31

CHAPTER 3

FUZZY MUTATED EVOLUTIONARY PROGRAMMING

FOR MULTI-OBJECTIVE REACTIVE POWER

OPTIMIZATION PROBLEM

3.1 INTRODUCTION

In this chapter a brief insight on EP and MOEP are reported and the

solution for MORPO problem using the MOEP is explained. The performance

of the MOEP is improved by incorporating fuzzy logic strategy in the

mutation process of EP which leads to an amendment termed as FMEP. The

MORPO problem with competing objectives namely, minimization of the real

power loss, minimization of voltage deviation, minimization of the L-index,

minimization of the investment cost of the compensating devices is solved

using the EP and FMEP based algorithm. Finally the optimal results of the

test system using the EP and FMEP based algorithms are reported along with

the convergences characteristics, Pareto fronts and analysis.

3.2 EVOLUTIONARY PROGRAMMING

EP was developed by Fogel (1962). It searches for the optimal

solution by evolving a population of feasible solutions over a number of

generations or iterations. It refers to a class of methods which apply a uniform

random mutation (through Gaussian distribution) to each member of a

population and generates a single offspring. In EP recombination, operators

are not entertained. After mutation, selection (Competitive selection) process

32

takes place in which half of the combined population of parent and offspring

enters the next generation. EP is so simple and robust.

The major steps involved in the evolutionary programming

approach are:

i) Initialization

The initial population of parent individuals is generated randomly

within the feasibility range in each dimension such that the distribution of the

initial trial parents is uniform. The parent individual is ( 1, 2,...... )pi pI pi N .

ii) Mutation (Creation of Offspring)

An offspring vector is generated from each parent vector by adding

a Gaussian random variable with zero mean and pre-selected standard

deviation to each element of parent individual. The offspring vector is

( 1, 2,......,2 )moi p p pI oi N N N . The Np parents create Np offspring which

leads to 2Np individuals in the competing pool.

iii) Competition and Selection

The fitness is evaluated for each individual in the competition pool.

Each individual in the competition pool compete with each other for the

selection. The selection process is probabilistic. The first Np individuals with

minimum fitness values are considered to be the parents of the next

generation.

iv) Stopping Criteria

Evolutionary Programming has no specific standard stopping

criterion. The process of mutation, competition and selection are repeated

33

until maximum number of iteration is reached. While choosing the maximum

number of iteration, it should not be very small as it leads to premature

convergence and it should not be very large as it will increase the

computation time. The value for the maximum number of iteration depends

on the nature of problem. The value is normally determined from trial studies.

A number of different values of maximum number of iterations is chosen and

for each chosen maximum number of iterations, 100 trial studies are made for

each problem.

3.3 MULTI-OBJECTIVE EVOLUTIONARY PROGRAMMING

The Multi-Objective Evolutionary Programming (MOEP) with

non-dominated sorting algorithm has the following steps:

Step 1: Initially Np number of trial solutions as parent solutions are

generated randomly.

Step 2: From the parent solution Np number of offspring solutions are

created.

Step 3: By combining the parent and offspring solution 2Np number of

solutions in a present population is created.

Step 4: The non-dominated solution are identified and the front number are

assigned.

Step 5: The 2Np solutions are sorted by rank sum sorting technique. In

rank-sum techniques each objective is divided into 100 ranks and

the corresponding rank for all the objectives are summed. A

population is assigned the rank-sum according to its position in the

search space.

34

Step 6: From 2Np number of sorted solutions, Np solutions are selected by

diversified selection. To keep the diversity of the population during

the selection process, the population is divided into preferential set

and backup set. The solution in the preferential set is used for

evolving the offspring and the backup set is used only if the

preferential set is not sufficient to evolve the offspring.

Step 7: The procedure from Step 2 is repeated until the maximum of

number of iteration is reached. If the maximum number of iteration

is reached then the present parent solutions are the pareto optimal

front solutions.

3.4 SOLUTION FOR MULTI-OBJECTIVE REACTIVE POWER

OPTIMIZATION PROBLEM USING MULTI-OBJECTIVE

EVOLUTIONARY PROGRAMMING

The various sequential steps for solving the Multi-Objective

Reactive Power Optimization problem using MOEP are as follow:

3.4.1 Initialization of Parent Population

An Np number of parent solutions are randomly generated within

the feasible range such that the distribution of the initial trial solutions is

uniform. The elements of each parent individual are the controllable

parameters namely, the voltage magnitude of voltage controllable buses,tap

setting of tap changing transformers and reactive power compensation by

capacitor banks. The initial population

Ipi = [Vpi1, Vpi2 piNv; Tpi1, Tpi2 piNT; Qpi1, Qpi2 QpiNC, P

(3.9)

35

The elements of Ipi are selected such that

Vpij = U( maxmin , ijij VV ) (3.10)

Tpik = U( maxmin , ikik TT ) (3.11)

Qpil = U( maxmin , ilil QQ (3.12)

U(x, y) denotes a uniform random variable between the limits x and y.

For each individual of the population the power flow equations are

solved by running the load flow using Newton Raphson method and the

fitness value is evaluated and the maximum fitness value is stored as ftmax.

The fitness value ft is calculated using the equation,

332211Fft i (3.13)

where, F is the weighted sum of the objective functions F1, F2, F3 and F4.

otherwiseVVVVVVVV

pipipipi

pipipipi

;0|;||;|

maxmax

minmin

1

otherwiseTTTTTTTT

pipipipi

pipipipi

;0|;||;|

maxmax

minmin

2

otherwiseQQQQQQQQ

pipipipi

pipipipi

;0|;||;|

maxmax

minmin

3

The values of the penalty factors 1, 2 and 3 are chosen by trial

and error. Initially a small value between 10 and 100 will be chosen. After the

36

investigation, if the constraint violated individuals have not been effectively

eliminated, then the penalty factor values will be increased until a converged

solution is reached with no constraint violations. If there is any constraint

violation then the fitness value corresponding to that parent will be

ineffective.

3.4.2 Mutation (Creation of Offspring)

An offspring population moiI is generated from Np parent individuals

as

moiI = [Voi1 V Voij; Toi1 T Toik; Qoi1 Q Qoil];

oi= Np+1, Np Np+Nm (3.14)

The elements of moiI are generated as,

VVijoij NjNrandVVij

,...2,1);,0( 2

(3.15)

TTikoik NkNrandTTik

,...2,1);,0( 2

(3.16)

NClNrandQQilQiloil ,...2,1);,0( 2

(3.17)

When the elements of moiI exceeds its corresponding minimum or

maximum limit, then the violating limit value is assigned to that element.

Nrand 2) represents a normal random variable with mean zero and 2. 2

ijV , 2ikT and 2

ilQ are the variances corresponding to each control

variable which decides the width of the normal distribution curve. The

variance of each variable are computed using the Equations (3.18) to (3.20) in

which ccording to the

relative fitness fti / ftmax so that the width of the normal distribution is small if

ift is small and vice versa. 2ijV , 2

ikT and 2ilQ are calculates as

37

(max

2

ftft i

Vi j)minmax

ijij VV VNj ,...2,1; (3.18)

(max

2

ftft i

Tkj)minmax

ikik TT TNk ,...2,1; (3.19)

(max

2

ftft i

Qi j)minmax

ilil QQ NCl ,...2,1; (3.20)

The fitness values corresponding to each offspring are calculated

using the fitness function Equation (3.13).

3.4.3 Combining the Parent and Offspring Solutions

The parent and the offspring solutions are combined and 2Np

solutions are created in the present population.

3.4.4 Identification of the Non-dominated Solutions

The non-dominated solutions are identified using the concept of

domination and each solution is specified with a front number. A solution X1

is said to dominate the other solution X2 if the both conditions stated below

are satisfied:

1. The solution X1 is no worse than X2 in all objectives.

2. The solution X1 is strictly better than X2 in at least one

objective.

If both solutions do not satisfy the above conditions, solution X1

and X2 are non-dominating each other. The steps to find the non-dominated

solutions are complicated and time consuming.

38

3.4.5 Rank-sum Sorting

The non-dominant solutions are sorted using rank-sum sorting. The

steps involved in the rank-sum sorting procedure are

(i) One unranked objective is selected.

(ii) The range of the objective is calculated based on the

maximum and minimum value of the objective.

(iii) The search range of the objective is divided into 100 fuzzy

ranks.

(iv) For every point in the search space, identify which grid it

belongs to.

(v) Assign the corresponding rank to the point for the selected

objective.

(vi) Repeat the steps (i) to (v) for all the objectives.

(vii) Calculate the rank-sum of the solution.

3.4.6 Selection

The preferential set and backup set is evolved in the selection

process. The steps involved in the process are:

i) One unselected objective is chosen.

ii) For the chosen objective a particular percentage of the rank is

scanned. For each rank a solution with the lowest rank-sum is

chosen as the preferential set.

iii) Repeat steps (i) and (ii) until all the objectives are chosen.

39

iv) The solutions which are not in the preferential set are chosen

as the backup set.

3.4.7 Stopping Criteria

The maximum number of iteration is considered as the stopping

criteria. The maximum number of iteration is identified as the one, if the

maximum number of iteration is decreased below that value then there would

not be a convergence at least in any one or more of the 100 trial studies. The

solutions at the end of the process are the pareto optimal front solutions.

3.5 FUZZY MUTATED MULTI-OBJECTIVE EVOLUTIONARY

PROGRAMMING

3.5.1 Need for Fuzzy Mutated Multi-Objective Evolutionary

Programming

In order to reach the global optimum the number of iteration

required is more in EP, due to which the computation time is larger. So in

order to minimize the computation time and to improve the convergence

characteristics, fuzzy logic is implemented in the EP algorithm.

2 depends

on three factors namely, maxftft i ,( )minmaxijij VV or ( )minmax

ikik TT or ( )minmaxilil QQ

maxftft i is the relative value of the fitness

function which has the major influence with 2 is

small then the width of the normal distribution will be small and vice versa.

( )minmaxijij VV or ( )minmax

ikik TT or ( )minmaxilil QQ is the search range. The search

range is constant throughout the process. But the value of it varies from

iteration to iteration accordingly to the control parameter in the current

40

convergence. Generally it is kept constant. If the va

It was found that the relations between the factors are arbitrary and

ambiguous. So a fuzzy logic strategy where the search criteria are not

precisely bound would be more appropriate than a crisp relationship.

The search range and the scaling factor need a control to obtain a

better convergence. The relation between them seems to be arbitory and

ambiguous. Hence, the fuzzy logic strategy where the search criteria are not

precisely bounded would be more appropriate than a crisp relation. Thus,

either an adaptive scaling factor or the variance can be obtained from the

fuzzy logic strategy. Therefore the inclusion of fuzzy logic strategy in the

mutation process of EP technique leads to an amendment termed as FMEP.

3.5.2 Overview of Fuzzy Logic

A fuzzy logic system is a nonlinear mapping of the input data

vector into the scalar output with some appropriate but partial information or

criteria. The fuzzy logic allows problems to be described and processed in

linguistic terms instead of precise mathematical models. The basic sequential

steps in developing a fuzzy logic system are as follows.

Fuzzification: The range of values that the inputs and output may take is

called the universe of discourse (Crisp value). The crisp values has to be

defined for all the inputs and output. Then the inputs have to be fuzzified

using the membership function µ(x) into linguistic labels or fuzzy sets. The

membership functions are triangular, trapezoidal, bell shaped etc. The most

commonly used membership function is the triangular membership function.

The fuzzy sets need a certain overlap with the adjacent sets.

41

Fuzzy Inference: After all physical input values have been converted into

fuzzy sets, conclusions are determined, or a hypothesis is generated, from the

given input state. This process is known as fuzzy inference. In a fuzzy logic

system, the rules define the dependencies between linguistically classified

input and output values. The commonly used fuzzy operators are the AND

and OR terms. The fuzzy AND operator means that the lesser value of the two

degree-of-membership values is used, while the fuzzy OR operator means that

the greater value of the two degree-of-membership values is used. Fuzzy

logic can employ one of these fuzzy operators and it imitates the human

decision-making strategies. The inference strategy is the min-max inference.

Fuzzy membership functions have to be set for the output. And a fuzzy rule

base is formulated on certain heuristic guidelines or through some reasonable

logic.

Defuzzification: The symbolic control action that results from fuzzy

inference cannot be used directly. The linguistically manipulated variables

must be defuzzified. This process of Defuzzification involves the calculation

of a crisp numerical value at the output based on the symbolic results. The

most common Defuzzification method is the Center-Of Area method (COA),

also known as the center-of-gravity method. Defuzzify the output to obtain a

crisp value.

3.5.3 FMEP based algorithm for Multi-Objective Reactive Power

Optimization problem

The steps involved in the implementation of fuzzy logic in the

mutation process of EP for solving the Multi-Objective Reactive Power

Optimization problem have the following steps:

i) The inputs and the outputs of the fuzzy logic system are

decided. The inputs are relative fitness value ( maxftft i ) and

42

the search range (( )minmaxijij VV or ( )minmax

ikik TT or ( )minmaxilil QQ ).

The output 2). The

control

logic.

ii) The range of values that the inputs and outputs may take is

called the universe of discourse (Crisp Value). Each input and

output has to be defined by the universe of discourse. Then the

inputs have to be fuzzified using the membership function

µ(x) into linguistic labels or fuzzy sets. Generally triangular,

trapezoidal and bell shaped membership functions are used.

The most commonly used membership function is the

triangular membership function. There should be a certain

overlap with the adjacent sets in the fuzzy sets.

Fuzzification of input and output using triangular membership is

done using five fuzzy linguistic sets as shown in figure 3.1.

Figure 3.1 Fuzzy membership function

iii) After Fuzzification of the input values the next step is the

fuzzy interference process. Fuzzy interference is the process

of determining conclusions or generating hypothesis from the

given input state. In a fuzzy logic system, the rules define the

43

dependencies between linguistically classified input and

output values. AND and OR operator are the commonly used

fuzzy operators which imitates the human decision making

strategies. The fuzzy AND operator means that the lesser

value of the two degree-of-membership values is used, while

the fuzzy OR operator means that the greater value of the two

degree-of-membership values is used. The inference strategy

is the min-max inference. Fuzzy membership functions have

to be set for the output. And a fuzzy rule base is formulated on

certain heuristic guidelines or through some reasonable logic.

The mutation scaling factor is resolved into the fuzzy control logic.

Fuzzy rule base is formulated based on their ranges in all possible

combinations and given in Table 3.1.

Table 3.1 Fuzzy rule base

Input 1 VerySmall Small Medium Large VeryLarge

Input 2 VerySmall VerySmall VerySmall Small Small Small Small VerySmall Small Small Medium Medium Medium VerySmall Small Medium Large Large Large Small Medium Large VeryLarge VeryLarge VeryLarge Small Medium Large VeryLarge VeryLarge

iv) The results from the fuzzy interference cannot be used as

such. It has to be defuzzified. Defuzzification is the process of

calculating a crisp numerical value at the output in accordance

with the symbolic result. Centre-of-Area (COA) or center-of-

gravity method is generally used for defuzzification.

Defuzzification of the output is done by the centroid method.

44

5

15

1

i ii

ii

x yC

y

where, xi is the mid-point of each fuzzy output set i and yi is its corresponding

membership function value. The centroid C is scaled (multiplied by its range)

to obtain the 2 value of each element in the parent individual.

3.6 RESULTS AND DISCUSSIONS

The ability of the proposed FMMOEP is proved by implementing it

over the standard IEEE 30-bus system. IEEE 30-bus system consists of 6

generating units, 41 lines, 4 shunt capacitor banks and 4 tap changing

transformers with a total demand of 283.4 MW and 12.6.2 MVAR. The buses

1, 2, 5, 8, 11 and 13 are the generator buses and the remaining buses are the

load buses. The lines connecting buses (6-9), (6-10), (4-12) and (27-18) have

the tap changing transformers. The shunt VAR compensators are at buses 10,

15, 19 and 24. The generator, load and line data are given in appendix 1.The

cost of the compensating capacitor banks are considered as 1000 $/MVAR

and the desired voltage magnitude of all the buses is 1 p.u. The algorithms

were programmed in MATLAB V 7.1 installed in a Pentium IV, 2.5 GHz

processor.

Initially the MORPO problem is solved using the MOEP algorithm

by choosing the population size as 25 and the scaling factor as 0.03. The

penalty factors 1, 2 and 3 of the fitness function are chosen by trial and

error method. Initially a small value is chosen. Then the penalty factors will

be increased based on the constraint violation until an acceptable solution is

reached. The fuzzy logic data for the FMEP based algorithm are given in

Tables 3.2 to 3.4.

45

Table 3.2 Fuzzy logic data for the mutation process of EP considering generator voltage

FUZZY SET maxi ftft ( )VV minij

maxij

VerySmall 0.00001 to 0.00004 0.95- 0.98 0.001 0.005 Small 0.00003 to 0.006 0.975- 0.99 0.004 - 0.06

Medium 0.005 to 0.05 0.985-1.0 0.04 0.08 Large 0.03 to 0.5 0.995-1.02 0.075 0.09

Verylarge 0.4 to 1 1.015-1.05 0.085 0.1

Table 3.3 Fuzzy Logic data for the mutation process of EP considering tap settings of tap changing transformers

FUZZY SET maxi ftft ( )TT minik

maxik

VerySmall 0.00001 to 0.00004 0.95-0.99 0.001 0.005 Small 0.00003 to 0.006 0.97-0.99 0.004 - 0.06

Medium 0.005 to 0.05 0.985-1.01 0.04 0.08 Large 0.03 to 0.5 0.995-1.02 0.075 0.09

Verylarge 0.4 to 1 1.015-1.1 0.085 0.1

Table 3.4 Fuzzy Logic data for the mutation process of EP considering reactive power compensation by capacitor banks

FUZZY SET maxi ftft ( )QQ minil

maxil

VerySmall 0.00001 to 0.00004 0- 0.072 0.001 0.005 Small 0.00003 to 0.006 0.068- 0.14 0.004 - 0.06

Medium 0.005 to 0.05 0.10-0.21 0.04 0.08 Large 0.03 to 0.5 0.18-0.27 0.075 0.09

Verylarge 0.4 to 1 0.25-0.36 0.085 0.1

46

The convergence characteristics of the EP and FMEP based

algorithm for MORPO are presented in Figure 3.2. The convergence

characteristic is drawn by considering the multi-objective problem as a single

objective optimization problem. The multi-objective optimization problem is

converted into a single objective problem by the linear combination of all the

objectives. The convergence characteristics are drawn by plotting the

minimum fitness value from the combined population across the iteration.

From the convergence characteristics it is observed that the fitness function

value converges without any abrupt oscillations. It is also observed that the

FMEP algorithm have much faster convergence than the EP algorithm.

Figure 3.2 Convergence characteristics of EP and FMEP algorithms

The comparative results of EP and FMEP in comparison with

SPEA are presented in Table 3.5. From Table 3.5 it is inferred that the

optimum value obtained by EP and FMEP algorithm are better than the SPEA

algorithm.

4.2

4.7

5.2

5.7

6.2

6.7

1 11 21 31 41 51 61 71 81 91

Fitn

ess V

alue

Iteration

FMEP

EP

x106

47

Table 3.5 Optimum results using EP and FMEP in comparison with SPEA for IEEE 30-bus system

SPEA EP FMEP

V1 (p.u) 1.05 1.05 1.05

V2 (p.u) 1.041 1.026 1.034

V5 (p.u) 1.018 1.027 1.029

V8 (p.u) 1.017 1.031 1.01

V11(p.u) 1.084 0.981 0.989

V13(p.u) 1.079 0.992 0.100

T(6-9) 1.002 1.046 1.020

T(6-10) 0.951 0.989 0.9887

T(4-12) 0.990 1.032 1.0

T(27-28) 0.940 0.932 1.055

F1(p.u) 0.54 0.052 0.051

F2 ($) 0.0178 0.0176 0.0154

F3 (p.u) 0.73x106 0.7x106 0.68 x106

F4 (p.u) 0.1418 0.132 0.130

The control parameters setting for EP and FMEP based algorithms

in case of Best F1, F2, F3 and F4 are given in Table 3.6. The best results

presented for an objective are obtained by giving more priority to anyone of

the objective and by weakening the others. From the Table 3.6 it is inferred

that the control parameters are within the limit. Hence, the proposed

algorithms are simple and efficient for MORPO problem and also the

proposed EP algorithm has the ability to obtain the optimal solution in a

single run.

48

Table 3.6 Control parameter settings for EP and FMEP based algorithms in case of Best F1, F2, F3 and F4

Best F1 Best F2 Best F3 Best F4 EP FMEP EP FMEP EP FMEP EP FMEP

V1 (p.u) 1.05 1.05 1.05 1.05 1.05 1.05 1.009 1.05

V2 (p.u) 1.026 1.034 1.042 1.036 1.056 1.037 1.006 1.028

V5 (p.u) 1.027 1.029 1.035 1.023 1.039 1.032 1.021 1.042

V8 (p.u) 1.031 1.01 1.033 1.023 1.031 1.030 0.998 1.043

V11(p.u) 0.981 0.989 0.987 0.978 0.985 0.989 1.066 1.032

V13(p.u) 0.992 0.100 0.995 0.989 0.993 1.003 1.051 1.095

T(6-9) 1.046 1.020 1.036 1.025 1.033 1.031 1.093 1.02

T(6-10) 0.989 0.9887 0.998 0.982 0.997 0.958 0.904 0.986

T(4-12) 1.032 1.0 1.021 1.024 1.023 0.999 1.002 1.022

T(27-28) 0.932 1.055 1.048 0.952 1.042 1.034 0.941 1.048

Qc10 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

Qc15 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

Qc19 0.45 0.05 0.25 0.25 0.35 0.25 0.25 0.05

Qc24 0.10 0.20 0.20 0.10 0.10 0.10 0.10 0.20

F1 (p.u) 0.049 0.0482 0.058 0.056 0.054 0.053 0.057 0.056

F2 (p.u) 0.0193 0.1928 0.017 0.0168 0.017 0.0177 0.0169 0.0167

F3 ($) 0.75x106 0.74x106 0.73x106 0.72x106 0.70x106 0.69x106 0.73x106 0.73x106

F4 (p.u) 0.154 0.16 0.162 0.154 0.167 0.157 0.132 0.129

Table 3.7 Average results obtained using EP and FMEP based algorithm

Objective EP FMEP

Ploss (F1 in p.u) 0.052 0.050

Voltage Deviation (F2 in p.u)

0.017 0.0154

Investment cost (F3 in $) 0.7x106 0.68 x106

L-index (F4 in p.u) 0.132 0.130

49

Table 3.8 Voltage magnitude at the buses

EP FMEP V1 (p.u) 1.05 1.05 V2 (p.u) 1.026 1.034 V3 (p.u) 1.006 1.012 V4 (p.u) 1.022 1.019 V5 (p.u) 1.027 1.029 V6 (p.u) 1.032 1.028 V7 (p.u) 1.012 1.006 V8 (p.u) 1.031 1.01 V9 (p.u) 1.022 1.017 V10 (p.u) 0.978 0.986 V11(p.u) 0.981 0.989 V12 (p.u) 0.978 0.964 V13(p.u) 0.992 1.010 V14 (p.u) 1.042 1.027 V15 (p.u) 1.032 1.028 V16 (p.u) 1.042 1.039 V17 (p.u) 1.018 1.021 V18 (p.u) 1.027 1.024 V19 (p.u) 1.032 1.029 V20 (p.u) 0.989 0.978 V21 (p.u) 1.048 1.037 V22 (p.u) 1.037 1.041 V23 (p.u) 1.029 1.025 V24 (p.u) 1.019 1.009 V25 (p.u) 1.026 1.024 V26 (p.u) 1.037 1.029 V27 (p.u) 1.047 1.039 V28 (p.u) 1.038 1.027 V29 (p.u) 1.024 1.018 V30 (p.u) 1.042 1.029

50

In Table 3.7 the average results of EP and FMEP based algorithm

for MORPO problem is presented. From the results it is inferred that the EP

and FMEP based algorithms have the ability to generate compromise

solutions. The Pareto fronts obtained using the EP and FMEP based

algorithms are given in Figures 3.3 to 3.8. From the Pareto fronts obtained,

the ability of the proposed algorithm to generate diversified solutions with in

the search space is clearly revealed. Further it shows the effectiveness of the

algorithm to generate Pareto solution in a well diverse manner.

0.015

0.017

0.019

0.021

0.023

0.025

0.027

0.029

0.031

0.05 0.051 0.052 0.053 0.054 0.055 0.056

Power Loss (p.u)

Vol

tage

Dev

iatio

n (p

.u)

51

Figure 3.3 Pareto set Power loss versus voltage deviation of EP based algorithm

Figure 3.4 Pareto set power loss versus voltage deviation of FMEP based algorithm

0.015

0.017

0.019

0.021

0.023

0.025

0.027

0.029

0.05 0.051 0.052 0.053 0.054 0.055 0.056

Power Loss (p.u)

Vol

tage

Dev

iatio

n (p

.u)

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.05 0.051 0.052 0.053 0.054 0.055 0.056Power Loss (p.u)

L-In

dex

52

Figure 3.5 Pareto set power loss versus L-index of EP based algorithm

Figure 3.6 Pareto set power loss versus L-index of FMEP based algorithm

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.05 0.051 0.052 0.053 0.054 0.055 0.056Power Loss (p.u)

L -I

ndex

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.015 0.02 0.025 0.03

L -

Inde

x

Voltage Deviation (p.u)

53

Figure 3.7 Pareto set voltage deviation versus L-index of EP based algorithm

Figure 3.8 Pareto set voltage deviation versus L-index of FMEP based algorithm

3.7 SUMMARY

A brief note about the Evolutionary Programming is explained and

the need for the fuzzy logic over the EP algorithm was demonstrated. Then

the MOEP and FMMOEP algorithm for the MORPO problem are developed.

The EP and FMEP algorithms are demonstrated with an IEEE 30-bus system.

The results were compared with the SPEA algorithm. The results obtained by

EP and FMEP algorithm are optimal. From the convergence characteristics it

is inferred that FMEP based algorithm converges faster than the EP based

algorithm. The proposed EP and FMEP algorithm have the ability to solve the

MORPO by generating diverse Pareto optimal solutions.

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.015 0.02 0.025 0.03

L-I

ndex

Voltage Deviation (p.u)