6-IJAEST-Vol-No.4-Issue-No.2-DIFFERENTIAL-EVOLUTION-METHOD-FOR-CAPACITOR-PLACEMENT-023-028 by ISERP ISERP

USHA REDDY V, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 023 - 028

DIFFERENTIAL EVOLUTION METHOD FOR CAPACITOR PLACEMENT OF DISTRIBUTION SYSTEMS DINAKARA PRASAD REDDY P2 M.Tech [PSOC] Sri Venkateswara University Tirupati, Andhra Pradesh, India pdinakarprasad@gmail.com2

Assistant professor Sri Venkateswara University Tirupati, Andhra Pradesh, India vyza_ushareddy@yahoo.co.in 1

USHA REDDY V1

Abstract- This paper presents a fuzzy and

differential Evolution (DE) method for the placement of capacitors on the primary feeders of the radial distribution systems to reduce the power losses and to improve the voltage profile. Fuzzy approach is used to find the optimal capacitor locations. Differential Evolution method is used to find the sizes of the capacitors. The proposed method is tested on 10-bus, 34-bus and 69-bus test systems and the results are presented. Keywords: capacitor placement, fuzzy, Differential Evolution method. I. INTRODUCTION

Shunt capacitors are very commonly used on the primary feeders of a radial distribution system to reduce the power losses and to improve the voltage profile of the system. The objective of the capacitor placement problem is to determine the location and size of the capacitor so that the real power loss minimized. Even though considerable amount of research work was done in the area of optimal Capacitor placement [1]-[8], there is still a need to develop more suitable and effective methods for the optimal capacitor placement. Although some of these methods to solve capacitor allocation problem are efficient, their efficacy relies entirely on the goodness of the data used. Fuzzy logic provides a remedy for any lack of uncertainty in the data. Fuzzy logic has the advantage of including heuristics and representing engineering judgments into the capacitor allocation optimization process.

ISSN: 2230-7818

Furthermore, the solutions obtained from a fuzzy algorithm can be quickly assessed to determine their feasibility in being implemented in the distribution system. Differential evolution (DE) method is proposed to find capacitor sizes. The capacitor placement is modeled with objective function which minimizes the real power loss. II. FUZZY BASED CAPACITOR LOCATION

This work presents a fuzzy approach to determine suitable locations for Capacitor placement. The two objectives are: (i) to minimize the real power loss and (ii) to maintain the voltage within the permissible limits. Voltages and Power loss indices of distribution system nodes are modeled by fuzzy membership functions. A fuzzy inference system (FIS) containing a set of rules is then used to determine the capacitor placement suitability of each node in the distribution system. Capacitors can be placed on the nodes with the highest suitability. In the first step, load flow solution for the original system is required to obtain the real and reactive power losses. Again, load flow solutions are required to obtain the power loss reduction by compensating the total reactive load at every node of the distribution system. The loss reductions are then, linearly normalized into a [0, 1] range with the largest loss reduction having a value of 1 and the smallest one having a value of 0. Power Loss Index value for nth node can be obtained using equation ( ) ( ) ( ) ( ) ( ) ( )

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USHA REDDY V, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 023 - 028

1 L

0.6 0.4 0.2 0

0.25

0.5

0.75

Figure 1. Membership function plot for P.L.I

0.25

0.5

0.75

0.8 0.6 0.4 0.2

Figure 3.Membership function plot for C.S.I For the capacitor allocation problem, rules are defined to determine the suitability of a node for capacitor installation. Such rules are expressed in the following form:

IF premise (antecedent), THEN conclusion (consequent) for determining the suitability of capacitor placement at a particular node, a set of multiple antecedent fuzzy rules has been established. The inputs to the rules are the voltage and power loss indices and the output is the suitability of capacitor placement. The rules are summarized in the fuzzy decision matrix in Table 3.1. The consequents of the rules are in the shaded part of the matrix. Table 1. Decision matrix for determining the optimal Capacitor locations

0.8

1.0

These power loss reduction indices along with the p.u. nodal voltages are the inputs to the Fuzzy Inference System (FIS), which determines the node more suitable for capacitor installation. In this paper, two input and one output variables are selected. Input variable-1 is power loss index (PLI) and Input variable-2 is the per unit nodal voltage (V). Output variable is capacitor suitability index (CSI). In this paper fuzzy logic tool box in Matlab 7.10 is used for finding the capacitor suitability index. The following figure shows the membership function plots for P.L.I, p.u. voltage and C.S.I. Here Y-axis represents degree of membership.

0.8 0.6 0.4

0.2

0.9

0.95

1.0

1.05

1.1

Figure 2. Membership function plot for p.u. Nodal voltage

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USHA REDDY V, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 023 - 028

ALGORITHM

x j ,i , 0  rand j (0,1).( b j ,U  b j , L )  b j , L (3) The random number generator rand j (0,1) returns a uniformly distributed random number from within the range (0, 1), i.e. 0 ≤ rand j (0,1) < 1. The subscript, j, indicates that a new random value is generated for each parameter. III. Mutation

Once initialized, DE mutates and recombines the population to produce a population of Np trial vectors. Equation (4) shows how to combine three different, randomly chosen vectors to create a mutant vector, vi,g

DE introduced by Storn and Price, is a branch of evolutionary algorithms for optimization problems over continuous domains. In DE, each variable’s value in the chromosome is represented by a real number. DE can be categorized into a class of floating-point encoded evolutionary algorithms. The theoretical framework of DE is very simple and DE is computationally inexpensive in terms of memory requirements and CPU times. Thus, nowadays DE has gained much attention and wide application in a variety of fields. DE starts with the random initialization of a population of individuals in the search space and works on the co-operative behaviors of the individuals in the population. It finds the global optima by utilizing the distance and direction information according to the differentiations among the population. However, the searching behavior of each individual is adjusted by dynamically altering the differentiation’s direction and step length. At each generation, the mutation and crossover operators are applied to individuals to generate a new population. Then, selection takes place and the population is updated. I. Population Structure

must be specified. bL and bU for which subscripts L and U indicate the lower and upper bounds, respectively. For example, the initial value (g = 0) of the jth parameter of the ith vector is

III. DIFFERENTIAL EVOLUTION

v i , g  x r 0, g  F .( xr1, g  x r 2, g )

(4)

The scale factor F  (0,1), is a positive real number that controls the rate at which the population evolves.

DE’s most versatile implementation maintains a pair of vector populations, both of which contain Np D-dimensional vectors of realvalued parameters. The current population, symbolized by Px is composed of those vectors, xi , g , that have already been found to be acceptable either as initial points, or by comparison with other vectors: Px , g  ( xi , g ) , i=0, 1…, Np  1,

g  0,1., g m ax

(2)

xi , g  ( x j ,i , g ), j  0,1....D  1.

The index, g = 0, 1... gmax, indicates the generation to which a vector belongs. In addition each vector is assigned a population index i which runs from 0 to Np − 1. Parameters within vectors are indexed with j, which runs from 0 to D − 1. II. Initialization Before the population can be initialized, both upper and lower bounds for each parameter

ISSN: 2230-7818

Fig 4.Diagram of Differential mutation IV. Crossover To complement the differential mutation search strategy, DE also employs uniform crossover. In particular DE crosses each vector with a mutant vector

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USHA REDDY V, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 023 - 028

(

{

)

( )

The crossover probability C r  [0,1] is a user-defined value that controls the fraction of parameter values that are copied from the mutant.

Step 7: Determine the best individual of the current new population with the best Objective value then updates best individual and its objective value. Step 8: If a stopping criterion is met, then output gives its bests and its objective value Otherwise go back to step 3. IV.RESULTS

If the trial vector u i , g has an equal or lower objective function value than that of its target vector xi , g it replaces the target vector in

{

(

)

(

)

the next generation (Eq.6). Otherwise the target retains its place in the population for at least one more generation. By comparing each trial vector with the target vector from which it inherits parameters DE more tightly integrates recombination and selection than do other EAs

The proposed method for loss reduction by capacitor placement is tested on 10 bus [1], 34 bus [3], 69 bus [3] radial distribution systems. The various constants used in the proposed algorithm are N=30, gmax=500, CR=0.9, F=0.8.The proposed method has been programmed using MATLAB 7.10 and run on an Intel core i5 personal computer. The test results are shown below.

V. Selection

( )

CASE I

The 10 bus test system with the proposed method is compared with the paper [1] in which better results are obtained.

Once the new population is installed, the process of mutation, recombination and selection is repeated until the optimum is located, or a pre specified termination criterion is satisfied, e.g., the number of generations reaches a preset maximum gmax.

Table 2: comparison of results

Algorithm to find capacitor sizes using DE

The basic procedure of DE is summarized as follows. Step 1: Randomly initialize the population of individual for DE. Step 2: Evaluate the objective values of all individuals, and determine the best individual. Step 3: Perform mutation operation for each individual according to Eq. 4 in order to obtain each individual’s corresponding mutant vector. Step 4: Perform crossover operation between each individual and its corresponding mutant vector according to Eq.5 in order to obtain each individual’s trial vector. Step 5: Evaluate the objective values of the trial vectors. Step 6: Perform selection operation between each individual and its corresponding trial Vector according to Eq.6 so as to generate the new individual for the next generation.

ISSN: 2230-7818

Compensated Items

uncompensated

Fuzzy Reasoning

Proposed method

[1] Total losses(kW) Minimum Voltage(p.u)

783.0289

704.88

696.44

0.8376

0.8590

0.8710

CASE II The 34 bus test system with the proposed method is compared with the paper [3], results obtaining are more promising.

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USHA REDDY V, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 023 - 028

Table 4: comparison of results Uncompensated

Items Total losses(kW) Minimum Voltage(p.u)

Compensated PSO Proposed [3] method

225.044 152.4 151.213 0.9096

0.9306

Table 3: comparison of results

Items Total losses(kW) Minimum Voltage(p.u)

CASE III

A new capacitor placement method that employs fuzzy and differential methods to reduce power losses and enhance voltage profile for primary distribution systems is presented. The method seeks the most effective buses to install compensation capacitors. The employed fuzzy and differential evolution algorithm to the buses is effective in reducing the total number of alternatives examined for finding the optimal solution.

Compensated

CONCLUSION

uncompensated PSO[3] Proposed method 221.7235

168.8

161.1265

0.9417

0.9510

The algorithm is tested with three Radial distribution systems consisting of 10, 34 and 69 buses and the results are presented. REFERENCES

The 69 bus test system with the proposed method is compared with the paper [3], results obtaining are more promising.

[1] A new fuzzy reasoning approach to optimum capacitor allocation for primary distribution systems, ching-tzong su, chih-cheng Tsai, IEEE international conference, 1996.

[2] A fuzzy – genetic algorithm for optimal capacitor placement in radial distribution systems, P.V. Prasad , S. Sivanagaraju and N.Sreenivasulu, ARPN Journal of Engineering and Applied Sciences, June 2007. [3] Particle swarm optimization based capacitor placement on radial distribution systems, K.Prakash and M.Sydulu, IEEE 2007. [4] S. H. Lee and J. J. Grainger, “Optimum placement of fixed and switched capacitors on primary distribution feeders”, IEEE Trans. on Power Apparatus and systems, vol. 100, pp. 345-352, Jan. 1981. [5] J. J. Grainger and S. H. Lee, “Capacity release by shunt capacitor Placement on distribution feeders: A new voltage-dependent model”, IEEE Trans. On Power Apparatus and Systems, vol. 101, pp. 1236-1244, May 1982.

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USHA REDDY V, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 023 - 028

[6] Y. Baghzouz and S. Ertem, ”Shunt capacitor sizing for radial Distribution feeders with distorted substation vo Itages”, IEEE Trans. on Power Delivery, vol. 5, pp. 650-657, Apr. 1990.

[8] S. Sundhararajan and A. Pahwa, “Optimal selection of capacitors for radial distribution systems using a genetic algorithm”, IEEE Trans. On Power Systems, vol. 9, pp. 1499-1507, Aug. 1994.

[9]C. T. Su, C. S. Lee, and C. S. Ho, “Optimal selection of capacitors in distribution systems”, in Proc. 1999 IEEE Power Tech Conference.

[7] J. L Bala, P. A. Kuntz, and R. M. Taylor, “Sensitivity-based optimal capacitor placement on a radial distribution feeder”, in Proc. 1995 Northcon 95 IEEE Technical Application Conf, pp. 225-230.

[10] K. Y. Lee and F. F. Yang, “Optimal reactive power planning using evolutionary algorithms: A comparative study for evolutionary programming, evolutionary strategy, genetic algorithm, and linear Programming”, IEEE Trans. on Power Systems, vol. 13, pp. 101-108, Feb. 1998.

[11] R. Storn and K. Price,”Minimizing the real functions of the ICEC’96 contest by differential evolution”, in Proc. 1996 IEEE on Evolutionary Computation Conf, pp. 842-844.

[12] K. V. Price, “Differential evolution vs. The functions of the Znd ICEO, in Proc. I9971EEE on Evolutionary Cornputation Conf, pp.153-157. [13] R. Storn, “System design by onstraint adaptation and differential evolution”, IEEE Trans. Evolutionary Computation, vol. 3, pp.2234, Apr. 1999.

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