When the distance between the conductors is affected significantly by the shortcircuit, the value of the plasticity factor shall be set to q = 1. If the conductor is
composed of sub-conductors the total bending stress, i.e., the sum of σm and σs as
per above has to be considered. The short-circuit strength is given, if
σtot = σm + σs ≤ q ∗ Rp0.2
(10.17)
If only limit values of the stress corresponding to the yield point rather than readings
are available, the minimal values should be used.
Quantities as per Equations (10.16) and (10.17) are
q
Rp0.2
Factor of plasticity
Stress corresponding to the yield point
More information on mechanical short-circuit stress on rigid conductors is included
in [47].
10.2.2.4 Forces on supports
The relevant force to be considered for short-circuit strength is the dynamic force Fd
to be calculated by
Fd = VF ∗ Vr ∗ α ∗ Fm
(10.18)
The three-phase short-circuit in three-phase systems, respectively the two-phase
short-circuit in two-phase systems, will cause the maximal force to be used for the
force Fm . Typical values for the factors Vr and VF are shown in Figures 10.8 and 10.9.
The factor α depends on the type and the number of supports and can be obtained
from Table 10.3.
The short-circuit strength of supports and fixing material is sufficient, if the
dynamic force Fd is below the rated force FrB as per manufacturer’s data.
Fd ≤ FrB
(10.19)
Standards for the short-circuit stress on foundations are actually under discussion.
Effects of short-circuit currents 207
3
Maximal value for
double-phase
short-circuits
Maximal value for
three-phase
short-circuits
VF
.
2
1.0
1
1.25
> 1.6
0
0.01
Figure 10.9
0.1
k
1
fc /f
10
Factor VF for the calculation of bending stress (according to Figure 4
of IEC 60865-1:1993)
10.2.2.5 Influence of conductor oscillation
In Sections 10.2.2.3 and 10.2.2.4, factors are explained which take account
of the function of time of the bending stress and forces. Deviations from values
for Vσ , Vr , Vσ s , Vrs and VF are permitted, if the mechanical natural frequency of the
arrangement is taken into account. It should be noted in this respect that the required
data are difficult to obtain.
The mechanical natural frequency fc of a conductor, either main or sub-conductor
is calculated by
γ
fc = 2 ∗
l
E∗J
m
(10.20)
The mechanical natural frequency of main conductors, composed of several
sub-conductors with rectangular cross-sections, can be calculated by
γ
fc = c ∗ 2 ∗
l
E ∗ Js
ms
(10.21a)
and for sub-conductors by
3.56
fcs = 2 ∗
l
E ∗ Js
ms
(10.21b)
208 Short-circuit currents
Quantities as per Equations (10.20) and (10.21) are:
Factor taking account of type and number of supports (Table 10.3)
Young’s modulus
Second mechanical moment of the conductor, respectively subconductor
Specific mass (mass per length), respectively sub-conductor
Conductor length, respectively sub-conductor length
Factor as per Equation (10.21c) taking account of stiffening elements
(Figure 10.10).
γ
E
J ; Js
m ; ms
l; ls
c
c=
cc
(10.21c)
1 + ξm (mz /(n ∗ l ∗ ms ))
where cc is the factor as per Table 10.4, ξm is the factor as per Table 10.4, mz the
specific mass of stiffening elements and n is the number of sub-conductors.
l
ls
k=1
ls
k=4
Figure 10.10
Calculation of mechanical natural frequency (Factor c). Arrangement
of distance elements and calculation equation (according to Figure 3
of IEC 60865-1:1993)
Number k and type of distance of stiffening elements, the ratio of the mass of
stiffening (distance) elements mz to the specific mass of the conductor ms l separately
for the two swing directions in parallel or rectangular to the side with largest crosssection and the second moments J and Js of the conductor, respectively the subconductor area, are duly considered. The Young’s modulus E and the specific mass m
depend on the construction and on the type of material. The remarks as per Annex A.3
of IEC 60865:1993 shall be observed.
The bending stress and the dynamic force on supports are calculated taking
account of the mechanical natural frequency fc and the factors Vσ , Vr , Vσ s , Vrs and VF
Effects of short-circuit currents 209
Table 10.4
Factors for the calculation of
mechanical natural frequency.
Swing is at right angle to the area
of sub-conductor
Stiffening or distance elements
Number k
Stiffening
element cc
Distance
element cc
0
1
2
3
4
1.0
1.0
1.48
1.75
2.14
1.0
1.0
1.0
1.0
1.0
ls /l
ξm
—
0.5
0.33
0.25
0.2
0.0
2.5
3.0
4.0
5.0
obtained from Figures 10.7 to 10.9. Parameters are the factor κ as per Chapter 4 and
the ratio of mechanical natural frequency fc to power frequency f (50 or 60 Hz).
Calculation equations are included in the Annex of IEC 60865-1:1993.
More information on the electromechanical effects and conductor oscillations can
be found in [46,49,51,53,55].
10.3
d.c. auxiliary installations
10.3.1
Substitute rectangular function
The calculation of thermal and electromagnetic effects of short-circuit currents in
d.c. auxiliary installations is carried out similar to the analysis in a.c. systems.
In general, two alternatives for the calculation are possible. The simplified approach
(first alternative) is based on the standard approximation function [41] of the
short-circuit current
1 − e−t/τ1
for 0 ≤ t ≤ tp
i1 (t) = ip ∗
1 − e−tp /τ1
Ik
Ik
i2 (t) = ip
∗ e−(t−tp )/τ2 +
1−
ip
ip
(10.22a)
for tp ≤ t ≤ Tk
(10.22b)
The electromagnetic effects are calculated using the peak short-circuit current ip of
the standard approximation function. The second alternative is based on the substitute
rectangular function, which achieves the same effects as the standard approximation
function [40,41]. Quantities as per Equations (10.22) are explained in Figure 10.11.
The substitute rectangular function is defined by IR2 and tR according to Figure 10.11
210 Short-circuit currents
(a) 1.2
1
1
ip
i1(t)
0.8
i2(t)
tp
0.6
i/ip
Ik
0.4
2
0.2
0
Tk
0
2
4
6
8
10
12
14
16
18
t
(b) 1.2
ip2
1
tp
0.8
i22(t)
i12(t)
Substitute rectangular function
IR
0.6
)2
(i/ip
0.4
tR
Ik2
0.2
0
Tk
0
Figure 10.11
2
4
6
8
t
10
12
14
16
18
Standard approximation function (a) and substitute rectangular
function (b) (according to Figure 4 of IEC 60660-2:1997). Not to scale
and calculated by
IR2 = 0.2887 ∗
tR = 3.464 ∗
A3i
Ig
Ig
Ai
(10.23a)
(10.23b)
whereas Ai and Ig are calculated by factors mθ 1 , mθ 2 , mg1 , mg2 , mIg1 and mIg2 ,
depending on the peak short-circuit current ip , the short-circuit duration Tk and the
Effects of short-circuit currents 211
time to peak tp
Ai = ip2 ∗ (tp ∗ mθ 1 + (Tk − tp ) ∗ mθ 2 )
3
t
(Tk − tp )3
2 p
Ig = ip
mIg1 +
mIg2 + mθ 1 ∗ tp ∗ (tg − mg1 ∗ tp )2
12
12
+ mθ 2 ∗ (Tk − tp ) ∗ (tp + mg2 ∗ (Tk − tp ) − tg )2
(10.24a)
(10.24b)
where
tg =
mθ 1 ∗ mg1 ∗ tp2 + mθ 2 ∗ (Tk − tp ) ∗ (tp + mg2 ∗ (Tk − tp ))
mθ 1 ∗ tp + mθ 2 ∗ (Tk − tp )
(10.24c)
or can be taken from diagrams in IEC 60865, which include calculation equations for
the factors mθ 1 , mθ 2 , mg1 , mg2 , mIg1 and mIg2 .
The mechanical natural frequency of either main or sub-conductors are to be taken
into account for the calculation of the substitute rectangular function. The mechanical
natural frequency is calculated by
γ
fc = 2 ∗
l
E∗J
m
(10.25a)
The factor γ takes account of the type and number of supports and can be
obtained from Table 10.3. For main conductors composed of several sub-conductors,
the mechanical natural frequency is to be calculated by
γ
fc = c ∗ 2 ∗
l
E ∗ Js
ms
(10.25b)
For the calculation of the bending stress of sub-conductors, the mechanical natural
frequency [43,44] is calculated by
3.56
fcs = 2 ∗
l
E ∗ Js
ms
(10.25c)
Quantities as per Equations (10.25) are:
γ
E
J ; Js
m ; ms
l; ls
c
Factor taking account of the type and number of supports (Table 10.3)
Young’s modulus
Second mechanical moment of the conductor, respectively subconductor
Specific mass (mass per length), respectively sub-conductor
Conductor length, respectively sub-conductor length
Factor taking account of stiffening elements (Equation 10.21c).
212 Short-circuit currents
The vibration period Tme of the main conductor, respectively Tmes of the subconductor is calculated by
Tme =
1
fc
(10.26a)
Tmes =
1
fcs
(10.26b)
where fc ; fcs are the natural mechanical frequency of the conductor, respectively
sub-conductors.
The parameters tR and IR2 of the substitute rectangular function are calculated
for the short-circuit duration Tk , in the case Tk ≤ 0.5Tme . In the case Tk > 0.5Tme
the parameters are calculated for the equivalent short-circuit duration being Tke =
MAX{0.5Tme ; 1.5tp }. In the case Tk = tp , i.e., the short-circuit current has a decreasing function only, the substitute rectangular function is calculated, independently
from the vibration period Tme for the total short-circuit duration Tk .
The calculation of the force for the main conductor is done by
FR max =
ls
μ0
∗
∗ IR2
2π am
respectively, in the case of sub-conductors (number n) by
μ0 ls
IRs 2
FRs max =
∗
∗
2π as
n
(10.27a)
(10.27b)
where IR ; IRs are the current of the substitute rectangular function, l; ls is the length
of conductor, respectively sub-conductor, am ; as is the effective distance between
conductor, respectively sub-conductor, and n is the number of sub-conductors.
10.3.2
Mechanical short-circuit strength of rigid conductors
10.3.2.1 Forces
The calculation of electromagnetic effects on rigid conductors is based on the substitute rectangular function as per Figure 10.11 and described in Section 10.3.1.
The substitute rectangular function leads to identical bending stress and forces
as the standard approximation function. The forces between main conductors are
calculated by
FR =
μ0
l
∗ IR2
∗
2π am
and between sub-conductors (number n) by using
IRs 2
μ0 ls
FRs =
∗
∗
2π as
n
(10.28a)
(10.28b)
Effects of short-circuit currents 213
Quantities as per Equations (10.28) are:
IR ; IRs
l; ls
am ; as
n
Current of the substitute rectangular function
Length of conductor, respectively sub-conductor
Effective distance between conductor, respectively sub-conductor
Number of sub-conductors
The bending stress and forces on supports for both main and sub-conductors are
calculated similar to a.c. installations as described in Sections 10.2.2.2 to 10.2.2.4.
The effective distance of main conductors am is calculated from the distance a by
am =
a
k12
(10.29)
with the correction factor k12 according to Figure 10.6.
The effective distance of sub-conductors as in the case of circular conductors is
calculated by
n
1
a1i
=
as
(10.30a)
i=2
respectively, in the case of rectangular cross-sections
n
k1i
1
=
as
a1i
(10.30b)
i=2
with the factor k1i according to Figure 10.6.
10.3.2.2 Bending stress
The bending stress on main conductors σm , respectively on sub-conductors σs ,
resulting from the bending forces is calculated using
FR ∗ l
8∗Z
FRs ∗ ls
σs = Vσ s ∗
16 ∗ Zs
σm = Vσ ∗ β ∗
(10.31a)
(10.31b)
where Z; Zs are the section moduli of the main, respectively sub-conductors, V is the
factor as explained below, l; ls is the length of conductor, respectively sub-conductor,
and β is the factor taking account of the type and number of supports (Table 10.3).
The forces FR and FRs are the forces as calculated for the substitute rectangular
function. Typical values of Vσ and Vσ s are outlined in Figure 10.12. IEC 61660-2:1997
indicates in Table 2 that Vσ and Vσ s should be lower than 1.
If the busbar consists of multiple bars supported in unequal distances, the maximal
supporting distance is to be considered. If a supporting distance is less than 20 per cent
of the distance of neighbouring bars, the bars have to be coupled by joints. Joints
between two supports are permitted, if the distance between the supporting points is
less than 70 per cent of the supporting distances of neighbouring bars.
214 Short-circuit currents
1.2
1
Vs 0.8
.
VsS
0.6
0.4
0.2
0
0
0.1
0.2
0.3
.
Figure 10.12
tR/Tme
0.4
0.5
Factors Vσ and Vσ s for the calculation of bending stress on conductors
(according to Figure 9 of IEC 61660-2:1997)
The conductor has sufficient electromagnetic strength with respect to bending
stress, if the bending stress σm is below the product of factor of plasticity and stress
corresponding to the yield point
σm ≤ q ∗ Rp0.2
(10.32a)
In the case of several sub-conductors the total bending stress, i.e., the sum of the
stresses σm and σs is to be considered
σtot = σm + σs ≤ q ∗ Rp0.2
(10.32b)
where q is the factor of plasticity and Rp0.2 is the stress corresponding to the yield
point.
Values for the factor of plasticity q for typical arrangements are included in
Table 4 of IEC 60660-2:1997. Values are between q = 1.19–1.83 for rectangular
cross-sections, U-, H- and I-shape profiles. For circular ring section type conductors
the factor of plasticity depends on the diameter and wall thickness. Reference is made
to IEC 61660-2:1997.
10.3.2.3 Forces on supports
The dynamical force Fd is the dominating parameter for the short-circuit strength of
supports to be calculated by
Fd = VF ∗ α ∗ FR
(10.33)
whereas the force FR is the force calculated on the basis of the substitute rectangular
function. Typical values of VF are outlined in Figure 10.13. The factor α takes account
of the type and number of supports and can be obtained from Table 10.3.
The requirements regarding the short-circuit strength of supports and fixing
material are fulfilled if the dynamical force Fd remains below the rated value FrB
Effects of short-circuit currents 215
.
2.2
2
1.8
1.6
VF 1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
Figure 10.13
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
tR/Tme
0.45
0.5
Factor VF for the calculation of forces on supports (according to
Figure 9 of IEC 61660-2:1997)
as per manufacturer’s data
(10.34)
Fd ≤ FrB
Standardisation of stresses on foundations is actually under consideration. More
information on the theoretical background can be found in [43–45,48].
10.3.3
Thermal short-circuit strength
The thermal short-circuit strength is analysed using the thermal equivalent short-time
current Ith and the short-circuit duration Tk .
Ai
Ith =
(10.35)
Tk
with Ai as per Equation (10.24a). The upper limit of the thermal equivalent short-time
current is defined as Ith = ip . IEC 61660-2 indicates that separate considerations for
equipment (e.g., transformers, transducers) and conductors (e.g., busbars, cables)
have to be taken.
The thermal short-circuit strength of equipment is fulfilled if the thermal
equivalent short-time current Ith is below the rated thermal equivalent short-time
current Ithr
Ith ≤ Ithr
for Tk ≤ Tkr
(10.36a)
with the rated short-circuit duration Tkr assumed equal to 1 s. If the short-circuit
duration Tk is longer than the rated short-circuit duration Tkr , the thermal short-circuit
216 Short-circuit currents
strength is given if
Ith ≤ Ithr ∗
Tkr
Tk
for Tk ≥ Tkr
(10.36b)
Permissible values for conductor material of cables are outlined in Table 10.1 and in
Figure 10.3.
In order to analyse the short-circuit strength of bare conductors, the thermal
equivalent short-time current density Jth is calculated using the thermal equivalent
current and the cross-section qn
Jth =
Ith
qn
(10.37)
where Ith is the thermal equivalent current and qn the cross-section of the conductor. The steel part of Al/St-conductors is not taken into account for the
cross-section.
Conductors have sufficient thermal strength if the thermal equivalent short-time
density Jth is below the rated thermal equivalent short-time density Jthr , whereas
short-circuit durations different from the rated short-circuit duration have to be
considered.
Tkr
(10.38)
Jth ≤ Jthr ∗
Tk
Data on the rated short-time current density of conductor material are given in
Table 10.1 and Figure 10.3. In non-insulated conductors, i.e., bare conductors and
busbars, the thermal equivalent short-time current density is allowed to be above the
rated short-time current density in the case Tk < Tkr .
10.4
10.4.1
Calculation examples (a.c. system)
Calculation of thermal effects
The thermal short-circuit strength of a cable N2XS2Y 240 6/10 kV is to be analysed.
Figure 10.14 indicates the equivalent circuit diagram of a power system. In the case
of near-to-generator short-circuit (three-phase) the maximal short-circuit current at
the busbars Q, respectively A, and the thermal equivalent short-circuit current Ith for
the short-circuit duration 0.1 s; 1.2 s; 2 s and 4 s are to be calculated as well as the
A
Q
N2XS2Y240 6/10 kV
Figure 10.14
Equivalent circuit diagram, data of equipment, resistance at 20◦ C
Effects of short-circuit currents 217
short-circuit current density Jthz permitted for a cable sheath of copper (initial temperature 80◦ C; temperature at the end of short-circuit 350◦ C; short-circuit duration
0.2 s; 1.2 s; 2 s).
Data of equipment are given below:
UnQ = 10 kV; SkQ
= 520 MVA
RK
= 0.11 /km; l = 2 km
= 0.0754 /km; XK
The maximal short-circuit current is given for short-circuits at the sending-end of
the cable (approximately identical to location Q) Ik3Q
= 30.03 kA. The maximal
= 13.28 kA. Table 10.1 indicates a value
short-circuit current at location A is Ik3A
of Jthr = 143 A/mm2 . The thermal equivalent rated current is Ithr = 34.32 kA.
The calculation results are outlined in Table 10.5.
Table 10.5
Results of calculation of thermal
equivalent currents
tk
(s)
Ithz
(kA)
IthQ
(kA)
IthA
(kA)
Jthz
(A/mm2 )
0.2
1.2
2
4
76.26
31.33
24.27
17.16
14.12
11.73
9.62
8.81
31.92
26.52
21.86
19.92
689
285
221
—
The short-circuit strength is sufficient if Ith ≤ Ithz . This is fulfilled if tk < 2 s for
short-circuits near busbar A and tk < 4 s for short-circuits near busbar Q.
10.4.2
Electromagnetic effect
A wind power plant is connected to the public supply system by four cables in parallel
laid on racks as indicated in Figure 10.15. The relevant parameters of the short-circuit
current and the maximal forces on the cables, respectively on the fixing material, are
to be calculated. The permissible distance of the fixing clamps is to be determined.
The maximal permissible force on the clamps is Fzul = 40 kN.
Q
T
F2
F1
4 × 3 Single-core cables
G
GS
3~
Figure 10.15
Equivalent circuit diagram of a power system with wind power plant
The data of equipment are given below:
= 1000 MVA; UnQ = 110 kV
SkQ
SrT = 2.5 MVA; ukrT = 6%; uRrT = 0.8%; trT = 20 kV/0.66 kV
218 Short-circuit currents
SrG = 2.5 MVA; UrG = 0.66 kV; cos ϕrG = 0.85; xdG
= 18 per cent
Each cable: XL = 9.84 m; RL = 10.82 m
Short-circuits are fed from the generator as well as from the power system feeder.
Short-circuit currents are given in Table 10.6.
Table 10.6
Results of short-circuit current
calculation
Parameter
Generator
Power system feeder
Short-circuit location
F1 (kA)
F2 (kA)
= 13.3
Ik3
= 12.17
Ik3
ip = 32.05
ip = 27.07
ip = 65.59
ip = 86.66
= 34.48
Ik3
= 36.43
Ik3
The maximal short-circuit current (peak short-circuit current) ip = 86.66 kA is
given for a short-circuit at location F2 and is taken as the basis for the analysis. The
force on the fixing material is Fs = 51.79 N/m. The distance of the fixing clamps
shall be below d ≤ 0.77 m.
10.5
Calculation examples (d.c. system)
10.5.1
Thermal effect
The thermal effect of the short-circuit current is to be calculated. The conductor of the
main busbar has a cross section qn = 2 × 400 mm2 (d × b = 10 mm × 40 mm; Cu),
ρ = 0.0173 mm2 /m, see Figure 10.16. The short-circuit duration is Tk = 100 ms.
Temperature at beginning of short-circuit is δ0 = 20◦ C and at the end δ1 = 250◦ C.
The short-circuit parameters are summarised below:
Peak short-circuit current
ip = 50.5 kA
Quasi steady-state short-circuit current Ik = 46.6 kA
Time-to-peak
tp = 12.1 ms
Rise-time constant
τ1 = 4.03 ms
Decay-time constant
τ2 = 17.3 ms
The time course of the short-circuit current is given by
i1 (t) = 47.99 kA ∗ (1 − e−t/4.03 )
for 0 ≤ t ≤ 12.1 ms
i2 (t) = 50.5 kA ∗ (0.077 ∗ e−(t−12.1)/17.3 + 0.923) for 12.1 ms ≤ t ≤ Tk
Effects of short-circuit currents 219
I
d
am
ls
Figure 10.16
as
a12
height b
Arrangement of busbar conductor (data, see text)
The thermal equivalent short-time current Ith as per Equation (10.35)
Ai
206.11 kA2
=
= 45.4 kA
Ith =
Tk
0.1 s
with Ai as per Equation (10.24a)
Ai = ip2 ∗ (tp ∗ mθ 1 + (Tk − tp ) ∗ mθ 2 ) = 50.52 kA2
∗ (0.0121 s ∗ 0.65 + (0.1 s − 0.0121 s) ∗ 0.83) = 206.11 kA2
Factors mθ 1 and mθ 2 are calculated as per IEC 61660-2
mθ 1 = 0.65
mθ 2 = 0.83
The thermal equivalent short-time current density Jth according to Equation (10.37)
is calculated as
Jth =
Ith
45(4) kA
= 56.75 A/mm2
=
qn
2(400) mm2
Conductors have sufficient thermal strength if the thermal equivalent short-time
density Jth is below the rated thermal equivalent short-time density Jthr taken from
Figure 10.3 (Jthr = 190 A/mm2 ).
Tkr
Jth ≤ Jthr ∗
Tk
1s
2
2
56.75 A/mm ≤ 190 A/mm ∗
= 600 A/mm2
0.1 s
The busbar-conductor has sufficient thermal short-circuit strength.
220 Short-circuit currents
10.5.2
Electromagnetic effect
The configuration as per Sections 9.7 and 10.5.1 is regarded. The conductor arrangement is outlined in Figure 10.16. The distances are: am = 75 mm; b = 40 mm;
as = 10 mm; l = 1050 mm; ls = 35 mm. The mechanical constants of the busbar are:
Stress corresponding to the yield point Rp0.2 = 340 N/mm2
Specific mass of subconductor
ms = 3.5 kg/m
Young’s modulus
E = 106 kN/mm2
Stiffening elements have dimension 40/40/10 mm.
10.5.2.1 Calculation of forces with simplified approach
Peak force between main conductors
FRm =
4π ∗ 10−7 ∗ (Vs/Am)
μ0
l 2
1.05 m
ip =
∗
∗
∗ 50.52 ∗ kA2
2π am
2π
0.0765 m
= 7000.7 N
Effective distance of main conductors am according to Equation (10.29)
am =
0.075 m
a
=
= 0.0765 m
k12
0.98
with the correction factor k12 = 0.98 according to Figure 10.6.
Peak force between sub-conductors
50.5 ∗ kA
4π ∗ 10−7 ∗ (Vs/Am) 0.35 m
μ0 ls ip 2
FRs =
∗
∗
∗
=
2π as n
2π
0.028 m
2
= 1593.9 N
Effective distance of sub-conductors as
as =
0.02 m
a12
=
= 0.028 m
k12
0.72
with the correction factor k12 = 0.72 according to Figure 10.6.
10.5.2.2 Calculation of forces with substitute rectangular function
Parameters of short-circuit current remain identical to those mentioned above.
ip = 50.5 kA;
tp = 12.1 ms;
Tk = 100 ms
Mechanical natural frequency of main conductor as per Equation (10.25)
E ∗ Js
γ
fc = c ∗ 2 ∗
ms
l
2
Effects of short-circuit currents 221
with c ≈ 1.44 as per Figure 10.10, factor γ as per Table 10.3 and the second
mechanical moment J of the conductor, respectively sub-conductor
d3 ∗ b
0.013 ∗ 0.04 4
=
m = 3.33 ∗ 10−9 m4
12
12
106 kN/mm2 ∗ 3.33 ∗ 10−9 m4
3.56
∗
fc = 1.44 ∗
= 46.7 Hz
2
2
3.5 kg/m
1.05 m
J = Js =
Vibration period Tme as per Equation (10.26)
Tme =
1
1
s = 21.4 ms
=
fc
46.7
Mechanical natural frequency of sub-conductors
3.56
E ∗ Js
106 kN/mm2 ∗ 3.33 ∗ 10−9 m4
3.56
fcs = 2 ∗
=
∗
ms
3.5 kg/m
ls
0.352 m2
= 291.8 Hz
Vibration period Tmes
Tmes =
1
1
s = 3.43 ms
=
fc
291.8
Substitute rectangular function to be calculated for the equivalent short-circuit
duration Tke , as Tk > 0.5Tme .
Tke = MAX{0.5 ∗ Tme ; 1.5 ∗ tp } = MAX{0.5 ∗ 21.4 ms; 1.5 ∗ 12.1 ms}
= 18.2 ms
Parameters of the substitute rectangular function (in this case identical for the
main conductor and for the sub-conductors) as per Equations (10.23) and (10.24)
A3i
IR2 = 0.2887 ∗
Ig
Ig
tR = 3.464 ∗
Ai
3
tp
(Tk − tp )3
mIg1 +
mIg2 + mθ 1 ∗ tp ∗ (tg − mg1 ∗ tp )2
Ig = ip2
12
12
+ mθ 2 ∗ (Tk − tp ) ∗ (tp + mg2 ∗ (Tk − tp ) − tg )2
Ai = ip2 ∗ (tp ∗ mθ 1 + (Tk − tp ) ∗ mθ 2 )
tg =
mθ 1 ∗ mg1 ∗ tp2 + mθ 2 ∗ (Tk − tp ) ∗ (tp + mg2 ∗ (Tk − tp ))
mθ 1 ∗ tp + mθ 2 ∗ (Tk − tp )
222 Short-circuit currents
Factors mθ 1 , mθ 2 , mg1 , mg2 , mIg1 and mIg2 are calculated according to IEC 61660-2
mθ 1 = 0.65;
mg1 = 0.63;
mIg1 = 0.42;
mθ 2 = 0.83
mg2 = 0.47
mIg2 = 0.86
Ai = (50.5 kA)2 ∗ (0.0121 s ∗ 0.65 + (0.0182 − 0.0121 s) ∗ 0.83)
= 32.97 kA2 s
tg = [0.65 ∗ 0.63 ∗ (0.0121 s)2 + 0.83 ∗ (0.0182 − 0.0121 s)
∗ (0.0121 + 0.47 ∗ (0.0182 − 0.0121 s))]
∗ [0.65 ∗ 0.0121 s + 0.83 ∗ (0.0182 − 0.0121 s)]−1
tg = 0.0105 s
tp3
(Tk − tp )3
mIg2 + mθ 1 ∗ tp ∗ (tg − mg1 ∗ tp )2
12
12
2
+ mθ 2 ∗ (Tk − tp ) ∗ (tp + mg2 ∗ (Tk − tp ) − tg )
Ig = (50.5 kA)
mIg1 +
⎤
(0.0182 − 0.0121 s)3
(0.0121 s)3
∗ 0.42 +
∗ 0.86 + · · · ⎥
⎢
12
12
⎥
⎢
⎥
⎢
2 ⎢· · · 0.65 ∗ 0.0121 s ∗ (0.0105 s − 0.63 ∗ 0.0121 s)3 + · · ·⎥
Ig = (50.5 kA) ⎢
⎥
⎥
⎢
⎥
⎢· · · 0.83 ∗ (0.0182 − 0.0121 s) ∗ (0.0121 s + 0.47
⎦
⎣
⎡
Ig = 457.2 A2 s3
∗(0.0182 − 0.0121 s) − 0.0105 s)2
The parameters of the substitute rectangular function
(32.97 ∗ 106 A2 s)3
IR2 = 0.2887 ∗
= 2556 ∗ 106 A2
3
457.2 A2 s
3
457.2 A2 s
tR = 3.464 ∗
= 12.9 ms
32.97 ∗ 106 A2 s
The standardised rectangular function and the approximated total short-circuit current
(see e.g., Section 9.7) are outlined in Figure 10.17.
Peak force between main conductors as per Equation (10.27)
FRm =
4π ∗ 10−7 ∗ (Vs/Am)
μ0
l
1.05 m
∗ IR2 =
∗
∗
∗ 2556 ∗ 106 A2
2π am
2π
0.0765 m
= 7016.5 N
Effective distance of main conductors am = 0.0765 m as per above.
Effects of short-circuit currents 223
60
IR
50
Current (A)
40
Approximated total short-circuit current Iappr
Tke
30
tR
20
Substitute rectangular function (IR; tR)
Tk
10
0
0
10
Figure 10.17
20
30
Time (ms)
40
50
60
Standardised rectangular function and approximated total shortcircuit current
Peak force between sub-conductors
μ0 ls IRs 2
4π ∗ 10−7 ∗ (Vs/Am) 0.35 m
=
∗
∗
FRs =
2π as n
2π
0.028 m
6
2
2556 ∗ 10 A
∗
= 1597.5 N
4
10.5.2.3 Bending stress
Bending stresses σm ; σs on main, respectively, sub-conductors (Equation (10.31)) are
calculated for the forces obtained by the substitute rectangular function only.
σm = Vσ ∗ β ∗
FRm ∗ l
7016.5 N ∗ 1.05 m
= 1.0 ∗ 0.73 ∗
8∗Z
8 ∗ 3.47 ∗ 10−6 N/mm2
= 193.7 N/mm2
σs = Vσ s ∗
1597.5 N ∗ 0.35 m
FRs ∗ ls
= 1.0 ∗
= 49.9 N/mm2
16 ∗ Zs
16 ∗ 0.667 ∗ 10−6 N/mm2
with Vσ and Vσ s equal One as per Figure 10.12 and the factor β = 0.73 as per
Table 10.3.
Total bending stress
σtot = σm + σs = (193.7 + 49.9) N/mm2 = 243.6 N/mm2 ≤ q ∗ Rp0.2
= 1.5 ∗ 340 N/mm2
224 Short-circuit currents
Conductors have sufficient electromagnetic strength as
σtot ≤ q ∗ Rp0.2 = 1.5 ∗ 340 N/mm2 = 510 N/mm2
and if
σs ≤ Rp0.2 = 340 N/mm2
10.5.2.4 Forces on supports
Forces on outer supports
Fd = VF ∗ α ∗ FRm = 2.0 ∗ 0.4 ∗ 7016.5 N = 5.61 kN
Forces on inner supports
Fd = VF ∗ α ∗ FRm = 2.0 ∗ 1.1 ∗ 7016.5 N = 15.4 kN
Chapter 11
Limitation of short-circuit currents
11.1 General
The expansion of electrical power systems by new power stations and new lines
(overhead transmission lines and cable circuits) results in an increase of short-circuit
currents due to an increase in sources feeding the short-circuit and due to a reduction
of system impedance. The improvement of existing installations and the replacement of equipment are necessary, in case the permissible short-circuit current will be
exceeded. Measures to limit the short-circuit currents can also be realised which might
be more economic than the replacement of equipment and installations. Different
measures have to be taken into account such as measures affecting the whole system
(higher voltage level), measures concerning installations and substations (separate
operation of busbars) and measures related to equipment (Ip-limiter).
All measures have an influence on the system reliability as well, which must be
guaranteed under outage conditions of equipment after the measures for limitation of
short-circuit currents are in operation. Measures for short-circuit current limitation
decrease the voltage stability, increase the reactive power requirement, reduce the
dynamic stability and increase the complexity of operation. Furthermore some measures to limit short-circuit currents will contradict requirements for a high short-circuit
level, e.g., to reduce flicker in the case of connection of arc-furnace.
The decision on location of power stations is determined beside other criteria
by the availability of primary energy (lignite coal fired power stations are build nearby
the coal mine), requirement of cooling water (thermal power stations are placed near
the sea or at large rivers), geological conditions (hydro power stations can only
be build if water reservoirs are available), requirements of the power system (each
power station requires a system connection at suitable voltage level) and the vicinity
to consumers (combined heat and energy stations need heat consumers nearby).
The connection of large power stations is determined by the branch short-circuit
current from the generators. Figure 11.1 outlines considerations to select the suitable
226 Short-circuit currents
kA
600
S ⬙kG/S ⬙kQ
0.025
500
0.1
0.15
380 kV
400
Un
Nuclear power station
1310 MW
300
220 kV
200
110 kV
Hardcoal power station 600 MW
Lignite coal power station 400 MW
100
Small hydropower station ∑ 220 MW
Gas turbine 120 MW
40
60
100
200
400
600
1000
2000
6000 MVA
SrG
Figure 11.1
Selection of suitable voltage level for the connection of power stations
voltage level for the connection of power stations to the power system. It is assumed
that more than one power station is connected to the system.
As generation of electrical energy without consumers is without any sense, a
suitable power system has to be planned and constructed accordingly. An increased
amount of small combined heat and power stations (distributed generation) with
connection to the medium-voltage and even to the low-voltage system requires additional considerations with respect to protection, operation and short-circuit level in
the different voltage levels [25].
11.2 Measures
11.2.1 Measures in power systems
11.2.1.1 Selection of nominal system voltage
A higher nominal system voltage by constant rated power of feeding transformers will
reduce the short-circuit level proportionally. The selection of nominal system voltage
must take into account the recommended voltages as per IEC 60038:1987 and the
common practice in the utility itself and maybe in the whole country. Table 11.1 lists
a selection of recommended voltages. The table also includes information on typical
applications in Europe.
The short-circuit current is directly proportional to the voltage level, respectively
to the voltage ratio of feeding transformers, if all other parameters are constant.
Limitation of short-circuit currents 227
Table 11.1
Selection of recommended voltage as per IEC 60038:1987
Nominal voltage Application
Low voltage (V)
400–230
500
Medium voltage (kV) 6
10
20
30
High voltage (kV)
Private consumers
According to
Small industrial consumers
IEC Table I
Motor connection in industry Not listed in IEC
HV-motors in industry,
auxiliary supply in power
stations
Urban distribution systems,
industrial systems
Industrial systems,
rural distribution systems
Electrolysis, arc furnace,
rectifiers
110
Urban transport systems
220
Transport system with
regional task
Transmission system
country-wide
380
Remarks
According to
IEC Table III
According to
IEC Table III
According to
IEC Table III
Not listed in IEC
According to
IEC Table IV
According to
IEC Table IV
According to
IEC Table V the
highest voltage
of equipment
Ubmax = 420 kV
is defined
The selection of a new nominal system voltage normally is only possible when new
electrification projects are considered. As the impedance voltage of transformers
increases with increasing voltage an additional positive effect on the reduction of
short-circuit currents is seen. As a by-effect it should be noted that the transmittable
power of overhead lines and cables is increased with increasing voltage without
increasing the cross-section of the conductor. On the other hand, the voltage drop of
the transformer is increased in the case of increase of impedance voltage.
11.2.1.2 Operation as separate subsystems
The power system is operated as several subsystems, which are connected at higher
voltage level. Figure 11.2 outlines the general structure of a 132-kV-cable system (total
load approximately 1500 MW). The system is supplied from the 400-kV-system and
by a power station connected to the 132-kV-level. Assuming a meshed system operation, i.e., the 132-kV-system is operated as one system with all breakers closed, the
short-circuit currents in the case of three-phase and single-phase short-circuits are
= 26.0–37.4 kA and I = 37.3–45.7 kA. Operating the 132-kV-system as two
Ik3
k1
separate subsystems coupled only on the 400 kV-level, the short-circuit currents will
be reduced to the values as outlined in Figure 11.2.
228 Short-circuit currents
GS
3~
∑ 2815 MW
2 × 500 MVA
132-kV-subsystem 1
I ⬙k3= 19.3–26.1 kA
∑ 520 MW
I ⬙1k = 24.2–32.8 kA
GS
3~
400-kV-system
132-kV-subsystem 2
GS
3~
I ⬙k3= 15.4–22.3 kA
I ⬙1k = 20.9–26.5 kA
∑ 440 MW
2 × 500 MVA
GS
3~
Figure 11.2
∑ 1090 MW
Schematic diagram of a 400/132-kV-system for urban load; values of
short-circuit currents in case of operation as two subsystems
Operating the 132-kV-system as two separate subsystems will require additional
cable circuits and an extension of the switchgear to fulfil the (n − 1)-criteria for a
reliable power supply.
11.2.1.3 Distribution of feeding locations
Power stations and system feeders from higher voltage levels are to be connected
to several busbars in the system. This measure was realised in the power system
Limitation of short-circuit currents 229
as per Figure 11.2, which is a by-effect to the system separation. A further example is
outlined in Figure 11.3. A power station of 395 MW is connected to a 132-kV-system,
which has a second supply from the 220-kV-system. The 132-kV-system is a pure
cable network and the shortest cable length between any two substations is 11.2 km.
In the case when the busbar-coupler K in the power station is closed, the three-phase
= 37.6 kA; the short-circuit currents at the
short-circuit current at the busbar is Ik3
= 33.5 kA. If the busbar-coupler K
busbars in the 132-kV-system remain below Ik3
is operated opened, the short-circuit currents at the busbar in the power station are
= 28.0 kA and I = 29.3 kA. For short-circuits at the busbars in the system itself
Ik3
k3
the short-circuit currents are reduced up to 4.1 kA.
220-kV-system
∑ 203 MW
GS
3~
132-kV-system
K
∑ 192 MW
Figure 11.3
GS
3~
Schematic diagram of a 132-kV-system with power station
The generators and the 132-kV-cables in the power stations need to be switchedon to the busbars in such a way that the generated power can be transferred to the
power system without overloading any of the cable even under outage conditions.
230 Short-circuit currents
11.2.1.4 Coupling of power system at busbars with low short-circuit level
Different parts of the power system shall be connected only at busbars with low shortcircuit level. Figure 11.4 outlines a 30-kV-system with overhead lines, which is fed
from the 110-kV-system by two transformers operated in parallel. The three-phase
= 10.09 kA. If the transformers are not operated in parallel
short-circuit current is Ik3
and the system is coupled at busbar K7 the short-circuit current at the feeding busbar
= 5.94 kA.
is Ik3
K8
Un = 30.000 kV
I 0k (L1) = 10.091 kA
S k0 (L1) = 524.341 MVA
(a)
K3
K2
K1
K7
Un= 30.000 kV
I 0k (L1) = 2.199 kA
S k0 (L1) =114.241 MVA
EL5
3200 MVA
K5
K4
K6
EL9
K9
Un = 30.000 kV
I k0 (L1) = 10.091 kA
S k0 (L1) = 524.341 MVA
K8
Un = 30.000 kV
I 0k (L1) = 5.935 kA
S k0 (L1) = 308.375 MVA
(b)
K3
K2
K1
K7
Un = 30.000 kV
I 0k (L1) = 2.193 kA
S k0 (L1) = 113.929 MVA
EL5
3200 MVA
K4
K5
K6
EL9
K9
Un = 30.000 kV
I 0k (L1) = 5.935 kA
S k0 (L1) = 308.375 MVA
Figure 11.4
Equivalent circuit diagram of a 30-kV-system with feeding 132kV-system: (a) Operation with transformers in parallel and (b) limitation of short-circuit current. Result of three-phase short-circuit
= 3.2 GVA; S = 40 MVA; u
current: SkQ
rT
krT = 12%; trT = 110/32;
OHTL 95Al; ltot = 56 km
It should be noted that the short-circuit level at busbar K7 is affected only to a
minor extent. If the transformers are loaded only up to 50 per cent of their rated power
Limitation of short-circuit currents 231
and if the lines have sufficient thermal rating, both system configurations have the
same supply reliability.
11.2.1.5 Restructuring of the power system
Restructuring of power systems is comparatively costly and complicated. In medium
voltage systems restructuring is in most cases only possible together with the commissioning of new primaries, loop-in and loop-out of cable (overhead line) circuits and
the operation of the system as a radial system. In a high voltage system, restructuring
requires a total different system topology. Figure 11.5 outlines the comparison of two
system topologies, i.e., ring fed system and radial fed system.
(a)
Each system
I 0k (L1) = 12.155 kA
S k0 (L1) = 8000.000 MVA
K12
Un =380.000 kV
I 0k (L1) =17.575 kA
S k0 (L1) =11567.2 MVA
K13
K14
K15
K16
K11
Un = 380.000 kV
I k0 (L1) = 23.517 kA
S k0 (L1) =15478.5 MVA
(b)
K17
K18
EL166
EL167
EL173
EL169
EL179
EL178
K20
Un =380.000 kA
I 0k (L1) = 22.668 kA
S k0 (L1) =14919.6 MVA
Figure 11.5
K19
Eaeh system
I 0k (L1) = 12.155 kA
S k0 (L1) = 8000.000 MVA
EL177
EL170
K21
Equivalent circuit diagram of a 380-kV-system and results of threephase short-circuit current calculation: (a) Radial fed system and
= 8 GVA; OHTL ACSR/AW 4 × 282/46;
(b) ring fed system. SkQ
li = 120 km
=
As can be seen from Figure 11.5 the short-circuit currents are reduced from Ik3
23.6 kA to Ik3 = 22.7 kA (3.8 per cent) with the new topology. The reduction of
the short-circuit currents is comparatively small, but will be more significant, if an
increased number of feeders (or generators) shall be connected [2].
232 Short-circuit currents
11.2.2 Measures in installations and switchgear arrangement
11.2.2.1 Multiple busbar operation
The connection of lines and feeders to more than one busbar per substation is
advantageous as compared with the operation of the substation with single busbar or
with busbar coupler closed. Figure 11.6 outlines the schematic diagram of a 110-kV
(a)
D
Un =110.000 kV
I k0 (L1) = 13.009 kA
S k0 (L1) = 2478.597 MVA
Spare busbar
SS1
Un = 110.000 kV
I k0 (L1) = 16.330 kA
S k0 (L1) = 3111.364 MVA
C
Un =110.000 kV
I 0k (L1) = 15.435 kA
S k0 (L1) = 2940.679 MVA
SS2
Un =110.000 kV
I 0k (L1) = 16.330 kA
S k0 (L1) = 3111.371 MVA
B
A
(b)
D
Un = 110.000 kV
I k0 (L1) = 12.779 kA
S k0 (L1) = 2434.703 MVA
Spare busbar
SS1
Un = 110.000 kV
I k0 (L1) = 14.932 kA
S k0 (L1) = 2844.867 MVA
SS2
Un = 110.000 kV
I k0 (L1) = 15.282 kA
S k0 (L1) = 2911.615 MVA
Figure 11.6
C
Un = 110.000 kV
I k0 (L1) = 15.434 kA
S k0 (L1) = 2940.645 MVA
A
B
Schematic diagram of a 110-kV-substation fed from the 220-kV-system:
(a) Operation with buscoupler closed and (b) operation with buscoupler
open. Result of three-phase short-circuit current calculation
Limitation of short-circuit currents 233
system. The 110-kV-substation is equipped with a double busbar and one additional
spare busbar. The substation is fed from the 220-kV-system; outgoing 110-kV-cables
are connected to each of the two busbars in operation.
The operation with two busbars reduces the three-phase short-circuit current from
= 16.3 kA to I = 14.9 kA (8.6 per cent ) at SS1 and I = 15.3 kA (6.1 per cent)
Ik3
k3
k3
at SS2. Each of the two busbars SS1 and SS2 can be switched-on to the spare busbar
without coupling the busbars.
11.2.2.2 Busbar sectionaliser in single busbar switchgear
Single busbars can be equipped with busbar sectionaliser, so that an operation mode
similar to double busbar operation is possible. The outgoing cables and the feeding
transformers need to be connected to the busbar section in such a way that the loading
of feeders is approximately equal. Figure 11.7 indicates an industrial system with
nominal voltage of 6 kV, which is fed from the 30-kV-system.
(a)
BB_110
K3
Un = 6.000 kV
I k0 (L1) = 11.353 kA
S k0 (L1) =117.986 MVA
M
BB_04_1
K4
Un = 6.000 kV
I k0 (L1) = 11.353 kA
S k0 (L1) =117.985 MVA
M
M
M
BB_110
K3
Un = 6.000 kV
I k0 (L1) = 9.482 kA
S k0 (L1) = 98.537 MVA
BB_04_1
M
K4
Un = 6.000 kV
I k0 (L1) = 9.478 kA
S k0 (L1) = 98.503 MVA
M
M
M
BB_6_2
M
Figure 11.7
M
BB_6_2
M
(b)
M
M
M
Equivalent circuit diagram of a 6-kV-industrial system. Results of threephase short-circuit current calculation: (a) Busbar sectionaliser closed
and (b) busbar sectionaliser open
234 Short-circuit currents
The short-circuit current at the feeding busbar is reduced by 16.8 per cent from
= 11.4 kA to I = 9.48 kA in the case when the busbar sectionaliser is kept open.
Ik3
k3
The outgoing feeders have to be arranged in such a way that the loading will be
approximately equal for both busbar sections K3 and K4.
11.2.2.3 Short-circuit current limiting equipment
Short-circuit current limiting equipment and fuses (medium voltage and low voltage
systems) can be installed to reduce the short-circuit level in parts of the installations.
In medium voltage installations, Ip-limiter can be installed. Figure 11.8 outlines the
schematic diagram of an industrial system. The existing switchgear A with low shortcircuit rating shall be extended with the busbar section B, which is fed by an additional
system feeder Q2. The maximal permissible short-circuit current IkAmax
of busbar
section A is exceeded by this extension.
Q1
Q2
T1
T2
i1; I1
i2; I2
A
B
i3 = i1+ i2
I3 = I1+ I2
Figure 11.8
Equivalent circuit diagram of switchgear with single busbar
The total short-circuit current from both system feeders shall be limited to the
permissible short-circuit current IkAmax
of busbar section A in the case of a short /I
circuit at busbar A. If the relation IkQ1
kQ2 depends on the ratio ZQ1 /ZQ2 of the
feeders Q1 and Q2 it is sufficient to measure the partial short-circuit current through
the Ip-limiter. The current ratio is
IkQ1
I1
=
I2
IkQ2
(11.1)
and the total short-circuit current
I3 = I2 ∗ 1 +
IkQ1
IkQ2
≤ IkAmax
(11.2)
Limitation of short-circuit currents 235
i
40 kA
i1 + i2
With Ip-limiter
i1 + i2
Without Ip-limiter
20 kA
i1
i2
t
Figure 11.9
Time course of short-circuit current in installations with and without
Ip-limiter
The threshold value I2an of the Ip-limiter is
I2an = IkA
max ∗
IkQ2
+ I
IkQ1
kQ2
(11.3)
When the permissible short-circuit currents IkAmax
of both busbar sections
and IkBmax
A and B are exceeded, the threshold value I1an of the Ip-limiter for short-circuits at
busbar section B is needed as well
I1an = IkB
max ∗
IkQ1
+ I
IkQ2
kQ1
(11.4)
The threshold value Ian of the Ip-limiter is set to the minimum of both values
Ian = MIN{I1an ; I2an }
(11.5)
The detailed design and determination of the settings are determined, besides other
factors, by different topologies of the power system, different phase angles of the
branch short-circuit currents and different rating of the switchgear in the system.
Figure 11.9 outlines the time curves of short-circuit currents at section A as per
Figure 11.8. The branch short-circuit current i2 from system feeder Q2 is switched off
by the Ip-limiter within 7 ms, thus reducing the peak short-circuit current significantly.
The technical layout of one phase of an Ip-limiter is shown in Figure 11.10.
Inside an insulating tube (1) the main current conductor (3) with a breaking element,
blown by a triggerable explosive loading, (2) is located. When the threshold value
is exceeded, the tripping circuit triggers the explosive loading; the arc inside the
insulating tube cannot be quenched and is commutated to the fuse element (4), which
236 Short-circuit currents
4
1 2
3
5
Figure 11.10
Cutaway view of an Ip-limiter support: (1) insulating tube, (2) explosive loading, (3) main conductor, (4) fuse element and (5) transducer
Source: ABB Calor Emag Schaltanlagen AG
is able to quench the short-circuit according to the fuse It-characteristic. The main
elements, i.e., the isolating tube with main conductor and fuse element, need to be
replaced after operation of the Ip-limiter. A measuring unit with tripping circuit is
needed to compare the actual current value with the threshold value.
Ip-limiters are nowadays available with thyristor technique. The short-circuit
current can be limited within 1–2 ms after initiation of the fault. The Ip-limiter is
back in operation after fault clearing; an exchange of main conductor and fuse is
not necessary. Additional operational functions, such as limitation of start-up current
of large motors can also be realised. Superconducting Ip-limiters are actually in
laboratory tests [7].
11.2.3 Measures concerning equipment
11.2.3.1 Impedance voltage of transformers
Transformers with high impedance voltage are reducing the short-circuit level,
however the reactive power losses are increased and the tap-changer needs to
be designed for higher voltage drops. Figure 11.11 indicates the equivalent circuit diagram of a 10-kV-system fed from a 110-kV-system by three transformers
SrT = 40 MVA. The system load is SL = 72 MVA, cos ϕ = 0.8. The short-circuit
= 2.2 GVA; the voltage at the 10-kV-busbar shall
power of the 110-kV-system is SkQ
be controlled within a bandwidth of ±0.125 kV around U = 10.6 kV.
(a)
110 kV
EL149
EL148
EL150
EL151 EL148
10-KV
Un = 10.000 kA
I k0 (L1) = 35.221 kA
S 0k(L1) = 610.042 MVA
I 0k(L1) = 11.740 kA
S 0k(L1) = 203.347 MVA
Each cable
I k0 (L1) = 0.000 kA
S k0 (L1) = 0.000 MVA
(b)
110 kV
EL149
EL148
EL150
EL151 EL148
I 0k(L1) = 9.719 kA
S 0k(L1) =168.333 MVA
10-kV
Un = 10.000 kV
I k0 (L1) = 29.156 kA
S k0 (L1) = 504.999 MVA
Each cable
I k0 (L1) = 0.000 kA
S 0k(L1) = 0.000 MVA
Figure 11.11
Equivalent circuit diagram of a 10-kV-system with incoming
feeder. Results of three-phase short-circuit current calculation:
(a) Impedance voltage 13% and (b) impedance voltage 17.5%
Table 11.2
UkrT
(%)
13
17.5
Result of loadflow and short-circuit analysis
as per Figure 11.11
Ik3
(kA)
Ik1
(kA)
Tap-changer
position
Reactive power losses
of one transformer
(Mvar)
35.2
29.2
22.5
20.7
+6
+8
2.61
3.58
238 Short-circuit currents
The relevant results of loadflow and short-circuit analysis are outlined in
Table 11.2. As can be seen the increase of the impedance voltage from
13 to 17.5 per cent reduces the short-circuit current, but increases the reactive power
losses and increases the number of steps at the tap-changer to control the voltage.
11.2.3.2 Short-circuit limiting reactor
The application of short-circuit limiting reactors can be defined as a measure related
to switchyards or a measure related to equipment. Figure 11.12 outlines the equivalent circuit diagram of a 10-kV-system in industry with direct connection to an
urban 10-kV-system. Two reactors are installed to limit the short-circuit currents.
The three-phase short-circuit current without local generation in the industrial system
= 20.43 kA.
at the coupling busbar between industry and utility is Ik3
I 0k(L1) = 21.29 kA
I 0k(L1) =14.43 kA
I 0k(L1) = 21.06 kA
I 0k(L1) = 21.06 kA
I k0 (L1) =19.28 kA
I 0k(L1) =19.28 kA
M I 0k(L1) = 1.75 kA
M I k0 (L1) = 1.60 kA
I 0k(L1) =19.24 kA
I k0 (L1) = 19.24 kA
M
I k0 (L1) = 1.32 kA Ik0(L1) = 1.80 kA
Figure 11.12
M
I k0 (L1) = 1.46 kA
Equivalent circuit diagram of a 10-kV-system with short-circuit
limiting reactors. Results of three-phase short-circuit current
calculation
The industrial system is connected to a heat and power plant with four generators
6.25 MVA each, three out of four are allowed to be in operation at the same time.
The short-circuit current is increased by this to 25.6 kA. To limit the short-circuit
≤ 21.5 kA reactors with I = 1600 A; u = 20% were installed. The
current to Ik3
n
k
= 21.3 kA.
short-circuit current is reduced to Ik3
11.2.3.3 Earthing impedances
Single-phase short-circuit currents can be reduced significantly by the installation
of earthing impedances in the neutral of transformers or at artificial neutrals
without affecting the three-phase short-circuit currents. Figure 11.13 represents
an 11.5-kV-system fed from the 132-kV-system. Each substation is equipped with
four transformers (Sr = 40 MVA, uk = 14%). The 132-kV-system has direct neutral
≈ 29.3 kA and I ≈ 37.3 kA.
earthing, the short-circuit currents are Ik3
k1
Limitation of short-circuit currents 239
S 0kQ = 6.7 GVA
Z0/Z1 = 0.454
UnQ = 132 kV
Figure 11.13
Un = 11.5 kV
Equivalent circuit diagram of 11.5-kV-system fed from the 132kV-system
The permissible short-circuit current in the 11.5-kV-system is 25 kA. The
= 15.04 kA when one
single-phase short-circuit currents at 11.5-kV-busbar are Ik1
transformer is in operation and Ik1 = 29.27 kA when two transformers are operated
in parallel.
In order to limit the single-phase short-circuit current on the 11.5-kV-side to
< 25 kA (two transformers in parallel), an earthing resistance of R = 0.31 or
Ik1
E
an earthing reactor of XE = 0.1 need to be installed in the 11.5-kV-neutral of each
of the transformers [3].
11.2.3.4 Increased subtransient reactance of generators
Generators are the direct sources for short-circuit currents; the contribution of one
generator to the short-circuit current is inversely proportional to the subtransient
reactance Xd when the voltage is not changed, see Chapters 3 and 4. An increased
subtransient reactance reduces the branch short-circuit current and by this the
total short-circuit current. Figure 11.14 indicates the results of short-circuit current
calculation of a power station. Generators of different make but identical rating
SrG = 150 MVA are installed. The three-phase branch short-circuit currents are in
= 2.32–2.75 kA depending on the subtransient reactance.
the range of Ik3
240 Short-circuit currents
EL144
400
EL142
EL143
EL143
Ik(L1) = 7.967 kA
Sk⬙(L1) = 1821.450 MVA
132K
Un =132.000 kV
Ik⬙(L1) = 31.569 kA
Sk⬙(L1) = 7217.597 MVA
GT4A1T
Ik⬙(L1) = 2.749 kA
Sk⬙(L1) = 628.456 MVA
GT2T
Ik⬙(L1) = 2.322 kA
Sk⬙(L1) = 530.987 MVA
GT3T
Ik⬙(L1) = 2.535 kA
Sk⬙(L1) = 579.469 MVA
Figure 11.14
GT1A1T
Ik⬙(L1) = 2.749 kA
Sk⬙(L1) = 628.456 MVA
GT1T
Ik⬙(L1) = 2.535 kA
Sk⬙(L1) = 579.469 MVA
GT2A1T
Ik⬙(L1) = 2.749 kA
Sk⬙(L1) = 628.456 MVA
Equivalent circuit diagram of a power station with 132-kV-busbar.
Results of three-phase short-circuit current calculation: SrG =
150 MVA; xd = 12–17.8%
High subtransient reactance of generators has a negative impact on the dynamic
stability performance of the generators. In the case of short-circuits on the transmission line with subsequent fault clearing the transmittable power from a power station
is reduced if the fault clearing time of the protection is kept constant, respectively, the
fault clearing time must be reduced to keep the transmittable power constant. Details
can be obtained from [2, 26].
11.3 Structures of power systems
11.3.1 General
It should be noted that some of the measures as per Section 11.2 to reduce short-circuit
currents can only be applied in certain power systems. Ip-limiters are only available
in low voltage and medium voltage systems. When only a single busbar is installed,
the operation with two busbars is not possible and in a radial fed system, no additional
feeding point is normally available. Within this section, the main structures of power
systems are introduced:
•
•
•
Radial system
Ring-main system
Meshed system
More details can be obtained from [26].
Limitation of short-circuit currents 241
11.3.2 Radial system
Radial systems represent the simplest topology of a power system and can usually
be found in low voltage systems. Figure 11.15 outlines the general structure, with
one feeding point and distributing of the lines in several branches. This structure is
suitable in the case of low load density but also for the connection of high bulk-supply
loads. The calculation of short-circuit currents is comparatively easy, as there are no
meshed lines in the system and only one topology has to be analysed.
MV/LV-transformer
Figure 11.15
General structure of a radial system with one incoming feeder
11.3.3 Ring-main system
In a ring-main system the receiving end of each line of a radial system is to be
connected either back to the feeding busbar or to an additional feeding busbar. Ringmain systems are most often planned for medium voltage systems, and in rare cases
for low voltage systems as well. Normally ring-main systems are operated with open
breaker or isolator in one primary as indicated in Figure 11.16(a). This enables an
operation similar to radial systems but with a switchable reserve for all consumers.
Feeding busbars can be planned at several locations of the system as indicated in
Figure 11.16(b). Short-circuit current calculation is more complicated than in radial
systems as several operating conditions, i.e., system topologies, are to be taken into
account to determine the minimal and the maximal short-circuit current.
11.3.4 Meshed systems
Meshed systems are normally applied only for high voltage systems or in industrial
supply systems for MV-level as well. With the consideration of consumer load and
242 Short-circuit currents
(a)
(b)
Switch (normally open)
Figure 11.16
MV/LV-transformer
General structures of ring-main systems: (a) Simple structure with one
feeding busbar and (b) structure with two feeding busbars (feeding
from opposite sides)
Interconnection lines to
neigbouring countries
GS
3~
GS
3~
380-kV
transmission
system
220-kV
subtransmission
system
(regional
distribution)
Industrial
systems
35-kV
20-kV
GS
3~
110-kV
(urban area
with
generation)
10-kV (20-kV)
distribution
system
Figure 11.17
110-kV
system
(rural areas)
10-kV
20-kV
(35-kV)
Principal structure of a high voltage system with different voltage levels
GS
3~
110-kV-system
(industry
with generation)
GS
3~
244 Short-circuit currents
capability of power stations, the system is planned, constructed and operated in such
a way as to allow the supply of consumers without overloading of any equipment and
without violating the permissible voltage profile even under outage of one ((n − 1)criteria) or more equipment at the same time. The calculation of maximal short-circuit
currents seems relatively simple, if all equipment are assumed to be in operation. As
far as different operation schedules of the power stations are considered, different
system topologies need to be considered. The calculation of the minimal short-circuit
current, however, is much more difficult, as a close cooperation between planning
and operation is required. A large number of different topologies in the meshed power
system have to be analysed to ensure that the calculated short-circuit current is the
minimal current. The principal structure of a meshed high voltage system is outlined
in Figure 11.17.
A special type of meshed systems is applied to low voltage systems; the principal
structure is outlined in Figure 11.18. The reliability of supply is comparatively high,
as reserve in the case of outage of any line or infeed is provided through the remaining
lines.
(a)
MV-cable
Switch
(normally open)
2
1
(b)
3
3
2
2
1
3
1
3
2
1
1; 2; 3 Connection to MV-cable No. 1; 2; 3
Figure 11.18
Principal structure of meshed low voltage system: (a) Single-fed
meshed system and (b) meshed system with overlapping feeding
Chapter 12
Special problems related to
short-circuit currents
12.1
12.1.1
Interference of pipelines
Introduction
Interference between overhead lines, communication circuits and pipelines is caused
by asymmetrical currents, which may be due to short-circuits, asymmetrical operation
or asymmetrical design of equipment, especially asymmetrical outline of overhead
line towers with respect to pipelines and communication circuits. This interference
is based on inductive, ohmic and capacitive coupling between the short-circuit path
(e.g., overhead line) and the circuit affected by interference (e.g., pipeline). Normal
operating currents, respectively voltages, cause magnetic as well as electric fields
which are asymmetrical in the vicinity of overhead lines which may cause interference
problems in the long-time range.
Short-circuit currents on overhead transmission lines and cables and short-circuit
currents through earth cause interference in the short-time range only when the shortcircuit is switched off after some seconds by the power system protection. Interference
problems may arise in cable and overhead line systems to a different extent depending
on the handling of the system neutral. The induced voltage in pipelines and communication circuits may endanger technical installations and safety of workers when
defined limits will be exceeded. Within the context of this book, only the short-time
interference, in the following called interference, is dealt with. Table 12.1 outlines
the needs for the analysis of interference problems.
Interference problems may occur in most of the cases by inductive and ohmic coupling in power systems with low-impedance earthing consisting mainly of overhead
lines, as can be seen from Table 12.1. In systems with isolated neutral or resonance
earthing interference problems have to be regarded only when the fault current is not
self-extinguishing (see Chapters 5 and 7). Capacitive coupling does not cause any
Table 12.1
Interference between power system, communication circuits and pipelines
Power system
Interference of communication circuits and pipelines by:
Handling of neutrals
Operating condition
Low-impedance
earthing
Without fault
OHTL
Cable
Short-circuit
OHTL
Cable
Without fault
OHTL
Cable
Short-circuit;
Earth fault
OHTL
Cable
No interference
Double earth
fault
OHTL
Cable
Present
No interference
Isolated neutral;
Resonance earthing
Inductive coupling
Capacitive coupling
Only if circuits on same tower
No interference problems
Present
Present in special cases
No interference
No interference
Only if circuits on same tower
No interference
Present
No interference
No interference
No interference
Ohmic coupling
No interference
Present
No interference
No interference
No interference
Present
No interference
Special problems related to short-circuit currents 247
severe problems in pipelines and communication circuits. It should be noted furthermore that interference of communication circuits is decreasing due to the decreasing
installations of overhead communication circuits, which are replaced by wireless
communication or by cable circuits, which can be protected easily against interference. The explanations on interference are therefore concentrated within this section
to the interference of pipelines.
Regulations on the permissible values for voltages induced in pipelines and communication circuits and/or for touch voltages exist in various countries. The main aim
is to protect any person likely to work on the pipeline or power circuit against electrocution hazard. According to an international survey carried out by CIGRE [20],
the maximum permissible touch voltage is defined in different countries in different
ways, ranging from 200 V up to 1500 V depending on the maximal fault duration
time. In Germany, the maximal permissible touch voltage and the maximal permissible voltage pipeline-to-earth are both limited to 1000 V for a fault-duration of 0.5 s.
Higher values are applied only in Australia (1500 V) and in Brazil (1700 V), whereas
the Brazilian regulation defines the admissible value of a touch current, which is converted for comparison into the voltage limit. Within this survey, only two US-utilities
have answered the questions on voltage limits. The limit for the touch voltage applied
there is given to be 500 V, whereas the voltage pipeline-to-earth should be less than
5 kV. It is unclear and could not be clarified in the CIGRE-survey, why low value for
touch voltage (500 V) as compared with other countries resulted in a comparatively
high value for the pipeline-to-earth voltage (5 kV).
According to [21] the maximal permissible voltage pipeline-to-earth for shorttime interference shall be below 1000 V. If ASME/IEEE-standard No. 80 is applied
a maximal permissible touch voltage is defined in relation to the fault duration time
for different body weight of the person involved. If the most severe restrictions
are applied, i.e., 50 kg body weight and fault duration (clearing time) of 150 ms, the
maximal permissible touch voltage Ut50 is 350 V. This is the recommended limit as
per IEEE-standard No. 80 item 6, where it is mentioned that the actual transferred
voltage should be less than the maximum allowable touch voltage Etouch to ensure
safety.
12.1.2
Calculation of impedances for inductive interference
In order to calculate the interference of pipelines the loop-impedances, coupling
impedances and self-impedances of the line conductor, earth conductors and the
pipeline itself are required. The loop-impedance of the pipeline with earth return is
Z P
=
RP
μ0
+
∗ω+j
8
μP
μ0
δ
+
∗ ω ∗ ln
2π
rP
4
(12.1)
where rP is the outer radius of the pipeline, μ0 is the absolute permeability, μP is the
fictitious relative permeability of the pipeline, δ is the depth of earth return path and
ω is the angular frequency.
248 Short-circuit currents
Equation (12.1) is composed of
RP
μ0
∗ω
8
δ
μ0
∗ ω ∗ ln
2π
rP
μP
μ0
∗ω∗
2π
4
Resistance of the pipeline per unit length
Resistance of the earth return path per unit length
Outer reactance of the loop with earth return path per
unit length
Internal (inner) reactance of the conductor (pipeline)
per unit length
The depth δ of the earth return path is given by Equation (12.2) with the resistivity
of soil ρ (specific soil resistance) according to Table 12.2.
δ=√
Table 12.2
Parameters
1.85
ω ∗ (μ0 /ρ)
(12.2)
Resistivity of soil ρ for different types of soil conditions
Type of soil
Alluvial
soil –
swamp
soil
Specific soil resistance 30
ρ (m)
σ = 1/ρ (µS/cm)
333
Depth of earth return
510
δ at 50 Hz (m)
Depth of earth return
465
δ at 60 Hz (m)
Clay
Limestone
clay –
farm soil
Wet
sand
Wet
gravel
Dry
sand
Dray
gravel
Stony
soil
50
100
200
500
1000
3000
20
660
10
930
5
1320
2
2080
1
2940
0.33
5100
600
850
1205
1900
2684
4655
The resistance RP of the pipeline can be calculated from the conductivity κP and the
thickness d of the pipeline wall taking eddy currents and the dissipation of the current
into the outer level of the pipeline wall into account.
3
(12.3)
RP = Rdc ∗ x + 0.25 +
64 ∗ x
The increase of the inner inductivity Xi is given by
3
3
∗ x−
+
Xi = Rdc
64 ∗ x
128 ∗ x 2
(12.4)
Special problems related to short-circuit currents 249
, the
with the parameter x = d/(2 ∗ δP ), the d.c. resistance of the pipeline wall Rdc
depth of current in the pipeline wall δP and d being the thickness of the pipeline wall:
1
δP = √
ω/2 ∗ (κP ∗ μP ∗ μ0 )
(12.5)
Comparing the loop-impedance as per Equation (12.1) with the inner inductance
of the pipeline as per Equation (12.4) the fictitious relative permeability of the pipeline
will be
μP = 4 ∗ Rdc
∗
x − 3/(64 ∗ x) + 3/(128 ∗ x 2 )
(ω ∗ μ0 )/2π
(12.6)
Coupling impedances [2] need to be calculated for the analysis of the interference
problems. For the individual distances and impedances reference is made to Figure
12.1 indicating a typical interference problem between a 380-kV-line with two earth
wires, counterpoise and a pipeline.
The coupling impedance Z LP of the loop conductor and pipeline with earth return
is obtained from
μ0
μ0
δ
Z LP =
(12.7)
∗ω+j
∗ ω ∗ ln
8
2π
dLP min
The coupling impedance Z EP of the loop earth conductor and pipeline with earth
return is obtained from
μ0
δ
μ0
Z EP =
∗ω+j
∗ ω ∗ ln
(12.8)
8
2π
dEP min
Coupling impedance Z LE of the loop earth conductor and conductor with earth return
is given by
μ0
δ
μ0
Z LE =
∗ω+j
∗ ω ∗ ln
(12.9)
8
2π
dLE
In case a second earth conductor is installed, the coupling impedance Z EP2 of the
loop second earth conductor and pipeline with earth return is calculated by
μ0
δ
μ0
Z EP2 =
(12.10)
∗ω+j
∗ ω ∗ ln
8
2π
dEP2 min
The coupling impedance Z LE2 of the loop second earth conductor and conductor with
earth return is obtained from
μ0
μ0
δ
Z LE2 =
(12.11)
∗ω+j
∗ ω ∗ ln
8
2π
dLE2
The coupling impedance Z E12 of the loop first and second earth conductor and
conductor with earth return is obtained from
μ0
δ
μ0
Z E12 =
(12.12)
∗ω+j
∗ ω ∗ ln
8
2π
dE12
250 Short-circuit currents
E2
E; E1
dE12
dLE
dLE2
s
L
dEP2min
dLBmin
dEPmin
dLPmin
P
B
hP
dBP
d
Figure 12.1
Outline and distances of a high-voltage transmission-line tower. B:
counterpoise; P: pipeline; L: conductor nearest to pipeline. E; first
earth conductor (nearest to pipeline), also named E1; E2: second earth
conductor
Special problems related to short-circuit currents 251
Furthermore the loop-impedance Z E of the earth conductor and earth return is given by
μ0
δ
μr
μ0
∗ω+j
∗ ω ∗ ln
+
(12.13)
Z E = R +
8
2π
r
4
Quantities as per Equations (12.7) to (12.13) are:
dLPmin
dEPmin
dLE
dEP2min
dLE2
dE12
r
R
Minimal distance between the pipeline and the lowest conductor
nearest to the pipeline
Minimal distance between the pipeline and the earth conductor
Distance between the earth conductor and the lowest conductor nearest
to the pipeline
Minimal distance between the pipeline and the second earth conductor
Distance between the second earth conductor and the lowest conductor
nearest to the pipeline
Distance between the first and second earth conductor
Radius of earth conductor
Resistance of earth wire per unit length.
Generally the minimal distances between the conductors and the pipeline have to
be considered. This includes considerations on the conductor sag and the conductor
swing under worst conditions. The mean effective height hS of the conductor may be
calculated from
hS = hL − 0.667 ∗ s̄
(12.14)
where hL is the conductor height at the tower and s̄ is the conductor sag.
In some cases a counterpoise parallel to the pipelines is used to reduce the induced
voltage into the pipeline. The coupling impedances with the conductor are needed in
these cases. The coupling impedance Z B of the loop counterpoise and earth return is
obtained from
μB
μ0
δ
μ0
+
∗ω+j
∗ ω ∗ ln
(12.15)
Z B = RB +
8
2π
rB
4
The coupling impedance Z LB of the loop conductor and counterpoise with earth return
is obtained from
μ0
δ
μ0
∗ω+j
∗ ω ∗ ln
(12.16)
Z LB =
8
2π
dLB min
The coupling impedance Z BP of the loop pipeline and counterpoise with earth return
is calculated from
μ0
μ0
δ
(12.17)
∗ω+j
∗ ω ∗ ln
Z BP =
8
2π
dBP
where rB is the radius of counterpoise, μB is the fictitious relative permeability of
counterpoise, RB the resistance of counterpoise per unit length, dLBmin is the minimal
252 Short-circuit currents
distance between counterpoise and the lowest conductor nearest to the pipeline, dBP
is the distance between counterpoise and pipeline.
12.1.3
Calculation of induced voltage
Based on the impedance calculations as per Section 12.1.2 the induced voltage into
the pipeline can be calculated for any configuration as follows:
U iP = −Z LP ∗ r ∗ I kE ∗ lPp ∗ w
(12.18)
where Z LP is the coupling impedance of the loop pipeline and conductor nearest
to earth with earth return, I kE is the initial short-circuit current (asymmetrical) of
the overhead line, lPp is the length of parallel exposure between pipeline and overhead line, r is the screening factor as per Equation (12.18) and w is a probability
factor taking into account that all worse conditions do not occur at the same time
instant [21].
The screening factor depends on the presence of earth wires, counterpoises and any
other compensation circuit capable to reduce the induced voltage into the pipeline.
If only one earth wire is present as compensation circuit the screening factor r is
calculated based on the coupling and loop-impedances by
r = 1 −
Z LE ∗ Z EP
Z E ∗ Z LP
(12.19)
If more than one earth conductor or additional compensation circuits are present,
additional considerations for the screening factor are required. The total screening
factor is then given by the difference of the individual factors of each earth conductor
and/or compensation circuit taking into account the correction factor which represents
the influence of each earth wire and compensation circuit on the current in the other
wire [22]. In the case of two earth conductors the total screening factor r tot is given by
r tot = 1 −
Z LE2 ∗ Z EP2
Z LE ∗ Z EP
−
∗
k
∗ k1
2
Z E ∗ Z LP
Z E2 ∗ Z LP
(12.20)
Correction factors k1 and k2 are calculated by
k1 =
1 − (Z LE ∗ Z E12 )/(Z E ∗ Z LE2 )
1 − (Z E12 ∗ Z E12 )/(Z E ∗ Z E2 )
(12.21a)
k2 =
1 − (Z LE2 ∗ Z E12 )/(Z E2 ∗ Z LE )
1 − (Z E12 ∗ Z E12 )/(Z E ∗ Z E2 )
(12.21b)
Special problems related to short-circuit currents 253
Correction factors for other arrangements and numbers of conductors can be obtained,
respectively calculated, as per [39]. Impedances as per Equations (12.19) to (12.21)
are calculated according to Section 12.1.2
Z LP
Z EP
Z LE
Z EP2
Z LE2
Z E12
ZE
Z E2
Coupling impedance of the loop conductor and pipeline with earth return
as per Equation (12.7)
Coupling impedance of the loop first earth conductor and pipeline with
earth return as per Equation (12.8)
Coupling impedance of the loop first earth conductor and conductor with
earth return as per Equation (12.9)
Coupling impedance of the loop second earth conductor and pipeline
with earth return as per Equation (12.10)
Coupling impedance of the loop first earth conductor and conductor with
earth return as per Equation (12.11)
Coupling impedance of the loop first and second earth conductor with
earth return as per Equation (12.12)
Loop-impedance of earth conductor and earth return as per Equation
(12.13)
Loop-impedance of second earth conductor and earth return as per
Equation (12.13)
For three and more compensation circuits, e.g., two earth conductors and a counterpoise, the screening factor needs to be calculated either from the individual
current distribution within the different compensation circuits or by means of the
multiplication method or other methods as outlined in [3].
12.1.4
Characteristic impedance of the pipeline
The induced voltage U iP is only an indication for the inductive interference but does
not take into account the earthing conditions and the conductivity of the pipeline
coating. In order to calculate step and touch voltages the voltage between pipeline
and earth is to be calculated taking account of the earthing and conductivity conditions of the pipeline. Based on the analysis of the system equivalent, i.e., faulted
phase conductor, presence of earth conductors, counterpoise, pipeline and earthing
conditions, the pipeline must be represented by means of its characteristic impedance
and its conductivity.
The conductivity of the pipeline against the surrounding earth is determined by
of the bare (uncoated)
the resistance RI of the pipeline coating and the resistance RPE
pipeline in earth (see Table 12.3). The total resistance is given by
= RI + RPE
RCtot
(12.22)
The specific reactance due to the capacitance of the pipeline can be neglected for
of
interference analysis as it is much smaller than the resistance. The resistance RPE
254 Short-circuit currents
Table 12.3
Resistance of pipeline coatings [23]
Type of coating
Thickness of
coating (mm)
Specific coating
resistance (k m2 )
Bitumen
Polyethylene
Epoxy resin
4–6
2–3
0.3–0.6
10
100
10
the bare (uncoated) pipeline in earth is calculated by
2∗l
ρ
∗ 2 ∗ ln
4π
d
=
RPE
+ ln
(2 ∗ hP )2 + (l/2)2 + l/2
(2 ∗ hP )2 + (l/2)2 − l/2
(12.23)
where hP is the depth of pipeline under ground, l is the total length of pipeline, d is
the outer diameter of pipeline and ρ is the resistivity of the soil as per Table 12.2.
It should be noted that the influence of the earth resistivity is comparatively low
for high resistance of pipeline coating.
The characteristic impedance Z W , the propagation constant γ and the characteristic length LK of the pipeline are required prior to the calculation of the voltage
pipeline-to-earth.
Z P ∗ RPE
ZW =
γ =
LK =
(12.24)
Z P
RPE
(12.25)
1
Re
Z P /RPE
(12.26)
where Z P is the loop-impedance of pipeline and earth return per unit length as per
is the resistance of the bare (uncoated) pipeline as per
Equation (12.1) and RPE
Equation (12.23).
12.1.5
Voltage pipeline-to-earth
The voltage pipeline-to-earth U PE (x) along the exposure length is calculated based
on the theory of line propagation for each individual location x of the pipeline. Using
the abbreviations
I ∗kE = r tot ∗ I kE ∗
Z LP
∗w
ZP
(12.27a)
Special problems related to short-circuit currents 255
and
b = 0.5 ∗
∗l
Z P ∗ RPE
(12.27b)
where l is the length of pipeline, Z P is the loop-impedance of pipeline and earth
return per unit length as per Equation (12.1), R PE is the resistance of the bare
(uncoated) pipeline as per Equation (12.23), r tot is the total reduction factor as per
Equation (12.20), Z LP the coupling impedance of the loop conductor and pipeline
with earth return as per Equation (12.7), I kE is the short-circuit current through earth
and w the probability factor.
The voltage pipeline-to-earth along the exposure length (parameter x) is calculated
taking account of the earthing resistances R1 and R2 at the end of the pipeline,
respectively at the end of section, under investigation by
U PE (x) = Z W ∗ I ∗kE ∗ {R1 (Z W + R2 )eb+γ x + R2 (Z W − R1 )e−b+γ x
− R1 (Z W − R2 )e−b−γ x − R2 (Z W + R+ )eb−γ x }
∗{(Z W +R1 ) ∗ (Z W + R2 )e2b − (Z W − R1 ) ∗ (Z W − R2 )e−2b }−1
(12.28)
Outside the exposure length (parameter y) the voltage pipeline-to-earth decays for
each individual location y of the pipeline according to
U PE (y) = U 1 ∗
+
RA − Z W
(RA + Z W )e2γ l + (RA − Z W )
(RA + Z W )e2γ l
(RA + Z W )e2γ l + (RA − Z W )
eγ l
e−γ l
(12.29)
where U 1 is the voltage at the end of exposure length, RA is the far-end impedance
of the pipeline, l is the total length of pipeline outside exposure length, γ is the
propagation constant and Z W is the characteristic impedance.
The method described above assumes constant parameters I ∗kE and for the effec
of the pipeline coating against earth and constant distance between
tive resistance RPtot
the pipeline and the overhead line. In other cases the analysis has to be carried out
for each individual section having constant or nearly constant parameters by separation of the exposure length into subdivisions for which constant parameters can be
assumed.
In case of oblique exposures or crossing between overhead line and pipeline the
exposure length has to be divided into different subsections having equal induced
voltages. An average distance ao between the overhead line and the pipeline has to
be calculated
ao =
√
a1 ∗ a 2
(12.30)
256 Short-circuit currents
(a)
Pipeline
lII
a3
lI
lIII
a2
lIV
OHTL
a1
(b)
L
L
Tower
dLPc
Tower
P
Figure 12.2
Oblique exposure and crossing of pipeline and overhead line. (a) Plot
plan and (b) elevation plan (detail from crossing location)
The parameters a1 and a2 are the distances at the ends of the oblique exposure,
see Figure 12.2 for details. The ratio a1 /a2 should not exceed the value of three
otherwise further subdivisions have to be selected. Distances of more than 1000 m
between overhead line and pipeline can be neglected for short-time interference.
Crossings shall be handled similarly to oblique exposures with the restriction that
the subdivisions in the vicinity of the crossing shall be selected as short as possible.
Special problems related to short-circuit currents 257
The distance between the overhead line and the pipeline at the location of crossing
shall be set equal to the minimal height of the overhead conductor above the pipeline
dLPc , taking account of the average conductor sag, see Figure 12.2(b). The average
distance ac between the overhead line and the pipeline is given by
ac =
a2 ∗ dLPc
(12.31)
The results for each subdivision have to be superimposed for each location of the
pipeline.
12.2
12.2.1
Considerations on earthing
General
The influence of neutral handling and to some extent of the earthing was outlined
already in Chapters 5 and 7. The handling of neutrals, however, requires additionally
the analysis of the earth itself, of earthing grids and rods and on step and touch voltages
related to earthing. With respect to the main task of this book, i.e., calculation and
analysis of short-circuit currents and their effects, the earthing in power systems are
only dealt with in relation to the impact of short-circuit currents on earthing. Special
problems such as corrosion of earthing material, influence of earthing on lightning
and on fast-front overvoltages are not explained here. More details can be found in
[5, 26].
Equipment and installations in power systems have to be designed and operated
in such a way to avoid impermissible conditions with respect to the health of human
beings and animals also taking into account reliable and sufficient operation of the
technical installations. Earthing in power systems is one of the main items to ensure
this safe and secure operation. Asymmetrical operation and short-circuits cause currents flowing through earth which may flow as well through the human body in the
case of contact of the body with earth or with installations connected to earth. The
earthing problem is determined by
•
•
•
•
•
Resistance of human body
Electrical conditions of the earth
Current through earth
Fault duration
Earthing impedance
12.2.2
Resistance of human body
The impedance of the human body, which is mainly a resistance, is determined by the
location of contacts to the electrical installations and depends on the touch voltage
as well. Figure 12.3 indicates the resistance of the body if measured between the two
hands, valid approximately 10 ms after initiation of currents through the body. As
258 Short-circuit currents
5
4.5
4
3.5
R (kΩ)
3
2.5
2
1.5
1
0.5
0
0
100
Figure 12.3
200
300
400
UB (V)
500
600
700
800
Impedance of the human body (hand-to-hand) depending on the touch
voltage
can be seen the resistance is between 3.5 k and 1.1 k decreasing with increasing
touch voltage. For other contacts the resistance is reduced, e.g.,
•
•
•
•
One hand to breast
One hand to knee
Two hands to breast
Two hands to knee
reduction to 45 per cent of resistance as per Figure 12.3
reduction to 70 per cent of resistance as per Figure 12.3
reduction to 23 per cent of resistance as per Figure 12.3
reduction to 45 per cent of resistance as per Figure 12.3
It should be noted in this respect that the current through the human body is the
critical physical phenomena causing uncontrolled operation of the heart (ventricle
flicker) or muscle convulsion. The critical current is a function of the exposure
time [24]. The duration of the permissible touch voltage UB therefore depends on
the exposure time as well (fault duration) and is outlined in Figure 12.4.
The permissible touch voltage is set in most of the standards to UB = 65 V, resulting in a negligible risk for ventricle flicker independent from the time of exposure.
If this voltage limit cannot be guaranteed the exposure time must be limited.
12.2.3
Soil conditions
The resistivity of the soil depends on the type of soil (swamp soil, stony soil) as
outlined in Table 12.2. The differences are determined by the humidity of the soil.
The value for swamp soil is between 10,000 m (humidity less than 10 per cent)
and 30 m (humidity 90 per cent). It is obvious that the resistivity of the soil varies
in a wider range if surface electrodes will be used for earthing as the soil humidity
Special problems related to short-circuit currents 259
800
700
600
UB (V)
500
400
300
200
100
0
0.1
Figure 12.4
1
Time of exposure (s)
10
Permissible touch voltage depending on the time of exposure
varies in a wider range throughout the year as compared with deep-ground earthing
by earthing rods. The resistivity for surface electrodes varies throughout the year in
a range of ±30 per cent, in the case of deep-earthing only in a range of ±8 per cent
of the average value, the highest value occurring in March and the lowest value in
August (European countries). It is therefore recommended to use surface electrodes
only in those installations where the soil humidity is nearly constant throughout the
year and to use deep-ground earthing for other conditions.
12.2.4
Relevant currents through earth
Currents through earth are only existing in the case of asymmetrical short-circuits with
earth connection. In most of the cases (X0 /X1 > 1), the single-phase short-circuit
is greater than the current through earth I
current Ik1
kE2E in case of a double-phase
short-circuit. Only in case of power systems (e.g., 115 and 132 kV) having high
amount of cables, power stations with high rating closely connected to the system under investigation and when all transformer neutrals are earthed through low
impedance, can the ratio X0 /X1 be below One. In these cases the double-phase
short-circuit current may exceed the single-phase short-circuit current.
Current through earth causes voltage drops UE at the impedance of the earth, at
the earthing itself and at the connection lines between the equipment (e.g., neutrals
of transformers, transformer tank, overhead line tower) and earth. The currents to be
taken into account as summarised in Table 12.4 depend on the type of neutral handling
in the system.
Table 12.4
Currents through earth for the design of earthing installations
Valid for
Type of neutral handling
Isolated neutral
Low-impedance
earthing
Earthing with
current limitation
Resonance earthing
Voltage at earthing
Touch voltage
ICE
Current through
earthing impedance
Current through
earthing impedance
2 + I2
With reactor: IrD
res
Without reactor: IRes
Thermal stress of earthing
and earthing connections
IkEE
or I
Ik1
kE2E
or I
Ik1
kE2E
IkEE
Earthing connections
Earthing voltage UE
Touch voltage UB
or I
or I
Ik1
Ik1
kE2E
kE2E
UE ≤ 125 V: no measures required
UE > 125 V then UB ≤ 65 V
UE > 125 V then UB according to
or equivalent measures
VDE 0141 depending on fault
according to VDE 0141
duration or fast switch-off
or equivalent measures
according to VDE 0141
ICE
IkEE
UE > 125 V then UB ≤ 65 V
or equivalent measures
according to VDE 0141
Special problems related to short-circuit currents 261
Table 12.5
Reduction factor for typical power system installations; distance of earth conductor to phase conductor
D ≈ 20 m; ρ = 100 m
Number of conductors
Type and cross-section
1 Earth conductor
St 50–90
Al/St 50/30
Al/St 120/70
Al/St 240/40
Al/St 120/70; counterpoise Cu 120
2 Earth conductors
Cable sheet
Al/St 95/55
NA2XS(2)Y; 150Al; 20 kV
Single-core cable; 110 kV
N2XS(SL)2Y; 240Cu; 110 kV
Cables in steel tube
Reduction
factor
0.98
0.78
0.7
0.65
0.52
0.5
0.5
0.1
0.27
<0.05
As mentioned already in Section 7.3, a part of the single- or double-phase earthfault current is flowing through earth and the relevant impedances of the earthing
installations, depending on the reduction factor of the earth conductor and/or of
cable screens and sheets connected to earth. Detailed investigations are required to
determine the reduction factor. For rough estimates the reduction factors for typical
installations are given in Table 12.5.
12.2.5
Earthing impedance
The impedance, i.e., the resistance of the earthing installations is determined by the
material of the earthing grid, electrodes and rods and by the presence of any connection of earth conductor, counterpoise, cable sheets and other earthed installations
in the vicinity of the earthing installation. The earthing resistance is proportional to
the resistivity of the soil and depends furthermore on the specific arrangements of
the earthing installations as outlined in Table 12.6.
The equations as per Table 12.6 for the calculation of the resistance of earthing
installation indicate that it is not recommendable to increase the number of meshes
in an earthing grid in order to reduce the earthing impedance, as the effect is only
marginal. Increasing the number of earthing rods on the same earthing installation is
highly recommendable as the total earthing impedance is approximately reciprocal to
the number of rods. Sufficient distance between the individual rods (at least more than
the rod-length) shall be provided in this case. Material, cross-sections and laying of the
earthing installations must comply with the relevant standards, e.g., ANSI C 33.8
(standard for safety grounding and bonding equipment).
262 Short-circuit currents
Table 12.6
Resistance of earthing installations REI for different types and
arrangement
Type of earthing
Earthing resistance
Single deep-ground rod
REI =
l
d
Multiple
deep-ground rod
d
1
2
n
a
Surface electrode
4l
ρ
∗ ln
2π ∗ l
d
ρ
4l
1
∗ ln
a≥l
REI ≈ k ∗ ∗
n 2π ∗ l
d
k = 1.2–1.5
n = 5: k ≈ 1.2
n = 10: k ≈ 1.25
d
REI =
2l
ρ
∗ ln
πl
d
REI ≈
4l
ρ
∗ 2.5 + ln
2π ∗ l
d
l
Crossed surface
electrode
Earthing grid: uniform
resistivity of soil
l
b
l
Earthing grid: two
layers of resistivity
Earthing grid
D1
r1
r2
12.3
12.3.1
ρ
ρ
4b ∗ l
REI ≈
+
π
2∗D
ltot
ltot total length
of earthing grid
ρ2
ρ ∗ D1
4b ∗ l
D=
REI ≈
+ 1
π
2∗D
b∗l
ρ1 : resistivity
of surface layer
ρ2 : resistivity of
deep layer
D=
Examples
Interference of pipeline from 400-kV-line
The exposure of a 32 -pipeline with an overhead line (400 kV) as outlined in
Figure 12.5 is analysed. The nearest distance of the pipeline is 27 m at tower No. 3,
increasing to 160 m at tower No. 7 over a length of approximately 1600 m, decreasing
to 120 m over a length of 1450 m and then crossing the overhead line at an angle of
90◦ between towers Nos 10 and 11. The elevation plan of the tower and the pipeline
is given in Figure 12.6.
Special problems related to short-circuit currents 263
1450 m
#7
#10
#11
1620 m
OHTL
140 m
#3
120 m
60 m
27 m
Pipeline
3074 m
Figure 12.5
Plot plan of the exposure length pipeline and transmission line
The pipeline parameters are outlined below. The pipeline is not buried into earth,
but laid directly on the surface. The total exposure length is 3074 m. At location
x = 0 at tower No. 2 the pipeline is equipped with an isolating flange. The average
sag of the line conductor and of the earth conductor is 10.2 m, the specific resistivity
of the soil is 4 m.
Type of pipeline
Diameter
Thickness of wall
Conductivity
Relative permeability
Specific resistance of pipeline coating
32 steel pipe
1524 mm
22.2 mm
5.56 Sm/mm2
200
20.3 /km
The induced voltage of the pipeline was calculated with the procedure explained in
the previous sections. The individual sections of the exposure length were chosen
according to the distances between the transmission towers. The specific field strength
and the specific induced voltage, both related to units of the short-circuit current, are
given in Figure 12.7.
The analysis of the voltage pipeline-to-earth indicated a maximum value of
U PE = 18.1 V/kA which was obtained by superposition of the result from each section.
The voltage U PE along the exposure length and a further 3 km outside the exposure
section is outlined in Figure 12.8.
Short-circuit current calculation for the 400-kV-system under investigation indicated that the single-phase short-circuit current will always give the maximal
asymmetrical short-circuit current. In order to cover future increase of short-circuit
level, the maximal permissible short-circuit current in the system Ik = 40 kA
was taken as a basis for the assessment of interference problems. The voltage
264 Short-circuit currents
9.15 m–9.33 m
12.6 m
5.6 m
58.0 m
12.6 m
7.0 m
6.45 m
25.8 m
0.4 m
27–140–120 m
Figure 12.6
Elevation plan of the overhead transmission tower and the pipeline
pipeline-to-earth will reach U PE = 724 V in this case. With respect to [21] the
voltage pipeline-to-earth will be below the maximal permissible voltage which is
U PEmax = 1000 V. If ASME/IEEE-standard No. 80 is applied (50 kg body weight
and fault duration 150 ms) the voltage pipeline-to-earth will be above the maximal
permissible voltage U t50 ≈ UPEmax = 350 V. Earthing at intermediate locations especially at the location where the voltage pipeline-to-earth is maximal must be done in
order to reduce the voltage. More details on this example can be found in [27].
12.3.2
Calculation of earthing resistances
The resistance of an earthing grid within a switchyard of 80 m × 110 m; grid width
10 m × 10 m; ρ = 100 m is
REI ≈
ρ
100 m
100 m
ρ
+
+
= 0.532
=
2∗D
ltot
2 ∗ 105.9 m
1950 m
Special problems related to short-circuit currents 265
(a) 160
Electric field strength in V/(km*kA)
140
120
100
80
60
40
20
0
2
3
4
5
6
7
Tower no.
2
3
4
5
6
8
9
10
11
8
9
10
11
(b) 350
Induced voltage in V/kA
300
250
200
150
100
50
0
Figure 12.7
7
Tow er no.
Specific electric field strength (a) and specific induced voltage (b) of
the pipeline between towers 2 and 11
Effective length
4b ∗ l
4 ∗ 80 m ∗ 110 m
D=
=
= 105.9 m
π
π
Total length of the earthing grid ltot = 9 ∗ 110 m + 12 ∗ 80 m = 1950 m.
If the grid width is reduced to 5 m × 5 m, the total length of the earthing grid is
ltot = 3550 m. The resistance of the earthing grid is REI = 0.5 m.
The same earthing grid (80 m × 110 m; grid width 10 m × 10 m) is now examined,
except that the soil resistivity is assumed to be in two layers, the surface layer with
a thickness D1 = 4 m with ρ1 = 400 m and the deeper layer with ρ2 = 600 m.
266 Short-circuit currents
20
18
Voltage pipeline to earth (V/kA)
16
14
12
10
8
6
4
2
0
0
400
800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 5600 6000 6400
Exposure length (m)
Figure 12.8
Voltage pipeline-to-earth along the exposure length (0–6400 m)
The earthing resistance is
ρ 1 ∗ D1
400 m
100 m ∗ 4 m
ρ2
+
=
+
= 1.93
2∗D
b∗l
2 ∗ 105.9 m
80 m ∗ 110 m
The resistance of an earthing with five deep-ground rods of length l = 15 m; diameter
d = 20 mm; distance a = 30 m; ρ = 100 m is given by
REI ≈
1
ρ
4l
1
100 m
4 ∗ 15 m
∗
∗ ln
= 1.2 ∗ ∗
∗ ln
= 2.04
n 2π ∗ l
d
5 2π ∗ 15 m
0.02 m
whereas the earthing impedance of one rod is
REI ≈ k ∗
4l
100 m
4 ∗ 15 m
ρ
∗ ln
=
∗ ln
= 8.5
2π ∗ l
d
2π ∗ 15 m
0.02 m
The earthing impedance of a surface electrode with length l = 60 m; diameter d =
4 mm; ρ = 100 m is
REI =
REI =
ρ
2l
100 m
2 ∗ 60 m
∗ ln
=
∗ ln
= 5.47
πl
d
π 60 m
0.004
Chapter 13
Data of equipment
13.1
Three-phase a.c. equipment
A summary of relevant data of equipment is given in IEC 60909-2:1992. The data
are based on a survey carried out by IEC TC 73. In some countries this document
does not have the character of a standard.
13.1.1
System feeders
Impedances of power system feeders, respectively, their initial short-circuit power
are difficult to determine as typical values, the structure of the power systems (cable
or overhead line system), the voltage levels or the application task, i.e., for rural,
urban or industrial power supply can vary in a wide range. Typical ranges of the
initial short-circuit power (three-phase short-circuit) are given in Figure 13.1 for a
power system with different voltage levels.
13.1.2
Transformers
Transformers are constructed with defined rated power with respect to their application. In low-voltage and to a certain extent in medium-voltage systems, transformers
are build with standard rated power and standard impedance voltage. Transformers
in high-voltage systems, sometimes also in medium-voltage systems, have to meet
special application conditions, such as the internal standard of the utility with special
defined values for rated power, impedance voltage and ohmic losses. It should be
noted in this respect that the minimisation of the impedance voltage is limited by
the minimal insulation thickness of the windings. Typical values for the impedance
voltage of transformers are outlined in Figure 13.2. Figure 13.3 gives data for the
short-circuit losses (ohmic losses). The relation of the impedance voltage (%-value)
268 Short-circuit currents
GS
3~
250–1300 MVA
S k⬙ = 20–50 GVA
380 kV
GS
3~
50–600 MVA
Sr = 630–1000 MVA
uk = 10–16%
220 kV
and 110 kV
GS
3~
S k⬙ = 2–5 GVA
< 60 MVA
Sr = 12.5–63 MVA
uk = 11–20%
S ⬙k = 100–500 MVA
10–30 kV
0.4 kV
Figure 13.1
Sr = 50–630 kVA
uk = 4%
Sr = 630–2.5 MVA
uk = 6%
S k⬙ = 2–5 MVA
< 1 MVA
GS
3~
Principal structure of a power supply system and typical values of
initial short-circuit power of public supply system [7]
to the rated power (MVA-value) as per IEC 60909-2:1992 is given below:
ukr = 8 + 0.92 ∗ ln(SrT )
(13.1)
Further data are included in Table 13.1.
The impedance voltage of auto-transformers is lower than that of full-winding
transformers.
Data of equipment 269
%
18
UrOS = 110 kV
16
14
12
UrOS > 220 kV
10
uk
8
2
6
1
4
UrOS < 30 kV
Low-voltage transformers
2
0
0
1
10
100
1000 MVA
SrT
Figure 13.2
Typical values for the impedance voltage of two-winding transformers
300
250
Losses (kW)
200
150
No-load losses type a
No-load losses type b
Short-circuit losses type a
100
Short-circuit losses type b
50
0
0
Figure 13.3
10
20
30
40
50
60
Rated power (MVA)
70
80
90
Typical values for the ohmic losses, no-load losses and no-load current
of two-winding transformers
270 Short-circuit currents
Table 13.1
Data of transformers
Voltage levels
Sr
(MVA)
MV/LV
Un < 1 kV
0.05–0.63
0.63–2.5
4
6
2.5–25
6–9
25–63
10–16
MV/MV
Un = 1–66 kV
HV/MV
Un > 66 kV
ukr
(%)
uRr
(%)
1–2
1–1.5
0.7–1
0.6–0.8
The ratio of positive to negative sequence impedance of transformers depends on
the vector group and is typically in the range of
Vector group YNd
Vector group Yzn
Vector group Yyn (three-limb core)
Vector group Yyn
(five limb core and three single-phase
transformers)
Vector group YNynd
13.1.3
X0 /X1 ≈ 0.8–1.0
X0 /X1 ≈ 0.1
X0 /X1 ≈ 3.0–10.0
X0 /X1 ≈ 10.0–100.0
X0 /X1 ≈ 1.5–3.2
Generators
The parameters such as rated power, rated voltage, power factor and subtransient
reactance and synchronous reactance are needed to calculate the impedance of
generators and by this the contribution to the short-circuit current.
Rated voltages within one power range may vary depending on the construction
type of the generator. The subtransient reactance of synchronous generators is typically in the range of 10–30 per cent depending on the rated power as mentioned in
IEC 60909-2. The synchronous reactance is between 100 and 300 per cent, whereas
salient pole generators normally have lower values than turbo generators. Typical
values are summarised in Table 13.2.
Power factor of generators with rated power below 20 MVA is approximately
cos ϕr = 0.8 and increases for high rated machines (>1000 MVA) to cos ϕr = 0.85 on
average. The ratio of saturated to unsaturated reactance xdsat /xd is between 0.8 and
0.9, whereas in the case of rated power below 100 MVA, the ratio can be between 0.65
and 1.0. Zero-sequence reactance of synchronous generators are x0 ≈ (0.4–0.8)xd
depending on the winding arrangement.
Data of equipment 271
Table 13.2
Typical data of synchronous generators (average values)
Rated power
SrG (MVA)
Below 4
4–20
20–200
Above 200
Synchronous motors
13.1.4
Rated voltage
UrG (kV)
Subtransient reactance
xd (%)
0.48–11.5
2.2–13.8
6–22
20–27
Up to 11.5
10–25
8–16
10–20
18–30
12–25
Synchronous reactance
xd (%)
110–170–230
120–180–210
160–205–260
220–230–240
Overhead lines
Impedances of overhead lines depend on the geometrical arrangement of phase
conductors, on the tower outline and on the number and type of conductors. The zerosequence impedance furthermore depends on the earth resistivity, on the arrangement
of earth conductors and on the design of the earthing system, including conductive
installations in earth, respectively, connected to earth, e.g., pipelines, counterpoise
and cable sheaths. The calculation of impedances of overhead lines in the positivesequence and the zero-sequence component are outlined below based on the tower
outline as per Figure 13.4. The equations are valid for overhead lines which are
symmetrically constructed and operated. All circuits are assumed to be in operation
except as noted.
(a)
(b)
E
B
E
b
B
Y
r
R
Figure 13.4
y
System b
Y
R
System a
Tower outline of high-voltage transmission lines. (a) Single-circuit line
and (b) double-circuit line
272 Short-circuit currents
The distances between phase wires RYB are named dRY , dYB and dBR for the
single-circuit line (and similar for the double-circuit line) and the average distances
for the calculation of impedances are
D=
3
dRY dYB dBR
(13.2a)
DmRy =
3
dRy dYb dBr
(13.2b)
DmRr =
3
dRr dYy dBb
(13.2c)
3
Dab =
DaE =
3
dRr dYy dBb ∗
3
(13.2d)
dRy dRb dYb
(13.2e)
dRq dYq dBq
Bundle-conductors of n conductors with radius r in a circular arrangement on the
radius rT must be represented by an equivalent radius rB
rB =
n
nr ∗ rTn−1
(13.3)
The impedance in the positive-sequence component is calculated for a single-circuit
line (μ0 = 4π 10−4 H/km)
R
μ0
D
1
ln
(13.4)
+
Z 1I = 1 + j ω
n
2π
rB
4n
The impedance of the double-circuit line is
R
D ∗ DmRy
μ0
1
ln
+
Z 1II = 1 + j ω
n
2π
rB ∗ DmRr
4n
(13.5)
The calculation of the impedance of the zero-sequence component has to take
account of the earth conductor. The impedance of the zero-sequence component of a
single-circuit line without earth conductor is given by
Z 0I =
R1
μ0
μ0
+ 3ω
+ jω
n
8
2π
3 ∗ ln
δ
3
rB
D2
+
1
4n
(13.6a)
and for operation with one earth conductor (index E)
Z 0IE =
R1
μ0
μ0
+ 3ω
+ jω
n
8
2π
3 ∗ ln
δ
3
rB D 2
+
1
4n
−3
(Z aE )2
Z E
(13.6b)
The loop-impedance Z E of the arrangement earth conductor and earth return, see
Equation (12.13), is
μ0
μr
μ0
δ
Z E = RE + ω
+
+ jω
ln
(13.7a)
8
2π
rE
4
Data of equipment 273
The coupling impedance Z aq of phase conductor and earth conductor (see
Equation (12.11), where Z aE is named Z LE2 )
Z aE = ω
μ0
μ0
δ
+ jω
∗ ln
8
2π
DaE
(13.7b)
The distance Daq is given by Equation (13.2e) and the depth δ of the earth return path
(see Equation (12.2)) is
1.85
δ=√
μ0 (ω/ρ)
(13.8)
The resistivity of the soil ρ is between 30 m (Swamp soil) and 3000 m (Stony
soil) as outlined in Table 12.2.
The impedance of the zero-sequence component of a double-circuit overhead line
without earth conductor is given by
Z 0II =
R1
μ0
μ0
+ 3ω
+ jω
n
8
2π
3 ∗ ln
δ
3
rB
D2
+
1
4n
+ 3Z ab
(13.9a)
and in the case of operation with earth conductor
Z 0IIE =
R1
μ0
μ0
+ 3ω
+ jω
n
8
2π
3 ∗ ln
δ
3
rB D 2
+
1
4n
+ 3Z ab − 6
(Z aE )2
Z E
(13.9b)
where Z aE is the coupling impedance of phase conductor and earth conductor according to Equation (13.7b), Z E is the impedance of the loop phase conductor and earth
return according to Equation (13.7a) and Z ab is the coupling impedance between the
systems a and b according to Equation (13.10).
Z ab = ω
μ0
μ0
δ
+ jω
∗ ln
8
2π
Dab
(13.10)
The relative permeability μr relevant for overhead lines is
Conductors from Cu or Al
Conductors from Al/St, cross-section ratio > 6
Conductors from Al/St, one layer of Al only
Conductors from Steel (St)
μr
μr
μr
μr
=1
≈1
≈ 5–10
≈ 25
Typical values for the impedances of MV-overhead lines are summarised in
Table 13.3. Table 13.4 shows the impedances of HV-overhead lines.
274 Short-circuit currents
Table 13.3
Table 13.4
Typical values of impedance of
the positive-sequence component
of MV-overhead lines
Conductor
Un
(kV)
Resistance
(/km)
Reactance
(/km)
50 Al
50 Cu
50 Cu
70 Cu
70 Al
95 Al
150/25 Al/St
2 ∗ 240/40 Al/St
4 ∗ 240/40 Al/St
10–20
10–20
10–20
10–30
10–20
20–30
110
220
380
0.579
0.365
0.365
0.217
0.439
0.378
0.192
0.06
0.03
0.355
0.355
0.423
0.417
0.345
0.368
0.398
0.3
0.26
Typical values of impedances of the positive- and zero-sequence component of HV-overhead lines (ρE = 100 m)
Conductor
Earth wire
AlSt 240/40
Al/St 2 ∗ 240/40
Al/St 4 ∗ 240/40
St 50
Al/St 44/32
Al/St 240/40
Al/St 44/32
Al/St 240/40
Al/St 240/40
Al/St 240/40
Un
(kV)
Positive-sequence
impedance
(/km)
110
0.12 + j 0.39
220
0.12 + j 0.42
400
0.06 + j 0.3
0.03 + j 0.26
Zero-sequence impedance
(/km)
One circuit
Two circuits
0.31 + j 1.38
0.32 + j 1.26
0.22 + j 1.1
0.3 + j 1.19
0.22 + j 1.1
0.16 + j 0.98
0.13 + j 0.091
0.5 + j 2.2
0.52 + j 1.86
0.33 + j 1.64
0.49 + j 1.78
0.32 + j 1.61
0.26 + j 1.49
0.24 + j 1.39
A detailed list of impedances of overhead lines of different voltage levels is given
in IEC 60909-2:1992.
The capacitances of overhead lines are only needed for special problems,
i.e., in case of isolated neutral or if the system is operated with resonance earthing
(see Chapter 5), or in case of double-circuit faults. The capacitance of the positivesequence component for single-circuit line is given by (ε0 = 8.854 ∗ 10−12 F/m)
=
C1I
2π ε0
ln(D/rB )
(13.11a)
Data of equipment 275
The capacitance of the positive-sequence component for a double-circuit line is
given by
2πε0
ln(DDmRy /rB DmRr )
=
C1II
(13.11b)
Similar to the calculation of the impedance of the zero-sequence component, the
earth conductor has to be considered for the calculation of capacitances of the zerosequence component. In the case of operation without earth conductor the capacitance
of the single-circuit line is calculated by
C0I
=
2πε0
(13.12a)
3 ∗ ln(2h/ 3 rB D 2 )
=
C0IE
√
3
3 ∗ (ln(2h/ r B
2πε0
2
D ) − (ln((h + h
2
q )/Daq )) / ln(2hq /rq ))
(13.12b)
The capacitances of the double-circuit line are calculated by
C0II
=
4h2
3 ∗ ln 2h ∗
= 2π ε0 / 3 ∗ ln
C0IIE
2h ∗
3
2πε0
3
/ rB D 2 ∗
2
+ DmRy
2
4h2 + DmRy
rB D 2 ∗
3
2
DmRr DmRy
3
2
DmRr DmRy
−2
(13.13a)
(ln ((h + hE )/DaE ))2
ln(2hE /rE )
(13.13b)
To take account of the conductor sag, the average height h of the conductor is used
h=
3
hR hY hB =
3
hr hy hb
(13.14)
The influence of the tower on the capacitance in the zero-sequence component is
considered by an increase of 6 per cent (overhead lines with nominal voltage 400 kV),
up to 10 per cent (overhead lines with nominal voltage 60 kV).
It should be observed that the capacitance in the positive-sequence component is
given by
C1 = 3CL + CE
(13.15a)
276 Short-circuit currents
and the capacitance in the zero-sequence component is identical to the line-to-earth
capacitance (see Section 13.2)
(13.15b)
C0 = CE
13.1.5
Cables
Impedances of cables differ very much depending on the type and thickness of insulation, the cable construction, cross-section of conductor, screening, sheaths and
armouring and on the type of cable laying, i.e., flat formation or triangle formation.
Sheaths and armouring have especially in low voltage cables a strong influence on
the impedance. The installation of other conductive installations, e.g., pipelines and
screening, armouring and sheaths of other cables have a strong influence on the zerosequence impedance, which therefore can only be given for simple arrangements.
Reference is made to [1], [2], [8], [9] and to data-sheets of manufacturers.
Due to the high permittivity εr of the insulation and the small distance between
phase conductor and sheeth, identical to earth potential, the capacitances of cables
are significantly higher as compared with overhead lines. Figures 13.5 and 13.6
indicate values for the capacitances and the capacitive loading currents of MV- and
HV-cables.
Figure 13.7 indicates typical values of reactances (positive-sequence system) of
cables of different construction.
mF/km
1.0
0.9
1
0.8
4
0.7
3
2
0.6
5
C⬘1 0.5
0.4
6
0.3
0.2
0.1
10
Figure 13.5
15
25 35
50 70 95 120 185
q
Capacitances MV-cables (Un < 20 kV )
1) Mass-impregnated cable NKBA 1kV
3) Three-core cable NEKEBY 10 kV
5) VPE-cable N2XSEY 10 kV
300
500 mm2
2) Mass-impregnated cable
NKBA 6/10 kV
4) PVC-cable NYSEY 10 kV
6) VPE-cable N2XSEYBY 20 kV
Data of equipment 277
(a) mF/km
0.6
(b) A/km
14
0.5
10
8
7
6
5
4
I⬘C
3
0.4
C⬘1
0.3
0.2
1
2
0.1
1
3
2
300
0
150
2
1400 mm
625
q
Figure 13.6
4
4
3
0
150
2
300
625
1400 mm2
q
Capacitances C1 (a) and capacitive loading current Ic (b) of HV-cables
1) Single-core oil-filled
2) single-core oil-filled
cable 110 kV
cable 220 kV
3) VPE-cable 110 kV
4) VPE-cable 220 kV
Ω/km
0.20
7
9
0.18
6 8
0.16
3
0.14
5
4
X ⬘1 0.12
0.10
2
0.08
1
0.06
10
Figure 13.7
15
25 35
50 70 95 120 185
q
500 mm2
Reactance (positive-sequence system) of three-phase
(Un ≤110 kV)
1) 0,6/1 kV, 4-conductor, NKBA
2) 0,6/1 kV, 4-conductor, NA2XY
3) Three-core cable with armouring 10 kV
4) PVC-cable NYFGby, 10 kV
5) VPE-cable NA2XSEY, 10 kV
6) Single-core oil-filled cable (triangle formation) 110 kV
7) Single-core oil-filled cable (flat formation) 110 kV
8) VPE-cable (triangle formation) 110 kV
cables
278 Short-circuit currents
13.1.6
Reactors and resistors
Short-circuit limiting reactors are constructed for all voltage levels, from low voltage
up to 750 kV. The reactors are manufactured with oil-insulated windings and as
air-insulated core-type reactors. Figure 13.8 shows a short-circuit limiting reactor
(air-insulated core-type; 10 kV; 630 A; 6 per cent).
Petersen-coils are constructed as reactor with fixed reactance, with tap-changer
and with continuous controllable reactors. The control range normally is limited to
1 : 2.5 for tap-changer control and 1 : 10 for continuous control. Standards as per
Figure 13.8
Arrangement of a short-circuit limiting reactor
Source: Mohndruck
Data of equipment 279
IEC 60289:1998 are applicable. The values as per below need to be specified:
√
Rated voltage
Ur (phase voltage) or Ur / 3 depending on application
Rated current
Ir (fixed reactance) or maximal current to be controlled
Rated frequency
50 or 60 Hz; for traction systems other
frequencies are used
Operating method
Continuous operation or short-time operation
(e.g., 2 h or some minutes)
Control range
Minimal and maximal adjustable current
Figure 13.9 shows a Petersen-coil adjustable in steps by tap-changer (oilinsulated).
Earthing resistors are designed individually for the special applications. They
are typically made from stainless steel, cast steel, NiO–Cr- or CuO–Ni-alloy. The
maximal permissible temperature, the temperature coefficient and the assumed cost
determine the selection of material. Stainless steel is an advantage compared with
cast steel due to low temperature coefficient. Low-impedance resistors are mainly
arranged from meandered wire elements, high-impedance resistors are arranged from
Figure 13.9
√
Adjustable Petersen-coil 21 kV/ 3; 4 MVAr; Ir = 70.1–330 A;
adjustable in 64 steps, 4.13 A each
Source: SGB Starkstromgerätebau
280 Short-circuit currents
steel plate grid or steel fabrics. Some characteristic parameters of resistor elements
are given in Table 13.5.
Rating and design of resistors are based on ANSI/IEEE 32:1972 and EN 60529,
creeping distances are determined in accordance to IEC 60815, insulation must
comply with IEC 60071 and high-voltage testing shall be carried out based on
Table 13.5
Characteristic parameters of resistor elements
Material
CuNi 44 or
NiCr 8020
Cast steel
CrNi-alloy steel
Arrangement
Resistance at 20◦ C ()
Rated current (A)
Thermal time constant (s)
Temperature coefficient (K−1 )
Maximal temperature (◦ C)
Wire elements
1500–0.5
<20
20–90
0.004
Cast element
0.2–0.01
25–125
240–600
0.075
400
Steel grid and fabric
0.75–0.04
25–250
120
0.05
760
Figure 13.10
Earthing resistor made from CrNi-alloy steel fabric for indoor
installation 3810 , 5 A for 10 s, 170 kV BIL, IP 00
Source: Schniewindt KG
Data of equipment 281
Figure 13.11
Earthing resistor made from meandering wire for outdoor installation
16 , 400 A for 10 s, 75 kV BIL, IP 20
Source: Schniewindt KG
IEC 60060. Earthing resistors can be suitably designed for indoor and outdoor
installations. Figures 13.10 and 13.11 show two different types of resistors.
13.1.7
Asynchronous motors
Data of asynchronous motors are included in IEC 60781:1989 (mentioned as A in
Table 13.6) , IEC 60909-2:1992, in [10] (mentioned as B in Table 13.6) and in IEC
60909-1:1991 (mentioned as C in Table 13.6). Table 13.6 outlines the relevant data
of asynchronous motors for LV- and MV-application.
13.2
d.c. equipment
Data of d.c. equipment are not documented in a similar way as for a.c. equipment.
Literature also presents less information. All data listed hereafter are based on
manufacturers’ data, information from calculation examples and incomplete data
from literature. They should be used for preliminary information only.
13.2.1
Conductors
Resistance of conductors in d.c. auxiliary installations is calculated from the crosssection of the cable or busbar and the material constant. The specific resistance of
282 Short-circuit currents
Table 13.6
Data of asynchronous motors
SrM
(kVA)
PrM
(kW)
UrM
(kV)
14.9
24
24
11
18.5
18.5
20
22
22
30
40
45
50
55
75
160
0.38
0.38
0.38
0.4
0.38
0.38
0.38
0.4
0.38
0.41
0.38
0.38
0.38
29.5
29.5
40
57.8
69.6
90
191.5
IrM
(A)
22.5
36.6
36.5
45
45
60.8
88
106
137
291
ηrM
cos ϕrM
IanM /IrM
RS
(m)
I0
(A)
0.89
0.83
0.85
0.85
370
a 1002
236.7
9
0.89
0.93
8.5
6.0
6
6
0.9
0.9
0.93
0.92
0.94
0.92
0.94
0.95
0.83
0.84
0.85
0.85
0.84
0.86
0.88
0.88
6
6.5
6
6.7
6
1.3
6.8
6.3
SrM
(MVA)
PrM
(MW)
UrM
(kV)
IrM
(A)
0.197
0.259
0.218
0.27
0.281
0.299
0.353
0.374
0.467
0.54
0.685
0.837
0.842
1.697
2.09
2.4
3.07
5.245
6.85
11.64
0.16
0.175
0.18
0.225
0.23
0.25
0.3
0.32
0.4
0.46
0.55
0.63
0.7
1.4
1.8
2.1
2.65
4.5
6
10
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
19
25
21
19
27
29
34
36
45
52
66
80.5
81
163
201
231
296
504
659
1120
ηrM
cos ϕrM
Pair of
poles
Ref.
13
2
2
2
a 813
160
91.67
16
22.6
3
3
2
50.34
31.0
2
36.67
22.97
7
38
43
74
2
2
2
B
C
B
A
C
B
B
A
B
B
B
B
B
IanM /
IrM
RS
()
I0
(A)
5.7
a 30.39
a 32.2
a 28.94
a 22.97
a 21.38
0.94
0.89
5.3
0.955
0.89
5.33
0.95
0.94
0.85
0.8
5.3
5.2
0.948
0.968
0.971
0.96
0.975
0.973
0.977
0.87
0.89
0.9
0.9
0.88
0.9
0.88
5
5.2
5.1
5
4.7
5.5
4
a Motor impedance instead of resistance.
1830
6.6
720
11.8
0340
806.1
a 7.13
169
98
100
73.26
27.3
11
9
21.5
27.4
a 16.43
a 16.04
a 9.52
44
50
58.6
60
94
138
154
Pair of
poles
Ref.
1
6
2
2
1
1
1
1
2
1
3
6
3
2
3
2
1
2
2
2
C
C
C
C
C
B
C
C
B
C
B
B
C
B
B
B
B
B
B
B
Data of equipment 283
materials at temperature of 20◦ C shall be used in accordance with IEC 61660-3:2000
as given below:
Copper
Aluminium
1 mm2
54 m
1 mm2
ρ=
34 m
ρ=
Resistance for other temperatures has to be calculated as given in Table 3.12.
Inductance of d.c. conductors installations depends on the arrangement of the
conductors and can only be calculated for simple layout as mentioned in Table 3.12.
13.2.2
Capacitors
Capacitors in d.c. auxiliary installations are installed up to some 10 mF for smoothing of the d.c. voltage. Typical values of the d.c. resistance and the a.c. resistance,
respectively, are summarised together with other relevant data in Tables 13.7 and 13.8.
According to information received from manufacturers, the inductance of capacitors is in the range of nano-Henry and can be neglected as compared with the
inductance of the connecting cables.
13.2.3
Batteries
The detailed data of batteries as requested in IEC 61660-1:1997 are not available from
data sheets of manufacturers, as some data, such as voltage of loaded and unloaded
Table 13.7
Typical values of MKP-capacitors; selfhealing dry insulation; different make of
capacitor can and fuse
Capacitor can
Capacity
(μF)
Nominal voltage
(V)
Resistance
(m)
Rectangular
12.000
9.000
4.000
500
200
1250
1.600
490
250
900
1000
1900
900
1100
440
690
440
250
0.8
0.5
0.5
2.5
3
<2
<2
<7.5
<3
Round
Prismatic
(Internal fuse)
Prismatic
(External fuse)
284 Short-circuit currents
Table 13.8
Table 13.9
Typical values of MKPcapacitors; resin insulation; round can
Capacity
(μF)
Nominal voltage
(V)
Resistance
(m)
500
1100
2000
500
750
1500
500
1000
1800
420
420
420
500
500
500
640
640
640
0.6
0.5
0.5
0.8
0.6
0.5
0.6
0.6
0.5
Resistance of loaded batteries (data from several manufacturers)
2-V-batteries
Capacity
RB ; manufacturer 1
RB ; manufacturer 2
12-V-batteries
Capacity
RB ; manufacturer 3
RB ; manufacturer 4
Ah 100 200 300 400 500 600 800 1000 2000 3000
m 0.9 0.8 0.75 0.7 0.65 0.63 0.6 0.55 0.5
0.45
m — 0.4 0.35 0.32 0.3 —
0.2 0.15 0.08 0.05
Ah 40
m 9.5
m 9.7
55
5.8
8.5
65
5.8
—
75
5.5
6.3
80
5.5
6.3
90
5.2
—
100 150
4.3 4.0
5
4.0
200
3.8
3.6
—
—
battery, depend on the layout and the operational requirements, e.g., required voltage
tolerance and voltage drops of connecting cables, of the whole battery plant.
A battery cell with UnB = 2.0 V is taken as an example. The voltage of the
loaded battery is 2.23 V/cell, which is contrary to IEC 61660-1:1997 stating a value
of EBge = 1.115 ∗ UnB instead of EBge = 1.05 ∗ UnB . When the minimal voltage
at the consumer inside a 220-V-installation shall be not less than Umin = 0.9 ∗ Un
the minimal permissible voltage at the battery plant is EBmin = 191.4 V taking
account of a voltage drop of 3 per cent at the connecting circuits. It is therefore
necessary to install 108 battery cells with a minimal voltage of EBun = 1.772 V/cell.
If the minimal voltage shall be set to EBun = 1.833 V/cell as recommended by some
manufacturers, only 105 cells need to be installed. The maximal voltage at the battery
plant is EBge = 240.8 V (108 cells) and EBge = 234.2 V (105 cells), respectively.
The number of cells and the minimal permissible voltage depend on the loading time
of the battery, the required capacity and the discharge time and vice versa.
Data of equipment 285
Values of the inductance of 12-V-batteries are in the range of LB = 1–10 μH and
LBZ = 0.17–1.7 μH/cell, respectively [11]. These values are in the same range as
inductances of conductors and cannot be neglected.
Values of the internal resistance of batteries are in the range of RB = 0.05–
70 m/cell. Batteries with high capacity have small value of resistance and batteries
with low capacity have high value of resistance. Typical values are outlined in
Table 13.9. The values of the resistance differ very much especially for 2-V-batteries.
It should be noted, that batteries with very low internal resistance are made from
lead-grid cathode and grid-anodes, and batteries with high internal resistance are made
from grid-anodes and lead-sheaths cathode sometimes also with additional stretched
copper grid.
Symbols, superscripts and subscripts
A detailed explanation of the quantities and symbols is included in the text for each
equation, table and figure. It cannot be avoided that some symbols and subscripts are
used for different physical quantities. It should be noted that e.g., using the symbol
‘J’ for the current density cannot be mixed up with the symbol ‘J’ for the second
mechanical moment, as both symbols will not occur in the same equations and even
not in a similar context.
Symbols for quantities
A
C
D
E
E
E
F
G
I
J
K
L
M
P
Q
Q
R
S
T
U
X
Y
Aperiodic current
Capacitance
Geometric mean distance, geometric factor
Young’s modulus
Matrix of unity
Voltage
Mechanical force
Conductance
Current
Current density
Factor
Inductance
Mutual impedance
Active power
Reactive power
Thermal heat
Resistance
Apparent power
Total time, time constant
Voltage
Reactance
Admittance
288 Symbols, superscripts and subscripts
Z
Z
a
a; a 2
c
d
d
f
h
k
l; l
m
n
p
q
q
q
r
r
s
t
t
v
w
α
ε
τ
ω
ϕ
ρ
μ
μ
λ
κ
δ
δ
δ
γ
γ
σ
Impedance
Section moduli
Distance
Complex operational phasor
Voltage factor
Distance, diameter
Damping factor
Frequency
Conductor height
Factor
Length
Mass
Number
Control range
Cross-section
Factor of plasticity
Factor
Radius
Reduction factor
Conductor sag
Time instant
Transformation ratio
Detuning factor
Probability factor
Temperature & permittivity coefficient
Permittivity
Time constant
Angular frequency
Phase angle
Resistivity
Factor according to IEC 60909
Permeability
Factor according to IEC 60909
Factor according to IEC 60909
Earth-fault factor (a.c.)
Decay coefficient (d.c.)
Depth of earth return path
Impedance angle
Propagation constant
Mechanical stress
Quantities (Example U)
U
U
Capital letter used for r.m.s-value
Underlined capital letter indicates phasor (vector)
Symbols, superscripts and subscripts 289
Û
|U |
u
u
u(t)
Ū
U
U
Peak value
Complex value
%/ MVA-value
Instantaneous value
Time course
Average value
Vector (matrix)
Complex vector (matrix)
Superscripts (Example U)
U∗
u
U
U
Conjugate–complex value
p.u.-value
Transient value
Subtransient value
Order of subscripts (Example U)
First order
Next order
Next order
Next order
Next order
Next order
UYbTmaxi
Component (R, Y, B or 0, 1, 2) UY
Type of operation (n; r; b)
UYb
Indication of equipment
UYbT
Number of equipment
UYbT4
Special condition
UYbTmax
Index
UYbTmaxi
Indicates voltage of phase Y before short-circuit, equipment is
T ransformer number 4 maximal value for alternative i
In most of the cases, positive-sequence system is used without subscript ‘1’
Subscripts, components, systems
0; 1; 2 Zero-, positive-, negative-sequence systems
ac
a.c. system
dc
d.c. system
R; Y; B Phases of three-phase a.c. system
Subscripts, type of operation
0
an
b
b
e
No-load value, eigenvalue
Locked rotor, starting
Before (prior to) fault, beginning
Breaking
End
290 Symbols, superscripts and subscripts
k
Short-circuit value
k1
Single-phase short-circuit
k2
Double-phase short-circuit
k3
Three-phase short-circuit
m
Highest value (IEC 60038)
n
Nominal value
r
Rated value
s; sat Saturated value
δ
At temperature δ
Subscripts, indication of equipment
B
B
Br
C
D
E
F
G
J
KW
L
M
M
N
P
Q
R
S
T
To
X
Y
Z
Battery
Counterpoise
Branch
Capacitor
Reactor, rectifier
Earth, earth wire
Field
Generator
Joint for connection
Power station
Line; load
Motor
Mutual value
Network (a.c. system)
Pipeline
System feeder
Resistance
Special earthing impedance
Transformer
Overhead-line tower
Reactance
Common branch (d.c. system)
Battery cell
Subscripts, special conditions
20
Value at 20◦ C
f
Fictitious
ge
Loaded
max Maximal value
min Minimal value
Symbols, superscripts and subscripts 291
o
res
s
th
tot
un
Without tap-changer
Residual
With tap-changer
Thermal
Total
Unloaded
Other subscripts
0; 1
1; 2; 3
a; b; c
B
c
d
HV; MV; LV
I
i; j
Mec
m
p
q
res
s
U
W
I
II
Condition at beginning or ending respectively
Different sides of a transformer
Index
Base or reference
Equivalent
Direct axis
High-, medium-, low-voltage side of a transformer
Current
Index
Mechanical
Main conductor
Pole pair
Quadrature axis
Equivalent resistance
Subconductor
Voltage
Characteristic impedance
One circuit in operation
Two circuits in operation
References
1 Schlabbach, J.: Electrical power system engineering (Elektroenergieversorgung).
2nd revised edition, VDE-Verlag. Berlin, Offenbach/Germany, 2003. ISBN
3-8007-2662-9.
2 Oeding, D. and Oswald, B.: Electrical systems and power stations (Elektrische
Kraftwerke und Netze). 6th edition, Springer-Verlag. Berlin, Heidelberg,
New York, 2004. ISBN 3-5400-0863-2.
3 Schlabbach, J.: Neutral handling (Sternpunktbehandlung). Systems engineering,
Vol. 15. VWEW-Energieverlag, Frankfurt/Germany, 2002. ISBN 3-8022-0677-0.
4 CCITT: Directives CCITT, Vol. V, Chapter 5. ITU, Geneva/Switzerland.
5 Niemand, T. and Kunz, H.: Earthing in power systems (Erdungsanlagen). Systems
engineering, Vol. 6. VWEW-Energieverlag, Frankfurt/Germany, 1996. ISBN
3-8022-0362-3.
6 Arbitration agency of VDEW: Technical recommendation No. 1 – Induced voltages in telecommunication circuits. VWEW-Energieverlag, Frankfurt/Germany,
1987.
7 Schlabbach, J., Blume, D., and Stephanblome, T.: Voltage quality in electrical
power systems. Power and Energy Series, No. 36. IEE-publishers, Stevenage,
UK, 2001. ISBN 0-85296-975-9.
8 ABB: Switchgear Manual. 9th edition, Cornelsen-Girardet, Essen/Germany,
1993. ISBN 3-464-48234-0.
9 VDEW: Cable-book (Kabelhandbuch). 6th edition, VWEW-Energieverlag,
Frankfurt/Germany, 2001. ISBN 3-8022-0665-7.
10 Scheifele, J.: Contribution of asynchronous motors to short-circuit currents
(Beitrag von Drehstrom-Asynchronmotoren zum Kurzschlussstrom). Ph.D. thesis, Technical University of Darmstadt/Germany, 1984.
11 Gretsch, R.: Design of electrical installations in automobiles (Ein Beitrag zur
Gestaltung der elektrischen Anlage in Kraftfahrzeugen). Dr. -Ing. habil. thesis,
University Erlangen-Nürnberg/Germany, 1979.
12 VDEW: Distributed generation in LV-systems (Eigenerzeugungsanlagen am
Niederspannungsnetz). 4th edition, VWEW-Energieverlag, Frankfurt/Germany,
2001. ISBN 3-8022-0790-4.
294 References
13 Balzer, G., Nelles, D., and Tuttas, C.: Short-circuit current calculation
acc. VDE 0102 (Kurzschlußstromberechnung nach VDE 0102). VDE-technical
reports, Vol. 77. VDE-Verlag, Berlin, Offenbach/Germany, 2001. ISBN
3-8007-2101-5.
14 Pistora, G.: Calculation of short-circuit currents and voltage drop (Berechnung
von Kurzschlussströmen und Spannungsfällen). VDE-technical reports, Vol. 118.
VDE-Verlag, Berlin, Offenbach/Germany, 2004. ISBN 3-8007-2640-8.
15 Gröber, H. and Komurka, J.: Transformation of zero-sequence voltage through
transformers (Übertragung der Nullspannung bei zweiseitig geerdeten Transformatoren). Technical report of FGH, Mannheim/Germany, 1973.
16 Balzer, G.: Double-side earthing of transformers (Beidseitige Sternpunktbehandlung von Transformatoren). In VDE: Neutral handling in 10-kV- to
110-kV-system. ETG-Report, Vol. 24. VDE-Verlag, Berlin, Offenbach/Germany,
1988, pp. 172–187.
17 SPEZIELEKTRA: Resonant earthing controller EZR2 (Erdschlußkompensationsregler EZR2). Operation manual N 9/4.88, Spezielektra, Linz/Austria,
1988.
18 Schäfer, D., Schlabbach, J., Gehrmann, A., and Kroll, R.: Increase of displacement voltage in MV cable systems with resonant earthing (Erhöhung der
Verlagerungsspannung in Mittelspannungs-Kabelnetzen mit Erdschlußkompensation). Elektrizitätswirtschaft, Vol. 93 (1994), VWEW-Energieverlag, Frankfurt/
Germany, pp. 1295–1298.
19 Fiernkranz, K.: MV-systems with isolated neutrals or resonant earthing (Mittelspannungsnetze mit isoliertem Sternpunkt oder Erdschlußkompensation).
ETG-Report, Vol. 24. VDE-Verlag, Berlin, Offenbach/Germany, 1988.
20 Kouteynikoff, P. and Sforzini, A.: Results of an international survey of the
rules limiting interference coupled into metallic pipes. CIGRE Committee 36.
ELECTRA, Geneva, Switzerland, 1986.
21 Arbitration agency of VDEW: Technical recommendation No. 3 – Measures
for construction of pipelines in the vicinity of HV/AC three-phase installations
(German). VWEW-Energieverlag, Frankfurt/Germany, 1982.
22 Arbitration agency of VDEW: Technical recommendation No. 5 – Principles of
calculation and measurement of reduction factor of pipelines and earth wires
(German). VWEW-Energieverlag, Frankfurt/Germany, 1980.
23 Arbitration agency of VDEW: Technical recommendation No. 7 – Measures
for the installation and operation of pipelines in the vicinity of three-phase
high-voltage installations (German). VWEW-Energieverlag, Frankfurt/Germany,
1985.
24 IEC 60479-1: Effects of currents passing through the human body. International
electrotechnical commission, Geneva, Switzerland, 1994.
25 Jenkins, N., Allan, R., Crossley, P., Kirschen, D., and Strbaq, G.: Embedded
generation. Power and Energy Series, No. 31. IEE Publishers, Stevenage, UK,
2000. ISBN 0-85296-774-8.
26 Schlabbach, J. (Ed.) and Metz, D.: Power system engineering (Netzsystemtechnik). VDE-Verlag, Berlin, Offenbach/Germany, 2005. ISBN 3-8007-2821-4.
References 295
27 Schlabbach, J.: Short-time interference of pipelines. Report on research and development No. 14. University of Applied Sciences, Bielefeld/Germany, 2000. ISBN
3-923216-52-1.
28 Cory, B. and Weedy, B.: Electrical power systems. 4th edition. John Wiley &
Sons Ltd., Chichester, England, 1998. ISBN 0-471-91659-5.
29 Funk, G.: System of symmetrical components (Symmetrische Komponenten).
Elitera-Verlag, Berlin, Germany, 1976. ISBN 3-8708-7087-7.
30 Kories, R. and Schmidt-Walter, H.: Electrotechnical handbook (Taschenbuch der
Eletrotechnik). 6th revised edition, Verlag Harri Deutsch, Frankfurt/Germany,
2004. ISBN 3-8171-1734-5.
31 Metz, D., Naundorf, U., and Schlabbach, J.: Handbook of electrotechnical
equations (Kleine Formelsammlung Elektrotechnik). 4th edition, Fachbuchverlag
Leipzig, Germany, 2003. ISBN 3-446-22545-5.
32 Wildi, T.: Units and conversion charts. IEEE Press, New York, USA, 1990. ISBN
0-87942-273-4.
33 Oeding, D. and Schünemann, H.: Calculation of short-circuit currents in
HV-systems using %/MVA-system (Berechnung der Kurzschlussströme in
Hochspannungsnetzen mit %/MVA-system). BBC-News, Mannheim/Germany,
1965.
34 Koglin, H.: The decaying d.c. component of short-circuit currents (Der anklingende Gleichstrom beim Kurzschluss in Elektroenergieversorgungsnetzen). Ph.D.
thesis, University of Darmstadt/Germany, 1971.
35 Pitz, V. and Waider, G.: Impedance correction factors of network transformers for short-circuit current calculation (Impedanzkorrekturfaktoren für
Netztransformatoren bei der Kurzschlussstromberechnung). Elektrie 47 (1993),
pp. 301–304.
36 Scheifele, J. and Waider, G.: Maximal short-circuit currents through linear optimisation (maximale Kurzschlussströme durch lineare Optimierung). etzArchiv 10
(1998), pp. 275–280.
37 Oeding, D. and Waider, G.: Maximal partial short-circuit currents of power
stations (Maximale Teilkurzschlussströme von Kraftwerksblöcken ohne Stufenschalter). etzArchiv 10 (1988), pp. 173–180.
38 Balzer, G. and Deter, O.: Calculation of thermal effects of equipment due to
short-circuit currents (Berechnung der thermischen Kurzschlussbeanspruchung
von Starkstromanlagen). etzArchiv 7 (1985), pp. 287–290.
39 Tuttas, C.: Approximation of reduction factors of complex conductor arrangement (Berechnung des Reduktionsfaktors komplizierter Leiteranordnungen).
AEG-report 54, Frankfurt, Germany, 1981, pp. 153–157.
40 Hosemann, G., Nietsch, C., and Tsanakas, D.: Short-circuit stress in d.c. auxiliary
systems. Cigre-report 23-104, Geneva, Switzerland, 1992.
41 Tsanakas, D.: Substitution function for the calculation of mechanical and thermal stress due to short-circuits in d.c. installations (Ersatzfunktion für die
Bestimmung der mechanischen und thermischen Kurzschlussbeanspruchung in
Gleichstromanlagen). etzArchiv 10 (1988), pp. 355–360.
296 References
42 Nietsch, C.: Calculation of short-circuit currents in d.c. installations. University
of Erlangen/Germany, EV-report F254, 1989.
43 Tsanakas, D., Meyer, W., and Safigianni, A.: Dynamical short-circuit stress in d.c.
installations (Dynamische Kurzschlussbeanspruchung in Gleichstromanlagen).
Archiv für Eletrotechnik 74 (1991), pp. 305–313.
44 Tsanakas, D. and Papadias, A.: Influence of short-circuit duration on dynamic
stresses in substations. IEEE Transactions PAS 102 (1983), pp. 492–501.
45 Hosemann, G. and Tsanakas, D.: Dynamic short-circuit stress of busbar structures
with stiff conductor. Parametric studies and conclusions. Electra 68, Geneva,
Switzerland, 1980, pp. 37–64.
46 Meyer, W.: Additional calculation acc. IEC 60865-1 for the determination of
short-circuit stress of lines (Ergänzung des Berechnungsverfahrens nach IEC
60865-1 zur Ermittlung der Kurzschlussbeanspruchung von Leitungsseilen mit
Schlaufen im Spannfeld). EE-report, University of Erlangen/Germany, 2002.
47 Meyer, W.: Mechanical short-circuit stress of rigid conductors in IEC
865-1 – Information on the norm (Mechanische Kurzschlussbeanspruchung
von biegesteifen Leitern in IEC 865-1 – Hintergründe zur Norm). EE-report,
University of Erlangen/Germany, 2002.
48 Tsanakas, D.: Dynamic stress in high-voltage structures by short-circuits of
short-duration. CIGRE-Symposium High currents in power systems. Proc. Report
500-01, Brussels, Belgium, 1985.
49 Rüger, W. and Hosemann, G.: Mechanical short-circuit effects of single-core
cables. IEEE Trans. PD, 4 (1989) pp. 68–74.
50 Nietsch, C. and Tsanakas, D.: Short-circuit currents in d.c. auxiliary installations
(Kurzschlussströme in Gleichstrom-Eigenbedarfsanlagen). Elektrie 46 (1992),
pp. 18–22.
51 Hosemann, C., Zeitler, E., Miri, A., and Stein, N.: The behaviour of droppers in
HV substations under short-circuit. Proceedings of the 5th International symposium on short-circuit currents in power systems. Proc. Report 3.2, Zlin, Czech
Republic, 1992.
52 Tsanakas, D., Meyer, W., and Nietsch, C.: Short-circuit currents of motors in
d.c. auxiliary installations in power plants and substations. Electromotion ’99,
Patras/Greece, pp. 489–496.
53 Stein, N., Meyer, W., and Miri, A.: Test and calculation of short-circuit forces and
displacements in HV substations with strained conductors and droppers. ETEP
10 (2000), pp. 131–138.
54 Herold, G. and Kunz, M.: Fast analytical short-circuit current calculation of
rectifier fed auxiliary subsystems. ETEP 13 (2003), pp. 151–159.
55 Pitz, V., Köster, H.-J., et al.: Short-circuit mechanical effects on outdoor HV substations with wide bundling. CIGRE-Session 2004. Proc. Report B3-107, Paris,
France.
56 Wessnigk, K. and Griesbach, P.: Digital calculation of short-circuit current in battery-fed d.c. installations (Digitale Berechnung des zeitlichen
Kurzschlussstromverlaufs in batteriegespeisten Gleichstromnetzen). Elektrie 43
(1989), pp. 379–381.
References 297
57 Albert, K., Apelt, O., Bär, G., and Koglin, H.-J. (Ed.): Electrical power supply (Elektrischer Eigenbedarf). VDE-Verlag, Berlin, Offenbach/Germany, 1993.
ISBN 3-8007-1586-4.
IEC-standards, EN-norms and other standards and norms mentioned within the
context of this book are listed in Tables 1.1 and 1.2.
Extracts from the norms DIN EN 60909-0 (VDE 0102) and DIN EN 61660-1 (VDE
0102 part 10) are permitted for this edition of the book by licensee agreement 392.004
of DIN (Deutsches Institut für Normung e.V.) and VDE (Verband der Elektrotechnik Elektronik Informationstechnik e.V.) on 16.12.2004. An additional permission is
required for other usages and editions.
Standards are only to be applied based on their actual issues, available from
VDE-Verlag GmbH, Bismarckstr. 33, D-10625 Berlin, Beuth-Verlag GmbH,
Burggrafenstr. 6, D-10787 Berlin or the national standard organisation. English
versions of the norms are also available through BSI in London/UK.
The English text of this book was neither checked by DIN (Deutsches Institut
für Normung e.V., Berlin) nor by VDE (Verband der Elektrotechnik Elektronik
Informationstechnik e.V., Frankfurt).
Index
%/MVA system semirelative units 33–4
conversion of quantities with p.u. and
ohm systems 35
example 40
correction factor 34
HV three-phase examples 93
impedance based on measurement
example 41–2
impedance calculations for a
three-winding transformer 37–40
impedance results of various
equipment 91
impedances calculation HV three-phase
example 92
a.c. systems, impedance: see impedance
calculations for a.c. equipment
a.c. systems, mechanical effects: see
conductors, a.c., mechanical effects
of short circuits
a.c. systems, thermal effects
about thermal effects 195
see also conductors, a.c., thermal effects
admittance/reactance/impedance 12
American National Standards Institute,
web address 4
asynchronous motors
data of 281–2
factor q for short-circuit breaking
current 162–3
impedance calculations 55, 63–4
auxiliary installations
supply of a power station calculations
example 94–6
see also batteries in d.c. auxiliary
installations; d.c. auxiliary
installations, effects of short circuits;
d.c. auxiliary installations,
short-circuit current calculations; d.c.
motors in auxiliary installations
batteries in d.c. auxiliary installations
example data 283–5
impedance calculations 60, 64, 65
short-circuit currents 170–2
breaking current with short-circuits: see
short-circuit breaking current
British Standards Institute, web address 4
busbar/switchgear short-circuit current
limitation measures
busbar sectionaliser in single busbar
switchgear 233–4
Ip-limiter usage 234–6
medium and low voltage limiting
equipment 234–6
multiple busbar operation 232–6
time curves of short-circuit currents 235
cables
capacitances 276–7
impedances 276
reactances 276–7
see also conductors
capacitors in auxiliary installations
factor κ 169–71
impedance calculations 59, 64, 65
short-circuit currents 169–70
typical values of capacity and resistance
283–4
causes of short circuit currents 1–3
communication circuit interference: see
pipelines/communication circuits,
interference with
complex calculations 11–14
300 Index
conductors, a.c., mechanical effects
of short circuits
about mechanical effects 201–2
conductor oscillation influence 207–9
bending stress and dynamic force on
supports 208–9
mechanical natural frequency 207–8
electromagnetic forces 202–4
fixing clamps calculations example
217–18
forces and stresses in rigid conductor
support calculations 204–7
conductors, a.c., thermal effects of short
circuits
bare conductors 198–9
cable calculation example 216–17
cable screening, armouring and sheath
199–201
maximal permissible short-circuit
current density 201
maximal permissible temperature
200–1
heat dissipation considerations,
factors m and n 197–8
IEC 60865–1 197
IEC 60909–0 197
impregnated paper-insulated cables
199–200
maximum permissible temperature and
current density table 196
overhead line conductors type Al/St 198
rated short-time current density 199
thermal equivalent short-time current
195–7
conductors, d.c. equipment,
impedance/resistance calculations
58, 64, 65, 281–3
converter fed drive, impedance calculations
56
correction factors
d.c. auxiliary installations, short-circuit
current calculations, σ 179
example 190–1
with HV three-phase systems examples
76, 91–4
impedance correction factor K G 154–6
examples 91–4
impedances of equipment and
short-circuit current example 154
%/MVA- or p.u.-system 34, 152–4
current limitation earthing power system
102–5
earth-fault factor δ 103, 104
d.c. auxiliary installations, effects of short
circuits
bending stress on conductors 213–14
bending stress of sub-conductors 211
calculation example 223–4
equivalent circuit diagrams and
short-circuit current time course 166
force calculation for main- and
sub-conductors 212–15
calculation example 222–3
forces on supports 214–15
mechanical natural frequency of main and
sub-conductors 211
standard approximation function
(simplified approach) 209
electromagnetic effect example 220
substitute rectangular function approach
209–16
calculation example 220–3
thermal effects calculation example
218–19
thermal short circuit strength 215–16
vibration period and natural mechanical
frequency of conductors 212
calculation example 220–1
d.c. auxiliary installations, short-circuit
current calculations
about auxiliary installations 165–8
correction factor σ 179
equivalent circuit diagram 180
example
battery short-circuit current 186
capacitor short-circuit current 185–6
correction factors and corrected
parameters 190–1
data and parameters 183–4
equivalent diagram 182
impedances of cable and busbar
conductors 184–5
partial short-circuit currents 191–3
rectifier short-circuit current 186–8
total short-circuit current 193–4
maximal short-circuit current calculations
168
minimal short-circuit current calculations
168
time function calculations 167
total short-circuit current 178–82
typical time curves 181
see also batteries in d.c. auxiliary
installations; capacitors in auxiliary
installations; d.c. motors in auxiliary
installations; impedance calculations
Index 301
for d.c. equipment; rectifiers in
auxiliary installations
d.c. motors in auxiliary installations
factor κ 175–8
IEC 61000–1:1997 175
IEC 61660–1:1997 175, 178
impedance calculations 62, 65, 66
with independent excitation, short-circuit
currents 174–8
factors k 175–8
short-circuit current calculations,
example 188–9
definitions and terms 30–2
Deutsches Institute für Normung,
web address 4
double earth-faults
about double-earth faults 139
fault in a 20-kV-system example 146–8
impedances 139–40
peak short-circuit current 143
power system configurations 140–2
steady-state short-circuit current 143
symmetrical short-circuit breaking
current 143
symmetrical short-circuit current 139–40
see also earth currents from short circuits
earth currents from short circuits
about short-circuits through earth 143–5
short-circuit at overhead-line tower
145–6
short-circuit inside a switchyard 144–5
single-phase short-circuit in a
110-kV-system example 148–9
earth faults: see double earth-faults; earth
currents from short circuits
earth-fault factor δ
and current limitation earthing 103, 104
and low impedance neutral earthing
effects 102
earthing considerations
about earthing 257
earthing grid resistance calculation
example 264–6
earthing impedance 261–2
resistance of different types of
earthing 262
human body resistance 257–9
relevant currents through earth 259–61
design information 260
soil conditions and resistivity 258–9
earthing rods 259
see also neutral earthing
earthing resistors
about earthing resistors 279–81
characteristic parameters 280–1
electromagnetic forces, on a.c. conductors
202–4
equivalent circuit diagrams for short circuits
24–9, 43
factors
about factors 151
factor k for d.c. motor short-circuit
currents 175–8
factor κ for peak short-circuit current
77–80, 156–8
with d.c. motors 175–8
factor q for short-circuit breaking current
of asynchronous motors 162–3
factor λ for steady-state short-circuit
current 82–3, 160–2
factor μ for symmetrical short-circuit
breaking current 81–2, 158–60
factors m and n, heat dissipation 197
IEC 60909–1:1991–10 151, 162
impedances of equipment and
short-circuit current example 152–3
induced voltage calculation, screening
factor 252
isolated neutral power system,
overvoltage factor 108
see also correction factors
faults, as causes of short circuit currents 1–3
fuses, and short circuit currents 2
generators
power factor data 270
rated voltage data 270–1
short circuit current limitation,
subtransient reactance influence
239–40
sutransient reactance data 270–1
synchronous reactance data 270–1
zero-sequence reactance data 270
human body resistance 257–9
HV a.c. three-phase system short circuit
current calculations
about HV short circuits 67–8
about parameter calculations 70–2
assumptions and methods 68–71
auxiliary supply of a power station
calculations example 94–6
decaying (aperiodic) component 80–1
302 Index
HV a.c. three-phase system short circuit
current calculations (continued)
far-from generator short-circuit 67–9
IEC 60909–0 71, 72–83
impedance correction factor examples
91–4
initial symmetrical short-circuit current
72–4
double-phase 73
far-from-generator 73
line-to-earth 73
three-phase 73
inside power plant short-circuit
currents 72–6
correction factors 76, 91–4
equivalent circuit diagrams 74–5
locations around transformers 74–6
line-to-earth (single-phase) short-circuit
example 87–8
maximal and minimal short-circuit
currents 71–2
meshed 110-kV-system example 89–91
minimal short-circuit currents 85–6
motor influences 84–5
near-to-generator short-circuit example
86–7
peak short-circuit currents 77–80
calculation example 88–9
equivalent frequency method 80
factor κ 77–80
meshed networks 79–80
non-meshed fed 77–8
ratio R/X method 80
single-fed 77
steady-state short-circuit current 82–3
factor λ 82–3
subtransient reactance 67
symmetrical short-circuit breaking current
80–2
factor μ 81–2
types of short circuit 67–9
IEC 60038 6, 226–7
IEC 60050 6, 7
IEC 60071 7
IEC 60265 7
IEC 60282 7
IEC 60479 7
IEC 60781 2, 5, 9, 131, 132
IEC 60865 2, 5, 9, 197
IEC 60896 7
IEC 60909 5, 6, 9, 71, 72–83, 131, 132,
139, 151, 162, 197
IEC 60949 7
IEC 60986 7, 8
IEC 61000 175
IEC 61071 7
IEC 61660 5, 9, 175, 178
IEC document list 5–9
impedance
impedance/admittance/reactance 12
impedances of a three-winding
transformer in MVA example 37–40
measurement for symmetrical components
20–4
in %/MVA-system for auxiliary supply 96
in %/MVA-system based on measurement
example 41–2
impedance calculations for a.c. equipment
about impedance for a.c. systems 45
asynchronous motors 55, 63–4
calculation examples 63–4
converter fed drive 56
for HV a.c. three-phase equipment under
short circuit examples 91
overhead transmission lines 53, 63–4
power-station unit 52, 63–4
short-circuit limiting reactor 54, 63–4
synchronous generators 51, 63–4
system feeder 46, 63
system loads 57, 63–4
three-winding transformer 48–50, 63–4
two-winding transformer 47, 63
impedance calculations for d.c. equipment
about impedance for d.c. systems 50
batteries 60, 64
calculation examples 64–6
capacitors 59, 64, 65
conductors 58, 64, 65
motors, d.c. 62, 65, 66
rectifiers 61, 63–4, 65–6
impedance correction factor KG 154–6
examples 91–4
impedance matrix, transformation of 19–20
inductive interference calculations 247–52
interference: see pipelines/communication
circuits, interference with
International Electrotechnical Commission,
web address 4
Ip-limiters 234–6
isolated neutral power system 105–8
capacitive earth-fault current 106–7
earth faults 105–8
earth-fault arcs 107–8
overvoltage factor 108
Index 303
limitation of short-circuit currents
about limitation measures 225–6
coupling of power system at busbars with
low short-circuit level 230–1
distribution of feeding locations 228–9
earthing impedances in neutral of
transformers 238
and generator subtransient reactance
239–40
meshed systems 241–4
nominal system voltage selection 226–7
IEC 60038:1987 recommended
voltages 226–7
operating subsystems separately 227–8
radial systems 241
restructuring power systems 231
ring-main systems 241–2
short-circuit limiting reactors 238
see also busbar/switchgear short-circuit
current limitation measures;
transformers
limiting value, definition 31
low-impedance neutral earthing power
system 98–102
earth-fault factor δ 102, 103, 104
single-phase short-circuit current 98–100
single/three-phase short circuit relation
100–1
low-voltage (LV) systems, short circuit
current calculations
about short-circuit currents in LV
systems 131
calculation methods 132
IEC 60781 131, 132
IEC 60909–0 131, 132
impedance calculations 132–3
approximations for generators and
rectifiers 133
initial symmetrical short-circuit current
133
maximal short-circuit current calculation
example 135–6
minimal short-circuit currents 134–5
calculation example 135, 137
motor influences 134
peak short-circuit current 133
steady-state short-circuit current 134
symmetrical short-circuit breaking current
133
types of faults 131
matrix equations 14–17
cyclic-symmetrical matrix 15–16
see also transformations/transformation
matrix
maximal and minmal short-circuit currents 3
meshed systems, limitation of short-circuit
currents 241–4
motors, influence for HV a.c. three-phase
systems 84–5
motors, d.c.: see d.c. motors
neutral earthing
about neutral earthing 97–8
characteristics of different fault types
summary 99
design of an earthing resistor connected to
an artificial neutral example 124
limitation of single-phase short-circuit
current by earthing through
impedance example 123–4
transformer HV-side and LV-side
considerations 116–19
compensation windings 116–18
voltages at neutral of a unit transformer
example 126–9
see also current limitation earthing power
system; isolated neutral power
system; low-impedance neutral
earthing power system; resonance
earthing power systems
(Peterson-coil)
nominal value, definition 31
norms
about technical standards and norms 4–5
international documents and norms 5–6
norms as referred in standards 6–9
ohm-system physical quantities
voltage/current/impedance 32
conversion of quantities with p.u. and
%/MVA systems 35
example 40
HV three-phase correction factor
examples 94
overhead transmission lines
capacitances 274–6
conductor sag considerations 275
coupling impedance 273
distance between phase wires 272
impedance calculations/data 53, 63–4,
271–6
impedance in the positive-sequence
component for a single and double
circuit line 272
304 Index
overhead transmission lines (continued)
impedance of the zero-sequence
component 272
loop impedance 272
relative permeability for 273
resistivity of soil 273
thermal effects 198
typical impedance values 274
parallel and series connection 27, 30
peak short-circuit currents 2–3
double earth faults 143
factor κ 77–80, 156–8, 175–8
low-voltage (LV) systems 133
see also HV a.c. three-phase system short
circuit current calculations
Peterson-coil: see resonance earthing power
systems (Peterson-coil)
phasor diagrams 11–14
pipelines/communication circuits,
interference with
about pipeline interference 245–7
characteristic impedance of the pipeline
253–4
pipeline coating resistance 254
coupling types summary table 246
induced voltage calculation 252–3
coupling impedances 253
screening factor 252
inductive interference calculations
247–52
coupling impedances 249–51
earth return path considerations 248–9
interference of pipeline from 400-kV-line
example 262–6
pipeline-to-earth voltage 254–7
oblique exposure 255–6
safety aspects 247
Potier’s reactance 160–1
power system short-circuit current
limitation: see limitation of
short-circuit currents
power-station unit, impedance calculation
52, 63–4
p.u.-system relative units 33–4
conversion of quantities with ohm and
MVA systems 35
example 40
correction factor 34
radial systems, limitation of short-circuit
currents 241
rated data, definition 32
rated value, definition 31
reactance/admittance/impedance 12
reactors, short-circuit current limiting: see
resonance earthing power systems
(Peterson-coil)
rectifiers in auxiliary installations
impedance calculations 61, 63–4, 65–6
short-circuit currents 172–4
relative quantities/p.u.-system 33–4
resonance earthing power systems
(Peterson-coil) 108–16, 278–9
20-kV-system example 124–5
about resonance earthing 108–11
alternate earthing of parallel transformer
neutrals 118, 121
basic principles 109–11
capacitive earth-fault current and residual
current calculation example 125–6
current limits 111
displacement voltage calculations 112–15
fault currents in MV-system from fault in
HV-system 118, 121
impedance calculations 54, 63–4
increase of displacement voltage example
119–23
capacitive asymmetry in a
20-kV-system 120, 122
resonance curve for detuning factors in
a 20-kV-system 122
Peterson-coils 278–9
tuning 115–16
resonance frequency/tuning 110–11
usage worldwide 108–9
see also neutral earthing
ring-main systems, limitation of short-circuit
currents 241–2
r.m.s. value of short-circuit currents 2–3
RYB-system
representation of a line example 42–4
see also symmetrical components
safety, interference with pipelines and
communication circuits 247
screening factor, pipelines/communication
circuits 252
semirelative quantities and %/MVA system:
see %/MVA system semirelative
units
series and parallel connection 27, 30
short-circuit breaking current
about breaking current value 2–3
double earth faults 143
Index 305
factor μ for symmetrical breaking 81–2,
158–60
low-voltage (LV) systems current
calculations 133
three-phase systems calculations 80–2
short-circuit current limitation: see
limitation of short-circuit currents
short-circuit currents
about short circuit currents 1–3
equivalent circuit diagrams 24–9
importance of 1–3
maximal and minimal 3
typical time course 2–3
see also busbar/switchgear short-circuit
current limitation measures; d.c.
auxiliary installations, short-circuit
current calculations; earth currents
from short circuits; HV a.c.
three-phase system short circuit
current calculations; limitation of
short-circuit currents; low-voltage
(LV) systems, short circuit current
calculations
short-circuit limiting reactors: see resonance
earthing power systems
(Peterson-coil)
soil conditions and resistivity 248, 258–9
standards
about technical standards and norms 4–5
American National Standards Institute 4
British Standards Institute 4
definitions and terms 30–2
Deutsches Institute für Normung 4
international documents and norms 5–6
International Electrotechnical
Commission 4
norms as referred in standards 6–9
VDE-Verlag 4
see also IEC
substitute rectangular function approach: see
d.c. auxiliary installations, effects of
short circuits
subtransient reactance of generators 67,
239–40
switchgear
and short circuit currents 2
see also busbar/switchgear short-circuit
current limitation measures
symmetrical components
example 34–7
impedance measurement of 20–4
interpretation of the system of 18–19
and representation of a line in the
RYB-system example 42–4
transformation of impedances 19–20
transformation matrix 14–17
see also ohm-system physical quantities
voltage/current/impedance; %/MVA
system semirelative units;
p.u.-system relative units
synchronous generators, impedance
calculation 51, 63–4
system feeders
impedance calculations 46, 63
ranges of initial short-circuit power 267,
268
system loads, impedance calculations 57,
63–4
terms and definitions 30–2
thermal effects of short circuits: see
conductors, a.c., thermal effects of
short circuits
thermal equivalent short-time current for
conductors 195–7
three-phase systems: see HV a.c. three-phase
system short circuit current
calculations
time course of short circuit currents 2
transformations/transformation matrix
14–17
complex rotational phasors 17
cyclic-symmetrical matrix 15–16
delta-star-transformation 31
reverse transformation 17
star-delta-transformation 30
transformation of impedances 19–20
transformers
impedance calculations in %/MVA 37–40
and impedance measurement for
symmetrical components 20–3
ratio of positive to negative sequence
impedance 270
short-circuit current limitation measures
236–8
earthing impedances in the neutrals
238–9
impedance voltage considerations
236–8
loadflow and short-circuit analysis
results 237
three-winding, impedance calculations
21–4, 48–50, 63–4
306 Index
transformers (continued)
two-winding, impedance calculations 47,
63
typical values two-winding transformer
impedance voltage 267–70
ohmic losses, no-load losses and
no-load current 267, 269
see also neutral earthing
transmission lines: see overhead
transmission lines
VDE documents 5–9
VDE-Verlag, web address 4
vectors/vector diagrams 11–15
example 34–7
web addresses, standards institutions 4
Short-circuitCurrents
Thecalculationofshort-circuitcurrentsisacentraltaskforPower
Systemengineers,astheyareessentialparametersforthedesignof
electricalequipmentandinstallations,theoperationofpowersystems
andtheanalysisofoutagesandfaults.
Short-circuitCurrentsgivesanoverviewofthecomponentswithin
powersystemswithrespecttotheparametersneededforshortcircuitcurrentcalculation.Italsoexplainshowtousethesystemof
symmetricalcomponentstoanalysedifferenttypesofshort-circuits
inpowersystems.Thethermalandelctromagneticeffectsofshortcircuitcurrentsonequipmentandinstallations,short-timeinterference
problemsandmeasuresforthelimitationofshort-circuitcurrentsare
alsodiscussed.Detailedcalculationproceduresandtypicaldataof
equipmentareprovidedinaseparatechapterforeasyreference,and
workedexamplesareincludedthroughout.
ProfessorDr-IngJürgenSchlabbachreceived
hisPhDin1982fromtheTechnicalUniversity
ofDarmstadt,Germany.Until1992heworked
asaConsultantEngineerwithresponsibilityfor
theplanninganddesignofpublicandindustrial
supplysystems.Since1992hehasworkedatthe
UniversityofAppliedSciencesinBielefeld,Germany
asProfessorforPowerSystemEngineeringand
UtilisationofRenewableEnergy.Hismaininterests
arepowersystemsplanning,faultanalysis,power
quality,interferenceproblemsandconnectionof
renewableenergysourcestopowersystems,and
hecarriesoutconsultancyworkinthesefields.Heis
amemberoftheIEEEandVDI.
TheInstitutionofEngineeringandTechnology
www.theiet.org
0-86341-514-8
978-0-86341-514-2