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CHAPTER – 4

GENETIC SEARCH FOR A MULTI-OBJECTIVE OPTIMAL POWER

FLOW SOLUTION FROM A HIGH DENSITY CLUSTER

4.0 INTRODUCTION

A new algorithm is developed for the solution of multi-objective

optimal power flow problem. Two individual objective functions are

chosen: 1) minimization of Fuel Cost and 2) minimization of Power

Loss. The proposed algorithm is the application of a multi-objective

genetic algorithm (MOGA), using the combination of High Density

cluster and continuous genetic algorithm. The OPF is modeled as a

nonlinear, non-convex and large scale constrained problem with

continuous variables. The algorithm uses a local search method for

the search of Global optimum solution. Binary coded Genetic

algorithm is replaced with continuous genetic algorithm that uses

real values of generation instead of binary coded data. An attempt is

made to reduce the chromosome length.

4.1 OBJECTIVES OF PROPOSED METHODOLOGY

Chapter – 4 presents need for alternative methodologies that

can avoid all the difficulties in the various approaches and provide a

better OPF solution. In this work, a Multi Objective Genetic algorithm

combining with high density cluster algorithm is proposed. This

methodology has the following objectives:

Aims for incorporating a local search method within a genetic

algorithm that can overcome most of the obstacles that arise as a

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result of finite population size.

Aims for better improvements in speed and accuracy of solution.

Aims for consideration of large varieties of constraints and system

nonlinearities.

Aims for avoiding the blind search, encountering with infeasible

strings, and wastage of computational effort.

Aims for reduction in population size, number of populations in

order to make the computational effort simple and effective,

considering population is finite in contrast to assuming it to be as

infinite.

Aims for testing other types of Genetic Algorithm methods instead

of conventional GA that uses binary coded chromosomes.

Aims for a suitable local search method that can achieve a right

balance between global exploration and local exploitation

capabilities. These algorithms can produce solutions with high

accuracy.

Aims for identification and selection of proper control parameters

that influence exploitation of chromosomes and extraction of

global optimum solution.

Aims for improvements in coding and decoding of Chromosome

that minimizes the population size.

Aims for undertaking multi-objective OPF problem. By integrating

objective functions, other than cost objective function, it can be

said economical conditions can be studied together with system

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security constraints and other system requirements.

The elitism was implemented to increase performance of algorithm

and prevent loss of good quality solutions found during the search.

4.2 SALIENT FEATURES IN THE PROPOSED METHODOLOGY

This work aims for examining several issues that need to be

taken into consideration when designing, a genetic algorithm that

uses another search method as a local search tool. These issues

include the different approaches for employing local search

information that is useful for genetic algorithm searches for global

optimum solution. The salient features in this work are depicted in

Fig 4.1.

The algorithm proposed and developed elitist is based on a

population that contains all the primary non-dominated solutions

over the generations. This is due to fact that GA does not begin by a

random generation of population but with parent chromosome which

Fig: 4.1 Salient Features of the work

Uses an existing popular method as a local search method to obtain a

Global optimum solution

Salient features of Proposed OPF Algorithm

Under takes Multi-Objective OPF problem for minimization of fuel

cost and power loss

Uses continuous Genetic Algorithm to develop population around the

suboptimal solution typically a high density cluster

Develops MOGA to search for an optimal OPF solution for the said

Multi-objective OPF problem.

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is a sub optimal solution, obtained through popular existing OPF

methodology.

The size of chromosome is restricted to minimal in this method.

This is due to fact that it does not contain control variables like

voltage magnitudes, tap position etc. As the size of chromosome is

short, population size can be minimum.

4.3 STRUCTURE OF PROPOSED OPF SOLUTION METHODOLOGY

Inspired by the results of EGA method and to overcome the

general difficulties in GA or EGA approaches, a novel method is

proposed in this work. The method uses high density cluster DBSCAN

and Continuous GA algorithms. The new technique for the solution of

OPF based on Genetic search from a High Density Cluster named in

short form as “GSHDC” is proposed in this work. The objective of

GSHDC is to retain advantages of Mathematical Programming

techniques and to encounter the difficulties of evolutionary methods

like GA and PSO Methods.

Proposed GSHDC has three stages for Single Objective OPF

solution and has four stages for Multi-Objective OPF solution. Stages

1 and 2 are common for both.

Stages in GSHDC Algorithm for Single Objective OPF Solution:

In the first stage a suboptimal solution for OPF problem is

obtained by the conventional analytical method such as Interior

Point method OR by PSO Methods that considers equality

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constraints, transmission loss B-Coefficients and penalty factors.

This solution for OPF problem is treated as an approximate,

owing to the limitations in the methods. However this solution

shall give a better insight in to the exact solution as the OPF is

solved with the regular proven methods.

In the second stage, with the help of a newly proposed

continuous data chromosome a population is formed

surrounding the suboptimal solution that is obtained in the first

stage by GA and two individual high density clusters one for

minimum fuel cost and another for minimum power loss are then

created by DBSCAN algorithm. The high density cluster consists

of several suboptimal solutions, one of which can be the exact

one.

In the third stage, a genetic search is carried out for finding the

exact solution. The solution in the last stage is the exact one, as

is confirmed by the best Fitness Value. The proposed GSHDC

technique in contrast to GA method avoids the blind search,

encountering with infeasible strings, and wastage of

computational effort.

Stages in GSHDC Algorithm for Multi- Objective OPF Solution:

As mentioned earlier, the first two stages in single objective

GSHDC algorithm are common in case of multi-objective OPF

solution. In these stages, two individual high density clusters are

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formed. They consist number of high density core points which are

having best fitness function values.

In the third stage, each core point in each high density cluster is

assigned a membership function value. In minimum fuel cost

cluster, the OPF solution with minimum fuel cost amongst the

others is assigned membership function value of 1 and relatively

maximum fuel cost is assigned 0. The same procedure is followed

for minimum power loss cluster points.

In the final stage, a search is carried out for the exact multi

objective optimal solution using Multi Objective Genetic Algorithm

(MOGA) based on Pareto- Optimal.

Above stages are explained in detail as below:

Common Stages in GSHDC Algorithm for Single Objective &

Multi-Objective OPF Solution:

Stage-1: In the first stage a suboptimal solution for OPF problem is

obtained by any of the following local search methods 1) Modified

Penalty Factor Method 2) Primal-Dual Interior Point method and 3)

Particle Swarm Optimization Method that considers Lagrange

multipliers, equality constraints, transmission loss B-Coefficients and

penalty factors. Owing to the limitations of the methods as discussed

in Chapter-3, this solution is taken only as suboptimal or local

optimal. However this solution shall give a better insight in to the

exact solution as the OPF is solved with the regular mathematical

programming / intelligent approaches. Because of this reason, this

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OPF solution cannot be taken as a global one. However due to

consideration of constraints of control parameters this solution gives

a core point in the high density cluster. The OPF solution for

minimum fuel cost and minimum power loss is show in Fig. 4.2 and

4.3, which are initial core points for the two individual high density

clusters for minimum fuel cost and power losses.

`

Stage-2: Owing to the limitations of the local search methods, in the

second stage, two independent High Density Clusters, which consists

of other suboptimal data points in the vicinity of the first are formed

by using Continuous Genetic Algorithm and DBSCAN algorithm. This

is also illustrated in Figs 4.2 and 4.3. The active power generation of

individual generating units is taken as continuous control variables

and the suboptimal solution obtained in the first stage, is first

encoded into a chromosome. This chromosome is treated as one of

Fig: 4.2 High density Cluster for minimum fuel cost

Initial Core Point obtained

through suboptimal solution methods

Cluster for Minimum cost

Other high density cluster points for

minimum-cost

Fig: 4.3 High density cluster for minimum power loss

Other high density cluster points for

minimum-loss

Cluster for Minimum-

power loss

Initial Core Point obtained

through suboptimal solution

methods

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the core points and the Fitness Function (FF) value of this

chromosome is calculated and is termed as Eps. Then the selection,

crossover and mutation processes are carried out for generating new

population consisting of other core points (or say other suboptimal

solutions), subject to FF values of these chromosomes are within Eps.

This forms a high density cluster and thoroughly avoids noise points

and border points which are regarded as infeasible solutions. It can

be stated here that, the length of chromosome in the proposed

method is reduced due to non consideration of certain control

parameters. This reduces the size of population of the high density

cluster to a large extent. However, the constraints of the control

parameters are considered in the third stage before arriving to the

exact optimal solution.

Further Stages in GSHDC Algorithm for Single Objective OPF Solution:

Stage-3: In the third stage, a genetic search is carried out for

finding the exact solution. The solution in the last stage is the exact

one, as is confirmed by the best Fitness Value.

Further Stages in GSHDC Algorithm for Multi-Objective OPF Solution:

Stage-3: In the third stage, each core point in each high density

cluster is assigned a membership function value. The basis for

assigning membership function values are assigned is discussed

below.

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Membership function values

The fuzzy sets are defined by equations called membership

functions. These functions represent the degree of the membership in

some fuzzy sets using values from 0 to 1. The membership value 0

indicates, incompatibility with the sets, while value 1 means full

compatibility. By taking account of the minimum and maximum

values of each objective function together with the rate of increase of

membership satisfaction, the decision maker must detect

membership function ( )iF in a subjective manner. Here it is

assumed that ( )iF is a strictly monotonic decreasing and

continuous function defined as:

min

maxmin max

max min

max

1 ;

( ) ;

0 ;

i i

i ii i i i

i i

i i

F F

F FF F F F

F F

F F

(4.1)

The value of membership function suggests how far (the scale

from 0 to 1) a non–inferior (non-dominated) solution has satisfied the

Fi objective. The sum of membership function values ( )iF

(i = 1, 2,.…M) for all the objectives can be computed in order to

measure the accomplishment of each solution in satisfying the

objectives.

Stage-4: In the final stage, a search is carried out for the exact multi

objective optimal solution using a Multi Objective Genetic Algorithm

(MOGA)[115,116]. The accomplishment of each non-dominated

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solutions can be read with respect to all the K non-dominated

solutions by normalizing its accomplishment over the sum of the

accomplishments of K non-dominated solutions as follows:

1

1 1

( )

( )

Mk

i

ik

D K Mk

i

k i

F

F (4.2)

The function D in Eq.(4.2) can be treated as a membership function

for non-dominated solutions, in a fuzzy set and represented as fuzzy

cardinal priority ranking of the non-dominated solution. The solution

that attains the maximum membership k

D , in the fuzzy set so

obtained can be chosen as the ‘best’ solution or the one having the

highest cardinal priority ranking.

Max {k

D : k = 1, 2,…., K} (4.3)

Based on the above procedure the best solution is obtained from Non-

domination sort data which has a membership function value close

to D , which also satisfies the constraints and convergence of load

flow.

GSHDC Method is implemented for two single objectives and one

multi-objective covering 3-Test cases (OPF Methods) and 3-Case Studies

(IEEE Test Systems) as described below:

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Suboptimal solution is obtained for two individual

objectives and one Multi-objective:

Objective-1: Minimum Fuel Cost

Objective-2: Minimum Power Loss

Using the OPF solutions obtained through objective-1 & 2 as

parent chromosomes, population is generated for the multi-objective

OPF problem. This is referred as:

Objective-3: Multi-Objective which includes Objective-1 & Objective-2.

Sub optimal OPF solutions for Minimum Fuel Cost and

Minimum Power Loss are obtained through following methods.

Test-1: Suboptimal Solution for Minimum Fuel Cost and Minimum

Power Loss is obtained through IP method

Test-2: Suboptimal Solution for Minimum Fuel Cost and Minimum

Power Loss is obtained through PSO method.

Test-3 In addition to above two test cases, GSHDC is also

implemented with suboptimal solution obtained through

modified penalty factor method to test its effectiveness. This

case is referred as Test-3.

Two individual High Density Clusters for Minimum Fuel

Cost and Minimum Power Loss are formed using continuous

Genetic algorithm for the following three case studies:

Case-1: IEEE 14-Bus System, Case-2: IEEE 30-Bus System

Case-3: IEEE 57-Bus System

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Summary of results are furnished as per the format given below:

4.4 PROPOSED GSHDC ALGORITHM FOR SINGLE & MULTI-

OBJECTIVE OPTIMAL POWER FLOW SOLUTION

The GSHDC involves the following steps:

Stage-1: Steps involved in stage-1 (for obtaining Sub-Optimal

Solution):

Step-1: Read Data namely: Cost coefficients of all units, B-

coefficients, convergence tolerance, error, step size and

maximum allowed iterations, Population Size, Probability of

Cross-over, Probability of mutation, λmin and λmax. System bus

data, load data.

Sl.No Method Description

1 Test-1. objective-1, Case-1/ 2/ 3

GSHDC-IP method, Min. Fuel Cost,14/30/57-Bus System

2 Test-1. objective-2

Case-1/ 2/ 3

GSHDC-IP method, Min. Power Loss,

14/30/57-Bus System

3 Test-1. objective-3, Case-1/ 2/ 3

GSHDC-IP method, Both Min Fuel Cost& MinPower Loss, 14/30/57- Bus System

4 Test-2. objective-1,

Case-1/ 2/ 3

GSHDC-PSO method, Min. Fuel Cost,

14/30/57-Bus System

5 Test-2. objective-2 Case-1/ 2/ 3

GSHDC-PSO method, Min.Power Loss, 14/30/57-Bus System

6 Test-2. objective-3,

Case-1/ 2/ 3

GSHDC-PSO method, Both Min Fuel Cost &

MinPower Loss,14/30/57- Bus System

7 Test-3. objective-1,

Case- 2

GSHDC-Penalty Factor method, Min. Fuel Cost,

30-Bus System

8 Test-3. objective-2,

Case- 2

GSHDC-Penalty Factor method, Min. Fuel Cost,

30-Bus System

9 Test-3. objective-3,

Case-2

GSHDC- Penalty Factor method, Both Min.Fuel

Cost& MinPower Loss,30- Bus System

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Step-2: Obtain Sub Optimal Power Flow solution that is, generation

schedule for minimum fuel cost and minimum power loss

through any one of the suboptimal solution methods (Sections

4.6, 4.7 and 4.8) for a given System load. This OPF solution

is the initial core point in the respective high density cluster.

(End of the Process for obtaining sub optimal OPF solution)

Stage-2: Steps involved in stage-2 (for High Density Cluster

Formation):

Step-3: Develop method for coding and decoding of chromosome

(string) for initial core point of high density cluster. (Section

4.9.1).

Step-4: Map cost of generation in to Fitness Function. (Section 4.9.2)

Step-5: Compute Fitness value of initial core point and assign it as

Eps (section 4.9.2)

(End of the Process for obtaining Initial Core Point in High

Density Cluster)

Steps involved in obtaining other Core Points in High Density

Cluster - generation of population

Step-6: Randomly generate population (other core points) in and

around initial core point of high density cluster (Section

4.9.3).

Step-7: Carryout Parent Selection Process from high density cluster

points (section 4.9.3)

Step-8: Carryout Cross over process. (Section 4.9.4)

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Step-9: Carryout Mutation Process. (Section 4.9.5)

(End of the Process for formation of High Density Cluster)

Step-10: For Single Objective OPF Solution GOTO Step-11

for Multi-Objective OPF solution GOTO Step-18

Single Objective OPF solution using individual high density clusters i.e.

minimum fuel cost and minimum power loss.

Stage-3: Steps involved in stage-3 (search for exact OPF solution)

are given below:

Step-10: Compute Fitness values of the newly generated population

(Section 4.9.2).

Step-11: Compare Fitness values of each chromosome with Eps.

Step-12: Store better fitness value chromosomes as high density

cluster points and place the rest in to noise cluster.

Steps 10 to 13 are repeated for all high density cluster points.

Exact OPF Solution

Step-13: Select the core point having highest fitness value, from a

high density cluster obtained from Step-13. (Section 4.9.5)

Step-14: For selected core point chromosome, run Load Flows to

check convergence and check equality and inequality

constraints satisfaction, limits for control parameters etc. as

per Eqns. (3.13) to (3.20). Also check for slack bus

generation limits.

Step-15: For the selected core point chromosome, if violation in limits

for constraints takes place, send this core point in to noise

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cluster and select another core point chromosome with next

highest fitness value. Go To Step-15. Else Go To next Step.

Step-16: The decoded core point chromosome is treated as OPF

solution.

Step-17: Print OPF results (Individual Generation Schedule for

Minimum Fuel Cost and Minimum Power Loss)

Multi – Objective OPF solution using individual high density

clusters i.e. minimum fuel cost and minimum power loss.

Stage-3: Steps involved in stage-3 are given below:

Step-18: Assign a membership function value to each core point in

each high density cluster (Section 4.3).

Stage-4: Steps involved in stage-4 are given below:

Step-19: Using MOGA Algorithm compute member ship function

values for k-non dominated solutions k

D as described in

Section 4.3.

Step-19: Obtain OPF solution for multi-objective problem, a

generation schedule having maximum membership function

value i.e. from Eq.4.3.

Max { k

D : k = 1, 2,…….., K.}

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4.5 MULTI-OBJECTIVE OPTIMAL POWER FLOW PROBLEM-

OBJECTIVE FUNCTIONS

The selected problem can be designated as a multi-criteria and

multi-objective optimization problem which requires simultaneous

optimization of two objectives with different individual optima.

Objective Function-1: Total Fuel Cost

Total Generation cost function is expressed as:

2

1

( )G

i i

N

G i i G i G

i

F P P p

(3.1)

The objective function is expressed as:

Min F (PG)= f1 (x,u)

Constraints are mentioned in the set of Eq. (3.7 -3.20)

Objective Function-2: Total Power Loss

The objective functions to be minimized are given by the sum of line

losses

1

lN

kL lk

P P (3.21)

Individual line losses 1kP can be expressed in terms of voltages and

phase angles as

2 2 2 cos( )k i j i j i jl

kP g V V VV (3.22)

The objective function can now be written as

Min 2 2

1

( 2 cos( )lN

L k i j i j i j

i

P g V V VV

(3.23)

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This is a quadratic form and is suitable for implementation

using the quadratic interior point method. The constraints are

equivalent to those specified in Section 3.1.1 for cost minimization,

with voltage and phase angle expressed in rectangular form.

4.6 SUB OPTIMAL SOLUTION FOR OPF PROBLEM USING

MODIFIED PENALTY FACTOR METHOD

The objective is to maximize profits and usage of the equipment in

service so as to achieve the greatest financial benefits.

Mathematically, the problem is defined as:

2

1

( )G

i i

N

T G G

ii i iC P P (3.1)

Subject to the energy balance equation

1 1( ) ( )G D

i i

N N

i G i D LP P P (3.12)

and the inequality constrains

min max , 1,...,i i iG G G GP P P i N (3.14)

Where i, i, i are the cost coefficients, PDi= load demand and PGi =

real power generation of ith Machine; NG = number of generation

buses and PL = transmission power loss. Most of the OPF problems

use a set of Lagrangian equations such as:

GG

G G

G G

NG

L L LG

G

dCdC dC1 1 1λ

P P PdP dP dP1 1 1

P P P

21

1 2

1 2

........

NG

NG

(4.4)

= incremental cost of received power units $/MWhr.

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The co-ordination equation of ith Machine is:

i

ii

G

dCL

dP

(i=1, 2, 3, 4. . . . . NG) (4.5)

Eq.(4.2) illustrates for economical operation of the system the

incremental cost equations for all the units in a plant should be equal

in value. The variable that describes this value is called Lambda (λ).

The penalty factor of ith Machine represented as Li is defined as :

1

1

i

iL

G

LP

P

(i=1, 2, 3, 4. . . . .NG) (4.6)

Where

i

L

G

P

P is the incremental transmission loss of ith Machine. The

total plant losses in terms of loss coefficients B as:

1 1

G G

i j

N N

L G ij G

i j

P P B P

(4.7)

The penalty factor represents the power losses incurred in

transmitting the power to the load demand buses. The operating cost

of some generating units, which are distantly located from a load, may

have a higher operating cost than some other more costly units

located nearby, due to transmission losses.

For NG generator bus system loss coefficient Bij – matrix is n x n

symmetric matrix given as:

11 12 1

21 22 2

1 2

n

n

n n nn

i j

B B B

B B BB

B B B

(4.8)

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The loss coefficients are used to calculate the penalty factors. Loss

coefficients describe the system losses for a given load.

The iterative method solves for each unit generation using the

incremental cost functions, multiplied by penalty factors. For a n-

generator bus system these set of n-equations are given by:

ii G iP (for i=1, 2,…….. NG) (4.9)

Set of linear coordinate equations can be written as:

( )ii Gi iL P (for i=1, 2,…….. NG) (4.10)

Eq.( 4.10) must satisfy for minimum cost of generation. The set of n-

coordinate equations (4.9) can be grouped in to n-1 equations by

eliminating λ from (4.9) and setting ith and i+1th equations equal to

each other as:

11 11 1( ) ( ) ( ) ( )

i ii G i G i ii i i iL P L P L L (4.11)

For NG – unit system the additional NG th equation is:

Gi

N

L Di 1

P P

G

P (4.12)

The complete set of linear equations in matrix form as:

1

2

2 2 1 1

1 1 2 2

3 3 2 22 2 3 3

1 11 1

0 0 0

0 0

0 0

1 1 1 1

G G G G

NG

G

G

NG NG NG NGN N N N

G

D

L LPL L

L LPL L

L LL L

PP

(4.13)

in condensed form Eq.(4.13) is:

[A][P] = [B] (4.14)

The new generation schedule can be obtained by using following

equation.

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[P] = [A]-1[B] (4.15)

The new generation schedule is used to calculate new penalty factors

using Eq. (4.6). Eqs. (4.6) and (4.15) are repeatedly solved until a

convergence criterion is satisfied. The algorithm for modified penalty

factors is presented below.

4.6.1. Algorithm Modified Penalty Factor Method

Step-1: Read System Line data comprising resistance, reactance and

charging admittance of various transmission lines.

Step-2: By representing loads as lumped admittances, formulate Bus

Admittance Matrix (YBus).

Step-3: Using partitioned YBus matrices, compute B-coefficients.

Step-4: Read NG -Generator cost coefficients βi and γi, Generator

Power limits, System load at various buses and total plant

load. Define convergence error € based on accuracy required.

Step-5: For a given system load, perform Load-Flow study by Newton-

Raphson method.

Step-6: Using the power generation schedule compute penalty factor

Li values.

Step-7: Using Li values and cost coefficients βi and γi for i=1, 2,…, NG,

compute A and B matrices.

Step-8: Compute new generation schedule matrix [P] that

containing,Gi new

P for i=1, 2,…, NG., using Eq.(4.15).

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Step-9: Check , ,max,minG G Gi new ii

P P P for i=1, 2,…, NG. If ,Gi new

P with in limit

then Go To Step-12. Else Go To Step-10.

Step-10: Set the limit violated unit generation to either,Gi new

P =,minGi

P or

,Gi newP =

,maxGiP as the case may be.

Step-11: Repeat from Step-6.

Step-12: Check for,Gi new

P -,Gi old

P ≤ €. If converged with in tolerance Go To

Step-14.

Step-13: Repeat from Step-6.

Step-14: Print Generation Schedule.

The mathematical programming method described above is

simple and capable of handling nonlinear incremental cost functions.

The number of iterations required in the solution ranges three to five.

Number of iterations are power system size independent but

dependent on tolerance value and non linearity of cost functions. The

developed method is fast and can be used with less CPU time and

memory. Complexities in representation of cost functions reduce

effectiveness of this method.

4.7 SUB OPTIMAL SOLUTION FOR OPF PROBLEM USING

PRIMAL DUAL INTERIOR POINT METHOD

OPF solution using Primal Dual Interior Point Method (PDIPM)

is presented in Section 3.3.5 in Chapter 3. For the sake of continuity,

the algorithm is reproduced below. The solution is obtained using

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PDIPM is only taken as sub-optimal, owing to the disadvantages

presented. However, in this work PDIPM is taken as local search

method as it is giving best insight for initial chromosome or core point

for a high density cluster.

4.7.1 Algorithm of PDIPM For The Solution Of OPF Problem

The PDIPM algorithm (Section 3.3.5.2) applied to the OPF

problem is summarized step-by-step as follows.

Step 1 Read relevant input data.

Step 2 Perform a base case power flow by a power flow

subroutine.

Step 3 Establish an OPF model.

Step 4 Compute Eqs. (3.105) – (3.107).

Step 5 Calculate search directions with Eqs. (3.112) – (3.114).

Step 6 Compute primal, dual and actual step-lengths with Eqs.

(3.115) – (3.118).

Step 7 Update the solution vectors with Eqs. (3.119) – (3.121).

Step 8 Check if the optimality conditions are satisfied by Eqs.

(3.98) – (3.110) and if μ ≤ ε (ε = 0.001 is chosen).

If yes, go to the next step. Otherwise go to step 4.

Step 9 Perform the power flow subroutine.

Step 10 Check if there are any violations in Eqs. (3.13) – (3.20). If

no, go to the next step; otherwise, go to step 4.

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Step 11 Check if a change in the objective function is less than or

equal to the prespecified tolerance. If yes, go to the next

step; otherwise, go to step 4.

Step 12 Print and display an optimal power flow solution.

Disadvantages of PDIPM

Limitation due to starting and terminating conditions

Infeasible solution if step size is chosen improperly

4.8 SUB OPTIMAL SOLUTION FOR OPF PROBLEM USING

PARTICLE SWARM OPTIMIZATION METHOD

OPF solution using Particle Swarm Optimization (PSO) method

is presented in Section 3.4.2 in Chapter 3. For the sake of continuity,

the algorithm is reproduced below. The solution is obtained using PSO

is only taken as sub-optimal, owing to the disadvantages presented.

However, in this work PDIPM is taken as local search method as it is

giving best insight for initial chromosome or core point for a high

density cluster.

4.8.1 PSO Algorithm

Description of basic elements required for the development of Solution

Algorithm is presented in section 3.4.2.2.

In order to make uniform search in the initial stages and very

local search in later stages, an annealing procedure is followed. A

decrement function for decreasing the inertia weight given as w(t)=

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w(t-1), is a decrement constant smaller than but close to 1 , is

considered here.

Feasibility checks, for imposition of procedure of the particle

positions, after the position updating to prevent the particles from

flying outside the feasible search space.

The particle velocity in the kth dimension is limited by some

maximum value, vk max. With this limit, enhancement of local

exploration space is achieved and it realistically simulates the

incremental changes of human learning. In order to ensure

uniform velocity through all dimensions, the maximum velocity in

the kth dimension is given as :

max max min( ) /k k kv x x N (3.112)

In PSO algorithm, the population has n particles and each particle is

an m – dimensional vector, where m is the number of optimized

parameters. Incorporating the above modifications, the computational

flow of PSO technique can be described in the following steps.

Step 1 (Initialization)

Set the time counter t = 0 and generate randomly n particles,

[ (0), 1,... ]jX j n , where , 1 ,(0) [ (0),..., (0)]j j j mX x x .

, (0)j kx is generated by randomly selecting a value with uniform

probability over the kth optimized parameter search space

min max[ , ]k kx x .

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Similarly, generate randomly initial velocities of all

particles,[ (0), 1,... ]jV j n , where , 1 ,(0) [ (0),..., (0)]j j j mV v v .

, (0)j kv is generated by randomly selecting a value with uniform

probability over the kth dimension max max[ , ]k kv v .

Each particle in the initial population is evaluated using the

objective function J.

For each particle, set *(0) (0)j jX X and * , 1,...,j jJ j nJ . Search

for the best value of the objective function bestJ .

Set the particle associated with bestJ as the global best, **(0)X , with

an objective function of **J .

Set the initial value of the inertia weight (0)w .

Step 2 (Time updating)

Update the time counter t = t + 1.

Step 3 (Weight updating)

Update the inertia weight ( ) ( 1)w t w t .

Step 4 (Velocity updating)

Using the global best and individual best of each particle, the

jth particle velocity in the kth dimension is updated according to the

following equation:

*

, , 1 1 , ,( ) ( ) ( 1) ( ( 1) ( 1))j k j k j k j kv t w t v t c r x t x t

**

2 2 , ,( ( 1) ( 1))j k j kc r x t x t (3.113)

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Where 1c and 2c are positive constants and 1r and

2r are uniformly

distributed random numbers in [0, 1]. It is worth mentioning that the

second term represents the cognitive part of PSO where the particle

changes its velocity based on its own thinking and memory. The third

term represents the social part of PSO where the particle changes its

velocity based on the social-psychological adaptation of knowledge. If

a particle violates the velocity limits, set its velocity equal to the limit.

Step 5 (Position updating)

Based on the updated velocities, each particle changes its

position according to the following equation:

, , ,( ) ( ) ( 1)j k j k j kx t v t x t (3.114)

If a particle violates its position limits in any dimension, set its

position at proper limit.

Step 6 (Individual best updating)

Each particle is evaluated according to its updated position. If

* , 1,...,j jJ J j n , then update individual best as *( ) ( )j jX t X t and

*

j jJ J and go to step 7; else go to step 7.

Step 7 (Global best updating)

Search for the minimum value minJ among *jJ , where min is the

index of the particle with minimum objective function, i.e.

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min { ; 1,..., }j j n . If **

minJ J , then update global best as

**

min( ) ( )X t X t and **

minJ J and go to step 8 ; else go to step 8.

Step 8 (Stopping criteria)

If one of the stopping criteria is satisfied then stop; else go to

step 2.

4.8.2 Disadvantages of PSO Method

The candidate solutions in PSO are coded as a set of real

numbers. But, most of the control variables such as transformer

taps settings and switchable shunt capacitors change in discrete

manner. Real coding of these variables represents a limitation of

PSO methods as simple round-off calculations may lead to

significant errors.

Slow convergence in refined search stage (weak local search

ability).

4.9 DESCRIPTION OF GSHDC ALGORITHM

The following sub sections provide the description of steps in

continuous GAs that are implemented in GSHDC algorithm.

4.9.1 Formation of Chromosome

Chromosomes in Binary Genetic Algorithms (BGAs) were

generated using binary representation of variables describing possible

solutions. Many studies recommend the use of real numbers, instead

of binary coded values to represent the possible solution

chromosomes for optimizing functions with inherently continuous

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domains such as OPF problem. In Continuous Genetic Algorithms

(CGAs) a real valued chromosome represents the parameters defining

solution in actual form.

Ex: If the chromosome has ‘n’ variables (an n-dimensional

optimization problem) given by1 2, ,....,

NGG G GP P P that is active power

generations of NG generators in power system, then the chromosome

is written as an array with (1×n) elements so that

Chromosome = [1 2, ,....,

NGG G GP P P ]

In this case, the variable values are represented as floating – point

numbers. Each chromosome has a cost found by evaluating the cost

function f as the variables1 2, ,....,

NGG G GP P P .

Cost = f (chromosome) = f (1 2, ,....,

NGG G GP P P ) =

Subject to constraints,min ,maxG G Gi i i

P P P . Since f is a function

of1 2, ,....,

NGG G GP P P , the clear choice for the variables is:

Chromosome = [1 2, ,....,

NGG G GP P P ]

For the present problem of OPF, The structure of the chromosomes

used in conventional BGAs such as Extended Genetic Algorithm

(EGA) method in [103] and in this proposed GSHDC method are

shown in the Figs.4.4 and 4.5 respectively. The interested reader can

refer [103] for complete information for string formation. The

difference in string lengths of chromosomes can be noted in both the

methods.

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4.9.2 Fitness and Cost Functions

The Cost Function is defined as: 2

1

( )G

i i

N

T G G

ii i iC P P

. Our

objective is to search GiP for i=1, 2,…,NG, generations in their

admissible limits so as to min( CT ) is obtained. The cost function of

initial core point in High Density Cluster is obtained through one of

the suboptimal solution methods. The value of the cost is then

mapped into a fitness value so as to fit in the genetic algorithm. The

fitness value of core point is taken as Eps. To minimize the cost is

equivalent to getting a maximum fitness value in the searching

process. A core point chromosome that has lower cost function

should be assigned a larger fitness value (f). The objective of the OPF

has to be changed to the maximization of fitness to be used in the

roulette wheel as follows:

1GP : GNGP

1GU : GNG

U t1 : t NG bs

h1 : bsh NG

Unit active Power Outputs

Generator bus voltage magnitudes

Transformer tap settings

Bus shunt admittances

Fig: 4.4 String Structure in Binary Genetic Algorithms [103]

1GP

2GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

GNGP

Unit active Power Outputs

Fig. 4.5 String Structure in GSHDC Method

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Fitness Value of ith chromosome fitness (i)= Eps - fi : if fi ≤; Eps ; for

i=1,2,…NG. ; = Zero other wise. Thus the GA tries to generate better

offspring to improve the fitness.

Because only active power generations are used in the fitness, the

reactive power levels and voltage constraints are scheduled in the

Power Flow Study. It can be understood that active power limits are

checked using GA procedure and the other constraints are checked

using an efficient power flow study.

4.9.3 Parent Selection

The sub optimal solution obtained through local search method

is the core point in the high density cluster. This OPF solution is a

chromosome and its structure is formed as discussed earlier. By

increasing or decreasing power generation for a constant load

demand, a population consists of certain number of chromosomes

having better FF values is then generated randomly. The

chromosomes in this population having better FF values are then

chosen as Parent Chromosomes for generating next off springs with

the help of Genetic operators. A Blending Method is used for crossover

operation and polynomial mutation is used to produce off springs.

4.9.4 Crossover

Crossover operator is believed to be the main operator that

creates the search of power of GAs in optimization problems [8].

Crossover operators have two important functions:

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First function is it searches the initial population that is initial

random strings containing problem variables to come up with a

good solution and

Second function is to combine good portions of these strings to

form even better solutions.

Since the use of real value representation is proposed in this work,

the search for a better crossover operator is on. Blending Method in

[113] used in this work. To generate a new offspring variable value,

PGinew from a combination of corresponding variable values in parent

chromosomes.

PGinew = PGi

male + (1- ) PGifemale for i=1,2,……n (4.16)

is the random variable chosen on the interval [0, 1] PGimale and

PGifemale are the incremented and decremented value respectively

around the suboptimal solution obtained through local search

method.

Using this blending method the offspring variables inherit that

property from their parent’s variables, and their value always fall

between the values in parent’s variables.

Ex: Let the suboptimal solution through a local search method for

two generations of 2-Generator problem gave PG1 = 100 MW and PG2 =

200 MW say (Initial Chromosome). The boundary for incremented

values PGimale (Male) are taken as 102 and 202 MWs and the boundary

for decremented values PGifemale (Female) are taken as 98 and 198

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MWs. Now the population can be generated using different values of

and using (4.1) as:

With = 0, Parent -1= (PG1, PG2) = (98 MW, 198 MW)

With = 0.1, Parent -2= (PG1, PG2) = (98.4 MW, 198.4 MW)

With = 0.25, Parent -3= (PG1, PG2) = (99 MW, 199 MW)

With = 0.5, Parent -4= (PG1, PG2) = (100 MW, 200 MW)

With = 0.75, Parent -5= (PG1, PG2) = (101 MW, 201 MW)

With =1 Parent -6= (PG1, PG2) = (102 MW, 202 MW)

Chromosome-1= [98 MW, {any PG2 value obtained from different values

of }]

Chromosome--2= [98.4 MW, {any PG2 value obtained from different

values of }]

…… … …. … … … … … … … … … …

….. …. … ….. …. …. ….. ….. ….. ….. ….. ….

Chromosome--n= [102 MW, {any PG2 value obtained from different

values of }]

Since, applying these blending methods does not result in introducing

values beyond extreme values of that variable in the initial

population, some extrapolating blending have been suggested in

the literature. However, these methods could generate value could

generate value outside the acceptable range for a parameter. Then the

offspring must be discarded and another selected.

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4.9.5 Mutation

Mutation Operator has two important roles:

The first role is to introduce unexplored genetic chromosomes to

the population.

Second role is to maintain the diversity of the chromosomes in a

population over the generations, preventing premature

convergence of the GA to suboptimal local optima.

Mutation in BGAs is rare compared to crossover, resulting in

lower probability of mutation. As the mutation probability for all the

bits in a binary string is the same, it will lead to higher probability of

small changes and lower probability of large changes.

One way to implement mutation in CGAs is make it to work

similar to binary BGAs. In this way if a variable PGi in the

chromosome has been chosen for the mutation, a random number F

is selected between [ 0,1] then the new variable PGi mutation would be:

PGi mutation = PGi crossover + F (PGi male - PGifemale ) for i =1, 2,…n (4.17)

Where PGi crossover is ith variable of the parent chromosome

selected for mutation, PGi male is the upper limit and PGifemale is the

lower limit for the PGi crossover.

Ex: Let F = 0.2; PGi crossover = Parent -3= (PG1, PG2) = (99 MW, 199 MW);

PGimale = [102,202] and PGi

female = [98,198].

PGi mutation = [{99, 199 MW}] + 0.2[{102-98}, {202-198}] = [99,199] +

[{0.8}, {0.8}] = [99.8, 199.8]

186

for different values of F different chromosomes for mutation can be

obtained.

4.9.6 GA Search for the exact OPF solution-Convergence Criterion

After crossover and mutation, Load flow study is run. If load

flow converges and slack bus generation is checked for limits for any

generation, the minimum generation cost amongst all (3.20). If any

violation of constraints, is observed the next core chromosome in the

list is selected. The Process is repeated until the desired chromosome

which satisfies all constraints is selected.

4.10 ADVANTAGES OF GSHDC OVER THE OTHERS

The advantages of GSHDC method over the other methodologies

are given below:

Length of Chromosome is reduced and hence the size of

population is reduced.

Number of generations is reduced. This makes the computational

effort simple and effective.

The problem of use of specific mutation or crossover operators is

avoided. This makes the OPF as another simple GA search

problem.

Blind search is avoided.

The process begins with no insignificant chromosomes.

System nonlinearities are somewhat considered as the initial

chromosome is obtained from the mathematical programming of

nonlinear equations.

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4.11 CONCLUSIONS

In this Chapter, a new algorithm for the solution of optimal

power flow problem was presented. The algorithm was unfolded into

three stages. In the first, a suboptimal solution was obtained by the

following local search methods: 1) Modified Penalty Factor method 2)

Primal Dual Interior Point 3) Particle Swarm Optimization method.

Owing to limitations in these methods, in the second, a High Density

Cluster, which consists of other suboptimal data points in the vicinity

of the first were formed by using Continuous Genetic Algorithm. In the

final stage, a search was carried out for the exact optimal solution

from a high density small size population High Density Cluster. The

final optimal solution thoroughly satisfies the well defined fitness

function. This work mainly undertakes the primary goal of OPF that

is to minimize the cost of generation for meeting a load demand while

maintaining the security of the system. System security can be

thoroughly maintained when each device of a power system, work in

desired operation range under steady state conditions.

This work aims for examining several issues that need to be

taken into consideration when designing genetic algorithm that uses

another search method as a local search tool. These issues include

the different approaches for employing local search information that

is useful for a genetic algorithm searches for global optimum solution.