Memetic Comp.DOI 10.1007/s12293-016-0208-z

REGULAR RESEARCH PAPER

Optimal placement and sizing of capacitor using Limaçon inspiredspider monkey optimization algorithm

Ajay Sharma1 · Harish Sharma1 ·Annapurna Bhargava1 · Nirmala Sharma1 · Jagdish Chand Bansal2

Received: 8 January 2016 / Accepted: 4 August 2016© Springer-Verlag Berlin Heidelberg 2016

Abstract The power system is a complex interconnectednetwork which can be subdivided into three components:generation, distribution, and transmission. Capacitors of spe-cific sizes are placed in the distribution network so that lossesin transmission and distribution isminimum.But the decisionof size and position of capacitors in this network is a complexoptimization problem. In this paper, Limaçon curve inspiredlocal search strategy (LLS) is proposed and incorporated intospidermonkeyoptimization (SMO)algorithm todeal optimalplacement and the sizing problem of capacitors. The pro-posed strategy is named as Limaçon inspired SMO (LSMO)algorithm. In the proposed local search strategy, the Limaçoncurve equation is modified by incorporating the persistenceand social learning components of SMO algorithm. The per-formance of LSMO is tested over 25 benchmark functions.Further, it is applied to solve optimal capacitor placementand sizing problem in IEEE-14, 30 and 33 test bus systemswith the proper allocation of 3 and 5-capacitors. The reportedresults are compared with a network without a capacitor (un-capacitor) and other existing methods.

B Ajay Sharmaajay_2406@yahoo.com

Harish Sharmahsharma@rtu.ac.in

Annapurna Bhargavaabhargava@rtu.ac.in

Nirmala Sharmansharma@rtu.ac.in

Jagdish Chand Bansaljcbansal@gmail.com

1 Rajasthan Technical University, Kota, India

2 South Asian University, New Delhi, India

Keywords Spider monkey optimization · Limaçon inspiredlocal search · Optimal capacitor placement · Capacitorsizing · Loss minimization

1 Introduction

The modern power distribution system is continuously fac-ing ever-growing load demand, resulting in increased burdenand reduced voltages. The voltages at buses or nodes reduceswhile moving away from a substation, due to an insufficientamount of reactive power. To improve this voltage profile,reactive compensation is required. The efficiency of powerdelivery is enhanced, and losses at distribution level arereduced by incorporating network reconfigurations, shuntcapacitor placement, etc. The optimal capacitor placementsupplies the part of reactive power demand which helps inreducing the energy losses, peak demand losses and improvesthe voltage profile, power factor (p f ) and system stability[14]. Therefore, specific size capacitors are required to beplaced at specific places in the distribution network to achievethe optimum reactive power.

To achieve this objective while maintaining the opti-mal economy, optimal placement of capacitor with propersizing should be decided by some conventional [6] or non-conventional strategy [9,14].

The shunt capacitor is a very common conventional strat-egy for distribution system. Further, the concept of lossminimization by a singly located capacitor was extended formultiple capacitors. Subsequently, combinational optimiza-tion strategywas developed to dealwith the discrete capacitorplacement problem. The described methods have their limi-tations of depending on the initial guess, lack of robustness,time-consuming, and many local optimal solutions for non-linear optimization problems [6].

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Swarm intelligence based meta-heuristics have impressedthe researchers to apply them for solving the capacitorplacement and sizing problems [24]. In past, the capacitorplacement problem was solved by applying fuzzy approxi-mate reasoning [18], genetic algorithm (GA) [29], artificialbee colony (ABC) algorithm [24], and particle swarm opti-mization (PSO) algorithm [23] etc. Recently, Bansal et al.[2] introduced a swarm intelligence based algorithm namely,spider monkey optimization (SMO) algorithm by takinginspiration from the social and food foraging behavior ofspider monkeys. It has been shown that the SMO is competi-tive to the ABC, PSO, DE, and covariance matrix adaptationevolution strategies (CMA-ES) algorithms [2].

Though SMO performs well still, due to the presence ofrandomcomponents (φ andψ) in the position update process,there is a chance of skip of the true solution. So, integration ofa local search strategy with SMO may improve the exploita-tion capability of the algorithmand, hence reduces the chanceof skipping true solution. Therefore, in this paper, a newlocal search algorithm is proposed by modifying the limaçoncurve equation and named as limaçon inspired local search(LLS). Further, the proposed local search is incorporatedwith SMO in expectation of improving exploitation capabil-ity. The proposed hybridized algorithm is named as Limaçoninspired spidermonkeyoptimization algorithm (LSMO).Theperformance of LSMO is tested through various numericalexperiments with respect to accuracy, reliability, and con-sistency. Then the LSMO is applied to solve the optimalplacement and sizing problem of capacitors in the distrib-ution network. The results are compared with un-capacitorand other existing methods for IEEE 14, 30 and 33 test buscaseswith 3 and 5-capacitor placement and sizing conditions.

The detailed description may be categorized as follows:Basic SMO is explained in Sect. 2. Section 3 describes abrief review on local search strategies. Limaçon inspiredlocal search strategy was proposed and incorporated to SMOin Sect. 4. In Sect. 5, the performance of proposed strategyis evaluated. Section 6 describes capacitor sizing and opti-mal allocation problem. Solution to the optimal placementand sizing problem of the capacitor is presented in Sect. 7.Finally, the conclusion of the work is given in Sect. 8.

2 Spider monkey optimization (SMO) Algorithm

SMO algorithm is based on the foraging behavior and socialstructure of spider monkeys [2]. Spider monkeys have beencategorized as a fission-fusion social structure (FFSS) basedanimals, in which individuals form small, impermanent par-ties whose member belongs to a larger community. Monkeyssplit themselves from larger to smaller groups and vice versabased on scarcity and availability of food.

2.1 Main steps of SMO algorithm

The SMO algorithm consists of six phases: Local leaderphase, Global leader phase, Local leader learning phase,Global leader learning phase, Local leader decision phase,and Global leader decision phase. Each of the phases isexplained as follows:

2.1.1 Initialization of the Population

Initially, SMO generates an equally distributed initial pop-ulation of N spider monkeys where each monkey SMi (i= 1, 2, . . . ,N) is aD-dimensional vector and SMi representsthe i th spider monkey (SM) in the population. SM representsthe potential solution of the problem under consideration.Each SMi is initialized as follows:

SMi j = SMmin j +U (0, 1) × (SMmax j − SMmin j ) (1)

where SMmin j and SMmax j are respectively lower and upperbounds of SMi in j th direction and U (0, 1) is a uniformlydistributed random number in the range [0, 1].

2.1.2 Local leader phase (LLP)

In this phase, each SM updates it’s current position based ongathered information from local leader as well as local groupmembers. The fitness value of so obtained new position iscomputed. If the fitness value of the new position is superiorto the old position, then the SMmodifies its position with thenew one. The position update equation for i th SM (which isa member of kth local group) in this phase is

SMnewi j = SMi j +U (0, 1) × (LLkj − SMi j )

+U (−1, 1) × (SMr j − SMi j ) (2)

where SMi j is the j th dimension of the i th SM, LLkj repre-sents the j th dimension of the kth local group leader position.SMr j is the j th dimension of the r th SMwhich is chosen arbi-trarilywithin kth group such that r �= i . U(0, 1) is a uniformlydistributed random number between 0 and 1. Algorithm 1shows position update process in the local leader phase. InAlgorithm 1, MG is the maximum number of groups in theswarm and pr is the perturbation rate which controls theamount of perturbation in the current position.

2.1.3 Global leader phase (GLP)

In this phase, all SMs update their positions using knowledgeof global leader and local group members experience. Theposition update equation for this phase is as follows:

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Algorithm 1 Position update process in Local Leader Phase:for each k ∈ {1, . . . , MG} do

for each member SMi ∈ kth group dofor each j ∈ {1, . . . , D} do

if U (0, 1) ≥ pr thenSMnewi j = SMi j +U (0, 1)×(LLkj −SMi j )+U (−1, 1)×(SMr j − SMi j )

elseSMnewi j = SMi j

end ifend for

end forend for

SMnewi j = SMi j +U (0, 1) × (GL j − SMi j )

+U (−1, 1) × (SMr j − SMi j ) (3)

whereGL j is the j th dimension of the global leader positionand j is the randomly chosen index. The positions of spidermonkeys (SMi ) are updated based on a probability probiwhich is a function of fitness. In this way, a better candi-date will have more chance to make it better. The probabilityprobi is calculated as shown in Eq. 4 [27].

probi = 0.9 × fitnessimax_fitness

+ 0.1, (4)

Here fitnessi is the fitness value of i th SMandmax_fitnessis the highest fitness in the group. The fitness of the newlygenerated SMs is calculated and compared with the oldone, and the better position is adopted. The position updateprocess of this phase is explained in Algorithm 2.

Algorithm 2 Position update process in global leader phase(GLP):

for k = 1 to MG docount = 1;GS = kth group size;while count < GS do

for i = 1 to GS doif U (0, 1) < probi thencount = count + 1.Randomly select j ∈ {1 . . . D}.Randomly select SMr from kth group s.t. r �= i .SMnewi j = SMi j +U (0, 1)× (GL j − SMi j )+U (−1, 1)×(SMr j − SMi j ).

end ifend forif i is equal to GS theni = 1;

end ifend while

end for

2.1.4 Global leader learning phase (GLLP)

In this phase, the position of the SM having best fitness in thepopulation is selected as the updated position of the globalleader using greedy selection. Further, the position of globalleader is checked whether it is updating or not and if not thenthe global limit count is incremented by 1.

2.1.5 Local leader learning phase (LLLP)

In this phase, the position of the SM having best fitness inthat group is selected as the updated position of the localleader using greedy selection. Next, if the modified positionof the local leader is comparedwith the old one and if the localleader is not updated then the local limit count is incrementedby 1.

2.1.6 Local leader decision phase (LLDP)

If any local leader is not updated up to a preset thresholdcalled local leader limit, then all the members of that minorgroup update their positions either by random initializationor by using combined information from global leader andlocal leader through Eq. 5.

SMnewi j = SMi j +U (0, 1) × (GL j − SMi j )

+U (0, 1) × (SMi j − LLkj ); (5)

It is clear from Eq. (5) that the updated dimension of thisSM is attracted towards global leader and repels from thelocal leader.

2.1.7 Global leader decision (GLD) phase

In this phase, the global leader is monitored, and if it is notupdated up to a preset number of iterations called globalleader limit, then the global leader divides the populationinto minor groups. Firstly, the population is divided into twogroups and then three groups and so on till the maximumnumber of groups (MG) are formed. After every division,LLL process is initiated to choose the local leader in thenewly formed groups. The case in which amaximum numberof groups are formed and even then the position of globalleader is not updated then the global leader combines all theminor groups to form a single group.

The SMO algorithm is better represented by pseudo-codein Algorithm 3.

3 Significant recent local search modifications

A local search is thought of as an algorithmic structure con-verging to the closest local optimum while the global search

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Algorithm 3 SMOInitialize parameters;while Termination criteria doStep 1: Local Leader Phase.Step 2: Global Leader Phase.Step 3: Local Leader Learning Phase.Step 4: Global Leader Learning Phase.Step 5: Local Leader Decision Phase.Step 6: Global Leader Decision Phase.

end whilePrint best solution.

should have the potential of detecting the global optimum.Therefore, to maintain the proper balance between explo-ration and exploitation behavior of an algorithm, it is alwayssuggested to incorporate a local search approach in the basicpopulation-based algorithm to exploit the identified region ina given search space. Therefore, the local search algorithmsare applied to the global search algorithms to improve theexploitation capability of the global search algorithm. Herethe main algorithm explores while the local search exploitsthe search space.

Researchers are constantlyworking in the field ofmemeticsearch approach. Natalio and Gustafson [11] discussedproofs of memetic concepts. Ong et al. [22], proposed atechnique to maintain a balance between genetic search andlocal search. Ong et al. [21], listed classification of memesadaptation by the mechanism used and the level of histor-ical knowledge on the memes employed. Lim presented avaluable discussion on memetic computing [15]. In the sameyear, Neri et al. [17], incorporated scale factor local searchto improve exploitation capability of DE. Further Nguyen etal. [19], presented a novel probabilistic memetic frameworkto model MAs as a process involving in finalizing separateactions of evolution or individual learning and analyzingthe probability of each process in locating the global opti-mum. Ong et al. [20] presented, an article to show severaldeployments of memetic computing methodologies to solvecomplex real world problems. In same year Mininno et al.[16] incorporated Memetic approach with DE in noisy opti-mization. Chen et al. [3], presented realization of memeticcomputing through memetic algorithm. Sharma et al. [26],included opposition based lévy flight local search with ABC.Sharma et al. [27], integrated lévy flight local search strat-egy with artificial bee colony algorithm. Recently, in 2016Sharma et al. presented power law based local search inSMO (PLSMO) algorithm [25]. In the above presented localsearch strategies, the direction and distance (step size) ofthe individuals, which are going to update, are based on theinter-individual distance among the solutions. Thismay forcethe individuals to move towards a specific direction. There-fore, development of local search strategy, which properlyexploit the identified search space is highly required. Anangular rotation based search process may reduce the chance

of trapping in a local optima. Therefore, in this paper limaçoncurve inspired local search strategy (LLS) is developed andincorporated with SMO algorithm. In the proposed LLS, thedirection and distance of the solutions are based on the fitnessof the solution (sign), the distance between the individual,and an angle of rotation.

4 Limaçon inspired local search strategy and it’sincorporation to SMO

The word limaçon is a Latin word meaning snail. During it’sevolution, two important features were adapted by Limaçonor snail naturally. In the first process called torsion, mostof the internal organs were twisted 180◦ anticlockwise. Theanother important feature is that the shell became more con-ical and then spirally coils. The shell is a line of defence forthe limaçon. The foot of limaçon allows it to move forwardand backward with muscle contracting and expanding move-ment with the help of mucus and slime. The limaçon’s basicspecifications are the height of shell, width of shell, heightof aperture, width of the aperture, the number of whorls, andapical angle. In this context, the height of the shell is it’smaximum measurement along the central axis. The width isthe maximum measurement of the shell at right angles to thecentral axis. The central axis is an imaginary axis along thelength of a shell, around which, in a coiled shell the whorlsspiral. The central axis passes through the columella, thecentral pillar of the shell. Normally the whorls are circularor elliptical, but from compression and other causes a vari-ety of forms can result. The spire can be high or low, broador slender according to the way the coils of the shells arearranged and the apical angle of the shell varies accordingly.The whorls overlap the earlier whorls, such that they may belargely or wholly covered by the later ones. When an angu-lation occurs, the space between it and the suture above itconstitute the area known as the shoulder of the shell. Theshoulder angle may be simple or keeled, and may sometimeshave nodes or spines. The limaçon and its single line diagramare shown in Fig. 1.

The proposed local search strategy is based on the limaçoncurve. The limaçon curve was introduced by Etienne Pascal(1588–1651) [5]. The Limaçon curve is a botanical curvewhich resembles the snail. Here, both rolling circles are hav-ing the same radius and the curve thus obtained “epicycloid”is the traces of a point P fixed to a circle that rolls aroundanother circle as shown in Fig. 2.

In literature, this curve already has been used in differentways [12]. But, in this paper, the first time the limaçon curveis used to develop a local search strategy and hybridizedwith the basic SMO to improve the exploitation capability ofSMO.

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Memetic Comp.

Fig. 1 a Limaçon (Snail) curve (this figure is accessed on Feb 2015 from http://entnemdept.ufl.edu/creatures/misc/whitegardensnail.htm), b singleline curve (this figure is accessed on Feb. 2015 from http://www.clipartpanda.com/categories/snail-clipart-black-andwhite)

Fig. 2 Epicycloid curve (this figure is accessed on Feb 2015 fromhttp://cf.ydcdn.net/1.0.1.42/images/main/epicycloid.jpg)

The basic equations of limaçon curve are shown in equa-tions 6 and 7 for vertical axis and horizontal axis curvesrespectively [30].

r = a ± bsinθ (6)

r = a ± bcosθ (7)

Here, r is the distance of the limaçon from the origin, a andb are constants and θ is an angle of rotation. The curve hastransient phases based on the value of b. From a circle, forb = 0 to cardioid for b = 1 and a noose on curve appears forb > a.

In this paper, this limaçon curve is used to form a localsearch strategy into SMO. In the proposed local search strat-egy, the distance r is used as a new position of a solutionwhich is going to update its position during the search process

in the given search region. The detailed description of the pro-posed limaçon curve based local search strategy, named aslimaçon local search (LLS) strategy is as follows:

In the proposed LLS, Eq. 6 of limaçon curve is adaptedwith some modifications as a position update equation of theproposed local search strategy. The modified equation is asfollows:

xnew = xi ± (xi − xk) × sinθ where,

k ∈ randomly selected solution, but k �= i. (8)

Here, a = xi is the solution which is going to update itsposition, xnew is the updated position of the xi , b = (xi − xk)is the social influence of the solution xi in the population andθ is an angle of rotation.

In this paper, in each iteration, only the best solution willbe allowed to update its position using the LLS strategy. Theposition update equation for the best solution is given by Eq.9.

xnew = xbest ± (xbest − xk) × sinθ (9)

where θ is calculated as

θ = π

2×

(1 − t

T

)(10)

Where, t = current iteration counter and T = total iterationsof local search. The pseudo-code of the proposed LLS isshown in Algorithm 4.

As the step size is also based on sine of an angle, resem-bling apical angle of limaçon, the step size reduces based ondecreasing angular values from θ = 90◦ to θ = 0◦ eitherin negative or positive direction. The higher height of spireshows a lesser apical angle for limaçon and vice versa. Sim-ilarly, the lesser angular step represents a small step size,while the higher angular step represents larger step size inthe LLS strategy. This implies that in early iterations larger

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step sizes are allowed while smaller step sizes are allowed inlater iterations.

Algorithm 4 Limaçon local search (LLS) strategy:Input optimization function Min f (x) and xbest ;Initialized iteration counter t = 0 and total iterations of LLS, T;while t < T doCalculate the value of θ using equation (10);Generate new solutions xnew1 using Sign=“−′′ and xnew2 usingSign=“+′′ by Algorithm 5.Calculate objective value f (xnew1) and f (xnew2).if f (xnew1) < f (xbest ) thenxbest = xnew1;

else if f (xnew2) < f (xbest ) thenxbest = xnew2;

end ifend whileReturn xbest .

In Algorithms 4 and 5, cr is a perturbation rate (a numberbetween 0 and 1) which controls the amount of perturbationin the best solution, U (0, 1) is a uniform distributed randomnumber between 0 and 1, D is the dimension of the problemand xk is a randomly selected solution within population. SeeSect. 5.1 for details of these parameter settings.

The proposed LLS strategy is incorporated with the SMOafter the global leader decision phase. The pseudo-code ofthemodified SMOnamed as limaçon inspired SMO (LSMO)algorithm is shown in Algorithm 6.

Algorithm 5 New solution generation:Input Sign and best solution xbest ;Randomly select a solution xk from the population such that best �= k;for j = 1 to D do

if U (0, 1) < cr thenxnew j = xbest j ;

elsexnew j = xbest j Sign (xbest j − xk j ) × sinθ ;

end ifend forReturn xnew

Algorithm 6 Limaçon inspired SMO:Initialize the parameters;while Termination criteria doStep 1: Local Leader phase.Step 2: Global Leader phase.Step 3: Local Leader Learning phase.Step 4: Global Leader Learning phase.Step 5: Local Leader Decision phase.Step 6: Global Leader Decision phase.Step 7: Apply Limaçon inspired Local Search (LLS) Strategy usingAlgorithm 4.

end whilePrint best solution.

5 Performance evaluation of LSMO algorithm

The performance of proposed LSMO algorithm is evaluatedon 25 different benchmark continuous optimization func-tions ( f1 to f25) having different degrees of complexity andmultimodality as shown in Table 1. The acceptable errors ofabove functions are set to see the clear difference among theconsidered algorithms in terms of success rate and numberof function evolutions. Here, the functions and acceptableerrors are adopted from the literature [1,2,27]. To check thecompetitiveness of LSMO , it is compared with SMO [2],ABC [10], DE [28], PSO − 2011 [4], CMA − ES [7]and one significant variant of ABC namely, Gbest-guidedABC (GABC) [31] as well as two local search variantsnamely, memetic ABC (MeABC) [1], and Lévy flight ABC(LFABC) [27] and one recent local search variant of SMOnamely, PLSMO [25]. The experimental setting is given inSect. 5.1.

5.1 Experimental setting

The experimental settings are as follows:

– Population Size N = 50;– MG = N/10.– GlobalLeaderLimit = 50,– LocalLeaderLimit = 1500,– pr (perturbation rate ofmainSMOalgorithm)∈[0.1, 0.4],

linearly increasing over iterations,

prG+1 = prG + (0.4 − 0.1)/MI R (11)

where, G is the iteration counter, MI R is the maximumnumber of iterations.,

– The stopping criteria is either maximum number of func-tion evaluations (which is set to be 200,000) is reached orthe acceptable error of test problem has been achieved,

– The number of simulations/run =100,– Parameter settings for the algorithm SMO , ABC ,GABC ,

MeABC , LFABC , PLSMO , CMA − ES, PSO −2011, and DE are similar to their legitimate researchpapers respectively.

– Themaximum number of iterations of LLS is set throughsensitivity analysis in terms of sum of success rate (SR).The performance of LSMO is measured for consideredtest problems on different values of T and results in termsof success are analyzed in Fig. 3. It is clear from Fig. 3that T = 20 gives better results (highest value of sum ofsuccess). Therefore in this paper maximum local searchiterations is set as T = 20.

– In order to investigate the effect of parameter cr (pertur-bation rate of local search), described by Algorithm 5 onthe performance of LSMO , its sensitivity with respect

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Memetic Comp.

Tabl

e1

Testproblems

Testproblem

Objectiv

efunctio

nSearch

range

Optim

umvalue

DAE

C

Sphere

f 1(x

)=

∑ D i=1x2 i

[−5.12,5

.12]

f(0)

=0

301.0E

−05

S,U

DeJong

f4f 2

(x)=

∑ D i=1i×

(xi)4

[−5.12,5

.12]

f(0)

=0

301.0E

−05

S,M

Cigar

f 3(x

)=

x 02+

100000

∑ D i=1x i

2[−

10,1

0]f(

0)=

430

1.0E

−05

S,U

brow

n3f 4

(x)=

∑ D−1

i=1

(xi2

(xi+

1)2

+1+

x i+1

2x i

2+1

)[−

1,4]

f(0)

=0

301.0E

−05

U,N

Schewel

f 5(x

)=

∑ D i=1|x i

|+∏ D i=

1|x i

|[−

10,1

0]f(

0)=

030

1.0E

−05

N,U

Axisparalle

lhyp

er-ellipsoid

f 6(x

)=

∑ D i=1i×

x2 i[−

5.12,5

.12]

f(0)

=0

301.0E

−05

U,S

Sum

ofdifferentp

owers

f 7(x

)=

∑ D i=1|x i

|i+1

[−1,

1]f(

0)=

030

1.0E

−05

S,M

Neumaier

3Problem

(NF3

)f 8

(x)=

∑ D i=1(x

i−

1)2−

∑ D i=2x ix i

−1[−

D2,D

2]

f min

=−(

D(D

+4)(D

−1))

610

1.0E

−01

U,N

Rotated

hyper-ellip

soid

f 9(x

)=

∑ D i=1∑ i j=

1x2j

[−65.536,6

5.536]

f(0)

=0

301.0E

−05

S,M

Levymontalvo1

f 10(x

)=

π D(10sin

2(π

y 1)+

∑ D−1

i=1

(yi−

1)2(1

+10

sin2

(πy i

+1))

+(y

D−

1)2),where

y i=

1+

1 4(x

i+

1)

[−10,1

0]f(−1

)=

030

1.0E

−05

N,M

Levymontalvo2

f 11(x

)=

0.1(sin2

(3πx 1

)+

∑ D−1

i=1

(xi−

1)2×

(1+

sin2

(3πx i

+1))

+(x

D−

1)2(1

+sin2

(2πx D

))

[−5,

5]f(

1)=

030

1.0E

−05

N,M

Ellipsoidal

f 12(x

)=

∑ D i=1(x

i−i)2

[−D,D

]f(1,2,

3,..

.,D

)=

030

1.0E

−05

U,S

Bealefunctio

nf 13(x

)=

[1.5

−x 1

(1−

x 2)]2

+[2.

25−

x 1(1

−x2 2

)]2+

[2.625

−x 1

(1−

x3 2)]2

[−4.5,

4.5]

f(3,0.5)

=0

21.0E

−05

N,M

Colville

functio

nf 14(x

)=

100[x

2−x

2 1]2 +

(1−x

1)2

+90(x 4

−x2 3)2

+(1−

x 3)2

+10

.1[(x

2−1)

2+

(x4−1)

2]+

19.8

(x2−1)

(x4−1)

[−10,1

0]f(

1)=

04

1.0E

−05

N,M

Kow

alik

f 15(x

)=

∑ 11 i=1[a i

−x 1

(b2 i+b

ix2)

b2 i+b

ix3+x

4]2

[−5,

5]f(0.192833

,0.190836,

0.123117

,0.135766)=

0.000307486

41.0E

−05

M,N

2DTripodfunctio

nf 16(x

)=

p(x 2

)(1

+p(x 1

))+

|(x1+

50p(x 2

)(1

−2p(x 1

)))|+

|(x2+

50(1

−2p(x 2

)))|

[−100,

100]

f(0,

−50)

=0

21.0E

−04

N,M

ShiftedSp

here

f 17(x

)=

∑ D i=1z2 i

+f bias,z

=x

−o,

x=

[x 1,x 2

,..

.x D

],o=

[o 1,o 2

,..

.oD]

[−100,

100]

f(o)

=f bias

=−4

5010

1.0E

−05

S,M

ShiftedAckley

f 18(x

)=

−20exp(

−0.2

√ 1 D

∑ D i=1z2 i

)−

exp(

1 D

∑ D i=1cos(2π

z i))

+20

+e

+f bias,

z=

(x−

o),x

=(x

1,x 2

,..

.x D

),o

=(o

1,o 2

,..

.oD)

[−32,3

2]f(o)

=f bias

=−1

4010

1.0E

−05

S,M

Easom

’sfunctio

nf 19(x

)=

−cosx 1cosx

2e(

(−(x

1−π

)2−(

x 2−π

)2))

[−10,1

0]f(

π,π

)=

−12

1.0E

−13

S,M

DekkersandAarts

f 20(x

)=

105x2 1

+x2 2

−(x2 1

+x2 2

)2+1

0−5(x

2 1+x2 2

)4[−

20,2

0]f(0,

15)=

f(0,

−15)

=−2

4777

25.0E

−01

N,M

123

Memetic Comp.

Tabl

e1

continued

Testproblem

Objectiv

efunctio

nsearch

Range

Optim

umValue

DAE

C

McC

ormick

f 21(x

)=

sin(x 1

+x 2

)+

(x1−x 2

)2−

3 2x 1

+5 2x 2

+1

−1.5

≤x 1

≤4,

−3≤

x 2≤

3f(

−0.547,

−1.547)=

−1.9133

301.0E

−04

N,M

Meyer

andRothProblem

f 22(x

)=

∑ 5 i=1(

x 1x 3t i

1+x 1t i+x

2vi−

y i)2

[−10,1

0]f(3.13

,15

.16,0.78

)=

0.4E

−04

31.0E

−03

U,N

Shubert

f 23(x

)=

−∑ 5 i=

1icos

((i+

1)x 1

+1)

∑ 5 i=1icos

((i+

1)x 2

+1)

[−10,1

0]f(7.0835

,4.8580

)=

−186

.7309

21.0E

−05

S,M

Sinusoidal

f 24(x

)=

−[A

∏ D i=1sin(x i

−z)

+∏ D i=

1sin(B(x

i−

z))],

A=

2.5,

B=

5,z

=30

[0,1

80]

f(90

+z)

=−(

A+

1)10

1.0E

−02

N,M

Moved

axisparalle

lhyp

er-ellipsoid

f 25(x

)=

∑ D i=15i

×x2 i

[−5.12,5

.12]

f(x)

=0;

x(i)

=5

×i,i=

1:D

301.0E

−15

U,S

Ddimensions,Ccharacteristic,U

unim

odal,M

multim

odal,S

separable,N

Non-separable,A

Eacceptableerror

5 10 15 20 252400

2420

2440

2460

2480

2500

Local search iterations (T)

Su

m o

f su

cces

s

Fig. 3 Effect of LLS termination criteria T on success rate

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.82400

2420

2440

2460

2480

2500

parameter cr

Su

m o

f su

cces

s

Fig. 4 Effect of parameter cr on success rate

to different values of cr in the range [0.1, 0.9], is exam-ined in the Fig. 4. It can be observed from Fig. 4 thatthe algorithm is very sensitive towards cr and value 0.6gives comparatively better results. Therefore cr = 0.6 isselected for the experiments in this paper.

5.2 Results comparison

The numerical results obtained are presented in Table 2 forsuccess rate (SR), average number of function evaluations(AFE), mean error (ME), and standard deviation (SD).

LSMO, SMO, ABC, DE, PSO–2011, CMA-ES and onesignificant variant of ABC namely, GABC, and two localsearch variants namely, LFABC, andMeABC, and one recentlocal search variant of SMO namely, PLSMO are comparedin terms of SR, AFE, ME, and SD as shown in Table 2. Theresults show that LSMO is competitive than SMO and otherconsidered algorithms for most of the benchmark test prob-lems irrespective of their nature either in termsof separability,modality and other parameters.

The considered algorithms are also compared throughMann–Whitney U rank sum test [27], acceleration rate andboxplot analysis. Mann–Whitney U rank sum test is appliedon average number of function evaluations. For all consid-

123

Memetic Comp.

Tabl

e2

Com

parisonof

theresults

oftestproblems

Testproblem

Measure

LSM

OSM

OLFA

BC

MeA

BC

GABC

ABC

PLSM

OCMA-ES

PSO-2011

DE

f 1SD

8.12E−0

78.37E−0

71.73E−0

67.87E−0

71.81E−0

62.02E−0

68.15E−0

77.98E−0

76.10E−0

78.24E−0

7

ME

7.78E−0

68.87E−0

68.39E−0

69.27E−0

68.11E−0

68.17E−0

68.96E−0

68.70E−0

69.33E−0

69.06E−0

6

AFE

10192.35

12642.3

16733.85

13626.8

14347.5

20409

13174.79

31604.8

38101.5

22444

SR100

100

100

100

100

100

100

100

100

100

f 2SD

1.37E−0

61.20E−0

63.02E−0

61.69E−0

62.72E−0

63.11E−0

69.49E−0

71.96E−0

68.62E−0

78.51E−0

7

ME

8.03E−0

68.49E−0

66.62E−0

68.26E−0

65.51E−0

64.90E−0

68.68E−0

67.37E−0

69.03E−0

69.01E−0

6

AFE

8266.64

10725.66

9556.12

5516.18

8388

9578.5

11198.07

20907.6

32596.5

20859.5

SR100

100

100

100

100

100

100

100

100

100

f 3SD

1.13E−0

68.98E−0

71.65E−0

68.48E−0

71.96E−0

62.20E−0

69.58E−0

79.97E−0

76.75E−0

77.51E−0

7

ME

8.70E−0

68.77E−0

68.84E−0

69.15E−0

68.10E−0

67.77E−0

68.86E−0

68.76E−0

69.28E−0

69.18E−0

6

AFE

19575.68

22596.75

24546.79

28443.32

23163

34887

23442.8

70154.2

68973

39676

SR100

100

100

100

100

100

100

100

100

100

f 4SD

8.87E−0

77.26E−0

71.58E−0

67.70E−0

71.88E−0

62.13E−0

68.37E−0

71.25E−0

67.31E−0

77.83E−0

7

ME

8.52E−0

69.02E−0

68.55E−0

69.21E−0

67.91E−0

67.84E−0

68.87E−0

68.39E−0

69.11E−0

68.99E−0

6

AFE

9961.85

12703.68

16111.3

13848.76

14072

20917

13246.78

32220.5

34906.5

21947.5

SR100

100

100

100

100

100

100

100

100

100

f 5SD

7.50E−0

75.45E−0

78.04E−0

73.75E−0

76.90E−0

71.03E−0

65.67E−0

75.08E−0

75.37E−0

74.90E−0

7

ME

9.13E−0

69.29E−0

69.34E−0

69.58E−0

69.30E−0

69.16E−0

69.33E−0

69.39E−0

69.50E−0

69.47E−0

6

AFE

21035.18

23300.64

30994.7

31451.16

27636

41646

24026.78

73331.4

70633.5

45014.5

SR100

100

100

100

100

100

100

100

100

100

f 6SD

7.09E−0

78.58E−0

71.70E−0

67.12E−0

71.98E−0

62.04E−0

69.77E−0

78.90E−0

77.22E−0

78.58E−0

7

ME

7.91E−0

69.01E−0

68.43E−0

69.20E−0

67.71E−0

67.96E−0

68.78E−0

68.88E−0

69.14E−0

69.03E−0

6

AFE

10692.31

14728.23

18093.08

16789.28

16084.5

22672

15275.92

38151.9

44316.5

25839.5

SR100

100

100

100

100

100

100

100

100

100

f 7SD

2.84E−0

61.72E−0

63.13E−0

62.87E−0

62.95E−0

62.65E−0

61.83E−0

63.27E−0

61.67E−0

61.87E−0

6

ME

6.37E−0

67.64E−0

65.86E−0

65.59E−0

65.62E−0

65.16E−0

67.99E−0

67.05E−0

68.35E−0

67.42E−0

6

AFE

3733.85

5153.94

7523.66

4527.08

9285

16229

5288.82

51777.2

9610

7867

SR100

100

100

100

100

100

100

83.33

100

100

f 8SD

8.68E−0

73.91E−0

66.84E−0

21.06E−0

21.04E+00

6.74E−0

18.60E−0

71.82E−0

64.12E−0

71.47E−0

6

ME

5.00E−0

61.06E−0

51.07E−0

18.88E−0

21.17E+00

9.22E−0

19.54E−0

67.12E−0

69.63E−0

68.05E−0

6

AFE

22010.61

169949.93

39650.81

22134.93

200000

200000

109047.36

11963

67937.5

17311.5

SR100

9195

100

00

100

100

100

100

123

Memetic Comp.

Tabl

e2

continued

Testproblem

Measure

LSM

OSM

OLFA

BC

MeA

BC

GABC

ABC

PLSM

OCMA-ES

PSO-2011

DE

f 9SD

1.08E−0

69.09E−0

71.99E−0

69.59E−0

71.93E−0

62.14E−0

67.86E−0

79.51E−0

77.00E−0

77.52E−0

7

ME

8.58E−0

68.74E−0

68.51E−0

69.11E−0

67.85E−0

67.80E−0

68.96E−0

68.56E−0

69.24E−0

69.08E−0

6

AFE

16202.15

18688.23

21192.18

22617.84

19475.5

28065

19427.76

42997.6

56112

32654

SR100

100

100

100

100

100

100

100

100

100

f 10

SD2.51E−0

21.33E−0

61.75E−0

68.84E−0

71.84E−0

62.19E−0

61.65E−0

68.08E−0

76.51E−0

78.76E−0

7

ME

4.16E−0

38.70E−0

68.27E−0

69.10E−0

68.10E−0

67.63E−0

68.69E−0

69.04E−0

69.31E−0

68.98E−0

6

AFE

17292.66

16804.91

15263.63

11797.24

13077

19747

24383.43

38128.4

35539.5

19806.5

SR97

100

100

100

100

100

100

100

100

100

f 11

SD1.53E−0

31.54E−0

31.81E−0

69.19E−0

71.65E−0

62.47E−0

62.16E−0

31.17E−0

63.15E−0

31.88E−0

3

ME

2.28E−0

42.29E−0

48.05E−0

69.07E−0

67.99E−0

67.11E−0

64.49E−0

48.78E−0

69.99E−0

43.39E−0

4

AFE

14794.98

18985

16857.9

12789.1

14295.5

21867.5

21667.18

32817.4

51964

25818

SR98

98100

100

100

100

96100

9197

f 12

SD1.02E−0

69.35E−0

71.97E−0

68.51E−0

71.94E−0

62.34E−0

67.74E−0

79.88E−0

77.68E−0

78.48E−0

7

ME

8.10E−0

68.97E−0

68.07E−0

69.14E−0

67.82E−0

67.44E−0

69.04E−0

68.60E−0

69.25E−0

69.03E−0

6

AFE

10792.78

15394.5

18653.59

17885.74

16683.5

24167

15915.57

40125.9

44293.5

27459.5

SR100

100

100

100

100

100

100

100

100

100

f 13

SD2.83E−0

63.00E−0

62.84E−0

62.96E−0

63.04E−0

62.25E−0

62.81E−0

62.44E−0

62.87E−0

62.67E−0

6

ME

4.03E−0

64.72E−0

67.52E−0

64.94E−0

65.61E−0

68.23E−0

65.36E−0

63.75E−0

64.92E−0

64.83E−0

6

AFE

1748.93

1537.47

3746.11

2573.53

9335.88

16098.58

1315.74

787.1

2716

1428

SR100

100

100

100

100

100

100

100

100

100

f 14

SD2.53E−0

42.12E−0

41.29E−0

32.31E−0

31.61E−0

21.20E−0

11.54E−0

42.07E−0

41.80E−0

44.49E−0

1

ME

6.84E−0

47.68E−0

49.19E−0

36.99E−0

31.58E−0

21.74E−0

18.66E−0

46.48E−0

48.49E−0

49.13E−0

2

AFE

19105.84

54551.33

65107.64

29780.95

197731.18

200000

17720.53

8173.3

51087.5

24687.5

SR100

100

100

100

50

100

100

100

90

f 15

SD8.35E−0

51.16E−0

41.79E−0

41.63E−0

53.80E−0

57.97E−0

51.16E−0

47.18E−0

54.03E−0

53.57E−0

4

ME

9.41E−0

51.06E−0

41.37E−0

48.46E−0

59.10E−0

51.75E−0

41.12E−0

42.50E−0

49.41E−0

53.02E−0

4

AFE

37661.35

43616.57

61386.26

41583.08

93336.22

181667.37

36332.34

151.6

38705

69677

SR99

9895

100

9020

97100

9967

f 16

SD9.47E−0

62.54E−0

52.37E−0

12.35E−0

52.61E−0

52.81E−0

52.47E−0

52.75E−0

73.01E−0

12.18E−0

1

ME

1.07E−0

65.93E−0

56.01E−0

26.09E−0

56.15E−0

56.51E−0

56.69E−0

54.87E−0

71.03E−0

15.01E−0

2

AFE

8925.42

14803.23

17885.73

8784.15

7949.1

8485.49

14405.7

1574

37910.5

13136.5

SR100

100

94100

100

100

100

100

8895

123

Memetic Comp.

Tabl

e2

continued

Testproblem

Measure

LSM

OSM

OLFA

BC

MeA

BC

GABC

ABC

PLSM

OCMA-ES

PSO-2011

DE

f 17

SD2.02E−0

61.73E−0

62.36E−0

61.89E−0

61.94E−0

62.50E−0

61.47E−0

61.91E−0

61.36E−0

61.74E−0

6

ME

6.98E−0

67.69E−0

67.27E−0

67.64E−0

67.64E−0

66.85E−0

67.63E−0

67.26E−0

68.44E−0

67.80E−0

6

AFE

6112.14

5948.91

6203.32

5587.6

5546.5

9074.5

6140.89

9665.3

15731.5

10397

SR100

100

100

100

100

100

100

100

100

100

f 18

SD9.31E−0

79.32E−0

71.34E−0

61.49E−0

61.60E−0

61.97E−0

61.14E−0

62.35E−0

68.20E−0

78.59E−0

7

ME

7.47E−0

68.66E−0

68.66E−0

68.71E−0

68.28E−0

67.79E−0

68.61E−0

65.37E−0

69.05E−0

68.99E−0

6

AFE

8825.98

9055.53

10934.63

10002.68

9305.5

16842

9500.74

17365

24686.5

15527.5

SR100

100

100

100

100

100

100

100

100

100

f 19

SD2.76E−1

42.78E−1

43.28E−1

42.08E−1

11.50E−1

28.06E−0

53.12E−1

48.17E−1

42.85E−1

42.96E−1

4

ME

4.58E−1

44.81E−1

45.60E−1

42.15E−1

22.03E−1

32.71E−0

54.85E−1

47.83E−1

45.08E−1

44.92E−1

4

AFE

14692.61

11829.51

14065.55

55466.86

43142.05

181234.08

11986.99

9612

9747.5

4773

SR100

100

100

9899

17100

100

100

100

f 20

SD5.05E−0

35.26E−0

35.68E−0

35.37E−0

34.84E−0

35.41E−0

35.88E−0

36.07E−0

35.16E−0

35.23E−0

3

ME

4.89E−0

14.90E−0

14.91E−0

14.89E−0

14.89E−0

14.89E−0

14.90E−0

17.91E−0

14.91E−0

14.90E−0

1

AFE

1413.94

1232.55

687.8

783.58

785

1432.53

1229.54

1725.5

4915

2113.5

SR100

100

100

100

100

100

100

100

100

100

f 21

SD6.69E−0

66.09E−0

66.96E−0

66.88E−0

66.38E−0

66.89E−0

67.07E−0

61.76E+00

6.36E−0

66.37E−0

6

ME

8.59E−0

58.70E−0

59.04E−0

58.70E−0

58.90E−0

58.93E−0

58.79E−0

52.05E−0

18.93E−0

58.70E−0

5

AFE

795.39

729.63

587.42

561.26

611.5

1204.01

724.83

258.433

1416

961

SR100

100

100

100

100

100

100

100

100

100

f 22

SD2.64E−0

62.96E−0

63.10E−0

62.87E−0

62.91E−0

62.85E−0

62.90E−0

68.99E−0

42.68E−0

61.78E−0

5

ME

1.94E−0

31.94E−0

31.95E−0

31.95E−0

31.95E−0

31.95E−0

31.95E−0

31.25E−0

21.95E−0

31.95E−0

3

AFE

2337.96

1947.07

3418.07

3886.93

4809.51

31742.53

1927.24

142128

3192

7823.5

SR100

100

100

100

100

100

100

87100

97

f 23

SD9.41E−0

46.67E−0

31.67E−0

32.39E−0

32.36E−0

31.85E−0

31.54E−0

36.27E−0

62.77E−0

12.41E−0

1

ME

1.20E−0

41.11E−0

28.35E−0

37.30E−0

37.50E−0

37.88E−0

38.37E−0

37.32E−0

63.94E−0

15.29E−0

1

AFE

15291.91

161340.38

22030.31

37251.9

48341.78

50666.58

27244.63

14262

180761

199693.5

SR100

66100

9999

100

100

100

211

f 24

SD1.05E−1

68.66E−1

71.09E−1

68.04E−1

76.06E−1

77.03E−1

79.47E−1

75.53E+00

5.85E−1

79.27E−1

7

ME

8.79E−1

68.97E−1

68.75E−1

69.09E−1

69.30E−1

69.31E−1

68.96E−1

68.36E−1

69.31E−1

68.90E−1

6

AFE

30109.1

34461.9

44903

45300.7

39749.5

62441.5

35773.96

199693.5

104853

59285.5

SR100

100

100

100

100

100

100

0100

100

f 25

SD9.21E−0

32.64E−0

41.62E+00

1.65E+00

3.88E+00

1.01E+01

1.17E−0

21.21E−1

63.12E−0

53.35E−0

5

ME

5.73E−0

45.88E−0

51.03E+00

2.20E+00

6.12E+00

1.74E+01

1.21E−0

31.50E−0

12.59E−0

52.46E−0

5

AFE

92344.7

110005.15

199313.43

200000

200000

200000

99717.72

91778.9

83559

66039.5

SR81

573

00

058

100

6971

123

Memetic Comp.

ered algorithms the test is performed at 5% significance level(α = 0.05) and the output results for 100 simulations are pre-sented in Table 3. In this table ‘+’ sign indicates that LSMOis significantly better than the other considered algorithmwhile ‘−’ sign represents that the other considered algorithmis better. The LSMO outperforms as compared to all otherconsidered algorithms for 10 test problems including f1, f3–f7, f9, f12, f18, and f24. LSMO performs better than basicSMO for 18 test problems, f1– f9, f11, f12, f14– f16, f18,and f23– f25. The LSMO shows better results for 24 testproblemswhen comparedwith basic ABC algorithm, f1– f15and f17– f25. The LSMO performs better for 21 test prob-lems, f1– f7, f9– f12, f14– f18, and f20– f24 in comparisonwith DE . The LSMO performs better for 23 test problems incomparison with PSO , f1– f18 and f20– f24. In comparisonwith CMA − ES, LSMO performs better on 15 functions,f1– f7, f9– f12, f17, f18, f20, f22, and f24. While comparingwith the variants of ABC , the LSMO performs better for 19test problems than GABC , f1– f9, f12– f15, f18, f19, f22–f25. The LSMO performs better than LFABC for 24 testproblems f1– f9 and f11– f25. In comparison with MeABC ,LSMO shows better results for 18 test problems f1, f3–f9, f11– f14, f17, f18, f22– f25. The LSMO shows betterresults for 18 test problems, f1– f12, f16– f18, f23– f25 whencompared with PLSMO algorithm. The above discussionrepresents that LSMO may be a competitive candidate inthe field of swarm intelligence.

Further, the convergence speed of considered algorithmsare compared by analysis of AFEs. There is an inverse rela-tion betweenAFEs and convergence speed, for smaller AFEsthe convergence speed will be higher and vice-versa. Forminimizing the effects of stochastic nature of algorithm, thereported AFEs are averaged for 100 runs for each consid-ered test problems. The convergence speed is compared usingacceleration rate (AR) for the considered algorithms. TheARwhich is calculated as follows:

AR = AFEALGO

AFELSMO, (12)

Here, ALGO ∈ {SMO,ABC,PSO,DE,CMA − ES,GbestABC,MeABC,LFABC,PLSMO} and AR > 1 repre-sents that LSMO is faster than the compared algorithm. TheAR results are shown in Table 4. The results in Table 4 showsthat for most of the considered benchmark test functions,LSMO converge faster than the considered algorithms.

The boxplots analyses have also been carried out for allthe considered algorithms for comparison regarding consol-idated performance. In boxplot analysis tool [27] graphicaldistribution of empirical data is efficiently represented. Theboxplots for LSMO and other considered algorithms arerepresented in Fig. 5. It is clear from this figure that LSMO

performs better than the considered algorithms as interquar-tile range, and the median is quite low.

6 Capacitor sizing and optimal placement problem

The placement of capacitors in the distribution networkis mainly needed, for improving power transfer capabil-ity, for properly serving to reactive loads, for the smoothworking of power transformers, and for secure and sta-ble transmission system in different network configurations.Further, these capacitors improve voltage profile and main-tain contractual obligations for electrical equipments. Thecapacitors also help in reducing the energy consumption ofvoltage-dependent sources aswell as technical losses [6]. Thecapacitors have been widely installed by utilities, to providereactive power compensation, to enhance the efficiency of thepower distribution, and to achieve deferral of construction.Economically, we can say that the capacitors installation indistribution network help in increasing, generation capacity,transmission capacity, and distribution substation capacity.Subsequently, it helps in increasing revenue generation. Butthe placement of capacitors exactly at required optimal posi-tion in a distribution system is a challenging task or can say adifficult problem for the distribution engineers. The objectiveof this problem is to minimize the energy losses while con-sidering the capacitor installation costs. In other words, thegoal is to achieve the optimal placement and sizing of capac-itors with the system constraints in the distribution network.The problem is defined as follows:

The total loss in a distribution system having n number ofbranches is given by

PLt =n∑

i=1

[I 2i ]Ri (13)

Here Ii and Ri are current magnitude and resistances respec-tively for the i th branch. The branch current obtained fromload flow solution has two components; active (Ia) and reac-tive (Ir ). In active and reactive branch currents, the associatedlosses are given by Eqs. 14 and 15 respectively.

PLa =n∑

i=1

[I 2ai

]Ri (14)

PLr =n∑

i=1

[I 2ri

]Ri (15)

In loss minimization technique of the capacitor placement, asingle capacitor is repetitively placed by varying its size fordetermining a sequence of nodes in viewof lossminimizationof the distribution system. The concept of loss minimization

123

Memetic Comp.

Tabl

e3

Com

parisonbasedon

Mann–Whitney

Urank

sum

testatsignificant

levelα

=0.05

andaveragenumberof

functio

nevolutions

TP

LSM

Ovs

SMO

LSM

Ovs

LFA

BC

LSM

Ovs

MeA

BC

LSM

Ovs

GABC

LSM

Ovs

ABC

LSM

Ovs

PLSM

OLSM

Ovs

CMA-ES

LSM

Ovs

PSO-2011

LSM

Ovs

DE

f 1+

++

++

++

++

f 2+

+−

++

++

++

f 3+

++

++

++

++

f 4+

++

++

++

++

f 5+

++

++

++

++

f 6+

++

++

++

++

f 7+

++

++

++

++

f 8+

++

++

+−

+−

f 9+

++

++

++

++

f 10

−−

−−

++

++

+

f 11

++

−−

++

++

+

f 12

++

++

++

++

+

f 13

−+

++

+−

−+

−f 14

++

++

+−

−+

+

f 15

++

++

+−

−+

+

f 16

++

−−

−+

−+

+

f 17

−+

−−

++

++

+

f 18

++

++

++

++

+

f 19

−+

++

+−

−−

−f 20

−+

−−

+−

++

+

f 21

−+

−−

+−

−+

+

f 22

−+

++

+−

++

+

f 23

++

++

++

−+

+

f 24

++

++

++

++

+

f 25

++

++

++

−−

−To

talN

o.of

+sign

1824

1819

2418

1722

21

TPtestprob

lem

123

Memetic Comp.

Tabl

e4

Com

parisonbasedon

acceleratio

nrate(A

R)

TP

LSM

Ovs

SMO

LSM

Ovs

LFA

BC

LSM

Ovs

MeA

BC

LSM

Ovs

GABC

LSM

Ovs

ABC

LSM

Ovs

PLSM

OLSM

Ovs

CMA-ES

LSM

Ovs

PSO-2011

LSM

Ovs

DE

f 11.24

1.64

1.34

1.41

21.29

3.1

3.74

2.2

f 21.3

1.16

0.67

1.01

1.16

1.35

2.53

3.94

2.52

f 31.15

1.25

1.45

1.18

1.78

1.2

3.58

3.52

2.03

f 41.28

1.62

1.39

1.41

2.1

1.33

3.23

3.5

2.2

f 51.11

1.47

1.5

1.31

1.98

1.14

3.49

3.36

2.14

f 61.38

1.69

1.57

1.5

2.12

1.43

3.57

4.14

2.42

f 71.38

2.01

1.21

2.49

4.35

1.42

13.87

2.57

2.11

f 87.72

1.8

1.01

9.09

9.09

4.95

0.54

3.09

0.79

f 91.15

1.31

1.4

1.2

1.73

1.2

2.65

3.46

2.02

f 10

0.97

0.88

0.68

0.76

1.14

1.41

2.2

2.06

1.15

f 11

1.28

1.14

0.86

0.97

1.48

1.46

2.22

3.51

1.75

f 12

1.43

1.73

1.66

1.55

2.24

1.47

3.72

4.1

2.54

f 13

0.88

2.14

1.47

5.34

9.2

0.75

0.45

1.55

0.82

f 14

2.86

3.41

1.56

10.35

10.47

0.93

0.43

2.67

1.29

f 15

1.16

1.63

1.1

2.48

4.82

0.96

01.03

1.85

f 16

1.66

20.98

0.89

0.95

1.61

0.18

4.25

1.47

f 17

0.97

1.01

0.91

0.91

1.48

11.58

2.57

1.7

f 18

1.03

1.24

1.13

1.05

1.91

1.08

1.97

2.8

1.76

f 19

0.81

0.96

3.78

2.94

12.34

0.82

0.65

0.66

0.32

f 20

0.87

0.49

0.55

0.56

1.01

0.87

1.22

3.48

1.49

f 21

0.92

0.74

0.71

0.77

1.51

0.91

0.32

1.78

1.21

f 22

0.83

1.46

1.66

2.06

13.58

0.82

60.79

1.37

3.35

f 23

10.55

1.44

2.44

3.16

3.31

1.78

0.93

11.82

13.06

f 24

1.14

1.49

1.5

1.32

2.07

1.19

6.63

3.48

1.97

f 25

1.19

2.16

2.17

2.17

2.17

1.08

0.99

0.9

0.72

TPtestprob

lem

123

Memetic Comp.

LSMO SMO LFABC MeABC GABC ABC PLSMO CMA−ES PSO DE

0

1

2

3

4

5

6

7

8 x 104

A

vera

ge

nu

mb

er o

f f

un

ctio

n e

valu

atio

ns

Fig. 5 Boxplots graphs for average number of function evaluation

by a singly located capacitor can be extended for multiplecapacitors [6].

Let us consider the following [13]:

– m = number of capacitor buses.– Ic = m dimensional vector consisting of capacitor cur-rents.

– α j = set of branches from the source bus to the j th capac-itor bus ( j = 1, 2, . . .m).

– D = a matrix of dimension n × m.

The elements of D are considered as

– Di j = 1; if branch i ∈ α

– Di j = 0; otherwise

When the capacitors are placed in the system, the newreactive component of branch currents is given by

[I newr ] = [Ir ] + [D][Ic] (16)

The loss associated with the new reactive currents in thecompensated system is

PcomLr =

n∑i=0

(Iri + Di j Icj

)2Ri (17)

The loss saving (S) is obtained by placing the optimalsize capacitors in the distribution network. The loss savingis calculated by taking the difference of the Eqs. 15 and 17and is shown as follows:

S = −n∑

i=1

⎡⎢⎣

⎛⎝2Iri

m∑j=1

Di j Icj +m∑j=1

Di j Icj

⎞⎠

2⎤⎥⎦ Ri (18)

For achieving themaximum loss saving, optimal capacitorcurrents can be obtained from the following equations:

δS

δ Ic1= 0

δS

δ Ic2= 0

− − −−− − −−δS

δ Ick= 0

(19)

After some mathematical manipulations, equation 19 can beexpressed by a set of linear algebraic equations as follows:

[A][Ic] = [B] (20)

Where A is a m ×m square matrix and B is a k-dimensionalvector. The elements of A and B are given by

A j j =∑iεα j

[Ri ] (21)

A j j =∑

iε(α j⋂

αm)

[Ri ] (22)

Bj =∑iεα j

[Iri Ri ] (23)

Only the branch resistances and reactive currents in theoriginal system are required to find the elements of A andB. The capacitor currents for the highest loss saving can beobtained from Eq. 20.

[Ic] = [A]−1[B] (24)

Once the capacitor currents are known, the optimal capac-itor sizes can be written as Qc in mega volt ampere reactive(MVAR) as equation 25

123

Memetic Comp.

Tabl

e5

Optim

alplacem

entand

sizing

ofcapacitorforIEEE14

bus3capacitorprob

lem

Sr.n

o.Optim

alplacem

ent

ofcapacitorson

bus

byGA

Optim

alplacem

ent

ofcapacitorson

bus

bySM

O

Optim

alplacem

ent

ofcapacitorson

busby

LSM

O

Size

ofcapacitors

byGA

(MVAR)

Size

ofcapacitors

bySM

O(M

VAR)

Size

ofcapacitors

byLSM

O(M

VAR)

Loss

uncapacitor

(MW)

Lossby

GA(M

W)

Lossby

SMO(M

W)

Lossby

LSM

O(M

W)

16

1413

14.642

1010

213

55

10.476

27.787

25.461

13.533

13.282

13.285196

13.2

76

39

99

41.771

2122.567

Tabl

e6

Optim

alplacem

entand

sizing

ofcapacitorforIEEE14

bus5capacitorprob

lem

Sr.n

o.Optim

alplacem

ent

ofcapacitorson

bus

byGA

Optim

alplacem

ent

ofcapacitorson

bus

bySM

O

Optim

alplacem

ent

ofcapacitorson

busby

LSM

O

Size

ofcapacitors

byGA

(MVAR)

Size

ofcapacitors

bySM

O(M

VAR)

Size

ofcapacitors

byLSM

O(M

VAR)

Loss

uncapacitor

(MW)

Lossby

GA(M

W)

Lossby

SMO(M

W)

Lossby

LSM

O(M

W)

114

1413

13.353

1010

210

33

46.426

23.855

25.510

310

22

17.356

12.770

10.071

13.533

13.318

13.275771

13.2

71

45

66

33.008

21.856

15.726

53

99

21.209

41.698

45.469

123

Memetic Comp.

Tabl

e7

Optim

alplacem

entand

sizing

ofcapacitorforIEEE30

bus3capacitorprob

lem

Sr.n

o.Optim

alplacem

ent

ofcapacitorson

bus

byGA

Optim

alplacem

ent

ofcapacitorson

bus

bySM

O

Optim

alplacem

ent

ofcapacitorson

busby

LSM

O

Size

ofcapacitors

byGA

(MVAR)

Size

ofcapacitors

bySM

O(M

VAR)

Size

ofcapacitors

byLSM

O(M

VAR)

Loss

uncapacitor

(MW)

Lossby

GA(M

W)

Lossby

SMO(M

W)

Lossby

LSM

O(M

W)

16

2224

40.204

2210.507

222

721

10.476

1016.051

17.944

17.531

17.480194

17.3

96

324

44

13.127

4135.083

Tabl

e8

Optim

alplacem

entand

sizing

ofcapacitorforIEEE30

bus5capacitorprob

lem

Sr.n

o.Optim

alplacem

ent

ofcapacitorson

bus

byGA

Optim

alplacem

ent

ofcapacitorson

bus

bySM

O

Optim

alplacem

ent

ofcapacitorson

busby

LSM

O

Size

ofcapacitors

byGA

(MVAR)

Size

ofcapacitors

bySM

O(M

VAR)

Size

ofcapacitors

byLSM

O(M

VAR)

Loss

uncapacitor

(MW)

Lossby

GA(M

W)

Lossby

SMO(M

W)

Lossby

LSM

O(M

W)

16

1717

17.569

1010

23

44

20.395

26.780

25.090

321

33

17.356

1010

17.944

17.489

17.355102

17.3

48

420

2424

13.008

11.292

11.320

513

2121

41.209

1010

123

Memetic Comp.

Tabl

e9

Optim

alplacem

entand

sizing

ofcapacitorforIEEE33

bus3capacitorprob

lem

Sr.n

o.Optim

alplacem

ent

ofcapacitorson

bus

byGA

Optim

alplacem

ent

ofcapacitorson

bus

bySM

O

Optim

alplacem

ent

ofcapacitorson

busby

LSM

O

Size

ofcapacitors

byGA

(MVAR)

Size

ofcapacitors

bySM

O(M

VAR)

Size

ofcapacitors

byLSM

O(M

VAR)

Loss

uncapacitor

(MW)

Lossby

GA(M

W)

Lossby

SMO(M

W)

Lossby

LSM

O(M

W)

112

2929

45.971

43.896

43.896

228

1313

43.734

44.000

44.000

178.735

165.840

165.422331

165.

42

313

1212

45.805

43.447

43.474

Tabl

e10

Optim

alplacem

entand

sizing

ofcapacitorforIEEE33

bus5capacitorprob

lem

Sr.n

o.Optim

alplacem

ent

ofcapacitorson

bus

byGA

Optim

alplacem

ent

ofcapacitorson

bus

bySM

O

Optim

alplacem

ent

ofcapacitorson

busby

LSM

O

Size

ofcapacitors

byGA

(MVAR)

Size

ofcapacitors

bySM

O(M

VAR)

Size

ofcapacitors

byLSM

O(M

VAR)

Loss

uncapacitor

(MW)

Lossby

GA(M

W)

Lossby

SMO(M

W)

Lossby

LSM

O(M

W)

130

2925

49.845

45.052

47

212

1211

40.395

45.000

47

313

1113

46.002

45.697

47178.735

158.052

157.668728

151.

287

422

1312

47.682

48.000

48

529

2529

47.175

44.035

46

123

Memetic Comp.

Qc = Vm × Ic (25)

Here Vm is the voltage magnitude vector of capacitorbuses. The saving in the compensated system can be esti-mated from Eq. 18 using the value of Ic given by Eq. 24.

The objective function may be formulated using Eq. 18 infollowing manner :

minf (xlocation, xsize) = S (26)

7 LSMO for optimal placement and sizing ofcapacitors

In this section, the LSMO and SMO algorithms are appliedto solve the optimal placement and sizing problem of capaci-tors in the distribution network. First in LSMO, the solutions

0 10 20 30 40 50 60 70 80 90 10013.25

13.3

13.35

13.4Optimal capacitor placement by LSMO for IEEE 14 bus system

Number of Generation

Lo

sses

Min

imu

m(M

W)

0 10 20 30 40 50 60 70 80 90 10013.27

13.28

13.29

13.3

13.31

13.32

13.33

13.34

13.35

13.36Optimal capacitor placement by LSMO for IEEE 14 bus system

Number of Generation

Lo

sses

Min

imu

m(M

W)

0 10 20 30 40 50 60 70 80 90 10017.38

17.4

17.42

17.44

17.46

17.48

17.5

17.52

17.54

17.56 Optimal capacitor placement by LSMO for IEEE 30 bus system

Number of Generation

Lo

sses

Min

imu

m(M

W)

0 10 20 30 40 50 60 70 80 90 10017.35

17.4

17.45

17.5

17.55

17.6 Optimal capacitor placement by LSMO for IEEE 30 bus system

Number of Generation

Lo

sses

Min

imu

m(M

W)

0 10 20 30 40 50 60 70 80 90 100165

166

167

168

169

170

171

172 Optimal capacitor placement by LSMO for IEEE 33 bus system

Number of Generation

Lo

sses

Min

imu

m(M

W)

0 10 20 30 40 50 60 70 80 90 100156

158

160

162

164

166

168 Optimal capacitor placement by LSMO for IEEE 33 bus system

Number of Generation

Lo

sses

Min

imu

m(M

W)

(a) (b)

Fig. 6 LSMO for Optimal placement of capacitor of IEEE−14, IEEE−30 and IEEE−33 bus system respectively for a 3 capacitor, b 5 capacitorproblems

are generated randomly in a given range i.e. capacitors ofgiven value are placed at random nodes in the distributionsystem. Here, each solution represents the size and locationof capacitors in the distribution network, for example for3-capacitor problem; a solution will be of six dimensionsof which first three will represent the size of the capaci-tors while remaining three will show the locations of thecapacitors. Here, it should be noted that the locations of thecapacitors are represented by discrete values while size bycontinues values. Therefore, the first three real values areconverted into discrete values by rounding off in the nearbyinteger value. In this way, a mixed representation of thesolution is prepared. As the capacitor placement and thesizing problem is non-separable and multimodal in nature,SMO and its proposed variant are applied to solve it. Inthis paper, the loss minimization is carried out by provid-ing the optimal size and location of the capacitors in the

123

Memetic Comp.

network. For testing, the performance of LSMO, it is appliedon IEEE−14, 30 and 33 test bus radial distribution system.The LSMO is used to update the bus data of the consid-ered bus systems iteratively for reducing the system losses.The reported results are compared with GA and SMO (Theparameter settings of GA and SMO are same as their legiti-mate research papers [2,8]) as shown in Tables 5, 6, 7, 8, 9,and 10. The better results are represented by bold values.The loss minimization curves are shown in Fig. 6. Fromthese tables, it is clear that the size of the capacitor isdetermined inMVAR (i.e. 106 VAR)while power loss ismea-sured in Mega Watts (i.e. 106 W). So, a little difference inpower loss and capacitor size affects the performance signif-icantly. The results show that the loss occurred using LSMOstrategy isminimumamong all the considered cases and algo-rithms. Therefore, the LSMO may be used for solving thecapacitor placement and sizing problem of the distributionsystem.

8 Conclusion

In this paper, a limaçon inspired local search (LLS) strat-egy is developed and hybridized with SMO. The pro-posed hybridized strategy is named as limaçon inspiredSMO (LSMO). The performance of LSMO has evaluatedover 25 well-known benchmark functions. Results indicatethat the proposed LSMO is a significant candidate amongmost promising swarm intelligence based global optimiza-tion algorithms. Further, a complex real-world optimizationproblem, optimal placement and sizing of capacitors indistributed network is solved with IEEE 14, 30, and 33bus test system using LSMO. Results have been comparedwith those of GA and SMO. It is observed that LSMOobtains minimum distribution and transmission losses whilemaintaining the minimum cost. This work may further beextended to an unbalanced radial system as a future researchperspective.

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