Performance Evaluation of Different Variants of Differential Evolution for Optimal capacitor Placement in Radial Distribution Systems

Abstract—Differential Evolution (DE) is one of the most recent population-based stochastic evolutionary optimization techniques. Differential Evolution is a very robust optimization technique and has been applied to various field of power systems such as economic load dispatch, hydro thermal scheduling, optimal reactive power planning, hybrid electric power systems etc. Premature convergence is one of the major difficulties in meta-heuristic techniques like DE. Since the introduction, several variants of DE have been applied by the researchers and scientists to address the problem of premature convergence. In the present paper, performance of different variants of DE for optimal capacitor placement in radial distribution systems has been studied and a comparative result is presented. Placement of capacitor of optimal sizes and at optimal locations can reduce the power losses and thereby can improve the performance of the electric power systems. The performance of several variants of DE is demonstrated on a sample test system. It is observed that they produce good quality solutions. The results obtained by different variants of differential evolution technique are compared with other evolutionary methods. It is seen that all the variants of differential evolution considered are capable of producing encouraging solutions where some were observed to perform better than the others in terms of cost or computation time.

Index Terms— Differential evolution, Distribution systems, Loss reduction, Performance evaluation, Voltage profile.

I. INTRODUCTION

A large amount of power losses takes place is distribution systems which reduces the distribution efficiency and thereby cost of energy supplied to the customers is increased. This is very important for deregulated electricity industry where competition is introduced and every supplying authority explores the all possibilities to reduce the system losses so that power can be supplied at a competitive price. Optimal reactive power compensation can play a major role in reducing system losses and capacitors are commonly used in distribution systems to provide reactive power compensation. Placement of capacitors of optimal sizes at optimal locations can reduce power losses, improve power factor and maintain voltage profile within

acceptable limits. The voltage profiles throughout the electric power system network are to be kept at acceptable levels to ensure network reliability. Thus, the benefits of compensation are highly governed by the optimal location, optimal size of the capacitors and associated cost. Therefore, the objective in distribution capacitor planning is to minimize system losses while satisfying various operating constraints under a certain load pattern.

The optimal distribution capacitor placement is a complex combinatorial optimization problem and several optimization techniques and algorithms have been applied over the years to solve it. Duran [1] presented dynamic programming technique considering the capacitor sizes as discrete variables to find the optimal solution. A heuristic numerical algorithm was proposed considering distorted substation voltage by Baghzouz et al. [2]. Nojavan et al. [3]

applied mixed integer non-linear programming technique for optimal sizing and placement of capacitor for radial as well as mesh distribution systems. An efficient heuristic algorithm was proposed by Segura et al. [4] to determine the optimal placement of capacitors for radial distribution systems. Ziari et al. [5] applied a new heuristic technique for optimal capacitor allocation and presented promising results.

Su et al. [6] proposed fuzzy-reasoning method to find the optimal location and size for capacitor addition. Lai et al. [7]

proposed the evolutionary programming to solve optimal reactive power planning problems. An algorithm based on particle swarm optimization technique was proposed by Yu et al. [8] for capacitor placement considering harmonic distortion. A new method based on plant growth algorithm was proposed by Wang et al. [9]. Chiou et al. [10] used variable scale differential evolution technique to find optimal solution for capacitor placement problem in large distribution systems. Opposition Based Differential Evolution Algorithm was proposed by Muthukumar et al.

[11] for Capacitor Placement on Radial Distribution System.

Rao et al. [12] presented a method using plant growth simulation algorithm for optimal capacitor placement in radial distribution system. Recently, a new algorithm based on gravitational search algorithm was proposed by Shuaib et

Performance Evaluation of Different Variants of Differential Evolution for Optimal capacitor Placement in Radial

Distribution Systems

S. Mandal¹, K.K. Mandal²

1Department of Electrical Engineering, Jadavpur University, Kolkata-32,

2 Department of Power Engineering, Jadavpur University, Kolkata-98, contact. [email protected]

al. [13] for optimal capacitor placement in radial distribution systems. Duque et al. [14] used a modified money search optimization technique for allocation of capacitor bank in distribution systems and presented promising results. Injeti et al. [15] presented encouraging results for optimal allocation of capacitor bank using bio-inspired optimization algorithms. Devabalaji et al. [16] applied bacterial foraging optimization algorithm for optimal location and sizing of capacitor placement in radial distribution system. More recently, a new method based on improved harmony algorithm was presented by Ali et al. [17] for optimal locations and sizing of capacitors in radial distribution systems.

DE is a powerful tool in solving optimization problems and in practice different variant of Differential Evolution exist. It is well accepted that individual performance of these variants depends on the specific problem under consideration. Differential evolution is comparatively new population based meta-heuristic optimization technique. In this paper, five different variants of DE are applied to the capacitor placement problem and their performances are compared. Further the results obtained by the different variants are compared with other evolutionary techniques. It is also seen that the all the variants of differential evolution considered are capable of producing encouraging solutions where some were observed to perform better that the others in terms of cost or computation time.

II. PROBLEM FORMULATION

The objective of optimal capacitor placement problem is to minimize the system losses and thereby to minimize the total annual cost of the system while satisfying several operating constraints under a certain load pattern. The mathematical model of optimal capacitor placement problem can expressed as follows:

   

1

min

minF COST

where COST includes cost of power loss and capacitor placement. The voltage magnitude at each bus must be maintained within its limits and is expressed as

 

2

max

min V V

V  _i 

where

V

_i is the voltage magnitude of i th bus,

V

_min^and

V

maxare the minimum and maximum bus voltage limits respectively.

A set of simplified feeder-line flow formulation is assumed for simplicity. Considering the one-line diagram shown depicted in Fig.1, the following set of equations may be used for power flow calculation [9].

Fig. 1 Single -line diagram of a main feeder

 

3

2 2 2 1 , 1

1 





 



 





 _{ }

 i Li ii i i i

i P P R P Q V

P

   

4

. ^{2 2 2}

1 , 1

1 

 





 _{ }

 i Li ii i i i

i Q Q X P Q V

Q



_{. .}

    _{ }

₅

2 ₂

2 2 2

1 , 2

1 , 1 , 1 , 2 2 1

i i i i i i i i i i i i i i

i V

Q X P

R Q X P R V

V_   _  _  _  _ 

where P_i and Q_i are the real and reactive powers flowing out of ith bus respectively. P_Li,Q_Liare the real and reactive load powers at the ith bus respectively. The resistance and reactance of the line section between buses i and i+1 are denoted by R_i,i1 and X_i,i1 respectively.

The power loss of the line section connecting buses i and i + 1 can be calculated as



^{, 1}



, 1^{. 2} ₂ ²

 

⁶

i i i i i Loss

V Q R P

i i

P 



 _

The total power loss of the feederP_T,Loss may then be determined by summing up the losses of all line sections of the feeder. The loss is given by



, 1

  

7

1

0

,



^





ⁿ

i Loss Loss

T P i i

P

The principle of placing compensating capacitor along distribution feeders is to lower the total power loss and keep the bus voltages within their specified limits while minimizing the total cost. Considering the practical capacitors, there exists a finite number of standard sizes which are integer multiples of the smallest sizeQ₀^c. Besides, the cost per kVAr varies from one size to another. In general, capacitors of larger size have lower unit prices. The available capacitor size is usually limited to

 

8

0

max c

c LQ

Q 

where L is an integer. Therefore, for each installation location, there are L capacitor sizes



^Q0^c,2^Q0^c,...,^LQ0^c



available. Let



L^c



c

c K K

K₁, ₂,..., be their corresponding equivalent annual cost per kVAr. Therefore, the total annual

cost function due to capacitor placement and power loss may be found as

 

9

1

,









n

i c i c i Loss T

PP K Q

K COST

where K_P is the equivalent annual cost per unit of power loss in $/ (kW – year) and, i = 1, 2, . . . , n are the indices of buses selected for compensation. The bus reactive compensation power is limited to

 

10

1





 ⁿ

i Li c

i Q

Q

III. OVERVIEWS OF SOME VARIANTS OF DIFFERENTIAL EVOLUTION (DE)

DE or Differential Evolution belongs to the class of evolutionary algorithms [18], [19] that include Evolution Strategies (ES) and conventional Genetic Algorithms (GA).

DE differs from the conventional genetic algorithms in its use of perturbing vectors, which are the difference between two randomly chosen vectors. DE is a scheme by which it generates the trial vectors from a set of initial populations.

In each step, DE mutates vectors by adding weighted random vector differentials to them. If the fitness of the trial vector is better than that of the target vector, the trial vector replaces the target vector in the next generation.

DE offers several strategies for optimization. They are classified according to the following notation such as DE/x/y/z, where x refers to the method used for generating parent vector that will form the base for mutated vector, y indicates the number of difference vector used in mutation process and z is the crossover scheme used in the cross over operation to create the offspring population [19]. The symbol x can be ‘rand‟ (randomly chosen vector) or „best‟

(the best vector found so far). The symbol y i.e. the number of difference vector, is normally set to be 1 or 2. For crossover operation, a binomial (notation: „bin‟) or exponential (notation: „exp‟) operation is used.

Based on the above notation, several variants of DE namely DE/rand/1/bin, DE/rand/2/bin, DE/best/1/bin, DE/best/2/bin, DE/current to best/1/bin, DE/current to best/2/bin are considered in the present work and they are described in the following sections.

A. Initialization

The optimization process in DE is carried with four basic operations: initialization, mutation, crossover and selection.

The algorithm starts by creating a population vector P of size N_P composed of individuals that evolve over G generation. Each individual X_i is a vector that contains as many elements as the problem decision variable. The population size N_P is an algorithm control parameter selected by the user. Thus,



^{( )},..., ^{( )}

  

10

)

( G

N G

i G

X P

X P 

  ^{ }

P G T

i D G

i G i

N i

X X

X

...

,...

1

11 ...

...

,... ⁽_,⁾

) (

, 1 ) (





The initial population is chosen randomly in order to cover the entire searching region uniformly. A uniform probability distribution for all random variables is assumed in the following as



^{max min}

  

¹²

min ) 0 (

,i j j j j

j X X X

X   

where i1,...N_P and j1,...D;

Here D is the number of decision or control variables, X^min_j and X^max_j are the lower and upper limits of the jth decision variables and _j



0,1



is a uniformly distributed random number generated anew for each value of j. Again

) 0 (

, i

Xj is the jth parameter of the ith individual of the initial population.

B. Mutation Operation

Several strategies of mutation have been discussed in the literature of DE [19]. The essential ingredient in the mutation operation is the vector difference. The mutation operator creates mutant vectors

 

Vi by perturbing a selected vector with the difference of other selected vectors.

Mutant vector can be created by using several mutation strategies. The strategies those are considered in the present work are as follows.

B.1 DE/rand/1/bin

In this case, the mutation operator creates mutant vectors

 

Vi by perturbing a randomly selected vector

 

Xk with the difference of two other randomly selected vectors



Xl and Xm



according to:



^{( ) ( )}

  

13

) ( )

( G

m G m l G

k G

i X f X X

V   

whereX , _k X _l and X_mare randomly chosen vectors 



1,...,NP



and klmi. In other words, the indices are mutually different including the running index i. The mutation factor f_m that lies within [0, 2] is a user chosen parameter used to control the perturbation size in the mutation operator and to avoid search stagnation.

B.2 DE/rand/2/bin

Here, mutant vector

 

Vi is created by perturbing a randomly selected vector



Xk



with the summation of the difference of four other randomly selected vectors



Xl and Xm



and



Xn andXp



according to:



^{( ) ( )}

 

^{( ) ( )}

 ^{ }

¹⁴

) ( )

( G

p G n m G m G l m G k G

i X f X X f X X

V     

where, X ,_k X ,_l X_m,X and_n X_p are randomly chosen vectors 



1,...,NP



and

. i p n m l

k    

B.3 DE/best/1/bin

In this case, mutant vector

 

Vi is created by perturbing the best vector



Xbest



from the current population with the difference of two other randomly selected vectors



Xl and Xm



according to:



^{( ) ( )}

  

¹⁵

) ( )

( G

l G m k G best G

i X f X X

V   

where, X_best^(G) is the vest vector in generation G.

B.4 DE/best/2/bin

Here, the mutation operator creates mutant vectors

 

Vi

by perturbing the best vector



Xbest



so far with the summation of the difference of four other randomly selected vectors



Xk and Xl



and



Xm and Xn



according to:



^{( ) ( )}

 

^{( ) ( )}

 ^{ }

¹⁶

) ( )

( G

n G m m G l G k m G best G

i X f X X f X X

V     

where, X_best^(G) is the best vector in generation G.

X ,k X ,_l X_m, and X are randomly chosen vectors _n



1,...,NP



 and klmni.

B.5 DE/current to best/1/bin

In this case mutant vector

 

Vi is created according to:



^{( ) ( )}

 

^{( ) ( )}

 ^{ }

¹⁷

) ( )

( G

m G l m G i G best m G i G

i X f X X f X X

V     

where, X is the current vector, _i X_best^(G) is the vest vector in generation G, X and _l X_m are randomly selected vectors



1,...,NP



 and lmi.

C. Crossover Operation

In order to extend further diversity in the searching process, crossover operation is performed. The crossover operation generates trial vectors

 

Ui by mixing the parameter of the mutant vectors with the target vectors. For each mutant vector, an index q



1,...NP



is chosen randomly using a uniform distribution and trial vectors are generated according to:

 

¹⁸

) , (

, ) , ( ) , (

,









  



otherwise G

i Xj

q j R or j C G if

i Vj G

i Uj



where i1,...,N_P and j1,...,D; _j is a uniformly distributed random number within [0, 1]

generated anew for each value of j. The crossover factor

 

^0,1

R

C is a user chosen parameter that controls the diversity of the population. AgainX⁽_j,^G_i⁾, V_j⁽_,^G_i⁾ and U⁽_j^G_,i⁾ are the jth parameter of the ith target vector, mutant vector and trial vector at G generation respectively.

D. Selection Operation

Selection is the operation through which better offspring are generated. The evaluation (fitness) function of an offspring is compared to that of its parent. The parent is replaced by its offspring if the fitness of the offspring is better than that of its parent, while the parent is retained in the next generation if the fitness of the offspring is worse than that of its parent. Thus, if f denotes the cost (fitness) function under optimization (minimization), then

   



 





 

 

19 ,

,

) (

) ( )

( )

( ) 1 (

otherwise X

X f U f if U X

G i

G i G

i G

i G

i

The optimization process is repeated for several generations. This allows individuals to improve their fitness while exploring the solution space for optimal values. The iterative process of mutation, crossover and selection on the population will continue until a user-specified stopping criterion, normally, the maximum number of generations allowed, is met. The other type of stopping criterion i.e.

convergence to the global optimum is possible if the global optimum of the problem is available. Keeping all these in consideration the DE technique has been applied to optimal capacitor placement problem.

IV. RESULTS AND DISCUSSIONS

The proposed algorithm was implemented using in house Matlab code on 3.0 GHz, 4.0 GB RAM PC. To demonstrate the effectiveness and feasibility of the proposed algorithm, it was applied on a sample test system. The test system [2], [6]

under consideration consists of a 23 kV, 9 section feeder as shown in Fig. 2.

Fig. 2 Single -line diagram of the test system TABLEI. FEEDERIMPEDANCE.

Line No.

From Bus, i

To Bus, i+1 R_i,i+1(Ω) X_i,i+1(Ω)

1 0 1 0.1233 0.4127

2 1 2 0.0140 0.6057

3 2 3 0.7463 1.2050

S

1 2 3 4 5 6 7 8 9

4 3 4 0.6984 0.6084

5 4 5 1.9831 1.7276

6 5 6 0.9053 0.7886

7 6 7 2.0552 1.1640

8 7 8 4.7953 2.7160

9 8 9 5.3434 3.0264

TABLEII.THREE-PHASESYSTEMLOAD.

Line No. PL(kW) QL(kVAr)

1 1840 460

2 980 340

3 1790 446

4 1598 1840

5 1610 600

6 780 110

7 1150 60

8 980 130

9 1640 200

Total 12368 4168

Table 1 shows the feeder impedance data, while three-phase load data is shown in Table 2. The equivalent unit cost per unit of power loss considered for the present problem is

$168/(kW-year) [2], [6]. Available capacitor size with their cost, possible capacitor size and other data are taken from [2], [6] and not reproduced here due to space limitation. The limits on bus voltages are as follows:

V_min = 0.90 p.u.

V_max = 1.10 p.u.

The following values of control parameters of DE were selected by parameter setting through trial and error for the present test system. Starting from the initial guess [18], [19]

several combinations of different parameters were used to test all the proposed algorithms. For each parameter combination several tests were performed and optimal combination was obtained. Several tests were also performed with increased number of iteration in step of 50 and no significant improvement was noticed beyond 300 iterations. The maximum iteration was therefore set at this value of 300. The parameters were selected for each of the variants of DE separately. The following values of parameters for different variants of DE have been found to provide optimum results.

DE/rand/1/bin (DE1) [15]: N_P = 40, f_m = 0.55 and C_R = 0.80 DE/rand/2/bin (DE2): N_P = 40, f_m = 0.55 and C_R = 0.90 DE/best/1/bin (DE3): N_P = 40, f_m = 0.66 and C_R = 0.80 DE/best/2/bin (DE4): N_P = 40, f_m = 0.60 and C_R = 0.90 DE/current to best/1/bin (DE5): N_P =40, f_m = 0.65, C_R = 0.9

It is considered that all the buses were available for compensation. The optimal annual cost, losses and computation time by different variants of DE are shown in Table 3. Table 3 also compares the results obtained by other population based meta-heuristic techniques like fuzzy reasoning [6], particle swarm optimization (PSO) [20], plant growth simulation algorithm (PGSA) [12].

TABLEIII.COMPARISONOFRESULTSOBTAINEDBYDIFFERENT VARIENTSOFDEANDOTHERMETHODS.

Methods Annual cost in ($/year)

Total Loss (MW)

Computation Time (Sec)

DE/rand/1/bin (DE1) 114,978 0.6780 50.30

DE/rand/2/bin (DE2) 114,990 0.6793 50.90

DE/best/1/bin (DE3) 114,950 0.6765 50.38

DE/best/2/bin (DE4) 114,930 0.6736 50.20

DE/current to best/1/bin (DE5)

114,970 0.6949 50.62

Fuzzy Reasoning [6] 119,420 0.7048 NA^#

PSO [20] 118,582 0.6763 NA

PGSA [12] 118,340 0.6949 NA

# NA: Not Available.

It is found from Table 3 that among the various variants of DE, DE/best/2/bin performs slightly better in terms of cost. However, computation time is almost same for all the variants of DE. From Table 3, it is clearly seen that proposed method can produce better results. The annual costs, system power loss both before and after compensation, capacitor addition at the desired location obtained by DE/best/2/bin (DE4) are shown in Table 4. It is seen from Table 4 that voltage profile for all the buses are with the system limits. The annual cost is $114,930 while the system power loss is 0.6736 MW in comparison with uncompensated cases where the annual cost is $131,675 and power loss is 0.7836 MW respectively. Fig.3 shows the convergence characteristics.

TABLEIV.RESULTSINCLUDINGVOLTAGEPROFILE,ANNUAL COST,CAPACITORANDPOWERLOSSBYDE/BEST/2/BIN (DE4) Bus No. Uncompensated

Voltage (p.u)

Placed (Qc) (kVar)

Compensated Voltage (p.u)

0 1 0 1

1 0.9929 150 1.0013

2 0.9874 3600 0.9997

3 0.9634 900 1.0040

4 0.9619 1800 0.9920

5 0.9480 300 0.9820

6 0.9072 600 0.9602

7 0.8890 300 0.9537

8 0.8587 450 0.9167

9 0.8375 450 0.9044

Total cap.

Size (Mvar)

8.55 Total Loss

(MW)

0.7837 0.6736

Annual cost in ($/year)

131,675 114,930

CPU time (sec)

50.20

0 50 100 150 200 250 300 1.14

1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32

1.34x 10⁵ Convergence Characteristics of Annual cost

No.of Iteration

cost in $

Fig. 3 Convergence characteristics for optimal annual cost V. CONCLUSIONS

Optimal sizing and allocation of capacitors is one of the important tasks in the operation of distribution systems. The basic objective is to reduce power losses as well as to improve voltage profile. . In this paper, different variants of differential evolution (DE) have been successfully implemented to solve capacitor placement problem and a comparison study of different variants of DE is made. To evaluate the performance of different variants of DE, they have been applied on a sample test system. The results obtained by the proposed method have been compared with other population based algorithms like fuzzy reasoning, PSO, PGSA. The results show that the proposed algorithm is indeed capable of obtaining good quality solution.

ACKNOWLEDGMENT

We would like to thank Jadavpur University, Kolkata, India for providing all the necessary help to carry out this work.

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