This paper presents a new hybrid particle swarm optimization (HPSO) method for solving multi-objective real power optimization problem. The objectives of the optimization problem are to minimize the losses and to maximize the voltage stability margin. The proposed method expands the original GA and PSO to tackle the mixed –integer non- linear optimization problem and achieves the voltage stability enhancement with continuous and discrete control variables such as generator terminal voltages, tap position of transformers and reactive power sources. A comparison is made with conventional, GA and PSO methods for the real power losses and this method is found to be effective than other methods. It is evaluated on the IEEE 30 and 57 bus test system, and the simulation results show the effectiveness of this approach for improving voltage stability of the system.

1. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011
Hybrid Particle Swarm Optimization for
Multi-objective Reactive Power Optimization with
Voltage Stability Enhancement
2
P.Aruna Jeyanthy1, and Dr.D.Devaraj
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 is used an objective for the voltage stability enhancement. It
optimization (HPSO) method for solving multi-objective real is a non- linear optimization problem and various mathematical
power optimization problem. The objectives of the techniques have been adopted to solve this optimal reactive
optimization problem are to minimize the losses and to power dispatch problem. These include the gradient method
maximize the voltage stability margin. The proposed method
[4, 5], Newton method [6] and linear programming [7].The
expands the original GA and PSO to tackle the mixed –integer
non- linear optimization problem and achieves the voltage gradient and Newton methods suffer from the difficulty in
stability enhancement with continuous and discrete control handling inequality constraints. To apply linear programming,
variables such as generator terminal voltages, tap position of the input- output function is to be expressed as a set of linear
transformers and reactive power sources. A comparison is made functions, which may lead to loss of accuracy. Recently, global
with conventional, GA and PSO methods for the real power optimization techniques such as genetic algorithms have been
losses and this method is found to be effective than other proposed to solve the reactive power optimization problem
methods. It is evaluated on the IEEE 30 and 57 bus test system, [8-15]. Genetic algorithm is a stochastic search technique based
and the simulation results show the effectiveness of this on the mechanics of natural selection [16].In GA-based RPD
approach for improving voltage stability of the system.
problem it starts with the randomly generated population of
Keywords: Hybrid Particle Swarm Optimization (HPSO), real points, improves the fitness as generation proceeds through
power loss, reactive power dispatch (RPD), Voltage stability the application of the three operators-selection, crossover
constrained reactive power dispatch (VSCRPD). and mutation. But in the recent research some deficiencies
are identified in the GA performance. This degradation in
I. INTRODUCTION efficiency is apparent in applications with highly epistatic
objective functions i.e. where the parameters being optimized
Optimal reactive power dispatch problem is one of the
are highly correlated. In addition, the premature convergence
difficult optimization problems in power systems. The sources
of GA degrades its performance and reduces its search
of the reactive power are the generators, synchronous
capability. In addition to this, these algorithms are found to
condensers, capacitors, static compensators and tap
take more time to reach the optimal solution. Particle swarm
changing transformers. The problem that has to be solved in
optimization (PSO) is one of the stochastic search techniques
a reactive power optimization is to determine the optimal
developed by Kennedy and Eberhart [17]. This technique
values of generator bus voltage magnitudes, transformer tap
can generate high quality solutions within shorter calculation
setting and the output of reactive power sources so as to
time and stable convergence characteristics than other
minimize the transmission loss. In recent years, the problem
stochastic methods. But the main problem of PSO is poor
of voltage stability and voltage collapse has become a major
local searching ability and cannot effectively solve the
concern in power system planning and operation. To enhance
complex non-linear equations needed to be accurate. Several
the voltage stability, voltage magnitudes alone will not be a
methods to improve the performance of PSO algorithm have
reliable indicator of how far an operating point is from the
been proposed and some of them have been applied to the
collapse point [1]. The reactive power support and voltage
reactive power and voltage control problem in recent years
problems are intrinsically related. Hence, this paper formulates
[18-20]. Here a few modifications are made in the original PSO
the reactive power dispatch as a multi-objective optimization
by including the mutation operator from the real coded GA.
problem with loss minimization and maximization of static
Thus the proposed algorithm identifies the optimal values of
voltage stability margin (SVSM) as the objectives. Voltage
generation bus voltage magnitudes, transformer tap setting
stability evaluation using modal analysis [2] is used as the
and the output of the reactive power sources so as to minimize
indicator of voltage stability enhancement. The modal
the transmission loss and to improve the voltage stability.
analysis technique provides voltage stability critical areas
The effectiveness of the proposed approach is demonstrated
and gives information about the best corrective/preventive
through IEEE-30and IEEE-57 bus system.
actions for improving system stability margins. It is done by
evaluating the Jacobian matrix, the critical eigen values/vector
[3].The least singular value of converged power flow jacobian
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2. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011
II PROBLEM FORMULATION N PQ is the set of number of PQ buses
Power systems are expected to operate economically
N b is the set of numbers of total buses
(minimize losses) and technically (good stability).Therefore
reactive power optimization is formulated as a multi-objective N i is the set of numbers of buses adjacent to bus i
search which includes the technical and economic functions. (including bus i )
A. Economic function: N o is set of numbers of total buses excluding slack bus
The economic function is concerned mainly to minimize N c is the set of numbers of possible reactive power
the active power transmission loss and it is stated as, since
source installation buses
reduction in losses reduces the cost.
Nt is the set of numbers of transformer branches
f ( x1 , x2 )  g (Vi 2  V j2  2ViV j cos  ij ) S l is the power flow in branch l the subscripts ‘min’
Min P =
loss k N E
k (1)
and “max” in Eq. (2-7) denote the corresponding lower and
upper limits respectively.
Subject to B. Technical function:
The technical function is to minimize the bus voltage
PGi  PDi  Vi  V j (Gij cos  ij  Bij sin  ij )
i  NB deviation from the ideal voltage and to improve the voltage
stability margin (VSM) and it is stated as
(2)
Max (VSM=max (min|eig (jacobi)) (8)
QGi  QDi  Vi  V j (Gij sin  ij  Bij cos  ij ) k  N PQ where jacobi is the load flow jacobian matrix , eig (jacobi)
returns all the eigen values of the Jacobian matrix,
(3) min(eig(Jacobi)) is the minimum value of eig (Jacobi) , max
Vi min
 Vi  Vi max
i  NB (4) ( min ( eig (Jacobi))) is to maximize the minimal eigen value in
the Jacobian matrix.
Tkmin  Tk  Tkmax k  NT
(5) III. PARTICLE SWARM OPTIMIZATION (PSO)
Q min  QGi  QGi
max A. Overview:
Gi
i  NG
PSO is a population based stochastic optimization
(6) technique developed by Kennedy and Eberhart [17]. A
Sl  Slmax l  Nl (7) population of particles exists in the n-Dimensional search
space. Each particle has a certain amount of knowledge, and
where f ( x1 , x 2 ) denotes the active po wer loss function of will move about the search space based on this knowledge.
the system. The particle has some inertia attributed to it and so it will
VG is the generator voltage (continuous) continue to have a component of motion in the direction it is
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
Qc is the shunt capacitor/ inductor (integer) direction such that it has additional components towards its
own best position, pbest and towards the overall best
VL is the load bus voltage position, gbest. The particle updates its velocity and position
QG is the generator reactive power with the following Equations (9) to (11)
k  (i , j ), i  N B , J  N i , g k is the conductance
of branch k.
 ij is the voltage angle difference between bus I &j
PGi is the injected active power at bus i
PDi is the demanded active power at bus i
Gij is the transfer conductance between bus i and j
: Velocity of particle i at the iteration
Bij is the transfer susceptance between bus i and j
Vi k : Velocity of particle i at the iteration k
QGi is the injected reactive power at bus i
S ik 1 : Position of particle i at the iteration k  1
QDi is the demanded reactive power at bus i
Sik : Position of particle i at the iteration k
N e is the set of numbers of network branches
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3. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011
C1 : Constant weighting factor related to pbest It provides a balance between adding variability and allowing
the particles to converge. Hence in this method it reduces
C2 : Constant weighting factor related to gbest the probability of getting trapped into local optima.
rand ( )1 : Random number between 0 and 1 C. HPSO Algorithm Procedure:
rand ( ) 2 : Random number between 0 and 1
Step 1: Initialization of the parameters
pbest i : pbest position of particle i Step 2: Randomly set the velocity and position
gbest i : gbest position of swarm of all the particles.
Step 3: Evaluate the fitness of the initial
Usually the constant weighting factor or the acceleration particles by conducting Newton-Raphson
coefficients C1 , C2  2 , control how far a particle moves in a power flow analysis results. pbest of e ach
single iteration. The inertia weight’ W’ is used to control the particle is set to initial position. The initial
convergence behavior of PSO. Suitable selection of the inertia best evaluation value among the particles is
weight provides a balance between global and local 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 solution. according to the equations (9) to (11).
In the PSO method, there is only one population in an iteration Step 5: Select the best particles come into mutation
that moves towards the global optimal point. This makes operation according to (12).
PSO computationally faster and the convergence abilities of Step 6: If the position of the particle violates the limit
this method are better than the other evolutionary computation of variable, set it to the limit value.
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
optimum, then it is not easy for the particles to escape from it. until the convergence criteria is met,
In addition, the performance of basic PSO is greatly affected usually a sufficiently good fitness or a
by the initial population of the particles, if the initial population maximum number of iterations.
is far away from the real optimum solution. A natural evolution
of the PSO can be achieved by incorporating methods that IV .HPSO IMPLEMENTATION OF THE OPTIMAL
have already been tested in other evolutionary computation REACTIVE POWER DISPATCH PROBLEM:
techniques. Many researchers have considered incorporating
selection, mutation and crossover as well as differential When applying HPSO to solve a particular optimization
evolution into the PSO algorithm. The main goal is to increase problem, two main issues are taken into consideration namely:
the diversity of the population by: preventing the particles (i) Representation of the decision variables and
to move too close to each other and collide, to self-adapt (ii) Formation of the fitness function
parameters such as constriction factor, acceleration constants These issues are explained in the subsequent section.
or inertia weight. As a result, hybrid versions of PSO have A. Representation of the decision variables
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. The mutation operator works by changing a particle capacitor are represented as integers. With this
position dimension using: representation the problem will look like the following:
S i  delta (iter , U  S i ) : rb  1 0. 0. ...1. 0. 0. ...1. 3.35 2.10 ...1.50
981 970 017 925 965 000
mutate( S i )   (12)            
S i  delta (iter , S i  L ) : rb  0
V1 V2 Vn t1 t2 tn Qc 1 Qc 2 Qcn
Where iter is the current iteration number, B. Formation of the fitness function
U is the upper limit of variable space
L is the lower limit of variable space In the optimal reactive power dispatch problem, the
rb is the randomly generated bit objective is to minimize the total real power loss while
delta (iter, y) return a value in the range [0: y] satisfying the constraints (14) to (20). For each individual,
the equality constraints are satisfied by running Newton-
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4. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011
Raphson algorithm and the constraints on the state variables Case A : RPD with loss minimization objective
are taken into consideration by adding penalty function to Here the PSO-based algorithm was applied to identify the
the objective function. With the inclusion of the penalty optimal control variables of the system .It was run with
function, the new objective function then becomes, different control parameter settings and the minimization
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 wmax : 0.9
wmin : 0.4
where w, KV , K q , K l are the penalty factors for the eigen
C1 :2
value,load bus voltage limit violation, generater reactive C2 :2
power limits violation and line flow limit violation respectively Maximum generations: 50
.In the above expressions Mutate rate : 0.1
Figure 1 illustrates the relationship between the best fitness
K V (Vi  Vi max ) 2 if Vi  Vi max values against the number of generations.

VPi  K V (Vi  Vi min ) 2 if Vi  Vi min
(14)
0 otherwise

 K q (Qi  Qimax ) 2 if Qi  Qimax


QPgi   K q (Qi  Qimin ) 2 if Qi  Qimin (15)

0

otherwise
 K ( S  S lmax ) 2 if S l  S lmax Figure . 1. Convergence characteristics
LPl   l l (16) From the figure it can be seen that the proposed algorithm
0 otherwise
converges rapidly towards the optimal solution. The optimal
Generally, PSO searches for a solution with maximum fitness values of the control variables along with the minimum loss
function value. Hence, the minimization objective function obtained are given in Table I for IEEE-30 bus system.
given in (17) is transformed into a fitness function ( f ) to be Corresponding to this control variable setting, it was found
maximized as, that there are no limit violations in any of the state variables.
To show the performance of the HPSO in solving this integer
f  K / F (17)
nonlinear optimization problem, it is compared to the well
where K is a large constant. This is used to amplify (1/F), the known conventional, GA &PSO techniques. But in HPSO the
value of which is usually small, so that the fitness value of best solution is achieved. This shows HPSO is capable of
the chromosomes will be in a wider range. reaching better solutions and is superior compared to other
methods. This means less execution time and less memory
V.SIMULATION RESULTS requirements.
In order to demonstrate the effectiveness and robustness T ABLE I
of the proposed technique, minimization of real power loss RESULTS OF PSO-RPD OPTIMAL CONTROL VARIABLES
under two conditions, without and with voltage stability
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.
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5. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011
Case B: Multi-objective RPD (RPD including voltage CONCLUSION
stability constraint)
This paper presents a hybrid particle swarm optimization
In this case, the RPD problem was handled as a multi- algorithm approach to obtain the optimum values of the
objective optimization problem where both power loss and reactive power variables including the voltage stability
maximum voltage stability margin of the system were constraint. The effectiveness of the proposed method for
optimized simultaneously. The optimal control variable RPD is demonstrated on IEEE-30 and IEEE-57 bus system
settings in this case are given in the last column of Table I. To with promising results. Simulation results show that the HPSO
maximize the stability margin the minimum eigen value should based reactive power optimization is always better than those
be increased. Here the VSM has increased to 0.2437 from obtained using conventional, GA and simple PSO methods.
0.2403, an improvement in the system voltage stability. For From this multi-objective reactive power dispatch solution
IEEE-57 bus system the minimum power loss obtained is the application of HPSO leads to global search with fast
25.6665 MW.The VSM has increased to 0.1568 from 0.1456. convergence rate and a feature of robust computation. Hence
To determine the voltage security of the system, contingency from the simulation work, it is concluded that PSO performs
analysis was conducted using the control variable setting better results than the conventional methods.
obtained in case A and case B. The eigen values
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