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Processes 2020, 8, 559; doi:10.3390/pr8050559 www.mdpi.com/journal/processes
Article
Two-Layer Optimization Model for the Siting
and Sizing of Energy Storage Systems in
Distribution Networks
Tao Sun
1
, Linjun Zeng
2
, Feng Zheng
3,
*, Ping Zhang
2
, Xinyao Xiang
2
and Yiqiang Chen
3
1
Shennongjia Power Supply Company, Wuhan 442400, China; syddgl@sohu.com
2
Shiyan Power Supply Company, Wuhan 442000, China; T17067@fzu.edu.cn (L.Z.);
zpsy5966@163.com (P.Z.); lut1989xxy@163.com (X.X.)
3
School of Electrical Engineering and Automation, Fuzhou University, Fuzhou 350116, China;
cyq447894172@163.com
* Correspondence: zf@fzu.edu.cn
Received: 21 March 2020; Accepted: 2 May 2020; Published: 9 May 2020
Abstract: One of the most important issues that must be taken into consideration during the
planning of energy storage systems (ESSs) is improving distribution network economy, reliability,
and stability. This paper presents a two-layer optimization model to determine the optimal siting
and sizing of ESSs in the distribution network and their best compromise between the real power
loss, voltage stability margin, and the application cost of ESSs. Thereinto, an improved bat algorithm
based on non-dominated sorting (NSIBA), as an outer layer optimization model, is employed to
obtain the Pareto optimal solution set to offer a group of feasible plans for an internal optimization
model. According to these feasible plans, the method of fuzzy entropy weight of vague set, as an
internal optimization model, is applied to obtain the synthetic priority of Pareto solutions for
planning the optimal siting and sizing of ESSs. By this means, the adopted fuzzy entropy weight
method is used to obtain the objective function’s weights and vague set method to choose the
solution of planning ESSs’ optimal siting and sizing. The proposed method is tested on a real 26-
bus distribution system, and the results prove that the proposed method exhibits higher capability
and efficiency in finding optimum solutions.
Keywords: optimal sizing and siting; energy storage system; multi-objective optimization; fuzzy
entropy weight; vague set
1. Introduction
In recent years, distribution networks using renewable energy such as wind power and
photovoltaic power have become one of the research hotspots at home and abroad. After the
increasing distributed generations (DGs) are connected to the distribution network, the randomness
and intermittency of renewable energy access will have a great impact on the safe and stable
operation of distribution networks. Therefore, incorporating ESSs within the distribution network
has a positive impact on improving system reliability, such as improved overall energy efficiency,
maximized voltage stability, and reduced real power losses [1,2]. Moreover, ESSs are of promising
technology with peak shaving, and they facilitate the integration of high penetration levels of
renewable energy sources. Despite the above advantages, if the site and size of ESSs are improperly
selected, it may cause insufficient voltage stability and economic benefits will decrease [3–5].
Therefore, one of the main problems associate with the use of ESSs in distribution networks is to find
their best site and size in order to maximize their impact on the grid.
Considering the multidimensional, constrained, and nonlinear characteristic of optimal siting
and sizing, the solution techniques for EESs are attained via optimization methods. Much research
Processes 2020, 8, 559 2 of 17
has proposed different methods such as analytic procedures as well as traditional optimized methods
or artificial intelligent methods to solve the problem. Rueda-Medina et al. [6] have introduced a
mixed-integer linear programming approach to solve DG siting and siting problems, along with
minimizing the annualized investment and operation costs. The works, presented in [7,8], have
established an optimal operating model for an integrated ESSs using mixed-integer linear
programming. However, increasing with the objective dimension, the shortcomings of poor
versatility, low efficiency, and increasingly complex problems prevented the above methods from
being widely used.
With the recent advancements in the field of artificial intelligent technique, many intelligent
optimization algorithms have been implemented to solve the siting and sizing problem of ESSs in
distribution networks. Chen C et al. [9] introduce the genetic algorithm (GA), which is used for the
optimal allocation of ESSs to minimize the operation costs of the targeted microgrid based on net
present value. A GA is used to optimize the integration of ESSs in [10]. The goals of the optimization
are to maximize the profits and minimize network losses. The authors of [11] presented a hybrid
method integrating sequential quadratic programming with GA to deal with the problem of optimal
allocation of ESSs in unbalanced three-phase low voltage microgrids. The goals of the planning
problem are the minimization of the total cost and power quality issue. Arabali et al. [12] presented
a technical assessment framework based on particle swarm optimization (PSO), which can effectively
minimize the sum of operation and congestion costs over a scheduling period. Mukherjee V [13] have
proposed a combined method that is based on symbiotic organisms search (SOS) and chaotic local
search (CLS) to find out the optimal location and sizes of real power DGs in a radial distribution
system, based on the power loss minimization and voltage profile improvement objective. A grey
wolf optimizer is used in [14] to determine the optimal size and location of ESSs in a distribution
network to minimize the total annual cost of a system comprising the cost of energy not supplied.
A limitation of the above-listed papers is represented by the fact that they integrated multiple
targets into a single target by introducing weights, which led to arbitrariness and subjectivity. In this
respect, multi-objective heuristic algorithms are the main means to solve multi-objective optimization
problems today because they can eliminate the error of weight. Wang Yongli et al. [15] use a non-
dominated sorting genetic algorithm-II (NSGA II) to get the Pareto set and TOPSIS to select the best
solution. The goals of the optimal design of the integrated energy system problem are the
minimization of the economic, technical, and environmental objectives. Sheng Wanxing et al. [16] and
Zhang Shuang et al. [17] apply an improved nondominated sorting genetic algorithm-II to optimal
planning of multiple DGs. They formulated their constrained nonlinear optimization problem by line
loss, voltage deviation, and voltage stability margin, then obtain the best compromise solution from
the Pareto-optimal set based on fuzzy set theory. In [18], a multiobjective particle swarm optimization
(MOPSO) has been used to minimize the economic and emission costs of the overall system. In [19],
an improved binary bat algorithm (IBBA) has been proposed, which is based on BBA and differential
evolution. In this capacity configuration optimization for stand-alone microgrid, the economic,
reliability, and environmental criteria should be maximized as the objectives. However, as basic
algorithms, the above algorithms have limited local optimality, and their optimal solutions were not
stable.
In this paper, three main factors associated with the procedure of ESSs siting and sizing are
studied through multi-objective optimization. An improved bat algorithm based on non-dominated
sorting (NSIBA) is proposed to optimal allocation and sizing of the ESSs in distribution networks.
Due to the iterative local search (ILS) strategy, stochastic inertia weight (SIW) strategy, and balance
strategy, NSIBA has significant advantages on optimization accuracy, solution speed, and
convergence stability. After obtaining the Pareto optimal set, in order to solve the multi-objective
decision problem, the fuzzy entropy weight of vague set as a traditional method was used to obtain
the best trade-off solutions from the Pareto optimal solution [20]. This method, however, did not
reflect the accuracy of the results because the score function could not fully reflect the relationship
between the support, opposition, and neutral target sets. For modifying the above method’s
shortcomings, a new score function was proposed to strengthen the influence of unknown
Processes 2020, 8, 559 3 of 17
information on decision-making. Based on the above, NSIBA and the method of fuzzy entropy weight
of vague sets are integrated into a two-layer optimization model that considers several constraints in
the process of ascertaining ESSs’ most cost-effective scheme and gives the planner the capability of
making the final decision. The effectiveness of the proposed algorithm is validated using the real 26-
bus system. To test the effectiveness and reliability of the proposed method, the results are compared
with NSGA II and NSPSO; moreover, it demonstrates and verifies that the proposed algorithm can
improve a distribution network’s economy and power quality. The rest of this paper is structured as
follows. The mathematic formulation is presented in Section 2. The outer layer optimization model
of the NSIBA optimization algorithm is constructed in Section 3. Section 4 presents the fuzzy entropy
weight of vague set as the internal optimization model. The simulation results are discussed in
Section 5, and the conclusion is given in Section 6.
2. Mathematical Problem Formulation
The main objective of current work is to find out the optimal siting and sizing of ESSs in
distribution networks, together with minimum network power loss, voltage stability margin, and the
application cost of ESSs. Each of the above factors can be considered as an objective function (OF)
that is subject to equality and inequality constraints as well as to boundary restrictions imposed by
the planner. In mathematical terms, a multi-objective optimization problem can be formulated as
follows.
() () ()
()
()
=
=
≤
1
min min , ,min
0
s.t. () 0
m
OF x OF x OF x
hx
gx
(1)
Here x = (x1, x2, …, xn) represents the solution to the n-dimensional problem; OF is the target
space to the m-dimensional problem; h(x) and g(x) are the equality and inequality constraints,
respectively. It is impossible to find a solution to minimize all objectives at the same time when the
OFs are in conflict; therefore, the dominance relationship and Pareto optimal solution set are
introduced as follows.
Dominance relationship: A solution x(1) is said to dominate another solution x(2) noted as
(1) (2)
xx
when (2), (3) are satisfied.
() ()
≤∀=
(1) (2)
,
1, 2, ,
ii
OF x OF x i m (2)
() ()
<∃=
(1) (2)
,
1, 2, ,
jj
OF x OF x j m (3)
Pareto optimal solution set: Pareto optimal solution is the set of all dominating solutions.
{
}
{
}
∗∗
==∈¬∃∈
(1) (2) (2) (1)
|:Px x x x xx x (4)
2.1. Objective Function
Considering the evaluation indexes of network power loss, voltage stability margin, and the
application cost of ESSs, the mathematical model of multi-objective optimization of ESSs is
established as follows:
()
=123
min ; ;FOFOFOF
(5)
It is impossible to guarantee that all objective functions can achieve a relative minimum at the
same time. Therefore, the Pareto optimal solution set is introduced to search the frontier solution in
the target region.
(1) Network power loss. In the process of ESSs’ siting and sizing, minimize the power loss as much
as possible is the first and foremost objective. The power losses depend on the impedances of
Processes 2020, 8, 559 4 of 17
line and transformer. Incorporating ESSs within the distribution network has a positive impact
on reducing the network power loss and improving the electric efficiency of the distribution
network. Therefore, under ESSs’ optimal configuration, the scheme with low total network
power loss should be considered.
The equivalent models of transformer and line are shown in Figure 1. The objective function of
network power loss can be expressed as
==+
1 loss tran line
OF P P P (6)
() ()
θθ θθ
==
=−++−−
line
11
cos( ) sin( )
NN
ij
ijij ij ij ijij
ij
ij
R
PPPQQQPPQ
UU (7)
Z
T
R
Y
T
LineTransformer
P
line
P
tran
Figure 1. The equivalent models of transformer and line.
Here, Ploss is the total losses of the distribution network. Ptran is the rated active power of the
transformer. Pline is the active loss of line. Rij is the resistance of the distribution line connecting the
ith and jth buses. Ui is the ith bus voltage. N is the total number of buses in the distribution network.
θi is the ith phase angle. Pi and Qi are the active and reactive powers of the ith bus, respectively.
(2) Voltage stability margin. The bus voltage of the network often experiences fluctuations with the
increase of load and DGs. ESSs connected to the network properly are conducive to improving
the overall voltage profile. The objective function of the voltage stability margin can be stated as
follows:
=
−
==
2
2dev
1
N
ie
ip
UU
OF D U (8)
Here, Ue is the rated voltage. Up is the rated voltage allowable deviation, namely, Up = 0.05.
(3) Application cost. The ESSs’ application cost includes investment and operation costs. In order
to pursue a higher economy, smaller operating and investment costs must be considered.
Therefore, the general formula of the objective function can be described as follows:
+
== ∗ + ∗
+−
33
(1 )
(1 ) 1
a
Oper
Invs ESSs
DG DG
a
rr
OF G C C S
r (9)
Here, r is the discount rate; a is ESSs’ lifetime; Ins
DG
C and Oper
DG
C are the investment cost and
operating cost of the unit ESSs, respectively; ESSs
S is the ESSs’ total investment capacity.
2.2. Equality Constraints
Due to the advantages of fewer iterations, simplicity, and flexibility, the back/forward sweep
method is widely adopted in distribution networks. Therefore, under operating frequency
conditions, the conserved formula of the active and reactive powers at a certain bus can be expressed
as
Processes 2020, 8, 559 5 of 17
θθ
=
+=+ +
1
(cos sin)
N
ESS
i i Li i j ij ij ij ij
j
PP P U UG B (10)
θθ
=
=+ −
1
(sin cos)
N
iLii jij ijij ij
j
QQ U UG B (11)
Here, ESS
i
P is ESSs’ output active and reactive powers of the ith bus, respectively. PLi and QLi
represent the active and reactive powers losses of the load, respectively. Gij and Bij are conductance
and susceptance, respectively. θij is the angle of the distribution line connecting the ith and jth buses.
2.3. Inequality Constraints
The node voltage, ESSs installed capacity, branch current, and SOC are bounded between two
extreme levels forced by physical limitations.
≤≤
,min ,maxiii
UUU
(12)
≤≤
,max
0ESSs ESSs
ii
SS
(13)
<<
,max
0ij ij
II (14)
Here, ESS
i
S is the total capacity of the ith ESSs. Iij is the current of the distribution line
connecting the ith and jth buses. In order to protect the service life of the ESSs, the SOC must be
limited to prevent the overcharge and overdischarge of ESSs.
+Δ −
<<
Δ
min max
t t t ESSi
SOC SOC E
PP
t (15)
<<20% 90%SOC (16)
Here, Pmax and Pmin are the maximum and minimum power of ESSs. In addition, in order to
extend the useful life of the ESSs, we set the usage range of the ESSs’ SOC as 20% to 90%.
3. Outer Layer Optimization Model
ILSSIWBA is a swarm intelligence optimization algorithm based on local iterative search and
random inertia weights proposed by Chao Gan [21]. The basic framework of ILSSIWBA is similar to
the previous bat algorithm (BA), which is popular in solving optimization problems. However,
differing from the BA, ILSSIWBA introduces the local optimal solution and the global inertia weight,
which can make the optimal solution more stable. This paper combines ILS strategy, SIW strategy,
and balance strategy with a non-dominated sorting strategy. NSIBA is proposed to obtain the Pareto
optimal front, which can solve the multi-objective optimization problem with faster convergence
speed. This algorithm is described in Figure 2.
Processes 2020, 8, 559 6 of 17
OF
1
initia l
populatio n
elite strategy
crossover and
mutation
crossover and
mutation
OF
2
ILS
ILS ILSSIW
SIW SIW
offspring
Pareto
optimal front
Figure 2. The main idea of an improved bat algorithm based on non-dominated sorting (NSIBA).
3.1. ILS Strategy
ILS strategy is introduced to make the target jump out of the local optimal solution [22], which
is defined as follows.
() ()
()
() ()
() ()
()
|
&
∗∗ ∗
∗∗
∗−−
∗
∗
=∗
>
** **
**
rand, if
=rand
,otherwise
OF x OF x
xx OFx OFx
xOF x OF x e
x
(17)
() ()
() ()
()
() ()
()
() ()
()
∗
∗∗∗
<+ <+ < ≥
**
** ** **
11 22 33
2
OF x OF x
if OF x OF x OF x OF x OF x OF x (18)
Here ∗
x is the current optimal solution; ∗∗
x is the perturbation solution. Because multi-
objective functions cannot be compared directly, the judgment condition is improved: it is equivalent
to single-objective comparison when there are at least two OFs values that are less than the
comparison value.
3.2. SIW Strategy
Stable solutions are obtained by introducing the SIW strategy, which updates the bat pulse
frequency, position, and velocity as follows [23].
()
=+ × −
min max min
rand(0,1)ff f f (19)
−
=+
1tt t
ii i
XX V
(20)
()
ω
−
=+−
1*tt t
ii i i
VV XXf
(21)
()
ωμ μ μ σ
=+ − × +×
min max min rand rand (22)
Here, f is the frequency of bat; fmax and fmin are the maximum and minimum value limits of the
frequency; t
i
Xand t
i
Vare the position and velocity of bat in the tth iterations, respectively; ω is the
random inertia weight; μmax and μmin are the maximum and minimum influencing factors of the inertia
weight, respectively; σ is the deviation coefficient; rand is a uniformly distributed random number.
This strategy introduces a weight coefficient for the bat speed update and adjusts the weight by
using random variables, which is beneficial to solve the problem of optimal solution instability in
traditional BA.
Processes 2020, 8, 559 7 of 17
3.3. Balance Strategy
In order to balance the local and global solutions, new emissivity and volume update formulas
are adopted, in which the pulse emissivity controls the search of the bat in local and global, and the
volume controls the acceptance of the new solution. The volume and pulse emission rate are updated
by
()
()
() ()
∞
∞
∞
∞
∗
−
=−+
−
−
=−+
−
<
<
0
max
max
0
max
max
1
1
rand
if
t
i
t
t
i
t
i
AA
A
tt A
t
rr
rttr
t
A
OF x OF x
(23)
Here, At is the bat volume in t iterations; r0 and A0 are the initial pulse emissivity and volume,
respectively; rꝏ and Aꝏ are the maximum values of initial pulse emissivity and volume, respectively.
3.4. Non-Dominant Sorting and Elite Preservation Strategy
Fast non-dominant is a key feature in stratifying the population according to the level of the
Pareto optimal solution set and making the target closer to the Pareto optimal front. The procedures
of non-dominant sorting and elite strategy are given as follows [24]:
(1) For each bat xi, set two parameters np and sp, where np is the number of solutions that dominate
the solution p; sp is a set of solutions that the solution p dominates.
(2) Find the non-dominated solution set with np = 0 and set the non-dominated rank as Rank1.
(3) For each individual xj in Rank1, check the corresponding sj. For each xk in sj, set the parameter np
= np−1; if np = 0, set xk as Rank2. Then repeat the above steps until the non-dominated ranks of all
bats are determined.
(4) Crowding distance is introduced to characterize the distance between two bats, to make the
distribution of bats more uniform in space, which can be formulated as follows:
() ()
+− −
=−
,max ,min
11
ii
x
ii
OF x OF x
DOF OF (24)
(5) Elite strategy. Elite preservation strategy combines the parents and individuals generated by
improved bat algorithm and genetic algorithm to form a set with the size of 3N and selects N
individuals with better performance to form the offspring generation. In the selection operation,
the solution with the smaller non-dominated Rank and the bigger crowding distance have
priority to be chosen than others when the non-dominated Rank is the same. The main idea of
elite strategy is given in Figure 3.
Processes 2020, 8, 559 8 of 17
Crowding
distance
Parent
Improved bat
algorithm
Genetic
algorithm
Non-
domina nt
sorting Abandon
inferior
solution
Offspring
Rank
1
Rank
2
Rank
3
Figure 3. The main idea of elite strategy.
4. Internal Layer Optimization Model
After the Pareto optimal front is obtained through the outer optimization model, the optimal
solution can be selected by setting different weights. Due to the difference and ambiguity of weight
caused by different design schemes, the fuzzy entropy weight of the vague set is used to reduce the
errors caused by decision making.
4.1. Analytic Hierarchy Process (AHP)
AHP is a subjective weight method, which determines the weight through the historical
experience or subjective preference of the decision made. The calculation process is given below.
(1) Construct a judge matrix. In order to make the scheme have a unified standard, the numbers 1–
9 and their inverses are used as scales to define the judge. The comparative important scale of
criteria is given in Table 1.
Table 1. Comparative important scale of criteria.
Scale Meaning
1 Two factors are equally important
3 One factor is weakly important than another
5 One factor is strongly important than another
7 One factor is demonstrably important than another
9 One factor is absolutely important than another
2, 4, 6, 8 The median of two adjacent judgments
(2) Arithmetic mean estimated weight vector.
ω
=
=
==
AHP
1
1
11,2,...,
mij
n
j
kj
k
fiN
nf
(25)
Here, ωAHP is the weight value corresponding to each OF; the elements in the judgment matrix
are normalized and averaged to obtain the weight vector.
4.2. Fuzzy Entropy Weight
Processes 2020, 8, 559 9 of 17
The entropy weight method uses information entropy to describe the objectivity of information,
and objectively determines the weight of attributes based on the degree of information difference in
the decision matrix. The calculation process is given below:
(1) Normalization is represented as follows:
() ()
−
=−
'max
max min
jji
ji
jj
OF OF x
OF x OF OF (26)
()
()
=
=
'
'
1
ji
ij N
ji
i
OF x
p
OF x
(27)
(2) The entropy is represented as follows:
()
=
=− = ≥
1
ln 1 / ln , 0
n
jijij j
i
eKpp K ne (28)
(3) The entropy method weight is represented as follows:
()
ω
=
−
=
−
1
1
1
j
eN
j
j
e
e
(29)
(4) The fuzzy entropy weight is represented as follows:
ωω
ω
ωω
×
=
×
AHP
AHP
iei
im
iei
i
(30)
4.3. Fuzzy Entropy Weight of Vague Set
Due to the ambiguity and uncertainty of the information, the linear weighting method often
loses some useful information in multi-objective decision making. Therefore, Gau proposed the
concept of the vague set based on the fuzzy set [25], which divides the membership function of each
element into support and opposition. The vague set of A can be formulated as follows:
{
}
ππ
=−∈
∈ ∈ ∈+≤++=
(),1 ()| ,
() 0,1, () 0,1, () , () () 1, () () () 1
Ai Ai i
Ai Ai Ai Ai Ai Ai Ai Ai
Atu fuuU
tu fu u Utu fu tu fu u (31)
Here, U is the universe of discourse; ui denotes a generic element of U; tA(ui) is lower bound on
the grade of membership on the evidence for ui and fA(ui) is a lower bound on the negation of ui
derived from the evidence against ui. πA is the degree of hesitation of the vague set A. The larger the
value of πA is, the more unknown information it may contain. The vague set can be interpreted as
tA(ui) and fA(ui) are the number of votes in favor and against; πA is the abstention.
As a generalization of fuzzy sets, vague fuzzy sets consider both membership and non-
membership information, so they can more fully express fuzzy information in multi-objective
decisions. The decision process is as follows:
(1) Determine the positive and negative ideal solutions P+, P−, and then calculate the solution of
vague set A = ([tAij, fAij])n×m.
{
}
{
}
+++ +−−− −
==
12 12
,,, ; ,,,
mm
PPP PPPP P
(32)
Processes 2020, 8, 559 10 of 17
−+
++
+− +−
+−
++
+− +−
−−
==
−−
−−
==
−−
ij j j ij
ij ij
j
jjj
j
ij ij j
ij ij
j
jjj
PP P P
tt
PP PP
PP PP
ff
PP PP
(33)
+− −+
==
ij ij ij ij ij ij
ttf ftf
(34)
Here, P+ and P− are the best and worst solution for each attribute to reach candidate solution,
respectively; the true and false membership of Pij relative to the positive ideal index P+ and the
negative ideal index P− are shown in (35). The comprehensive vague membership is shown in (36).
(2) Determine the comprehensive vague value of each ideal solution in the Pareto optimal front,
which can be described as follows:
()
ωω
==
===
11
,
,1,2,,
mm
Ai i i j Ai i ij
ii
ttf fi n (35)
(3) Sort the solutions by a new score function to choose the best solution.
π
−
=+1
Ai Ai
i
Ai
tf
S (36)
The new score function not only emphasizes the difference between true and false membership
degrees but also strengthens the influence of unknown information on decision-making. Obviously,
the greater Si is, the more credible of the scheme is. The flowchart based on the two-layer optimization
model is shown in Figure 4.
Yes
Star
Define the parameters of two-
layer methodology
Initialize the population of bat
Inpu t the ne twork
descriptive data
ILS strategy
j=j+1
j≤N
j=1
i≤tmax
i=0
i=i+1
SIW strategy
Balance between local
and global search
Crossover and mutation
Non-dominant sorting
and elite strategy
The output of Pa reto
optimal front
Calculate
the objective
weight
Calculate the
subjective
weight
Calculate the fuzzy entropy
weight
AHP
Fuzzy entropy weight of vague set
End
The best sol ution
No
Yes
No
Outer lay er
optimization
model
Internal layer
optimization
model
Economic optimization
configuration of solution
Processes 2020, 8, 559 11 of 17
Figure 4. The flowchart based on a two-layer optimization model.
When the optimal configuration analysis of the obtained scheme is carried out, the
comprehensive evaluation mainly based on the system’s operating economy has attracted much
attention. Therefore, the three OFs constructed in the study are converted into economic indicators,
and the economic optimization configuration of ESSs are analyzed by calculating the following costs
[26–28]:
=
=Δ
365 24
1ep loss,
11
t
t
GaC tP (37)
ω
=
=Δ
365 24
2ep dev,
11
t
t
GaC tD (38)
Here, G1 is the cost of network power loss; Cep is the cost of unit network power loss; Ploss,t is the
network power loss at time t; Δt is equal to 1 h; G2 is the cost of the voltage stability margin; Ddev,t
is the voltage stability margin at time t; ω is the conversion coefficient between the total line loss and
voltage stability margin.
ESSs in distribution networks not only have fast corresponding speed but also have the
advantages of “spread arbitrage”, which can alleviate the power supply tension during the peak of
power consumption. Therefore, in conjunction with peak and valley power prices, it can reduce
economic costs.
=
365
4ebuy
11
iESSs
loss i
GanC S (39)
Here G4 is the benefits of economical recycling; Cebuy is the peak-valley unit price spread; the loss
of the ESSs in its life cycle is expressed as nloss, which is equal to 0.95.
5. Simulation Results and Discussion
The proposed algorithm is implemented on a 26-node system of a 10-KV substation that is
depicted in Figure 5. The system contains 12 transformers, where the transformer nodes are set as
candidate nodes for ESSs.
Processes 2020, 8, 559 12 of 17
｛0｝
｛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｝
PV
PV
WG WG
Figure 5. The structure drawing of a distribution network.
5.1. Algorithm Performance Analysis
NSGA II, NSPSO, and NSIBA are applied in this paper to compare the performance for solving
multi-objective optimization problems. Algorithm parameters are set as follows: The maximum
iteration is 300; the bat population is 50; the initial and maximum value of the pulse emissivity are
0.1 and 0.7, respectively; the initial and minimum values of the volume are 0.9 and 0.6, respectively;
maximum and minimum values of inertia weight are 0.9 and 0.4, respectively; the coefficient of
deviation is 0.2; lifetime a = 10 year; discount rate r = 10%; investment cost Ins
DG
C = 172 USD/kW and
operating cost Oper
DG
C= 257 USD/kW; the cost of unit network power loss Cep= 0.07 USD/(kW h); the
peak-valley unit price spread Cebuy = 0.083 USD/(kW h).
The convergence curves of OFs are depicted in Figure 6. As shown in Figure 6, OF1 and OF2 of
NSIBA have a sharp decline in the first 50 iterations. It can be affirmed that NSIBA has quicker
convergence speed and better search accuracy than other algorithms.
Generati ons Generati ons Generations
OF
3
/×10
4
$
OF
2
/pu.
OF
1
/
kW
0 100 200 300
3.1
3.15
3.2
0 100 200 300
0.17
0.195
0.22
0.245
0.27
010020030
0
0
2
4
6
8
NSIBA
NSPSO
NSGAII
（a）（
b
）（c）
Figure 6. The convergence property for OFs of NSIBA and other methods; (a) minimum network
power loss, (b) minimum voltage stability margin, and (c) total cost of investment and operation.
In order to explain the performance of NSIBA further, Figure 7 shows the Pareto optimal front
obtained by different approaches. Comparing the OF values, the solutions by NSIBA has a significant
advantage than other algorithms, which not only distributes more uniformly in the Pareto optimal
Processes 2020, 8, 559 13 of 17
front but also has a wider range of solution set distribution. Therefore, the NSIBA algorithm can
search for more possible solutions and avoid falling into a local optimum.
3.15 3.2 3.25 3.3 3.35 3.4
0
5
1
0
15
20
3.236 3.238 3.24
10
10.2
10.4
10.6
10.8
0 5 10 15
0.
1
20
0.15
0.2
0.25
0.3
0.203
0.202
8.6
0.201
8.5
0.2
8.4
0.199 8.3
3.1 3.15 3.2 3.25 3.3
0.1
3.35
0.15
0.2
0.25
0.3
3.15
4
3.15
6
3.15
8
3.1
6
3.16
2
0.12
0.122
0.124
0.126
0.128
f1
OF
3
OF
1
OF
2
OF
3
OF
2
OF
1
(b) (c) (d)
(a)
OF
2
OF
3
OF
1
NSGAII
NSPSO
NSIBA
Figure 7. Pareto optimal fronts: (a) three-dimensional diagram, (b) vertical view, (c) vront view, (d)
end view.
5.2. Scheme Economic Analysis
In the actual operation of a distribution network, affordability is gradually increasing as the load
increases. Therefore, this section compares the economics of ESSs with capacity expansion equipment
such as transformers and analyzes the optimal configuration of the ESSs, which is listed in Table 2.
Processes 2020, 8, 559 14 of 17
Table 2. Analysis of economic optimization configuration of an expansion-transformed energy
storage system.
Scenario Buses Retrofit Scheme Cost ($) Revenue
Expense ($)
G1 G2 G3 G4
transformer
expansion 7,17,22,25 315 kVA to 500
kVA 2076.93 11.086 6857.14 / −8945.16
ESSs 7,17,22,25
7(0.3319 MWh)
2072.54 11.087 62,018.57 66,714.20 2612.22
17(0.2571 MWh)
22(1 MWh)
25(0.6477 MWh)
In order to realize the comparison in the same conditions, it supposes that the capacity of the
transformer is increased from 315 to 500 kVA on the bus where the ESSs are installed. The expansion
of the transformer is generally increased by 50% of the original equipment capacity. The transformer
upgrade cost is about USD 6857.14, which is represented by G3 in Table 2. From the comparison
results, we can see that G1 and G2 within ten years has a small difference in the two scenarios.
Although the investment cost G3 of ESSs is higher than that of traditional transformer expansion, the
cost-saving through “ESSs high storage and low generation” is enough to make up for investment
cost and achieve capital recovery. Therefore, in comparison to the expansion of traditional
transformers, the configuration of ESSs can bring more economic benefits to the network.
5.3. Scheme Effectiveness Analysis
In order to enrich the background of the example, 200 kW photovoltaic systems are connected
at bus 6 and 7, and a 200-kW wind power system is connected at bus 24 and 25. The typical daily
characteristic curves of load, photovoltaic, and wind power are shown in Figure 8.
LOAD/kW
POWER/kW
0:00 04:00 08:00 12:00 16:00 20:00 24:00
2400
2800
3200
3600
4000
4400
Load
Photovolatic system
Wind power system 0
40
80
120
160
200
Time/h
Figure 8. The typical daily characteristic curves of load, photovoltaic, and wind power.
In order to verify the effectiveness of the scheme, three scenarios are established for comparison.
Scenario 1: The network without DGs and ESSs.
Scenario 2: The network with DGs only.
Scenario 3: The network with both DGs and ESSs.
The typical daily system voltage at each bus is shown in Figure 9. Comparing Scenario 1 with
Scenario 2, due to the randomness of power output by DGs, the system voltage has a large deviation,
and the total line loss of the system has increased greatly. Comparing Scenario 2 with Scenario 3, the
adjustability of ESSs can greatly suppress the voltage fluctuation, which is of great significance to the
Processes 2020, 8, 559 15 of 17
safe and stable operation of the system. Thus, the ESSs performs very well in terms of suppressing
bus voltage fluctuation and load fluctuation.
Voltage(k V)
Voltage(k V)
Voltage(k V)
(a) (b)
(c)
Figure 9. Voltage in different scenarios: (a) Scenario 1 (b) Scenario 2 (c) Scenario 3.
The voltage and current of each bus at 20 h are shown in Figure 10. The voltage amplitude
increases toward the value of the rated voltage, and the current amplitude is smaller than the
corresponding value in Scenario 1. The voltage and current quality of the network with ESSs have
been significantly improved, which not only help improve the accommodation ability of the network,
but also verify the practicability of the proposed two-layer optimization model.
5 10152025
5.76
5.765
5.77
5.775
0
20
40
60
Bus Number
Voltage/kV
Current/kA
Node voltage with ESSs Node voltage without ESSs
Node current with ESSs Node current without ESSs
Figure 10. The voltage and current amplitude of each node.
6. Conclusions
According to the network power loss, voltage stability margins, and application costs of ESSs,
this paper proposes a two-layer optimization model for the optimal siting and sizing of ESSs. Based
on a theoretical analysis and simulation verification, the following conclusions can be drawn:
(1) This paper establishes a two-layer optimization model by integrating NSIBA and fuzzy entropy
weight of the vague set. The proposed NSIBA can keep the population evolving during the
search and jump out of the local optimal solution due to the integration of ILS strategy, SIW
strategy, and balance strategy within it. The proposed fuzzy entropy weight of the vague set can
strengthen the influence of unknown information on decision-making due to the integration of
the new score function. The two-layer optimization model can fairly and reasonably give
decision-makers a comprehensive plan.
(2) The proposed method is tested on a real 26-bus distribution system. The convergence curve and
Pareto optimal front show that NSIBA has the quicker convergence speed and better search
Processes 2020, 8, 559 16 of 17
accuracy than NSGA II and NSPSO. The simulation in different scenarios demonstrates that the
proposed scheme not only can achieve arbitrage of USD 2612.22 by “ESSs high storage and low
generation”, but also improve the power quality when accessing the DGs. Hence, it may be
concluded that the proposed two-layer optimization model is a good choice over the other
algorithm for determining the optimal siting and sizing of ESSs in distribution networks. It will
be very useful for decision-making in the optimal configuration of ESSs using simulation tools.
Author Contributions: F.Z. and T.S. conceived and designed the study. L.Z. performed the simulation. P.Z. and
X.X. provided the simulation case. Y.C. and T.S. wrote the paper. F.Z. and T.S. reviewed and edited the
manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the science and technology project of State Grid Hubei Electric Power Co.
LTD through grant number 5215C018002H and the Natural Science Foundation of Fujian Province, China,
through grant number 2019J01249.
Conflicts of Interest: The authors declare no conflict of interest.
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