capacitor placement of distribution systems using particle swarm optimization approaches

Electrical Power and Energy Systems 64 (2015) 839–851

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Capacitor placement of distribution systems using particle swarmoptimization approaches

http://dx.doi.org/10.1016/j.ijepes.2014.07.0690142-0615/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Industrial and Systems Engineering GraduateProgram (PPGEPS), Pontifical Catholic University of Parana (PUCPR), ImaculadaConceição, 1155, Zip code 80215-901, Curitiba, PR, Brazil.

Chu-Sheng Lee a, Helon Vicente Hultmann Ayala b, Leandro dos Santos Coelho b,c,⇑a Department of Electrical Engineering, National Formosa University, 64, Wen-Hua Road, Huwei, Yunlin 632, Taiwanb Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, Zip code 80215-901, Curitiba,PR, Brazilc Department of Electrical Engineering, Federal University of Parana (UFPR), Rua Cel. Francisco Heraclito dos Santos, 100, Zip code 81531-980, Curitiba, PR, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 July 2013Received in revised form 19 July 2014Accepted 23 July 2014

Keywords:Capacitor placementChaos theoryGaussian probability distributionParticle swarm optimization

Capacitor placement plays an important role in distribution system planning and operation. In distribu-tion systems of electrical energy, banks of capacitors are widely installed to compensate the reactivepower, reduce the energy loss in system, voltage profile improvement, and feeder capacity release. Thecapacitor placement problem is a combinatorial optimization problem having an objective functioncomposed of power losses and capacitor installation costs subject to bus voltage constraints. Recently,many approaches have been proposed to solve the capacitor placement problem as a mixed integerprogramming problem. This paper presents a new capacitor placement method which employs particleswarm optimization (PSO) approaches with operators based on Gaussian and Cauchy probabilitydistribution functions and also in chaotic sequences for a given load pattern of distribution systems.The proposed approaches are demonstrated by two examples of application. Simulation results show thatthe proposed method can achieve an optimal solution as the exhaustive search can but with much lesscomputational time.

� 2014 Elsevier Ltd. All rights reserved.

Introduction

The problem of capacitors placement in distribution systems ofelectrical energy involves the determination of the number, loca-tion, type and size of the capacitors to be placed on the distributionfeeders such that the total cost of installation and operation of thesystem is minimum with respect to the load on the system. Thisproblem is combinatorial in nature and the application of discretecapacitors will be considered to solve it in this paper. That is, thereare a total of (L + 1)J possible solutions, where J is the bus numberand L is the number of available capacitor sizes. It is worth men-tioning that (L + 1)J possibilities will become a large number toevaluate exhaustively even for a medium-size distribution system.

In this context, considerable efforts have been devoted to solvethe capacitor placement problem. A variety of solution approachesbased on mathematical programming techniques and gradient-based algorithms [1–5] have been developed to solve the capacitorplacement problem over the years. In contrast to these analytical

optimization techniques, meta-heuristics of natural computingfield have been recently proposed in literature [6–11].

The purpose of developing such meta-heuristics is to decreasethe exhaustive search space, while providing as the final result(objective function) an optimal or near-optimal value. Moreover,classical gradient-based algorithms have disadvantages, such as(i) requirement of continuous and differentiable objective func-tions, (ii) difficulty in escaping local minima, and (iii) difficulty tohandle discrete control variables.

To overcome these disadvantages, flexible meta-heuristics, suchas particle swarm optimization (PSO) approaches, have been stud-ied. The PSO algorithm is a stochastic algorithm. It does not needgradient information about the objective function, as the gradi-ent-based algorithm does. PSO uses the analogy of swarming andcollaboration principles and is one powerful method for solvingunconstrained and constrained nonlinear global optimization ofoptimal capacitor placement.

The framework of PSO is simple. PSO is easy to be implementedwithin a personal computer and is inexpensive in terms of memoryrequirements and computational time. Moreover, the PSOtechnique can generate a high quality solution within shortercalculation time and stable convergence characteristics than otherstochastic methods.

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Fig. 1. Single-lime diagram of a main feeder.

Fig. 2. Main calculation procedures of the proposed method.

840 C.-S. Lee et al. / Electrical Power and Energy Systems 64 (2015) 839–851

This paper presents an optimal capacitor placement methodthat applies PSO approaches. The PSO originally developed by Ken-nedy and Eberhart in 1995 [12,13] is a population-based algorithmof the swarm intelligence. Similarly to genetic algorithms [14], anevolutionary algorithm approach, PSO is an optimization toolwhere each member is seen as a particle, and each particle is apotential solution to the problem under analysis. Each particle inPSO has a randomized velocity associated to it, which movesthrough the space of the problem. However, unlike geneticalgorithms, PSO does not have operators such as crossover andmutation. PSO does not implement the survival of the fittest indi-viduals; rather, it implements the simulation of social behavior.PSO approaches have been utilized in optimization problems inpower systems area, such as power flow [15], loss powerminimization [16], reactive power dispatch [17], economic dis-patch [18–20], stabilizers tuning [21], expansion planning [22],unit commitment problem [23], reactive power and voltage control[24], and so on. A survey of the PSO applications in electrical powersystems is presented in [25].

In PSO, a uniform probability distribution to generate randomnumbers is used. However, the use of other probability distribu-tions may improve the ability to fine-tune or even to escape fromlocal optima. In the meantime, the use of the Gaussian and Cauchyprobability distributions has been proposed to generate randomnumbers to update the velocity equation [26,27] inspired by stud-ies of mutation operators in fast evolutionary programming[28,29]. All these approaches attempted to improve the perfor-mance of the standard PSO, but the amount of parameters of thealgorithm to tune remained the same.

This paper employs the Gaussian and the Cauchy probabilitydistributions and also chaotic sequences in PSO approaches tosearch the size and location of capacitors to be installed on a radialdistribution feeder. In other words, the main contribution of thepaper can be summarized as using different distribution to gener-ate random numbers required by the PSO. During each iterativeprocedure referred to as a generation, a new set of particles is pro-duced using rules of evolution with improved performance. Thefitness value for each particle of PSO is composed of the totalpower losses and the cost of capacitors added for that correspond-ing configuration. To demonstrate the effectiveness, the proposedapproach is applied to two application example systems.

The remaining sections of this paper are organized as follows.Section ‘Problem description and formulations’ describes the prob-lem and its formulations. Section ‘Particle swarm optimization’then describes the Gaussian, Cauchy and chaotic sequences forPSO approaches adopted here, while Sections ‘Computational pro-cedures and The application examples’ discuss the computationalprocedures and analyze the results applied to two examples,respectively. Lastly, Section ‘Conclusions and future research’presents our conclusions and future research work.

Problem description and formulations

This work discusses the capacitor placement problem of distri-bution systems. The objective is to minimize the annual cost of the

system, subject to operating constraints under a certain loadpattern. Mathematically, the objective function of the problemcan be described as min F = min (COST), where COST includes thecost of power loss and capacitor placement, and will be furtherdiscussed later. The voltage magnitude at each bus must be main-tained within its limits expressed as follows

Vmin 6 jVij 6 Vmax ð1Þ

where |Vi| is voltage magnitude of bus i, Vmin and Vmax are busminimum and maximum voltage limits, respectively.

A set of feeder-line flow formulations to avoid the complex iter-ation process for power flow analysis is applied. Considering thesingle-line diagram depicted in Fig. 1, the following set of recursiveequations is used for power flow computation [7–9,11]:

Piþ1 ¼ Pi � PLiþ1 � Ri;iþ1½ðP2i þ Q 2

i Þ=jVij2� ð2ÞQiþ1 ¼ Q i � QLiþ1 � Xi;iþ1½ðP2

i þ Q2i Þ=jVij2� ð3Þ

jViþ1j2 ¼ jVij2 � 2ðRi;iþ1Pi þ Xi;iþ1Q iÞ þ ðR2i;iþ1 þ X2

i;iþ1ÞðP2

i þ Q2i Þ

jVij2ð4Þ

Fig. 3. A 9-section feeder.

Table 1Feeder impedance data at 60 Hz.

From bus i To bus i + 1 Ri, i+1 (X) Xi, i+1 (X)

0 1 0.1233 0.41271 2 0.0140 0.60512 3 0.7463 1.20503 4 0.6984 0.60844 5 1.9831 1.72765 6 0.9053 0.78866 7 2.0552 1.16407 8 4.7953 2.71608 9 5.3434 3.0264

Table 2Three-phase load data.

Bus no. PL (kW) QL (kvar)

1 1840 4602 980 3403 1790 4464 1598 18405 1610 6006 780 1107 1150 608 980 1309 1640 200

Total 12,368 4186

Table 3Available three-phase capacitor sizes and annual cost.

Size (kvar) 150 300 450 600 900 1200Cost ($/kvar) 75 97.5 114 132 165 204

Table 4Possible choices of capacitor size and cost/kvar.

j 1 2 3 4 5 6 7Qc

j 150 300 450 600 750 900 1050

j 8 9 10 11 12 13 14Qc

j 1200 1350 1500 1650 1800 1950 2100

j 15 16 17 18 19 20 21Qc

j 2250 2400 2550 2700 2850 3000 3150

j 22 23 24 25 26 27 –Qc

j 3300 3450 3600 3750 3900 4050 –

C.-S. Lee et al. / Electrical Power and Energy Systems 64 (2015) 839–851 841

where Pi and Qi are the real and reactive powers flowing out of bus i,PLi and QLi are the real and reactive load powers at bus i. The resis-tance and reactance of the line section between buses i and i + 1 aredenoted by Ri,i+1 and Xi,i+1, respectively. The power loss of the linesection connecting between buses i and i + 1 may be computed as

PLossði; iþ 1Þ ¼ Ri;iþ1P2

i þ Q 2i

jVij2: ð5Þ

The total power loss of the feeder, PT,Loss, may then be deter-mined by summing up the losses of all line sections of the feeder,which is given by

PT;Loss ¼Xn�1

i¼0

PLossði; iþ 1Þ: ð6Þ

The purpose of placing compensating capacitors along distribu-tion feeders is to lower the total power loss and bring the busvoltages within their specified limits while minimizing the totalcost. Considering shunt capacitors, there exists a finite number ofstandard sizes which are integer multiples of the smallest sizeQ c

0. Besides, the cost per kvar varies from one size to another. Ingeneral, capacitors of larger size have lower unit prices. The avail-able capacitor size is usually limited to

Q cmax ¼ LQc

0 ð7Þ

where L is an integer. Therefore, for each installation location, thereis L capacitor sizes fQc

0;2Qc0; . . . ; LQc

0g available. Let the correspond-ing equivalent annual cost per kvar for the L capacitor sizes,fKc

1;Kc2; ldots;Kc

Lg, be given, then the equivalent annual capacitorinstallation cost for each compensation bus can be determined.Let this cost of bus i be denoted as Cc

i , then the total cost of (1)due to capacitor placement and power loss change is written as

COST ¼ KPPT;Loss þXn

i¼1

Kci Q c

0

¼ KPPT;Loss þXn

i¼1

Cci

ð8Þ

where KP is the equivalent annual cost per unit of power loss in $/(kW-year), $ is a fictional monetary unit, and i = 1, 2, . . ., n are theindices of buses selected for compensation. The constant Kc

i is theannual unit capacitor installation cost. The bus reactive power com-pensation is limited to

Q ci 6

Xn

i¼1

Q Li ð9Þ

where Qci is the reactive power compensation at bus i. This problem

mentioned above is a nonlinear optimization problem.

Particle swarm optimization

Fundamentals

The proposal of PSO algorithm was put forward by severalscientists who developed computational simulations of the move-ment of organisms such as flocks of birds and fish schools. Such

simulations were heavily based on manipulating the distancesbetween individuals, i.e., the synchrony of the behavior of theswarm was seen as an effort to keep an optimal distance betweenthem. Sociobiologist Edward Osbourne Wilson outlined a link ofthese simulations for optimization problems [30].

Each particle in PSO keeps track of its coordinates in the prob-lem space, which are associated with the best solution (fitness) ithas achieved so far. This value is called pbest (personal best).Another ‘‘best’’ value that is tracked by the global version of theparticle swarm optimizer is the overall best value and its locationobtained so far by any particle in the population. This location iscalled gbest (global best).

The PSO concept consists of, at each time step, changing (accel-erating) the velocity of each particle flying toward its pbest andgbest locations (global version of PSO). Acceleration is weightedby random terms, with separate random numbers being generatedfor acceleration toward pbest and gbest locations, respectively. Theprocedure for implementing the global version of PSO is given bythe following steps [26,31]:

Table 5Results of PSO approaches for case 1 (100 runs). The best results are in bold letter.

Annual cost ($) Generations for obtain the minimum annual cost Mean time

Type of PSO Maximum Minimum Mean Standard deviation Maximum Minimum Mean Standard deviation (in seconds) a

(1) 119451.271 118538.530 118890.601 321.031 36 15 23 7 3.90(2) 119694.524 118538.530 118952.112 338.122 48 11 27 12 4.08(3) 119773.052 118538.530 118977.894 347.843 56 14 27 13 4.10(4) 119123.586 118538.530 118851.077 185.484 45 11 25 10 4.27(5) 119876.965 118538.530 118952.254 370.730 50 9 24 11 4.11(6) 119287.171 118538.530 118818.363 269.905 38 8 28 8 4.11(7) 119628.169 118683.181 118962.112 291.273 42 20 29 7 4.28(8) 119628.160 118538.530 119035.710 306.213 48 14 31 11 4.29(9) 119485.560 118538.530 118809.104 297.640 31 13 22 6 4.28(10) 121474.308 118538.530 118963.234 364.159 14 1 27 11 3.92(11) 120304.614 118538.530 118979.946 333.371 51 1 27 11 3.91(12) 121526.121 118538.530 119013.115 382.299 51 1 24 12 4.10(13) 121276.678 118538.530 119051.985 466.554 43 1 22 10 4.09(14) 122139.667 118538.530 119183.858 535.691 51 1 25 11 4.11(15) 122539.231 118538.530 119191.123 515.138 53 1 27 10 4.10

a Mean time of each run in a PC-compatible with AMD 1.10 GHz processor and 112 MB RAM using Matlab 6.5.

Table 6Annual cost and total power loss for case 1.

Items Uncompensated Compensated

Heuristic [4] Genetic algorithm Tabu search PSO(1)–(15) Exhaustive search

Total loss (kW) 787.778 704.3 701.478 700.567 698.140 698.140Annual cost ($) 131,675 119,513 118,916 118,865 118,538 118,538Net savings ($) – 12,162 12,759 12,810 13,137 13,137Compensation

Qc4 ðkvarÞ 0 2700 3300 4050 4050 4050

Qc5 ðkvarÞ 0 2850 1800 1350 1650 1650

Qc9 ðkvarÞ 0 900 900 900 750 750

Max |Vi| (p.u.) 0.9929 (bus 1) 1.000 0.9992 (bus 2) 0.9979 (bus 2) 1.0001 (bus 2) 1.0001 (bus 2)Min |Vi| (p.u.) 0.8375 (bus 9) 0.905 0.9007 (bus 9) 0.9009 (bus 9) 0.9000 (bus 9) 0.9000 (bus 9)


Step 1. Initialization: initialize a population (array) of particleswith random positions and velocities in the n dimensionalproblem space using a uniform probability distribution.Step 2. Evaluation: evaluate the fitness value of each particle.Step 3. Comparison 1: compare each particle’s fitness with theparticle’s pbest. If the current value is better than pbest,then set the pbest value equal to the current value and thepbest location equal to the current location in n-dimensionalspace.Step 4. Comparison 2: compare the fitness with the population’soverall previous best. If the current value is better than gbest,then reset gbest to the current particle’s array index and value.Step 5. Updating: change the velocity and position of the parti-cle according to Eq. (10) and (11), respectively [32,33]:

miðt þ 1Þ ¼ w � miðtÞ þ c1 � ud � ðpiðtÞ � xiðtÞÞ þ c2 � Ud

� ðpgðtÞ � xiðtÞÞ ð10Þ

xiðt þ 1Þ ¼ xiðtÞ þ Dt � miðt þ 1Þ: ð11Þ

where t = 1, 2, . . ., indicates the iterations, w is a parametercalled the inertia weight. vi = [vi1, vi2, . . ., vin]T stands for thevelocity of the i-th particle, xi = [xi1, xi2, . . ., xin]T stands for theposition of the i-th particle, and pi = [pi1, pi2, . . ., pin]T representsthe best previous position of the i-th particle; Positive constantsc1 and c2 are the cognitive and social components, respectively,which are the acceleration constants responsible for varyingthe particle speed towards pbest and gbest, respectively. Indexg represents the index of the best particle among all the particlesin the swarm. Variables ud and Ud are two random numbersgenerated in the range [0, 1]. Eq. (11) represents the position

update, according to its previous position and its velocity, con-sidering Dt = 1.Step 6. Stopping criterion: Loop to Step (2) until a stopping crite-rion is met, usually a fitness sufficiently good or a maximumnumber of iterations.The use of variable w was proposed by Shi and Eberhart [32].

This parameter is responsible for dynamically adjusting the speedof the particles, so it is responsible for balancing between local andglobal search, hence requiring fewer iterations for the algorithm toconverge. A low value of inertia weight implies a local search,while a high value leads to a global search.

Applying a high inertia weight at the start of the algorithm andmaking it decay to a low value through the PSO execution makesthe algorithm search globally at the beginning of the search, andsearch locally at the end of the execution. The following weightingfunction w is used in (10):

w ¼ wmax �wmax �wmin

tmaxt ð12Þ

Eq. (12) shows how the inertia weight is updated, considering tmax isthe maximum iteration number, t is the current iteration number,and wmax and wmin are the initial and final weights, respectively[33].

The first part in Eq. (10) is the momentum part of the particle.The inertia weight w represents the degree of the momentum ofthe particles. The second part is the ‘cognition’ part, whichrepresents the independent thinking of the particle itself.

In this work, new approaches to PSO, named fast PSO areproposed which are based on the studies of [26,27,31]. The aimis to modify the Eq. (10) of the conventional PSO (case 1) with udand Ud based on uniform distribution to use it with Gaussian,Cauchy distributions and/or chaotic sequences in the range [0, 1].

(a) PSO(1) (b) PSO(2) (c) PSO(3)

(d) PSO(4) (e) PSO(5) (f) PSO(6)

(g) PSO(7) (h) PSO(8) (i) PSO(9)

(j) PSO(10) (k) PSO(11) (l) PSO(12)

(m) PSO(13) (n) PSO(14) (o) PSO(15)

Fig. 4. Mean behavior (100 runs) of PSO approaches for case 1.





(1) 125622.909 115677.376 117407.182 1403.013 135 9 38 24 4.13(2) 120565.022 115813.749 117254.163 905.369 117 6 37 21 4.66(3) 125607.108 115843.027 117505.255 1486.147 88 9 36 19 4.83(4) 121218.045 115689.240 117568.817 992.197 118 8 36 22 5.22(5) 123278.418 115990.997 117438.265 1130.731 176 1 39 26 4.70(6) 121710.507 115811.523 117235.209 849.660 94 8 36 18 4.72(7) 119936.503 115593.400 117168.279 837.616 111 1 37 23 5.23(8) 122956.824 115820.811 117476.927 1026.892 120 12 37 21 5.24(9) 125490.397 115781.097 117675.557 1461.583 146 1 38 24 5.23(10) 120565.021 115739.260 117290.542 868.306 113 8 40 22 4.17(11) 124275.737 115630.020 117637.896 1570.516 114 9 45 24 4.16(12) 123542.172 115934.811 117610.092 1175.079 129 8 41 25 4.70(13) 124583.210 115827.716 117702.906 1499.768 112 8 41 22 4.69(14) 125827.143 115867.534 117553.323 1400.160 120 11 39 21 4.70(15) 123542.172 115681.667 117815.630 1433.244 156 9 42 27 4.70


Table 8The voltage profile and capacitors added for case 2.

Busno.

Uncompensatedvoltage (p.u.)

Fuzzy reasoning [7] Genetic algorithm Tabu search PSO(7)

Placed Qc

(kvar)Compensatedvoltage (p.u.)

Placed Qc


Placed Qc


Placed Qc

(kvar)Compensatedvoltage (pu)

– 1.000000 – 1.000000 – 1.000000 – 1.000000

1 0.9929 0 0.997162 1050 1.001220 0 0.999836 0 1.0008712 0.9874 0 0.997890 3450 1.006657 3600 1.004442 4050 1.0069933 0.9634 1050 0.986671 600 0.997261 1200 0.992984 1800 0.9976194 0.9619 1050 0.976671 3000 0.988841 2400 0.982776 2100 0.9878145 0.9480 1950 0.957769 1650 0.967715 900 0.958441 1500 0.9656476 0.9072 0 0.950215 0 0.959485 0 0.949889 0 0.9571637 0.8890 0 0.935827 0 0.944152 0 0.934035 0 0.9414358 0.8587 0 0.914346 750 0.920053 4500 0.908812 0 0.9164089 0.8375 900 0.902869 0 0.900347 1500 0.889819 600 0.900431


New PSO approaches

Coelho and Krohling [26] proposed the use of a truncatedGaussian and Cauchy probability distribution to generate randomnumbers for the velocity updating equation of PSO. In this context,Coelho and Lee [18] presented a chaotic Gaussian approach tosolve economic load dispatch problems in power systems.

In this paper, new approaches to PSO are proposed which arebased on Gaussian and Cauchy probability distributions linkedwith chaotic sequences. Firstly, random numbers are generatedusing the Gaussian or the Cauchy probability distribution orchaotic sequences in the interval [�1, 1], and then mapped tothe interval [0, 1]. The use of Cauchy distribution in PSO could beuseful to escape the local minima, while the Gaussian distributioncould provide a faster convergence in local searches.

Chaos, apparently disordered behavior that is nonethelessdeterminist, is a universal phenomenon that occurs in many sys-tems in several areas of science [34]. The chaos phenomena reflectsthe inner mechanism of complex nonlinear system that includestime-variation and nonlinear. The chaos is mathematically definedas ‘‘randomness’’ generated by simple determinist systems. Therandomness is a result of the sensitivity of chaotic systems to theinitial conditions. A chaotic movement can go through every statein a certain area according to its own regularity, and every state isobtained only once. Chaotic optimization approaches are based onergodicity, stochastic properties and irregularity [35]. In this work,the logistic map [36] for chaotic PSO approach was adopted. Thelogistic map is given by:

yðkÞ ¼ l � yðk� 1Þ � ½1� yðk� 1Þ� ð13Þ

where k is the sample, and l is a control parameter, 0 6 l 6 4. Thebehavior of the system of Eq. (13) is greatly influenced by thevariation of l. The value of l determines whether y stabilizes at aconstant size, oscillates between a range, or behaves chaotically inan unpredictable pattern. Eq. (13) is deterministic, displaying cha-otic dynamics when l = 4 and y(1) R {0, 0.25, 0.50, 0.75, 1}. In thiscase, y(k) is distributed in the range [0, 1] provided the initialy(1) e [0, 1] and thus we adopt y(1) = 0.48 in the present work.PSO approaches are proposed with combination of chaoticsequences based on logistic map and Gaussian and Cauchy proba-bility density functions. In this work, Sd and sd sequences adoptedin PSO are equal for the results of Eq. (13) for y(k) in the range [0, 1].

We hereafter propose 14 different types of PSO by the modifica-tion of the velocity Eq. (10) (refer to conventional PSO as Type 1),according to the type of random distribution used:

Type 2: PSO uses numbers generated with Cauchy distribution,cd, in the interval [0, 1] for the ‘‘cognitive part’’:

miðtþ1Þ¼w �miðtÞþc1 �cd � ðpiðtÞ�xiðtÞÞþc2 �Ud � ðpgðtÞ�xiðtÞÞð14Þ

Type 3: PSO uses numbers with Cauchy distribution, Cd, in theinterval [0, 1] are generate for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �ud � ðpiðtÞ�xiðtÞÞþc2 �Cd � ðpgðtÞ�xiðtÞÞð15Þ

Type 4: PSO uses numbers generated with Cauchy distributions,cd and Cd, in the interval [0, 1] are used for the ‘‘cognitive part’’and ‘‘social part’’:

Table 9The annual cost and total power loss for case 2.


Fuzzy reasoning [7] Genetic algorithm Tabu search PSO(7)

Total loss (kW) 783.778 704.8829 683.3326 672.7619 677.0202Annual cost ($) 131,675 119,420 115,819.9 115,795.4 115,593.4Net savings ($) – 12,255 (9.31%) 15,855.1 (12.04%) 15,879.6 (12.06%) 16,081.6 (12.21%)


miðtþ1Þ¼w �miðtÞþc1 � cd � ðpiðtÞ�xiðtÞÞþc2 �Cd � ðpgðtÞ�xiðtÞÞð16Þ

Type 5: PSO uses numbers generated with Gaussian distribu-tion, gd, in the interval [0, 1] for the ‘‘cognitive part’’:

miðtþ1Þ¼w �miðtÞþc1 �gd � ðpiðtÞ�xiðtÞÞþc2 �Ud � ðpgðtÞ�xiðtÞÞð17Þ

Type 6: PSO uses numbers generated with Gaussian distribu-tion, Gd, in the interval [0, 1] are adopted for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �ud � ðpiðtÞ�xiðtÞÞþc2 �Gd � ðpgðtÞ�xiðtÞÞð18Þ

Type 7: PSO uses numbers generated with Gaussian distribu-tion, gd and Gd, in the interval [0, 1] for the ‘‘cognitive part’’and ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �gd � ðpiðtÞ�xiðtÞÞþc2 �Gd � ðpgðtÞ�xiðtÞÞð19Þ

Type 8: PSO uses numbers generated with Gaussian distributionin the interval [0, 1], gd, for the ‘‘cognitive part’’ and with Cau-chy distribution for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �gd � ðpiðtÞ�xiðtÞÞþc2 �Cd � ðpgðtÞ�xiðtÞÞð20Þ

Type 9: PSO uses numbers generated with Cauchy distributionin the interval [0, 1], cd, for the ‘‘cognitive part’’ and with Gauss-ian distribution for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 � cd � ðpiðtÞ�xiðtÞÞþc2 �Gd � ðpgðtÞ�xiðtÞÞð21Þ

Type 10: PSO uses numbers generated by chaotic sequences, Sd,in the interval [0, 1] for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �ud � ðpiðtÞ�xiðtÞÞþc2 �Sd � ðpgðtÞ�xiðtÞÞð22Þ

Type 11: PSO uses numbers generated by chaotic sequences, sd,in the interval [0, 1] for the ‘‘cognitive part’’:

miðtþ1Þ¼w �miðtÞþc1 � sd � ðpiðtÞ�xiðtÞÞþc2 �Ud � ðpgðtÞ�xiðtÞÞð23Þ

Type 12: PSO uses numbers generated in the interval [0, 1] withGaussian distribution, gd, for the ‘‘cognitive part’’ and thechaotic sequences, Sd, in the interval [0, 1] for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �gd � ðpiðtÞ�xiðtÞÞþc2 �Sd � ðpgðtÞ�xiðtÞÞð24Þ

Type 13: PSO uses numbers generated in the interval [0, 1] withGaussian distribution, Gd, for the ‘‘social part’’ and the chaoticsequences, sd, in the interval [0, 1] for the ‘‘cognitive part’’:

miðtþ1Þ¼w �miðtÞþc1 � sd � ðpiðtÞ�xiðtÞÞþc2 �Gd � ðpgðtÞ�xiðtÞÞð25Þ

Type 14: PSO uses numbers generated in the interval [0, 1] withCauchy distribution, cd, for the ‘‘cognitive part’’ and the chaoticsequences, Sd, in the interval [0, 1] for the ‘‘social part’’:

miðtþ1Þ¼w �miðtÞþc1 �cd � ðpiðtÞ�xiðtÞÞþc2 �Sd � ðpgðtÞ�xiðtÞÞð26Þ

Type 15: PSO uses numbers generated in the interval [0, 1] withCauchy distribution, Cd, for the ‘‘social part’’ and the chaoticsequences, sd, in the interval [0, 1] for the ‘‘cognitive part’’:

miðtþ1Þ¼w �miðtÞþc1 � sd � ðpiðtÞ�xiðtÞÞþc2 �Cd � ðpgðtÞ�xiðtÞÞð27Þ

Discrete optimization using PSO approaches

In this work, optimal capacitor placement is formulated as anonlinear programming model and considers the capacitor sizesas discrete variables. In this case, the classical canonical PSOmethod needs modification to the particles move in a discrete deci-sion space.

The canonical PSO algorithm was originally proposed for con-tinuous problems, and attempts have been made recently toextend it to discrete optimization problems. Kennedy and Eberhart[37] proposed the first discrete version and Clerc [38] has shownpromising results on variants of the PSO specialized for some con-strained optimization problems such as the traveling salesmanproblem.

Fukuyama and Yoshida [39] have shown that the PSO is effec-tive at optimizing both continuous and discrete variables simulta-neously. The velocity update equation can be adapted for the usewith discrete variables by simply discretizing the values beforeusing them in the velocity update step. The position of the particleis also discretized after being updated.

Another approach was proposed by Miranda and Fonseca [40]using probabilistic rounding. Instead of rounding the number tothe next integer, they round with probabilities proportional tothe distance of the number to each of the integers. They reportgood results using that method applied to electric power systems.In this context, other approaches of discrete PSO have been alsoproposed in [41,42].

However, the determination of location and sizes for capacitorsin the PSO approaches tested in this work are based on Laskariet al. [42]. In this case, the solution vectors are composed by thedecision variables, which are rounded (truncated) to the nearestinteger, when computing the annual cost. The truncation of realvalues to integers seems not to affect significantly the performanceof PSO approaches tested in this work.

Probability distributions in PSO

Metaheuristic methods for optimization are based on stochasticnumbers at a certain step, which may be generated according tosome probability function. For example, genetic algorithms resortto the crossover and mutation operators, the standard PSO usesuniform random numbers in the range [0,l] to randomly weightthe cognitive and social behaviors, the well-known differentialevolution [43] also performs crossover given some probability.Krohling [27] show that, by using a uniform random distribution,setting the parameters of the PSO algorithm was made more

(a) PSO(1) (b) PSO(2) (c) PSO(3)

(d) PSO(4) (e) PSO(5) (f) PSO(6)

(g) PSO(7) (h) PSO(8) (i) PSO(9)

(j) PSO(10) (k) PSO(11) (l) PSO(12)

(m) PSO(13) (n) PSO(14) (o) PSO(15)

Fig. 5. Mean behavior (100 runs) of PSO approaches for case 2.





(1) 26894.809 25042.811 25495.398 324.899 104 10 21 24 14.83(2) 27662.922 25041.182 25466.453 418.856 104 12 33 23 16.64(3) 27752.750 25060.455 25564.269 458.704 103 11 36 21 16.85(4) 27552.043 25037.564 25448.432 369.813 103 12 36 19 18.65(5) 27144.458 25062.148 25456.521 418.682 112 15 41 19 16.96(6) 26232.022 25061.051 25419.172 287.044 104 14 42 19 17.04(7) 26736.820 25049.034 25432.043 348.759 100 9 42 17 18.85(8) 26844.558 25071.486 25466.153 384.923 100 9 42 15 18.95(9) 28239.471 25045.930 25468.815 448.247 84 8 41 14 18.91(10) 27137.136 25029.168 25508.115 413.346 83 9 40 14 15.06(11) 28854.957 25041.527 25723.140 687.689 86 9 41 14 15.41(12) 28678.453 25031.915 25549.464 558.320 83 8 42 13 16.74(13) 27708.898 25047.263 25821.366 625.277 83 8 42 14 16.48(14) 27765.177 25069.640 25566.520 507.502 84 10 44 14 16.45(15) 29093.051 25065.358 25763.351 771.647 82 16 42 14 16.61


Table 11The annual cost and total power loss of best results of PSO approaches (100 runs) for the second example. The best results are in bold letter.

Type of PSO Max |Vi| (p.u.) Min |Vi| (p.u.) Total loss (kW) Annual cost ($) Net savings ($)

(1) 1.0000 (bus 0) 0.9500 (bus 592) 158.952 25042.811 8184.189(2) 1.0000 (bus 0) 0.9501 (bus 592) 160.050 25041.182 8185.818(3) 1.0000 (bus 0) 0.9501 (bus 592) 159.673 25060.455 8166.545(4) 1.0000 (bus 0) 0.9501 (bus 592) 159.366 25037.564 8189.360(5) 1.0000 (bus 0) 0.9500 (bus 592) 159.089 25062.148 8164.852(6) 1.0000 (bus 0) 0.9501 (bus 592) 159.074 25061.051 8165.949(7) 1.0000 (bus 0) 0.9500 (bus 592) 159.663 25049.034 8177.966(8) 1.0000 (bus 0) 0.9501 (bus 592) 159.847 25071.486 8155.514(9) 1.0000 (bus 0) 0.9501 (bus 592) 159.276 25045.930 8181.070(10) 1.0000 (bus 0) 0.9505 (bus 592) 159.039 25029.168 8197.832(11) 1.0000 (bus 0) 0.9505 (bus 592) 159.370 25041.527 8185.473(12) 1.0000 (bus 0) 0.9500 (bus 592) 159.248 25031.915 8195.085(13) 1.0000 (bus 0) 0.9501 (bus 592) 159.606 25047.263 8179.737(14) 1.0000 (bus 0) 0.9501 (bus 592) 159.625 25065.358 8161.642(15) 1.0000 (bus 0) 0.9514 (bus 592) 159.620 25065.358 8161.642

Table 12The annual cost and total power loss for the second example (best result using PSO(10) approach).


Genetic algorithm Tabu search PSO(10)

Max |Vi| (p.u.) 1.0000 (bus 0) 1.0000 (bus 0) 1.0000 (bus 0) 1.0000 (bus 0)Min |Vi| (p.u.) 0.9417 (bus 592) 0.9503 (bus 592) 0.9504 (bus 592) 0.9505 (bus 592)Total loss (kW) 221.51 159.193 159.193 159.039Annual cost ($) 33,227 25089.03 25078.92 25029.68Net savings ($) – 8137.97 (24.49%) 8148.08 (24.52%) 8197.32 (24.67%)


difficult to escape from local minima, if compared to the case ofGaussian distribution. Several heuristics have been proposedaccording to other probability distributions, like cuckoo search,firefly algorithm [44] and flower pollination algorithm [45] thatare based on Lévy flights – which is one of the aspects pointedfor their success on solving difficult problems, also in the multiob-jective context [46,47].

When using Gaussian and Cauchy probability distributions, asimplemented in the PSO types above stated, the algorithm is lesslikely to be prone to local minima. This assertion is supported bythe results presented in Section ‘The application examples’ andcomes from the fact that by using uniform distributions all valuesfor a given parameter are equally divided when actually, for certainapplications, it may be better that the values present a differentdistribution along the desired range. This observation is in linewith previous applications of Cauchy and Gaussian [26,27,48–52]probability distributions applied in PSO. The greater is the standard

deviation of the random number, so is the chance of escaping localminima [27]. However, care should be taken when parameterizingthe random number generator, so that the jumps in the searchspace do not make the swarm explode avoiding thus convergence.This is the reason why we keep the values of the random genera-tors inside the range [0,1], which improved the original PSOalgorithm (see Section ‘The application examples’).

Computational procedures

The computational procedures of the proposed method aremainly composed of power loss computation, bus voltage determi-nation and PSO algorithmic calculations. The overall computationalprocedure finds configurations with the addition of differentcapacitors such that the objective function value is successivelyreduced.

(a) PSO(1) (b) PSO(2) (c) PSO(3)

(d) PSO(4) (e) PSO(5) (f) PSO(6)

(g) PSO(7) (h) PSO(8) (i) PSO(9)

(j) PSO(10) (k) PSO(11) (l) PSO(12)

(m) PSO(13) (n) PSO(14) (o) PSO(15)

Fig. 6. Mean behavior (100 runs) of PSO approaches for the second example.



The solution procedures start off with performing a feeder-lineflow study to calculate the bus voltages, line losses, and annualcost. Then, to determine the location and sizes for capacitors sittingby randomly selecting the positions. The results obtained from therandom selection serve as one of the initial strings for the subse-quent PSO application. The PSO employs its steps to search theoptimal solution. Since voltages along the feeder are required tomaintain between their upper and lower limits, we penalizesolutions whose voltage profiles are not kept within its limits.The solution procedures of the proposed method may be statedusing a flowchart as shown in Fig. 2.

The application examples

The proposed method has been programmed using MATLABand run on an IBM compatible personal computer. To illustratethe application and demonstrate the effectiveness of the proposedmethod, two application examples are presented. The real solutionvalues of PSO approaches are obtained by rounding to the nearestinteger, in order to calculate the practical size.

For the optimization problems studied in this paper, the popu-lation size was set to 10 particles for PSO approaches. The numberof generations is chosen to be 200. The particles are randomlyinitialized according to a uniform probability distribution and thevalues of the variables were initialized within the boundaries foreach run. The inertia weight of each PSO is linearly decreased overthe course of each run, starting from wmax = 0.9 and ending atwmin = 0.4. Based on previous experience with particle swarm opti-mization we set the acceleration constants c1 and c2 equal to 2.05.Each run is terminated if the maximum number of cycles elapsed.The results presented are averaged over 100 runs.

The optimization results of GA [6,14,53] and tabu search (TS)[54,55] with same number of evaluations of cost function (bestresult in 100 runs) of PSO approaches for the purpose of comparingare also presented for the two application examples in this section.

A total of 2000 cost function evaluations was done by GA andTS. Other particular parameters and design procedures used inthese optimization methods are fixed empirically and given below:

� GA: roulette wheel selection with elitism mechanism, andcrossover and mutation probabilities are set to 0.80 and 0.05,respectively.� TS: number of neighbors is set to 4, frequency and recency fac-

tor were 2 and 0.2, respectively.

The first application example described in the Refs. [1–4,7–9] isa 23 kV, 9-section feeder as given in Fig. 3. Table 1 shows theimpedance for the 9 sections of the feeder. Load data for the feederare given in Table 2. The equivalent annual cost per unit of powerloss, KP, is selected to be 168$/(kW-year) [4,7–9], and the limits onthe bus voltages are

Vmin ¼ 0:90 p:u:

Vmax ¼ 1:10p:u:

Available capacitor sizes with their corresponding annual costsare listed in Table 3 [4,7–9]. Table 4 shows the 27 possible capac-itor sizes. Two cases are investigated:

Case 1: It is restricted only 3 locations (buses 4, 5 and 9) avail-able for placement of capacitors [4,7–9]. Table 5 presents theresults of PSO approaches for case 1. Table 6 shows calculationresults of using different methods. The annual cost, power loss,minimum and maximum voltage profiles are also shown in thesame table. Comparing the last two columns of Table 6, it is

observed that the computational results of the proposed methodare exactly the same as that of the Exhaustive Search.

The best result (minimum annual cost in $) was obtained by allPSO(1) to PSO(15) approaches from Table 5. We can see that PSO(9)finds the best mean convergence in annual cost terms. The PSO(9)approach finds a solution very close to the optimal solution whichis much better than the solution found using PSO(6).

Fig. 4 shows the best fitness of 30 runs and the mean of best fit-ness convergence evolutions of independent runs by all PSO(1) toPSO(15) approaches in case 1.

Case 2: All buses are available for placement of capacitors.Table 7 presents the results of PSO approaches for case 2. Table 8shows voltage profile before and after compensation and requiredcapacitor addition obtained by using different methods. The powerlosses and total annual cost before and after compensation areshown in Table 9.

The best solution in terms of annual cost, for the 100 indepen-dent runs, was found by PSO(7) approach (see Table 7). In this case,the PSO(7) approach presented better convergence property withbest values of maximum, mean, and standard deviation of annualcost. On the other hand, observing the results from the Table 7, itcan be seen that the solution found by PSO(7) presents a perfor-mance (minimum annual cost) slightly better than the best resultobtained by PSO(11).

Fig. 5 shows the best fitness of 30 runs and the mean of best fit-ness convergence evolutions of independent runs by all PSO(1) toPSO(15) approaches in case 2.

It is worth noting that in case 1, the total number of enumera-tion required for Exhaustive Search is (27 + 1)3 = 28,192. However,in case 2, the total number of enumeration becomes289 = 1.058 � 1013. It is too large to conduct the Exhaustive Search.

The second example is shown in [5]. The system is a 11 kV with4 laterals and 33 section feeders. All buses are also available forplacement of capacitors. The annual cost of the capacitor is 0.5$/kvar for size up the 300 kvar. The equivalent annual cost per unitof power loss KP, is selected to be 150$/(kW-year), and the limitson the bus voltages are

Vmin ¼ 0:95p:u:

Vmax ¼ 1:05p:u:

The PSO applied its steps to solve this example. Table 10 showscalculation results of using different PSO methods for the secondexample. Table 11 presents the details about the best results ofPSO approaches.

The best solution in terms of minimum annual cost, for the sec-ond example, was found by PSO(10) approach (see Table 10). Inthis case, the PSO(6) approach presented better convergence prop-erty with best values of maximum, mean, and standard deviationof annual cost. The power losses and total annual cost before andafter compensation of best results using PSO(10) are shown inTable 12.

Fig. 6 shows the best fitness of 30 runs and the mean of best fit-ness convergence evolutions of independent runs by all PSO(1) toPSO(15) approaches in second example.

Conclusions and future research

A new capacitor placement method that employs PSO as well asfeeder-line flow formulations to reduce power losses and enhancevoltage profile for primary distribution systems is presented. Themethod seeks the most effective buses to install compensationcapacitors so that a maximum annual cost saving is attained.

From the first application example, it is observed that the com-putational results obtained by the proposed method are exactly the


same as that obtained using an exhaustive search method. FromTable 5, it can be seen also that the convergence of PSO(4) foundpresents a very small standard deviation (185.484) and the lowermaximum annual cost of tested approaches, which demonstratesthe robustness of the PSO method.

From the second application example, it is observed that thebest computational result obtained by the PSO(10) approach is24.67% better than the uncompensated method.

From Table 10, it can be seen also that the convergence ofPSO(6) found the best values in terms of mean, maximum, andstandard deviation. However, the best minimum value of annualcost was obtained by PSO(10).

The proposed PSO approaches tested in this paper are relativelysimple, reliable and efficient for capacitor placement in distribu-tion systems applications. Future research will address the ‘opti-mal’ design of control parameters of PSO approaches and use ofdifferent distributions in different search stages for other capacitorplacement problems. Furthermore, the PSO approaches will becompared with other typical optimization methods for capacitorplacement problems.

Acknowledgments

This study is supported by National Council of Scientific andTechnologic Development of Brazil (CNPq) under Grants 479764/2013-1 and 307150/2012-7/PQ. Furthermore, the authors wish tothank the editor and anonymous referees for their constructivecomments and recommendations, which have significantlyimproved the presentation of this paper.

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