1. No category

ACADEMIE ROYALE DES SCIENCES D OUTRE-MER BULLETIN DES SEANCES
Vol. 4 No. 3 June 2015 pp. 16-22
ISSN: 0001-4176
Application of particle swarm optimization algorithm for optimal capacitor placement
problem on radial networks
RezaSeyedi Marghaki, Alimorad Khajezadeh
Electrical Department of Azad Kerman University, Kerman, Iran
Abstract: Power generated in generating bus is transmitted in transmission network and fed to the loads
through distribution terminals. The generated power distributed into the power network has some losses,
which is greater in distribution system as compared to transmission section. This issue could be sited by
locating capacitor at valuable terminals due to which the kW power loss may be minimized and the net
savings may be improved. This article introduces an intelligent system by particle swarm optimization
(PSO) algorithm for the placement of capacitors on the radial distribution systems to minimize the power
losses and to enhance the voltage profile simultaneously. The proposed optimization approach suggested in
this paper has been applied successfully in a real world standard system. Results achieved using computer
simulation of real system, is described to validate the proposed solution approach. The computer simulation
results prove that the suggested intelligent system has good performance.
Key words: Capacitor PSO Optimization Power loss Voltage profile




systems and the usage of non-continuous capacitors
will be observed to solve it in this study.
A pied of solution techniques based on classical
approaches and gradient based methods [1–5] have
been introduced to solve the optimal capacitor
placement issue in previous years. In contrast to these
classical optimization methods, soft computing
algorithms science has been newly suggested in papers
[6–11].
In one of the initial studies [12], the issue was divided
into primary and secondary problem and solved in two
independent stages. The first stage optimizes the
capacitor terminals, as the second stage assigns the
capacitor size values. In reference [13], authors present
the optimal capacitor sizing solution using non-linear
approach. In [14] authors propose the new fitness
function that was classified as a non-differentiable
adding more challenges to the optimization technique
and simulating annealing was applied as solution
algorithm. Also, more nature based optimization
algorithms were applied: immune system algorithm
[15], GA [16], fuzzy genetic algorithms version [17],
ant colony algorithm (ACO) [18], and PSO [19]. In last
decade, direct search algorithm method [20], cuckoo
search algorithm [21], self-adaptive harmony search
algorithm [22], ABC [23,24] and a new optimization
technique known as teaching learning based
INTRODUCTION
In An electric power distribution system is the last
stage in the delivery of electric power; it carries
electricity from the transmission system to individual
loads. Distribution substations connect to the
transmission system and lower the transmission voltage
to medium voltage ranging between 2 kV and 35 kV
with the use of transformers. First distribution lines
carry this medium voltage power to distribution
transformers placed close to the customer's premises.
Distribution transformers again lower the voltage to the
utilization voltage of household instruments and
typically feed several loads through secondary
distribution lines at this constant voltage. Commercial
and residential loads are connected to the secondary
distribution lines through service drops. Loads
demanding a much larger amount of power can be
connected straightly to the initial distribution level or
the sub-transmission level.
The issue of capacitors placement in power distribution
systems of electrical energy consists the assignment of
the number, location, kind and size of the capacitors to
be placed on the distribution feeders such that the total
charge of setting and performance of the power system
is minimum with respect to the consumers on the
system. This issue is combinatorial in real world
Corresponding Author: Seyedi, Azad Kerman University, Kerman, Iran
16
optimization [25] and many works [26- 28] are also
applied in papers.
The rest of paper is organized as follows. Section 2
presents the problem formulation. Section 3 presents
the optimization method. Section 4 introduces the
proposed method. Section 5, depicts some computer
simulation results and finally Section 6 concludes the
article.
J
Min F=K p Ploss   K cj Qcj
(1)
j 1
Which:
PROBLEM FORMULATION
V min  V j V max
(2)
THD j  THD max
(3)
Qcj  Qcmax
(4)
Here
Qcmax  L Qc0
The problem of optimal capacitor placement has many
parameters consist the applied capacitor size, terminal
number, capacitor charge, voltage magnitude and
harmonic limits on the power system. The optimal
capacitor placement is defined in this section by details.
Just the lowest standard size of capacitors and several
of this standard size are permitted to be installed at the
terminals to have more realistic optimal out. The
capacitor sizes are considered as non-continuous
parameters and the charge of the capacitor is not
linearity symmetric to the capacitor size, this makes the
mathematical problem a complex issue.
The target of the capacitor placement issue is to lessen
the general power losses of the electrical system while
striving to minimize the charge of capacitors placed in
the power system. The fitness function includes of two
main parts. The first is the charge of the capacitor
installation and the second one is the charge of the
general power losses.
The charge related with capacitor installation is made of
a fixed installation charge, a purchase charge and
operational charge (keeping and depreciation). The
charge mathematical formulation described in this
approach is a step-like function rather than a
continuously differentiable function since capacitors in
practice are grouped in banks of standard discrete
capacities with charge not linear symmetric to the
capacitor bank size.
It should be hinted that since the fitness function is nondifferentiable, all nonlinear optimization methods
become unskillful to use. The second part in the fitness
function represents the general charge of power losses.
This part is gained by summing up the annual real
power losses for the network.
Voltages along the feeder are wanted to remain within
maximum and minimum constraints after the addition
of capacitors on the feeder. Voltage limits may be taken
into account by specifying the maximum and minimum
ranges of the amplitude of the voltages. The oscillation
of voltage is considering by specifying for maximum
total harmonic distortion (THD) of voltages and the
maximum number of banks to be placed in one terminal
is taken into account.
The optimal capacitor placement issue is defined
mathematically as given follow:
(5)
More details regarding the applied system can be found
in [29]. The parameters of fitness function are presented
in Table 1.
F
Table 1: The parameters of fitness function
The total annual cost function.
Kp
Annual cost per unit of power losses.
Ploss
The total power losses. (Result from ETAP
PowerStation Harmonic Load Flow Program).
Number of buses.
J
K cj
The capacitor annual cost/kvar
Qcj
The shunt capacitor size placed at bus j.
Vj
The rms voltage at bus j. (Result from ETAP
PowerStation Harmonic Load Flow Program).
V min
Minimum permissible rms voltage.
V max
Maximum permissible rms voltage.
THD j
The total harmonic distortion at bus j. (Result
from ETAP PowerStation Harmonic Load Flow
Program).
THD max
Maximum permissible total harmonic distortion.
Qcmax
L
Qc0
Maximum permissible capacitor size.
An integer.
Smallest capacitor size.
PSO
In soft computing science, particle swarm optimization
(PSO) is an intelligent calculation approach that
optimizes a problem by iteratively effort to enhance
nominate with regard to a given measure of quality.
PSO optimizes an issue by having a population of
nominate solutions, here dubbed birds, and moving
these birds about in the search-boundary according to
easy mathematical formulae over the particle's position
and velocity. Each bird's motion is affected by its local
best known location but, is also guided toward the best
known location in the search-space, as are updated as
better locations are discovered by other birds. This
strategy moves the crowd toward the best outs.
17
PSO is primary introduced to Kennedy, Eberhart and
Shi [30] and was first intended for simulating social
behavior, as a stylized representation of the motion of
organisms in a bird population.
PSO is a met heuristic as it makes few or no
assumptions about the issue being optimized and may
investigate numerous spaces of nominate solutions. But,
met heuristics such as PSO do not guarantee an optimal
solution is ever discovered. Furthermore, PSO is a
pattern search approach which does not apply the
gradient of the problem being improved, which means
PSO does not need that the optimization issue be
differentiable as is essential by traditional optimization
techniques such as gradient descent and quasi-newton
algorithms. PSO may therefore also be applied on
optimization cases that are partially irregular, noisy,
nonlinear, oscillate over time, etc.
A main variant of the PSO works by having a crowd
(called a swarm) of nominate solutions (called
particles). These birds are moved about in the searchspace according to a few easy strategies. The motions
of the birds are governed by their own best known
location in the search-space as well as the entire
swarm's best known location. When enhanced location
are being found these will then come to guide the
movements of the population. The procedure is
repeated and by doing so it is hoped, but not
guaranteed, that a good solution will finally be found.
is added in the terminal(i). Thus a particle may be
described as shown in Fig. 1:
Terminal
number
Particle
1
2
3
….
….
m-2
3
0
1
….
….
4
Fig. 1: Sample of particle in PSO-CPP
m-1
m
1
2
SIMULATION RESULTS
The benchmark standard power system applied in this
article is IEEE 37 buses test feeder. One line diagram of
this standard power network has been depicted in Fig.
2. This system is a real world feeder placed in USA.
Some features of the feeder are as presented here:
 Three wire delta working at a rated voltage of 4.8kV.
 All cables are underground cables.
 All consumers are “spot” consumers and include of
fixed PQ, fixed current and fixed impedance value.
 The consumers are significantly unbalanced [32].
It is considered that nonlinear consumers are placed at
buses 2, 15, 22 with the weight factor of 0.5
furthermore the node 35 with the weight factor of 0.25.
Weight factor of nonlinear consumers at a special
terminal means the ratio of nonlinear consumers to
general consumer of that terminal. In this paper we
apply 25 KVAR capacitors banks.
PROPOSED METHOD
Because of its effectiveness, generality, powerfulness
and ability to cope with practical limits, a PSO has been
suggested to solve the general optimal capacitor
placement problem (OCPP).The following notes shed
some guidance on the draft faces of the proposed
method as applied to the OCPP:
 The number of particles (n) is a constant value and
the same is determined empirically by trial and error
way.
 The fitness function itself is applied to generate
fitness values of the newly generated outs.
 The proposed method is designed to stop after
nominative number of iterations.
Based on the above texts, a PSO-based solution strategy
used to the OCPP has been applied. The proposed
approach implementation (PSO-CPP) may be described
as follows:
In this study the representation by means of strings of
integers was selected. Each variable (represent terminal
numbers) of the bird (its length is equal to the total
number of the system terminals (m)) may save a zero,
which indicates absence of capacitors on the
corresponding terminal or an integer different from zero
that indicates the number of placed capacitor sizes that
Fig. 2: One line diagram of IEEE 37 terminals standard
benchmark network
WITHOUT CAPACITOR PLACEMENT
Table 1 indicates the described indices before capacitor
placement in investigated power network. Voltage
amplitude profile of all terminals in three phases,
without capacitor placement, is illustrated in Fig. 3.
18
Furthermore voltage THD of all terminals in three
phases, without capacitor placement, is illustrated in
Fig. 4.
0.075
0.07
Table 1: Described indices without Capacitor placement
Index
Value
62.4659
Active Power Loss
0.8242
Voltage Unbalancing Index
1.1737
Voltage Profile Index
5.6729
Voltage THD Index
1.005
0.065
THD
0.06
0.055
0.05
Phasea
Phaseb
Phasec
1
Phasea
Phaseb
Phasec
0.045
0.995
0.04
0
5
10
15
0.99
20
NodeNo.
25
30
35
40
Fig. 4: Voltage THD of used network without capacitor
installation
0.98
PERFORMANCE AFTER OPTIMIZATION
Voltagemagnitude(P.U)
0.985
In this experiment, the suggested method is used in test
standard benchmark power network. Table 2 depicts the
PSO variables. Table 3 shows the defined indices with
capacitor installation in test standard benchmark power
network. Voltage value profile of all terminals in three
phases, with capacitor installation, is illustrated in Fig.
5. Furthermore voltage THD of all terminals in three
phases, with optimal capacitor installation, is depicted
in Fig. 6.
0.975
0.97
0
5
10
15
20
NodeNo.
25
30
35
40
Fig. 3: Voltage amplitude profile of applied network
Table 2: Parameters used in the PSO
Number of particles, n
20
1.6
2
100
C1
C2
Number of iterations, R
It may be noticed from the Tables 1 and 3, the common
active power loss in power network is decreased
significantly from 62.46 KW to 52.42 KW, the voltage
profile index is decreased significantly from 1.1737 to
0.933, voltage unbalancing index is gained to 0.785
from 0.8242 and voltage THD index is decreased to
5.012 from 5.6729. Voltage amplitude profile of all
terminals in three phases, with capacitor installation, is
illustrated Fig. 5. Furthermore voltage THD of all
terminals in three phases, with capacitor installation, is
illustrated in Figs 6.
1.005
Phasea
Phaseb
Phasec
1
Voltagemagnitude(P.U)
0.995
0.99
0.985
0.98
Table 3: Defined indices for standard benchmark power network
with capacitor installation
Index
Value
52.42
Active Power Loss
0.785
Voltage Unbalancing Index
0.933
Voltage Profile Index
5.012
Voltage THD Index
0.975
0
5
10
15
20
NodeNo.
25
30
35
40
Fig. 5: Voltage magnitude profile of all terminals in three phases,
with capacitor installation
19
0.065
1
Phasea
Phaseb
Phasec
GA
IBA
0.9
0.06
0.8
0.055
Normalizedfitnessvalue
0.7
THD
0.05
0.6
0.5
0.045
0.4
0.04
0
0
5
10
15
20
NodeNo.
25
30
35
40
1
0.9
Normalizedfitnessvalue
0.7
0.6
0.5
0.4
50
Iteration
60
70
80
90
100
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Optimal capacitor installation problem is an issue that
helps towards discovers a lowest of defined fitness
function through solving a combinatorial problem in
which the placement and size values of capacitor banks
are to be selected. There are many of articles where
independent mathematical formulations and definition
of the optimal capacitor allocation issue along with
solution approaches have been introduced. The target in
common, is to discover optimal terminal numbers and
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real power/energy losses and that of parallel capacitors
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network. This paper investigated the optimal capacitor
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enhanced from iteration 0 to 100, and exhibited no
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The obtained curves depict the average of 50
independent runs for both optimization algorithms. As
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velocity of convergence compared with GA.
20
30
CONCLUSION
INVESTIGATION OF PROPOSED METHOD IN
SEVERAL RUNS
10
20
Fig. 8: Operation comparisons of GA and PSO
Fig. 6: Voltage THD of all terminals in three phases, with
capacitor installation
0
10
60
70
80
90
100
Fig. 7: Investigation of proposed methods in several runs
20
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