This paper proposes a hybrid particle swarm optimization (HPSO) method to solve the multi-objective reactive power dispatch problem of minimizing transmission losses and maximizing voltage stability. The HPSO method expands on PSO and genetic algorithms to handle mixed integer and nonlinear optimization problems with both continuous and discrete control variables. The method is evaluated on the IEEE 30-bus and 57-bus test systems. Results show the HPSO approach improves voltage stability compared to conventional, genetic algorithm, and particle swarm optimization methods.

1. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010
Hybrid Particle Swarm Optimization for
Multi-objective Reactive Power Optimization with
Voltage Stability Enhancement
P.Aruna Jeyanthy1, and Dr.D.Devaraj 2
1
N.I.C.E ,Kumarakoil/EEE Department,Kanyakumari,India
Email: arunadarwin@yahoo.com
2
Kalasingam University/EEE Department, Srivillipithur,India
Email: deva230@yahoo.com
Abstract —This paper presents a new hybrid particle swarm It is a non- linear optimization problem and various
optimization (HPSO) method for solving multi-objective real mathematical techniques have been adopted to solve this
power optimization problem. The objectives of the
optimal reactive power dispatch problem. These include the
optimization problem are to minimize the losses and to
maximize the voltage stability margin. The proposed method
gradient method [4, 5], Newton method [6] and linear
expands the original GA and PSO to tackle the mixed –integer programming [7].The gradient and Newton methods suffer
non- linear optimization problem and achieves the voltage from the difficulty in handling inequality constraints. To apply
stability enhancement with continuous and discrete control linear programming, the input- output function is to be
variables such as generator terminal voltages, tap position of expressed as a set of linear functions, which may lead to loss
transformers and reactive power sources. A comparison is made of accuracy. Recently, global optimization techniques such
with conventional, GA and PSO methods for the real power as genetic algorithms have been proposed to solve the
losses and this method is found to be effective than other reactive power optimization problem [8-15]. Genetic algorithm
methods. It is evaluated on the IEEE 30 and 57 bus test system,
is a stochastic search technique based on the mechanics of
and the simulation results show the effectiveness of this
approach for improving voltage stability of the system.
natural selection [16].In GA-based RPD problem it starts with
the randomly generated population of points, improves the
Keywords: Hybrid Particle Swarm Optimization (HPSO), real fitness as generation proceeds through the application of
power loss, reactive power dispatch (RPD), Voltage stability the three operators-selection, crossover and mutation. But
constrained reactive power dispatch (VSCRPD).
in the recent research some deficiencies are identified in the
GA performance. This degradation in efficiency is apparent
I. INTRODUCTION
in applications with highly epistatic objective functions i.e.
Optimal reactive power dispatch problem is one of the where the parameters being optimized are highly correlated.
difficult optimization problems in power systems. The sources In addition, the premature convergence of GA degrades its
of the reactive power are the generators, synchronous performance and reduces its search capability. In addition to
condensers, capacitors, static compensators and tap this, these algorithms are found to take more time to reach
changing transformers. The problem that has to be solved in the optimal solution. Particle swarm optimization (PSO) is
a reactive power optimization is to determine the optimal one of the stochastic search techniques developed by
values of generator bus voltage magnitudes, transformer tap Kennedy and Eberhart [17]. This technique can generate high
setting and the output of reactive power sources so as to quality solutions within shorter calculation time and stable
minimize the transmission loss. In recent years, the problem convergence characteristics than other stochastic methods.
of voltage stability and voltage collapse has become a major But the main problem of PSO is poor local searching ability
concern in power system planning and operation. To enhance and cannot effectively solve the complex non-linear equations
the voltage stability, voltage magnitudes alone will not be a needed to be accurate. Several methods to improve the
reliable indicator of how far an operating point is from the performance of PSO algorithm have been proposed and some
collapse point [1]. The reactive power support and voltage of them have been applied to the reactive power and voltage
problems are intrinsically related. Hence, this paper formulates control problem in recent years [18-20]. Here a few
the reactive power dispatch as a multi-objective optimization modifications are made in the original PSO by including the
problem with loss minimization and maximization of static mutation operator from the real coded GA. Thus the proposed
voltage stability margin (SVSM) as the objectives. Voltage algorithm identifies the optimal values of generation bus
stability evaluation using modal analysis [2] is used as the voltage magnitudes, transformer tap setting and the output
indicator of voltage stability enhancement. The modal of the reactive power sources so as to minimize the
analysis technique provides voltage stability critical areas transmission loss and to improve the voltage stability. The
and gives information about the best corrective/preventive effectiveness of the proposed approach is demonstrated
actions for improving system stability margins. It is done by through IEEE-30and IEEE-57 bus system.
evaluating the Jacobian matrix, the critical eigen values/vector
[3].The least singular value of converged power flow jacobian
is used an objective for the voltage stability enhancement.
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2. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010
II PROBLEM FORMULATION N o is set of numbers of total buses excluding slack bus
Power systems are expected to operate economically N c is the set of numbers of possible reactive power
(minimize losses) and technically (good stability).Therefore source installation buses
reactive power optimization is formulated as a multi-objective
search which includes the technical and economic functions. N t is the set of numbers of transformer branches
A. Economic function: S l is the power flow in branch l the subscripts
‘min’ and “max” in Eq. (2-7) denote the corresponding
The economic function is concerned mainly to minimize
lower and upper limits respectively.
the active power transmission loss and it is stated as, since
reduction in losses reduces the cost. B. Technical function:
2 2
Min P = f ( x1 , x2 )  k g k (Vi  V  2ViV j cos  ij )
loss N
j (1) The technical function is to minimize the bus voltage
E
deviation from the ideal voltage and to improve the voltage
Subject to stability margin (VSM) and it is stated as
PGi  PDi  Vi  V j (Gij cos  ij  Bij sin  ij ) Max (VSM=max (min|eig (jacobi)) (8)
i  NB (2) where jacobi is the load flow jacobian matrix , eig (jacobi)
returns all the eigen values of the Jacobian matrix,
QGi  QDi  Vi  V j (Gij sin ij  Bij cos ij ) k  N min(eig(Jacobi)) is the minimum value of eig (Jacobi) , max
PQ (3)
( min ( eig (Jacobi))) is to maximize the minimal eigen value in
Vi min  Vi  Vi max i  NB (4) the Jacobian matrix.
Tkmin  Tk  Tkmax k  NT III. PARTICLE SWARM OPTIMIZATION (PSO)
(5)
A. OVERVIEW:
Q min  QGi  QGi
Gi
max
i  NG PSO is a population based stochastic optimization
(6) technique developed by Kennedy and Eberhart [17]. A
population of particles exists in the n-Dimensional search
Sl  Slmax l  Nl (7) space. Each particle has a certain amount of knowledge, and
where f ( x1 , x 2 ) denotes the active power loss function of will move about the search space based on this knowledge.
The particle has some inertia attributed to it and so it will
the system.
continue to have a component of motion in the direction it is
VG is the generator voltage (continuous) moving. It knows where in the search space, it will encounter
Tk is the transformer tap setting (integer) with the best solution. The particle will then modify its
direction such that it has additional components towards its
Qc is the shunt capacitor/ inductor (integer) own best position, pbest and towards the overall best
VL is the load bus voltage position, gbest. The particle updates its velocity and position
with the following Equations (9) to (11)
QG is the generator reactive power
Vik1 W*Vik C1 *rand)1 *( pbestSik )C2 *rand)2 *(gbest Sik ) (9)
( (
k  (i , j ), i  N B , J  N i , g k is the conductance of branch k. i i
ij is the voltage angle difference between bus I &j Wmax  W min
W  Wmax  * iter (10)
PGi is the injected active power at bus i itermax
PDi is the demanded active power at bus i
Gij is the transfer conductance between bus i and j
Bij is the transfer susceptance between bus i and j Vi k 1 : Velocity of particle i at the iteration k  1
QGi is the injected reactive power at bus i Vi k : Velocity of particle i at the iteration k
QDi is the demanded reactive power at bus i S ik 1 : Position of particle i at the iteration k  1
N e is the set of numbers of network branches Sik : Position of particle i at the iteration k
N PQ is the set of number of PQ buses C1 : Constant weighting factor related to pbest
Nb is the set of numbers of total buses C2 : Constant weighting factor related to gbest
N i is the set of numbers of buses adjacent to bus i rand ( )1 : Random number between 0 and 1
(including bus i )
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3. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010
rand ( ) 2 : Random number between 0 and 1 C. HPSO Algorithm Procedure:
pbesti : pbest position of particle i Step 1: Initialization of the parameters
gbesti : gbest position of swarm Step 2: Randomly set the velocity and position
of all the particles.
Usually the constant weighting factor or the acceleration Step 3: Evaluate the fitness of the initial
coefficients C1 , C2  2 , control how far a particle moves in particles by conducting Newton-Raphson power flow
a single iteration. The inertia weight’ W’ is used to control analysis results.pbest of e ach particle is set to initial
the convergence behavior of PSO. Suitable selection of the position. The initial best evaluation value among the
inertia weight provides a balance between global and local particles is set to gbest.
exploration and exploitation of results in lesser number of Step 4: Change the velocity and position of the particle
iterations on an average to find a sufficient optimal according to the equations (9) to (11).
solution. In the PSO method, there is only one population Step 5: Select the best particles come into mutation
in an iteration that moves towards the global optimal point. operation according to (12).
This makes PSO computationally faster and the Step 6: If the position of the particle violates the limit
convergence abilities of this method are better than the of variable, set it to the limit value.
other evolutionary computation techniques such as GA. Step 7: Compute the fitness of new particles. If the
fitness of each individual is better than the
B. Proposed Algorithm: previous pbest; the current value is set to
The main drawback of the PSO is the premature pbest value. If the best pbest is better than
convergence. During the searching process, most particles gbest, the value is set to be gbest.
contract quickly to a certain specific position. If it is a local Step 8: The algorithm repeats step 4 to step 7 until the
optimum, then it is not easy for the particles to escape from it. convergence criteria is met, usually a sufficiently good
In addition, the performance of basic PSO is greatly affected fitness or a maximum number of iterations.
by the initial population of the particles, if the initial population
is far away from the real optimum solution. A natural evolution IV .HPSO IMPLEMENTATION OF THE OPTIMAL
of the PSO can be achieved by incorporating methods that REACTIVE POWER DISPATCH PROBLEM:
have already been tested in other evolutionary computation
When applying HPSO to solve a particular optimization
techniques. Many researchers have considered incorporating
problem, two main issues are taken into consideration namely:
selection, mutation and crossover as well as differential
evolution into the PSO algorithm. The main goal is to increase (i) Representation of the decision variables and
the diversity of the population by: preventing the particles (ii) Formation of the fitness function
to move too close to each other and collide, to self-adapt These issues are explained in the subsequent section.
parameters such as constriction factor, acceleration constants
A. Representation of the decision variables
or inertia weight. As a result, hybrid versions of PSO have
been created and tested in different applications. In the While solving an optimization problem using HPSO, each
proposed approach, mutation which is followed in genetic individual in the population represents a candidate solution.
algorithm is carried out. Mutation is one of the effective In the reactive power dispatch problem, the elements of the
measures to prevent loss of diversity in a population of solution consists of the control variables namely; Generator
solution, which can cover a greater region of the search bus voltage (Vgi), reactive power generated by the capacitor
space.Hence in this algorithm the addition of mutation into (QCi), and transformer tap settings (tk).Generator bus voltages
PSO will expand its global search space, add variability into are represented as floating point numbers ,whereas the
the population and prevent stagnation of the search in local transformer tap position and reactive power generation of
optima. capacitor are represented as integers. With this
The mutation operator works by changing a particle representation the problem will look like the following:
position dimension
S i  delta (iter , U  S i ) : rb  1
using: mutate( S i )  S  delta (iter , S  L ) : rb  0
 i i
B. Formation of the fitness function
(12)
Where iter is the current iteration number, In the optimal reactive power dispatch problem, the
U is the upper limit of variable spac objective is to minimize the total real power loss while
L is the lower limit of variable space satisfying the constraints (14) to (20). For each individual,
rb is the randomly generated bit the equality constraints are satisfied by running Newton-
delta (iter, y) return a value in the range [0: y] Raphson algorithm and the constraints on the state variables
It provides a balance between adding variability and allowing are taken into consideration by adding penalty function to
the particles to converge. Hence in this method it reduces the objective function. With the inclusion of the penalty
the probability of getting trapped into local optima. function, the new objective function then becomes,
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4. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010
N PQ Ng Ni solution was obtained with the following parameter setting:
Min F  Ploss  wEig max   VPi   QPgi   LPl (13) Population size : 30
i 1 i 1 l 1
where w, KV , K q , K l are the penalty factors for the eigen wmax : 0.9
value,load bus voltage limit violation, generater reactive wmin : 0.4
power limits violation and line flow limit violation respectively
.In the above expressions C1 :2
C2 :2
K V (Vi  Vi max ) 2 if Vi  Vi max
 Maximum generations: 50
VPi  K V (Vi  Vi min ) 2 if Vi  Vi min
(14) Mutate rate : 0.1
0 otherwise
 Figure 1 illustrates the relationship between the best fitness
values against the number of generations.
Figure . 1. Convergence characteristics
Generally, PSO searches for a solution with maximum fitness From the figure it can be seen that the proposed algo-
function value. Hence, the minimization objective function rithm converges rapidly towards the optimal solution. The
optimal values of the control variables along with the mini-
given in (17) is transformed into a fitness function ( f ) to be
mum loss obtained are given in Table I for IEEE-30 bus sys-
maximized as, tem. Corresponding to this control variable setting, it was
f  K / F (17) found that there are no limit violations in any of the state
variables. To show the performance of the HPSO in solving
where K is a large constant. This is used to amplify (1/F), the
this integer nonlinear optimization problem, it is compared to
value of which is usually small, so that the fitness value of
the well known conventional, GA &PSO techniques. But in
the chromosomes will be in a wider range.
HPSO the best solution is achieved. This shows HPSO is
capable of reaching better solutions and is superior compared
V.SIMULATION RESULTS to other methods. This means less execution time and less
In order to demonstrate the effectiveness and robustness memory requirements.
of the proposed technique, minimization of real power loss TABLE I
under two conditions, without and with voltage stability RESULTS OF PSO-RPD OPTIMAL CONTROL VARIABLES
margin (VSM) were considered. The validity of the proposed
PSO algorithm technique is demonstrated on IEEE- 30and
IEEE-57 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 IEEE 57-bus system has 7
generator buses, 50 load buses and 80 transmission lines of
which 17 branches are with tap setting transformers. The real
power settings are taken from [1]. 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, 0.05 p.u. for the PQ buses and the
reference bus for IEEE 30-bus system. The PSO –based
optimal reactive power dispatch algorithm was implemented
using the MATLAB programmed and was executed on a
Pentium computer.
Case A : RPD with loss minimization objective
Here the PSO-based algorithm was applied to identify the
optimal control variables of the system .It was run with
different control parameter settings and the minimization
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5. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010
Case B: Multi-objective RPD (RPD including voltage stability from the simulation work, it is concluded that PSO performs
constraint) better results than the conventional methods.
In this case, the RPD problem was handled as a multi-
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