30 Bus System 2 Capacitor

742 IEEE SYSTEMS JOURNAL, VOL. 7, NO. 4, DECEMBER 2013

Multiobjective Optimal Reactive Power Dispatchand Voltage Control: A New Opposition-Based

Self-Adaptive Modified GravitationalSearch Algorithm

Taher Niknam, Mohammad Rasoul Narimani, Rasoul Azizipanah-Abarghooee, and Bahman Bahmani-Firouzi

Abstract—This paper presents a novel opposition-based self-adaptive modified gravitational search algorithm (OSAMGSA)for optimal reactive power dispatch and voltage control in power-system operation. The problem is formulated as a mixed integer,nonlinear optimization problem, which has both continuous anddiscrete control variables. In order to achieve the optimal value ofloss, voltage deviation, and voltage stability index, it is necessaryto find the optimal value of control variables such as the tappositions of tap changing transformers, generator voltages, andcompensation capacitor. Therefore, this complicated problemneeds to be solved by an accurate optimization algorithm.This paper solves the aforementioned problem by using thegravitational search algorithm (GSA), which is one of the noveloptimization algorithms based on the gravity law and mass in-teractions. To improve the efficiency of this algorithm, the tuningof its parameters is accomplished using random generation, andby applying the self-adaptive parameter tuning scheme. Also, theproposed OSAMGSA of this paper employs the opposition-basedpopulation initialization and self-adaptive probabilistic learningapproach for generation jumping and escaping from local optima.Since the proposed problem is a multiobjective optimizationproblem incorporating several solutions instead of one, weapplied the Pareto optimal solution method in order to find allPareto optimal solutions. Moreover, the fuzzy decision method isused for obtaining the best compromise solution between them.

Index Terms—Gravitational search algorithm (GSA), multiob-jective optimization, opposite numbers, optimal reactive powerdispatch, self-adaptive probabilistic learning approach, voltagecontrol.

Nomenclature

Indiceseq Equality constraint index.k Transmission line index.a Mutation method index.t Iteration index.

Manuscript received April 5, 2012; revised September 8, 2012; accepted Oc-tober 25, 2012. Date of publication January 30, 2013; date of current versionSeptember 25, 2013. (Corresponding author: R. Azizipanah-Abarghooee.)

T. Niknam is with the Department of Electrical Engineering, Shiraz Univer-sity of Technology, Shiraz 13876-71557, Iran (e-mail: [email protected]).

M. R. Narimani and R. Azizipanah-Abarghooee are with the Departmentof Electrical Engineering, Marvdasht Branch, Islamic Azad University, Marv-dasht 465, Iran (e-mail: [email protected]; [email protected]).

B. Bahmani-Firouzi is with the Sharif University of Technology, Tehran,Iran (e-mail: bahman−[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSYST.2012.2227217

ueq Inequality constraint index.Constants

iterationmax Maximum iteration.gk Conductance of the kth line.L1, L2 Penalty factors for respective equality

and inequality constraints.n Number of buses.Na Number of agents that choose the

mutation method a.Nparam Number of control variables in each

agent.Npop Number of agents.Nobj Number of objective functions.NC Number of compensator capacitors.Neq Number of equality constraints.Ng Number of generation units.NL Number of transmission lines.NLoad Number of load buses.Nueq Number of inequality constraints.NT Number of tap transformers.Pgi max Active power capacity of unit i (MW).Pij max Maximum real power flows through

the branch between buses i and j.Pgi min Minimum active power output of unit

i (MW).Qgi max Reactive power capacity of unit i

(MVAr).Qmax

Ci Reactive power capacity of the ithcompensator capacitor (MVAr).

QminCi Minimum reactive power output of the

ith compensator capacitor (MVAr).Qgi min Minimum reactive power output of

unit i (MVAr).rand(), rand()1, rand()2 Random function generator in the

range [0, 1].tmaxi Maximum tap position of the ith

transformer.tmini Minimum tap position of the ith

transformer.V ref

j Prespecified reference value of thevoltage magnitude at the jth load buswhich is usually set to one p.u.

1932-8184 c© 2013 IEEE

NIKNAM et al.: MULTIOBJECTIVE OPTIMAL REACTIVE POWER DISPATCH AND VOLTAGE CONTROL 743

Vi max, Vi min Maximum and minimum validvoltages for bus i, respectively.

ε A constant small value.

Variablesai(t), vi(t), xi(t) Acceleration, velocity, and position of

agent i in iteration t.Floss Total active power loss (MW).Fvoltage deviation Total voltage deviation (p.u.).Fbest(t) Fitness function value of the best

solution in iteration t.Fworst(t) Fitness function value of the worst

solution in iteration t.fi(t) Fitness function value of agent i in

iteration t.L Voltage stability index.f min

i , f maxi Minimum and maximum acceptable

level of objective function i,respectively.

G(t) Gravitational constant in iteration t.Mi(t) Mass of agent i in iteration t.Proba Probability of mutation method a.PDi Active power demand of the ith bus

(MW).Pgi Active power generation output of unit

i (MW).PGi Active power generation of the ith bus

(MW).Pij Real power that flows between buses

i and j.QCi Reactive power of the ith compensator

capacitor (MVAr).QDi Reactive power demand of the ith bus

(MVAr).Qgi Reactive power generation output of

unit i (MVAr).QGi Reactive power generation of the ith

bus (MVAr).Rij(t) Euclidean distance between two

agents i and j in iteration t.ti Tap of the ith transformer.Vgi Voltage magnitude of the ith genera-

tion unit (p.u.).Vi, Vj Voltage amplitude of the ith and jth

buses, respectively (p.u.).V1,k, V2,k Voltage amplitude of two ending buses

1 and 2 of the arbitrary branch k.(p.u.).

Xbest Best solution between all agents.Xworst Worst solution between all agents.θij Voltage angle of two ending buses i

and j of an arbitrary branch.θi, θj Voltage angle of respective buses i and

j.θ1,k, θ2,k Voltage angle of two ending buses 1

and 2 of the arbitrary branch k.μi Membership function of objective

function i.

SetsKbest Set of first K agents with the best fitness value

and biggest mass.αL Set of load buses.

I. Introduction

THE OPTIMAL power flow problem, which was proposedby Carpentier about 50 years ago, is one of the major

issues in power-systems operation [1]–[3]. This problem canbe divided into two subproblems, optimal reactive powerdispatch (ORPD) and optimal real power dispatch [4]. ORPDin power systems is concerned with the security and economyof the power system operation. The reactive power dispatch isrelated to the allocation of reactive power generation, whichis used in order to minimize real power transmission lossesand keep all voltages within the limits, which satisfies anumber of equality and inequality constraints. Therefore, theORPD problem is a highly large-scale nonlinear constrainedoptimization problem.

Within the past few years, many researchers have tendedtoward optimization-based approaches [5]–[8]. The enhancedbee swarm optimization algorithm and self-adaptive modifiedfirefly algorithm are implemented in [5] and [6], respectively,for solving the complex dynamic economic dispatch problem.In [7], the bacterial foraging algorithm is used for the optimalcontrol of a doubly fed induction generator (DFIG) system.The self-adaptive evolutionary programming method is pre-sented in [4] for solving the nonlinear optimal power flowproblem. Gonzalez-Longatt et al. [8] describe a new modi-fication model of the genetic algorithm for optimal electricnetwork design of a large offshore wind farm. In the ORPDarea, a wide variety of conventional optimization techniquessuch as decomposition approach [9], heuristic approach basedon binary search [10], interior point method [11], modifiedsimple genetic algorithm [12], evolutionary programming [13],PSO using multiagent [14], cooperative co-evolutionary dif-ferential evolution [15], differential evolution [16], learningparticle swarm optimization [17], self-adaptive real codedgenetic algorithm [18], dynamic programming method [19],and nonlinear programming [20], were developed to solvethe ORPD problem. Generally, these techniques suffer fromalgorithmic complexity and insecure convergence as well asbeing sensitive to the initial search point. By comparing theobtained results with other approaches, it is clearly confirmedthat the presented algorithm can overcome all of the previouslymentioned problems.

As previously mentioned, multiobjective optimal reactivepower dispatch (MORPD) is a mix integer problem, which hasa lot of local optima; therefore, it is necessary to solve thisproblem with a precise algorithm. In recent years, evolutionaryalgorithms have been used in order to solve complicated op-timization problems. In fact, conventional algorithms, such asNewton–Raphson-based techniques, nonlinear programming,interior point algorithms, and quadratic programming havefailed to manage nonconvexity optimization problems. Oneof the newest evolutionary algorithms is gravitational searchalgorithm (GSA), presented by Rashedi et al. in [21]. GSA is


applied on the optimization problem with different objectivefunctions in [21]. The obtained results are compared withparticle swarm optimization (PSO) and residual gas analyzer(RGA), and the obtained results are shown to be better thanthose obtained by the PSO and RGA algorithms [21]. GSApresents no difficulty in solving mixed integer problems;hence, it is highly suitable for the reactive power optimizationproblem. The GSA concept is simple and is known as apowerful optimization algorithm. Despite its various privilegesit has some demerits such as being trapped in local optimain some cases. To overcome these problems, two parametersof the GSA algorithm are tuned randomly, and considered asa control variable during the optimization process. Throughthis modification, this algorithm can escape from the localoptima and converge to the global optima in proper time.Moreover, the concept of opposition-based learning (OBL)is presented in [22]. The main goal of the OBL idea is thatan opposite point could be estimated for each particle in theinitial population in order to achieve a better approximation forthe existing candidate solution. In this paper, OBL has beenimplemented to accelerate the convergence rate of the originalGSA. Moreover, in order to improve the search ability of thisalgorithm, a new self-adaptive learning mechanism (SALM) isimplemented in this paper. This new strategy has some mutantrules and profits a probabilistic procedure for selecting each ofthese rules. This modification expands the search ability of theproposed algorithm. Hence, our proposed technique is calledopposition-based self-adaptive modified gravitational searchalgorithm (OSAMGSA). One objective function considered inthis paper is active power transmission loss. By decreasingthe loss in power systems we can reduce the total generationand diminish the generation cost leading to increased socialwelfare. Another considered objective is voltage deviation,which plays an important role in power-system control. Allequipment is designed for specific voltage range in powersystems, and any deviation from this range can damage thesedevices. Therefore, it is important to keep all voltages in anacceptable range; therefore, voltage deviation is considered asan objective function.

With the expansion of power networks and the increasingdemand for electrical power, network security has becomemore important than before. For a secure operation of thepower system, it is also important to maintain a required levelof stability; therefore, the voltage stability index (VSI) relatedto bus voltage magnitudes is considered as an objective func-tion and is optimized in order to increase the secure operationof the power system. For the previously mentioned conditions,voltage deviation and voltage stability enhancement index areconsidered as objective functions besides the active power-loss objective function. Hence, we are encountered with amultiobjective problem which needs to be solved with amultiobjective method.

There are several techniques mentioned in articles to solvemultiobjective problems such as reducing the multiobjectiveproblem into a single objective one by treating other objectivesas a constraint, or uniting all the objectives into a singleobjective function. There are some drawbacks associated withthese methods, such as the limitation of the available choices

and their prior selection need of weights or targets for eachof the objective functions. A fuzzy multiobjective optimizationtechnique is a method that is used to solve multiobjective prob-lems, its produced solutions are suboptimal and the algorithmdoes not provide a systematic framework to direct the searchtoward the Pareto-optimal front.

The control variables in the proposed problem are thegenerator buses voltage, the tap positions of transformers, andthe reactive power of the compensator capacitor. Since theobjectives are in conflict with each other in multiobjectiveproblems, it is usual to obtain a set of solutions instead ofone. The considered multiobjective problem is solved using theOSAMGSA that is capable of finding nondominated solutionsthat represent the best possible tradeoff among the objectives.The decision maker can choose any of the Pareto-optimalsolutions based on his/her own preference. Furthermore, arepository is considered in this paper to save all nondom-inated solutions in each iterate after they are sorted usingthe fuzzy decision making method to determine the bestsolution [23]. The 30-bus IEEE test system is used to illustratethe proposed problem, and for more validation the obtainedresults are compared with other evolutionary methods in theliterature.

The main contributions and novelties of this approach areas follows: 1) the MORPD and voltage control problemis formulated including three important objective functions;2) a new Pareto-based OSAMGSA method, which involvessome novel metaheuristic approaches is proposed to solve themultiobjective problem to achieve the Pareto-optimal set; and3) the performance of the proposed approach is successfullyvalidated with simulation results.

II. Problem Objectives

A. Active Power Loss

With the expansion of power networks and the increasingdemand for power, the amount of wasted power in powerlines is increased too; making power system loss an im-portant concern in the power sector. We have consideredpower transmission loss as an objective function as follows[24]:

Floss =NL∑k=1

gk[V 21,k − V 2

2,k − 2V1,kV2,k cos(θ1,k − θ2,k)]. (1)

B. Voltage Deviations

Another important function in power systems is voltagedeviation. Electrical equipment is designed for optimum oper-ation at their nominal voltage. Any deviation from this ratedvoltage can result in decreased efficiency, damaged electronicequipment, and severely reduced life of electrical control andutilization equipment. Voltage deviation is to optimize the volt-age profile of the power system and defined by minimizationof the sum voltage deviations at load buses. This function isdefined as follows:

Fvoltage deviation =NLoad∑

j=1

∣∣Vj − V refj

∣∣ . (2)


C. Voltage Stability Index

Among the different indexes for voltage stability and voltagecollapse prediction, a fast indicator of voltage stability, L-index, presented by Kessel and Glavitsch [25], is chosen as theobjective for the voltage stability index in this paper; describedas follows:

Lj =

∣∣∣∣∣1 −Ng∑i=1

Fji

V i

Vj

∣∣∣∣∣ , j = Ng + 1, . . . , n. (3)

All quantities within the sigma in (3) are complex quantities.The matrix F is computed as follows:

[F ] = −[YLL]−1[YLG] (4)

where [YLL] and [YLG] are submatrices of the Y bus matrix.The L indices for the given load condition are computed

for all load buses and the maximum of L-indices gives theproximity of the system to voltage collapse.

For stable situations the condition 0 ≤ Lj ≤ 1 must notbe violated for any of the nodes j. A global indicator (L)describes the stability of the whole system which is calculatedby the following equation:

L = max(Lj) j ∈ αL. (5)

In practice, L must be lower than a given threshold value.The predetermined threshold value is specified depending onthe system configuration and on the utility policy regardingservice quality and allowable margin [26]. Therefore, theminimization of the L value is considered as an objectivefunction.

D. Constraints

The proposed problem has some equality and inequalityconstraints. Equality constraints reflect the physics of thepower system, expressed as

Pi = PGi − PDi =n∑

j=1

ViVj(Gij cos θij + Bij sin θij) (6)

Qi = QGi − QDi =n∑

j=1

ViVj(Gij sin θij − Bij cos θij) (7)

where θij = θi − θj .Inequality constraints reflect the limits of physical devices

in the power system as well as the limits created to ensuresystem security, expressed as

Pgi min ≤ Pgi ≤ Pgi max , i = 1, 2, . . . , Ng (8)

Qgi min ≤ Qgi ≤ Qgi max (9)

∣∣Pij

∣∣ ≤ Pij max (10)

Vi min ≤ Vi ≤ Vi max , i = 1, 2, . . . , NLoad (11)

QminCi ≤ QCi ≤ Qmax

Ci , i = 1, 2, . . . , NC (12)

tmini ≤ ti ≤ tmax

i , i = 1, 2, . . . , NT . (13)

III. Multiobjective Optimization Fundamental

and Method

A. Fuzzy Modeling for Normalizing Objective Functions

In this section, a fuzzy method is proposed to calculatethe normalized form of the objective. Since the objectivefunctions are not in the same range, they are formulated asfuzzy sets, which are generally formulated by a membershipfunction (μi). The fuzzy membership function substitutes eachobjective function as a value between 0 and 1. The ithobjective function of fi is depicted by a membership functionsμi and defined as (14) [27]

μi =

⎧⎪⎨⎪⎩

1, if fi ≤ f mini

f maxi

−fi

f maxi

−f mini

, if f mini < fi < f max

i

0, if fi ≥ f maxi .

(14)

B. Pareto Optimal Solution

Multiobjective optimization refers to the simultaneous op-timization of multiple conflicting objectives, which produce aset of alternative solutions called the Pareto optimal solutions.The Pareto method determines a set of solution for multiobjec-tive problems using the dominance concept. Each of the threepreviously described objective functions f1, f2, and f3 canhave one of the two attained possibilities: either one dominatesthe other, or neither dominates the other. A vector Fa is saidto dominate another vector Fb, as follows [24]:

Fa ≺ Fb, iff fa,i ≤ fb,i ∀i = {1, 2, . . . , M}and ∃j ∈ {1, 2, ......, M} where fa,j ≺ fb,j

(15)

where M is the number of control variables (parameters). Inother words, a vector dominates another one when it is lessthan or equal to it with respect to all of its components, andit is at least strictly less than one of them. The solutions thatare nondominated during the entire search space are denotedas Pareto optimal and constitute the Pareto-optimal set.

IV. Gravitational Search Algorithm (GSA)

A. Original GSA Algorithm

The gravitational search algorithm is based on the lawof gravity and the notion of mass interactions. The GSAalgorithm uses the Newtonian physics theory and its searcheragents are the collection of masses. In physics, gravitation is atendency by which objects with mass accelerate toward eachother. In the Newton gravitational law, each particle attractsother particles with a force known as the gravitational force[28].

For describing the proposed algorithm, consider a systemwith n masses in which the location of ith mass is

Xi = (x1i , x

2i , . . . , x

ni ). (16)

Objective functions are initially calculated for all agents.Then, among all achieved results, the best and worst results in


Fig. 1. Acceleration vector and force vector.

each iteration are considered as Fbest(t) and Fworst(t), respec-tively. In order to compute mass for each agent it is necessaryto calculate the following function for each existing agent:

Qi(t) =fi(t) − Fworst(t)

Fbest(t) − Fworst(t). (17)

Mass of each agent is computed as follows:

Mi(t) =Qi(t)∑nj=1 Qj(t)

. (18)

After obtaining the mass for each agent, it is possible tocompute the acceleration for each individual mass. The totalforce from a set of heavier masses that is applied on eachagent is computed as follows based on Newton rules:

Fi(t) =∑

j∈Kbest j �=i

randj()G(t)Mj(t)Mi(t)

Rij(t) + ε(xj(t) − xi(t)). (19)

According to Newton rules in classic mechanics, the accel-eration of each agent can be computed as follows:

ai(t) =Fi(t)

Mi(t)=

∑j∈Kbest j �=i

rand1()G(t)Mj(t)Mi(t)

Rij(t) + ε(xj(t)−xi(t)).

(20)Acceleration vector and force vector are shown in Fig. 1.

For updating the position in the proposed optimizationalgorithm we should calculate the velocity of each agent. Bycalculating the velocity we can obtain the position of eachagent according to the classic mechanical equation as follows:

vi(t + 1) = rand2() × vi(t) + ai(t) (21)

xi(t + 1) = xi(t) + vi(t + 1). (22)

Kbest is the squad of the first K agents which have bestfitness value and biggest mass, the value of K is decreasedin each iteration (in the original GSA at the first iteration K

equals the number of agents and this value decreases to 1linearly). G has a trait like K, i.e., at first it is consideredas G0, afterward it decreases by each iteration. For a betterillustration of the GSA process, its pseudocode is given inAlgorithm 1.

In GSA, like particle swarm optimization (PSO) optimizedsolution is obtained by agent movement in the search space,but the procedure of these movements is different. In the PSOalgorithm the direction of the movement is calculated only by

Algorithm 1 Original GSA

1: Generate the initial population2: while the termination criterion is not satisfied do3: for i =1 to Npop do4: Evaluate the fitness function for each individual

and update Xbest, Xworst and the mass of each agentusing (18).

5: Update the acceleration, velocity, and the positionof each agent according to (20)–(22).

6: end for7: end while

using Pbest and Gbest, but in the GSA, movement is calculatedusing all forces obtained by all other agents. Another differ-ence between GSA and PSO is that in GSA existing distancebetween solutions is affecting the new position updating, butin PSO existing distance between solutions does not affect thenew position updating.

V. Opposition-Based Self Adaptive Modified

Gravitational Search Algorithm (OSAMGSA)

A. Self-Adaptive Parameter Tuning

There are two tuning parameters: G (gravitational constant)and K (number of best fitness), that greatly affect the GSAalgorithm performance. These previously discussed parametersare often held constant for the entire run of the GSA, althoughthis approach will not produce optimal results in many cases.In this paper, G and K are updated with a new approach,called self-adaptive, in each iterate.

It is not quite clear how to change these parameters duringthe run to achieve better results. In this paper, G and K areinput variables of the GSA algorithm and are generated withinGmin ≤ G ≤ Gmax and Kmin ≤ K ≤ Kmax. They are alsoupdated with the GSA algorithm at each iterate. In this mannerK and G are obviously stochastic variable. This manner iscalled self-adaptive, and helps achieve better results comparedto when G and K are constant or vary linearly. The controlvector is changed as follows:

Xi =[Vgi , G, K , Qci , tapi

]. (23)

B. Opposition-Based Initialization

Let Xi = [xi,1, ..., xi,n] be a solution in the n-dimensionalsearch space and corresponding apposite point of the Xi is�

Xi= [�xi,1, ...,

�xi,n] where xi,j ∈ [LBi,j, UBi,j] and LBi,j, UBi,j

are lower and upper bounds of the jth control variables inparticle i. Accordingly,

�xi,j is defined as follows:

�xi,j= LBi,j + UBi,j − xi,j. (24)

Now, the opposite point that is defined in the aforemen-tioned text can be implemented to generate the initial popula-tion. Thus, the opposition-based optimization is defined in thefollowing form.

In the multiobjective optimization, if�

Xi dominates the Xi,then point Xi can be replaced with

�

Xi; otherwise, we continue


with Xi. Hence, the point and its opposite point are evaluatedsimultaneously in order to continue with the fitter one.

C. Self-Adaptive Learning Mechanism (SALM)

The objective of designing a SALM approach is to improvethe effectiveness or robustness of the GSA to achieve goodresults in the optimization procedure with diverse character-istics. The basic idea behind this novel methodology is toadaptively choose several effective mutation methods based ontheir previous experiences of producing promising solutions.Therefore, at different steps of the optimization procedure,multiple strategies may be assigned a different probabilitybased on their success rate in generating improved solutionswithin a certain number of previous generations. We used twomutation strategies in the OSAMGSA to diverse the proposedcomplex multiobjective optimization problem. These mutationoperators can be formulated in the following form.

1) Mutation method 1

Xi,1 = Xr1 + rand1(·)(Xr2 − Xr3) + rand2(·)(Xr4 − Xr5), i = 1, ..., N1. (25)

2) Mutation method 2

Xi,2 = Xr1 +rand(·)(Xbest −Xworst) , i = 1, ..., N2. (26)

In this regard, the vectors Xr1, Xr2, Xr3, Xr4, and Xr5 areselected randomly from the existing population in order touniformly cover the algorithm search domain. All particlesin the population will have a chance to mutate, controlledby the probability of their methods of mutating. Based ona probability model, each particle selects one of these twomethods. Denote Proba = 0.5, a= 1, 2 as the initial probabilityof implementing ath mutation strategy. The Roulette wheelmechanism (RWM) selection method is used to choose theath method for each of the particle in the population. Proba,new

is updated after the LP iterations according to the followingform:

Proba,new =sra

2∑a=1

sra

; a = 1, 2 (27)

where sra represents the success rate of the trial solutionsgenerated by the ath mutation strategy and successfully entersthe next step within the previous LP iterations with respect tothe tth iteration. Thus, sra can be formulated as follows:

sra =

t−1∑it=t−LP

nsita

t−1∑it=t−LP

(nsit

a + nf ita

) + δ; a = 1, 2; k > LP (28)

where nsita and nf it

a are the respective numbers of the trialsolutions generated by the ath mutation strategy, which remainor fail in the selection process in the last LP iterations. Thesmall constant value δ = 0.01 is used to avoid the possible nullvalues for sra. Equation (27) can be used in order to ensure

that the summing of the probabilities of choosing the strategiesis always equal to one. It can be expected that the larger thevalue of the sra is, the larger the probability of implementingit to generate the trial solutions at the current iteration t is. Tothis end, the RWM selection method is applied to choose theath mutation strategy for each particle.

VI. Application of OSAMGSA to Proposed Problem

To implement the OSAMGSA on the proposed multiobjec-tive problem, it is necessary to execute the following steps:

Step 1) Define the input data.In this step, the input data including the generatorbus voltages, transformer tap settings, and reactivepower of switchable VAR sources are defined.

Step 2) Convert the constraint multiobjective problem to anunconstraint one using the following equation:

J(X) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

f1(X) + L1

(Neq∑eq=1

(heq(X))2

)+ L2

(Nueq∑ueq=1

(max [0, −gueq(X)])2

)

f2(X) + L1

(Neq∑eq=1

(heq(X))2

)+ L2

(Nueq∑ueq=1


)

f3(X) + L1

(Neq∑eq=1

(heq(X))2

)+ L2

(Nueq∑ueq=1


)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

f1(X), f2(X) and f3(X) are the objective functionsdescribed in (1)–(3). heq(X) and gueq(X) are theequality and inequality constraints, respectively, andL1 and L2 are the penalty factors. Since the con-straints should be met, the values of the parametersshould be high. We considered values of 10 000.

Step 3) An initial population Xi which must meet con-straints, is generated randomly as follows:

Population =

⎡⎢⎢⎣

X1

X2

...

XNpop

⎤⎥⎥⎦ (30)

Xi = [xi,1, xi,2, ..., xi,Nparam] (31)

Xi =[Vgi , G, K , QCi , ti

](32)

where

Vgi = [Vgi,1, Vgi,2, . . . , Vgi,Ng]1×Ng (33)

ti = [ti,1, ti,2, . . . , ti,Nc]1×NT (34)

QCi = [Qci,1, Qci,2, ..., Qci,Nc]1×NC (35)

xi,j = rand (.) ∗ (xj,max − xj,min

)+ xj,min

j = 1, 2, 3, ..., Nparam ; i = 1, 2, 3, ..., Npop(36)

where xi,j is the position of the jth state variable.For each target vector the opposite point is gen-erated based on the opposition-based optimization.


If the new solution dominates the candidate’s one,the opposite solution replaces with the target vector,else the algorithm continues with the target vector.

Step 4) Calculate the objective functions values and normal-ize them by the fuzzy decision maker in accordancewith (14). The augmented objective function (29) isevaluated by using the load flow result. Also, foreach individual (Xi) the membership values of allthe different objectives are computed.

Step 5) Apply the Pareto method in order to obtain thenormalized objective function of the previous stepand save nondominate solutions in the repository.Compute the weight factor for all nondominatesolutions.

Step 6) Sort all agents according to their weight factors andselect the best (Xbest) and worst (Xworst) agent.

Step 7) Calculate mass for all agents according to (17) and(18).

Step 8) Compute the force that is applied to each agent andits acceleration according to (19) and (20), respec-tively. The velocity of each agent is computableusing its acceleration according to (21).

Step 9) Update the position of each agent according to (22).Step 10) Select the ath strategy by the RWM based on

Section V-C for all the existing solutions.Step 11) If any element of each agent breaks its inequality

constraints then the position of the individual isfixed at its maximum/minimum operating point.Therefore, this can be formulated as

Xk+1i,j =

⎧⎨⎩

xk+1i,j , if xj,min〈 xk

i,j 〈 xj,max

xj,min, if xki,j 〈 xj,min

xj,max, if xki,j 〉 xj,max.

(37)

Step 12) Objective functions are calculated for new agents.For each target vector, if the new solution dominatesthe candidate’s one, the new solution replaces withthe target vector, else the algorithm continues withthe target vector.

Step 13) Update the probability for each mutation strategy,i.e., Proba ; a = 1, 2 using (27) and (28).

Step 14) If the current iteration number (iterationmax) reachesthe predetermined maximum iteration number, thesearch procedure stops, otherwise it goes to Step 6.

Step 15) The last achieved Xbest is the solution of the prob-lem.

The flowchart of the proposed OSAMGSA is given in Fig. 2.

VII. Simulation Results

A. Validation of the GSA and Proposed OSAMGSA

The first study of this section is an optimization problemwith two decision variables mx1 and x2 on generalized Schwe-fel’s function, presented in [40]. Fig. 3 shows the objectivefunction of the problem including 3-D and contour graphs,which should be minimized in terms of the decision variablesx1 and x2 . The optimum position of X = [x1, x2] is [420.9867,420.9867] and the optimum value for this test function at this

Fig. 2. Flowchart of the proposed OSAMGSA.

point is zero. In this figure, different areas from the worstarea to the best one are indicated with red, heavy yellow, lightyellow, and blue, respectively.

As shown, it is a nonlinear optimization problem withmultiple local minima. The complexity of this test functionis because of its deep local optima being far from the globaloptimum. It will be hard to find the global optimum if manyparticles fall into one of these local optima. Fig. 4 illustratesthe enhanced exploitation and exploration capability of theproposed OSAMGSA with respect to GSA. The iterationnumber and number of population are taken 50 and 12 for the2-D, respectively. The results of Fig. 4 were obtained after the


Fig. 3. (a) 3-D graph of generalized Schwefel’s function. (b) Contour graphof generalized Schwefel’s function.

Fig. 4. (a.1), (a.2) Obtained results for the presented test function afterthe first iteration for respective GSA and proposed OSAMGSA. (b.1), (b.2)Obtained results for the presented test function after the final iteration forrespective GSA and proposed OSAMGSA.

first and final iteration of each optimization method. Althoughthe global search capability of the original GSA is very good,it is clear that after the final iteration, all agents concentrateon the global optimum only for the proposed technique.

B. Case Studies

To illustrate the efficiency of the proposed algorithm, fourcase studies were performed. In the first three case studies,loss, voltage deviation, and voltage stability index are op-timized individually. In the fourth case, multiobjective opti-mizations involve two and three objectives. All simulationsare done on 30-bus IEEE test system. The topology and thecomplete data of the IEEE 30-bus can be found in [29], andFig. 5 shows the one line diagram of this system.

All tap settings are considered within [0.9, 1.1] and theavailable reactive power of capacitor banks are within [0, 30]MVAR which are connected to busses 10 and 24. Voltagesare considered within the range of [0.9, 1.1]. The simula-tions search space has 14 dimensions, namely six generatorvoltages, four transformer taps, two capacitor banks, and twotuning parameters of the OSAMGSA algorithm. In TablesI–III, the bold values depict the optimum value of eachobjective function for single objective optimization cases. Fourcase studies for a typical system are proposed to show theperformance of the proposed approach:

Fig. 5. One line diagram of 30-bus IEEE test system.

Fig. 6. Convergence plots for GSA and OSAMGSA algorithms in case ofload voltage deviation objective function.

Case 1: Minimization of the load voltage deviation: Amajor operating task for power systems is to maintain theload bus voltages within the limits for high quality consumerservice. This target is defined by voltage deviation in powersystems, and computed for the 30-bus IEEE test system(Table I). As shown in Table I, it is clear that the proposedmethod is practically applicable to real systems and convergesto lower value in comparison with EA [30], PSO [32], CA[32], and original GSA algorithms.

Fig. 6 shows the convergence plot of load voltage deviationobtained by the proposed OSAMGSA and original GSAalgorithms. As shown, the proposed algorithm has a betterconvergence property compared with the original GSA. Thus,the proposed OSAMGSA method is faster and has a betterconvergence trend compared with the original GSA. Moreover,the value of the objective function settles at the minimum pointafter about 25 iterations, and does not vary thereafter; whilethe original GSA converges to global point in about at least38 iterations.

Case 2 Minimization of the power loss: Power system op-erators are always faced with the problem of how to minimizethe transmission loss. There is a number of ways for achievingthis goal. One of the best ways is reactive power supports ap-plied to the power system for minimizing transmission loss by


TABLE I

Results of Competitors in Voltage Control

of IEEE 30-Bus System

Initial EA PSO CA GSAOSAMGSA

setting [30] [32] [32] GSAVG1 1.05 1.037 1.05 1.005 0.9649 0.9714VG2 1.045 1.027 1.05 0.95 1.1 1.1VG5 1.01 1.013 0.95 1.05 1.0022 1.0022VG8 1.01 1.008 0.95 1.05 1.0199 1.02VG11 1.05 1.03 1.05 1.0021 1.0238 1.0206VG13 1.05 1.007 1.0156 1.0279 0.9855 0.9792t6–9 0.978 1.054 1.0335 1.0287 1.0059 1.0053t6–10 0.969 0.907 0.9532 0.95 0.9396 0.9382t4–12 0.932 0.928 0.9941 0.9929 0.9454 0.932t27–28 0.968 0.945 1.0222 1.0248 0.9595 0.9646Qc10 0.19 0.19 0.11131 0.00467 0.1694 0.15739Qc24 0.043 0.43 0.00734 0.00636 0.0436 0.0652Voltage

0.4993 0.1477 0.1393 0.1245 0.1133 .1126deviation

TABLE II

Results of Competitors in Power Loss of IEEE 30-Bus System

Initial EA PSO CA GSAOSAMGSA

setting [30] [32] [32]VG1 1.05 1.05 1.0408 0.95 1.05 1.05VG2 1.045 1.044 1.05 0.95 1.0079 1.0107VG5 1.01 1.024 0.95 0.95 0.9637 0.95VG8 1.01 1.026 0.95 0.9622 0.9542 0.9791VG11 1.05 1.093 1.05 0.9753 0.9661 0.95VG13 1.05 1.085 1 1.05 1.05 1.05t6–9 0.978 1.078 1.0329 0.9966 1.0988 0.9121t6–10 0.969 0.906 1.0132 1.05 1.0992 0.9t4–12 0.932 1.007 1.0007 1.0006 0.9 0.9263t27–28 0.968 0.959 1.0069 1.0073 1.0533 0.9222Qc10 0.19 0.19 0.18938 0.25 0.2476 0.27432Qc24 0.043 0.043 0.06281 0.06253 0.0852 0.06284Real power

5.3786 5.1065 5.0938 5.0933 5.0924 5.0713loss (MW)

optimizing compensator capacitors, transformers taps, and thevoltage of generator buses. The proposed algorithm is appliedto 30-bus IEEE test system to obtain the best transmissionloss by optimizing the control variables (Table II). For morevalidation they are compared with those in other literature.By comparing the obtained results in Table II, the proposedalgorithm can converge to lower value of real power loss withrespect to EA [30], PSO [32], CA [32], and original GSAalgorithms.

Moreover, the performance of metaheuristic search-basedoptimization algorithms is judged through many trials withdifferent initial populations to measure the robustness. Todiscuss the robustness of the proposed technique, Table IIshows that the mean, best, and the worst results of theproposed approach are 5.0713 MW, 5.0713 MW, and 5.0713MW, respectively, which are superior to the results of the bestsolution in Table II (i.e., CA [32] with best solution 5.0933).As shown, the OSAMGSA method produces better resultsmost consistently.

Case 3 Minimization of the voltage stability index: Tomaintain the voltage stability and move away from voltage

TABLE III

Result of Voltage Stability Index Objective Function

for Different Method

Initial EGA-DQLF PSO FAPSOOSAMGSA

setting [33] [33] [34] [34]VG(1) 1 1.0618 – – 1.05VG(2) 1 1.053 – – 1.049VG(5) 1 1.053 – – 1.048VG(8) 1 1.014 – – 1.05VG(11) 1 1.025 – – 1.03VG(13) 1 1.046 – – 1.048T6–9(p.u) 1 0.9125 – – 0.93T6–10(p.u) 1 0.9 – – 0.91T4–12(p.u) 1 0.9 – – 0.92T27–28(p.u) 1 0.925 – – 0.91Voltage stability

0.23 0.104020.130 0.123

0.1036index 7 8

collapse point, the maximum value of L-index among loadbuses should be minimized. This case considers the voltagestability index as an objective function. The obtained resultsare shown in Table III and are compared with EGA-DQLF[33], PSO [34], and FAPSO [34] for more validation. Asshown in Table III, the proposed algorithm can achieve betterresults than other approaches in the literature. The voltagestability index is an important factor in power system stability;therefore, its minimization is important for power systemdecision making. The proposed approach can be useful forpower system programmers and decision makers.

For this case study, we used the error analysis to depictthe robustness of the proposed OSAMGSA method. The erroris the average difference between the obtained best solutionand the global solution which indicates the ability of eachtechnique for reaching the global optima or near to the globaloptimum solution. To show the superiority of the proposedapproach, the error index is implemented and calculated foreach optimization method. The corresponding error of thePSO [34], FAPSO [34], EGA-DQLF [33], and proposedOSAMGSA method are 0.0271, 0.0202, 0.00042, and zero,respectively. The same justification can be applied to the othertest cases.

Computational efficiency: Table IV depicts the CPU execu-tion time of different algorithms for solving different cases.According to this table, the proposed algorithm can convergeto its global optima or near its global solution in a lowercomputation time. In other words, the proposed algorithm hasa higher convergence speed. Once again the superiority ofthe proposed algorithm in solving optimization problems isproved.

Case 4: Minimization of different objectives: simultane-ously having choices that satisfy our purpose could help selectan economical and practical choice. The Pareto-optimal solu-tion, which can obtain a squad of results is a very significantand acceptable method for the multiobjective optimizationproblem. In this paper, Pareto-optimal fronts are obtained forthe proposed objective functions and the best compromisesolution among all attained solutions is shown in red. In eachPareto front, the solutions that are nondominated during theentire search space are denoted as Pareto optimal. In general,


TABLE IV

Comparison of CPU Time of Different Algorithms

for Different Cases

Solution techniqueCase Studies

CPU time CPU time of CPU time ofof Case 1 (s) Case 2 (s) Case 3 (s)

EA [30] NA NA NAH-PSO [31] NA NA NACA [32] 0.76 0.91 NAEGA-DQLF [33] NA NA NAPSO [34] 11.3 11.3 NAFAPSO [34] 13.6 13.6 NAGSA 3.4 4.1 6.6Proposed OSAMGSA 4.0 4.9 7.8

NA: not available in the literature.

the target of a multiobjective optimization algorithm is notonly guiding the search toward the Pareto-optimal front, butalso obtaining a set of nondominated solutions. Therefore, inthe proposed algorithm we defined a repository to save allnondominated solutions in each iteration. Afterward, savedsolutions in the repository for all iterations are sorted by atype of decision making. Sorted solutions are Pareto-optimalset and we can select the best solution set by choosing thetop solutions in this assortment. The used decision makingfunction in this paper is defined as follows [27]:

Nμ(j) =

n∑k=1

ωk × μjk

m∑j=1

n∑k=1

ωk × μjk

(38)

where ωk is the weight for the kth objective functions and m

is the number of nondominated solutions.Since the objectives conflict with each other in multiobjec-

tive optimization, by optimizing the value of one objective,the value of another objective is increased. For example, inFig. 7(a), minimizing voltage deviation causes the systemvoltage stability index to increase. In this case, three objectivefunctions are optimized two by two simultaneously and theirPareto fronts are shown. From the Pareto-ptimal fronts, it canbe perceived that the proposed OSAMGSA method is ableto give well distributed solutions. Fig. 7(a) depicts the Paretooptimal front for voltage deviation and voltages stability indexobjective functions; the best compromise solution obtainedis 0.1379 and 0.1082 for deviation and voltages stabilityindex, respectively. Also, Fig. 7(b) displays the obtained Paretooptimal front for loss and voltage stability index minimization;the best obtained compromise solution is 5.303 and 0.1176MW. The loss and voltage stability index are both very closeto the optimized values using single objective optimization,which shows that these objectives do not conflict with eachother. Fig. 7(c) shows the Pareto-optimal front for loss andvoltages deviation objective functions, and the best obtainedcompromise solution is 0.1174 and 6.87 MW.

In the following sections, we present three competing objec-tives. These three objectives are optimized simultaneously bythe proposed OSAMGSA method, Fig. 8 shows their obtained

Fig. 7. 2-D Pareto-optimal fronts for different objective functions.

Fig. 8. Pareto-optimal front for voltage deviation, loss, and voltage stabilityindex functions.

Pareto-optimal set and the best compromise is depicted withred.

Performance of multiobjective optimization: In this pa-per, an experiment is presented in order to evaluate theOSAMGSA’s performance by looking for the compromisesolutions of Case 4 for 2 and 3-D Pareto-optimal fronts.Hence, the average satisfactory degree (ASD) of the decisionmaker with compromise solution is calculated according to themaximum and minimum of each objective function using (14).For a compromise solution, Xbest, its ASD value is calculated


TABLE V

Comparison of ASD for Different Dimensional of Pareto

Optimal Fronts of GSA and Proposed OSAMGSA

Solution techniqueASD

L-VSI VD-VSI VD-L L-VSI-VDGSA 0.7106 0.6795 0.6157 0.6598Proposed OSAMGSA 0.9315 0.8218 0.7812 0.8113

as follows:

ASD =1

Nobj

Nobj∑i=1

μi. (39)

Table V compares the obtained results of the proposedOSAMGSA technique for the compromise solution withthe obtained results of the original GSA for 2-D Pareto-optimal fronts like loss-voltage stability index (L-VSI), voltagedeviation-voltage stability index (VD-VSI), voltage deviation-loss (VD-L) and 3-D Pareto-optimal front, i.e., loss-voltagestability index-voltage deviation (L-VSI-VD). As shown inTable V, the OSAMGSA method attains the best compromisesolutions with respect to the original GSA technique. There-fore, the OSAMGSA algorithm is more efficient than GSAin providing a better compromise solution for multiobjectiveoptimization problems.

To sum up, in this paper the OSAMGSA is used to solvethe MORPD. Tables I–III present the obtained optimal controlsettings and related objective function values while the pro-posed problem is assumed as a single objective optimizationproblem for voltage deviation, transmission loss, and voltagestability index, respectively. The MORPD is then solved usingthe OSAMGSA algorithm (see Figs. 7 and 8). The resultsreveal that the OSAMGSA method provides well-distributedPareto-optimal solutions and it proves that this algorithm issuitable for solving the multiobjective optimization problem.Also, these results are not negligible due to the continuousoperations of reactive power dispatch throughout the years aswell as the numerous power plants in worldwide.

VIII. Conclusion

This paper presented a multiobjective OSAMGSA for re-active power and voltage control considering load voltagedeviation, transmission loss, and voltage stability index. Thecontinuous and discrete control variables involving generatorbus voltage, tap positions of transformers and reactive powerof compensation capacitor were considered simultaneously forsolving the proposed problem. The proposed method, based ona nondominated solution, was used to find the Pareto-optimalsolutions for the multiobjective problem. This approach pro-vides the decision maker with a selection opportunity, wherehe/she can consider the applicable and economical subjectsand choose the most appropriate solution. Finally, a fuzzydecision approach was used to identify the best compromisesolution. The obtained results guaranteed that the OSAMGSAis a stochastic optimization algorithm which can solve com-plicated problems as well as other evolutionary algorithms

such as the PSO, GA, and EA. Furthermore, it can obtainbetter results and it is also suitable for solving multiobjectiveproblems as well single objective optimization problems.

References

[1] T. Niknam, M. R. Narimani, and R. Azizipanah-Abarghooee, “A newhybrid algorithm for optimal power flow considering prohibited zonesand valve point effect,” Energy Convers. Manage., vol. 58, pp. 197–206,Jun. 2012.

[2] K. Zehar and S. Sayah, “Optimal power flow with environmental con-straint using a fast successive linear programming algorithm: Applicationto the Algerian power system,” Energy Convers. Manage., vol. 49, no.11, pp. 3361–3365, 2008

[3] T. Niknam, M. R. Narimani, J. Aghaie, and M. Nayeripour, “Modifiedhoney bee mating optimization to solve dynamic optimal power flowwith considering generator constraints,” IET Gener. Transm. Distrib.,vol. 5, no. 10, pp. 989–1002, 2011.

[4] L. Shi, C. Wang, L. Yao, Y. Ni, and M. Bazargan, “Optimal power flowsolution incorporating wind power,” IEEE Syst. J., vol. 6, no. 2, pp.233–241, Jun. 2012.

[5] T. Niknam and F. Golestaneh, “Enhanced bee swarm optimiza-tion algorithm for dynamic economic dispatch,” IEEE Syst. J., doi10.1109/JSYST.2012.2191831.

[6] T. Niknam, R. Azizipanah-Abarghooee, and A. Roosta, “Reserve con-strained dynamic economic dispatch: A new fast self-adaptive modifiedfirefly algorithm,” IEEE Syst. J., vol. 6, no. 4, pp. 635–646, Dec. 2012.

[7] Y. Mishra, S. Mishra, and F. Li, “Coordinated tuning of DFIG-basedwind turbines and batteries using bacteria foraging technique for main-taining constant grid power output,” IEEE Syst. J., vol. 6, no. 1, pp.16–26, Mar. 2012.

[8] F. M. Gonzalez-Longatt, P. Wall, P. Regulski, and V. Terzija, “Optimalelectric network design for a large offshore wind farm based on amodified genetic algorithm approach,” IEEE Syst. J., vol. 6, no. 1, pp.164–172, Mar. 2012.

[9] N. Deeb and S. M. Shaidepour, “Linear reactive power optimization ina large power network using the decomposition approach,” IEEE Trans.Power Syst., vol. 5, no. 2, pp. 428–435, May 1990.

[10] J. R. S. Monlovani and A. V. Garcia, “A heuristic method for reactivepower planning,” IEEE Trans. Power Syst., vol. 11, no. 1, pp. 68–74,Feb. 1996.

[11] S. Granville, “Optimal reactive dispatch through interior point meth-ods,” IEEE Trans. Power Syst., vol. 9, no. 1, pp. 136–146,Feb. 1994.

[12] K. Y . Lee and Y . M. Park, “Optimization method for reactive powerplanning by using a modified simple genetic algorithm,” IEEE Trans.Power Syst., vol. 10, no. 4, pp. 1843–1850, Nov. 1995.

[13] L. L. Lai and J. T. Ma, “Application of evolutionary programming toreactive power planning approach,” IEEE Trans Power Syst., vol. 12,no. 1, pp. 198–206, Feb. 1997.

[14] B. Zhao, C. X. Guo, and Y. J. Cao, “A multiagent based particle swarmoptimization approach for reactive power dispatch,” IEEE Trans. PowerSyst., vol. 20, no. 2, pp. 1070–1078, May 2005.

[15] C. H. Liang, C. Y. Chung, K. P. Wong, and X. Z. Duan, “Parallel optimalreactive power flow based on cooperative co-evolutionary differentialevolution and power system decomposition,” IEEE Trans. Power Syst.,vol. 22, no. 1, pp. 249–257, Feb. 2007.

[16] M. Varadarajan and K. S. Swarup, “Differential evolution approach foroptimal reactive power dispatch,” Appl. Soft Comput., vol. 8, no. 4, pp.1549–1561, 2008.

[17] K. Mahadevan and P. S. Kannan, “Comprehensive learning particleswarm optimization for reactive power dispatch,” Appl. Softw. Comput.,vol. 10, no. 2, pp. 641–652, 2010.

[18] P. Subbaraj and P. N. Rajnarayanan, “Optimal reactive power dispatchusing self-adaptive real coded genetic algorithm,” Electric. Power Syst.Res., vol. 79, no. 2, pp. 374–381, 2009.

[19] F. C. Lu and Y. Y. Hsu, “Reactive power/voltage control in a distributionsubstation using dynamic programming,” IEEE Proc. Gen. Transm.Distrib, vol. 142, no. 6, pp. 639–645, Nov. 1995.

[20] M. O. Mansour and T. M. Abdel-Rahman, “Non-linear VAR optimiza-tion using decomposition and coordination,” IEEE Trans. Power Syst.,vol. 9, no. 2, pp. 597–598, May 1994.

[21] E. Rashedi, H. Nezamabadi-pour, and S. Saryazdi, “GSA: A gravita-tional search algorithm,” Info. Sci., vol. 179, no. 13, pp. 2232–2248,2009.


[22] B. Shaw, V. Mukherjee, and S. P. Ghoshal, “A novel opposition-basedgravitational search algorithm for combined economic and emissiondispatch problems of power systems,” Electric. Power Energy Syst., vol.35, no. 1, pp. 21–33, 2012.

[23] T. Niknam, M. R. Narimani, M. Jabbari, and A. R. Malekpour, “Amodified shuffle frog leaping algorithm for multiobjective optimal powerflow,” Energy, vol. 36, no. 11, pp. 6420–6432, 2011.

[24] T. Niknam, M. R. Narimani, J. Aghaei, and R. Azizipanah-Abarghooee,“Improved particle swarm optimization for multiobjective optimal powerflow considering the cost, loss, emission, and voltage stability index,”IET Gener. Transm. Distrib., vol. 6, no. 6, pp. 515–527, Jun. 2012.

[25] P. Kessel and H. Glavitsch, “Estimating the voltage stability of a powersystem,” IEEE Trans. Power Del., vol. 1, no. 3, pp. 346–354, Jul. 1986.

[26] W. Zhang and Y. Liu, “Multiobjective reactive power and voltage controlbased on fuzzy optimization strategy and fuzzy adaptive particle swarm,”Electric. Power Energy Syst., vol. 30, no. 9, pp. 525–532, 2008.

[27] T. Niknam, F. Golestaneh, and M. Sha Sadeghi, “θ-multiobjectiveteaching–learning-based optimization for dynamic economic emissiondispatch,” IEEE Syst. J., vol. 6, no. 2, pp. 341–352, Jun. 2012.

[28] D. Halliday, R. Resnick, and J. Walker, Fundamamentals of Physics.New York: Wiley, 1993.

[29] The University of Washington Electrical Engineering. (2012). PowerSystem Test Case Archive, the IEEE 30-Bus Test System Data [Online].Available: http://www.ee.washington.edu/research/pstca/pf30/pg tca30bus.htm

[30] M. A. Abido and J. M. Bakhashwain, “Optimal VAR dispatch usinga multiobjective evolutionary algorithm,” Int. J. Electr. Power EnergySyst., vol. 27, no. 1, pp. 13–20, 2005.

[31] A. A. A. Esmin, G. Lambert-Torres, and A. C. Z. De Souza, “A hybridparticle swarm optimization applied to loss power minimization,” IEEETrans Power Syst., vol. 20, no. 2, pp. 859–866, May 2005.

[32] J. G. Vlachogiannis and K. Y. Lee, “Coordinated aggregation particleswarm optimization applied in reactive power and voltage control,” inProc. IEEE Power Eng. Soc. General Meeting, 2006, paper 06GM0780.

[33] M. Sailaja Kumari, S. Maheswarapu, “Enhanced genetic algorithm basedcomputation technique for multiobjective optimal power flow solution,”Electric. Power Energy Syst., vol. 32, no. 6, pp. 736–742, 2010.

[34] W. Zhang and Y. Liu, “Multiobjective reactive power and voltage controlbased on fuzzy optimization strategy and fuzzy adaptive particle swarm,”Electric. Power Energy Syst., vol. 30, no. 9, pp. 525–532, 2008.

Taher Niknam was born in Shiraz, Iran. He re-ceived the B.S., M.S., and Ph.D. degrees from ShirazUniversity, Shiraz, Iran, and the Sharif University ofTechnology, Tehran, Iran.

He is currently a Faculty Member with the Depart-ment of Electrical Engineering, Shiraz University ofTechnology. His current research interests includepower system restructuring, impact of distributedgenerations on power systems optimization methods,and evolutionary algorithms.

Mohammad Rasoul Narimani was born in Ker-manshah, Iran. He received the B.Sc. degree inelectric power engineering from Razi University,Kermanshah, Iran. He is currently pursuing the M.S.degree with the Shiraz University of Technology,Shiraz, Iran.

He is currently teaching at the Department ofElectrical Engineering, Marvdasht Branch, IslamicAzad University, Marvdasht, Iran. His current re-search interests include power system operation andevolutionary algorithms.

Rasoul Azizipanah-Abarghooee was born in Yazd,Iran, in 1987. He received the B.Sc. and M.S. de-grees in electric power engineering from the IsfahanUniversity of Technology, Isfahan, Iran, in 2008, andthe M.S. degree in electrical engineering from theShiraz University of Technology, Shiraz, Iran, wherehe is currently pursuing the Ph.D. degree in electricalengineering.

Currently, he is teaching at the Department ofElectrical Engineering, Marvdasht Branch, IslamicAzad University, Marvdasht, Iran. His current re-

search interests include power system operation and restructuring, microgrid,unit commitment, optimization methods, and evolutionary algorithms.

Bahman Bahmani-Firouzi was born in 1975. Hereceived the B.Sc. degree in power engineering fromShiraz University, Shiraz, Iran, in 1999, and theM.Sc. degree in power engineering from the SharifUniversity of Technology, Tehran, Iran, in 2002,where he is currently pursuing the Ph.D. degree inpower engineering.

His current research interests include power sys-tem planning, computational intelligence applica-tions, and power system protection.

Documents

30 Bus System 2 Capacitor