A chaotic particle swarm optimization (cpso) algorithm for solving optimal reactive power dispatch problem

Industrial Engineering Letters www.iiste.org
ISSN 2224-6096 (Paper) ISSN 2225-0581 (online)
Vol.4, No.3, 2014
11
A Chaotic Particle Swarm Optimization (CPSO) Algorithm for
Solving Optimal Reactive Power Dispatch Problem
K. Lenin1*
, Dr.B.Ravindranath Reddy2
, Dr.M.Surya Kalavathi3
1.Research Scholar, Jawaharlal Nehru Technological University Kukatpally, Hyderabad 500 085, India
2.Deputy Executive Engineer, Jawaharlal Nehru Technological University Kukatpally, Hyderabad 500 085, India
3.Professor of Electrical and Electronics Engineering, Jawaharlal Nehru Technological University
Kukatpally,Hyderabad 500 085, India
* E-mail of the corresponding author: gklenin@gmail.com
Abstract
This paper presents a chaotic particle swarm algorithm for solving the multi-objective reactive power dispatch
problem. To deal with reactive power optimization problem, a chaotic particle swarm optimization (CPSO) is
presented to avoid the premature convergence. By fusing with the ergodic and stochastic chaos, the novel
algorithm explores the global optimum with the comprehensive learning strategy. The chaotic searching region
can be adjusted adaptively. In order to evaluate the proposed algorithm, it has been tested on IEEE 30 bus
system and simulation results show that (CPSO) is more efficient than other algorithms in reducing the real
power loss and maximization of voltage stability index.
Keywords:chaotic particle swarm optimization, Optimization, Swarm Intelligence, optimal reactive power,
Transmission loss.
1.Introduction
One of the major problems faced by power system operators is the reactive power dispatch imposed on electric
power utilities for a continuous and reliable supply of energy. Major power loads require a significant amount of
reactive power that has to be supplied while maintaining load bus voltages within their permissible operating
limits. In order to maintain desired levels of voltages and reactive flows under various operating conditions and
system configurations, power system operators may utilize a number of control tools such as switching var
sources, changing generator voltages, and by adjusting transformer tap settings. By an optimal adjustment of
these controls, the redistribution of the reactive power would minimize transmission losses.Various
mathematical techniques have been adopted to solve this optimal reactive power dispatch problem. These
include the gradient method (O.Alsac et al.1973; Lee K Yet al.1985), Newton method (A.Monticelli et al.1987)
and linear programming (Deeb Net al.1990; E. Hobson1980; K.Y Lee et al.1985; M.K. Mangoli 1993) .The
gradient and Newton methods suffer from the difficulty in handling inequality constraints. To apply linear
programming, the input- output function is to be expressed as a set of linear functions which may lead to loss of
accuracy. Recently Global Optimization techniques such as genetic algorithms have been proposed to solve the
reactive power flow problem (S.R.Paranjothi et al 2002;D. Devaraj et al 2005) . In recent years, the problem of
voltage stability and voltage collapse has become a major concern in power system planning and operation. To
enhance the voltage stability, voltage magnitudes alone will not be a reliable indicator of how far an operating
point is from the collapse point (C.A. Canizareset al.1996). The reactive power support and voltage problems are
intrinsically related. Hence, this paper formulates the reactive power dispatch as a multi-objective optimization
problem with loss minimization and maximization of static voltage stability margin (SVSM) as the objectives.
Voltage stability evaluation using modal analysis is used as the indicator of voltage stability. In recent years,
several new optimization techniques have emerged. The evolutionary algorithms (EAs) for reactive power
optimization problem have been extensively studied. Several global optimization algorithms such as differential
evolution (DE) (Dib.N et al .2010; Lin. C et al.2010), genetic algorithm (GA) (Zhang et al.2009 ;
Vaitheeswaran.S et al.2008) , simulated annealing (SA) (Ferreira, J. A et al.1997 ), Ant colony optimization
(ACO) (Hosseini.S. A et al 2008 ), particle swarm optimization (PSO) (Perez Lopez et al.2009; Liu, D et al.2009;
Li, W.-T et al.2010 ; Goudos, S et al.2010 ; Shavit, R et al.2005) are used for reactive power optimization
problem. However, these methods present certain drawbacks with the possibility of premature convergence to a
local optimum. In this paper, a novel chaotic PSO algorithm (CPSO) is proposed. Based on the ergodicity,
regularity and pseudo-randomness of the Chaotic variable, chaotic search is used to explore better solutions. The
performance of (CPSO) has been evaluated in standard IEEE 30 bus test system and the results analysis shows
that our proposed approach outperforms all approaches investigated in this paper.
2. Voltage Stability Evaluation
2.1 Modal analysis for voltage stability evaluation
Modal analysis is one of the methods for voltage stability enhancement in power systems. The linearized
steady state system power flow equations are given by.

Vol.4, No.3, 2014
12
∆P
∆Q
=
J J
J J
(1)
Where
∆P = Incremental change in bus real power.
∆Q = Incremental change in bus reactive
Power injection
∆θ = incremental change in bus voltage angle.
∆V = Incremental change in bus voltage Magnitude
Jpθ , J PV , J Qθ , J QV jacobian matrix are the sub-matrixes of the System voltage stability is affected by
both P and Q. However at each operating point we keep P constant and evaluate voltage stability by
considering incremental relationship between Q and V.
To reduce (1), let ∆P = 0 , then.
∆Q = J − J J J ∆V = J ∆V (2)
∆V = J − ∆Q (3)
Where
J = J − J J JPV (4)
J is called the reduced Jacobian matrix of the system.
2.2 Modes of Voltage instability
Voltage Stability characteristics of the system can be identified by computing the eigen values and eigen
vectors
Let
J = ξ˄η (5)
Where,
ξ = right eigenvector matrix of JR
η = left eigenvector matrix of JR
∧ = diagonal eigenvalue matrix of JR and
J = ξ˄ η (6)
From (3) and (6), we have
∆V = ξ˄ η∆Q (7)
or
∆V = ∑
!
"# ∆Q (8)
Where ξi is the ith column right eigenvector and η the ith row left eigenvector of JR.
λi is the ith eigen value of JR.
The ith modal reactive power variation is,
∆Q$% = K%ξ% (9)
where,
K% = ∑ ξ%'(' − 1 (10)
Where
ξji is the jth element of ξi
The corresponding ith modal voltage variation is
∆V$% = *1 λ%⁄ -∆Q$% (11)
It is seen that, when the reactive power variation is along the direction of ξi the corresponding voltage
variation is also along the same direction and magnitude is amplified by a factor which is equal to the magnitude
of the inverse of the ith eigenvalue. In this sense, the magnitude of each eigenvalue λi determines the weakness
of the corresponding modal voltage. The smaller the magnitude of λi, the weaker will be the
corresponding modal voltage. If | λi | =0 the ith modal voltage will collapse because any change in that
modal reactive power will cause infinite modal voltage variation.
In (8), let ∆Q = ek where ek has all its elements zero except the kth one being 1. Then,
∆V = ∑
ƞ . ξ
λ% (12)
ƞ /
k th element of ƞ
V –Q sensitivity at bus k
0 1
0 1
= ∑
ƞ . ξ
λ% = ∑ .
λ% (13)
3.Problem Formulation
The objectives of the reactive power dispatch problem considered here is to minimize the system real power loss
and maximize the static voltage stability margins (SVSM).

Vol.4, No.3, 2014
13
3.1Minimization of Real Power Loss
It is aimed in this objective that minimizing of the real power loss (Ploss) in transmission lines of a power
system. This is mathematically stated as follows.
P23445 ∑ g/( (
8 9
(
: 9 ;<= θ 9
)
?
/5
/5(%,')
(14)
Where n is the number of transmission lines, gk is the conductance of branch k, Vi and Vj are voltage
magnitude at bus i and bus j, and θij is the voltage angle difference between bus i and bus j.
3.2 Minimization of Voltage Deviation
It is aimed in this objective that minimizing of the Deviations in voltage magnitudes (VD) at load buses. This
is mathematically stated as follows.
Minimize VD = ∑ |V/ − 1.0|?2
/5 (15)
Where nl is the number of load busses and Vk is the voltage magnitude at bus k.
3.3 System Constraints
Objective functions are subjected to these constraints shown below.
Load flow equality constraints:
DEF – DHF − IF ∑ JK
LM
KN
OFP cos TFP
+VFP sin TFP
= 0, Y = 1,2 … . , ] (16)
ÊF − ^HF − IF ∑ JK
LM
KN
OFP cos TFP
+VFP sin TFP
= 0, Y = 1,2 … . , ] (17)
where, nb is the number of buses, PG and QG are the real and reactive power of the generator, PD and QD are the
real and reactive load of the generator, and Gij and Bij are the mutual conductance and susceptance between bus i
and bus j.Generator bus voltage (VGi) inequality constraint:
IEF
_F`
≤ IEF ≤ IEF
_bc
, Y ∈ e (18)
Load bus voltage (VLi) inequality constraint:
IfF
_F`
≤ IfF ≤ IfF
_bc
, Y ∈ g (19)
Switchable reactive power compensations (QCi) inequality constraint:
^hF
_F`
≤ ^hF ≤ ^hF
_bc
, Y ∈ i (20)
Reactive power generation (QGi) inequality constraint:
ÊF
_F`
≤ ÊF ≤ ÊF
_bc
, Y ∈ e (21)
Transformers tap setting (Ti) inequality constraint:
jF
_F`
≤ jF ≤ jF
_bc
, Y ∈ k (22)
Transmission line flow (SLi) inequality constraint:
lfF
_F`
≤ lfF
_bc
, Y ∈ g (23)
Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.
4. Principle of CPSO
4.1Principle of CPSO
Inspired by the social behaviours of animal, bird flocking and fishing, PSO was developed by (Kennedy et al .
1995) . The particle is endowed with two factors: velocity and position which can be regarded as the potential
solution in the D dimension problem space. In basic PSO, they can be updated by following formulas:
mFn(k + 1) = omFn(k) + i p n DFn(k) − qFn(k) + i:p:n rDsn(k) − qFn(k)t (24)
qFn(k + 1) = qFn(k) + mFn(k + 1) (25)
Where i = 1,….., N, d = 1,…,D, N is the number of particles. o is the inertia weight factor to control the
exploration and exploitation. r1d and r2d are two random numbers within the range [0, 1]. vid(t) and xid(t) are the
velocity and position of the current particle i at time step t in the dth-dimensional search space respectively.
When vid(t) and xid(t) are beyond the boundary, the solution may be illegal. So, the treatment of boundaries in the
PSO method is important in order to prevent the swarm from explosion (Xu.S et al .2007 ). In many practical
problems, the search range xid is in [Xmin;Xmax]D. vid should be clamped to a maximum magnitude Vmax. pi is the
previous best position of particle I which is also called “personal best”, and its dth-dimensional part is pid. The
global best" pg is the best position found in the whole particles, and its dth-dimensional part is pgd. c1, c2 are the
acceleration constants which change the velocity of a particle towards the pi and pg.
4.2Modification Techniques in CPSO
The basic PSO uses pg as neighbourhood topology. Each particle learns from its pi and pg. Restricting the social
learning part to pg can make basic PSO converge quickly. However, because all particles in the swarm learn

Vol.4, No.3, 2014
14
from the pg even if the current pg is far from the global optimum, particles may easily be attracted to the area and
trapped in a local optimum. Furthermore, the fitness value of a particle is determined by all dimensions. A
particle that has discovered the region corresponding to the global optimum in some dimensions may have a low
fitness value because of the poor solutions in other dimensions ( Liang, J.-Jet al 2006 ) . In order to acquire more
beneficial information from the entire swarm, we define pc as “comprehensive best position”.
uv = w
∑ xy
z
yN
{
,
∑ xy(
z
yN
{
, . . ,
∑ xy|
z
yN
{
, . . ,
∑ xy}
z
yN
{
~ (26)
Where i = 1,..,N. Thus Equation (1) is modified as
mFn(k + 1) = omFn(k) + i p n DFn(k) − qFn(k) + i:p:n Dvn(k) − qFn(k) (27)
where pcd is the dth-dimensional part of pc. By using pc instead of pg, all particles' pi can potentially be used as
the exemplars to guide their flying direction. The comprehensive learning strategy yields a larger potential
search space than that of the basic PSO. On the other hand, a particle can learn from pg, as well as its personal
best and the other particles' best, so that the particle can learn from particle itself, the elite and other particles.
The strategy can increase the initial diversity and enable the swarm to overcome premature convergence problem.
Basic PSO has shown some important advances by providing high speed of convergence in specific problems.
However it does exhibit some shortages (Modares.H et al.2010). During the process of evolution, sometimes
particles lose their abilities of exploration and will be stagnated. When some particles' velocity is be close to zero,
other particles will quickly fly into the region near the inactive particles position that guided by pi and pg.
Because of the particles randomicity in initialization and evolution process, the updating sometimes looks
aimless. As a result, when pg is trapped in a local optimum, the whole swarm becomes premature convergence,
and the exploration performance will not be improved. Optimization algorithms based on the chaos theory are
stochastic search methodologies that differ from any of the existing evolutionary algorithms. Due to the non-
repetition of chaos, it can carry out overall exploration at higher velocities than stochastic and ergodic searches
that depend on probabilities (Coelho, L. Det al.2009) . Chaotic PSO can be divided into two types. In the first
type, chaos is embedded into the velocity updating equation of PSO. In (Modares.H et al.2010) , c1 and c2 are
generated from the iterations of a chaotic map instead of using the rand function. In (Wang, Y., et al. 2010 ), a
chaotic map is used to determine the value of o during iterations. In the second type, chaotic search is fused with
the procedures of PSO. This type is a kind of multi-phase optimization technique that chaotic optimization and
PSO can switch to each other according to certain conditions (Wu.Q 2011 ) . Therefore, this paper provides a
new strategy, which not only introduces chaotic mapping with certainty, ergodicity and stochastic property into
PSO algorithm, but also proposes multi-phase optimization integrated by chaotic search and PSO evolution. The
multi-phase optimization of chaotic PSO includes: vid and xid are updated by basic PSO with comprehensive
learning strategy. If the swarm is stagnated, chaotic disturbance would be introduced. Here, variance •:
demonstrates the converge degree of all particles.
•:
= ∑ €F − €b•s €⁄
:{
F5 (28)
€ = ‚ƒq „1, ‚ƒq…†€F − €b•s†‡ˆ (29)
Where fi is the fittness of the ith particle; favg is the average fitness value; f is the factor of fitness value. The
bigger •:
is the broader ith particles will spread. Otherwise, they will almost converge. The chaotic sequence
can be generated by the logistic map introduced by Robert May in 1976. It is often cited as an example of how
complex behaviour can arise from a simple dynamic system without any stochastic disturbance (He, Y.-Y., et al.
2009) The equation is the following
‰Fn(k + 1) = Š‰Fn(k) 1 − ‰Fn(k) (30)
Where ‰Fn(k) ∈ (0,1), Y = 1, . . , ‹ , Œ = 1, . . , •. Š is usually set to 4 obtain ergodicity of ‰Fn(k + 1) ŽYkℎY (0,1).
When the initial value ‰Fn(0) ∈ •0.25.0.5.0.75“ using equation (30) we can obtain chaotic sequences. In order
to increase the population diversity and prevent premature convergence, we add adaptively chaotic disturbance
Dv at the time of stagnation. Thus, Dv Y” ‚•ŒY€Y–Œ ƒ” Dv
′
.
uvn
— (k + 1) = uvn(k) + ˜Fn 2™Fn(k) − 1 (31)
˜Fn = š|uvn(k) − uFn(k)| (32)
Where š is the region scale factor. Because‰Fn ∈ (0,1), the second part of Equation (31) is in the range of
(−|˜Fn|, |˜Fn|) that would restrict the searching area around pc. In addition, the searching range can be
adaptively adjusted by the distance between pi and pc. If pc is surrounded with the previous best positions pi, it
means that a good region may have been found, and it is reasonable to search elaborately in a small area. On the
contrary, if pi is far from pc, this probably suggests that a good area has not yet been found. For better solution,

Vol.4, No.3, 2014
15
searching region should be enlarged (Lin, C.et al.2007).
Thus in CPSO, the new position can be expressed as
mFn(k + 1) = omFn(k) + i p n uFn(k) − qFn(k) + i:p:n uvn
— (k) − qFn(k) (33)
Different from Equation (24), pg is replaced by uv
—
at the time of stagnation when •:
is less than the stagnation
factor ›. Chaotic search is restricted into a small range to obtain high performance in local exploration.
Additionally, the algorithm keeps a dynamic balance between global and local searches due to its adaptive
mechanism. With the new updating rule, different exemplars are used in different dimensions to explore a larger
search space than the basic PSO. In addition, chaotic disturbance is embedded in different dimensions to
maintain the diversity which plays an important role in avoiding early convergence.
5.Simulation Results
The validity of the proposed Algorithm technique is demonstrated on IEEE-30 bus system. The IEEE-30 bus
system has 6 generator buses, 24 load buses and 41 transmission lines of which four branches are (6-9), (6-10) ,
(4-12) and (28-27) - are with the tap setting transformers. The lower voltage magnitude limits at all buses are
0.95 p.u. and the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all the PQ buses and the reference bus.
Table 1. Voltage Stability under Contingency State
Sl.No Contigency ORPD
Setting
Vscrpd
Setting
1 28-27 0.1400 0.1422
2 4-12 0.1658 0.1662
3 1-3 0.1784 0.1754
4 2-4 0.2012 0.2032
Table 2. Limit Violation Checking Of State Variables
State
variables
limits
ORPD VSCRPD
Lower upper
Q1 -20 152 1.3422 -1.3269
Q2 -20 61 8.9900 9.8232
Q5 -15 49.92 25.920 26.001
Q8 -10 63.52 38.8200 40.802
Q11 -15 42 2.9300 5.002
Q13 -15 48 8.1025 6.033
V3 0.95 1.05 1.0372 1.0392
V4 0.95 1.05 1.0307 1.0328
V6 0.95 1.05 1.0282 1.0298
V7 0.95 1.05 1.0101 1.0152
V9 0.95 1.05 1.0462 1.0412
V10 0.95 1.05 1.0482 1.0498
V12 0.95 1.05 1.0400 1.0466
V14 0.95 1.05 1.0474 1.0443
V15 0.95 1.05 1.0457 1.0413
V16 0.95 1.05 1.0426 1.0405
V17 0.95 1.05 1.0382 1.0396
V18 0.95 1.05 1.0392 1.0400
V19 0.95 1.05 1.0381 1.0394
V20 0.95 1.05 1.0112 1.0194
V21 0.95 1.05 1.0435 1.0243
V22 0.95 1.05 1.0448 1.0396
V23 0.95 1.05 1.0472 1.0372
V24 0.95 1.05 1.0484 1.0372
V25 0.95 1.05 1.0142 1.0192
V26 0.95 1.05 1.0494 1.0422
V27 0.95 1.05 1.0472 1.0452
V28 0.95 1.05 1.0243 1.0283
V29 0.95 1.05 1.0439 1.0419
V30 0.95 1.05 1.0418 1.0397
Table 3. Comparison of Real Power Loss

Vol.4, No.3, 2014
16
Method Minimum loss
Evolutionary programming[11] 5.0159
Genetic algorithm[12] 4.665
Real coded GA with Lindex as
SVSM[13]
4.568
Real coded genetic algorithm[14] 4.5015
Proposed CPSO method 4.2031
6. Conclusion
A large-scale power system should supply power to the customers in a reliable and economic way while keeping
system voltages in permissible limits. The purpose of the optimal reactive dispatch problem is to improve system
voltage profile by minimizing power system losses. In this research paper a chaotic particle swarm has been
utilized to sole the optimal reactive power dispatch problem. The performance of the proposed algorithm has
been tested in standard IEEE test system and simulation results shows the best performance of the proposed
system and the real power loss has been considerably reduced .
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