Applied Energy 195 (2017) 974–990
Contents lists available at ScienceDirect
Applied Energy
journal homepage: www.elsevier.com/locate/apenergy
Network-aware approach for energy storage planning and control in the
network with high penetration of renewables
Khashayar Mahani ⇑, Farbod Farzan, Mohsen A. Jafari
Rutgers, The State University of New Jersey, USA
h i g h l i g h t s
Approximate solution for energy storage (ES) sizing and operation planning.
Approximate solution significantly reduces the complexity of problem.
Approximate solution for complex network with multiple applications of ESs.
A rule-based control scheme for the near real-time operation of complex ES network.
The control schema has been developed by mining the I/O statistical relationship.
a r t i c l e
i n f o
Article history:
Received 6 December 2016
Received in revised form 23 March 2017
Accepted 24 March 2017
Keywords:
Network-aware planning and control
Energy storage network
Data-driven control
Optimal planning
Community level micro-grid
a b s t r a c t
In this paper, we consider multiple energy storage nodes distributed over a power distribution network,
and are purposed for multiple applications. The research problems of interests are to optimally locate
these nodes over the distribution network and to create day-ahead plans according to planned applications. The two problems are formulated as stochastic optimization problems, and hourly and timeaggregated approximate solutions are presented. The approximation identifies time periods where load
and generation patterns demonstrate low variability, and marks the whole period as a single time zone,
thus significantly reducing the number of decision variables and the overall problem size. We show that
aggregate and hourly planning solutions are close. The planning problem can handle any number of storage nodes with general topology and load connections, and deterministic or stochastic capacities. In this
paper, we focus on network of static energy storages with deterministic capacity. Finally, we build a novel
rule based control scheme for the near real time operation of the storage network by mining the statistical relationship between input and optimal charge and discharge patterns.
Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction
Energy storage (ES) has the potential to offer a new means of
added flexibility on the electricity distribution systems. This flexibility can be used in a number of ways, including adding value
towards asset management, power quality and reliability. An
important factor in evaluating the feasibility of ES technology is
the application(s) for which the storage is used for [1]. ES can provide local level services such as, peak shaving and renewable integration [2,3], and network level services, such as voltage and
frequency control [4]. It can also be utilized for loss minimization
and deferral of network infrastructure upgrades. With the use of
energy storage in a distribution networks for multiple applications,
however, comes the challenge of determining how best to control
⇑ Corresponding author.
E-mail address: km723@scarletmail.rutgers.edu (K. Mahani).
http://dx.doi.org/10.1016/j.apenergy.2017.03.118
0306-2619/Ó 2017 Elsevier Ltd. All rights reserved.
these storage units under load and system state uncertainties. For
example, with increasing number of Electrical Vehicles (EVs) the
uncertainty in the electricity demand rises due to EV charging
demand [5–7]. But, on the other hand, Vehicle-to-Grid (V2G) technology, while mitigating some of this uncertainly, can add system
dynamics complexities to the network [8–10].
Han et al. [11] and Wong et al. [12] provide control algorithms
to maximize EV owner’s profit, which comes from selling power to
grid and participating in the frequency regulation market. They
formulate the problem as a discrete-time Markov decision process
and solve it by introducing an online learning algorithm which
iterates every hours based on available information. Koutsopoulos
et al. [13] study the optimal energy storage control problem by taking the point of view of a utility operator and focuses on arbitrage
application of energy storage. The authors show that the model can
be extended to account for a renewable source that feeds the storage device. The same problem was considered in [14], where the
K. Mahani et al. / Applied Energy 195 (2017) 974–990
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Nomenclature
Ens;max
Ps;max
time index
static storage node index
demand node index
renewable node index
temporal zone index
day index
year index
scenario (cluster) index
tree index in tree-bagging method
number of clusters
annual inflation rate (%/year)
annual discount rate (%/year)
storage unit s energy capacity (kWh)
energy storage s rated capacity (kW)
Inv Cap
s
investment unit cost on storage capacity ($/kWh)
Inv PR
s
LSc ðj; iÞ
investment unit cost on power rating ($/kW)
total electricity demand during zone i at demand node j
for cluster Sc (kWh)
total renewable generation during zone i at renewable
node k for cluster Sc (kWh)
demand matrix for day ‘‘d” in year ‘‘y”
renewable generation matrix for day ‘‘d” in year ‘‘y”
electricity price matrix for day ‘‘d” in year ‘‘y”
representative demand matrix for cluster Sc
representative renewable generation matrix for cluster
Sc
representative electricity price matrix for cluster Sc
t
s
j
k
i
d
y
Sc
b
CL
c
a
RSc ðk; iÞ
Ld;y ð:Þ
Rd;y ð:Þ
Pr d;y ð:Þ
LSc ð:Þ
RSc ð:Þ
PrSc ð:Þ
ech;g
s;i;Sc
ech;r
s;k;i;Sc
eds;j;i;Sc
total energy charged from grid during zone i in storage
unit s for cluster Sc (kWh)
total energy charged from renewable node k during
zone i in storage unit s for cluster Sc (kWh)
total energy discharged during zone i from storage s to
demand node j for cluster Sc (kWh)
cost of energy is minimized subject to both user demands and
prices using a Markov Decision Process. Dufo-Lopez et al. [15] consider the energy storage in private facility to reduce the electricity
bill. They conclude that electricity price variation has a great effect
on the profitability of storage system. Renewable resource integration is an important application of energy storage, and chargedischarge control policy of energy storage to serve this application
is presented by Wang et al. [16]. Renewable energy sources are
considered by Teleke et al. [17] too, where an open-loop optimal
control scheme was developed which incorporates the operating
constraints of battery energy storage. They use the battery energy
storage in a smoothing application where a wind farm is dispatched on an hourly basis based on the forecasted wind
conditions.
Earlier works on component sizing or optimal operation employ
different approaches, which are differentiated by decisional variables. Studies that take into account both sizing and scheduling
problems are generally scarce. Ru et al. in [18] determine the optimal size of a grid-connected PV-battery system which is used in an
arbitrage application. Their objective is to minimize the net power
purchase cost plus battery capacity loss, without considering any
initial capital investment. Khalilpour et al. [19] introduce a decision support tool for sizing and operation of PV-battery system
in a single facility, with the objective of maximizing the net present
value generated by bill reduction. Zhang et al. [20] introduce a rule
based charge and discharge strategy which simultaneously optimizes the battery sizing and operation in a bill management
edem;g
j;i;Sc
total energy from grid during zone i to demand node j
for cluster Sc (kWh)
edem;r
total energy from renewable k during zone i to demand
j;k;i;Sc
node j for cluster Sc (kWh)
ci;j
configuration number between nodes i and j
Effch;s
energy storage ‘‘s” charging efficiency
energy storage ‘‘s” discharging efficiency
Effdis;s
average electricity whole sale price during the hours of
Pr w
Sc ðiÞ
zone i for cluster Sc ($/kWh)
Pnsub
penalty for damage to substation due to reverse flow of
power ($/kWh)
Dem
demand charge for peak demand ($/kW)
storage s energy level at the end of zone i (kWh)
SOC s;i
dri
duration of temporal zone i
Enmax
maximum energy reservoir capacity
Pmax
maximum power rating
safety reserve capacity for storage unit s
SF s
ESL
Storage - Demand Eligibility Matrix
ERS
Renewable - Storage Eligibility Matrix
ERL
Renewable - Demand Eligibility Matrix
ST t
network state vector at time t
ps
control policy for storage s
control action of storage s at time t
ast
rwst (ST, ast Þ reward function for storage s when action ast is taken
in state ST
s
Vp
value storage s under control policy ps
Y
classification response vector (Control action vector)
LR
level of on-site renewable generation
LD
level of demands
EP
electricity price
X
classification feature matrix
B
number of bags
s
memory window in control module
dn
nth digit in control action code
application. The introduced rule-based approach works well for a
single PV-battery system with in the facility, however the interaction between multiple battery units in more complex distribution
network has not been investigated. The similar problem was considered by Brekken et al. [21], where sizing and control methodologies for a battery-based energy storage system is presented for
wind farm applications. The sizing problem of distributed generator and energy storage system (single application – electricity cost
reduction) for demand response applications in smart households
has been studied in [22,23]. Andreotti et al. [24] consider a network of renewable generation units and formulate a singleobjective optimization problem whose objective function is power
loss minimization while satisfying constraints on active and reactive power at the interconnection bus. Nick et al. [25] studied the
optimal allocation of storage systems in an active distribution network by defining a multi-objective optimization problem. The
application of renewable generation integration is also considered
in [26–29]. Van de ven et al. [30] present a battery control policy,
which minimizes the total discounted costs, taking into account
arbitrage application of energy storage. Jayawarna et al. [31] studied the energy storage power reliability application and present the
concept of using central energy storage system as the main fault
current source in micro-grid islanded mode.
To the best of our knowledge, there is a major gap in understanding how multiple storage units programmed for multiple
applications should operate in a distribution network. This paper
intends to fill this gap by developing simple but verifiable control
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
strategies, which directly take into account system characteristics
and states. The proposed approach is applicable to connected
energy storage units in distribution networks with multiple applications. More specifically, we consider a system with multiple storage nodes distributed over an arbitrary power distribution
network, and given that there are infrastructure limitations on
the use of energy storage over this power network. Capacity of
each node is assumed to be static and deterministic. Stochastic
energy storage nodes are also possible, and will be examined in
another paper under preparation by the same team. A good example of energy storage with dynamic capacity is a parking lot with
multiple spaces for EV and V2G connections, where arrival and
departure of vehicles are random and only a random portion of
parked vehicles can serve vehicle to grid flow [32]. In this paper,
energy storage is considered as a node with two main parameters,
namely; energy capacity (in kWh) and rated capacity (in kW). The
behavior of storage nodes is deterministic. We also assume that the
voltage of nodes across the distribution network will be maintained in the proper feasible region by network operator.
Fig. 1 gives an example of a power distribution network with
multiple loads or demand nodes, storage nodes, and renewable
generation nodes as well as connectivity to a macro power grid.
We are interested in the following problems: (i) Where to locate
static storage nodes and how much capacity to allocate to each
node for optimal sizing and operation; (ii) How to day-ahead plan
for the charge and discharge of these nodes, and (iii) How to control their operation in a near real time basis. In this paper, we use a
network-aware distributed planning and control approach to solve
these problems. Network-aware planning and control is an
approach for near real time control of assets in an arbitrary interconnected network. Such a network has multiple nodes with different functionality and criticality. Planning and real time control
decisions need to be made in reference to the characteristics and
state of constituent elements (nodes and arcs) of the network
and the overall control objective. The state description of an element includes, among other attributes, the element’s availability
and efficiency factors. We will present two models: (i) A model
for optimal location and capacity planning that also solves for
day-ahead operational plan, and (ii) A model for optimal charge
and discharge control of storage nodes in a near real time basis.
The paper is structured as follows. In Section 2, we describe the
first model. Some illustrative case studies for planning phase are
demonstrated in Section 3. Section 4 gives a data-driven
network-aware control scheme with a number of case studies. To
investigate the impact of behavior changes of stochastic input processes on the network planning and control we include sensitivity
analysis in section 5. Conclusion and future work are explained in
Section 6.
2. Optimal planning
In this study, we consider a power distribution network, which
could be utility-owned or a community level micro-grid, with high
penetration of renewable resources, such that renewable output
may exceed system load from time to time (see Fig. 1). The reverse
flow of power, resulting from the high level of renewable output
and inability to absorb excessive power at loads, could damage
the distribution network infrastructure. The storage nodes absorb
this excessive power and mitigate the damages (Renewable
reverse flow reduction). The energy charged from excessive renewable output can be used to reduce the cost of purchased energy
from grid during peak hours, given that renewable peak and price
peak do not coincide (Time of Use). The energy charged from
renewable during off-peak hours can be utilized during on-peak
hours to shave the peak demand as well (Peak Reduction).
Here we formulate a stochastic optimal planning problem that
takes into account the long-term cost of investment on a network
of storage nodes and their short-term operation costs calculated on
the basis of day-ahead planning schemes. The formulation is general in such a way that time units can vary from sub-hourly to
hourly to more aggregate temporal zones that are constructed on
the basis of stochastic pattern changes of some or all of the input
sources (e.g., electricity price and power demand). The aggregation
is done in a way that the designated input processes have small
variations within these temporal zones. Fig. 2 illustrates a simple
example of the temporal aggregation scheme over three stochastic
input processes, namely, renewable generation, demand (in this
specific example our network has two demand nodes) and electricity price. Coefficient of variation is used to measure the variation of
the stochastic processes. In order to aggregate several hours into
one temporal zone, we start from the first time slot (hour 1). The
consecutive time slots (hours) are aggregated into a single temporal zone as long as the coefficient of variation (CV) for each input
parameter (namely, demand, on-site power generation and electricity price) remains less than the selected threshold (0.3 in this
example) for that temporal zone. If adding the next hour increases
the inputs’ CVs for the current temporal zone to more than 0.3 (in
either of the input parameter), that hour is then considered for the
Fig. 1. An example circuit with multiple storage units, renewable energy resources and demand nodes.
K. Mahani et al. / Applied Energy 195 (2017) 974–990
977
Fig. 2. Temporal zone aggregation example.
next temporal zone. It is intuitive that decreasing (or increasing)
the threshold value for coefficient of variation increases (or
decreases) the number of zones in the aggregate model.
Fig. 2(a)–(c) shows the example patterns of stochastic input
processes. We note that within each temporal zone the input patterns are fixed at their average values (Fig. 2d, e and f). The advantage of the temporal aggregation over common hourly mixed
integer programming is that, in the aggregate model, the planning
period is decomposed into zones (usually in several hours), and the
only variable which is moved from one zone to the next is the state
of charge at the end of the zone. By the virtue of this decomposition, a problem with long planning periods and a complex network
configuration does not suffer from excessive computational times.
Hereafter, we will use the term temporal zone to commonly
refer to a time unit (sub-hourly or hourly). The objective function
is the total operation and investment cost defined over these temporal zones, and energy storage capacity (both energy capacity and
rated capacity), aggregate charge and discharge amounts within
each zone (in kWh) are the decision variables. Multiple applications of energy storage are considered, namely, ‘‘Time of Use”,
‘‘Renewable Reverse Flow Reduction” and ‘‘Peak Shaving”. We borrow ideas from the classical multi-period inventory control problem [33], where each temporal zone represents a single period
and the remaining energy in storage at the end of each period
defines the storage state of charge at the beginning of the next
period.
The total cost function is measured in net present value and
includes the present value of investment costs and operation costs
of the installed nodes during a lifetime period. That is,
MinimizeðOBJ Inv þ OBJ op Þ
ðO:1Þ
For the investment part we have
OBJInv ¼
X
ðCAPS Inv Cap
þ PS Inv PR
s
s Þ
ðO:2Þ
s
where Inv Cap
is investment cost related to the capacity of storage
s
node s (measured in $/kWh) and Inv PR
s is cost related to power rating of the node (measured in $/kW). For the short-term operation
costs, we must cover the uncertainties over the planning horizon.
We assume that input daily profiles of demand, renewable power
generation and electricity price each follow a stochastic pattern,
and together they can be clustered into groups of profiles over the
planning horizon. We note that one can always form such clusters
using historical data over the subject distribution network.
Total operation cost (OBJop ) of network will be the sum of daily
operation costs over the planning horizon. Daily operation cost is
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
stochastic because of the uncertainty in daily profile of demand,
renewable power generation and electricity price.
OBJ op ¼
"
X 1 þ c y
1
1þa
y
#
365
X
DailyOperationCostðd; yÞ
ðO:3Þ
d¼1
For the above problem, stochastic scenarios are generated over
the stochastic inputs namely, demand, electricity price and renewable power generation, and on the basis of historical data from several years (we choose 3 for illustration). To reduce the
computational complexity, we apply a high-dimensional data clustering method [34] to group these profiles and reduce the number
of scenarios [25]. Daily demand profile, electricity price and renewable power generation are considered as features in this clustering.
Suppose that a total of ‘‘CL” clusters of input profiles exist. In each
cluster we consider the representative average profile (over a cluster) for each demand node, renewable resources and electricity
price. Now, depending on the size of each cluster the chance of
its occurrence is calculated (ProbSc Þ:
OBJ op ¼
"
X 1 þ c y
y
1
1þa
CL
X
365 ProbSc DailyOperationCostðScÞ
Sc¼1
#
ðO:4Þ
where c and a are annual inflation and discount rates (%/year), ‘‘Sc”
is the index of scenarios according to the input data and ProbSc is the
probability of scenario ‘‘Sc”. DailyOperationCostðScÞ is the daily operation cost given scenario ‘‘Sc”, which is the representative scenario
for cluster ‘‘Sc”. Since multiple applications are considered, the daily
operation cost includes multiple sections as described below:
1. Within each zone a portion of electricity at each demand node
is served by the renewable nodes, which are connected to it.
Also, a percentage of demand at that demand node is served
by discharging energy from its connected energy storage nodes.
The rest of the demand plus any amount to be stored in storage
nodes are supplied by purchasing electricity from the grid.
2. The owner of the distribution system is usually charged for peak
demand. High peak demand could also cause more depreciation
of distribution devices. Charging storage nodes during off-peak
hours and discharging during on-peak hours reduce the peak.
Demand charge is usually defined in a monthly term. Here,
we use the same concept in daily planning. In order to consider
the hourly peak demand in time aggregate approach we assume
that the aggregate power flow is distributed uniformly within
each zone.
3. It is assumed that the remaining power from renewable creates
a reverse flow at the substation if it is not absorbed by storage.
Cost of damage to substation due to the reverse power is estimated by multiplying the remaining renewable output by a
penalty factor (Pnsub Þ.
þ w3 Obj 3ðScÞÞ
Obj 1ðScÞ ¼
i
j
(
Obj 3ðScÞ ¼
"
X
i
"
X
Pnsub
RSc ðk; iÞ
k
ðO:5Þ
!#
1 X dem;g X ch;g
ej;i;Sc þ
es;i;Sc
dr i
s
j
Obj 2ðScÞ ¼ Dem max
i
s
!)
X dem;r X ch;r
ej;k;i;Sc þ
es;k;i;Sc
j
0 6 P s 6 Pmax ;
s
!##
8s
ðC:1Þ
8s
ðC:2Þ
Operation Constraints: Operation constraints are valid in every
scenario (Sc). The total amount of inflow and outflow electricity for
each static storage node is limited due to its power rating,
ech;g
s;i;Sc þ
X
ech;r
s;k;i;Sc þ
X
eds;j;i;Sc 6 Ps dr i;Sc
8s; i; Sc
ðC:3Þ
j
k
As we mentioned before, storage level (state of charge) is
moved from one temporal zone to the next. Storage level at the
end of a zone is calculated based on the amount of energy charged
and discharged during that zone. It is obvious that storage level
cannot exceed its maximum capacity. Also, SFs as a safety reserve
capacity is considered for storage nodes. Since, according to
(O.5), daily operation cost is calculated in objective function the
initial state of charge in each individual day will be important.
For daily optimization, we assume that all nodes have the same initial state of charge (e.g., 80% of maximum energy capacity ðCAPS Þ).
Furthermore, the SOC of each storage at the end of a day is the
same as its daily initial state.
SOC s;i;Sc ¼ SOC s;i
1;Sc
þ Effch;s
ech;g
s;i;Sc
þ
X
ech;r
s;k;i;Sc
k
X
ed
j s;j;i;Sc
8s; i; Sc
Effdis;s
SF s CAPs 6 SOC s;i 6 CAP s
!
ðC:4Þ
8s; i
ðC:5Þ
We assume that the electricity load at each demand node has to
be satisfied, so:
LSc ðj; iÞ ¼
X dem;r X
ej;k;i;Sc þ
eds;j;i;Sc þ edem;g
j;i;Sc
8j; i; Sc
ðC:6Þ
s
k
Electricity generated by a renewable unit is used to serve
demand nodes and charge the storage nodes which are connected
to it. The remaining generation from renewable creates a reverse
flow of power at the substation.
X dem;r X ch;r
ej;k;i;Sc þ
es;k;i;Sc
j
DailyOperationCostðScÞ ¼ ðw1 Obj 1ðScÞ þ w2 Obj 2ðScÞ
X dem;g X ch;g
PrSc ðiÞ
ej;i;Sc þ
es;i;Sc
0 6 CAP S 6 Enmax ;
RSc ðk; iÞ P
The daily operation cost is then given by:
"
X
where PrSc ðiÞ is the average electricity price during the hours of
zone i for cluster Sc. w1 ; w2 and w3 represent the importance of different storage applications in the planning phase. The constraints of
the above problem are defined in two categories: ESS installation
constraints and operation constraints. ‘‘j”, ‘‘k” and ‘‘s” representing
sets of demand nodes, renewable nodes and static storage
respectively.
Installation Constraints: Constraints 5 and 6 show the maximum capacity (energy and power rating) of ESS that can be
installed on possible node ‘‘s”.
8k; i; Sc
ðC:7Þ
s
The power distribution network has ‘‘S” possible nodes for static
ESS installation, ‘‘J” demand nodes and ‘‘K” renewable resources.
Power can flow between two nodes if they are connected physically. The topology of the network is defined by configuration
numbers (ci1 ;i2 Þ. Configuration numbers have binary value; 1 means
physical connection exists. ci1 ;i2 ¼ 1 means that nodes i1 and i2 are
connected. Following equations illustrate the network configuration constraints:
0 6 edem;r
8j; k; i; Sc
j;k;i;Sc 6 M c k;j ðiÞ
ðC:8Þ
0 6 eds;j;i;Sc 6 M cSs ;d ðiÞ 8j; s; i; Sc
ðC:9Þ
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
0 6 ech;r
8s;k; i; Sc
s;k;i;Sc 6 M c k;Ss ðiÞ
ðC:10Þ
where ‘‘M” is a very big number (e.g. 10 millions).
Solution methodology is illustrated using the following
example.
3. Illustrative example
A modified IEEE 13 node test feeder as a community level distribution system with high penetration of renewable resources is
used as a case study (see Fig. 3). We start with the clustering of
stochastic input profiles and grouping of days in the planning horizon. Then we show optimal capacities for different network configurations with different applications. Finally, the day-ahead
operation results will be demonstrated.
As shown in Fig. 3 eight demand nodes are considered in this
network. Four residential sectors are considered in nodes D2, D5,
D7 and D8. The other four demand nodes (D1, D3, D4 and D6)
assumed to be commercial facilities. Four PV solar systems with
generation rated capacity of, respectively, 1500 kW, 1600 kW,
800 kW and 600 kW are assumed as renewable resources at nodes
R1 - R4. The hourly power generation of these nodes is related to
hourly solar intensity (radiation). We assume the same solar radiation at the locations of solar systems. Four nodes with renewable
resources are considered as the potential nodes for energy storage
installation.
Following eligibility matrices are assumed in this case study:
Fig. 3. Distribution network.
5. c = 2%/year and a = 10%/year are assumed as annual inflation
and discount rate [35].
Three years data for residential and commercial sectors, solar
radiation and electricity price are grouped into several clusters.
High Dimensional Data Clustering (HDDC) method is used to allocate input data into smaller number of groups [34]. This method
is available in CRAN server and can be used in R. By applying
HDDC algorithm, three years input data are grouped into fortyeight (48) clusters with each cluster associated with daily operation cost that varies in a small range. Fig. 4 shows the input data
with hourly mean value in one of these 48 clusters. More information about these clusters can be found in Appendix A. The
cluster shown in Fig. 4 (cluster 13) has population size of 41.
All of these 41 individual input data are within a heating season,
specifically from December to February. Since the total sample
size is 1095 (3 365), the probability of having inputs similar
to data in cluster 13 is 41=1095 ¼ 0:0374.
Next, we present our results for the optimal capacity planning
with different weights for different storage applications. This will
be followed by the optimal operation planning for some sample
clusters.
3.1. Capacity planning
As an example based on the defined eligibility matrices, storage
at node S2 can be charged from renewable R2. Furthermore, this
storage node can serve electricity demand in nodes D3 to D8.
The following assumptions are considered in this case study:
1. Sodium-Sulfur energy storage with 15 years lifetime is candidate technology to install as energy storage system.
2. According to [35] 350 $/kWh and 350 $/kW are considered as
investment unit cost related to ESS energy capacity and power
rated capacity (Inv Cap
and Inv PR
s
s ).
3. Both charging and discharging efficiency are assumed at 90%.
4. 5% power loss is assumed for long distance transmission. Note
that nodes shown in Fig. 3 are grouped into four classes, class
1 includes D1, D2, R1 and Storage 1, class 2 contains D3, D4,
R2 and Storage 2. D5, D6, R3 and storage 3 are in class 3 and class
4 includes the other nodes. Transferring power between two different classes is considered as long distance transmission.
The capacity planning optimization problem is solved using
both hourly and aggregate temporal zones. The same optimal
capacity plans were obtained from both cases; the reason being
that the variation of stochastic input processes within each temporal zone is small, as per approximation scheme. Multiple applications of storage with weights w1, w2 and w3 are considered in
both cases. Note that each storage application has its own cost elements, and using these weights we are able to obtain a weighted
objective function that includes multiple applications. Table 1
shows four different combinations of storage applications with different weights. In the first three cases a single application is considered in planning and in the last case (Case 4) multiple
applications are investigated. This table also shows the optimal
capacity for these cases.
Clearly, capacity plans depend on the intended storage applications. For instance, for renewable reverse flow reduction application installing storage system in nodes S3 and S4 is sufficient.
The reason is that renewable R1 and R2, which respectively can
charge storage S1 and S2, are serving more electricity demands,
therefor there exist less excessive reverse power flow in these
two nodes. In the following section the day-ahead planning
results will be demonstrated for some sample clusters. The
hourly model is used for the day-ahead planning problem. The
results from the aggregate model are then compared to the
results from the hourly model.
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 4. Input data in cluster 13.
Table 1
Capacity planning results.
Case
1
2
3
4
(W1, W2,
W3)
Storage 1
Capacity
(kWh)
Power Rate
(kW)
Capacity
(kWh)
Power Rate
(kW)
Capacity
(kWh)
Power Rate
(kW)
Capacity
(kWh)
Power Rate
(kW)
(0,
(0,
(1,
(1,
17,000
0
6200
10,000
1500
0
2000
1500
700
0
6600
2000
90
0
2200
500
290
80
1000
900
40
20
300
160
690
1500
600
1800
90
300
120
300
1,
0,
0,
1,
0)
1)
0)
1)
Storage 2
Storage 3
3.2. Aggregate model validation
In the aggregate model electricity flows from the main grid and
DER resources are distributed uniformly within each temporal
zone. Hence, the aggregate state of charge at storage nodes and
the overall objective function are expected to closely approximate
the hourly results. Moreover, one would expect that the computational complexity of the aggregate model to be significantly lower
than the exact hourly model especially for large networks and
higher temporal resolution. Next, we compare the results of aggregate model to the exact hourly model. Note that both aggregate
and hourly models produce the same optimal sizing. We illustrate
results from one of the representative clusters that we will later
use for daily planning, as shown in Table 2. Fig. 5 compares the
state of charge of ESSs in hourly and aggregate models when the
optimal configuration of case 3 (EBM application) is installed in
the distribution network and inputs come from cluster 47.
As demonstrated in the above figure the state of charges at the
end of each temporal zone in aggregate model are close to the
hourly model. Table 2 compares the two models in different cases.
As illustrated in Table 2 the values of objectives in aggregate and
hourly model are close.
Recent table compares the objective values for different cases
for one specific cluster. Analyzing the results from aggregate model
Storage 4
shows the little error for all input clusters. Following Table 3 illustrates the mean value and standard deviation of error over all input
clusters for each objective in different applications.
Table 3 verifies that proposed aggregate model is a good
approximation for exact hourly model in day-ahead planning.
Using aggregate model lowers the computational complexity of
optimization problem significantly. For instance, in our example
case (Fig. 3 – Distribution network, which is defined by configuration matrices ESL, ERS and ERL; there are 56 possible directions for
power flow in the distribution network. Also as mentioned earlier
48 input clusters exist based on historical input data. Using hourly
model results in 56 48 24 ¼ 64512 operational decision variables in the capacity planning problem, however, using the aggregate model reduces the decision variables to 56 672 ¼ 37632
(almost % 45 reduction). Reduction in the number of decision variables has a significant value especially in the complex and large
networks.
3.3. Daily operation planning
In this section, daily operation planning for different capacity
configuration (shown in Table 1) are discussed for some sample
input clusters. For each capacity arrangement the impact of different application weighting values on daily operation planning are
Table 2
Aggregate model vs. Hourly model (inputs from cluster 47).
Case
Application
Electricity cost ($)
Hourly
1
2
3
4
Peak shaving
Reverse flow reduction
EBM
Bundle
46,246
46,940
Peak value (kW)
Aggregate
47,498
48,044
Total reverse flow (kWh)
Hourly
Aggregate
7331
7849
7696
7912
Hourly
Aggregate
0
0
0
0
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 5. Aggregate model vs. Hourly model (SOC), Application: EBM.
Table 3
Percentage error over all input clusters (Aggregate model vs. Hourly model).
% error for each objective for different applications (average over all clusters)
Case
Application
Electricity cost ($)
Mean value
1
2
3
4
Peak shaving
Reverse flow reduction
EBM
Bundle
% 3.5
%3
Peak value (kW)
Standard deviation
% 0.8
% 0.6
studied. For daily operation planning analysis, we focus on three
different clusters of input data (clusters 44, 46 and 47) – Fig. 6
shows the mean value of input data for these clusters. Cluster 44
has low level residential demand, but a high-level commercial
demand. It also represents days with medium-level of solar intensity. Cluster 46 has low levels of residential and commercial
demand with a high-level of solar intensity. Cluster 47 represents
days with a high mean value and variance in hourly electricity
price. Both residential and commercial nodes have a high-level of
demand in this cluster.
In each cluster, a 24-h planning horizon is divided into temporal
zones and used to find the optimal amount of charge and discharge
for a given cluster representative profiles. Table 4 shows the number of zones and their duration (in hours) for each input cluster.
In this example, we assume that ESS units are installed for TOU
application to reduce the cost of purchasing electricity from the
main grid. So according to the capacity planning results, shown
in Table 1, ESSs with capacities given by case 3 are installed in
the subject distribution network. Based on capacity planning
results, this configuration is optimal when reducing the cost of
purchasing electricity from grid is the only goal for ESS installation.
Total reverse flow (kWh)
Mean value
Standard deviation
%6
% 1.5
% 5.5
% 1.2
Mean value
Standard deviation
% 0.2
% 0.05
% 0.2
% 0.05
Figs. 7–9 show the optimal power dispatch of DER assets (renewable resources and energy storages) to demand nodes 1 and 8
when the input data, respectively, come from clusters 44, 46 and
47. Note that aggregate model is used to find the optimal dispatch
and aggregated amount of power is depicted uniformly within
each temporal zone in these figures.
As illustrated in the above figures the optimal dispatch of DER
resources is sensitive to input profile. For instance, in all the three
clusters a major portion of demands are satisfied from renewable
resources during the noon time when sunlight is available, and
stored energy in ESSs is utilized in peak price periods. As shown
in Fig. 6 peak price times are different in the three representative
clusters. In cluster 44 peak price is at 1:00 AM and as illustrated
in Fig. 7 during that period stored power in ESS is discharged to
supply electricity demand and reduce the network electricity cost.
The above figures also show that the configuration of network
affects the dispatch of DERs. For example, demand 8 is connected
to all storage and renewable resources, but because of the higher
power loss in long transmission, R4 and S4 are used to serve this
demand node. As an another example, D1 could be served only
by R1 and S1, so these two resources are used to provide electricity
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 6. Average Input data in clusters 44, 46 and 47.
Table 4
Local zones’ duration for different clusters (Clusters 44, 46 and 47).
Cluster
number
Number of temporal
zones
Duration of temporal zones
44
46
47
13
16
12
[1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 7]
[3, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
[6, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 4]
Fig. 8. DER optimal dispatch with input data from cluster 46.
4. Network-aware real-time control
Fig. 7. DER optimal dispatch with input data from cluster 44.
to D1. The optimal dispatch of DERs to other six demand nodes are
presented in Appendix B. Finally Table 5 illustrates the interaction
between storage application and optimal operation for input cluster 47.
The above Day-ahead planning was performed based on the
forecast of stochastic input processes. But what happens if the
real-time inputs are not close to the forecast values? For real time
control, one could always update the forecast values of input
patterns and repeatedly carry online optimization by solving the
optimization problem at some predefined time steps [36,37] (online optimization). Running on-line optimization for complex networks with large number of components and inputs is not practical, however. Here, we propose a rule-based control approach that
determines the optimal real-time actions for each storage node by
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
renewable nodes, (iii) state of charge at each storage unit, and
(iv) electricity price trend. That is,
ST t ¼ ½LDd;t s ; . . . ; LDd;t ; LRk;t s ; . . . ; LRk;t ; EPt s ; . . . ; EP t ; SOC i;t T
2 SS
ð1Þ
8d 2 Demand nodes; 8k 2 Renewable nodes; 8i 2 Storage units
Fig. 9. DER optimal dispatch with input data from cluster 47.
Table 5
peak value, total reverse flow and total electricity cost when ESSs are used for
different applications.
Case
Weights
([w1,w2,
w3])
Application
Peak
value
(kW)
Total
reverse flow
(kWh)
Electricity
cost ($)
1
2
[0, 1, 0]
[0, 0, 1]
7849
9313
955
0
48,499
50,376
3
[1, 0, 0]
8559
374
47,498
4
[1, 1, 1]
Peak shaving
Reverse flow
reduction
EBM (Energy
Bill
Management)
Bundle
7912
0
48,044
monitoring the state of components and inputs across the network.
We adopt a reinforcement learning approach and build a control
policy, which maps the actions and network state, by a supervised
learning classification model.
In the proposed Network-aware control model the near optimal
actions are taken in each individual storage unit based on the partial knowledge of the whole network state. The state space SS of
the distribution network is defined by: (i) electricity consumption
trend at demand nodes, (ii) renewable power generation trend at
where LDd;t represents the level of electricity demand at node d at
time step t, LRk;t represents the level of power generation at renewable k at time step t, EP t represents the electricity price at time step
t and SOC i;t represents the level of energy at storage node i at the
beginning of time t. In order to investigate time series impact of
these variables we define a new variable ‘‘s” for a time window.
At each time step t, control agent s (for storage s) receives state
STt of the network and selects an action ast 2 As (STt) # As , where As
(STt) is the set of actions available for storage s in state STt and As is
the set of all possible actions for storage s. As is defined by the
amount and direction of power at any given state. Since As has a
finite possible actions the amount of power has to be discretized.
Power directions are: charging from grid, charging from different
connected renewable nodes and discharging to different demand
nodes. We note that depending on the network topology, storage
nodes will have different action spaces. For illustration, storage
unit S4 in our example case study is connected to two demand
nodes (D7 and D8), renewable node R4 and also main electricity
grid (4 directions). If the amount of power in each direction discretized into three levels, then 43 will be the maximum number
of actions in action set A4 . We should note that some of these
actions won’t be feasible because of rated capacity limit on storage
unit, hence will be removed from the action set.
Since our focus in this section is electricity bill management as a
primary application of storage units, for each individual storage s,
the reward function rwst ðST t ; ast Þ during time interval t, is the saving
in network electricity cost as a result of storage s. Since action ðast Þ
at time t affects the state of network at the next time step (ST tþ1 Þ, ast
should be taken in a way to maximize rewards during that and all
future time intervals. For storage s we seek a control policy ps ,
s
such that as
t ¼ p (STt), which minimizes the overall network electricity cost. Value of the storage s under the control policy ps when
network starts in state ST at time t is defined as:
ps
V ðSTÞ ¼
rwst ðST; ast Þ
þE
(
1
X
i
e
i¼1
Fig. 10. Stochastic Input pattern when charging from grid is the optimal action for ESS1.
rwstþi ðST tþi ; astþi Þ
)
ð2Þ
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 11. Stochastic Input pattern when discharging to D1 is the optimal action for ESS1.
where
X
rwst ðST; ast Þ ¼ PrðtÞ
eds;j;t
j
ech;g
s;t
!
ð3Þ
ech;g
s;t is the amount of power flow from grid to the storage s during
P
time interval t and j eds;j;t represents the total amount of power flow
discharged from storage s to serve connected demand nodes. These
quantities are determined according to action ast . Note that future
states and rewards are not only dependent on the action taken by
storage unit s, but also are dependent on the actions of the other
agents. This brings uncertainty to the future rewards, hence,
expected value of the value function is used. A threshold value
0 < e < 1 is introduced to ensure convergence. The optimal policy
ps then maximizes the value of storage s, so that,
s
ps ¼ ArgmaxðV p ðSTÞÞ
actions. After a predefined time window (usually a day),
the state space and action spaces will be updated and rules
will be reconstructed (repeat steps (i) and (ii)). This will
ensure that the control model is robust to the changes in
the input patterns.
ð4Þ
ps
The above optimal control problem is solved using the following three steps:
(i) Compute the optimal hourly charging and discharging
actions for each storage node in the network using the above
hourly optimization model. Several points are in order: (a)
Hourly model is used since control actions are to be made
at the top of each hour; (b) The optimization model is run
for 365 days per year times the number of years for which
historical data exist. In practice the sample size (N S Þ can be
very large. For illustration, we will use a sample size
N S ¼ 3 365 24 hourly data each described by a state vector STt and a set of corresponding actions ast for storage unit s.
For the illustrative distribution network, there are four
storage units, so in each sample data there exist four sets
of actions (ast ; 8sf1; 2; 3; 4g).
(ii) Construct simple rules that characterize the optimal actions
as a function of network state and its stochastic input patterns. The output of this step is ps for all storage units
(8sf1; 2; 3; 4gÞin illustraive case. The construction is carried
out using a classification algorithm, which will be discussed
next in details.
(iii) Monitor and match the real-time values of inputs and the
network state to the most similar stored patterns computed
in step (i). Use rule set from Step (ii) to take the optimal
Algorithm 1 (Offline classifier construction algorithm).
Estimate the optimal control policy ps
For (8s) repeat: /storage index/
1.1 – Construct the response vector: Y = [as1 , as2 ; ; asN S ]T
1.2 – Construct the feature matrix: X = [ST 1 , ST 2 ; ; ST N S ]T
For ð8bÞ repeat: /tree index in tree bagging method (denoted
by ‘‘B”)/
1.3 – Calculate classifier psb (ST) based on bootstrapped
training data set (X b , Y b )
End
1.4 – Calculate ps ðSTÞ ¼ B1
End
PB
b¼1
psb ðSTÞ ð5Þ
As described in ‘‘Algorithm 1” a classification technique is used
to characterize the optimal actions as a function of state of the network components and stochastic inputs. The hourly optimization
model is utilized to generate the required data for classification.
The hourly decision variables of the day-ahead optimization problem, namely, charge and discharge quantities for each storage
node, are the designated response variables in the classification
(optimal control actions). A unique class label is then assigned to
each action for the classification purposes. We note that the number of classes changes according to the configuration and topology
of distribution network. The response of the classification for each
storage system is a vector with labels associated with optimal
actions for that storage unit. Tree bagging technique [38] is used
to build a classification model to predict the optimal action in each
individual storage node as a function of stochastic input values and
the SOC of ESSs in the entire network. Tree bagging creates and
ensembles decision trees for predicting response variable (optimal
action at each time step) as a function of predictors (network
state). Given a training set of predictors with corresponding
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Table 6
Storage 4 control action representation.
a
Value of each digit (dn ; n = 1, 2, 3 and
4)
d1 (level of charging from
grid)
d2 (level of charging from
R4)
d3 (level of discharging to
D7)
d4 (level of discharging to
D8)
0
1
2
0% of rated powera
50% of rated power
100% of rated power
0% of rated power
50% of rated power
100% of rated power
0% of rated power
50% of rated power
100% of rated power
0% of rated power
50% of rated power
100% of rated power
In the focused case in control problem (case 3) rated power is 120 kW for storage 4.
responses, tree bagging repeatedly selects a random sample with
replacement of the training set and fits trees to these samples
(Algorithm 1 – step 1.3). Number of trees ‘‘B” is not a critical
parameter with bagging; using a very large number of ‘‘B” will
not lead to overfitting [38]. We used a sufficiently big number
(e.g., 500) to achieve a good performance with low out-of-bag
error. After training, predictions for unseen samples may be created by taking the average of predictions from all the individual
trees (Eq. (5) in step 1.4). Since the classification model is built
based on the optimization model, which has been solved for all
available data in historical data set, this classifier is expected to
be a good estimation of optimal control policy. However, this control policy is updated continuously according to the new occurred
network state as described below in ‘‘Algorithm 2”.
Algorithm 2 (Online monitoring and classification).
Figs. 10 and 11 confirm the correlation between charge/discharge controls of electricity storage and input patterns. According
to Fig. 10a and d, charging from grid actions of S1 should follow
decreasing trends in electricity price and solar radiation. This is
expected since electricity storage application is intended to reduce
the electricity bill. Fig. 11a–d illustrate that discharging to
commercial demand D1 follow the increasing trends in demand
profiles and electricity price.
According to the illustrative network configuration, the possible
directions for power flow in storage ‘‘S4” are: discharge to demand
nodes D7 and D8, charge from renewable R4 and charge from grid.
In each of these directions the amount of electricity flow is discretized into three levels based on power rating: at 0%, between
0% and 50% and greater than 50%. As a result, there are 34 (=81)
total possible actions for storage ‘‘S4” during a given time step. A
four-digit number in turnary (the base-3 numeral) system
Assign the optimal action ast
For (8d) repeat: /day index/
For (8t) repeat: /time interval index/
2.1 – Monitor the state of the network; ST t
For (8s) repeat: /storage index/
2.2 – Assign action ast ¼ ps ðST t Þ to storage unit s
End
End
2.3 – Solve the exact optimization problem for day ‘‘d” and
update the state space SS and action space
As ; 8sðstorage indexÞ
2.4 – Repeat the steps in ‘‘Algorithm 1” with updated
response vector and feature matrix and update the optimal
policy ps ðSTÞ8sðstorage indexÞ
End
Fig. 12. Daily cost deviation (%) – (a) s = 0 – (b) s = 4.
Next, the proposed control algorithm is applied to our example
distribution network. The configuration defined in ‘‘Case 3” with
electricity cost reduction application is considered for the illustration. To devise a classifier, we explore the relationship between
stochastic patterns of input variables and optimal actions. This is
followed by a discretization process according to the network configuration and topology. We also discuss the accuracy of the control model using sensitivity analysis.
Our hypothesis is that there exists a strong correlation between
the stochastic input patterns and optimal charge and discharge
actions. Here we investigate this hypothesis by analyzing input
patterns in a time window defined by (t-s;t). In Figs. 10 and 11
(with s = 4) we show the average patterns for these inputs with
two different actions at storage node 1. We assume that the solar
radiation at renewable nodes are close (due to the same geographical zone), therefore, there is no need to consider separate levels of
renewable electricity generation for different PV systems in the
classification phase. Hence, instead of analyzing the level of generation at the renewable nodes we look at the solar intensity
patterns.
Fig. 13. Example day electricity price and outputs profile – (a) Purchased electricity
from grid, (b) electricity price.
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 14. Hourly demand profile boxplot (Heating season and cooling season).
Table 7
Capacity planning for different electricity demand growth.
Growth in elec.
demand
Storage 1
Storage 2
Storage 3
Capacity
(kWh)
Power rate
(kW)
Capacity
(kWh)
Power rate
(kW)
Capacity
(kWh)
Power rate
(kW)
Capacity
(kWh)
Power rate
(kW)
Baseline (0%)
5%
10%
15%
20%
6200
6500
6900
7200
7500
2000
2200
2300
2500
2600
6600
6800
7200
7500
7800
2200
2300
2400
2500
2600
1000
1000
1100
1150
1200
300
300
350
350
400
600
600
600
650
650
120
120
130
130
130
Fig. 15. Deviation from the daily optimal cost when elec. Demand increase by 5% –
s = 4.
ðd1 d2 d3 d4 Þ3 is used to represent each action for this storage. Each
digit represents the amount of electricity flow in specific direction.
The following Table 6 shows the meaning of each digit for control
actions in storage 4.
For example, action number 3, which is represented as ð0 0 1 0Þ3
in turnary system, represents discharging to D7 with 50% power
rating. Control actions for other storage units are defined similarly.
It is obvious that some of these actions (for instance, multiple
charging and discharging during same time step) are not feasible
because of the limitation on the power rating (see Eq. (C.3)).
Finally, we note that, in addition to state of charge of ESSs at
time ‘‘t”, the level of renewable generation, electricity demands
and electricity price from time ‘‘t-s” to ‘‘t” will be considered as
Storage 4
classification features. Furthermore, we expect that higher values
of s will improve the misclassification and control errors. Next,
we present the above methodology for our case study network
with two different values of s.
First, we assume s = 0, which means that no memory is considered for the control algorithm. Therefore, only the current state of
charge of ESS nodes, electricity generation in renewable nodes,
electricity demand and electricity price are taken into account.
Two-third (2/3) of the sample data (the same three years historical
data) is used to train the classification model. The daily cost is calculated for each individual day and the deviation from optimal
daily cost is also calculated. Fig. 12a shows the histogram chart
of daily cost deviation when s = 0. The mean value of the cost deviation from optimal case (with exact information) is 6%, which
means that by using approximation rule-based control (with
zero-hour memory); on the average, the daily operation cost will
be 6% more than the optimal cost.
Repeating the above for s = 4 demonstrates that deviation from
the optimal cost reduces significantly with the size of time memory window (Fig. 12b).
The mean value of the cost deviation from optimal case (with
exact information) is 4% when memory window of 4 h considered
for control algorithm. As expected, by increasing the memory window the error of model decreases. In the following Fig. 13a the
amount of electricity required to be purchased from the main grid
as a result of executing control actions (for both s = 0 and s = 4)
during an example day are illustrated. Fig. 13b shows the electricity price this example day.
Fig. 13a shows that by increasing the memory window the rulebased control assigns charge and discharge actions to storage
nodes in a way to reduce the purchasing electricity from grid
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
during the peak hours. Note that, none of controllers (s = 0 and
s = 4) has information about future and they assign control actions
according to their prediction ability. But it seems that higher sizes
of the memory window improve the future prediction capability of
patterns, which results in less daily cost and better performance of
the control model.
any changes in network components’ behavior and increase the
robustness of control module.
Appendix A
Following table shows the population size and corresponding
probability value for each cluster.
5. Sensitivity analysis on stochastic parameter changes
Thus, far, the planning and real-time controlling models have
been made based on the available historical data. In this section,
we conduct a sensitivity analysis and investigate the impact of load
changes on planning decisions and control actions. We will experiment with data of Fig. 14 which gives the hourly profile boxplot
for two demand nodes in our distribution network during cooling
season and heating seasons.
We fit a multi-variate log-normal distribution to each demand
data and compute mean value and covarriance matrix. We consider 5%, 10%, 15% and 20% growth in demand for each node and
investigate the impact of these increases on capacity planning
and control for distribution network. We focuse on energy bill
management application.
The following Table 7 confirms that increasing in electricity
demand results in higher value of investment in energy storage,
therefore capacity planning model suggests higher capacity of
energy storages.
Now let’s move to the proposed network aware control model.
Capacity of storage units are same as baseline (no growth in electricity demand). The initial control policy is determined based on
base-line demand scenario (Step ‘‘A” in Algorithm 1). Now consider
that the level of demand is increasing with 5% growth. This results
in new state vectors which do not exist in the initial state space. As
described in steps B-3 and B-4 in Algorithm 1, the control policy
will be updated continuously according to the new occurred states
to adjust control actions with new states and maintain the performance of control module. Following Fig. 15 demonstrates the deviation from optimal cost as a result of applying proposed network
aware control algorithm on the new state space with %5 growth
in electricity demand. The mean value of the cost deviation from
optimal case is %4.5.
6. Conclusion
The paper proposed ‘‘Network-aware planning and control
approach” for multiple energy storage nodes distributed over
power distribution network. We were interested in optimally
locate these storage units over distribution network and to create
day-ahead plans according to planned application. An approximation model has been proposed for planning purpose, which lowers
the computational complexity of optimization problem by 45% in
our illustrative example (in terms of number of decision variables).
Two types of storage system with static and dynamic capacity are
considered in the planning model.
Furthermore, a novel rule based control scheme for the near real
time operation of the storage network has been built. Tree-bagging
classification technique is utilized to determine the near optimal
control policy, which maximizes the value of storage nodes. In
the proposed Network-aware control model the near optimal
actions at a time ‘‘t” are taken in each individual storage unit based
on the partial knowledge of the whole network state from time ‘‘ts” to ‘‘t. Comparing the daily costs of approximate control model
with the exact optimal case when s = 0 shows 6% difference in
average. Increasing the memory (s) of the model from 0 h to 4 h
reduces this deviation to 4%. In the proposed model the control
policy is being updated frequently to adjust the control rules with
Cluster number
Population size
Cluster probability
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
11
43
10
14
14
9
11
42
34
12
33
19
41
12
11
21
40
12
49
11
45
49
12
13
13
41
52
45
14
11
12
9
15
28
44
22
12
9
11
22
11
41
16
40
18
16
13
12
0.01005
0.03927
0.00913
0.01279
0.01279
0.00822
0.01005
0.03836
0.03105
0.01096
0.03014
0.01735
0.03744
0.01096
0.01005
0.01918
0.03653
0.01096
0.04475
0.01005
0.04110
0.04475
0.01096
0.01187
0.01187
0.03744
0.04749
0.04110
0.01279
0.01005
0.01096
0.00822
0.01370
0.02557
0.04018
0.02009
0.01096
0.00822
0.01005
0.02009
0.01005
0.03744
0.01461
0.03653
0.01644
0.01461
0.01187
0.01096
Appendix B
Optimal dispatch to different demand nodes in different example clusters (case study without parking-lot).
Optimal dispatch to different demand nodes in an example
cluster (case study with parking-lot): (see Figs. 16–19).
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 16. Optimal dispatch to 8 demand nodes (Cluster 44).
Fig. 17. Optimal dispatch to 8 demand nodes (Cluster 46).
K. Mahani et al. / Applied Energy 195 (2017) 974–990
Fig. 18. Optimal dispatch to 8 demand nodes (Cluster 47).
Fig. 19. Optimal dispatch to 8 demand nodes (With dynamic energy storage) – (Cluster 47).
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K. Mahani et al. / Applied Energy 195 (2017) 974–990
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