Optimal Sizing and Locations of Capacitors Using Slime Mould
Algorithm
E. S. ALI1, S. M. ABD ELAZIM2
1Electric Department, Faculty of Engineering, Jazan University,
Jazan, KINGDOM OF SAUDI ARABIA
2Computer Science Department, Faculty of Computer Science and Information Technology,
Jazan University, KINGDOM OF SAUDI ARABIA.
Abstract- A new and powerful approach called Slime Mould Algorithm (SMA) is suggested in this paper, for
optimal siting and sizing of capacitors for an IEEE distribution network. First, the most nominee buses for
installing capacitors are developed using various indices. Loss Sensitivity Factors (LSF), Voltage Stability Index
(VSI), and Power Loss Index (PLI) are employed to determine the selected buses. Then the proposed SMA is
used to deduce the size of capacitors and their positions from the picked buses. The objective function is
introduced to minimize the net cost and then, increase the total saving per year. The developed approach is tested
on the IEEE distribution network. The obtained results are compared with others to highlight the advantages of
the developed approach. Also, the results are presented to confirm its influence in minifying the losses, and net
cost and to improve the voltage profile and total saving for a radial distribution network.
Key-Words: Slime Mould Algorithm; Loss Sensitivity Factors; Voltage Stability Index; Power Loss Index;
Optimal Capacitor Locations; Distribution System.
Received: October 26, 2021. Revised: September 25, 2022. Accepted: October 27, 2022. Published: November 23, 2022.
1. Introduction
Transmission and distribution system losses are
considered the main consumption in any power
network. Due to the growth in the load, and
environmental limits, the transmission, and
distribution networks are being worked under
overloaded situations, and losses in the distribution
systems have become the main concern. To attain
economic advantages, the fundamental conditions to
get agreeable power quality and enhanced efficiency
have formed a very auspicious environment for the
matter of loss minimization approaches and using
recent operational practices. Power loss reduction is
the only alternative to enhance the efficacy of the
distribution network. Therefore, it is noted that in the
last few decades many researchers have concentrated
on distribution network loss reduction and voltage
stability. There are many helpful techniques in the
literature for distribution network loss reduction [1].
However, the most often used mechanisms like (a)
capacitor siting, (b) network restructure [2] (c) DG
siting [3], (d) DSTATCOM sitting and its mixed
versions to realize upper potential interests are (e)
simultaneous restructure and capacitor sitting, (f)
simultaneous restructure and DG sitting, (g)
simultaneous DG and DSTATCOM siting, and (h)
simultaneous restructure, capacitor, and DG siting
are presented in [1]. Conventionally, loss reduction
has focused mainly on network restructure
optimizing or capacitor siting for reactive power
policy. Since installing capacitors are the simplest
and most famous solution, they are getting steadily to
be important components of the distribution network
[1].
Pending last years, diverse algorithms are presented
to find the proper locations and optimal sizes of shunt
capacitors. Simulated Annealing (SA) [4], Tabu
Search (TS) [5], Genetic Algorithm (GA) [6], Mixed
Integer Nonlinear Programming Approach (MINPA)
[7], Direct Search Algorithm (DSA) [8], Teaching
Learning Based Optimization (TLBO) [9], Plant
Growth Simulation Algorithm (PGSA) [10],
Heuristic Algorithm [11], Cuckoo Search Algorithm
(CSA) [12], Particle Swarm Optimization (PSO)
[13], Fuzzy Genetic Algorithm (FGA) [14],
Differential Evolution (DE) [15], Flower Pollination
Algorithm (FPA) [16-17], Improved Harmony
Search (IHS) [18], Mine Blast Algorithm (MBA)
[19], and Moth Swarm Algorithm (MSA) [20] are
developed to deal with the capacitor placement task.
However, these techniques may drop to compass the
optimum cost. To conquer these abuses, the SMA is
chosen in this article to treat the process of optimum
capacitor placement.
SMA is a vigorous population-based optimizer based
on the oscillation mode of slime mould in nature [21-
23]. It proves its effectiveness in many applications
[24-25]. It is suggested here as a modern optimization
approach to minify the net active power losses, the
net cost and to promote the voltage profiles for an
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DOI: 10.37394/232016.2022.17.38
E. S. Ali, S. M. Abd Elazim
E-ISSN: 2224-350X
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Volume 17, 2022
IEEE distribution network. The stations of the shunt
capacitors procedure are acquired firstly by
inspecting the points according to many indices.
Then SMA is submitted to deduct the optimum siting
and sizing of capacitors from specific nodes. The
validation of the suggested algorithm in progressing
the voltage profile and lowering resistive losses is
given for an IEEE distribution system. The results of
the SMA are compared with several algorithms to
assert its superiority.
The rest of the paper is constituted as follows:
Section 2 shows the different indices for capacitor
installing techniques. Section 3 presents the cost
function and limits. Section 4 introduces Slime
Mould Algorithm. Section 5 examines the results on
voltage profiles and power loss. Section 6 gives the
conclusion and future works to treat the distribution
network optimization process.
2. Various Indices
In this section, three different indices with their
equations are introduced.
2.1 Power Loss Index
In this article, PLI is utilized to specify the nominee
points for capacitors. The region of inspection is
diminished greatly and then wasted time in the
optimization procedure. The demerit of this index is
the pivotal computations. It is desired to execute load
flow and define the attenuation in power losses by
intromission reactive power at every bus except the
slack one [26-27]. The PLI is given as the following
equation.
PLI(i) =
l
lr
u
lr
l
lrilr
)(
(1)
Where
u
lr
: The upper attenuation in actual power losses.
l
lr
: The lower attenuation in actual power losses.
: The attenuation in actual power losses at the
bus
i
.
The buses of greater PLI will have the primacy to be
the nominee bus to constitute compensator apparatus.
2.2 Loss Sensitivity Factors
LSFs are appointed to specify the nominee buses to
install capacitors [28]. Figure 1, shows a transmission
line '
l
' linked between '
i
' and '
k
' buses.
The actual power loss is obtained by
ik
R
l
I2
in this
line that can be computed by
2
)(
)
22
(
k
V
ik
R
k
Q
k
P
lossik
P
(2)
The LSFs can be given from the following equation:
2
)(
*2
k
V
ik
R
k
Q
k
Q
lossik
P
(3)
The typical voltages are specified by dividing the
base voltages by 0.95. If these voltages are lower than
1.01 they can be suggested as nominee nodes for
capacitor installation. It is account note that the LSFs
judge the series in which nodes are to be prepared to
install capacitors.
2.3 Voltage Stability Index
VSI amount is close to 1 so the least VSI amounts,
the mightily sensitive points to voltage collapse.
Thus, VSI is utilized to elect the lowest points that
have more probability of voltage collapse along all
points. VSI amount is designated as the following
equation [29-31]:
2
4
2
4
4
)(
i
V
ik
X
k
Q
ik
R
k
P
ik
R
k
Q
ik
X
k
P
i
VkVSI
(4)
where
i
V
:
The magnitude of the voltage at the bus
i
.
3. Cost Function
The developed cost function of the optimum
capacitor position task is to lessen the net cost which
is planned as the following equation:
CB
o
K
CB
iCi
Q
C
KCB
I
KD
T
Loss
P
P
KCost
**
**
(5)
Where the parameters are given as [17].
P
K:
The price per KW-Hours equalizes
to 0.06 $/KW-Hours,
Loss
P:
The net losses after compensation,
T:
The time in Hours equalizes to 8760,
D
:
The depreciation agent equalizes to 0.2,
)(ilr
Fig. 1. Radial distribution network equivalent circuit.
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E. S. Ali, S. M. Abd Elazim
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CB :
The number of compensated points,
C
K:
The price per Kvar equalizes to 25 $/Kvar,
I
K:
The price per inauguration equalizes to 1600$,
Ci
Q:
The value of inaugurated reactive power in Kvar,
o
K
:
The working price equalizes to 300 $/year/position.
The overhead equation is constricted whereas
accepting the following equality and inequality
limits.
3.1 Equality Limit
Load flow limit
Conventional methods cannot be utilized in
distribution networks due to ill conditions. The
forward sweep method has been presented in [29] to
treat the load flow process of distribution networks.
The equality limit is shown by the following
equation:
L
i
N
q
qPdi
Lineloss
P
Slack
P
1 1
)()(
(6)
L
i
N
q
qQdi
Lineloss
Q
CB
b
b
C
Q
Slack
Q
1 1
)()(
1
)(
(7)
where
Slack
P
:
The active power of the slack node,
Slack
Q
:
The reactive power of slack node,
L
:
The size of the transmission line in a
distribution network,
qPd )(
:
The request for active power at bus
q
,
qQd )(
:
The request for reactive power at bus
q
,
N
:
The size of total nodes.
3.2 Inequality Limits
Voltage Limit
The magnitude of the voltage at every node must be
constrained by the following limit:
V 05.190.0
(8)
Compensation Limit
The injected reactive power at every nominee node
should be lower than its efficient reactive power.
Power Factor Constraint
Power Factor
)(PF
should override the lower
amount and less than the upper amount as given by
the following limit.
PFPFPF maxmin
(9)
Total Reactive Power Limit
Remarkably, the net injected reactive power is lower
than 0.7 of the net reactive power request to extend
the operating of the power system with lagging PF
and prohibition the leading one.
N
q
qQd
CB
b
b
C
Q
)(7.0
1
)(
(10)
4. Conventional SMA
Physarum polycephalum has been named slime
mould because it is considered a fungus [21].
4.1 Approach food
SMA is illustrated by the mathematical equations of
[21,22]. The equations represent the contraction
mechanism as follows:
pr if tXvc
pr if t
B
Xt
A
XWvbt
b
X
tX
)(.
)()(..)(
)1(
(11)
Where,
vb
is a value
aa,
,
vc
is reduced linearly from 1 to 0,
t
is the
th
t
iteration,
b
X
is the individual location with the most
concentrated scent,
X
is the location of slime mould (solution),
A
X
and
B
X
are the random selection of two
individuals from the population,
W
is the weight solution,
p
is computed as:
DFiSp )(tanh
(12)
)(,,,3,2,1 iSni
is to the competence of
X
,
DF
is the most competence calculated over all
iterations.
vb
is computed as follows:
aavb ,
(13)
)1
max_
(arctan
t
t
ha
(14)
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W
is computed as follows:
others
wFbF
iSbF
t
condition
wFbF
iSbF
t
lSmellIndex
W
1
)(
log.1
1
)(
log.1
))((
(15)
)(ssortsmellIndex
(16)
Where
)(iS
ranks the first half of the given solutions,
r
a random number
1,0
,
bF
is the optimal competence exist in the current
iteration,
wF
is the worst competence exist in the current
iteration,
smellIndex
is the sequence of the sorted
competence values.
Fig. 2 shows the impacts of (11).
4.2 Wrap food
When the search area extends to an area with a rather
low concentration of food, its importance will
decrease and the food group will go to explore other
areas. Fig. 3 shows the competence evaluation
functions for slime mould [23-24].
The solution location is updated according to the
following equation:
pr if tXvc
pr if tXtXWvbtX
zrand LBLBUBrand
XBAb
)(.
)()(..)(
).(
*
(17)
Where,
LB
is the lower limit of the search space,
UB
is the upper limit of the search space,
r
is a random value
1,0
.
4.3 Grabble food
vb
is a random vector between
aa,
and regularly
accessed zero as the repetitions advance.
The
vc
values lay between
1,1
and are driven to
zero in the end. Synergistic cooperation between
vb
and
vc
imitate the specific manner of slime mould.
Although sticky mould is a better source of nutrition,
it is preferable to distribute the organic matter to
explore other areas to find the best food source rather
than pouring it all into one source to find a more
reliable source of food. The Pseudo-code of the SMA
algorithm is shown below [25].
S
S
S
Best competence
Evaluate W
Evaluate W
Figure 3. Evaluation of the competence function
Figure 2. Potential location in 2_dimensional and
3_dimensional
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Pseudo-code of SMA
Initialize
size_population
,
dim
,
LB
,
UB
,
z
,
iterMax_
;
Initialize a set of Slime Mould random locations
),,2,1( niXi
;
While (
iterMaxt _
)
Compute and sort the competence of all Slime
Mould;
Update the best and the worst competence
Calculate the weight of Slime Mould (
W
);
Update the best competence, the best location
b
X
For each search agent
Update
p
,
vb
,
vc
Update the position of search agent
End For
1 tt
End While
Return the best competence, best location
b
X
5. Results and Discussion
The prevalence of the suggested SMA is applied to
an IEEE distribution systems. The results of 15 radial
distribution systems are offered below in detail. The
suggested algorithm has been completed via Matlab
[32]. Simulations were performed under the Matlab
environmental (release 2013 a) and done on a Lenovo
laptop with Intel core i7 CPU 2.90 GHz processor
with 4 GB RAM and a 64-bit operating system.
15 Bus Test System
The tested case is 15 bus system as given in Figure
(4). The system data are displayed in [29]. The total
load for this system is 1752 KVA with PF=0.7. The
losses without compensation are 60.5844 KW.
Figure (5) gives the nominee buses according to their
PLI. The ordered of these buses are 15, 11, 4, 7, 6,
12, 14, 3, 8, 13, .. 2. The nominee buses are obtained
in Figure (6) according to LSF. The ordered of buses
are 6, 3, 11, 4, 12,…Figure (7) shows the nominee
buses according to VSI values. The improvements in
system voltages due to install one and two capacitors
are shown in Figures (8, 9) respectively. A
comparison between various indices is performed
and shown in Table (1, 2) for installing one and two
capacitors respectively. It is clear that, PLI gives
better results than VSI and LSF for this system.
Based on these results, SMA is proposed with PLI to
give the better response in terms of cost and losses.
The notability of the suggested SMA is demonstrated
compared with other algorithms in [13, 14, 15, 33,
and 34]. The value of installed capacity of reactive
power is 850 KVAR. The minimum voltage is
increased from 0.9424 to 0.9679 p.u. The losses with
compensation are decreased to 32.2499 KW due to
capacitors installation as given in Table (3). The
percentage reduction in losses is increased to be
46.768%. Moreover, the value of total cost due to the
proposed objective algorithm is 23060.54 $ which is
the smallest one. Also, the net saving with the
proposed SMA is improved to 29.183 % which is the
maximum one compared with other algorithms.
Fig. 5. PLI for the 15 system.
0
0,2
0,4
0,6
0,8
1
1,2
123456789101112131415
PLI
Bus number
Fig. 4. The connection diagram of the 15-point
system.
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0
0,002
0,004
0,006
0,008
0,01
0,012
0,014
0,016
0,018
2 3 4 5 6 7 8 9 10 11 12 13 14 15
LSF
Bus
Fig. 6. LSF of the first system.
Table (1) Comparison of various indices for injection of one
capacitor.
Items
Un-
compensated
Compensated
PLI
LSF
VSI
Total losses (Kw)
60.5844
40.2634
43.8035
47.365
Loss reduction (%)
-
33.54
27.698
21.819
Minimum voltage
0.9459
0.9619
0.9550
0.9601
Optimal location
and size in Kvar
-
700@
Bus15
750@
Bus 6
400@
Bus 13
Total Kvar
-
700
750
400
Annual cost($/year)
32563.4
25282.44
27393.12
27515.04
Net saving ($/year)
-
7280.96
5170.28
5048.36
% saving
-
22.36
15.877
15.5
Fig. 7. VSI for the 15 system.
2 4 6 8 10 12 14
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Bus
VSI
VSI
Figure. 8. Effect of installing one capacitor
on system voltage.
Figure. 9. Effect of installing two capacitors
on system voltage.
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6. Conclusion
In this paper, SMA has been applied successfully to
solve the tasks of optimum positions and sizing of
capacitors in distribution system that have been
established as an objective optimization task with
competing power losses, cost of installation,
operation and injected vars. The superiority of the
suggested approach is clarified by using IEEE test
system. Also, the results have been compared with
those obtained using recent optimization techniques.
Moreover, it provides a promising and preferable
performance over other approaches in terms of
voltage profiles, active power losses, net cost, and
total saving. Implementation of the network
reconfiguration and distributed generation with the
most novel optimization algorithm to enhance the
voltage profile and to reduce the active losses is the
future scope of this work.
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Total losses (KW)
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1132
900
850
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32563.4
24599.8
24339.6
24387.1
24496.8
24429.9
23060.54
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-
7963.6
8223.8
8176.3
8066.4
8133.5
9502.86
% saving
-
24.46
25.26
25.11
24.77
24.98
29.183
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Volume 17, 2022
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