Electrical Power and Energy Systems 43 (2012) 562–572

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Electrical Power and Energy Systems

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Optimal allocation of shunt Var compensators in power systems usinga novel global harmony search algorithm

Reza Sirjani ⇑, Azah Mohamed, Hussain ShareefDepartment of Electrical, Electronic and Systems Engineering, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor, Malaysia

a r t i c l e i n f o

Article history:Received 13 March 2012Received in revised form 25 May 2012Accepted 29 May 2012Available online 7 July 2012

Keywords:Shunt Var compensatorsPower lossModal analysisVoltage profileHarmony search algorithm

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.05.068

⇑ Corresponding author. Tel.: +60 166191510.E-mail address: sirjani@eng.ukm.my (R. Sirjani).

a b s t r a c t

In this paper, a novel global harmony search algorithm (NGHS) is used to determine the optimal locationand size of shunt reactive power compensators such as shunt capacitors, static Var compensators (SVCs),and static synchronous compensators (STATCOMs) in a transmission network. The problem is decom-posed into two subproblems. The first deals with the optimal placement of shunt Var compensationdevices using the modal analysis method. The second subproblem is the optimization of the load flowusing the NGHS algorithm. A multi-criterion objective function is defined to enhance the voltage stability,improve the voltage profile, and minimize power loss while minimizing the total cost. The results from a57-bus test system show that the NGHS algorithm causes lower power loss and has better voltage profileand greater voltage stability than the improved harmony search algorithm (IHS) and particle swarm opti-mization (PSO) techniques in solving the placement and sizing problem of shunt Var compensators.Finally, a comparison of the convergence characteristics of three optimization methods demonstratesthe greater accuracy and higher speed of the proposed NGHS algorithm in finding better solutions com-pared with PSO and IHS.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Shunt compensation is used to influence the natural electricalcharacteristics of transmission lines to increase the steady-statetransmittable power and control the voltage profile along the line[1]. Providing adequate reactive power support at the appropriatelocation not only leads to a reduction in the power loss andimprovement in the voltage profile, but also solves voltage insta-bility problems. Many reactive compensation devices are used bymodern electric power utilities for this purpose, and each devicehas its own characteristics and limitations. At present, utilitiesaim to achieve this purpose using the most beneficial compensa-tion device [2].

Traditionally, shunt capacitors are installed in power networksto compensate for reactive power. They are used for many pur-poses, such as power loss reduction, voltage profile improvement,and increasing the maximum transmitted power in cables andtransformers [3]. Among the reactive compensation devices, shuntflexible AC transmission system (FACTS) devices play an importantrole in controlling the flow of reactive power to the power net-work, thereby affecting the system voltage fluctuations and stabil-ity [4]. The static Var compensator (SVC) is the most widely usedshunt FACTS device in power networks because of its low cost

ll rights reserved.

and good performance in system enhancement. It is a shunt-connected static Var generator or absorber with an adjustableoutput, which allows the exchange of the capacitive or inductivecurrent so as to provide voltage support. When installed at a prop-er location, the SVC can also reduce power losses [5]. The staticsynchronous compensator (STATCOM) is also a shunt compensatorand one of the important members of the FACTS family that areincreasingly being used in long transmission lines in modernpower systems. STATCOMs can have various applications in theoperation and control of a power system, such as in power flowscheduling, reducing the number of unsymmetrical componentsthat damp the power oscillations, and enhancing the transientstability [3]. The shunt capacitor, SVC, and STATCOM increase thestatic voltage stability margin and power transfer capability. How-ever, SVC and STACOM perform better in terms of reducing the lossand improving the voltage profile [2]. The increase in losses when ashunt capacitor is used under lightly loaded conditions is due topoor voltage profile. Overall, SVC and STACOM behave better thana simple shunt capacitor; however, these controllers are moreexpensive [2]. The benefits of reactive power compensation greatlydepend on the placement and size of the added compensators. Theinstallation of shunt controllers in all buses is impossible andunnecessary because of economical considerations. Identifyingthe best location for Var compensators involves the calculation ofsteady-state conditions for the network. However, the problem be-comes highly complex because of the nonlinearity of the load flow

Table 1Cost comparison of shunt controllers (US $/kVar).

Shunt controller Cost (US $/kVar)

Shunt capacitor 8SVC 40 (controlled portion)STATCOM 50 (controlled portion)

R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572 563

equations, and an extensive investigation has to be undertaken inorder to solve it. Several studies on the use of these controllersfor voltage and angle stability applications have been conductedand reported in literature. A variety of techniques are used to opti-mize the allocation of these devices in power systems. At present,the shunt Var compensator placement problem is generally solvedusing evolutionary programming methods.

Numerous techniques for solving the optimal capacitor place-ment problem in power systems have been reported. These tech-niques may be classified into the following categories: analytical,numerical programming, heuristic, and artificial intelligence-basedtechniques [6]. Of these, the heuristic-based techniques have beenwidely applied in solving the optimal capacitor placement problem[7]. Several methods and approaches to determining the optimallocation of SVC in the power system have been reported in litera-ture, and different techniques, such as the genetic algorithm(GA), simulated annealing (SA), artificial immune system (AIS),and particle swarm optimization (PSO), have been used [8]. A solu-tion algorithm based on SA is used to determine the location, types,and sizes of Var sources, as well as their settings at different load-ing conditions [9]. The purchase, installation, and total costs of en-ergy loss over the life of Var sources are minimized by consideringthe operational constraints [9]. In [10], GA is used to determine thebest location of only one SVC within a power system, in which theobjective function is to reduce the power loss, voltage deviation,and cost. An AIS technique is used to minimize the total loss andimprove the voltage in a power system [11] by determining thecorrect SVC placement. The well-known PSO is explored in [12]to obtain the optimal locations of SVCs in the IEEE 30 bus system.In [13], a new method is proposed to optimally locate the STAT-COMs in distribution networks. The suggested approach uses sen-sitivity analysis and a genetic algorithm (GA). A step-by-stepsensitivity analysis approach is utilized to determine the optimalplacement of compensators. The objective function takes into ac-count the voltage stability, the reduction of active losses, and thereduction of the reactive power of the network. The use of PSOfor the sizing and location of a STATCOM in a power system whileconsidering voltage deviation constraints is demonstrated in [14].Results from their illustrative example show that the PSO algo-rithm can find the best size and location, with statistical signifi-cance and a high degree of convergence, when evaluating theminimum, maximum, average, and standard deviation values ofthe voltage deviation metric. The performance of the enhancedPSO algorithm is compared with classical optimization approachesin [15] using a case study of the optimal allocation of STATCOM de-vices while considering steady-state and economic criteria. Theapplication of PSO and the continuation power flow (CPF) for theoptimal allocation of multiple STATCOMs is also introduced in[4]. The goal of the optimization is to improve the voltage profile,minimize the total power system loss, and maximize the systemloadability while considering the size of the STATCOMs.

This paper uses a new optimization technique, referred to as thenovel global harmony search (NGHS) algorithm, to find the opti-mum placement and sizing of shunt Var compensators in powersystems. The harmony search (HS) algorithm is a metaheuristicoptimization method that is inspired by musicians adjusting thepitches of their instruments to find better harmony [16]. It has sev-eral advantages over other methods, one of which is that it doesnot require initial value settings for the decision variables, and thatit can handle both discrete and continuous variables.

The HS algorithm is successfully applied to a wide range of opti-mization problems, especially in electrical engineering. In [17], aharmony search approach is presented for annual reconfigurationin electric distribution networks considering annual load leveland switching costs. A multi-objective harmony search (MOHS)algorithm for optimal power flow (OPF) problem is proposed in

[18]. The HS algorithm is implemented to the optimal reactivepower dispatch problem for determination of the global or nearglobal optimum solution [19]. In [20], the concept of opposition-based learning is employed to accelerate the HS algorithm. The po-tential of the proposed algorithm is assessed by means of an exten-sive comparative study of the solution obtained for four standardcombined economic and emission dispatch problems of power sys-tems [20]. The application of a hybrid harmony search algorithm tothe Spread-Spectrum Radar Polyphase (SSRP) codes design is pre-sented in [21]. The results have demonstrated that the proposedhybrid harmony search algorithm is a robust algorithm capableto outperform previous heuristic approaches tailored for this spe-cific optimization problem. A harmony search approach has beenpresented for centralized and distributed spectrum channel assign-ment in cognitive wireless networks [22]. In the proposed algo-rithm, a novel single-parametric logarithmic progression of theparameters has been employed which allowed balancing the trade-off between the explorative and exploitative behavior of the heu-ristic allocation procedure [22].

Recently, the novel global harmony search (NGHS) algorithm isproposed to solve complex problems [23,24]. The NGHS is a mod-ified version of the HS algorithm, and is inspired by the swarmintelligence of the PSO algorithm. The NGHS is a simple but prac-tical optimization algorithm. In previous works, the authors ofthe present paper have successfully applied the improved harmonysearch (IHS) algorithm for determining the optimal location andsize of shunt capacitors in a radial distribution network [7]. Fur-thermore, a multi-objective Improved Harmony Search (IHS) algo-rithm has been employed to optimal placement and sizing of onlySVCs in a transmission system [5]. In this paper, a novel global har-mony search (NGHS) algorithm is used to optimal placement andsizing of multi-Var compensators in power systems.

In the present paper, the suitable buses are first identified usingmodal analysis. The NGHS algorithm is then used to determine theamount of shunt compensation required to minimize loss and im-prove the voltage and voltage stability with respect to the totalcost. The results obtained using the proposed algorithms in a 57-bus test system are compared with that of other optimizationmethods for validation.

2. Shunt capacitor, SVC, and STATCOM

Shunt compensation can be used to provide reactive powercompensation. Traditional shunt capacitors or newly introducedFACTS controllers can be used for this purpose. FACTS controllersare very expensive; Table 1 gives an idea of the cost of variousshunt controllers [25,26]. Descriptions of each of these controllersare given in the next subsections [2].

2.1. Shunt capacitor

Shunt capacitors are relatively inexpensive to install and main-tain, and installing them in the load area or at the point where theyare needed increases the voltage stability. However, They havepoor voltage regulation and, beyond a certain level of compensa-tion, a stable operating point is unattainable. Furthermore, thereactive power delivered by the shunt capacitor is proportional

564 R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572

to the square of the terminal voltage. Var support drops in low-voltage conditions, thereby compounding the problem [27].

2.2. SVC

The SVC is a shunt-connected static Var generator/load whoseoutput is adjusted to exchange capacitive or inductive current soas to maintain or control a specific power system variable [28,29].

Typically, the power system control variable is the terminal busvoltage. SVCs have two popular configurations. One configurationconsists of a fixed capacitor (FC) and a thyristor-controlled reactor(TCR), and the other consists of a thyristor-switched capacitor(TSC) and TCR. In the limit of minimum or maximum susceptance,SVC behaves like a fixed capacitor or an inductor. The choice of theappropriate size is one of the important issues in the application ofSVCs in voltage stability enhancement [2].

2.3. STATCOM

STATCOM is a voltage-source, converter-based device that con-verts a DC input voltage into an AC output voltage to compensatefor the active and reactive needs of the system. STATCOMs have bet-ter characteristics than SVCs. When the system voltage drops suffi-ciently to force the STATCOM output to its ceiling, its maximumreactive power output is not affected by the voltage magnitude.Therefore, the STATCOM exhibits constant current characteristicswhen the voltage is low under the limit [29].

A schematic diagram of an SVC and a STATCOM are shown inFig. 1. Fig. 2 shows the terminal characteristics of the shunt capac-itor, SVC, and STATCOM [2].

Fig. 1. Basic structure of (a) a static Var compensator (SVC);

Fig. 2. Terminal characteristics of (a) the shu

3. Modal analysis

The modal analysis method provides an accurate estimate of thesystem proximity to instability using the system eigenvalues andidentifies the elements of the power system that have the highestcontributions to the incipient voltage instability (i.e., critical loadbuses, branches, and generators) [30]. In this method, the Jacobianmatrix of the operating point of the power system is calculated[27]. For this purpose, the power flow equation linearized aboutthe operating point is used as follows:

DP

DQ

� �¼

JPh JPV

JQh JQV

" #Dh

DV

� �ð1Þ

Here, DP is the incremental change in the bus active power, DQ isthe incremental change in the bus reactive power, Dh is the incre-mental change in the bus voltage angle, and DV is the incrementalchange in the bus voltage magnitude. Jph, JPV, JQh, and JQV are the ele-ments of the Jacobian matrix that represent the sensitivity of thepower flow to bus voltage changes.

System voltage stability is affected by both P and Q; however, ateach operating point, we may keep P constant and evaluate thevoltage stability by considering the incremental relationship be-tween Q and V.

If the active power P is kept constant in Eq. (1), then DP = 0, and

DQ ¼ JRDV ð2Þ

where JR is the reduced Jacobian system matrix and is given by

JR ¼ ½JQV � JQhJ�1Ph JPV � ð3Þ

and (b) a static synchronous compensator (STATCOM).

nt capacitor; (b) SVC; and (c) STATCOM.

R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572 565

The modes of the power network can be defined by the eigenvaluesand eigenvectors of JR. Assume that

JR ¼ nKg ð4Þ

where n is the right eigenvector matrix of JR, g is the left eigenvectormatrix of JR, and k is the diagonal eigenvalue matrix of JR

J�1R ¼ nK�1g ð5Þ

The relationship between the incremental changes in the reactivepower and voltage is expressed by Eq. (2). Substituting Eq. (5) intoEq. (2), we obtain

DV ¼ ðnK�1gÞDQ ð6Þ

or

DV ¼X

i

nigi

ki

� �DQ ð7Þ

where ki is the ith eigenvalue, ni is the ith column right eigenvectorof JR, and gi is the ith row left eigenvector of JR. ki; ni, and gi definethe ith mode of the system.

Therefore, the ith modal reactive power variation is

DQ mi ¼ jini ð8Þ

where ki is a normalization factor, such that

j2i

XJ

n2ij ¼ 1 ð9Þ

and nij is the jth element of ni.Therefore, the ith modal voltage var-iation can be written as

DVmi ¼ 1=kiDQ mi ð10Þ

From Eq. (10), the stability of a mode i with respect to reactivepower changes is defined by the modal eigenvalue ki. Large valuesof ki imply small changes in the modal voltage for reactive powerchanges. As the system becomes stressed, the value of ki becomessmaller and the modal voltage becomes weaker. If the magnitudeof ki is equal to zero, the corresponding modal voltage collapses be-cause it undergoes an infinite change for a finite reactive powerchange. A system is therefore defined as voltage-stable if all theeigenvalues of JR are positive. The bifurcation, or voltage stabilitylimit, is reached when at least one eigenvalue reaches zero, i.e.,when one or more modal voltages collapse. If any of the eigen-values are negative, the system is unstable. The magnitude of theeigenvalues provides a relative measure of the proximity of thesystem to instability. The critical modes (which are associated withthe minimum eigenvalues) are of major importance in the voltagestability analysis [30].

The left and right eigenvectors corresponding to the criticalmodes in the system can provide information on the mechanismof voltage instability by identifying the elements participating inthese modes. The bus participation factor measuring the participa-tion of the kth bus in the ith mode can be defined as

Pki ¼ nkigik ð11Þ

Bus participation factors corresponding to the critical modescan predict areas or nodes in the power system that are susceptibleto voltage instability. Buses with large participation factors in thecritical mode are the most critical system buses.

4. Optimization problem formulation

A multi-objective function consisting of shunt Var compensatorsize was considered in searching for an optimal solution. This mul-

ti-objective function, which not only maximizes the voltage stabil-ity margin but also minimizes voltage deviation, active-power loss,and cost, is explained below [5,10].

Active power loss minimization: The total active power loss in anelectric power system is given by

Ploss ¼Xn

l¼1

RlI2l

¼Xn

l¼1

Xn

j¼1;i–j

V2i þ V2

j � 2ViVj cosðhi � hjÞh i

Yii cos uij; ð12Þ

where n is the number of lines, Rl is the resistance of line l, Il is thecurrent through line l, Vi and hi are the voltage magnitude and angleat node i, respectively, and Yij and uij are the magnitude and angleof the line admittance, respectively.

Voltage deviation minimization: The voltage improvement indexfor a power system is defined as the deviation from unity of thevoltage magnitudes of all the buses. Thus, for a given system, thevoltage improvement index is defined as

Lv ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

Viref � Vi

Viref

� �2s

ð13Þ

where n is the number of buses, Viref is the reference voltage at bus i,and Vi is the actual voltage at bus i.

Voltage stability margin maximization: From the voltage stabilityviewpoint, the critical modes with the lowest eigenvalues are ex-tremely important. The minimum eigenvalue should be increasedto maximize stability margin [31].

Cost minimization: The cost of shunt controllers in US$/kVar areshown in Table 1. The total Var compensator cost is given by

Costshunt ¼XN

k¼1

CostðQ kÞ ð14Þ

where Cost(Qk) is the cost of the kth installed shunt Var compensa-tor in kVar.

4.1. Operational constraints

The objective of using Var compensators is to control the sys-tem variables, such as the real and reactive line power flows andbus voltages. Hence, the following constraints are considered:

Power flow balance equations: A balance between active andreactive powers must be attained in each node. The power balancewith respect to a bus can be formulated as follows:

PGi � PLi ¼ Vi

Xb

j¼1

½Vj½G0ij cosðdi � djÞ þ B0ij sinðdi � djÞ�� ð15Þ

QGi � Q Li ¼ Vi

Xb

j¼1

½Vj½G0ij sinðdi � djÞ � G0ij sinðdi � djÞ�� ð16Þ

where PGi and QGi are the generated active and reactive powers,respectively, and PLi and QLi are the active and reactive power loadsat node i, respectively. The conductance G0ik and the susceptance B0ikrepresent the real and imaginary components of element Y 0ij of the½Y 0bb� matrix, respectively, which are obtained by modifying the ini-tial nodal admittance matrix during STATCOM introduction.

Power flow limit: The apparent power transmitted through abranch l must not exceed a limiting value Slmax, which representsthe thermal limit of the line or transformer in a steady-stateoperation:

Sl 6 Sl max ð17Þ

566 R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572

Bus voltage limits: The bus voltages must be maintained withinlimits around the nominal value because of several factors, such asstability and power quality. This limit is given by

Vi min 6 Vinom 6 Vi max ð18Þ

In practice, the accepted deviations can be up to 10% of the nominalvalues [5,10].

Compensation device size: Commercially available Var compen-sators have constraints. This study is performed under the restric-tion that the injected Q does not exceed 50 MVar. The types ofsuitable Var compensators are assumed based on the injected Q,as shown in Table 2.

4.2. Fitness function

The fitness function for solving the optimal placement problemof shunt Var compensators is calculated using Eqs. (12)–(14). Theconstraints of this problem do not explicitly contain the variables.Therefore, the effect of the constraints must be included in the fit-ness function value. The constraints are separately checked, andthe violations are handled using a penalty function approach.Incorporating all of the constraints in the same mathematical func-tion is impossible because the three objectives are different. Thus,an overall fitness function, in which each objective function is nor-malized with respect to the base system without a Var compensa-tor, is considered. This fitness function is given by

f ðxÞ ¼ x1PlossP

DLossbaseþx2

LvPDVbase

þx3kCrticalðbaseÞ

kCriticalþx4

Costshunt

CostMax

� v1 �Xnr

i¼1

bali � v2 �Xn

k¼1

thermalk � v3 �Xn

k¼1

voltagek ð19Þ

Subject to:

x1 þx2 þx3 þx4 ¼ 10 < x1;x2;x3;x4 < 1

ð20Þ

where Ploss, Lv, kCritical, and Costshunt are the total active power loss,voltage deviation index, smallest eigenvalue, and total Var compen-sator cost, respectively. x1, x2, x3, and x4 are the coefficients ofthe corresponding objective functions,

PDLossbase is the total active

power loss in the network of the base system,P

Vbase is the totalvoltage deviation of the base system, kCritical(base) is the smallesteigenvalue of the base case, and CostMax is the maximum cost.

The bali element is a factor equal to 0 if the power balance con-straint at bus i is not violated; otherwise, bali is equal to 1. The sumof these violations represents the total number of buses in the net-work that do not follow constraints (15) and (16). This sum is mul-tiplied by a penalty factor that increases the fitness function to anunacceptable value, which results in an unfeasible solution thatmust be discarded. The second and third sums in the fitness func-tion represent the total number of violations of constraints (17)and (18), respectively, and are also multiplied by penalty factors.The last three sums in this fitness function are the measures ofunfeasibility for each candidate solution. The penalty factors usedin this study are v1, v2, and v3, which are each set to 100 [5,10].

Table 2Types of suitable Var compensators based on the injected Q.

Shunt controller Qmin (MVar) Qmax (MVar)

Capacitor �10 10SVC �30 30STATCOM �50 50

5. Harmony search algorithm

The HS algorithm is a metaheuristic optimization algorithm in-spired by the playing of music. It uses rules and randomness toimitate natural phenomena. Inspired by the cooperation withinan orchestra, the HS algorithm achieves an optimal solution byfinding the best ‘‘harmony’’ among the system components in-volved in a process. Just as discrete musical notes can be playedbased on a player’s experience or on random processes in improv-ization, the optimal design variables in a system can be obtainedwith certain discrete values based on computational intelligenceand random processes [16]. Musicians improve their experiencebased on aesthetic standards, whereas design variables can be im-proved based on an objective function. The HS algorithm looks forthe vector or the path of X that reduces the computational cost orshortens the path. The computational procedure for the HS algo-rithm is as follows [32]:

Step 1: Initialization of the optimization problem

Consider an optimization problem that is described by

Minimize FðxÞ subject to xi 2 Xi; i ¼ 1;2;3; . . . N ð21Þ

where F(x) is the objective function, x is the set of design variables(xi), Xi is the range set of the possible values for each design variable(xLi < Xi < xUi), and N is the number of design variables.

The following HS algorithm parameters are also specified: theharmony memory size (HMS) or number of solution vectors inthe harmony memory; the harmony memory considering rate(HMCR); the pitch adjusting rate (PAR); the number of decisionvariables (N); the number of improvisations (NI), and the stoppingcriterion.

Step 2: Initialization of the harmony memory

The harmony memory (HM) matrix shown in Eq. (22) is filledwith randomly generated solution vectors for HMS and sorted bythe values of the objective function f (x).

HM ¼

x11 x1

2 . . . x1N�1 x1

N

x21 x2

2 . . . x2N�1 x2

N

� � � � �� � � � �� � � � �xHMS�1

1 xHMS�12 . . . xHMS�1

N�1 xHMS�1N

xHMS1 xHMS

2 . . . xHMSN�1 xHMS

N

2666666666664

3777777777775

) f ðxð1ÞÞ) f ðxð2ÞÞ� �� �� �) f ðxðHMS�1ÞÞ) f ðxHMSÞÞ

ð22Þ

Step 3: Improvisation of a new harmony from the HM setA new harmony vector, x0 ¼ ðx01; x02; . . . ; x0nÞ, is generated based

on three criteria: random selection, memory consideration, andpitch adjustment.

� Random Selection: To determine the x01 values for the new har-mony x0 ¼ ðx01; x02; . . . ; x0nÞ, the HS algorithm randomly picks avalue from the total value range with a probability of (1-HMCR).Random selection is also used for memory initialization.� Memory Consideration: To determine a x01 value, the HS algo-

rithm randomly picks a value xj1 from the HM with a probability

of HMCR, where j = {1, 2,. . ., HMS}.

x01 x1

i ıfx0i; x2i ; . . . ; xHMS

1 g with probablity HMCRx01 2 X1 with probablity ð1�HMCRÞ

( )

ð23Þ

R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572 567

� Pitch Adjustment: Every component of the new harmony vectorx0 ¼ ðx01; x02; . . . ; x0nÞ is examined to determine whether it shouldbe pitch-adjusted. After the value x0i is randomly picked fromthe HM in the memory consideration process, it can be furtheradjusted by adding a certain amount to the value with probabil-ity PAR. This operation uses the PAR parameter, which is theprobability of pitch adjustment, given as follows:

Fig. 4. .Schematic diagram of position updating [23].

x01 Yes with probablity PAR

No with probablity ð1� PARÞ

� �ð24Þ

The (1-PAR) value indicates the probability of doing nothing. Ifthe pitch adjustment decision for x0i is yes, then x0i is replaced asfollows:

x0i x0i � bw; ð25Þ

where bw is the arbitrary distance bandwidth for a continuous de-sign variable. In this step, pitch adjustment or random selection isperformed on each variable of the new harmony vector.

Step 4: Updating the HMIf the new harmony vector x0 ¼ ðx01; x02; . . . ; x0nÞ is better than the

worst harmony in the HM, as determined from the objective func-tion value, the new harmony is inserted into the HM, and the worstharmony is removed from the HM [32].

Step 5: Checking the stopping criterion

If the stopping criterion, which is based on the maximum num-ber of improvizations, is satisfied, the computation is terminated.Otherwise, Steps 3 and 4 are repeated.

5.1. Improved harmony search algorithm

The IHS algorithm was developed by Mahdavi et al. [33] and hasbeen successfully applied to various benchmarking tests and stan-dard engineering optimization problems. Numerical results haveproven that the improved algorithm can find better solutions thanthe conventional HS algorithm and other heuristic or deterministicmethods. The key difference between the IHS and traditional HSmethods is in the manner by which PAR and bw are adjusted.The IHS algorithm uses variable PAR and bw values in the improv-ization step (Step 3) to improve the performance of the HS algo-rithm and eliminate the drawbacks associated with using fixedPAR and bw values. The PAR values change dynamically with thegeneration number and expressed as follows [33]:

PARðgnÞ ¼ PARmin þðPARmax � PARminÞ

NI� gn ð26Þ

where PAR is the pitch adjustment rate for each generation, PARmin

is the minimum pitch adjustment rate, PARmax is the maximumpitch adjustment rate, NI is the number of solution vector genera-

Fig. 3. Pseudo code of the modified impro

tions, and gn is the generation number. bw changes dynamicallywith the generation number and is defined as follows:

bwðgnÞ ¼ bwmaxexpðc � gnÞ ð27Þ

c ¼Ln bwmax

bwmin

NI

ð28Þ

where bw(gn) is the bandwidth at each generation, and bwmin andbwmax are the minimum and maximum bandwidths, respectively.

5.2. Novel global harmony search algorithm

An NGHS based on the swarm intelligence of particle swarms isproposed by Zou et al. (2010) to solve optimization problems[23,24]. The NGHS algorithm includes two important operations,namely, position updating and genetic mutation with a small prob-ability. The new approach modifies the improvisation step of theHS, such that the new harmony can mimic the global best harmonyin the HM. The NGHS and HS differ in three aspects, as follows [23]:

(1) In Step 1, the HMCR and PAR are excluded from the NGHS,and the genetic mutation probability (pm) is included inthe NGHS.

(2) In Step 3, the NGHS modifies the improvisation step of theHS and works as the pseudo code in Fig. 3.

Here, ‘‘best’’ and ‘‘worst’’ are the indices of the global best har-mony and the worst harmony in HM, respectively. r1i, r2i, and r3i

are all uniformly generated random numbers in [0,1]. r1i is usedfor position updating, r2i determines whether the NGHS shouldperform a ‘‘genetic mutation,’’ and r3i is used for the ‘‘genetic muta-tion.’’ Fig. 4 illustrates the principle of position updating [23].

The equation, stepi ¼j xbesti � xworst

i j, is defined as the adaptivestep of the ith decision variable. The region between P and R is de-fined as the trust region for the ith decision variable, which actu-

vization step in the NGHS algorithm.

568 R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572

ally near the global best harmony. A reasonable explanation is asfollows [24].

In the early stage of optimization, all solution vectors in thesolution space are sporadic. Thus, most adaptive steps are largeand most trust regions are wide, which is beneficial to the globalsearch of the NGHS. In the late stage of optimization, all non-bestsolution vectors tend to move to the global best solution vector;thus, most solution vectors are close to one other. In this case, mostadaptive steps are small, and most trust regions are narrow, whichis beneficial to the local search of the NGHS. A reasonable designprocedure can guarantee that the proposed algorithm has strongglobal search ability in the early stage of optimization, as well asa strong local search ability in the late stage of optimization. Thedynamically adjusted stepi maintains a balance between the globaland local searches [23,24].

A genetic mutation operation with a small probability is per-formed for the worst harmony of HM after position updating be-cause it can increase the probability of the proposed algorithm todeviate from the local optimum.

Fig. 5. .Procedures of th

(3) In Step 4, the NGHS replaces the worst harmony xworst in HMwith the new harmony x0 even if x0 is worse than xworst.

Fig. 5 shows a schematic diagram of the procedures of the NGHSalgorithm.

5.3. Application of modal analysis and the NGHS algorithm for theoptimal placement and sizing of shunt Var compensators

The most critical system buses (i.e., buses with large participa-tion factors), which are considered suitable for shunt Var compen-sator installation, are first identified using the modal analysismethod. The optimal sizing of shunt controllers at the suitablebuses is then determined using the NGHS algorithm, in whichthe optimal Var compensator set fQCOMP

1;... ;QCOMP1;... ;QCOMP

n g leads to amaximum power loss reduction, minimum voltage deviation, max-imum eigenvalue in critical mode, and highest cost savings. Theprocedures for implementing the proposed method for the optimalplacement and sizing of shunt Var compensators are as follows:

e NGHS algorithm.

R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572 569

i. The system parameters such as bus, branch, and generatordata, are entered.

ii. The Jacobian matrix and eigenvalues for the base system arecalculated.

iii. Eigenvectors and bus participation factors for the smallesteigenvalue are calculated.

iv. Buses with large participation factors are determined andconsidered suitable for shunt Var compensator installation.

v. QCOMP is randomly injected as additional Var compensatorsat suitable buses. The total power loss, voltage deviation,smallest eigenvalue, and total cost are calculated. Each Varcompensator set is considered a harmony vector. The HMarrays are randomly initialized using Eq. (22). The numberof columns in the HM is equal to the number of buses inthe test system. The optimal parameters of the test system,xLi and xUi, are assumed to have minimum and maximumvalues of 0 and 50 MVar, respectively. The HMS value isassumed to be 10.

vi. A new harmony is improvised through position updatingand genetic mutation. In this step, the genetic mutationprobability is assumed as pm = 0.2.

vii. The power flow program is run to calculate power loss, volt-age deviation, and the eigenvalues.

viii. The total cost with regard to the type and the cost of eachadded Var compensator is calculated using Tables 1 and 2and Eq. (14).

Fig. 6. Procedures of the proposed method for solv

ix. The objective function is calculated using Eq. (19).x. The worst harmony in HM is replaced with the new

harmony.xi. Whether the stopping criterion (maximum genera-

tion P 5000) is satisfied is assessed. If not, step (vi) isrepeated.

xii. The optimal Var compensator set (i.e., the best harmony)that provides the maximum power loss reduction, minimumvoltage deviation, maximum voltage satiability enhance-ment, and minimum total cost is determined.

Fig. 6 shows a schematic diagram of the procedures used insolving the optimal shunt Var compensator placement and sizingproblem using modal analysis and the NGHS algorithm.

6. Case study and results

The proposed method is tested on the 57-bus system shown inFig. 7. The network consists of 7 generators (one of which is theslack node), 50 load buses, and 80 lines. The system data can befound in [34]. The base configuration system load is 12.508 pu,and the system power loss is 0.2845 pu.

The eigenvalues of the reduced Jacobian matrix are first gener-ated to obtain the relative proximity of the system to instability.The bus participation factors for the critical mode are then gener-ated to predict the critical buses in the system (i.e., those with the

ing the Var compensator placement problem.

Fig. 7. .Network configuration of the IEEE 57-bus power system.

Table 4Shunt Var compensator placement and sizing results using three optimizationmethods.

570 R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572

smallest margins to instability). The smallest eigenvalue for thebase configuration of this system is calculated as 0.2344. The larg-est bus participation factors for the smallest eigenvalue are shownin Table 3.

Table 3 shows that the buses with the high participation factorshave the low reactive stability margins in the system, and thereforecorrectly correspond to the critical system buses. In this study, theten buses with high participation factor values that are consideredsuitable for Var compensator installation are 24, 25, 30–36, and 40.The power flow program is used to calculate the minimum powerloss, Ploss, as the reactive power injection independently varies at0.01 steps per unit at each load bus. The reduction in Ploss causedby the Q injection at the load bus is recorded along with the busnumber.

Table 3Largest bus participation factors for the smallest eigenvalue of the test system in thebase case.

BusNo.

Participationfactor

BusNo.

Participationfactor

BusNo.

Participationfactor

31 0.1833 40 0.0177 56 0.009233 0.1744 36 0.0176 23 0.007232 0.1704 24 0.0165 42 0.00730 0.133 39 0.0142 21 0.006725 0.1004 37 0.014 22 0.006734 0.0306 26 0.0138 38 0.005735 0.0229 57 0.0114 20 0.0043

Since in this study, the power loss reduction, voltage improve-ment and voltage stability enhancement are equally important,the coefficients for the first three objectives of Eq. (19) are consid-ered same (x1 = x2 = x3). The coefficients for the objectives are de-fined as x1 = 0.3, x2 = 0.3, x3 = 0.3, and x4 = 0.1 based on thelower significance of the total cost objective function comparedwith the power loss reduction, voltage profile improvement, andvoltage stability enhancement functions. The penalty factors aredefined as v1 = 100, v2 = 100, and v3 = 100.

The proposed NGHS algorithm is implemented to determine theoptimal location and size of the Var compensation devices in the

BusNo.

IHS PSO NGHS

Injected Q(MVar)

Type ofdevice

Injected Q(MVar)

Devicetype

Injected Q(MVar)

Devicetype

24 1.53 C 1.79 C 1.59 C25 6.78 C 17.14 SVC 1.34 C30 8.70 C 2.13 C 3.55 C31 15.93 SVC 20.08 SVC 32.14 STATCOM32 13.28 SVC 19.74 SVC 7.52 C33 33.98 STATCOM 12.03 SVC 11.80 SVC34 3.76 C 6.22 C 6.25 C35 4.34 C 4.91 C 9.20 C36 1.98 C 7.49 C 8.55 C40 1.84 C 1.71 C 1.69 C

Fig. 8. Convergence characteristics of different optimization methods for theoptimal placement and sizing of shunt Var compensators.

Table 7Monte Carlo numerical results of different optimization techniques for the optimalplacement and sizing of shunt Var compensators.

The fitness function values IHS PSO NGHS

Minimum 0.6609 0.6581 0.6302Maximum 0.6615 0.6782 0.6302Mean 0.6612 0.6689 0.6302Standard deviation 2.65 � 10�4 6.51 � 10�3 3.04 � 10�5

R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572 571

network. The results obtained from the NGHS algorithm is com-pared with those obtained using the IHS algorithm and the PSOmethod in Table 4. In order to find the optimal parameters ofPSO and IHS, the best parameters which produce lowest fitnessfunction value with suitable convergence are selected. For PSO,the population size is set to 50 which is the best value in range[2,500] and the acceleration constant is set to 1.49 which is thebest value in range [0.1,5]. Consequently, the optimal parametersof IHS are selected as follows: HMCR = 90%, PARmax = 0.99, PAR-min = 0.4, bwmax = 1, bwmin = 0.0001 and NI = 5000.

This study is conducted with the restriction that the injected Qdoes not exceed 50 MVar. The place, size, and selected type ofshunt controllers that use the three optimization methods areshown in Table 4. The maximum sizes of the reactive power com-pensators that use the NGHS algorithm are installed at buses 31and 33, which have the largest bus participation factors for thesmallest eigenvalue of the test system in the base case. The maxi-mum shunt Var compensator sizes determined using PSO are in-stalled at buses 33, 31, and 32, and the largest reactive controllersizes using the improved HS algorithm are installed at buses 31,32, 25, and 33.

The types of suitable Var compensators according to size arechosen from Table 2. The different installed Var compensation de-vices that use the NGHS algorithm are 1 STATCOM, 1 SVC, and 8capacitors. Only 4 SVCs and 6 capacitors that use PSO are installedon the network, whereas 1 STATCOM, 2 SVCs, and 7 capacitors in-stalled on the 57-bus system as shunt Var compensators use theIHS algorithm.

Table 5 shows the results of a more detailed analysis of the totalpower loss, voltage deviation, smallest eigenvalue, and total costafter the installation of the Var compensation devices. Shunt con-troller placement using the NGHS algorithm leads to a lower fit-ness function value, lower total power loss, greater eigenvalue incritical mode, lower total cost, and slightly less voltage deviationcompared with the IHS and PSO techniques.

Although the SVC and STATCOM are expensive compared withthe shunt capacitor, they exhibit better performances in terms ofloss reduction, voltage profile, and voltage stability [2]. This cost-benefit analysis justifies the economic viability of the SVC andSTATCOM.

A comparison of the total Var compensator size using the threeoptimization methods are shown in Table 6. The total size of theinstalled Var compensators that use the NGHS algorithm is lowerthan that of the installed Var compensators installed that use thePSO and IHS algorithms. Therefore, the total cost of using the NGHSalgorithm is lower than those of the PSO and IHS, even if SVC de-vices are installed instead of the STATCOM units in Table 4. EitherSVC or STATCOM can be installed in the weakest buses after theoptimal placement and sizing of the Var compensators, with theircost being the determining factor.

Table 5Comparison of the objective functions after installation of shunt Var compensators.

Objective function IHS PSO NGHS

Total power losses (MW) 26.49 26.11 25.84Total voltage deviation (pu) 0.1884 0.1851 0.1804Smallest eigenvalue 0.495 0.471 0.503Total cost (1000 USD) 3098.7 2953.3 2396.4

Table 6Comparison of the total Var compensator sizes.

Optimization methods IHS PSO NGHS

Total compensator size (MVar) 92.12 93.24 83.63

The convergence characteristics of the different optimizationmethods for the optimal placement and sizing of shunt Var com-pensators are shown in Fig. 8. The NGHS algorithm not only per-forms better in finding better solutions, it also converges fastercompared with the IHS and PSO.

For a more detailed analysis of the results, consider Table 7which shows the Monte Carlo results of the aforementioned threealgorithms. The minimum, maximum, mean and standard devia-tion of the fitness function values from 50 trials of the IHS, PSOand NGHS algorithm are obtained. The number of iterations is as-sumed as 5000.

From Table 7, it is observable that the fitness function valuesand their standard deviation, using NGHS algorithm are lower thanusing IHS and PSO. The significant difference between standarddeviations, can demonstrate that the accuracy of the proposedNGHS algorithm for the optimal placement and sizing of shuntVar compensators in 57-bus test system is better than PSO andIHS algorithm.

The NGHS notably consumes less average time than the IHSalgorithm for each problem [23,24]. The proposed NGHS algorithmhas demonstrated a stronger capacity for solving problems in spaceexploration compared with the other approaches [24]. The resultsshow that the NGHS can be an efficient alternative for determiningoptimal allocation problems for Var sources.

7. Conclusions

The application of the NGHS algorithm as a new metaheuristicoptimization method for determining the optimal location and sizeof reactive power compensation devices in a transmission networkhas been presented. The algorithm is easy to implement and canfind multiple optimal solutions to constrained multi-objectiveproblems, thereby providing more flexibility in making decisionsabout the locations of shunt Var compensator units. The extentsof reduction in the power system loss, improvement in the busvoltage profile voltage stability, and reduction in the total costare used as measures of power system performance in the optimi-zation algorithm. The proposed multi-objective NGHS algorithm isvalidated on a 57-bus transmission network. The results show that

572 R. Sirjani et al. / Electrical Power and Energy Systems 43 (2012) 562–572

the NGHS algorithm produces greater voltage stability enhance-ment and greater reductions in power loss, voltage deviation, andtotal cost than the IHS and PSO methods. Moreover, the NGHS algo-rithm not only performs better in finding better solutions but alsoconverges faster compared with the IHS and PSO.

References

[1] Saranjeet K. Evolutionary algorithm assisted optimal placement of FACTScontrollers in power system. Master thesis. Thapar University; 2009.

[2] Sode-Yome A, Mithulananthan N. Comparison of shunt capacitor, SVC andSTATCOM in static voltage stability margin enhancement. Int J Electr Eng Educ2004;41(2):158–71.

[3] Sirjani R, Mohamed A, Shareef H. Optimal capacitor placement in a radialdistribution system using harmony search algorithm. J Appl Sci2010;10(23):2996–3006.

[4] Azadani EN, Hosseinian SH, Hasanpor P. Optimal placement of multipleSTATCOM for voltage stability margin enhancement using particle swarmoptimization. Electr Eng 2008;90(7):503–10.

[5] Sirjani R, Mohamed A, Shareef H. Optimal placement and sizing of static varcompensators in power systems using improved harmony search algorithm.Prz Elektrotechniczny 2011;87(7):214–8.

[6] Ng H, Salama M, Chikhani A. Classification of capacitor allocation techniques.IEEE Trans Power Deliv 2000;15(1):387–92.

[7] Sirjani R, Mohamed A, Shareef H. An improved harmony search algorithm foroptimal capacitor placement in radial distribution systems. In: 5th Int powereng and optimization conf; 2011.p. 323–28.

[8] Minguez R, Milano F, Zarate-Minano R, Conejo A. Optimal network placementof SVC devices. IEEE Trans Power Syst 2007;22(4):1851–60.

[9] Hsiao YT, Liu CC, Chiang HD. A new approach for optimal VAR sources planningin large scale electric power systems. IEEE Trans Power Syst 1993;8(3):988–96.

[10] Pisica I, Bulac C, Toma L, Eremia M. Optimal SVC placement in electric powersystems using a genetic algorithms based method. IEEE Power Technol Conf2009:1–6.

[11] Ishak S, Abidin AF, Rahinan TKA. Static var compensator planning usingartificial immune system for loss minimisation and voltage improvement.National Power Energy Conf Proc 2004:41–6.

[12] Sundareswaran K, Hariharan B, Parasseri FP, Antony DS, Subair B. Optimalplacement of static var compensators (SVC’s) using particle swarmoptimization. Int Conf Power Control Embed Syst 2010:1–4.

[13] Samimi A, Golkar MA. A novel method for optimal placement of STATCOM indistribution networks using sensitivity analysis by DIgSILENT software. Asia-Pacific Power Energy Eng Conf 2011:1–5.

[14] Del Valle Y, Hernandez JC, Venayagamoorthy GK, Harley RG. OptimalSTATCOM sizing and placement using particle swarm optimization. IEEE/PESTrans Distr Conf Expo 2006:1–6.

[15] Del Valle Y, Venayagamoorthy GK, Harley RG. Comparison of enhanced-PSOand classical optimization methods: a case study for STATCOM placement. In:15th Int conf on intel syst appl to power syst; 2009. p. 1–7.

[16] Geem Z, Kim J, Loganathan A. A new heuristic optimization algorithm:harmony search. Simulation 2001;76(2):60–8.

[17] Shariatkhah MH, Haghifam MR, Salehi J, Moser A. Duration basedreconfiguration of electric distribution networks using dynamicprogramming and harmony search algorithm. Int J Electr Power2012;41(1):1–10.

[18] Sivasubramani S, Swarup KS. Multi-objective harmony search algorithm foroptimal power flow problem. Int J Electr Power 2011;33(3):745–52.

[19] Khazali AH, Kalantar M. Optimal reactive power dispatch based on harmonysearch algorithm. Int J Electr Power 2011;33(3):684–92.

[20] Chatterjee A, Ghoshal SP, Mukherjee V. Solution of combined economic andemission dispatch problems of power systems by an opposition-basedharmony search algorithm. Int J Electr Power 2012;39(1):9–20.

[21] Gil-Lopez S, Del Ser J, Salcedo-Sanz S, Perez-Bellido AM, Cabero JM, Portilla-Figueras JA. A hybrid harmony search algorithm for the spread spectrum radarpolyphase codes design problem. Expert Syst Appl 2012;39(12):11089–93.

[22] Del Ser J, Matinmikko M, Gil-Lopez S, Mustonen M. Centralized and distributedspectrum channel assignment in cognitive wireless networks: a harmonysearch approach. Appl Soft Comput 2012;12(2):921–30.

[23] Zou D, Gao L, Wu J, Li S, Li Y. A novel global harmony search algorithm forreliability problems. Comput Ind Eng 2010;58(2):307–16.

[24] Zou D, Gao L, Li S, Wu J, Wang X. A novel global harmony search algorithm fortask assignment problem. J Syst Software 2010;83(10):1678–88.

[25] GE. Guide for economic evaluation of flexible AC transmission systems (FACTS)in open access environment. EPRI-TR 108500; 1997.

[26] Coevering JV, Stovall JP, Hauth RL, Tatto PJ, Railing BD, Johnson BK. The nextgeneration of HVDC- needed R&D, equipment costs, and cost comparisons.Proc EPRI Conf Future Power Deliv 1996:9–10.

[27] Kundur P. Power system stability and control. EPRI power system engineeringseries. McGraw-Hill; 1994.

[28] Thukaram D, Lomi A. Selection of static VAR compensator location and size forsystem voltage stability improvement. Electr Power Syst Res2000;54(2):139–50.

[29] Mithulananthan N, Cañizares CA, Reeve J. Comparison of PSS, SVC andSTATCOM controllers for damping power system oscillation. IEEE TransPower Syst 2003;18(2):786–92.

[30] Sharma C, Ganness MG. Determination of power system voltage stability usingmodal analysis. IEEE Int Conf Power Eng Energy Electr Driv 2007:381–7.

[31] Amgad AED, Youssef KMH, EL-Metwally MM, Osman Z. Optimum VAR sizingand allocation using particle swarm optimization. Electr Power Syst Res2007;77:965–72.

[32] Kazemi A, Parizad A, Baghaee H. On the use of harmony search algorithm inoptimal placement of FACTS devices to improve power system security. ProcIEEE EUROCON 2009:570–6.

[33] Mahdavi M, Fesanghary M, Damangir E. An improved harmony searchalgorithm for solving optimization problems. Appl Math Comput2007;188(2):1567–79.

[34] Power systems test case archive. University of Washington. <http://www.ee.washington.edu/research/pstca/>.