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Electrical Power and Energy Systems 60 (2014) 34–44
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Optimal location and sizing of DSTATCOM in distribution systems
by immune algorithm
Seyed Abbas Taher ⇑, Seyed Ahmadreza Afsari
Department of Electrical Engineering, University of Kashan, Kashan, Iran
a r t i c l e
i n f o
Article history:
Received 16 March 2013
Received in revised form 28 January 2014
Accepted 18 February 2014
Available online 21 March 2014
Keywords:
Optimal location
Sizing
DSTATCOM
Immune algorithm (IA)
Objective function (OF)
Distribution system
a b s t r a c t
In this paper, optimal location and sizing of DSTATCOM for the sake of power loss reduction, and
improvement of current and voltage profile in distribution networks are investigated. An effective biologically inspired algorithm (Immune Algorithm) is used to search the best location and determine the size
of DSTATCOM. By keeping voltage and current profile improvements in mind, minimum cost of installation of DSTATCOM and maximum power loss reduction are integrated into an objective function through
appropriate weighting factors. Comparative results are obtained on two standard distribution systems
(IEEE 33-bus and IEEE 69-bus) with encouraging optimization as far as location and size of DSTATCOM
and objective function minimization are concerned.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Distribution networks have increasingly found more importance in recent years as they play essential roles in power system
quality and planning. By introducing deregulation in power systems, it has become necessary to use advanced equipments for
quality improvement of the distribution networks. Complete utilization of lines capacity in these networks is not possible for several
reasons including allowable voltage sag and restrictions of stability. These lead to power loss increase, slower response time and
decreasing of power flow limits [1,2]. Modern techniques and
power electronic devices such as flexible AC transmission systems
(FACTS) have improved in these regards considerably. Development of such networks requires provision of reactive power by
appropriate sources and additive usage of FACTS compensators.
Desirable control methods for improving the distribution system
are possible with the aid of power electronic technology such as
custom power (CP) devices. These are elements that are used for
power quality improvement, loss reduction and voltage profile
improvement in distribution networks and include equipments
such as SSTS (Solid State Transfer Switch), DVR (Dynamic Voltage
⇑ Corresponding author. Address: Department of Electrical Engineering, University of Kashan, Kashan 87317-51167, Iran. Tel.: +98 9131614352; fax: +98
3615559930.
E-mail address: [email protected] (S.A. Taher).
http://dx.doi.org/10.1016/j.ijepes.2014.02.020
0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
Restorer), DSTATCOM (Distribution STATic COMpensator) and
UPQC (Unified Power Quality Conditioner) [3–5].
DSTATCOM as a shunt connected voltage source converter (VSC)
is frequently employed to compensate power quality concerns.
During the voltage sag and over loading, the load voltage of the
bus in which the DSTATCOM is connected can be regulated by
injection of compensating current into the system [6]. The injected
current is evaluated using bus voltage and reactive power required
for regulating/balancing voltage in the desired value. The design of
a prototype DSTATCOM for voltage sag mitigation in an unbalanced distribution system is presented by [7]. A new cascade loop
control strategy to regulate and balance the voltage at a distribution bus using a DSTATCOM is proposed by [8]. Design and operating fundamentals and characteristics of a DSTATCOM as well as its
basic control strategy are described in [9]. Implementation and
experimental details of a real-time digital simulator designed for
representing a DSTATCOM interfaced with a digital controller is
also described by [10]. Balancing of source currents, power factor
correction and harmonic mitigation in three-phase, three-wire distribution system supplying delta connected load under various
source voltage conditions is done by using DSTATCOM in [11]. In
[12] performance of the DSTATCOM using an control strategy
based on improved instantaneous active and reactive current component theory for generating reference currents has been evaluated under various source and load conditions. The performance
of the proposed control strategy has been evaluated in terms of
load balancing, reactive power compensation, compensator rating
35
S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Nomenclature
Vi
voltage of ith bus
IDSTATCOM current of DSTATCOM
Ri + jXi
impedance between ith and i + 1th bus
Pi + jQi
local loads are connected in ith and i + 1th bus
Ii
current between ith and i + 1th bus
hi
angle of Vi
di
angle of Ii
V 0i
voltage of ith bus after installing DSTATCOM
h0i
angle of Vi after installing DSTATCOM
QDSTATCOM reactive power injecting to the network by DSTATCOM
Ke
energy cost of losses
Ti
time duration of ith load level
Kci
time duration proportion of ith load level to the total
time duration
Ploss i
power losses in ith load level
nbus
number of buses
CostDSTATCOM year annual cost of DSTATCOM
CostDSTATCOM cost of investment in the year of allocation
and harmonic mitigation. Ref. [13] analyses the negative sequence
equivalent circuit of DSTATCOM in unbalanced distribution networks. A comparative evaluation of three different control techniques (instantaneous reactive power, symmetrical component,
and improved instantaneous active and reactive current component) for DSTATCOM installed in three-phase four-wire distribution system has been presented in [14]. In [15] a three-phase
three-wire DSTATCOM which is fed by Photovoltaic array or battery operated DC–DC boost converter and controlled by I Cos/
algorithm is proposed for power quality improvement in the distribution system. A number of works [16–18] have been carried out
on optimal location of STATCOM using techniques such as particle
swarm optimization (PSO) and genetic algorithm (GA).
Immune algorithm (IA) is also considered as one of the best evolutionary algorithms (EA) [19–28] and is widely used to solve the
optimization problems in general as well as in power system
[29–36]. Ref. [37] presents the application of immune algorithm
(IA) to find optimal location of unified power flow controller
(UPFC) to achieve optimal power flow (OPF) and congestion management. Then in [38] the application of hybrid immune algorithm
such as immune genetic algorithm and immune particle swarm
algorithm to find optimal location of UPFC to achieve optimal
power flow has been investigated. Ref. [39] proposes a two-stage
immune algorithm that embeds the compromise programming to
perform multi-objective optimal capacitor placement. A new approach using immune algorithms to solve thermal generation
scheduling problems is proposed in [40]. Ref. [41] propose the
application of artificial immune systems as bio-inspired optimization technique to reconfigure radial distribution networks and to
minimize the total energy losses in these systems. In [42] an integrated algorithm proposed for forecasting annual electrical energy
consumption based on artificial immune system, genetic algorithm
and particle swarm optimization. Artificial immune system method with the clonal selection algorithm shows satisfactory results
when applied with simulated data and has been selected as the
preferred method.
This paper proposes an application of IA approach to solve the
optimal location and sizing of DSTATCOM in distribution systems
to reduce power and energy losses, improve voltage profile and decrease the line current. A newly defined objective function is introduced for this optimization, which includes installation cost of
DSTATCOM and cost of energy losses.
NDSTATCOM longevity of DSTATCOM
B
asset rate of return
OC
line over current factor
OV
voltage stability index for buses
Imax
maximum current that can flow in the network lines
k
small positive constant
l
small positive constant
OF
objective function
TCS
total cost saving
Pi
total power loss before installation of DSTATCOM
PDSTATCOM
total power loss after installation of DSTATCOM
i
Pr
probability of replacement
Pc
probability of cloning
Pm
probability of mutation
Rand
a random number
Abi
ith anti-body
IMAX
maximum allowable current of lines
2. DSTATCOM structure and modeling in distribution load flow
DSTATCOM is a shunt device that injects or absorbs both active
and reactive current at a point of common coupling connection.
DSTATCOM is also a DC/AC converter consisting of a dc-link capacitor or a dc energy storage device that provides constant dc-link
voltage, and a 3-phase PWM voltage source converter (VSC) bridge.
All these are usually connected to the network via a coupling transformer [43]. A DSTATCOM can work as a synchronous voltage
source with a variable magnitude and phase angle. Hence, it is
capable of controlling its bus voltage and correcting the power factor. Fig. 1 shows a bus in a distribution system equipped with a
proposed DSTATCOM. Switch changing can absorb or generate
the current by considering control strategy and depend on voltage
of common coupling bus.
In steady-state operation with heavy loading or some short-circuit events, DSTATCOM typically injects appropriate compensating
current to the point of coupling connection, and thus voltage at the
load bus regulated by the DSTATCOM will be lifted close to the
nominal or a given value [1,44–46].
Generally, DSTATCOM has the ability of exchanging active and
reactive power simultaneously. The amount of active power
exchanging depends on the capacity of energy source. In this paper,
only DSTATCOM application for reactive power exchanging is
considered and exchange of active power with the network is
neglected.
Bus i
V
i
IDSTATCOM
VSC
Energy
Storage
Fig. 1. A typical DSTATCOM connected to bus i.
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S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Fig. 2. Single line diagram of two consecutive buses of a distribution system.
2.1. System model
Backward/forward sweep load flow calculations are used in this
work, in conjunction with a suitable steady state model for
DSTATCOM as presented by [47]. Many distribution networks have
radial structures which only feed from one side. A section of a
sample distribution network is shown in Fig. 2 [48,49] which assumes that three phase radial distribution network is in balance.
Impedance between buses i and i + 1 are shown with Ri + jXi. Local
loads are connected in buses i and i + 1 are named Pi + jQi and
Pi+1 + jQi+1, respectively. Vi and Vi+1 are voltages of these buses.
The phasor diagram for Fig. 2 is presented in Fig. 3. KVL equation
can be stated as below:
V iþ1 \hiþ1 ¼ V i \hi ðRi þ jX i ÞIi \d
ð1Þ
Values of variables are derived from load flow. Usually in traditional
networks, the buses voltage is less than 1 pu, in which case, one can
assume that voltage of bus i + 1 is also less than 1 pu. A DSTATCOM
device is installed in this study in order to compensate voltage of
bus i + 1 to a desired value. As noted earlier, DSTATCOM is used
for voltage regulation and loss reduction in steady state condition
and can inject only reactive power to the system. Consequently,
IDSTATCOM must be kept in quadrature with respect to the system
voltage. As shown in Figs. 4 and 5, by installing DSTATCOM in bus
i + 1, currents Ii and IDSTATCOM flow in branch simultaneously.
\IDSTATCOM ¼
V 0iþ1 \h0iþ1
¼
p
2
þ h0iþ1
V 0i \h0i
p
ðRi þ jX i Þ Ii \d þ IDSTATCOM \ þ h0iþ1
2
ð2Þ
ð3Þ
By separating the real and imaginary parts of Eq. (3) and some computation we have:
pffiffiffiffi
B D
x¼
2A
Fig. 4. DSTATCOM installation in bus i + 1 of proposed distribution system.
ð4Þ
That:
x ¼ sin h0iþ1
ð5Þ
A ¼ ða1 a3 a2 a4 Þ2 þ ða1 a4 þ a2 a3 Þ2
ð6Þ
B ¼ 2ða1 a3 a2 a4 Þ V 0iþ1 ðRi Þ
ð7Þ
2
C ¼ V 0iþ1 R ða1 a4 þ a2 a3 Þ2
ð8Þ
Fig. 5. Phasor diagram of voltage and current of system shown in Fig. 4.
a1 ¼ Real V 0i \h0i RealððRi þ jX i Þ ðIi \dÞÞ
ð9Þ
a2 ¼ Imag V 0i \h0i ImagððRi þ jX i Þ ðIi \dÞÞ
ð10Þ
a3 ¼ X i
ð11Þ
a4 ¼ Ri
ð12Þ
As seen from Eq. (4), there are two roots for the variable x, and
therefore two values are calculated for \IDSTATCOM and |IDSTATCOM|.
In order to determine the correct answer, the boundary conditions
are examined in these roots.
V 0iþ1 ¼ V iþ1 )
IDSTATCOM ¼ 0
h0iþ1 ¼ hiþ1
pffiffiffi
D
Results show that x ¼ Bþ
is the correct answer of Eq. (4). There2A
fore \IDSTATCOM can be determined as below:
\IDSTATCOM ¼
p
2
þ h0iþ1 ¼
p
2
1
þ sin
x
ð13Þ
Current magnitude of DSTATCOM can be calculated by the appropriate separated real and imaginary equations as below:
jIDSTATCOM j ¼
V 0iþ1 cos h0iþ1 a1
a4 sin h0iþ1 a3 cos h0iþ1
ð14Þ
Thus IDSTATCOM and voltage of common coupling bus are calculated
using Eqs. (13), (14), and (3).
Finally reactive power injected to the network by DSTATCOM
for voltage correction of connected bus up to V 0iþ1 can be expressed
as:
p
jQ DSTATCOM ¼ V 0iþ1 \h0iþ1 IDSTATCOM \ þ h0iþ1
2
Fig. 3. Phasor diagram of voltage and current of the system shown in Fig. 2.
Such that, symbol ‘‘*’’ denotes complex conjugate.
ð15Þ
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S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
2.2. Installing model in load flow
suggested answers loop. Details of calculation for each component
of Eq. (16) are expressed in below.
To evaluate load flow considering DSTATCOM in a distribution
network with any iteration in forward sweep, voltage magnitude
of the compensated node can be assumed to be any desired value
(for example 1 pu). The phase angle of the compensated node,
reactive power and also the size of DSTATCOM can now be
calculated using above equations. The updated voltage and
injected reactive power by DSTATCOM are used for continuing
forward sweep in order to determine load currents in the next
backward sweep load flow. This procedure is repeated until load
flow converge [48,50,51].
3. Problem formulation
In this work the optimal location (bus number) and reactive
power (kV Ar) of DSTATCOM in a distribution system are obtained,
in a steady state condition. Minimizing the size of DSTATCOM and
power loss in distribution network, are considered in the objective
function (OF). The voltage and current constraints are formulated
as a penalty function to the OF.
3.2. Cost of DSTATCOM
Cost of investment can be extracted from cost of DSTATCOM as
Eq. (19) [54]:
CostDSTATCOMyeari ¼ CostDSTATCOMi
OC ¼
OV ¼
"
3
3
X
X
O:F: ¼ K e ðT i P lossi Þ þ
ðK ci Cost DSTATCOMyeari Þ
i¼1
i¼1
! 3
4 OC OV 5
i¼1 j¼1
j¼1
i
2
3 Y
#
nl
Y
nb
Y
ð16Þ
where i, nb and nl indicate the number of load level, the number of
bus and number of lines respectively. Ke corresponds to the energy
cost of losses, Ti is the time duration of ith load level and Kci is the
proportion of ith load level time duration to the total time duration
[52], determined as
Ti
K ci ¼ P3
i¼1 T i
:
Plossi ¼
nl
X
Rj jIj j2 :
8
< 1;
if Ij 6 Imax
Ij : exp k1 I ; if Ij > Imax
max
j¼1
A straight forward approach is to convert a constraint optimization
problem into a non-constrained optimization problem by adding
penalty for violation of constraints [38]. The first term in the
R.H.S. of Eq. (16) corresponds to the total costs (in US$) of power
loss and DSTATCOM installations, needs to be minimized. The second term deals with the voltage and current limitations of the network, whose quantity should be bounded within desired limits. The
latter act as a penalty factor assigned in a constant ratio to the first
term, in response to the deviation from specific boundary conditions, and is equals to 1 when all limitations are secured for buses
voltages and lines currents.
The ultimate goal is to minimize proposed OF so that all boundary conditions be satisfied. Loss minimization and minimum size of
DSTATCOM utilization are represented as their cost in first part of
OF which should be minimized. Over current and over voltage
boundaries are represented as a coefficient for the first part of
OF. Violation from proposed constraints by the suggested candidates for best answer, greatly increase the value of related OF
and thus lead to exclusion of inappropriate candidate from
if V min 6 V b 6 V max
ð20Þ
ð21Þ
where Ij is the current magnitude flows in jth line and Imax is the
maximum current that can flow in the network lines, k and l are
small positive constants, and Vb is the voltage magnitude of the
bth bus. If all lines currents are less than Imax, OC will then be equal
to 1, and if all buses voltages are within the desired boundaries, OV
would equal unity. In all other conditions, OC or OV will acquire a
value (larger than unity) that acts as a penalty factor in OF.
Total cost saving (TCS) is the difference between total energy
loss cost before installation and total energy loss cost and annual
cost of DSTATCOM after installation in three load levels and is given by Eq. (22).
TCS ¼ K e
3
3
X
X
T i Plossi K e T i PWith
lossi
i¼1
ð18Þ
1;
expðlj1 V b jÞ; otherwise
ð17Þ
The power losses in ith load level (P lossi ), can be described as Eq. (18)
[53].
ð19Þ
where CostDSTATCOMyeari is the annual cost of DSTATCOM in ith load
level and CostDSTATCOMi is the cost of investment in the year of allocation in ith load level. nDSTATCOM is the longevity of DSTATCOM and B
is the asset rate of return.
Minimizing the deviation of node’s voltage and line’s current
are formulated in the second term of OF. OC denotes line over current factor and OV denotes voltage stability index for buses [55]
defined as
3.1. Objective function
The proposed OF has been mathematically formulated as
expressed by Eq. (16)
ð1 þ BÞnDSTATCOM B
:
ð1 þ BÞnDSTATCOM 1
DSTATCOM
i¼1
3
X
K ci CostDSTATCOMyeari
ð22Þ
i¼1
4. Immune algorithm
Artificial immune systems (AIS), have been defined as ‘‘adaptive
systems, inspired by theoretical immunology and observed immune functions, principles and models, which are applied to problem solving’’ [56,57].
The immune system is considered to provide both defense and
maintenance of the body. When an animal is exposed to an Ag (foreign invader) (Ag), some of it’s bone marrow derived cells respond
by producing antibodies (Ab) [19]. The immune system has a fundamental ability to produce new types of Ab or find the best-fitting
Ab to attack an invading Ag [36]. Many properties of the immune
system (IS) are of great interest for computer scientists and engineers, such as: uniqueness, recognition of foreigners, anomaly
detection, distributed detection, imperfect detection (noise tolerance), and reinforcement learning and memory [58]. Immune
response reflects on how antibodies learn antigenic structure pattern and eliminate Ags ultimately [59]. Pattern recognition is carried out by white blood cells (lymphocytes) of two types: B-cells,
and T-cells. Both B-cells and T-cells have receptors, present on
their surfaces that are responsible for recognition antigenic
patterns. However, B- and T-cells detect different features of
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S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
pathogens [60], and function quite differently: B-cells can recognize isolated Ags from outside the Ag-cell, while T-cells operate
on Ag-cell complex and can recognize parts of the complex presented by organic molecules [20].
The size of subpopulations of these cells is controlled by a process termed clonal selection. After successful recognition, cells
with capability of binding with non-self Ags are cloned. The elements of this subpopulation also undergo mutations resulting in
a subpopulation of cells that are slightly different. In addition, in
thymus the cells which recognize self-Ags are removed (negative
selection) [20]. The Ag can be regarded as a problem to be solved
and the Ab a solution vector that best fits to solve the problem.
In this article, an objective function is defined aiming to be optimized in its minimum value. This OF is however assigned to the
IA as an Ag and IA calculates the optimized OF as an Ab to fit the
problem as best as possible.
Start
Input Data
(Generating first random
Population)
(Location and Three Desired
Voltage)
Calculate
population affinity
(Run Load Flow and
Calculating OF.)
iteration
< max
iteration
N
Output
(Best Population With
Minimum OF.)
End
4.1. Initialize repertoire
Y
An Ag or an Ab, can be represented as an attribute string m [61].
At first a population of Abs is randomly selected (strings of real or
binary forms) as candidate for the solution result.
Random
value
> Pr
4.2. Evaluation
Random
value
> Affinity
(Abi)
N
Y
Degree of binding between Ag and Ab string can be described as
affinity. In this paper the inverse of Objective Function (OF) is defined as affinity. Greater affinity means the better Ab for solving the
problem and minimizing OF [20].
Y
Replace
Antibody by
maximum affinity in
new population
Replace this
Antibody in
new population
4.3. Adaptation
In this section, n attribute strings with highest affinity are selected to proliferate by cloning, hyper-mutation and replacing.
Three variables are tuned for these purposes: probability of
replacement Pr, probability of cloning Pc and probability of mutation Pm which are defined as constant values in IA. For any Ab (i),
an affinity value (affinity (Abi)) is defined, equal to inverse of OF.
Any Ab with higher affinity (i.e. lower OF) is a better candidate to
remain in repertoire.
A random number (e [0, 1]) is generated and compared to Pr. If
rand 6 Pr, hence a new Ab is generated and placed in new repertoire. Else if rand > Pr, an Ab from the current repertoire is selected
for cloning. If randnew 6 affinity(Abi), the Ab is cloned and put into
the new generation with probability of cloning Pc. If randnew 6 affinity(Abi) but Abi has not been cloned due to the stochastic
character of the cloning process, the Ab is submitted to hyper
mutation. In hyper-mutation each attribute of Ab string replaced
with a new randomly-generated value by the probability of mutation (Pm) [20].
This procedure is repeated until a complete new repertoire has
been created. In this way high-affinity (low OF) Abs can be cloned
and/or mutated several times and low affinity (high OF) Abs can be
skipped several times and be implicitly replaced by newly generated ones. The algorithm proceeds iteratively until a stopping criterion is met. Fig. 6 shows the flowchart of immune algorithm.
Artificial immune systems provide superior performances in
dealing with multi-modal, combinatorial, time dependent, and
inventory optimization problems [62]. According to [17,18] the
capability of IA method for pattern recognition and memorization
does provide a more efficient way to solve the optimization problem as compared to the GA. In [17] by comparing the proposed IA
with the standard EA, is stated that IA (clonal selection) can reach
a diverse set of local optima solutions, while the conventional EA
tends to bias the whole population of individuals toward the best
candidate solution. Essentially, their encoding schemes and
N
Y
Random
value
> Pc
N
Mutate this
Antibody
Y
Random
value >
[Pm/NormAffi,1]
N
N
Complete
new
population
Y
Fig. 6. Flowchart of immune algorithm.
evaluation functions are similar, but their evolutionary search processes differ from the viewpoint of inspiration, vocabulary, and sequence of steps. It is also remarkable to note that the IA (clonal
form) version for optimization implicitly accounts for the search
of multiple solutions. Thus, its computational cost is reduced when
compared to that has to explicitly evaluate the degree of similarity
among the individuals of the population [17]. Another important
aspect of IA model, when compared with the standard EA, is the fact
that the IA takes into account the cell affinity, corresponding to the
fitness of the individuals, in order to define the proliferation and
mutation rates to be applied to each member of the population. It
is shown in [63] that IA is superior to the other algorithms in most
of cases except smooth shaped functions and IA will be useful for
wide optimization aspects. Both IA and GA are immanent in parallel
action and being random and both of them have the ability to escape from local optima [64]. The learning process of IA is implemented by an evolutionary process which is similar to GA.
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S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Here, at first a random population, each consisting location and
three desired voltages (for three load levels) is produced and assigned to IA as the first repertoire. IA, then runs load flow for each
Ab to evaluate the OF and associated affinity of each Ab candidate
within the population (repertoire). IA will finally run the internal
calculation to obtain the best Ab defined by its minimum OF (or
maximum affinity) based on the presented flowchart (see Fig. 6).
5. Implementation of IA for finding optimal location and size of
DSTATCOM
The purpose of the IA in this paper is to determine the size of
DSTATCOM at the candidate locations for each load level in order
to minimize power loss, size, cost of DSTATCOM, and deviation of
voltages/currents from the desired values. Hence, DSTATCOM is located and tuned to achieve the minimum OF. In other words, attempts were made to decrease the total power loss and size of
installing DSTATCOM and maintaining voltages and currents of
network in desired boundaries as much as possible.
For examine applicability of the proposed approach, it was applied to two standard sample distribution networks. Examined
IEEE networks have 33 and 69 bus that work at 12.66 kV and have
radial structure [50,51]. The single line diagrams are shown in
Figs. 7 and 8, respectively. Line data and nominal load data of study
systems are given in Appendix A.
In order to model the annual load profile, three load levels are
selected (Light, Medium and Peak). Table 1 shows the time duration and total load for each load level.
In order to evaluate the effectiveness of IA, its performance is
compared with GA, run on the same basis. To find the optimum
set of parameters for IA (and GA), 50 trials are performed for each
possible set of parameters. For each trial, the optimum OF is recorded and the appropriate statistical measures are compared in
continue.
Maximum iteration and initial population size are set to 100
and 50, respectively. Initial binary strings (chromosomes) are randomly produced, containing a number of selected bus for compensation as well as three values for voltages of compensated node in
three load levels. The IA and GA settings are shown in Table 2.
After running the load flow in three levels, OF is calculated for
each chromosome. The OF parameters applied are indicated in
Table 3 [65,66].
It should be noted that the constraint of injected reactive power
by DSTATCOM, voltage of buses and current of lines are as
indicated below in steady state:
Ar
kV Ar
0 6 Q kV
DSTATCOM 6 10; 000
ð23Þ
0:9pu 6 V pu 6 1:1pu
ð24Þ
Table 2
Parameters setting of the IA and GA methods.
IA
GA
Fig. 7. Single line diagram of IEEE 33-bus distribution system.
Pr
Pc
Pm
Mutation rate
Selection rate
0.3
–
0.9
–
0.1
–
–
0.3
–
0.5
Fig. 8. Single line diagram of IEEE 69-bus distribution system.
Table 1
Load level and load duration time in 33 and 69 bus IEEE test systems.
Load level
Light
Medium
Peak
Duration (h)
Total load (kV Ar)
2000
3715 + j2300
3802.2 + j2694.6
202.68
225
5260
4829.5 + j2300
4942.8 + j2694.6
305.86
342.99
1500
5944 + j2300
6083.5 + j2694.6
442.41
502.52
Total power loss (kW)
33-bus
69-bus
33-bus
69-bus
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S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Table 3
OF parameters setting for optimization.
CostDSTATCOM US($/kV Ar)
nDSTATCOM (year)
B
Ke US ($/kW h)
k
l
50
30
0.1
0.06
2
1
IMax 6 520A
ð25Þ
6. Simulation results
The effectiveness of proposed approach is illustrated using IEEE
33-bus, and IEEE 69-bus test systems. It is tried to obtain the optimal solution using the objective function given by Eq. (16) in two
sample networks; the simulation results are described in the following sections.
6.1. IEEE 33-bus system
Table 4 shows the placement and size of DSTATCOM, and also
the minimum objective function using IA and GA methods in 33
buses distribution system in three load levels.
As can be seen, compared with GA, IA offers an improved optimal solution with its lower size of DSTATCOM and lower OF. In this
system, the 12th bus is selected for DSTACOM installation by both
IA and GA methods for compensating in light, medium and heavy
load levels. It is worth noting that the OF value is reduced by
22.5% and size of DSTATCOM is also reduced by at least 13.62%
in three load levels. For IA it is multi-focused and can implement
Table 4
Comparison results of OF value, optimal location and sizing of DSTATCOM in IEEE 33bus test system.
OF
IA
GA
IA
GA
Location
Size (kV Ar)
IA
GA
2.5
2.4
Light
962.49
1114.21
149149.1
192456.04
12
12
Medium
1008.18
1376.97
Peak
1222.66
1845.48
Kci
Load level
1
2
3
0.22831
0.60046
0.17123
search for multiple optimal points with certain decentralization
and independence concerning for its search objectives so IA covers
global and local search simultaneously. Meanwhile, GA focuses on
one optimal solution due to singleness and exclusion concerning
its search objective and emphasizes global search while ignores local search [64].
For IA, it is running based on memory units and guarantees its
convergence whereas in GA, it is running based on parent population and in some cases standard GA cannot guarantee its convergence [64]. The proposed method has been implemented on a
quad computer with 2.8 GHz CPU. The convergence curves for
the OF obtained by GA and IA algorithms in the IEEE 33-bus network are presented in Fig. 9. As shown after 50th iteration, solution
convergence takes place in IA quicker than that of GA, even after
75th iteration. Also, IA has reached a better answer (i.e. lower OF
and nearer to the global optima) compared with GA. For IA, it
incarnates the self-adjustment of immune response by accelerating and suppressing the generation of antibodies and so it can
guarantee the diversity of individuals. While for GA, it only chooses
individuals from parent generation according to fitness and does
not adjust the diversity of individuals [64]. Table 5 shows CPU time
for each algorithm. In this optimization, CPU time is reduced by
12.2% using IA compared with GA method. Therefore, optimal solution found by the IA was better than the GA method.
Table 6 presents comparison of power loss, annual cost of
DSTATCOM, no. of under voltage buses and no. of over current lines
pre and post installation using proposed method in three different
load levels, in 33 buses distribution system.
Table 5
CPU time for optimization in IEEE 33-bus test system.
Algorithm
IA
GA
CPU time (s)
21,220
24,157
x 106
OF Convergence by IA (33 bus)
OF Convergence by GA (33 bus)
2.2
2
1.8
1.6
OF
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
Iteration
Fig. 9. Comparison of convergence between GA and IA.
41
S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Results indicate a power loss reduction in all three load levels,
and in general, the total power loss is reduced by 10.9%, while all
voltages and currents are within the desired limits.
Table 7 presents the annual results of economic evaluations. As
can be seen, total cost saving is around 115,209 ($).
Voltage of buses pre and post DSTATCOM installation using proposed method are presented in Table 8. It can be seen that all voltages are within the desired limits (Vmin = 0.9 pu and Vmax = 1 pu).
Table 6
Summary results of IEEE 33-bus test system.
Load level
Total power loss (kW)
CostDSTATCOM year ($)
No. of under voltage buses
No. of over current lines
B.I.
A.I.
A.I.
B.I.
A.I.
B.I.
A.I.
Light
Medium
Peak
202.68
171.81
5105.1
0
0
0
0
305.86
272.04
5347.4
7
0
0
0
442.41
407.71
6485
14
0
2
0
B.I., Before installation; A.I., After installation.
Table 7
Comparison of annual costs of IEEE 33-bus test system.
Total
Total
Total
Total
energy loss cost before installation ($)
energy loss cost after installation ($)
annual cost of DSTATCOM ($)
cost saving ($)
160,670
143,160
5989.1
115,209
6.2. IEEE 69-bus system
Table 9 shows the placement and size of DSTATCOM, as well as
the minimum objective function using IA and GA methods in 69
buses system in three load levels.
Similar to the first case study, IA offers better optimal solution
here (i.e. lower DSTATCOM size and lower OF) compared to GA.
In this 69 buses distribution system, the 61st bus is selected for
DSTATCOM installation by both IA and GA methods for compensating in light, medium and heavy load levels. OF value is evidently
reduced by 6.54% and size of DSTATCOM is also reduced by at least
9.58% in three load levels. For IA, it often integrates affinity with
concentration to evaluate the quality of an individual, thus reflects
the diversity of real immune system. While for GA, it simply uses
fitness as the only standard to evaluate the quality of an individual
[64].
Table 10 shows CPU time for each algorithm. It can be seen that
IA has a faster convergence than GA and optimal solution found by
the IA is better than the GA method.
Table 11 shows the results for the 69 buses distribution system.
Minimum voltage and maximum current of the system has
improved and the system losses are reduced in each load level.
Generally total power loss is reduced by 18%.
Annual results of economic evaluation for the 69 bus system, is
presented in Table 12 with total cost saving of 21,972 ($).
Table 9
Comparison results of OF value, optimal location and sizing of DSTATCOM in IEEE 69bus test system.
OF
IA
GA
IA
GA
Location
Table 8
Voltages of 33 bus distribution network before and after DSTATCOM installation.
Medium
Size (kV Ar)
IA
GA
Bus no.
Light
Peak
B.I.
A.I.
B.I.
A.I.
B.I.
A.I.
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
1.0000
0.9970
0.9829
0.9754
0.9680
0.9496
0.9461
0.9413
0.9350
0.9292
0.9283
0.9268
0.9207
0.9185
0.9170
0.9157
0.9136
0.9130
0.9965
0.9929
0.9922
0.9915
0.9793
0.9726
0.9693
0.9477
0.9451
0.9337
0.9255
0.9219
0.9177
0.9168
0.9165
1.0000
0.9973
0.9848
0.9786
0.9724
0.9585
0.9590
0.9556
0.9541
0.9530
0.9526
0.9519
0.9460
0.9438
0.9424
0.9411
0.9391
0.9385
0.9968
0.9932
0.9925
0.9918
0.9812
0.9746
0.9713
0.9566
0.9541
0.9428
0.9346
0.9311
0.9270
0.9261
0.9258
1.0000
0.9963
0.9787
0.9695
0.9603
0.9381
0.9341
0.9279
0.9200
0.9127
0.9116
0.9097
0.9021
0.8993
0.8975
0.8958
0.8933
0.8925
0.9956
0.9913
0.9904
0.9897
0.9743
0.9661
0.9620
0.9357
0.9326
0.9192
0.9096
0.9053
0.9002
0.8991
0.8988
1.0000
0.9966
0.9808
0.9728
0.9649
0.9475
0.9477
0.9431
0.9403
0.9381
0.9375
0.9364
0.9290
0.9263
0.9246
0.9229
0.9205
0.9198
0.9959
0.9916
0.9907
0.9900
0.9764
0.9682
0.9641
0.9452
0.9421
0.9289
0.9194
0.9152
0.9101
0.9090
0.9087
1.0000
0.9955
0.9744
0.9633
0.9522
0.9260
0.9215
0.9138
0.9043
0.8955
0.8941
0.8917
0.8825
0.8792
0.8770
0.8749
0.8720
0.8711
0.9948
0.9896
0.9886
0.9878
0.9692
0.9594
0.9546
0.9232
0.9195
0.9041
0.8930
0.8880
0.8819
0.8806
0.8802
1.0000
0.9959
0.9769
0.9673
0.9578
0.9375
0.9380
0.9324
0.9290
0.9265
0.9257
0.9245
0.9156
0.9124
0.9103
0.9083
0.9054
0.9045
0.9951
0.9900
0.9890
0.9881
0.9716
0.9619
0.9571
0.9348
0.9311
0.9159
0.9050
0.9001
0.9000
0.9000
0.9000
158500.15
169600.36
61
61
Light
1704.42
1918.39
Medium
1911.23
2223.28
Peak
2606.83
2883.00
Table 10
CPU time for optimization in IEEE 69-bus test system.
Algorithm
IA
GA
CPU time (s)
32,305
45,588
Table 11
Summary results of IEEE 69-bus test system.
Load level
Total power loss (kW)
CostDSTATCOM year ($)
No. of under voltage buses
No. of over current lines
B.I.
A.I.
A.I.
B.I.
A.I.
B.I.
A.I.
Light
Nominal
Peak
225
157.50
9040.2
0
0
0
0
343
274.4
10137.1
6
0
0
0
502.5
472
13826.5
8
0
4
0
B.I., Before installation; A.I., After installation.
Table 12
Comparison of annual costs of IEEE 69-bus test system.
Total
Total
Total
Total
energy loss cost before installation ($)
energy loss cost after installation ($)
annual cost of DSTATCOM ($)
cost saving ($)
180,470
147,980
10,518
21,972
42
S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Table 13 shows voltage of buses pre and post installation of
DSTATCOM in each load level in 69-bus distribution network, in
which all voltages are again within the upper and lower bounds.
Table 13
Voltages of 69 bus distribution network before and after DSTATCOM installation.
Bus no.
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
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
Light
Medium
Peak
B.I.
A.I.
B.I.
A.I.
B.I.
A.I.
1.0000
0.9999
0.9999
0.9998
0.9990
0.9900
0.9807
0.9785
0.9774
0.9724
0.9713
0.9681
0.9652
0.9623
0.9594
0.9589
0.9580
0.9580
0.9575
0.9572
0.9568
0.9568
0.9567
0.9565
0.9564
0.9563
0.9563
0.9999
0.9998
0.9997
0.9997
0.9996
0.9993
0.9990
0.9989
0.9999
0.9997
0.9995
0.9995
0.9997
0.9990
0.9987
0.9987
0.9986
0.9985
0.9985
0.9997
0.9985
0.9946
0.9941
0.9785
0.9785
0.9746
0.9714
0.9669
0.9625
0.9400
0.9290
0.9247
0.9197
0.9123
0.9120
0.9116
0.9097
0.9091
0.9712
0.9712
0.9678
0.9678
1.0000
0.9999
0.9999
0.9999
0.9994
0.9926
0.9856
0.9839
0.9831
0.9781
0.9770
0.9739
0.9710
0.9681
0.9652
0.9647
0.9638
0.9638
0.9634
0.9631
0.9626
0.9626
0.9625
0.9623
0.9622
0.9621
0.9621
0.9999
0.9998
0.9997
0.9997
0.9996
0.9993
0.9990
0.9989
0.9999
0.9997
0.9996
0.9995
0.9997
0.9990
0.9987
0.9987
0.9987
0.9986
0.9986
0.9998
0.9986
0.9947
0.9942
0.9839
0.9839
0.9813
0.9793
0.9765
0.9738
0.9579
0.9500
0.9470
0.9434
0.9392
0.9389
0.9385
0.9367
0.9361
0.9770
0.9770
0.9735
0.9735
1.0000
0.9999
0.9999
0.9998
0.9988
0.9877
0.9762
0.9734
0.9720
0.9657
0.9643
0.9604
0.9567
0.9530
0.9494
0.9487
0.9476
0.9476
0.9470
0.9467
0.9461
0.9460
0.9460
0.9458
0.9455
0.9455
0.9454
0.9999
0.9998
0.9996
0.9996
0.9995
0.9992
0.9987
0.9987
0.9999
0.9997
0.9995
0.9994
0.9996
0.9988
0.9985
0.9984
0.9984
0.9983
0.9983
0.9997
0.9983
0.9941
0.9935
0.9734
0.9734
0.9685
0.9645
0.9589
0.9534
0.9249
0.9108
0.9054
0.8989
0.8897
0.8893
0.8888
0.8865
0.8857
0.9643
0.9643
0.9600
0.9600
1.0000
0.9999
0.9999
0.9998
0.9992
0.9906
0.9816
0.9794
0.9784
0.9721
0.9708
0.9668
0.9631
0.9595
0.9559
0.9553
0.9542
0.9541
0.9536
0.9532
0.9526
0.9526
0.9525
0.9523
0.9521
0.9520
0.9520
0.9999
0.9998
0.9997
0.9996
0.9995
0.9992
0.9988
0.9987
0.9999
0.9997
0.9995
0.9995
0.9997
0.9988
0.9985
0.9985
0.9985
0.9983
0.9983
0.9998
0.9984
0.9941
0.9935
0.9794
0.9794
0.9760
0.9734
0.9697
0.9661
0.9450
0.9347
0.9307
0.9260
0.9204
0.9200
0.9195
0.9172
0.9165
0.9707
0.9707
0.9664
0.9664
1.0000
0.9999
0.9999
0.9997
0.9986
0.9852
0.9713
0.9680
0.9663
0.9587
0.9571
0.9523
0.9478
0.9434
0.9390
0.9382
0.9368
0.9368
0.9361
0.9357
0.9349
0.9349
0.9348
0.9346
0.9343
0.9342
0.9342
0.9999
0.9998
0.9996
0.9996
0.9994
0.9990
0.9985
0.9984
0.9998
0.9996
0.9994
0.9994
0.9996
0.9987
0.9983
0.9982
0.9982
0.9981
0.9981
0.9997
0.9982
0.9935
0.9928
0.9680
0.9680
0.9621
0.9572
0.9504
0.9438
0.9088
0.8915
0.8848
0.8769
0.8657
0.8653
0.8647
0.8618
0.8609
0.9570
0.9570
0.9518
0.9518
1.0000
0.9999
0.9999
0.9998
0.9992
0.9890
0.9784
0.9759
0.9746
0.9671
0.9655
0.9607
0.9563
0.9519
0.9476
0.9467
0.9454
0.9454
0.9447
0.9442
0.9435
0.9435
0.9434
0.9432
0.9429
0.9428
0.9428
0.9999
0.9998
0.9996
0.9996
0.9994
0.9991
0.9986
0.9985
0.9999
0.9997
0.9995
0.9994
0.9996
0.9987
0.9983
0.9983
0.9982
0.9981
0.9981
0.9998
0.9983
0.9936
0.9929
0.9758
0.9758
0.9719
0.9688
0.9646
0.9605
0.9355
0.9233
0.9187
0.9131
0.9070
0.9066
0.9060
0.9033
0.9024
0.9654
0.9654
0.9602
0.9602
7. Conclusion
This paper presents a new approach for optimization problem
of DSTATCOM placement and sizing in radial distribution systems using IA method for obtaining minimum power loss and
minimizing cost of DSTATCOM installation with pre-determined
voltage and current constraints in network. Energy and power
losses due to installed DSTATCOM as well as its associated cost
are used to define the objective function. For system solution,
backward/forward sweep load flow is applied. Simulation results
show that utilizing DSTATCOM reduce objective function. Using
IA method, the optimal location and size of DSTATCOM is obtained in order to decrease power loss, cost of DSTATCOM and
current profile, and improve voltage of buses. Compared with
GA, IA technique provides minimum DSTATCOM size, CPU time,
and objective function. Installation of DSTATCOM by the proposed approach leads to 10.9% and 18% power loss reductions,
in 33 and 69 buses distribution systems, respectively. All buses
voltage and current of lines are within the desired boundaries.
Total energy loss cost reduction were 10.89% and 18% in 33
and 69 buses distribution network, respectively. Total cost saving
as a result of this exercise are estimated to be of the order of
7.17% and 12.17%, in 33 and 69 buses distribution network,
respectively.
Appendix A
Tables A1 and A2 show the line and bus data in IEEE-33 and 69
bus distribution networks in light loading.
Table A1
IEEE 33-bus distribution network data.
Bus
Send
Receive
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
2
19
20
21
3
23
24
6
26
27
28
29
30
31
32
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
P kW
Receive
Ar
Q kV
Receive
R (X)
X (X)
100
90
120
60
60
200
200
60
60
45
60
60
120
60
60
60
90
90
90
90
90
90
420
420
60
60
60
120
200
150
210
60
60
40
80
30
20
100
100
20
20
30
35
35
80
10
20
20
40
40
40
40
40
50
200
200
25
25
20
70
600
70
100
40
0.0922
0.4930
0.3660
0.3811
0.8190
0.1872
0.7114
1.0300
1.0440
0.1966
0.3744
1.4680
0.5416
0.5910
0.7463
1.2890
0.7320
0.1640
1.5042
0.4095
0.7089
0.4512
0.8980
0.8960
0.2030
0.2842
1.0590
0.8042
0.5075
0.9744
0.3105
0.3410
0.0470
0.2511
0.1864
0.1941
0.7070
0.6188
0.2351
0.7400
0.7400
0.0650
0.1238
1.1550
0.7129
0.5260
0.5450
1.7210
0.5740
0.1565
1.3554
0.4784
0.9373
0.3083
0.7091
0.7011
0.1034
0.1447
0.9337
0.7006
0.2585
0.9630
0.3619
0.5302
S.A. Taher, S.A. Afsari / Electrical Power and Energy Systems 60 (2014) 34–44
Table A2
IEEE 69-bus distribution network data.
Bus
Send
Receive
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
3
28
29
30
31
32
33
34
3
36
37
38
30
40
41
42
43
44
45
4
47
48
49
8
51
9
53
54
55
56
57
58
59
60
61
62
63
64
11
66
12
68
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
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
43
References
P kW
Receive
Ar
Q kV
Receive
R (X)
X (X)
0
0
0
0
2.6
40.4
75
30
28
145
145
8
8
0
45.5
60
60
0
1
114
5.3
0
28
0
14
14
26
26
0
0
0
14
19.5
6
26
26
0
24
24
1.2
0
6
0
39.22
39.22
0
79
384.7
384.7
40.5
3.6
4.35
26.4
24
0
0
0
100
0
1244
32
0
227
59
18
18
28
28
0
0
0
0
2.2
30
54
22
19
104
104
5.5
5.5
0
30
35
35
0
.6
81
3.5
0
20
0
10
10
18.6
18.6
0
0
0
10
14
4
18.55
18.55
0
17
17
1
0
4.3
0
26.3
26.3
0
56.4
274.5
274.5
28.3
2.7
3.5
19
17.2
0
0
0
72
0
888
23
0
162
42
13
13
20
20
0.0005
0.0005
0.0015
0.0251
0.3660
0.3811
0.0922
0.0493
0.8190
0.1872
0.7114
1.0300
1.0440
1.0580
0.1966
0.3744
0.0047
0.3276
0.2106
0.3416
0.0140
0.1591
0.3463
0.7488
0.3089
0.1732
0.0044
0.0640
0.3978
0.0702
0.3510
0.8390
1.7080
1.4740
0.0044
0.0640
0.1053
0.0304
0.0018
0.7283
0.3100
0.0410
0.0092
0.1089
0.0009
0.0034
0.0851
0.2898
0.0822
0.0928
0.3319
0.1740
0.2030
0.2842
0.2813
1.5900
0.7837
0.3042
0.3861
0.5075
0.0974
0.1450
0.7105
1.0410
0.2012
0.0047
0.7394
0.0047
0.0012
0.0012
0.0036
0.0294
0.1864
0.1941
0.0470
0.0251
0.2707
0.0619
0.2351
0.3400
0.3450
0.3496
0.0650
0.1238
0.0016
0.1083
0.0696
0.1129
0.0046
0.0526
0.1145
0.2475
0.1021
0.0572
0.0108
0.1565
0.1315
0.0232
0.1160
0.2816
0.5646
0.4873
0.0108
0.1565
0.1230
0.0355
0.0021
0.8509
0.3623
0.0478
0.0116
0.1373
0.0012
0.0084
0.2083
0.7091
0.2011
0.0473
0.1114
0.0886
0.1034
0.1447
0.1433
0.5337
0.2630
0.1006
0.1172
0.2585
0.0496
0.0738
0.3619
0.5302
0.0611
0.0014
0.2444
0.0016
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