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MULTI-OBJECTIVE, SHORT-TERM

HYDROTHERMAL SCHEDULING BASED ON

TRADITIONAL AND HEURISTIC SEARCH METHODS

A THESIS

Submitted by

ABRAHAM GEORGE

for the award of the degree

of

DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL & ELECTRONICSENGINEERING

Dr. M.G.R. EDUCATIONAL AND RESEARCH INSTITUTE

UNIVERSITY(Declared U/S 3 of the UGC Act, 1956)

CHENNAI - 600 095July 2011

ii

Dr. M. Channa Reddy Phone : Res: 080-26595566Director Cell : + 91 9448355966Vemana Institute of Technology, e-mail:[email protected]# 1, Mahayogi Vemana Road,3rd Block, Koramangala,Bangalore – 560 034

BONAFIDE CERTIFICATE

Certified that this thesis titled “MULTI-OBJECTIVE SHORT-TERM

HYDROTHERMAL SCHEDULING BASED ON TRADITIONAL AND

HEURISTIC SEARCH METHODS”, is the bonafide work of Mr. Abraham

George who carried out the research under my supervision. Certified further, that

to the best of my knowledge the work reported herein does not form part of any

other thesis or dissertation on the basis of which a degree or award was conferred

on an earlier occasion on this or any other candidate.

Dr. M. CHANNA REDDY

SUPERVISOR

iii

DECLARATION

This is to certify that the thesis titled “MULTI-OBJECTIVE SHORT-TERM

HYDROTHERMAL SCHEDULING BASED ON TRADITIONAL AND

HEURISTIC SEARCH METHODS” submitted by me to the Dr. M.G.R.

Educational and Research Institute University for the award of the degree of

Doctor of Philosophy is a Bonafide record of research work carried out by me

under the supervision of Dr. M Channa Reddy. The contents of this thesis, in

full or in parts, have not been submitted to any other Institute or University for

the award of any degree or diploma.

ABRAHAM GEORGE

iv

ABSTRACT

This investigation proposes a new algorithm for the solution of multi-objective,

short-term hydrothermal scheduling problem. In the traditional algorithm using

‘Weighting method’, a set of weight vectors are fed in, for each weight vector

objective function values during the optimization interval and the corresponding

fuzzy membership functions are computed and the best compromise solution is

identified from the set of non-inferior solutions obtained. In the proposed

algorithm, a set of weight vectors are genetically generated and the best solution is

identified as described above. Further the weight vectors are genetically modified

and the process is repeated to identify the best solution using the modified weight

vectors. The process is continued till change in the fitness value of the best

solution is marginal.

The investigation further proposes three methods for determining objective

function values in the optimization interval, which is a major sub-process in the

above algorithm. These are; the Newton-Raphson method, a method involving a

genetic search and a method involving a random search. In the genetic search

method search space reduction technique is incorporated to speed-up the search

and fuzzy multipliers are used to reduce the number of iterations to satisfy hydro

and coal constraints. The above two modifications are incorporated in random

search method also, but search space reduction technique is essential for random

search method. A coal-constrained thermal plant is introduced to make the

scheduling problem more universal. N-R method solves the problem in minimum

time but complexity is the most. Genetic search method is simple, but very slow

due to numerous operations to be performed on binary strings. Random search

method is much faster than genetic search method and the simplest of all.

v

ACKNOWLEDGEMENT

I wish to express my sincere gratitude to my research supervisor Dr. M. CHANNA

REDDY, Director, Vemana Institute of Technology, Bangalore for his continuous

support, encouragement and invaluable guidance during my research work. I was

benefited a lot from his research and teaching experience, constructive

suggestions, disciplines and principles to accomplish timely completion of my

research work.

I would like to express my heartfelt thanks to Dr. A.Y. SIVARAMAKRISHNAN,

Dean, Department of EEE and Dr. S. RAVI, Head of the Dept. of ECE for their

continuous support and guidance extended to me during the course of my research

work.

Also I would like to extend my thanks to Mr. L RAMESH, Head of the Dept. of

EEE for all the help extended to me for the successful completion of my research

work.

ABRAHAM GEORGE

vi

CONTENTS

LIST OF TABLES xi

LIST OF FIGURES xvi

LIST OF SYMBOLS xviii

LIST OF ABBREVIATIONS xx

1 INTRODUCTION 1-13

1.1 Genetic Algorithm 3

1.2 Fuzzy Logic 5

1.3 Combinatorial Optimization 6

1.4 Popular Metaheuristics 8

1.5 Multi-objective GA 12

2 LITERATURE SURVEY 14-17

3 OBJECTIVE OF THE INVESTIGATION 18-20

3.1 Multi-objective, short-term hydrothermal scheduling 18

3.2 Objective of the investigation 19

4 PROPOSED INVESTIGATION 21-46

4.1 Major steps in the proposed algorithm 22

4.1.1 Genetic generation of weight vectors 22

4.1.2 Determining objective function values in the

optimization interval

23

4.1.3 Determining fitness of weight vectors 25

4.1.4 Discarding dominated solutions 26

4.1.5 Convergence criteria 26

4.1.6 Modifying the genetic population 26

4.2 Problem Formulation

vii

28

4.2.1 Assumptions/Approximations 28

4.2.2 Problem Objective 29

4.2.3 Constraints 29

4.2.4 The Lagrangian 30

4.2.5 Optimality conditions 30

4.3 Objective function values in the optimization interval 31

4.3.1 By N-R Method 31

4.3.1.1 Elements of H, Δz and J 31

4.3.1.2 Computation of initial values 35

4.3.1.2.1 Initial values of power

allocation and λk

35

4.3.1.2.2 Initial values of φ and µ 36

4.3.2 By a method using a genetic search 37

4.3.2.1 Search space reduction technique 38

4.3.2.2 To find the roots of a quadratic

equation

40

4.3.2.3 Modifying the Lagrange multipliers 43

4.3.3 By a method using a random search 45

5 TEST SYSTEMS 47-51

5.1 Test System -1 47

5.2 Test System-2 50

6 RESULTS AND DISCUSSIONS 52-153

6.1 Preliminary Investigations 52

6.1.1 Short-term hydrothermal scheduling : cost

minimization objective

viii

52

6.1.2 Short-term hydrothermal scheduling : NOx

emission minimization objective

58

6.1.3 Short-term hydrothermal scheduling : SO2


64

6.1.4 Short-term hydrothermal scheduling : CO2


70

6.1.5 Multi-objective, short-term hydrothermal

scheduling : determining the best compromise

solution considering 24 sub-intervals

76




80




82


scheduling with random generation of weight

vectors : trial -1

84

6.1.8.1 Generation of random weight vector of

four weights

84


scheduling with random generation of weight

vectors : trial -2

88

6.1.10 Effectiveness of search space reduction

technique

ix

91


scheduling with 161 user-fed weight vectors

93


scheduling with random generated weight

vectors

111

6.2 Advanced Investigations 115

6.2.1 Multi-objective, short term hydrothermal

scheduling with genetically generated weight

vectors for 24 sub-intervals- trial 1

115



vectors for 24 sub-intervals- trial 2

122



vectors for 72 sub-intervals

129


scheduling : optimization interval –execution

time characteristic

139


scheduling : cost –total emission characteristic

141


scheduling : objective function values

determined by the three different methods

144

6.2.6.1 Determining objective function values

by N-R method

x

144


by a genetic search

146


by a random search – trial 1

148


by a random search – trial 2

150

6.2.6.5 Comparison of genetic and random

search methods

151

6.2.6.6 Comparison of execution time 152

7 CONCLUSION AND SCOPE FOR FUTURE WORK 154-155

7.1 Conclusion 154

7.2 Scope for future work 155

REFERENCES

LIST OF PUBLICATIONS

xi

LIST OF TABLES

Table

No.

Title of Table Page

no.

4.1 Search space reduction in Generation 1 41



5.1 Features of hydro units 48

5.2 Features of thermal units 49

5.3 Loss coefficients 50

5.4 Features of thermal units 51

5.5 Loss coefficients 51

6.1a Objective function values in optimization sub-intervals 53

6.1b λ, power allocations and losses in optimization sub-intervals 55







6.5a Weight vectors and objective function values in the


77

6.5b Membership function values and fitness 77

6.6 Optimum solution 77

xii

LIST OF TABLES CONTINUED

Table

No.

Title of Table Page

no.



80





82





85





89



6.15 Effectiveness of search space reduction technique 91



94



6.18a Optimum solution –trial 1 112

6.18b Optimum solution –trial 2 112

xiii


Table

No.

Title of Table Page

no.

6.18c Optimum solution –trial 3 113

6.18d Optimum solution –trial 4 113

6.18e Optimum solution –trial 5 114

6.19 Highest fitness in 10 generations 116

6.20 Set of weight vectors and fitness values in the last generation 117

6.21 Objective function values corresponding to the weight

vectors given in Table 6.20

118

6.22 Fuzzy membership functions corresponding to the weight


119

6.23 Power demand, power allocations , losses and λ in each sub-

interval corresponding to the highest fitness solution

120





125



126



127





132

xiv


Table

No.

Title of Table Page

no.



133



134

6.34 Optimization interval- execution time 139

6.35 Table 6.31 arranged in the ascending order of cost 142

6.36a Weight vectors and fitness values 145

6.36b Objective function values 145

6.36c Fuzzy membership functions 145

6.37 No. of iterations, water withdrawals, coal-consumption

corresponding to each weight vector

146






147






149



xv


Table

No.

Title of Table Page

no.




151

6.44 Execution time for the three methods 152

xvi

LIST OF FIGURES

Figure

No.

Title of Figure Page

no.

3.1 A commonly used algorithm 18

4.1 The proposed algorithm 21

4.2 Two sample strings, each divided into ‘ob’ sub-strings 22

4.3Determining objective function values in the optimization

interval24

4.4 First ‘ob’ weight vectors in the population 25

4.5 A typical crossover with sites towards left half of the string 27

4.6a Elements of Hessian 32

4.6b Elements of Hessian 32

4.6c Elements of Hessian 33

4.6d Elements of Hessian 33

4.7 Elements of Δz 34

4.8 Elements of Jacobian 34

4.9 Variation of error 39

4.10 Sample crossover restricted to right half 40

4.11 Search space reduction in 3 generations 43

4.12 Desired output surface for the FIS 44

4.13 Generating a random population 45

6.1 Variation of highest fitness value in ten generations 116

6.2 Power demand, total thermal and total hydro generations

corresponding to Table 6.23

121

xvii

LIST OF FIGURES CONTINUED

Figure

No.

Title of Figure Page

no.



corresponding to 24 sub-intervals

128



corresponding to 72 sub-intervals

138

6.7 Optimization interval-execution time characteristic 140

6.8 Cost-total emission characteristic 143

xviii

LLIISSTT OOFF SSYYMMBBOOLLSS

Symbol Expansion

ob number of objectives

n total number of thermal units

n1 number of coal-constrained thermal units

h number of hydro units

a1i , b1i, c1i cost coefficients of ith thermal unit

a2i , b2i, c2i NOx emission coefficients of ith thermal unit

a3i , b3i, c3i SO2 emission coefficients of ith thermal unit

a4i , b4i, c4i CO2 emission coefficients of ith thermal unit

wi weight assigned to ith objective

PD k Power Demand during kth sub-interval

PLk Real power Loss during kth sub-interval

Pik Power output of ith unit during kth sub-interval

Pimax, Pi

min maximum/minimum power output of ith unit

I total number of sub-intervals

dn+i,k discharge of ith hydro unit or (n+i)th unit during kth sub-int.

Van+i allocated volume of water for ith hydro unit or (n+i)th unit

µ ,φ Lagrange multipliers

λk Lagrange multiplier for sub-interval k

αn+i, βn+i, γn+i discharge coefficients of ith hydro unit or (n+i)th unit

QTi Total coal allocation for ith coal-constrained thermal unit

α1i, β1i, γ1i coal discharge coefficients of ith coal constrained thermal unit

H Hessian

J Jacobian

xix

LLIISSTT OOFF SSYYMMBBOOLLSS CCOONNTTIINNUUEEDD

Symbol ExpansionΔz vector of mismatches

R symbol used for currency

xx

LIST OF ABBREVIATIONS

Abbreviation Expansion

GA Genetic Algorithm

NP Non-deterministic Polynomial

CPU Central Processing Unit

SA Simulated Annealing

TS Tabu Search

GLS Guided Local Search

ILS Iterated Local Search

SS Scatter Search

GRASP Greedy Randomized Adaptive Search Procedure

N-R Newton-Raphson

1

CHAPTER 1

INTRODUCTION

Electric power plays a major role in modern society and in the

development of various sectors of economy. This trend has led to increase in

number of power stations, increase in number of transmission lines, expansion of

existing power stations and many other additions to power system. Even today,

major portion of energy required for the whole world is supplied by fossil-fired

units and hence the pollution caused by them is a matter of concern. Hence to meet

the future demand, immediate attention is to be paid to energy production from

hydro, fossil and nuclear resources in the best possible technical, economic and

environmental conditions.

In the present set-up of large power systems with mainly hydro and

thermal stations, the integrated operation of them is inevitable with due

consideration to economic and environmental aspects. The operating cost of a

thermal plant is high but its capital cost is low and the operating cost of a hydro

plant is low but its capital cost is high. A hydro plant has higher reliability, greater

speed of response and can take up fluctuating loads. Hence it is economical and

convenient to have both thermal and hydro plants on the same grid.

As far as the aspect of pollution is concerned, fossil-fired plants pollute

air, soil and water. The combustion of fossil fuels gives rise to particulate

materials and gaseous pollutants apart from discharge of heat to water courses.

The contamination of air is contributed mainly by the gaseous pollutants such as

2

oxides of Carbon, Nitrogen and Sulphur. The usual practice is to reduce offensive

emission through post-combustion cleaning system or using a fuel with a low

emission potential. Hence in addition to achieving minimum operating cost it has

become necessary to minimize the emission of gaseous pollutants in the operation

of fossil-fired units. Thus the goal of multi-objective hydrothermal scheduling is

not only to minimize the system operating cost, but also to minimize the quantity

of gaseous pollutants emitted by thermal units.

The problem of multi-objective optimization comes up when hydro and

thermal units are to be operated together with simultaneous minimization of cost

and emission of pollutants. Decades of research in various disciplines has

contributed to many solution methods for multi-objective optimization problems.

There are two general approaches to multi-objective optimization. One approach is

to combine various objective functions into a single composite objective function

and the other is to consider one objective as prime and the remaining as

constraints. Difficulty in accurately determining the weights assigned to objectives

is the drawback of the first approach. The second approach has the major

drawback of establishing constraining values, which can be rather arbitrary. The

first approach is often preferred due to its superiority over the other.

In most of the real life problems, objectives are conflicting in nature. In such

cases, optimization with respect to a single objective yields results which are

unacceptable to other objectives and hence simultaneously minimizing

/maximizing all the objectives is impossible. A way to solve the multi-objective

problem is to find a set of solutions, referred to as Pareto-optimal solutions, each

solution of which satisfies all the objectives to a certain degree, without being

3

dominated by any other solution. But, identifying all feasible, non-dominated

solutions is practically difficult due to its size. If all the objective functions are for

minimization, a feasible solution ‘x’ is said to dominate another feasible solution

‘y’, if and only if, zi(x)≤ zi(y) i=1,2,…,ob and zj(x)<zj(y) for at least one objective

function j . Hence a set of weight combinations is provided by the user and the

best compromise solution is identified form the set of Pareto-optimal solutions

obtained. Multi-objective hydrothermal scheduling problem is a non-linear

optimization problem with equality and inequality constraints. Traditionally this

problem can be solved by Method of Lagrange Multipliers, where the objective is

to minimize fuel cost and emission of pollutants subject to a set of constraints.

1.1 GENETIC ALGORITHM

The concept of GA was developed by Holland and his colleagues in

1960s. Genetic algorithm is inspired by theory of evolution explaining the origin

of species. In nature weak and unfit are faced with extinction by natural selection.

The strong ones have better opportunity to pass their genes to future generations

via reproduction. In the long run, species carrying the correct combination in their

genes become dominant in their population. Sometimes during the slow process of

evolution, random changes may occur in genes.

In GA terminology, a solution vector is called an individual or a

chromosome. Chromosomes are made up of discrete units called genes. Each gene

controls one or more features of the chromosome. In the original implementation

4

of GA by Holland, genes are assumed to be binary digits. In later implementations,

more varied gene types have been introduced. Normally, a chromosome

corresponds to a unique solution in the solution space. This requires a mapping

mechanism between the solution space and the chromosomes. This mapping is

called encoding. In fact, GA works on the encoding of a problem, not on the

problem itself. GA operates with a collection of chromosomes, called a

population. The population is randomly initialized. As the search evolves, the

population includes more fit solutions, and eventually it converges, which means

that it is dominated by a single solution. Holland also presented a proof of

convergence to the global optimum where chromosomes are binary vectors. GA

uses two operators to generate new solutions from existing ones, crossover and

mutation. The crossover operator is the most important operator in GA. In

crossover, genetically two chromosomes, called parents, are combined together to

from two new chromosomes, called off-springs. The parents are selected from

existing chromosomes in the population with preference towards fitness so that

offspring is expected to inherit good genes. By iteratively applying crossover

operator, genes of good chromosomes are expected to appear more frequently in

the population, eventually leading to convergence to an overall good solution. The

mutation operator introduces random changes into characteristics of the

chromosome. Mutation is generally applied at the gene level. In typical GA

implementations, the mutation rate is very small and depends on the length of the

chromosome. Therefore the new chromosome will not be much different from the

original one. Mutation plays a critical role in GA. Mutation reintroduces the

genetic diversity back into the population and helps the search to escape from

local optima. Reproduction involves selection of chromosomes for the next

5

generation. In most general cases, the fitness of an individual determines the

probability its survival to the next generation. There are different selection

procedures in GA depending on how the fitness values are used. Roulette-wheel

selection and tournament selection are the most popular selection procedures.

1.2 FUZZY LOGIC

Fuzzy logic is a form of multi-valued logic derived from fuzzy set

theory to deal with reasoning that is approximate rather than precise. In contrast

with crisp logic, fuzzy logic variables may have a truth value that ranges between

0 and 1 and is not constrained to the two truth values of classic propositional logic.

Furthermore, when linguistic variables are used, these degrees may be managed by

specific functions.

Fuzzy logic emerged as a consequence of the 1965 proposal of fuzzy

set theory by Lotfi Zadeh. Though fuzzy logic has been applied to many fields,

from control theory to artificial intelligence, it still remains controversial among

many statisticians, who prefer Bayesian logic, and some control engineers, who

prefer traditional two-valued logic.

Fuzzy logic and probabilistic logic are mathematically similar, both

have truth values ranging between 0 and 1, but conceptually different, due to

different interpretations. Fuzzy logic corresponds to degrees of truth, while

probabilistic logic corresponds to probability or likelihood; as these differ, fuzzy

logic and probabilistic logic yield different models of the same real-world

situations.

6

Both degree of truth and probability, range between 0 and 1, and hence

may seem similar at first. It is essential to realize that fuzzy logic uses truth

degrees as a mathematical model of vagueness while probability is a mathematical

model of randomness.

There are many research papers in which fuzzy set theory has been

applied to power system operation e.g. for load forecasting and unit commitment.

In certain experiment, forecasted hourly load has been taken as fuzzy set notations

and has been proved to be superior to the conventional practice of assuming that

hourly loads are exactly known and there exists no error in forecasted loads. There

have been applications of fuzzy theory in optimal generation dispatch also where

membership functions have been introduced to measure generation-load balance,

fuel cost and time to stay in a zone. The value of a membership function can be

anywhere from 0 to 1 and this range is what makes it different from the crisp set.

The closer the values of the membership function to 1, the more that member

belongs to the set or group and hence a fuzzy set has no sharp boundary.

1.3 COMBINATORIAL OPTIMIZATION

Combinatorial optimization problems are intriguing because they are

often easy to state but difficult to solve. Many of the problems arising in

applications are NP-hard, that is, it is strongly believed that they cannot be solved

to optimality within polynomially bounded computation time. Hence to practically

solve large instances one often has to use approximate methods which return near

optimal solutions in a relatively short time. Algorithms of this type are loosely

called heuristics. There are two basic strategies for heuristics: divide and conquer

7

and iterative improvement. In the first, one divides the problem into sub-problems

of manageable size, and then solves the sub-problems. The solutions to the sub-

problems must then be patched back together. For this method to provide very

good solutions, the sub-problems must be naturally disjoint, and the division made

must be an appropriate one, so that errors made in patching do not offset the gains

obtained in applying more powerful methods to the sub-problems. In the iterative

improvement, one starts with the system in a known configuration. A standard

rearrangement operation is applied to all parts of the system in turn, until a

rearrangement configuration that improves the cost functions is discovered. The

rearranged configuration then becomes the new configuration of the system, and

the process is continued until no further improvements can be found. Iterative

improvement consists of a search in this co-ordinate space for rearrangement steps

which lead downhill. Since this search usually gets stuck in a local but not a global

optimum, it is customary to carry out the process several times, starting from

different randomly generated configurations and save the best result. They often

use some problem-specific knowledge to either build or improve solutions.

Recently many researchers have focused their attention on a new class of

algorithms, called metaheuristics. A metaheuristic is a set of algorithmic concepts

that can be used to define heuristic methods applicable to a wide set of different

problems. The use of metaheuristics has significantly increased the ability of

finding very high quality solutions to hard, practically relevant combinatorial

optimization problems in a reasonable time.

Combinatorial optimization problems involve finding values for

discrete variables such that the optimal solution with respect to a given objective

function is found. Many optimization problems of practical and theoretical

8

importance are of combinatorial nature. A combinatorial optimization problem is

either maximization or a minimization problem which has a set of problem

instances. Formally, an instance of a combinatorial optimization problem is a

triple (S, f, Ω), where S is a set of candidate solutions, f is the objective function,

and Ω is a set of constraints. The solutions belonging to the set of candidate

solutions that satisfy the constraints Ω are called feasible solutions. The goal is to

find a globally optimal feasible solution.

A straightforward approach to the solution of combinatorial

optimization problems would be exhaustive search, that is, the enumeration of all

possible solutions and the choice of the best one. Unfortunately, in most cases,

such a naïve approach becomes rapidly infeasible because the number of possible

solutions grows exponentially with the instance size n.

1.4 POPULAR METAHEURISTICS

The world of metaheuristics is rich and multifaceted and a number of

metaheuristics are available in the literature. Some of the best known and most

widely applied metaheuristics are given below. It is interesting to note that, for all

metaheuristics, there is no general termination criterion. In practice a number of

rules of thumb are used: the maximum CPU time elapsed, the maximum number

of solutions generated, the percentage deviation from a lower/upper bound from

the optimum, the maximum number of iterations without improvement in quality

are examples of such rules.

9

· Simulated Annealing: Simulated Annealing (Cerny, 1985; Kirkpatrick et

al., 1983) is inspired by an analogy between the physical annealing of

solids(crystals) and combinatorial optimization problems. In the physical

annealing process a solid is first melted and then cooled very slowly,

spending a long time at low temperature, to obtain a perfect lattice structure

corresponding to minimum energy state. SA transfers this process to local

search algorithms for combinatorial optimization problems.

· Tabu Search: Tabu Search (Glover, 1989, 1990; Glover & Laguna, 1997)

relies on the systematic use memory to guide the search process. TS uses a

local search, that at every step, makes the best possible move from current

solution to a neighbor solution even if the new solution is worse than the

current one; in the latter case, the move that least worsens the objective

function is chosen. To prevent the local search from immediately returning

to a previously visited solution TS can explicitly memorize recently visited

solutions and forbid moving back to them.

· Guided Local Search: One alternative possibility to escape from local

optima is to modify the evaluation function while searching. Guided local

search (Voudouris 1997; Voudouris & Tsang, 1995) is a metaheuristics that

makes use of this idea. It uses an augmented function which consists of the

original objective function plus additional penalty terms associated with

each solution feature. GLS uses the augmented function for choosing local

search moves until it gets trapped in a local optimum. At this point, a utility

value is computed for each feature. The utility values are scaled to avoid

10

the same features from getting penalized over and over again and the search

trajectory from becoming too biased. Then, the penalties of the features

with maximum utility are incremented and the augmented function is

adapted by using the new penalty values. Last the local search is continued

which will no longer be optimal with respect to the new augmented

function.

· Iterated Local Search: Iterated Local Search (Lourenco et al. 2002;

Martin, Otto & Felten, 1991) is a simple and powerful metaheuristic, whose

working principle as follows. Starting from an initial solution s, a local

search is applied. Once the local search is stuck, the locally optimal

solution s* is perturbed by a move in a neighborhood different from the one

used by the local search. This perturbed solution s* is the new starting

solution for the local search that takes it to the new local optimum s*’.

Finally, an acceptance criterion decides which of the two locally optimal

solutions to select as a starting point for the next perturbation step. The

main motivation for the ILS is to build a randomized walk in a search space

of the local optima with respect to some local search algorithm.

· Greedy Randomized Adaptive Search Procedure: Greedy Randomized

Adaptive Search Procedures (Feo & Redende, 1989, 1995) randomize

greedy construction heuristics to allow the generation of a large number of

different starting solutions for applying a local search. GRASP is an

iterative procedure which consists of two phases, a construction phase and a

local search phase. In the construction phase a solution is constructed from

search, adding one solution component at a time. At each step of the

construction heuristic, the solution components are ranked according to

some greedy function and the number of best-ranked components are

11

included in a restricted candidate list; typical ways of deriving the restricted

candidate list are either to take the best γ% of the solution components or to

include all solution components that have a greedy value within some δ%

of the best-rated solution component. Then, one of the components of the

restricted list is chosen randomly, according to a uniform distribution. Once

a full candidate solution is constructed, this solution is improved by a local

search phase.

· Evolutionary Computation: Evolutionary computation has become a

standard term to include problem-solving techniques which use design

principles inspired from models of the natural evolution of species.

Historically, there are three main algorithmic developments within the field

of EC: evolution strategies (Rechenberg, 1973; Schwefel, 1981),

evolutionary programming (Fogel et al., 1966) and genetic algorithms

(Holland 1975, Goldberg 1989). Common to these approaches is that they

are population based algorithms that use operators inspired by population

genetics to explore the search space. The most typical genetic operators are

reproduction, crossover and mutation.

· Scatter Search: The central idea of scatter search (SS), first introduced by

Glover(1977), is to keep a small population of reference solutions, called a

reference set, and to combine them to create new solutions. A basic version

of SS proceeds as follows. It starts by creating a reference set. This is done

by first generating a large number of solutions using diversification

generation method. Then these solutions are improved by a local search

procedure. From these improved solutions the reference set is built. The

12

solutions to be put in the reference set are selected by taking into account

their solution quality and their diversity. Then, the solutions in the

reference set are used to build a set c-cand of subsets of solutions. The

solutions in each subset, which can be of size 2 in the simplest case, are

candidates for combination. Solutions within each subset of c-cand are

combined; each newly generated solution is improved by local search and

possible replaces one solution in the reference set. The process of subset

generation, solution combination and local search is repeated until the

reference set does not change anymore.

1.5 MULTI-OBJECTIVE GA

Being a population-based approach, Genetic Algorithms are well-suited

to solve multi-objective optimization problems. A genetic, single-objective GA

can be modified to find a set of multiple non-dominated solutions in a single run.

The ability of GA to simultaneously search different regions of a solution space

makes it possible to find a diverse set of solutions for difficult problems with non-

convex, discontinuous and multi-modal solution spaces. The crossover operator of

GA may exploit structures of good solutions with respect to different objectives to

create new non-dominated solutions in unexpected parts of the Pareto front. In

addition, most multi-objective Genetic Algorithms do not require the user to

prioritize, scale or weigh objectives. Therefore GA has been the most heuristic

approach to multi-objective design and optimization problems.

13

The first multi-objective GA, called vector evaluated GA (or VEGA),

was proposed by Schaffer. Afterwards, several multi-objective evolutionary

algorithms were developed including Multi-objective Genetic Algorithm(MOGA),

Niched Pareto Genetic Algorithm(NPGA), Weight-based Genetic

Algorithm(WBGA), Random Weighted Genetic Algorithm(RWGA), Non-

dominated Sorting Genetic Algorithm(NSGA), Strength Pareto Evolutionary

Algorithm(SPEA), Pareto-Archived Evolutionary Strategy(PAES), Pareto

Envelope-based Selection Algorithm(PESA), Region-based Selection in

Evolutionary Multi-objective Optimization(PESA-II), Fast Non-dominated Sorting

Genetic Algorithm(NSGA-II), Multi-objective Evolutionary Algorithm(MEA),

Rank Density- based Genetic Algorithm(RDGA) and Dynamic Multi-objective

Evolutionary Algorithm(DMOEA). Note that although there are many variations

of multi-objective GA in the literature, these cited GA are well-known and

credible algorithms that have been used in many applications and their

performances were tested in several comparative studies.

Several survey papers have been published on evolutionary multi-objective

optimization. Generally, multi-objective GA differs based on their fitness

assignment procedure, elitism or diversification approaches. Most survey papers

on multi-objective evolutionary approaches introduce and compare different

algorithms. Many researchers who applied multi-objective GA to their problems

have preferred to design their own customized algorithms by adapting strategies

from various multi-objective GA. This observation is another motivation for

introducing the components of multi-objective GA rather than focusing on several

algorithms.

14

CHAPTER 2

LITERATURE SURVEY

In recent years there has been an increase in research in multi-objective

optimization methods. Decisions with multiple objectives are quite prevalent in

real life problems. Researchers from a wide variety of disciplines have contributed

to the solution of multi-objective optimization problems. There are many

traditional methods for finding the solution of multi-objective problems, but lately,

research has been more oriented towards heuristic search methods as these

methods reduce problem complexity considerably. Some recent works are

discussed below.

A new model to deal with the short-term generation scheduling

problem for hydrothermal systems was proposed by Esteban Gil (2003). The

model simultaneously handled the sub-problems of short-term hydrothermal

coordination, unit commitment and economic load dispatch. Future cost curves of

hydro generation, obtained from long and mid -term models, had been used to

optimize the amount of hydro energy to be used during the scheduling horizon.

The model was implemented using the Genetic Algorithm.

M.A Abido (2003) presented a new multi-objective evolutionary

algorithm for environmental/economic load dispatch. The problem was formulated

as a nonlinear, constrained multi-objective optimization problem. The Strength

Pareto Evolutionary Algorithm based approach was proposed to handle

economic/environmental dispatch with competing and non-commensurable

15

objectives. The approach also employed a diversity preserving mechanism to

overcome premature convergence.

M.Basu (2006) developed an algorithm for economic-emission load

dispatch for thermal plants with non-smooth fuel cost and emission level functions

in co-ordination with fixed head hydro units through an interactive fuzzy

satisfying method. Assuming that ‘decision maker’ had fuzzy goals for each of the

objective functions; the multi-objective load dispatch problem was transformed

into a mini-max problem, which was then handled by Simulated Annealing

technique. The solution methodology offered global optimum or near-global

optimum non-inferior solutions for the ‘decision maker’.

A tutorial on multi-objective optimization using genetic algorithms was

presented by Abdulla Konak (2006) in which a variety of meta-heuristic search

methods such as Vector Evaluated Genetic Algorithm, Niched Pareto Genetic

Algorithm, Strength Pareto Evolutionary Algorithm etc. were discussed along

with their design issues and components.

A short-term hydrothermal scheduling algorithm based on a hybrid

evolutionary and a conventional technique was proposed by Nallasivan (2006). In

the algorithm the thermal units on the system were represented by an equivalent

unit. The power balance constraints, total water discharge constraints, reservoir

volume constraints and constraints on operating limits of the equivalent thermal

and hydro units were taken into account. The proposed method was developed in

such a way that a standard adaptive evolutionary programming method was acting

as the base level search and it directed the search towards the optimum region.

16

A heuristic search technique based on binary successive approximation

using stochastic models was proposed by J.S Dhillon(2007). They considered five

objectives; economy, emission of SO2, emission of NOx, emission of CO2 and

variance of power. Normally the input system data is believed to be deterministic,

but practically it is bound to vary depending on the uncertainties and hence it is

worthwhile to assume the system data to be uncertain for a more realistic

approach. The new algorithm was tested on a small system and realistic results

were obtained.

Operation planning of integrated hydrothermal and natural gas system

was proposed by Clodomiro (2007). In their work the natural gas network model

including storages and pipelines were integrated with hydrothermal systems to

optimize short term operation of both the systems simultaneously. The model

considered the constraints at the hydrothermal system, natural gas extraction,

natural gas storage operation and pipeline. Result analysis of a case study showed

that the interdependency could be affected by physical characteristics and

capabilities of gas pipelines and electric power systems.

An algorithm based on an evolutionary particle swarm optimization for

solving the pumped-storage scheduling problem was proposed by Po-Hung Chen

(2008). The proposed approach combined a basic particle swarm optimization

with binary encoding/decoding techniques as well as a mutation operation. The

binary encoding/decoding techniques were adopted to model the discrete

characteristics of a plant. The mutation operation was applied to accelerate

convergence and escape local optima.

17

A short-term hydro scheduling study was carried out by J.P.S Catalao

(2008,2009) on a system with seven head dependent cascaded reservoirs with the

goal to maximize the value of total hydroelectric power generation throughout the

time horizon considered, satisfying all physical and operational constraints, and

consequently to maximize the profit of the hydroelectric utility from selling

energy into the market.

A co-evolutionary algorithm based on the Lagrangian method (Ruey-

Hsun Liang, 2009) was proposed for hydrothermal generation scheduling in which

a genetic algorithm was successfully incorporated into the Lagrangian method.

The genetic algorithm searched out the optimum using multiple-path techniques

and had the ability to deal with continuous and discrete variables.

Decades of research in various disciplines had contributed to solution

methods for multi-objective optimization problems. More literature is available in

the field of multi-objective optimization and heuristic search techniques, a few of

which are mentioned in the reference.

18

CHAPTER 3

OBJECTIVE OF THE INVESTIGATION

3.1 MULTI-OBJECTIVE HYDRO THERMAL SCHEDULING

A commonly used algorithm for multi-objective, short-term

hydrothermal scheduling is shown in Figure 3.1. A set of weight vectors are fed in

and for each weight vector objective function values in the optimization interval

are computed. Furthermore, fuzzy membership functions are evaluated for the

objective function values corresponding to each weight vector and the best

compromising solution is picked from the non-inferior solution set.

Figure 3.1 A commonly used algorithm

19

Determining objective function values in an optimization interval is the most

important and time consuming sub-process in the algorithm. Many methods are

used for this. But each method has its own merits and drawbacks with respect to

computational complexity and time requirements.

The major weaknesses of this approach are the following:

1. The best compromising solution is identified from the set of non-inferior

solutions corresponding to the weight combinations fed in, but there can be

many other weight combinations that can yield better solutions than the one

obtained.

2. To get a solution with a high overall satisfaction, many weight

combinations need to be tested.

3. Objective function values in an optimization interval can be determined by

many methods of which no method is simple as well as fast.

3.2 OBJECTIVE OF THE INVESTIGATION

The objective of this investigation is to modify the above algorithm incorporatingthe following:

1. To find a high fitness weight combination in minimum time and attempts.

2. To introduce a method for determining objective function values in an

optimization interval which is simple as well as fast.

3. To make the scheduling problem more universal by imposing coal

constraint on the composite objective function.

20

As we know, coal resources are depleting at a high rate. Certain power stations

already started experiencing coal shortage and the problem is likely to intensify in

future. Hence the periodical availability of coal is to be distributed over the period

depending on energy forecast data. The new constraint being introduced in this

algorithm is stated as: ‘quantity of coal allocated to a thermal unit for the

optimization interval is to be fully utilized’. This is similar to the hydro constraint.

21

CHAPTER 4

PROPOSED INVESTIGATION

Four objectives are considered in the investigation, which are

minimization of cost and gaseous pollutants NOx, SO2 and CO2 subject to the

constraints: power balance in sub-intervals, generation limits, hydro constraints,

weight constraint and coal constraints. A new algorithm is proposed for the

solution of multi-objective, short-term hydrothermal scheduling problem which is

shown in Figure 4.1.

Figure 4.1 The proposed algorithm

22

4.1 MAJOR STEPS IN THE PROPOSED ALGORITHM

4.1.1 GENETIC GENERATION OF WEIGHT VECTORS

Weight combinations are infinite and feeding in all of the weight

combinations is almost impossible. Moreover, there are many weight

combinations that give solutions with fitness values falling into a very narrow

margin. Hence this is a search for global optimum. Genetic algorithms are ideally

suited for this occasion, as they are efficient global optimum locators.

L binary strings are generated, each with a length of l bits. Each

string is split into ob sub-strings of length l/ob bits and hence, a combination of

ob weights, or a weight vector, can be derived from a string. Two sample stings

are shown in Figure 4.2.

1 2 ob

1 0…0 1 | 1 0 …1 0 | …………….| 1 0…0 0

0 0…1 0 | 0 1 …1 1 | …………….| 1 1…1 0

Figure 4.2 Two sample strings, each divided into ‘ob’ sub-strings

Each sub-string of a binary string is converted to its equivalent decimal

value. Each decimal value is divided by sum of ob decimal values. This results in

a combination of ob weights with a sum of 1.0 satisfying the weight constraint. In

similar way the remaining weight vectors are generated.

23

4.1.2 DETERMINING OBJECTIVE FUNCTION VALUES IN THE OPTIMIZATION

INTERVAL

Objective function values in an optimization interval can be determined

by three different methods which are the following:

1. Newton-Raphson method

2. A method using a genetic search

3. A method using a random search

The common flow diagram for second and third methods is shown in

Figure 4.3. For the third method, instead of a genetic population a random

population is used.

24

Figure 4.3 Determining objective function values in the optimization interval

25

4.1.3 DETERMINING FITNESS OF WEIGHT VECTORS

When a set of L weight vectors are generated genetically, the first ob

weight vectors should be pre-defined, of the format shown in Figure 4.4. This is

necessary for computing Fimax , Fi

min (i=1,2,…,ob) , the extreme values of objective

functions.

1 2 3 … ob

1 1 0 0 .. 0

2 0 1 0 .. 0

3 0 0 1 .. 0

.. .. .. .. .. ..

ob 0 0 .. .. 1

Figure 4.4 First ob weight vectors in the population

The membership function an objective is evaluated using Eq. (4.1).

max max min min max) / where( ) ( ( )i i i i i i i im F F F F F F F F- -= £ £ (4.1)

min( ) 1i i im F for F F= £ for i=1,2,.....,obmax( ) 0i i im F for F F= ³ for i=1,2,.....,ob

The fitness function of a weight combination is evaluated using Eq. (4.2).

1( ) /

ob

ii

fitness m F ob=

=å (4.2)

26

4.1.4 DISCARDING DOMINATED SOLUTIONS

Each solution is checked for dominancy. A feasible solution x is said to

dominate another feasible solution y, if and only if , Fi(x) ≤ Fi(y) for i=1,2,…,ob

and Fj(x) < Fj(y) for at least one objective function j. A solution is Pareto optimal

if it is not dominated by any other solution in the solution space. A Pareto optimal

solution cannot be improved with respect to any objective without worsening at

least one other objective.

A weight vector corresponding to a dominated solution is replaced with

another weight vector corresponding to a non-dominated solution having the

nearest fitness value.

4.1.5 CONVERGENCE CRITERIA

The process can be terminated after a definite number of generations.

Alternately, if the improvement in the highest fitness value over a certain number

of generations is marginal, the process can be terminated.

4.1.6 MODIFYING THE GENETIC POPULATION

After computing fitness of each weight vector in the population and

checking for dominancy, the population is sorted in the descending order of

fitness. The population is then divided into two halves of L/2 individuals. First

string from the upper half and a random one from lower half are selected for

27

mating. Two random numbers between 1 and l/ (2*ob), are generated, which

provides the crossover site as well as the number of bits to be replaced. This is a

search for global optimum as there are many different weight combinations with

fitness values lying in a very narrow range. The crossover site is restricted mainly

to the left half of a sub-string. Only the lower half sub-strings are modified and the

upper half sub-strings are unchanged. This retains the high fitness individuals in

the upper half. No selection is used in this process. Mutation probability is kept

higher to maintain higher genetic diversity.

A typical crossover is shown in Figure 4.5. Crossover is happening

between ob pairs of sub-strings. Second mating pair consists of second string from

the upper half and a random one from lower half, other than the one selected

already. Many low fitness individuals in the lower half become stronger after

genetic operations and a few very efficient ones will be pushed to the upper half

when the population is sorted next.

1 0 1 …1 0 | 0 1 0 …1 0 | …….. | 0 1 0 ….1 1 upper half string

0 0 1 …0 0 | 1 1 0 …1 1 | …….. | 1 0 0 ….0 1 lower half string

Figure 4.5 A typical crossover with sites towards left half of the string

28

4.2 PROBLEM FORMULATION

4.2.1 ASSUMPTIONS/APPROXIMATIONS

1. All generators are operating at all times.

2. Equation for transmission losses is approximated to PL =1 1

n h n h

i i j ji j

P B P+ +

= =å å MW

3. The optimization sub-interval is taken as one hour.

4. The Glimn-Kirchmayer model represents all hydro units.

5. Fuel cost function is approximated to a quadratic of power generation,

neglecting valve-point effects or neglecting the sinusoidal function.

6. Emission functions and coal-discharge function are also expressed as quadratic

of power generation.

The above are widely followed in most of the simulation studies. Unit

commitment problem is not taken up in this study and hence all generators are

assumed to be operating all times. Transmission losses can be approximated as

indicated in second statement above as it gives reasonably accurate results. For

almost all simulation studies optimization interval is taken as 1 hour. Glimn-

Kirchmayer model is widely preferred in hydrothermal scheduling to represent a

hydro unit and hence used in this study. Fuel cost function is normally assumed as

a quadratic neglecting the sinusoidal function.

29

4.2.2 PROBLEM OBJECTIVE

The objective is to minimize

ob

m mm=1

w Få (4.3)

where

2

1 1( )

I n

m mi mi ik miikk i

F = a P b P c+

= =

+åå

4.2.3 CONSTRAINTS

1. Power balance constraint1

0 ,n h

Dk Lk iki

P P P k = 1,2,..., I+

=

+ - =å

2. Generation limit constraint min maxi i iP P P i = 1,2,.....,n + h£ £

3. Weight constraint1

( )ob

m mm

w 1 w is zero / +ive=

=å

4. Hydro constraint

2 3

1 /

I

ik ai ik i ik i ik ik

d V i = n+1,...,n h where d P P m ha b g=

= + = + +å

5. Coal constraint

2 1 1 1

1 /

I

ik Ti 1 ik i ik i ik ik

Q Q k = 1,2,...,n where Q P P kg ha b g=

= = + +å

30

4.2.4 THE LAGRANGIAN

The approach used for solution of the above problem is to find the conditions for

optimality by application of calculus. By the method of Lagrange Multipliers, the

composite objective function given by Eq. (4.3), when augmented by equality

constraints, can be expressed as Eq. (4.4).

1 1

1 1 1 1 1 1 1 1 1( ) +

ob I n h I n h n h I n n

m m k Dk Lk ik i ik i i i ik i Tim k i k i n i n k i i

L w F P P P d Va Q Ql m m j j+ + +

= = = = = + = + = = =

= + + - + - -å å å å å å åå å

(4.4)

4.2.5 OPTIMALITY CONDITIONS

Differentiating Eq. (4.4) with respect to decision variables gives the optimality

conditions, numbering six, which are the following.

1( / ) ( / 1) ( / ) 0,

ob

m m ik k Lk ik i ik ik 1m

w F P P P Q P i =1,..n , k =1,...,Il j=

¶ ¶ + ¶ ¶ - + ¶ ¶ =å

(4.5)1

( / ) ( / 1) 0ob

m m ik k Lk ik 1m

w F P P P i = n +1,... n, k = 1,...,Il=

¶ ¶ + ¶ ¶ - =å

(4.6) k( / )+ ( / - )=0i ik ik Lk ikd P P P 1 i = n + 1,..n + h, k = 1,2,...Im l¶ ¶ ¶ ¶

(4.7)

10

n h

D k L k iki

P P P k = 1 , .. ..I+

=

+ - =å (4.8)

1

0I

ik ik

d V a i = n + 1 , .., n + h=

- =å (4.9)

11

0 ,I

ik T ik

Q Q i = 1,2 ,... n=

- =å (4.10)

31

4.3 OBJECTIVE FUNCTION VALUES IN THE OPTIMIZATION

INTERVAL

4.3.1 BY NEWTON-RAPHSON METHOD

Optimality conditions described by Equations (4.5) – (4.10) can be simplified to

Newton-Raphson equations and expressed as Eq. (4.11).

[ ] [ z]=[J]H D (4.11)

In Eq. (4.11) H is the Hessian, ∆z is the vector of mismatches and J is the

Jacobian.

4.3.1.1 Elements of H, ∆z and J

For illustrative purpose, a system having two thermal and two hydro

units is considered. Units 1 and 2 are thermal units; also unit 1 is coal-constrained.

Units 3 and 4 are hydro units. For two optimization sub-intervals, the size of the

Hessian is 13x13 and size of the Jacobian is 13x1. Figure 4.6a - 4.6d show the

Hessian in parts. Figure 4.7 shows the vector of mismatches and Figure 4.8 shows

the Jacobian.

Eq. (4.11) is solved iteratively for power allocations in all the

optimization sub-intervals. Limit violations of generators are taken care of by

applying Kuhn-Tucker conditions. If any generation violates the specified upper or

lower limit, the output of the particular generator is fixed as the limit it violated, it

is taken out of allocation process for the sub-interval and power demand in the

sub-interval is modified.

32

Once the solution is converged, the objective function values which are cost and

emission of NOx, SO2 and CO2 for all the sub-intervals are added to find the total

cost and emissions for the optimization interval.

1 2 3 4 5

1 Σ2 wm am,1+2λ1B1,1 +2φ1α11

2λ1B1,2 2λ1B1,3 2λ1B1,4 Σ2B1,jPj,1 -1

2 2λ1B2,1Σ2 wm am,2+2λ1B2,2

2λ1B2,3 2λ1B2,4 Σ2B2,jPj,1 -1

3 2λ1B3,1 2λ1B3,2 2µ3α3 +2λ1B3,3 2λ1B3,4 Σ2B3,jPj,1 -1

4 2λ1B4,1 2λ1B4,2 2λ1B4,3 2µ4α4 +2λ1B4,4 Σ2B4,jPj,1 -1

5 Σ2B1,jPj,1 -1 Σ2B2,jPj,1 -1 Σ2B3,jPj,1 -1 Σ2B4,jPj,1 -1 0

Figure 4.6a Elements of Hessian

6 7 8 9 10

6 Σ2 wm am,1+2λ2B1,1 +2φ1α11

2λ2B1,2 2λ2B1,3 2λ2B1,4 Σ2B1,jPj,2 -1

7 2λ2B2,1Σ2 wm am,2+2λ2B2,2

2λ2B2,3 2λ2B2,4 Σ2B2,jPj,2 -1

8 2λ2B3,1 2λ2B3,2 2µ3α3 +2λ2B3,3 2λ2B3,4 Σ2B3,jPj,2 -1

9 2λ2B4,1 2λ2B4,2 2λ2B4,3 2µ4α4 +2λ2B4,4 Σ2B4,jPj,2 -1

10 Σ2B1,jPj,2 -1 Σ2B2,jPj,2 -1 Σ2B3,jPj,2 -1 Σ2B4,jPj,2 -1 0

Figure 4.6b Elements of Hessian

33

1 2 3 4 5 6 7 8 9 10 11 12 13

11 2α3P3,1+β32α3P3,2+

β3

12 2α4P4,1+β4 2α4P4,2+β4

13 2α11P1,1+β11

2α11P1,2+β11

Figure 4.6c Elements of Hessian

11 12 13

1 2α11 P1,1+β11

2

3 2α3P3,1+β3

4 2α4P4,1+β4

5

6 2α11 P1,2+β11

7

8 2α3P3,2+β3

9 2α4P4,1+β4

10

11

12

13

Figure 4.6d Elements of Hessian

Note : In Figures 4.6a and 4.6b

1. in the element Σ2 wm am,1 m = 1,2,…,ob

2. in the element Σ2B1,jPj,1 -1 j=1,2,…,n+h

34

1 ∆P1,1

2 ∆P2,1

3 ∆P3,1

4 ∆P4,1

5 ∆λ1

6 ∆P1,2

7 ∆P2,2

8 ∆P3,2

9 ∆P4,2

10 ∆λ2

11 ∆µ3

12 ∆µ4

13 ∆φ1

Figure 4.7 Elements of ∆z

1 -∑(2wm am,1 P1,1+bm,1) –λ1 (∑2B1,jPj,1-1)-φ1(α11P1,1+β11) j=1,..,4, m=1,..,ob

2 -∑(2wm am,2 P2,1+bm,2) –λ1 (∑2B2,jPj,1-1) j=1,..,4, m=1,..,ob

3 -μ3(2α3P3,1+β3) –λ1 (∑2B3,jPj,1-1) j=1,..,4

4 -μ4(2α4P4,1+β4) –λ1 (∑2B4,jPj,1-1) j=1,..,4

5 -(PD1+PL1-∑Pi,1) i=1,..,4

6 -∑(2wm am,1 P1,2+bm,1) –λ2 (∑2B1,j Pj,2-1)-φ1(α11P1,2+β11) j=1,..,4, m=1,..,ob

7 -∑(2wm am,2 P2,2+bm,2) –λ2 (∑2B2,j Pj,2-1) j=1,..,4, m=1,..,ob

8 -μ3(2α3P3,2+β3) –λ2 (∑2B3,j Pj,2-1) j=1,..,4

9 -μ4(2α4P4,2+β4) –λ2 (∑2B4,j Pj,2-1) j=1,..,4

10 -(PD2+PL2-∑Pi,2) i=1,..,4

11 -(d31+d32-Va3) hydro unit 1

12 -(d41+d42-Va4) hydro unit 2

13 -(Q11+Q12-QT1) coal-constrained thermal unit 1

Figure 4.8 Elements of Jacobian

35

4.3.1.2 Computation of initial values

4.3.1.2.1 Initial values of power allocations and λk

For solving the problem iteratively, a set of initial values are required,

the computation of which is as follows.

For each hydro generator, the total quantity of water available for the

optimization interval is allocated for each sub-interval proportional to the power

demand in each sub-interval. The allocated volume of water for each sub-interval,

for each generator can be found using Eq. (4.12).

*1

, / ( )I

n i k n i Dk Dkk

V Va P P i =1,..,h, k =1,...,I+ +

=

= å (4.12)

Initial power allocation for each hydro generator, in each sub-interval

can be found by solving Eq. (4.13).

Now, 2 * *, + + =n i n i k n i n +i n +i n +i, kP P V i=1,..,h, k=1,...,Ia b g+ + + (4.13)

Let the remainder power demand be Pdk (k=1,…,I)

Now consider coal-constrained thermal generators. For each coal-

constrained thermal generator, the total quantity of coal available for the

optimization interval is allocated for each sub-interval proportional to Pdk as given

by Eq. (4.14).

* 11

, / ( )I

i k Ti dk dkk

Q Q P P i = 1,..,n , k = 1,...,I=

= å (4.14)

Initial power allocation for each coal-constrained thermal generator, in

each sub-interval can be found by solving Eq. (4.15).

36

2 * * , ,, + + =1i i k 1i i k 1i i k 1P P Q i = 1,..,n , k = 1,...,Ia b g (4.15)

The remainder power demand (Pbk , k=1,…,I), after allocating for coal-constrained

thermal generators, is optimally allocated among the remaining thermal generators

assuming losses equal to zero. The initial values of λk are computed using Eq.

(4.16).

2 1 11 1

( ( / )) / ( / )n n

k bki i

P t t 1 t k = 1,..., Il= =

= +å å (4.16)

1 1 1i 2 2 i ob o biw here t = 2w a + 2w a + ...........+ 2w a

2 1 1 i 2 2 i o b o b it = w b + w b + . . . . . . . . . . .+ w b

Substituting the values of λk in Eq. (4.17), initial values of power allocations forother thermal generators for each sub-interval are computed.

1 1( ) / 2 ,

ob ob

ik k m mi m mi 1m m

P w b w a i = n +1,....,n k = 1,2,...Il= =

= -å å (4.17)

4.3.1.2.2 Initial values of φ and µ

Initial values of φi , i=1,..n1 can be computed using Eq. (4.5), assuming

losses equal to zero. Simplification of Eq. (4.5) yields Eq. (4.18).

1 (

ob

i k m mi ik + mi 1i ik 1im

w (2 a P b )) / (2 P + )j l a b=

= -å (4.18)

Similarly the initial values of µi , i=n+1,..,n+h can be computed using

Eq. (4.7), assuming losses equal to zero. Simplification of Eq. (4.7) yields Eq.

(4.19).

k i i= /(2 + )i ikP i = n+1,..,n+hm l a b (4.19)

37

4.3.2 BY A METHOD USING A GENETIC SEARCH

Initial values of power allocations and Lagrange multipliers are to be

computed using equations described in sub-section 4.3.1.2.

Optimality conditions described by Equations (4.5) – (4.7) can be

simplified to equations for power output of coal-constrained thermal units, other

thermal units and hydro units respectively.

k k 1 11,1 1

=( - 2 ) / ( 2 2 2 )ob obn h

ik m mi ij jk i i m mi k ii i i 1j j im m

P w b B P w a B i=1,2,...nl l j b l ja+

- += ¹= =

- +åå å (4.20)

k k1,1 1

=( - 2 ) / ( 2 2 )ob obn h

ik m mi ij jk m mi k ii 1j j im m

P w b B P w a B i = n +1,.....,nl l l+

= ¹= =

- +åå å (4.21)

k k1,

=( - 2 ) / (2 2 )n h

ik i i ij jk i i k iij j i

P B P B i = n+1,n+ 2,...n+ hl m b l m a l+

= ¹- +å (4.22)

Figure 4.3 details the sub-algorithm. The power allocations for a sub-

interval, given by Equations (4.20) - (4.22), which satisfy power balance

constraint in the sub-interval, correspond to an optimum λk from a genetic

population. The search incorporates ‘search space reduction technique’. For

generation limit violations, the same procedure as mentioned in sub-section 4.3.1.1

is followed. At the end of an optimization interval water withdrawals for the hydro

units and coal-consumption for the coal-constrained thermal units are computed. If

hydro and coal constraints are not satisfied the associated Lagrange multipliers are

modified.

38

4.3.2.1 Search space reduction technique

This technique can be applied to a wide range of single variable

algebraic equations of higher order to locate their real roots by repeated search.

The gradient of the function to be optimized is equated to zero and its real roots

are found by exploring the search space several times. Searching for roots is found

to be easier than directly searching for the function optimum. An x-y plot of a 5th

order algebraic equation is shown in Figure 4.9. ‘y’ has values of opposite sign on

either side of a root in its vicinity, which can be taken advantage of, in locating

the roots of the equation in the search space.

Let there be a population of ‘N’ individuals, xopt be the optimum

solution looked for and xmax , xmin be the upper and lower limits of the search

range respectively. The decoded binary strings are mapped between xmax and xmin ,

normally a few above and the rest below xopt. Value of ‘y’ is computed for each

value of ‘x’. The values of ‘x’ corresponding to the minimum positive and

minimum negative values of ‘y’ will be the new xmax and xmin in the next

generation where the new xmax will be less than the present xmax and new xmin will

be greater than the present xmin thus reducing the search range. For example, 6

being one of the roots of the equation displayed in figure, when searched for it in

the range 5.5 - 6.5, one will get a positive y for 5.92 and a negative y for 6.21,

where 5.92 and 6.21 are two mapped values of x most close to 6 on either side.

Hence it can be concluded that the optimum value searched for, lies in between

5.92 - 6.21 and the search range can be reduced to 5.92 - 6.21 in the next

generation.

39

In case the gap between the new maximum/minimum and the optimum

is too small, there are chances for no number to be mapped to that region. In such

a case all the error values will be of the same sign and the search range is to be

reset to their initial values. Even in such a case the algorithm works faster than a

normal genetic search.

When the modified strings, which moved closer to the optimum during

the first round of genetic operations, are decoded and mapped to the new,

narrower search range, move closer to the optimum and the chance of a string

coinciding with optimum is more. This is the efficiency of search space reduction

technique and can be applied to a variety of equations with one variable. Also, this

technique is highly useful when search range is not definitely known. If the answer

searched for, is somewhere between 0 and 10000, then locating the same by a

conventional genetic search may take thousands of iterations consuming a lot of

time.

Figure 4.9 Variation of error

40

The initial values of λk in each sub-interval (k=1,2,…,I) can be

determined assuming losses equal to zero. The final value of λk is in the vicinity

of its initial value as only transmission losses are incorporated in the final solution

stage. Hence search for λk is a search for local optimum. Roulette-wheel selection

and a special crossover technique are used in this. Mutation probability is kept

low.

As this is a search for local optimum, crossover is restricted only to the

right half of the binary string. The binary digits on the right half has comparatively

lower equivalent decimal values and hence the decimal values of off-springs will

not deviate much from the decimal values of selected parent strings. This helps in

exploring the search space closer to the present location of the parent strings. Fig

4.10 shows a sample crossover. λk for each sub-interval is located by a genetic

search incorporating search space reduction technique.

1 0 0 1 …………|………….1 0 0 0

1 1 1 0 …………|………….0 1 1 0

Fig. 4.10 Sample crossover, restricted to right half

4.3.2.2 To find the roots of a quadratic equation

Let the quadratic equation be x2 – 3x - 10 = 0 which has a positive root equal to 5.

A population of five individuals is formed with string length of four

bits each. Search range is selected as 0 - 10. The binary strings are mapped

41

between 0 and 10 using the standard mapping rule. Error = x2 – 3x – 10 is

evaluated using each mapped number. Minimum negative error is obtained for

x=4.666 and minimum positive error is obtained for x=8 and hence it can be

concluded that the root, which we are searching for, lies between 4.666 and 8. In

the next generation, search range can be selected as 4.666-8. Table 4.1 lists the

values.

Table 4.1 Search space reduction in Generation 1

String DecimalNumber

Mapped NumberBetween 0 and 10

Error Remarks

0100 4 2.666 -10.89

0111 7 4.666 -2.226 Minimum negative error

0010 2 1.333 -12.222

1100 12 8.0 30.0 Minimum positive error

1110 14 9.333 49.1

Basic genetic operations can be performed on the strings of first

generation to obtain the second generation strings. These are converted to their

corresponding decimal values and mapped between 4.666 and 8. Roulette wheel

selection employed, neglected 4th and 5th strings, made 3 copies of second string

and 1 copy each of 1st and 3rd strings. Strings in the second generation are mapped

to the new search range and corresponding errors are computed. Table 4.2 shows

that the search range has reduced to 4.666-5.1105. Table 4.3 shows the search

range reduction in third generation to 4.9623 - 5.0808.

42


Matingpool

Secondgeneration

(after crossoverand mutation)

Decimalnumber

Mapped numberbetween 4.666 and

8

Error Remarks

0100 0000 0 4.6660 -2.226 Minimumnegative error

0111 0101 5 5.7773 +6.0450

0111 0011 3 5.3328 +2.4396

0111 0110 6 5.9996 +7.9955

0010 0010 2 5.1105 +0.7857 Minimumpositive error


Matingpool

Thirdgeneration

(after crossoverand mutation)

Decimalnumber

Mapped numberbetween 4.666 and

5.1105

Error Remarks

0000 0001 1 4.6962 -2.0341

0000 0010 2 4.7259 -1.8438

0010 0100 4 4.7851 -1.4579

0010 1010 10 4.9623 -0.2581 Minimumnegative error

0010 1110 14 5.0808 +0.5721 Minimumpositive error

At the end of 3 generations the search range is 4.9623 - 5.0808. Figure

4.11 shows that the upper limit moves from 10 to 8, then to 5.1105 and then to

5.0808. The lower limit moves from 0 to 4.666 and then to 4.9623. The major

advantage of this technique is that, when all the individuals of the population are

43

mapped to a narrower range, the chances of a number coinciding with the

optimum is very high.

10 8.0

5.1105 5.0808

5.0

4.666 4.666 4.9623

0

Figure 4.11 Search space reduction in 3 generations

This technique is highly useful when search range is very large. If

solution looked for is somewhere between 0 and 10000, then locating the same by

a conventional genetic search may take hundreds of iterations consuming a lot of

time.

4.3.2.3 Modifying the Lagrange multipliers

At the end of an optimization interval water withdrawals for hydro

plants and coal consumption for coal-constrained thermal plants are computed. If

hydro and coal constraints are not satisfied the associated Lagrange multipliers are

modified using Equations (4.15) and (4.16).

44

1 ( ) /i i i i i i i i iwhere k V Va Va i = n+1,...,n+hm m m m m¬ +D D = - (4.15)

2 ( ) /i i i i i i i Ti Ti 1where k Q Q Q i = 1,2,...nj j j j j¬ +D D = - (4.16)

k1 and k2 are multipliers to which the execution is very sensitive. Normally an

arbitrary multiplier is selected and maintained constant during the entire iterative

procedure irrespective of the magnitude of mismatch.

The number of iterations for satisfying hydro and coal constraints can

be reduced considerably if the multipliers are made proportional to magnitudes of

mismatch. A multiplier proportional to the magnitude of the mismatch for each

hydro unit and coal-constrained thermal unit (k1i and k2i), for each iteration, can be

determined by application of fuzzy logic and hence the multipliers become

variables rather than constants. In many trial simulations best results have been

obtained when output surface for the FIS has an approximate shape as shown in

Figure 4.12. Fuzzy acceleration factors were successfully applied to load flow

studies by Jayadeep Chakravorty (2009).

Figure 4.12 Desired output surface for the FIS

45

4.3.3 BY A METHOD USING A RANDOM SEARCH

This method is exactly same as the method described in sub-section

4.3.2. The only difference is that, for locating optimum λk, instead of a genetic

search, a search with a random population is used. In this method search space

reduction technique compensates for genetic operations. Figure 4.13 gives the

flow diagram for generating a random population in the search range.

Figure 4.13 Generating a random population

46

In a normal genetic search, the individuals move towards the optimum

due to genetic operations in each generation and the search range remains the

same. In this case, the search is always random and the search range reduces in

each generation. When the search range becomes very narrow, one of the values of

xmi coincides with the optimum λk.

The step-by-step procedure, which incorporates search space reduction

technique, for checking power balance constraint in sub-interval k is given below.

Let xmi (i=1,2,..,L) be the mapped values of xi (i=1,2,…,L).

1. Start population count i

2. Assign, λk = xmi

3. Compute Pjk (j=1,2,..,n+h) and PLk using λk

4. Compute1

( )n h

Dk Lk jkj

error i P P P+

=

= + -å

5. If |error (i)| < tolerance then stop, else next step.

6. Increment i. If i ≤ L, then step 2, else nest step.

7. Locate the values of xm corresponding to minimum positive error and

minimum negative error. Assign these values as the new limits of search

range and map xi ,(i=1,2,..L) to the new search range. If all error values are

of the same sign, then reset the search range with initial upper and lower

limits and map xi ,(i=1,2,..L) to the initial search range.

8. Go to step 1.

47

CHAPTER 5

TEST SYSTEMS

Two test systems are described in this chapter. The test systems give

information about coal-constrained thermal units, other thermal units, hydro

units, number of sub-intervals in the optimization interval, cost coefficients, NOx,

SO2 and CO2 emission coefficients, hydro discharge coefficients, coal discharge

coefficients, loss coefficients, generation limits of hydro and thermal units, water

allocation for hydro plants for the optimization interval, coal allocation for coal-

constrained thermal plants for the optimization interval and power demand during

sub-intervals.

5.1 TEST SYSTEM 1

Number of coal constrained thermal units 1

Total number of thermal units 2

Number of hydro units 2

Number of sub-intervals 24 to168

Table 5.1 lists the features of hydro units, Table 5.2 lists features of

thermal units and Table 5.3 lists the loss coefficients.

48

Table 5.1 Features of hydro units

Feature Hydro unit 1 Hydro unit 2

α [m3/MW2-h] 0.06 0.065

β [m3/MW-h] 20 22.5

γ [m3/h] 140 150

Generation (max.) MW 600 500

Generation (min.) MW 50 55

Water allocation[m3] for 24sub-intervals

100000 110000

49

Table 5.2 Features of thermal units

Feature Thermal unit 1 Thermal unit 2 Coefficients

a1 [R/MW2-h] 0.0025 0.0008

Fuel costb1 [R/MW-h] 3.2 3.4

c1 [R/h] 25 30

a2 [kg/MW2-h] 0.006483 0.006483

NOx emissionb2 [kg/MW-h] -0.79027 -0.79027

c2 [kg/h] 28.82488 28.82488

a3 [kg/MW2-h] 0.00232 0.00232

SO2 emissionb3 [kg/MW-h] 3.84632 3.84632

c3 [kg/h] 182.2605 182.2605

a4 [kg/MW2-h] 0.084025 0.084025

CO2 emissionb4 [kg/MW-h] -2.944584 -2.944584

c4 [kg/h] 137.7043 137.7043

α1 [kg/MW2-h] 0.059

Coal dischargeβ1 [kg/MW-h] 75.14

γ1 [kg/h] 594.7

Generation (max.)MW

800 1000

Generation (min.)MW

60 80

Coal allocation [kg]for 24 sub-intervals

676000

50Table 5.3 Loss coefficients

1 2 3 4

1 0.00014 0.00001 0.000015 0.000015

2 0.00001 0.00006 0.00001 0.000013

3 0.000015 0.00001 0.000068 0.000065

4 0.000015 0.000013 0.000065 0.00007

5.2 TEST SYSTEM 2

Number of thermal units 6

Power demand 2700 MW

Table 5.4 lists the features of thermal units and Table 5.5 lists the loss

coefficients.

51

Table 5.4 Features of thermal units

FeatureThermal

Unit 1

Thermal

Unit 2

Thermal

Unit 3

Thermal

Unit 4

Thermal

Unit 5

Thermal

Unit 6

Coeffi-

cient

a1 [R/MW2-h] 0.002035 0.003866 0.002182 0.001345 0.002182 0.005963

Fuel costb1 [R/MW-h] 8.43205 6.41031 7.4289 8.30154 7.4289 6.91559

c1 [R/h] 85.6348 303.778 847.1484 274.2241 847.1484 202.0258

a2 [kg/MW2-h] 0.006323 0.006483 0.003174 0.006732 0.003174 0.006181NOx

emissionb2 [kg/MW-h] -0.38128 -0.79027 -1.36061 -2.39928 -1.36061 -0.39077

c2 [kg/h] 80.9019 28.8249 324.1775 610.2535 324.1775 50.3808

a3 [kg/MW2-h] 0.001206 0.00232 0.001284 0.000813 0.001284 0.003578SO2

emissionb3 [kg/MW-h] 5.05928 3.84624 4.45647 4.97641 4.45647 4.14938

c3 [kg/h] 51.3778 182.2605 508.5207 165.3433 508.5207 121.2133

a4 [kg/MW2-h] 0.26511 0.140053 0.105929 0.106409 0.105929 0.403144CO2

emissionb4 [kg/MW-h] -61.0194 -29.9522 -9.55279 -12.7364 -9.55279 -121.981

c4 [kg/h] 5080.148 3824.77 1342.851 1819.625 1342.851 11381.07

Gen.(max.) MW 50 50 50 50 50 50

Gen.(min.) MW 1000 1000 1000 1000 1000 1000

Table 5.5 Loss coefficients

1 2 3 4 5 6

1 0.0002 0.00001 0.000015 0.000005 0 -0.00003

2 0.00001 0.0003 -0.00002 0.000001 0.000012 0.00001

3 0.000015 -0.00002 0.0001 -0.00001 0.00001 0.000008

4 0.000005 0.000001 -0.00001 0.00015 0.000006 0.00005

5 0 0.000012 0.00001 0.000006 0.00025 0.00002

6 -0.00003 0.00001 0.000008 0.00005 0.00002 0.00021

52

CHAPTER 6

RESULTS AND DISCUSSIONS

6.1 PRELIMINARY INVESTIGATIONS

6.1.1 SHORT-TERM HYDROTHERMAL SCHEDULING: COST MINIMIZATION

OBJECTIVE

Weight vector [1 0 0 0]

Method to determine objective function values N-R Method

Test system 1

Sub-intervals 24

The short-term hydrothermal scheduling problem is solved with only

cost minimization objective and hence the corresponding weight vector is fed-in as

[1 0 0 0].

The solution converged in 5 iterations and 0.343 s. Objective function

values which are cost, emission of NOx, emission of SO2 and emission of CO2 in

each sub-interval are listed in Table 6.1a. Values of λ, power allocations and

losses in sub-intervals are listed in Table 6.1b.

53

Table 6.1a Objective function values in optimization sub-intervals

Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

1 600 1549.19 283.89 2146.50 6214.06

2 750 2067.18 608.92 2805.73 11386.34

3 590 1515.59 267.18 2104.52 5933.94

4 675 1804.91 428.85 2469.15 8571.43

5 700 1891.60 484.94 2579.78 9458.55

6 525 1299.94 173.52 1837.58 4307.41

7 550 1382.31 206.50 1939.04 4893.29

8 900 2611.80 1076.57 3521.54 18416.79

9 1230 3907.76 2629.58 5304.23 40713.60

10 1250 3990.78 2747.69 5421.78 42377.44

11 1350 4413.78 3380.67 6026.37 51249.65

12 1400 4630.28 3724.02 6339.32 56035.68

13 1200 3784.21 2457.67 5129.98 38286.25

14 1250 3990.78 2747.69 5421.78 42377.44

15 1250 3990.78 2747.69 5421.78 42377.44

16 1270 4074.32 2868.61 5540.45 44077.93

17 1350 4413.78 3380.67 6026.37 51249.65

18 1470 4939.09 4235.27 6789.55 63134.34

19 1330 4328.11 3248.37 5903.20 49400.94

54

Table 6.1a continued

Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

20 1250 3990.78 2747.69 5421.78 42377.44

21 1170 3661.84 2292.02 4958.21 35940.50

22 1050 3183.86 1691.26 4295.67 27363.20

23 900 2611.80 1076.57 3521.54 18416.79

24 600 1549.19 283.89 2146.50 6214.06

55

Table 6.1b λ, power allocations and losses in optimization sub-intervals

Sub-Int.

λ (R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

1 3.88 194.58 217.37 101.57 100.37 13.9

2 4.10 229.21 314.23 114.56 113.42 21.42

3 3.87 192.29 210.95 100.72 99.51 13.47

4 3.99 211.85 265.68 108.03 106.86 17.42

5 4.02 217.63 281.84 110.20 109.03 18.69

6 3.78 177.40 169.30 95.20 93.96 10.87

7 3.81 183.12 185.30 97.32 96.09 11.83

8 4.32 264.19 412.08 127.88 126.77 30.91

9 4.84 342.45 630.95 158.40 157.32 59.13

10 4.88 347.26 644.38 160.31 159.23 61.17

11 5.05 371.38 711.82 169.95 168.85 72.01

12 5.13 383.51 145.73 174.84 173.73 77.80

13 4.80 335.26 610.84 155.55 154.47 56.13

14 4.88 347.26 644.38 160.31 159.23 61.17

15 4.88 347.26 644.38 160.31 159.23 61.17

16 4.91 352.07 657.83 162.22 161.14 63.26

56

Table 6.1b continued

Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

17 5.05 371.38 711.82 169.95 168.85 72.01

18 5.26 400.56 793.40 181.76 180.63 86.35

19 5.01 366.54 698.30 168.01 166.91 69.76

20 4.88 347.26 644.38 160.31 159.23 61.17

21 4.75 328.09 590.78 152.72 151.64 53.22

22 4.55 299.53 510.94 141.53 140.45 42.26

23 4.32 264.19 412.08 127.88 126.77 30.91

24 3.88 194.58 217.37 101.57 100.37 13.90

The total cost, emissions, water withdrawal and coal-consumption for the 24 sub-

intervals are as given below.

Total cost = R 75583. 64

Total NOx emission = 45789.7 kg

Total SO2 emission =103072.37 kg

Total CO2 emission = 720774.15 kg

Total water withdrawal (unit 3) = 100000.0 m3


Total coal consumption (unit 1) = 676000.0 kg

No. of iterations = 5

57

Time of execution = 0.343 s

Discussion

The total cost of fuel for 24 sub-intervals is R 75583.64, which is the minimum

possible. For any other weight combination the total cost would be more.

58

6. 1.2 SHORT-TERM HYDROTHERMAL SCHEDULING: NOX EMISSION

MINIMIZATION OBJECTIVE



Test system 1

Sub-intervals 24


NOx emission minimization objective and hence the corresponding weight vector

is fed-in as [0 1 0 0].





59


Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

1600 1916.08 596.56 2659.77 10976.03

2 750 2455.14 1004.15 3350.94 17222.44

3 590 1874.18 569.06 2606.43 10542.67

4 675 2237.04 827.58 3070.21 14549.44

5 700 2346.70 914.40 3211.17 15869.02

6 525 1606.95 408.85 2267.58 7968.64

7 550 1708.69 466.68 2396.31 8909.14

8 900 2844.95 1356.33 3856.22 22457.30

9 1230 3763.20 2351.85 5063.06 36863.40

10 1250 3821.62 2422.37 5140.60 37869.83

11 1350 4118.64 2793.14 5536.11 43139.84

12 1400 4270.25 2990.06 5738.83 45925.82

13 1200 3676.17 2248.31 4947.73 35383.08

14 1250 3821.62 2422.37 5140.60 37869.83

15 1250 3821.62 2422.37 5140.60 37869.83

16 1270 3880.37 2494.09 5218.65 38892.00

17 1350 4118.64 2793.14 5536.11 43139.84

18 1470 4486.03 3278.92 6028.32 49998.75

19 1330 4058.58 2716.54 5455.95 42053.83

20 1250 3821.62 2422.37 5140.60 37869.83

60


Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

21 1170 3589.86 2147.44 4833.54 33937.72

22 1050 3251.79 1770.24 4388.13 28500.23

23 900 2844.95 1356.33 3856.22 22457.30

24 600 1916.08 596.56 2659.77 10976.03

61


Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

1 3.83 166.35 342.82 50.00 55.00 14.16

2 4.90 224.24 416.99 67.42 63.99 22.64

3 3.75 161.70 336.93 50.00 55.00 13.63

4 4.47 201.23 387.31 50.00 55.00 18.54

5 4.68 212.88 402.27 50.00 55.00 20.15

6 3.22 131.56 298.89 50.00 55.00 10.45

7 3.42 143.14 313.47 50.00 55.00 11.61

8 5.66 264.11 468.97 100.93 97.79 31.80

9 7.45 352.44 586.63 176.17 173.54 58.78

10 7.57 357.82 593.92 180.80 178.19 60.73

11 8.14 384.78 630.61 204.06 201.57 71.03

12 8.43 398.29 649.13 215.78 213.33 76.53

13 7.28 344.37 575.74 169.24 166.57 55.92

14 7.57 357.82 593.92 180.80 178.19 60.73

15 7.57 357.82 593.92 180.80 178.19 60.73

16 7.68 363.21 601.22 185.43 182.85 62.71

62


Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

17 8.14 384.78 630.61 204.06 201.57 71.03

18 8.85 417.24 675.25 232.26 229.88 84.63

19 8.02 379.38 623.23 199.39 196.88 68.89

20 7.57 357.82 593.92 180.80 178.19 60.73

21 7.12 336.31 564.89 162.33 159.62 53.15

22 6.46 304.15 521.87 134.87 131.98 42.87

23 5.66 264.11 468.97 100.93 97.79 31.80

24 3.83 166.35 342.82 50.00 55.00 14.16

The total cost, emissions, water withdrawal and coal-consumption for

the 24 sub-intervals are as given below.

Total cost = R 76250. 78


Total SO2 emission =103243.45 kg





No. of iterations= 5

63

Time of execution= 0.407 s

Discussion

The total NOx emission for 24 sub-intervals is 43369.68 kg, which is the minimum

possible. For any other weight combination the NOx emission would be more.

64

6. 1.3 SHORT-TERM HYDROTHERMAL SCHEDULING: SO2 EMISSION




Test system 1

Sub-intervals 24


SO2 emission minimization objective and hence the corresponding weight vector

is fed-in as [0 0 1 0].





65


Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

1 600 1631.64 406.16 2292.82 7980.13

2 750 2123.10 727.73 2918.22 13049.84

3 590 1599.82 388.45 2252.64 7690.77

4 675 1874.04 553.76 2600.17 10342.98

5 700 1956.32 608.80 2704.99 11206.77

6 525 1395.83 284.51 1996.00 5955.72

7 550 1473.71 322.20 2093.78 6593.26

8 900 2641.49 1156.53 3587.10 19518.51

9 1230 3879.56 2495.57 5218.83 38910.52

10 1250 3959.02 2594.88 5325.07 40322.63

11 1350 4364.04 3123.66 5869.25 47803.40

12 1400 4571.43 3408.45 6149.59 51809.93

13 1200 3761.34 2350.58 5061.09 36844.28

14 1250 3959.02 2594.88 5325.07 40322.63

15 1250 3959.02 2594.88 5325.07 40322.63

16 1270 4038.99 2696.32 5432.16 41762.57

17 1350 4364.04 3123.66 5869.25 47803.40

18 1470 4867.31 3830.42 6551.44 57722.37

19 1330 4282.00 3013.57 5758.67 46250.77

66


Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

20 1250 3959.02 2594.88 5325.07 40322.63

21 1170 3644.27 2210.34 4905.26 34839.96

22 1050 3187.37 1696.21 4300.89 27433.42

23 900 2641.49 1156.53 3587.10 19518.51

24 600 1631.64 406.16 2292.82 7980.13

67


Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

1 5.44 142.21 294.58 88.83 87.28 12.90

2 5.85 194.60 365.79 105.85 104.48 20.71

3 5.42 138.72 289.88 87.70 86.14 12.45

4 5.65 168.40 330.04 97.29 95.83 16.56

5 5.71 177.13 341.92 100.13 98.70 17.89

6 5.24 116.04 259.41 80.46 78.82 9.73

7 5.31 124.76 271.10 83.24 81.63 10.73

8 6.28 247.03 438.18 123.28 122.06 30.55

9 7.27 362.54 601.80 163.11 162.17 59.62

10 7.33 369.54 611.92 165.59 164.67 61.72

11 7.66 404.58 662.84 178.13 177.27 72.83

12 7.82 422.11 688.53 184.48 183.65 78.77

13 7.18 352.03 586.67 159.40 158.44 56.54

14 7.33 369.54 611.92 165.59 164.67 61.72

15 7.33 369.54 611.92 165.59 164.67 61.72

16 7.40 376.55 622.06 168.08 167.18 63.87

68


Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

17 7.66 404.58 662.84 178.13 177.27 72.83

18 8.05 446.65 724.74 193.46 192.67 87.52

19 7.59 397.57 652.61 175.61 174.74 70.53

20 7.33 369.54 611.92 165.59 164.67 61.72

21 7.08 341.52 571.60 155.71 154.73 53.55

22 6.72 299.51 511.80 141.12 140.04 42.47

23 6.28 247.03 438.18 123.28 122.06 30.55

24 5.44 142.21 294.58 88.83 87.28 12.90



Total cost = R 75765.51


Total SO2 emission = 102742.33 kg





No. of iterations= 5

69


Discussion

The total SO2 emission for 24 sub-intervals is 102742.33 kg, which is the

minimum possible. For any other weight combination the SO2 emission would be

more.

70

6. 1.4 SHORT-TERM HYDROTHERMAL SCHEDULING: CO2 EMISSION




Test system 1

Sub-intervals 24


CO2 emission minimization objective and hence the corresponding weight vector

is fed-in as [0 0 0 1].





71


Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

1 600 1916.13 595.59 2659.44 10963.40

2 750 2416.92 971.28 3301.27 16729.43

3 590 1874.23 568.07 2606.10 10529.86

4 675 2237.13 826.71 3069.93 14538.29

5 700 2295.84 872.78 3145.37 15239.61

6 525 1606.96 407.77 2267.21 7954.70

7 550 1708.71 465.64 2395.95 8895.62

8 900 2818.79 1330.62 3821.85 22079.65

9 1230 3766.28 2355.37 5067.06 36913.93

10 1250 3826.59 2428.27 5147.13 37954.15

11 1350 4133.27 2811.98 5555.68 43406.78

12 1400 4289.83 3016.04 5765.15 46292.66

13 1200 3676.44 2248.40 4947.97 35384.63

14 1250 3826.59 2428.27 5147.13 37954.15

15 1250 3826.59 2428.27 5147.13 37954.15

16 1270 3887.24 2502.44 5227.74 39011.03

17 1350 4133.27 2811.98 5555.68 43406.78

18 1470 4512.69 3315.65 6064.35 50515.12

19 1330 4071.25 2732.66 5472.87 42282.43

72Table 6.4a continued

Sub-Int.

PowerDemand(MW)

Cost(R)

NOxEmission

(kg)

SO2Emission

(kg)

CO2Emission

(kg)

20 1250 3826.59 2428.27 5147.13 37954.15

21 1170 3587.35 2144.25 4830.08 33892.31

22 1050 3238.46 1755.50 4370.40 28287.17

23 900 2818.79 1330.62 3821.85 22079.65

24 600 1916.13 595.59 2659.44 10963.40

73


Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

1 57.28 166.78 342.38 50.00 55.00 14.17

2 70.21 220.55 411.50 71.68 68.63 22.36

3 56.19 162.15 336.49 50.00 55.00 13.63

4 65.55 201.60 386.95 50.00 55.00 18.55

5 67.07 207.82 395.00 61.92 55.00 19.75

6 49.21 132.06 298.39 50.00 55.00 10.45

7 51.88 143.62 312.99 50.00 55.00 11.62

8 80.54 261.69 465.29 103.68 100.95 31.61

9 104.64 352.77 586.97 175.62 173.45 58.80

10 106.17 358.31 594.50 180.05 177.91 60.76

11 113.91 386.08 632.42 202.32 200.31 71.12

12 117.86 399.99 651.55 213.53 211.59 76.66

13 102.37 344.45 575.71 168.99 166.78 55.93

14 106.17 358.31 594.50 180.05 177.91 60.76

15 106.17 358.31 594.50 180.05 177.91 60.76

16 107.70 363.86 602.04 184.48 182.38 62.76

74


Sub-Int.

λ(R/MWh) P1(MW)

P2(MW)

P3(MW)

P4(MW)

Losses(MW)

17 113.91 386.08 632.42 202.32 200.31 71.12

18 123.47 419.49 678.54 229.32 227.45 84.81

19 112.35 380.52 624.80 197.85 195.82 68.98

20 106.17 358.31 594.50 180.05 177.91 60.76

21 100.11 336.15 564.49 162.38 160.12 53.14

22 91.26 302.99 520.01 136.11 133.66 42.78

23 80.54 261.69 465.29 103.68 100.95 31.61

24 57.28 166.78 342.38 50.00 55.00 14.17



Total cost = R 76212.06


Total SO2 emission = 103193.89 kg





No. of iterations = 5

75


Discussion

The total CO2 emission for 24 sub-intervals is 691183.05 kg, which is the

minimum possible. For any other weight combination the CO2 emission would be

more.

76

6. 1.5 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING:

DETERMINING THE BEST COMPROMISE SOLUTION CONSIDERING 24 SUB-

INTERVALS

Four weight vectors are fed in and the best compromise solution is identified.

Weight vectors 4


Test system 1

Sub-intervals 24

F1 represents cost, F2 - emission of NOx, F3 - emission of SO2 and F4

- emission of CO2. µ(F1) , µ(F2), µ(F3) and µ(F4) are the corresponding fuzzy

membership functions. Four weight vectors are fed in and the best compromise

solution is identified.

Execution results are listed in Table 6.5a and 6.5b. Minimum values of

objective functions are highlighted in Table 6.5a. Optimum solution, which is the

solution with the highest fitness, is given in Table 6.5b (highlighted) and Table

6.6.

77

Table 6.5 a Weight vectors and objective function values in the optimization interval

Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

1 1.0 0.0 0.0 0.0 75583.64 45789.70 103072.37 720774.15

2 0.0 1.0 0.0 0.0 76250.78 43369.68 103243.45 691241.83

3 0.0 0.0 1.0 0.0 75765.51 44339.13 102742.33 702307.73

4 0.0 0.0 0.0 1.0 76212.06 43372.02 103193.89 691183.05

Table 6.5b Membership function values and fitness

Sl.No. m(F1) m(F2) m(F3) m(F4) fitness

1 1.00000 0.00000 0.34140 0.00000 0.33535

2 0.00000 1.00000 0.00000 0.99801 0.49950

3 0.72739 0.59941 1.00000 0.62405 0.73771

4 0.05804 0.99903 0.09891 1.00000 0.53900

Table 6.6 Optimum solution

Sl.No. w1 w2 w3 w4

F1(R)

F2(kg)

F3(kg)

F4(kg)

3 0.0 0.0 1.0 0.0 75765.51 44339.13 102742.33 702307.73


3 0.72739 0.59941 1.00000 0.62405 0.73771

78


Discussion

In Table 6.5a F1, F2, F3 and F4 indicate the total cost, total NOx emission, total

SO2emission and total CO2 emission in the optimization interval corresponding to

a weight vector. The maximum and minimum values of the above objective

functions are given below.

F1max = 76250.78 corresponding to sl. no. 2 in Table 6.5a

F1min = 75583.64 corresponding to sl. no. 1 in Table 6.5a







The fuzzy membership functions of objective function values can be computed

using Eq. (4.1).

For example, for sl. no. 2

m(F1) = (76250.78 - 76250.78) / (76250.78 – 75583.64) = 0

m(F2) = (45789.70 - 43369.68)/ (45789.70 - 43369.68) = 1

79

m(F3) = (103243.45- 103243.45)/(103243.45-102742.33) = 0

m(F4) = (720774.15-691241.83)/(720774.15-691183.05)= 0.99801

fitness = 1.99801/4= 0.49950

m(F1) and m(F3) are zero which indicate that they are totally incompatible with

the corresponding sets, while m(F4) is highly compatible and m(F2) indicates total

compatibility. The average of membership function values, termed as fitness, is an

indication of accomplishment of each solution in satisfying the objectives. The

fitness value obtained for the third weight combination, 0.73771, is the highest

among all and hence the best compromising solution out of the four weight vectors

tested.

80



INTERVALS

Weight vectors 4


Test system 1

Sub-intervals 72

Water allocation[m3] for 72 sub-intervals Unit 3 - 300000

Unit 4- 330000

Coal allocation [kg] for 72 sub-intervals 2028000

Four weight vectors are fed in and the best compromise solution is

identified.

Execution results are listed in Table 6.7a and 6.7b. Optimum solution,

which is the solution with the highest fitness, is given in Table 6.8.


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

1 1.0 0.0 0.0 0.0 226750.92 137369.09 309217.11 2162322.44

2 0.0 1.0 0.0 0.0 228752.35 130109.05 309730.35 2073725.48

3 0.0 0.0 1.0 0.0 227296.53 133017.38 308226.98 2106923.18

4 0.0 0.0 0.0 1.0 228636.18 130116.07 309581.66 2073549.16

81

Table 6.7 b Membership function values and fitness


1 1.00000 0.00000 0.34140 0.00000 0.33535

2 0.00000 1.00000 0.00000 0.99801 0.49950

3 0.72739 0.59941 1.00000 0.62405 0.73771

4 0.05804 0.99903 0.09891 1.00000 0.53900


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

3 0.0 0.0 1.0 0.0 228752.35 130109.05 309730.35 2073725.48


3 0.72739 0.59941 1.00000 0.62405 0.73771


Discussion

Results listed in Tables 6.7a , 6.7b and 6.8 are similar to the results given in sub-

section 6.1.5. Water and coal allocatins are increased to meet power demand for

72 sub-intervals. For 72 sub-intervals execution time has increased from 1.016 s

to 5.125 s.

82



INTERVALS

Weight vectors 4


Test system 1

Sub-intervals 168

Water allocation[m3] for 168 sub-intervals Unit 3 - 700000

Unit 4 - 770000

Coal allocation [kg] for 168 sub-intervals 4732000

Four weight vectors are fed in and the best compromise solution is

identified.

The solution converged in 45.641 s. Execution results are listed in

Table 6.9a and 6.9b. Optimum solution, which is the solution with the highest

fitness, is given in Table 6.10.


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

1 1.0 0.0 0.0 0.0 529085.48 320527.87 721506.58 5045419.03

2 0.0 1.0 0.0 0.0 533755.48 303587.79 722704.15 4838692.80

3 0.0 0.0 1.0 0.0 530358.58 310373.88 719196.30 4916154.08

4 0.0 0.0 0.0 1.0 533484.43 303604.15 722357.20 4838281.38

83



1 1.00000 0.00000 0.34140 0.00000 0.33535

2 0.00000 1.00000 0.00000 0.99801 0.49950

3 0.72739 0.59941 1.00000 0.62405 0.73771

4 0.05804 0.99903 0.09891 1.00000 0.53900


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

3 0.0 0.0 1.0 0.0 530358.58 310373.88 719196.30 4916154.08


3 0.72739 0.59941 1.00000 0.62405 0.73771


Discussion

Results are similar to the cases given in sub-section 6.1.5 and 6.1.6. This

simulation is mainly carried out to test the efficiency of the software developed.

With 168 sub-intervals, maximum size of Hessian matrix would be 843 x 843.

Still, the program has sucessfully converged with very good acuracy satisfying all

the constraints.

84

6.1.8 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH

RANDOM GENERATION OF WEIGHT VECTORS: TRIAL 1

Weight vectors 10 (4 fixed, 6random generated


Test system 1

Sub-intervals 24

First four weight vectors are of the format shown below which is

necessary for determining extreme values of objective functions. Remaining six

weight vectors are random generated.

1 0 0 00 1 0 00 0 1 00 0 0 1

The solution converged in 0.766 s. Execution results are given in

Table 6.11a and 6.11b. Optimum solution, which is the solution with the highest

fitness, is given in Table 6.12.

6.1.8.1 Generating a random weight vector of four weights

Geneate four random numbers xi (i=1,..,4)

Find4

1i

isum x

=

=åWeight vector: 1 2 3 4[ / , / , / , / ]w x sum x sum x sum x sum=

85


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

1 1.0 0.0 0.0 0.0 75583.64 45789.70 103072.37 720774.15

2 0.0 1.0 0.0 0.0 76250.78 43369.68 103243.45 691241.83

3 0.0 0.0 1.0 0.0 75765.51 44339.13 102742.33 702307.73

4 0.0 0.0 0.0 1.0 76212.06 43372.02 103193.89 691183.05

5 0.22 0.35 0.11 0.33 76189.24 43375.77 103167.33 691182.34

6 0.08 0.05 0.18 0.70 76200.13 43373.84 103179.00 691179.17

7 0.04 0.17 0.65 0.14 76129.85 43400.77 103092.35 691357.97

8 0.21 0.12 0.28 0.40 76182.17 43377.74 103157.96 691190.06

9 0.46 0.40 0.08 0.06 76112.65 43405.02 103090.96 691407.92

10 0.01 0.14 0.39 0.47 76188.81 43376.35 103164.20 691183.93

86



1 1.00000 0.00000 0.34140 0.00000 0.33535

2 0.00000 1.00000 0.00000 0.99788 0.49947

3 0.72739 0.59941 1.00000 0.62397 0.73769

4 0.05804 0.99903 0.09891 0.99987 0.53896

5 0.09225 0.99748 0.15190 0.99989 0.56038

6 0.07593 0.99828 0.12861 1.00000 0.55071

7 0.18128 0.98716 0.30153 0.99396 0.61598

8 0.10285 0.99667 0.17059 0.99963 0.56744

9 0.20706 0.98540 0.30429 0.99227 0.62225

10 0.09289 0.99725 0.15815 0.99984 0.56203


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

3 0.0 0.0 1.0 0.0 75765.51 44339.13 102742.33 702307.73


3 0.72739 0.59941 1.00000 0.62397 0.73769


87

Discussion

Six weight vectors are randomly generated and fitness values are computed for

all the ten weight vectors. Out of the ten weight vectors , the third weight vector

yielded highest fitness value in this case. This simulation is meant for testing the

efficiency of random generation of weight vectors.

88


RANDOM GENERATION OF WEIGHT VECTORS: TRIAL 2

Weight vectors 10 (4 fixed, 6random generated


Test system 1

Sub-intervals 24


necessary for determining extreme values of objective functions. This is the

second trial of the problem described in sub-section 6.1.8 using the same

parameters.

1 0 0 00 1 0 00 0 1 00 0 0 1

The solution converged in 0.77s. Execution results are listed in Table

6.13a and 6.13b. Optimum solution, which is the solution with the highest fitness,

is given in Table 6.14.

89


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

1 1.0 0.0 0.0 0.0 75583.64 45789.70 103072.37 720774.15

2 0.0 1.0 0.0 0.0 76250.78 43369.68 103243.45 691241.83

3 0.0 0.0 1.0 0.0 75765.51 44339.13 102742.33 702307.73

4 0.0 0.0 0.0 1.0 76212.06 43372.02 103193.89 691183.05

5 0.05 0.36 0.27 0.32 76189.40 43376.09 103165.48 691182.99

6 0.13 0.29 0.05 0.53 76201.94 43373.38 103181.83 691178.50

7 0.26 0.15 0.18 0.41 76185.05 43376.84 103162.10 691186.33

8 0.54 0.11 0.29 0.07 76043.98 43460.18 103013.64 691951.29

9 0.43 0.09 0.39 0.09 76067.99 43440.43 103033.70 691743.35

10 0.38 0.19 0.41 0.02 75935.90 43607.51 102899.80 693566.47



1 1.00000 0.00000 0.34140 0.00000 0.33535

2 0.00000 1.00000 0.00000 0.99786 0.49947

3 0.72739 0.59941 1.00000 0.62396 0.73769

4 0.05804 0.99903 0.09891 0.99985 0.53896

5 0.09201 0.99735 0.15558 0.99985 0.56120

6 0.07322 0.99847 0.12296 1.00000 0.54866

7 0.09853 0.99704 0.16233 0.99974 0.56441

8 0.30998 0.96261 0.45860 0.97389 0.67627

9 0.27400 0.97076 0.41856 0.98091 0.66106

10 0.47199 0.90172 0.68577 0.91931 0.74470

90


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

10 0.38 0.19 0.41 0.02 75935.90 43607.51 102899.80 693566.47


10 0.47199 0.90172 0.68577 0.91931 0.74470

Discussion

Six weight vectors are randomly generated and fitness values are computed for all

the ten weight vectors. Out of the ten weight vectors, the tenth weight vector

yielded highest fitness value this time. As weight vectors are randomly generated,

higher fitness values are possible in subsequent trials. Also it can’t be predicted

that how many trials are required to obtain a solution with an expected value of

fitness.

91

6.1.10 EFFECTIVENESS OF SEARCH SPACE REDUCTION TECHNIQUE

Weight vectors 161 (user fed)Test system 2

Table 6.15 illustrates the effectiveness of search space reduction

technique. For a normal genetic search, out of 161 executions corresponding to

161 weight combinations, solutions for 4 weight vectors converged in 1-5

generations, solutions for 12 weight vectors converged in 6-10 generations and

solutions for 145 weight vectors took more than 10 generations to converge. For

genetic and random searches, both with search space reduction technique

incorporated, the corresponding figures are also listed in the table. For a normal

genetic search with convergence tolerance 0.0001 the number of generations

exceeded the upper limit.

Table 6.15 Effectiveness of search space reduction technique

Sl.

No

Type of search Generations

1-5

Generations

6-10

Generations

>10

Convergence

Tolerance

Time

(s)

1 Normalgenetic search

4 12 145 0.1 513

2 Genetic search withsearch space reductiontechnique incorporated

102 47 12 0.0001 116

3 Random search withsearch space reductiontechnique incorporated

3 60 98 0.0001 2

92

Discussion

As far as the number of generations are concerned, genetic search with search

space reduction technique incorporated is found the most efficient. Tabulated

results indicate that, in genetic search method with search space reduction

technique incorporated, solutions for 102 weight vectors out of 161 converged in 5

generations whereas in random search method solutions for only 3 weight vectors

out of 161 converged in 5 generations. This is because in genetic search method

with search space reduction technique incorporated, the individuals move towards

the optimum due to genetic operations and at the same time search space is also

getting reduced in each generation. In random search method only search space

reduction is taking place, but the individuals are generated randomly. Hence

genetic search method takes less generations to find the optimum. But the time of

execution is much higher for genetic search method comparing with random

search method. Hence, of all the three, the random search method stands out.

93


161 USER- FED WEIGHT VECTORS

Weight vectors 161 (user fed)Method to determine objective function values N-R Method

Test system 1

Sub-intervals 24

1 0 0 00 1 0 00 0 1 00 0 0 1

First four weight vectors are of the format shown above which is

necessary for determining the extreme values of objective functions. The

remaining 157 weight vectors cover almost all intermediate weight combinations

in discrete steps. Hence the best solution determined using the 161 weight vectors

will be closer to global optimum.

The solution converged in 12.67 s. Execution results are listed in Table

6.16a and 6.16b. Optimum solution is given in Table 6.17.

94


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

1 1.0 0.0 0.0 0.0 75583.64 45789.70 103072.37 720774.15

2 0.0 1.0 0.0 0.0 76250.78 43369.68 103243.45 691241.83

3 0.0 0.0 1.0 0.0 75765.51 44339.13 102742.33 702307.73

4 0.0 0.0 0.0 1.0 76212.06 43372.02 103193.89 691183.05

5 0.9 0.1 0.0 0.0 75635.73 44676.78 102846.46 706654.50

6 0.9 0.0 0.1 0.0 75588.77 45429.87 102970.65 716158.38

7 0.9 0.1 0.0 0.1 76068.65 43435.18 103050.29 691707.94

8 0.8 0.2 0.0 0.0 75733.95 44102.85 102818.15 699528.79

9 0.8 0.0 0.2 0.0 75601.36 45157.28 102898.27 712669.77

10 0.8 0.0 0.0 0.2 76132.68 43394.56 103111.01 691314.39

11 0.8 0.1 0.1 0.0 75654.41 44532.63 102815.77 704823.10

12 0.8 0.1 0.0 0.1 76083.67 43423.67 103064.83 691591.73

13 0.8 0.0 0.1 0.1 76057.01 43445.67 103034.66 691809.55

14 0.7 0.3 0.0 0.0 75843.43 43768.98 102868.47 695501.72

15 0.7 0.0 0.3 0.0 75618.43 44948.26 102846.60 710001.61

16 0.7 0.0 0.0 0.3 76161.82 43383.07 103140.30 691224.53

17 0.7 0.0 0.1 0.2 76132.92 43394.78 103109.18 691313.92

18 0.7 0.0 0.2 0.1 76057.95 43445.89 103031.67 691807.01

19 0.7 0.1 0.0 0.2 76143.96 43389.59 103122.35 691273.26

20 0.7 0.2 0.0 0.1 76106.50 43408.34 103087.18 691442.16

21 0.7 0.1 0.2 0.0 75673.86 44419.26 102793.69 703386.40

22 0.7 0.2 0.1 0.0 75750.76 44042.16 102808.15 698762.98

95

Table 6.16 a continued

Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

23 0.6 0.4 0.0 0.0 75947.43 43572.72 102947.33 693221.51

24 0.6 0.0 0.4 0.0 75638.09 44786.50 102809.84 707942.37

25 0.6 0.0 0.0 0.4 76176.10 43378.80 103154.61 691197.23

26 0.6 0.0 0.3 0.1 76058.94 43446.35 103028.83 691807.68

27 0.6 0.0 0.1 0.3 76161.87 43383.27 103138.93 691224.64

28 0.6 0.3 0.0 0.1 76125.12 43398.02 103105.16 691346.73

29 0.6 0.1 0.0 0.3 76169.11 43380.75 103147.77 691209.11

30 0.6 0.1 0.3 0.0 75693.56 44329.78 102778.08 702255.67

31 0.6 0.3 0.1 0.0 75854.78 43745.26 102862.93 695199.54

32 0.6 0.2 0.2 0.0 75767.16 43994.08 102801.03 698157.65

33 0.6 0.2 0.0 0.2 76156.73 43384.74 103135.84 691237.23

34 0.6 0.2 0.2 0.0 75767.16 43994.08 102801.03 698157.65

35 0.6 0.2 0.1 0.1 76107.38 43408.34 103084.60 691437.69

36 0.6 0.1 0.2 0.1 76084.95 43424.29 103058.63 691588.32

37 0.6 0.1 0.1 0.2 76144.07 43389.87 103120.42 691273.25

38 0.5 0.5 0.0 0.0 76031.55 43468.07 103022.19 692063.74

39 0.5 0.0 0.5 0.0 75659.13 44660.45 102783.97 706342.67

40 0.5 0.0 0.0 0.5 76185.16 43376.56 103163.85 691185.91

41 05 0.4 0.1 0.0 75953.51 43565.11 102941.04 693116.65

42 0.5 0.1 0.4 0.0 75713.17 44259.07 102767.37 701365.01

43 0.5 0.4 0.0 0.1 76143.11 43389.75 103123.07 691276.49

44 0.5 0.1 0.0 0.4 76181.69 43377.36 103160.45 691189.72

96


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

45 0.5 0.0 0.1 0.4 76176.05 43378.98 103153.47 691197.44

46 0.5 0.0 0.4 0.1 76059.97 43447.04 103026.15 691811.39

47 0.5 0.3 0.2 0.0 75865.79 43727.19 102858.63 694969.08

48 0.5 0.2 0.3 0.0 75783.09 43956.21 102796.19 697682.19

49 0.5 0.3 0.0 0.2 76166.78 43381.37 103146.07 691213.77

50 0.5 0.2 0.0 0.3 76176.29 43378.70 103155.20 691196.96

51 0.5 0.0 0.2 0.3 76161.94 43383.51 103137.60 691225.19

52 0.5 0.0 0.3 0.2 76133.48 43395.43 103105.69 691315.72

53 0.5 0.3 0.1 0.1 76125.63 43398.25 103102.34 691344.62

54 0.5 0.1 0.3 0.1 76085.67 43429.90 103055.76 691590.88

55 0.5 0.1 0.1 0.3 76169.10 43380.96 103146.36 691209.29

56 0.5 0.2 0.2 0.1 76108.31 43408.54 103082.16 691435.83

57 0.5 0.2 0.1 0.2 76156.86 43384.99 103134.01 691237.08

58 0.5 0.1 0.2 0.2 76144.21 43390.21 103118.55 691274.11

59 0.4 0.6 0.0 0.0 76095.75 43414.78 103083.43 691515.01

60 0.4 0.0 0.6 0.0 75680.77 44561.78 102766.16 705094.76

61 0.4 0.0 0.0 0.6 76194.51 43374.73 103174.34 691181.91

62 0.4 0.3 0.3 0.0 75876.48 43713.82 102855.39 694798.62

63 0.4 0.3 0.0 0.3 76183.36 43376.90 103162.60 691187.84

64 0.4 0.0 0.3 0.3 76162.02 43383.77 103136.29 691226.15

65 0.4 0.4 0.2 0.0 75959.61 43560.24 102935.68 693047.15

66 0.4 0.2 0.4 0.0 75798.54 43926.67 102793.17 697312.63

97


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

67 0.4 0.4 0.0 0.2 76176.62 43378.52 103156.23 691196.58

68 0.4 0.2 0.0 0.4 76187.21 43376.06 103166.26 691183.99

69 0.4 0.0 0.4 0.2 76133.80 43395.85 103104.02 691317.93

70 0.4 0.0 0.2 0.4 76176.01 43379.18 103152.34 691197.91

71 0.4 0.5 0.1 0.0 76034.62 43466.00 103015.19 692025.82

72 0.4 0.1 0.5 0.0 75732.46 44203.26 102760.41 700664.71

73 0.4 0.5 0.0 0.1 76161.72 43382.83 103142.40 691225.33

74 0.4 0.1 0.0 0.5 76189.69 43375.55 103168.63 691181.94

75 0.4 0.0 0.1 0.5 76185.07 43376.72 103162.87 691186.09

76 0.4 0.0 0.5 0.1 76061.05 43447.94 103023.60 691817.97

77 0.4 0.3 0.2 0.1 76126.19 43398.66 103099.66 691344.87

78 0.4 0.3 0.1 0.2 76166.79 43381.65 103144.16 691213.81

79 0.4 0.2 0.3 0.1 76109.29 43408.93 103079.85 691436.46

80 0.4 0.1 0.3 0.2 76144.37 43390.61 103116.74 691275.82

81 0.4 0.2 0.1 0.3 76176.22 43378.92 103153.75 691197.16

82 0.4 0.1 0.2 0.3 76169.11 43381.21 103144.98 691209.89

83 0.4 0.2 0.2 0.2 76157.01 43385.30 103132.24 691237.75

84 0.3 0.7 0.0 0.0 76140.02 43390.74 103125.88 691293.61

85 0.3 0.0 0.7 0.0 75702.50 44484.36 102754.39 704119.56

86 0.3 0.0 0.0 0.7 76198.76 43373.92 103178.66 691179.58

87 0.3 0.6 0.1 0.0 76098.09 43413.91 103077.36 691493.58

88 0.3 0.1 0.6 0.0 75751.31 44159.40 102756.34 700116.72

98


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

89 0.3 0.1 0.0 0.6 76198.03 43374.04 103178.02 691179.91

90 0.3 0.6 0.0 0.1 76177.35 43378.16 103158.45 691196.14

91 0.3 0.0 0.1 0.6 76191.32 43375.35 103169.44 691180.83

92 0.3 0.0 0.6 0.1 76062.17 43449.03 103021.20 691827.28

93 0.3 0.5 0.2 0.0 76037.91 43465.52 103009.07 692009.10

94 0.3 0.2 0.5 0.0 75813.48 43903.97 102791.60 697030.03

95 0.3 0.5 0.0 0.2 76186.26 43376.14 103166.32 691185.05

96 0.3 0.2 0.0 0.5 76197.03 43374.21 103177.15 691180.41

97 0.3 0.0 0.2 0.5 76184.99 43376.88 103161.90 691186.44

98 0.3 0.0 0.5 0.2 76134.13 43396.32 103102.41 691320.97

99 0.3 0.4 0.3 0.0 75965.48 43558.09 102930.88 693012.07

100 0.3 0.3 0.4 0.0 75886.86 43704.42 102853.04 694678.58

101 0.3 0.4 0.0 0.3 76193.43 43374.84 103174.01 691182.72

102 0.3 0.3 0.0 0.4 76195.62 43374.45 103175.91 691181.22

103 0.3 0.0 0.3 0.4 76175.97 43379.40 103151.23 691198.63

104 0.3 0.0 0.4 0.3 76162.11 43384.07 103135.02 691227.53

105 0.3 0.4 0.2 0.1 76143.50 43390.66 103117.10 691277.18

106 0.3 0.4 0.1 0.2 76176.52 43378.81 103154.23 691196.68

107 0.3 0.2 0.4 0.1 76110.30 43409.48 103077.67 691439.43

108 0.3 0.1 0.4 0.2 76144.55 43391.06 103114.98 691278.36

109 0.3 0.1 0.2 0.4 76181.52 43377.74 103158.12 691190.37

110 0.3 0.2 0.1 0.4 76187.08 43376.25 103165.06 691184.16

99


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

111 0.3 0.3 0.2 0.2 76166.83 43381.98 103142.30 691214.64

112 0.3 0.2 0.3 0.2 76157.19 43385.66 103130.52 691239.23

113 0.3 0.2 0.2 0.3 76176.17 43379.17 103152.32 691197.77

114 0.3 0.5 0.1 0.1 76161.81 43383.20 103139.43 691224.64

115 0.3 0.1 0.5 0.1 76086.09 43427.68 103049.00 691613.12

116 0.3 0.1 0.1 0.5 76189.57 43375.71 103167.63 691182.09

117 0.2 0.8 0.0 0.0 76180.22 43377.17 103166.85 691198.71

118 0.2 0.0 0.8 0.0 75723.99 44423.62 102747.18 703358.02

119 0.2 0.0 0.0 0.8 76202.01 43373.36 103181.98 691178.48

120 0.2 0.7 0.1 0.0 76140.36 43391.14 103118.54 691285.52

121 0.2 0.1 0.7 0.0 75769.62 44125.17 102754.53 699691.57

122 0.2 0.7 0.0 0.1 76195.26 43374.25 103177.94 691182.34

123 0.2 0.1 0.0 0.7 76201.80 43373.38 103181.85 691178.53

124 0.2 0.0 0.7 0.1 76063.32 43450.32 103018.93 691839.16

125 0.2 0.0 0.1 0.7 76198.71 43374.03 103177.98 691179.65

126 0.2 0.6 0.2 0.0 76100.67 43414.16 103072.04 691487.24

127 0.2 0.2 0.6 0.0 75827.92 43886.92 102791.22 696819.20

128 0.2 0.6 0.0 0.2 76198.35 43373.80 103179.79 691180.03

129 0.2 0.2 0.0 0.6 76201.52 43373.41 103181.69 691178.61

130 0.2 0.0 0.6 0.2 76134.49 43396.86 103100.84 691324.82

131 0.2 0.0 0.2 0.6 76191.22 43375.49 103168.59 691181.09

132 0.2 0.5 0.3 0.0 76041.40 43466.44 103003.74 692010.98

100


133 0.2 0.3 0.5 0.0 75896.94 43698.36 102851.46 694601.08

134 0.2 0.5 0.0 0.3 76199.78 43373.61 103180.64 691179.27

135 0.2 0.3 0.0 0.5 76201.14 43373.45 103181.46 691178.73

136 0.2 0.0 0.5 0.3 76162.22 43384.39 103133.77 691229.32

137 0.2 0.0 0.3 0.5 76184.91 43377.06 103160.94 691186.96

138 0.2 0.4 0.4 0.0 75971.28 43558.06 102926.76 693004.46

139 0.2 0.4 0.0 0.4 76200.61 43373.51 103181.14 691178.92

140 0.2 0.0 0.4 0.4 76175.95 43379.64 103150.14 691199.60

141 0.2 0.4 0.2 0.2 76176.44 43379.16 103152.29 691197.53

142 0.2 0.2 0.4 0.2 76157.39 43386.08 103128.85 691241.47

143 0.2 0.2 0.2 0.4 76186.97 43376.45 103163.87 691184.57

144 0.2 0.6 0.1 0.1 76177.16 43378.58 103155.27 691195.69

145 0.2 0.1 0.6 0.1 76086.68 43429.10 103046.20 691625.65

146 0.2 0.1 0.1 0.6 76197.97 43374.16 103177.24 691179.98

147 0.2 0.5 0.2 0.1 76161.96 43383.72 103136.59 691225.95

148 0.2 0.5 0.1 0.2 76186.05 43376.44 193164.23 691185.08

149 0.2 0.2 0.5 0.1 76111.36 43410.20 103075.60 691444.64

150 0.2 0.1 0.5 0.2 76144.75 43391.58 103113.27 691281.69

151 0.2 0.1 0.2 0.5 76189.46 43375.88 103166.64 691182.41

152 0.2 0.2 0.1 0.5 76196.98 43374.34 103176.24 691180.47

153 0.2 0.4 0.3 0.1 76143.79 43391.35 103114.33 691280.68

154 0.2 0.4 0.1 0.3 76190.14 43375.56 103168.40 691181.63

101


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

155 0.2 0.3 0.4 0.1 76127.48 43399.94 103094.70 691351.98

156 0.2 0.1 0.4 0.3 76169.16 43381.79 103142.30 691212.32

157 0.2 0.1 0.3 0.4 76181.45 43377.96 103156.98 691191.07

158 0.2 0.3 0.1 0.4 76195.56 43374.61 103174.81 691181.28

159 0.1 0.9 0.0 0.0 76218.76 43370.62 103208.41 691191.47

160 0.1 0.0 0.9 0.0 75745.04 44376.10 102743.43 702765.49

161 0.1 0.0 0.0 0.9 76210.28 43372.28 103192.13 691183.15

102



1 1.00000 0.00000 0.34140 0.00000 0.33535

2 0.00000 1.00000 0.00000 0.99786 0.49946

3 0.72739 0.59941 1.00000 0.62396 0.73769

4 0.05804 0.99903 0.09891 0.99985 0.53896

5 0.92192 0.45988 0.79220 0.47708 0.66277

6 0.99231 0.14869 0.54437 0.15596 0.46033

7 0.27300 0.97293 0.38545 0.98211 0.65337

8 0.77469 0.69704 0.84870 0.71785 0.75957

9 0.97344 0.26133 0.68881 0.27384 0.54935

10 0.17704 0.98972 0.26429 0.99541 0.60661

11 0.89392 0.51945 0.85345 0.53897 0.70145

12 0.25049 0.97769 0.35644 0.98604 0.64267

13 0.29044 0.96860 0.41665 0.97868 0.66359

14 0.61059 0.83500 0.74828 0.85392 0.76195

15 0.94786 0.34700 0.79192 0.36399 0.61287

16 0.13335 0.99447 0.20584 0.99844 0.58303

17 0.17667 0.98963 0.26794 0.99542 0.60742

18 0.28904 0.96851 0.42262 0.97876 0.66473

19 0.16011 0.99177 0.24166 0.99680 0.59759

20 0.21627 0.98403 0.31185 0.99109 0.62581

21 0.86476 0.56629 0.89750 0.58751 0.72902

22 0.74949 0.72212 0.86866 0.74373 0.77100

103

Table 6.16 b continued


23 0.45471 0.91610 0.59092 0.93097 0.72317

24 0.91839 0.41454 0.86528 0.43357 0.65795

25 0.11194 0.99623 0.17728 0.99937 0.57120

26 0.28756 0.96832 0.42828 0.97874 0.66572

27 0.13327 0.99438 0.20857 0.99844 0.58367

28 0.18837 0.98829 0.27597 0.99431 0.61174

29 0.12242 0.99543 0.19093 0.99896 0.57694

30 0.83523 0.60327 0.92866 0.62572 0.74822

31 0.59358 0.84480 0.75934 0.86413 0.76546

32 0.72492 0.74199 0.88286 0.76418 0.77849

33 0.14098 0.99378 0.21474 0.99801 0.58688

34 0.72492 0.74199 0.88286 0.76418 0.77849

35 0.21494 0.98402 0.31699 0.99124 0.62680

36 0.24858 0.97744 0.36882 0.98615 0.64525

37 0.15995 0.99166 0.24550 0.99680 0.59848

38 0.32861 0.95934 0.44153 0.97009 0.67489

39 0.88685 0.46663 0.91690 0.48762 0.68950

40 0.09836 0.99716 0.15885 0.99975 0.56353

41 0.44559 0.91924 0.60347 0.93451 0.72570

42 0.80584 0.63249 0.95004 0.65581 0.76104

43 0.16139 0.99171 0.24022 0.99669 0.59750

44 0.10357 0.99683 0.16563 0.99962 0.56641

104



45 0.11202 0.99616 0.17956 0.99936 0.57178

46 0.28601 0.96804 0.43364 0.97861 0.66658

47 0.57708 0.85227 0.76792 0.87192 0.76730

48 0.70103 0.75764 0.89251 0.78025 0.78286

49 0.12591 0.99517 0.19433 0.99881 0.57856

50 0.11166 0.99628 0.17610 0.99938 0.57085

51 0.13317 0.99429 0.21123 0.99842 0.58428

52 0.17582 0.98936 0.27491 0.99536 0.60887

53 0.18760 0.98820 0.28159 0.99439 0.61294

54 0.24749 0.97718 0.37454 0.98607 0.64632

55 0.12244 0.99534 0.19375 0.99896 0.57762

56 0.21355 0.98394 0.32186 0.99130 0.62766

57 0.14078 0.99368 0.21839 0.99802 0.58772

58 0.15975 0.99152 0.24923 0.99677 0.59932

59 0.23239 0.98136 0.31931 0.98863 0.63042

60 0.85441 0.50740 0.95244 0.52979 0.71101

61 0.08435 0.99791 0.13791 0.99988 0.55502

62 0.56106 0.85779 0.77439 0.87768 0.76773

63 0.10106 0.99702 0.16134 0.99968 0.56478

64 0.13305 0.99418 0.21383 0.99839 0.58486

65 0.43645 0.92126 0.61417 0.93686 0.72718

66 0.67789 0.76984 0.89854 0.79273 0.78475

105



67 0.11116 0.99635 0.17405 0.99939 0.57024

68 0.09530 0.99737 0.15404 0.99981 0.56163

69 0.17535 0.98919 0.27824 0.99529 0.60952

70 0.11209 0.99607 0.18181 0.99934 0.57233

71 0.34201 0.96020 0.45549 0.97137 0.67777

72 0.77692 0.65555 0.96393 0.67947 0.76897

73 0.13350 0.99457 0.20165 0.99842 0.58203

74 0.09158 0.99757 0.14930 0.99988 0.55958

75 0.09850 0.99709 0.16081 0.99974 0.56403

76 0.28440 0.96766 0.43871 0.97839 0.66729

77 0.18675 0.98803 0.28693 0.99438 0.61402

78 0.12590 0.99506 0.19815 0.99881 0.57948

79 0.21209 0.98378 0.32647 0.99128 0.62841

80 0.15951 0.99135 0.25286 0.99671 0.61111

81 0.11176 0.99618 0.17900 0.99937 0.57158

82 0.12243 0.99524 0.19650 0.99894 0.57828

83 0.14056 0.99355 0.22192 0.99800 0.58851

84 0.16603 0.99130 0.23461 0.99611 0.59701

85 0.82184 0.53939 0.97592 0.56274 0.72497

86 0.07798 0.99825 0.12929 0.99996 0.55137

87 0.22887 0.98172 0.33143 0.98935 0.63285

88 0.74868 0.67367 0.97203 0.69799 0.77309

106



89 0.07908 0.99820 0.13057 0.99995 0.55195

90 0.11007 0.99650 0.16961 0.99940 0.56890

91 0.08913 0.99766 0.14769 0.99992 0.55860

92 0.28272 0.96721 0.44351 0.97808 0.66788

93 0.31907 0.96040 0.46771 0.97193 0.67978

94 0.65549 0.77922 0.90167 0.80228 0.78467

95 0.09671 0.99733 0.15392 0.99978 0.56194

96 0.08056 0.99813 0.13231 0.99993 0.55273

97 0.09862 0.99703 0.16274 0.99973 0.56453

98 0.17485 0.98899 0.28146 0.99519 0.61012

99 0.42765 0.92215 0.62373 0.93805 0.72789

100 0.54550 0.86168 0.77908 0.88174 0.76700

101 0.08596 0.99787 0.13858 0.99986 0.55557

102 0.08269 0.99803 0.13478 0.99991 0.55385

103 0.11214 0.99598 0.18402 0.99932 0.57287

104 0.13291 0.99406 0.21638 0.99834 0.58542

105 0.16080 0.99133 0.25213 0.99666 0.60023

106 0.11132 0.99623 0.17804 0.99939 0.57214

107 0.21057 0.98356 0.33083 0.99118 0.62903

108 0.15924 0.99116 0.25637 0.99663 0.60085

109 0.10382 0.99667 0.17028 0.99960 0.56759

110 0.09548 0.99729 0.15644 0.99981 0.56226

107


111 0.12585 0.99492 0.20185 0.99878 0.58035

112 0.14029 0.99340 0.22535 0.99795 0.58925

113 0.11185 0.99608 0.18185 0.99935 0.57228

114 0.13337 0.99441 0.20758 0.99844 0.58345

115 0.24686 0.97604 0.38803 0.98531 0.64906

116 0.09175 0.99751 0.15130 0.99988 0.56011

117 0.10577 0.99691 0.15286 0.99932 0.56371

118 0.78962 0.56449 0.99032 0.58847 0.73322

119 0.07311 0.99848 0.12266 1.00000 0.54856

120 0.16552 0.99114 0.24927 0.99638 0.60058

121 0.72123 0.68782 0.97564 0.71235 0.77426

122 0.08323 0.99811 0.13072 0.99987 0.55298

123 0.07343 0.99847 0.12292 1.00000 0.54870

124 0.28099 0.96668 0.44804 0.97768 0.66835

125 0.07805 0.99821 0.13065 0.99996 0.55172

126 0.22501 0.98162 0.34206 0.98957 0.63456

127 0.63385 0.78627 0.90245 0.80941 0.78299

128 0.07859 0.99830 0.12704 0.99995 0.55097

129 0.07385 0.99846 0.12325 1.00000 0.54889

130 0.17432 0.98877 0.28458 0.99506 0.61068

131 0.08928 0.99760 0.14938 0.99991 0.55905

132 0.31386 0.96002 0.47835 0.97187 0.68103

108



133 0.53039 0.86418 0.78223 0.88435 0.76529

134 0.07644 0.99838 0.12533 0.99997 0.55003

135 0.07441 0.99844 0.12370 0.99999 0.54914

136 0.13275 0.99392 0.21888 0.99828 0.58596

137 0.09874 0.99695 0.16465 0.99971 0.56501

138 0.41896 0.92216 0.63197 0.93830 0.72785

139 0.07521 0.99842 0.12435 0.99999 0.54949

140 0.11218 0.99589 0.18620 0.99929 0.57339

141 0.11143 0.99608 0.18191 0.99936 0.57220

142 0.13999 0.99322 0.22868 0.99787 0.58994

143 0.09566 0.99720 0.15880 0.99979 0.56286

144 0.11036 0.99632 0.17597 0.99942 0.57052

145 0.24598 0.97545 0.39362 0.98489 0.64998

146 0.07916 0.99815 0.13213 0.99995 0.55235

147 0.13314 0.99420 0.21324 0.99840 0.58475

148 0.09703 0.99721 0.15808 0.99978 0.56302

149 0.20899 0.98326 0.33495 0.99101 0.62955

150 0.15894 0.99095 0.25978 0.99651 0.60154

151 0.09191 0.99744 0.15327 0.99987 0.56062

152 0.08065 0.99807 0.13413 0.99993 0.55320

153 0.16037 0.99105 0.25767 0.99655 0.60141

154 0.09089 0.99757 0.14976 0.99989 0.55953

109



155 0.18482 0.98750 0.29683 0.99414 0.61582

156 0.12235 0.99500 0.20184 0.99886 0.57951

157 0.10393 0.99658 0.17255 0.99957 0.56816

158 0.08278 0.99796 0.13697 0.99991 0.55441

159 0.04800 0.99961 0.06993 0.99956 0.52928

160 0.75807 0.58413 0.99780 0.60849 0.73712

161 0.06071 0.99893 0.10241 0.99984 0.54047

110


Sl.No.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

66 0.4 0.2 0.4 0.0 75798.54 43926.67 102793.17 697312.63


66 0.67789 0.76984 0.89854 0.79273 0.78475


Discussion

The 161 weight vectors fed-in covers the entire range of weight combinations in

discrete steps. This is yet another test for proving the efficiency of the software

developed. This simulation proves that the program is capable of yielding solution

for any weight combination. Also the fitness value of the optimum solution

obtained can be in the vicinity of global optimum.

111


RANDOM GENERATED WEIGHT VECTORS

Weight vectors Random generated

Method to determine objective function valuesN-R Method

Test system 1

Sub-intervals 24

In these trials weight vectors are continuously generated until the

fitness value has reached close to 0.78475, the one obtained in sub-section 6.1.11.

Table 6.18 gives the simulation results of Test system-1 with random

generated weight vectors. First four weight vectors are of the format shown below

which is necessary for determining extreme values of objective functions.

1 0 0 00 1 0 00 0 1 00 0 0 1

Table 6.18a shows the results of the first trial. A fitness value of

0.78429 has been obtained when the number of attempts reached 257, which is

close to 0.78475, obtained in the case of 161 user-fed weight vectors.

The trial has been repeated 4 times and the results are listed in Table

6.18 b, c, d and e. Time of execution in each case is also given.

112

Table 6.18a Optimum solution- trial 1

Atte-mptNo.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

257 0.530 0.032 0.422 0.016 75787.64 43954.84 102790.65 697655.55

Atte-mptNo.

m(F1) m(F2) m(F3) m(F4) fitness

257 0.69422 0.75820 0.90358 0.78114 0.78429


Table 6.18b Optimum solution – trial 2

Atte-mptNo.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

382 0.440 0.007 0.533 0.020 75805.35 43917.06 102791.67 697191.58

Atte-mptNo.


382 0.66768 0.77381 0.90154 0.79682 0.78496


113

Table 6.18c Optimum solution – trial 3

Atte-mptNo.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

318 0.294 0.167 0.537 0.002 75804.86 43937.15 102784.12 697425.88

Atte-mptNo.


318 0.66841 0.76551 0.91660 0.78890 0.78486


Table 6.18d Optimum solution – trial 4

Atte-mptNo.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

58 0.468 0.031 0.482 0.019 75810.25 43894.45 102798.28 696924.52

Atte-mptNo.


58 0.66033 0.78316 0.88835 0.80584 0.78442


114

Table 6.18e Optimum solution – trial 5

Atte-mptNo.

w1 w2 w3 w4 F1(R)

F2(kg)

F3(kg)

F4(kg)

367 0.305 0.153 0.538 0.004 75808.37 43924.12 102786.85 697270.03

Atte-mptNo.


367 0.66315 0.77090 0.91116 0.79417 0.78484


Discussion

In this test random weight vectors are continuously generated till a fitness value,

close to the one in the case of 161 weight vectors, is obtained. As weight vectors

are generated randomly, the expected solution cannot be guaranteed in a definite

number of attempts. The five trial simulations took 257, 382, 318, 58 and 367

attempts respectively to yield a solution with the expected fitness value. Hence it

can be concluded that in this method the number of attempts for a solution with an

expected fitness value is not definite.

115

6.2 ADVANCED INVESTIGATIONS


GENETICALLY GENERATED WEIGHT VECTORS FOR 24 SUB-INTERVALS: TRIAL 1

Weight vectors Geneticallygenerated


Test system 1

Sub-intervals 24


necessary for determining the extreme values of objective functions.

1 0 0 00 1 0 00 0 1 00 0 0 1

The remaining 12 weight vectors are genetically generated in the first

trial. Fitness is found for all the 16 weight vectors and arranged in the descending

order of fitness. These weight vectors are genetically modified, fitness computed

for each and again arranged in the descending order of fitness. The trials can be

continued a definite number of times or till the change in highest fitness value is

marginal over a few number of generations. In this case the trials are repeated ten

times.

Table 6.19 lists the highest fitness value obtained in each generation.

The highest fitness value obtained in the first generation has been 0.7393. The

116

highest fitness value has reached 0.7868 in the tenth generation. Fig 6.1 shows the

corresponding plot.

Table 6.19 Highest fitness in ten generations

Generations Highest fitness1 0.7393

2 0.7637

3 0.7838

4 0.7838

5 0.7859

6 0.7859

7 0.7862

8 0.7862

9 0.7862

10 0.7868

Fig. 6.1 Variation of highest fitness value in ten generations

117

Table 6.20 lists the weight vectors and fitness values obtained in the

last generation. Table 6.21 lists the objective function values corresponding to the

weight vectors given in Table 6.20 and Table 6.22 lists the corresponding fuzzy

membership functions. Table 6.23 lists the power demand, power allocations,

losses and λ in each sub-interval corresponding to the highest fitness solution in

the last generation, given against sl. no. 1 in Table 6.20. Fig 6.2 shows the

variations in power demand, total thermal generation and total hydro generations

over the 24 sub-intervals.

Table 6.20 Set of weight vectors and corresponding fitness values in the last generation

Sl.no. w1 w2 w3 w4 fitness

1 0.4317 0.2196 0.3487 0.0 0.7868

2 0.4747 0.1939 0.3313 0.0 0.7862

3 0.4464 0.2261 0.3276 0.0 0.7862

4 0.4505 0.2291 0.3204 0.0 0.7859

5 0.4672 0.2064 0.3227 0.0038 0.7847

6 0.4519 0.2308 0.3154 0.0019 0.7838

7 0.4602 0.2159 0.3201 0.0038 0.7835

8 0.4578 0.2004 0.3360 0.0059 0.7831

9 0.4579 0.2211 0.3170 0.0039 0.7826

10 0.0913 0.0 0.8935 0.0152 0.7821

11 0.0590 0.0 0.9225 0.0185 0.7818

12 0.0582 0.0255 0.9018 0.0145 0.7818

13 0.0442 0.0241 0.9157 0.0161 0.7814

14 0.0607 0.0179 0.9000 0.0214 0.7800

15 0.0899 0.0150 0.8689 0.0262 0.7767

16 0.4341 0.1899 0.3624 0.0136 0.7714

118

Table 6.21 Objective function values corresponding to the weight vectors given in Table 6.20

Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

1 75808.35 43887.91 102803.99 696854.01

2 75782.48 43963.74 102792.44 697768.46

3 75811.24 43876.21 102807.95 696716.76

4 75813.04 43870.03 102809.79 696643.80

5 75822.94 43842.12 102816.49 696311.49

6 75829.44 43824.70 102820.91 696104.57

7 75831.32 43819.92 102822.08 696047.73

8 75835.70 43810.22 102823.82 695931.27

9 75836.54 43806.29 102825.85 695886.32

10 75820.36 43962.97 102772.90 697724.41

11 75839.39 43903.15 102784.57 697007.56

12 75832.98 43926.25 102779.67 697283.63

13 75841.14 43901.40 102784.89 696986.53

14 75860.95 43831.00 102802.58 696149.90

15 75878.10 43774.90 102819.92 695488.85

16 75882.45 43704.53 102857.00 694686.84

119

Table 6.22 Fuzzy membership functions corresponding to the weight vectors given in Table 6.20

Sl.no. m(F1) m(F2) m(F3) m(F4)

1 0.67004653 0.78570873 0.88189486 0.80939827

2 0.70803369 0.75438024 0.90401047 0.77845571

3 0.66580514 0.79054292 0.87431057 0.81404265

4 0.66315509 0.79309721 0.87077631 0.81651143

5 0.64861528 0.80462989 0.85795312 0.82775585

6 0.63908044 0.81182612 0.84948132 0.83475753

7 0.63631893 0.81380021 0.84724082 0.83668085

8 0.62988659 0.81780600 0.84390862 0.84062156

9 0.62864626 0.81943288 0.84001396 0.84214265

10 0.65241514 0.75469908 0.94143913 0.77994629

11 0.62445881 0.77941311 0.91908557 0.80420264

12 0.63388040 0.76987147 0.92846782 0.79486118

13 0.62190067 0.78013681 0.91847240 0.80491424

14 0.59281272 0.80922119 0.88459118 0.83322350

15 0.56762432 0.83240144 0.85137133 0.85559190

16 0.56123312 0.86147433 0.78034824 0.88272989

120Table 6.23 Power demand, Power allocations, losses and λ in each sub-interval corresponding to the highest fitness solution given in Table 6.20

Sl.No.

PowerDemand

MW

P1MW

P2MW

P3MW

P4MW

LossesMW

λR/MWh

1 600 163.45 300.69 75.52 73.74 13.40 4.37

2 750 208.80 370.29 96.81 95.29 21.19 4.80

3 590 160.43 296.09 74.11 72.32 12.95 4.35

4 675 186.11 335.37 86.11 84.47 17.06 4.58

5 700 193.67 346.98 89.67 88.06 18.38 4.66

6 525 140.81 266.26 65.02 63.10 10.20 4.17

7 550 148.36 277.71 68.51 66.64 11.22 4.23

8 900 254.26 440.89 118.49 117.22 30.86 5.24

9 1230 354.66 599.91 167.70 166.86 59.13 6.27

10 1250 360.76 609.71 170.75 169.94 61.16 6.34

11 1350 391.29 659.05 186.13 185.42 71.90 6.67

12 1400 406.58 683.90 193.90 193.24 77.62 6.84

13 1200 345.51 585.24 163.14 162.27 56.15 6.18

14 1250 360.76 609.71 170.75 169.94 61.16 6.34

15 1250 360.76 609.71 170.75 169.94 61.16 6.34

16 1270 366.86 619.54 173.81 173.02 63.24 6.40

17 1350 391.29 659.05 186.13 185.42 71.90 6.67

18 1470 428.00 718.91 204.87 204.26 86.04 7.08

19 1330 385.18 649.14 183.04 182.31 69.67 6.60

20 1250 360.76 609.71 170.75 169.94 61.16 6.34

21 1170 336.36 570.61 158.60 157.69 53.26 6.08

22 1050 299.83 512.53 140.60 139.54 42.50 5.70

23 900 254.26 440.89 118.49 117.22 30.86 5.24

24 600 163.45 300.69 75.52 73.74 13.40 4.37

121

Fig 6.2 Power demand, total thermal and total hydro generations corresponding to the

results given Table 6.23

Discussion

Referring to Table 6.19, the highest fitness values obtained after five and ten

generations are respectively 0.7859 and 0.7868. The population has 16 members

and after five generations the number of weight combinations tested is 80 and after

10 generations it is 160. The results shows that just after testing 80 weight

combinations a fitness value higher than the one in the case of 161 weight

combinations is obtained. Hence, when weight vectors are generated and

modified genetically a better solution is normally possible in less number of

attempts, but not always. The fitness value further improved to 0.7868 after testing

160 weight combinations.

122


GENETICALLY GENERATED WEIGHT VECTORS FOR 24 SUB-INTERVALS: TRIAL 2



Test system 1

Sub-intervals 24

This is the second trial of the problem executed in sub-section 6.2.1.

Table 6.24 lists the highest fitness values obtained in each of the ten generations

and Fig. 6.3 shows the corresponding plot.




membership functions. Table 6.28 lists the power demand, power allocations,

losses and λ in each sub-interval corresponding to the highest fitness solution

obtained in the last generation. Fig 6.4 shows the variations in power demand,

total thermal generation and total hydro generations over the 24 sub-intervals.

123



2 0.7815

3 0.7815

4 0.7815

5 0.7815

6 0.7825

7 0.7835

8 0.7835

9 0.7835

10 0.7837

Fig. 6.3 Variation of highest fitness value in ten generations

124

Table 6.25 Set of weight vectors and fitness values in the last generation


1 0.1429 0.0298 0.8095 0.0179 0.7837

2 0.1128 0.1487 0.7333 0.0051 0.7835

3 0.1280 0.0366 0.8232 0.0122 0.7835

4 0.0947 0.1526 0.7526 0 0.7825

5 0.1297 0.0216 0.8270 0.0216 0.7815

6 0.0556 0.0486 0.8854 0.0104 0.7811

7 0.0406 0.0148 0.9299 0.0148 0.7810

8 0.0955 0.1348 0.7697 0 0.7808

9 0.0304 0.0076 0.9430 0.0190 0.7807

10 0.1016 0.0321 0.8449 0.0214 0.7800

11 0.0848 0.0141 0.8905 0.0106 0.7792

12 0.0894 0.1117 0.7989 0 0.7772

13 0.0338 0.0075 0.9323 0.0263 0.7759

14 0.1311 0.0984 0.7486 0.0219 0.7712

15 0.0309 0.0116 0.9575 0 0.7448

16 0.0914 0 0.9086 0 0.7387

125


Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

1 75841.24 43871.11 102792.88 696627.22

2 75840.24 43877.98 102791.04 696708.68

3 75817.79 43957.41 102774.40 697658.47

4 75815.46 43973.91 102771.23 697856.36

5 75856.79 43826.67 102804.39 696099.65

6 75824.18 43960.56 102773.08 697694.99

7 75831.26 43939.49 102776.94 697442.10

8 75805.78 44014.32 102764.81 698343.21

9 75848.55 43880.04 102789.76 696731.76

10 75863.56 43812.94 102808.08 695936.90

11 75805.22 44028.33 102762.53 698511.79

12 75794.95 44067.69 102757.70 698988.55

13 75879.85 43781.73 102817.39 695568.64

14 75896.88 43717.92 102841.16 694823.97

15 75763.22 44316.05 102742.50 702023.49

16 75746.82 44372.55 102743.24 702721.39

126



1 0.62174353 0.79265140 0.90316206 0.81707246

2 0.62321514 0.78981238 0.90668688 0.81431589

3 0.65617789 0.75699774 0.93856446 0.78217753

4 0.65960784 0.75017786 0.94464465 0.77548119

5 0.59892005 0.81101268 0.88112135 0.83492407

6 0.64680631 0.75569544 0.94110216 0.78094181

7 0.63640738 0.76439897 0.93370374 0.78949896

8 0.67382162 0.73348259 0.95694089 0.75900753

9 0.61101476 0.78896295 0.9091427 0.81353488

10 0.58897087 0.81668502 0.87405433 0.84043091

11 0.67464322 0.72769817 0.96130589 0.75330316

12 0.68971234 0.71143501 0.97056135 0.73717087

13 0.56504925 0.82957764 0.85622374 0.85289216

14 0.54004163 0.85594062 0.81069474 0.87808989

15 0.73631430 0.60882876 0.99966892 0.63447589

16 0.76039227 0.58548477 0.99824777 0.61086057

127

Table 6.28 Power demand, Power allocations, losses and λ in each sub-interval corresponding to the highest fitness solution

Sl.No.

PowerDemand

MW

P1MW

P2MW

P3MW

P4MW

LossesMW

λR/MWh

1 600 153.87 308.95 76.12 74.33 13.27 6.07

2 750 202.43 375.77 97.19 95.67 21.06 6.65

3 590 150.64 304.53 74.73 72.93 12.82 6.03

4 675 178.14 342.23 86.60 84.95 16.93 6.35

5 700 186.24 353.38 90.12 88.51 18.25 6.45

6 525 129.62 275.93 65.73 63.81 10.08 5.78

7 550 137.70 286.91 69.18 67.31 11.09 5.88

8 900 251.06 443.65 118.68 117.40 30.78 7.26

9 1230 358.26 596.87 167.50 166.66 59.28 8.67

10 1250 364.76 606.33 170.53 169.71 61.33 8.76

11 1350 397.31 653.96 185.80 185.09 72.17 9.22

12 1400 413.59 677.98 193.52 192.86 77.96 9.45

13 1200 348.50 582.71 162.97 162.09 56.27 8.54

14 1250 364.76 606.33 170.53 169.71 61.33 8.76

15 1250 364.76 606.33 170.53 169.71 61.33 8.76

16 1270 371.27 615.81 173.57 172.77 63.42 8.85

17 1350 397.31 653.96 185.80 185.09 72.17 9.22

18 1470 436.40 711.83 204.42 203.82 86.47 9.78

19 1330 390.80 644.39 182.73 182.00 69.93 9.13

20 1250 364.76 606.33 170.53 169.71 61.33 8.76

21 1170 338.74 568.60 158.46 157.55 53.35 8.41

22 1050 299.75 512.62 140.60 139.53 42.50 7.89

23 900 251.06 443.65 118.68 117.40 30.78 7.26

24 600 153.87 308.95 76.12 74.33 13.27 6.07

128

Fig 6.4 Power demand, total thermal and total hydro generations over the 24 sub-intervals

Discussion

In this case the highest fitness value obtained after ten generations is 0.7837,

slightly less than in the case of 161 weight vectors fed in. The fitness value

obtained is slightly lower than in the case of 161 user-fed weight vectors, but

fairly high. Every trial simulation normally yields a weight vector with fairly high

fitness value.

.

129


GENETICALLY GENERATED WEIGHT VECTORS FOR 72 SUB-INTERVALS



Test system 1

Sub-intervals 72

Table 6.29 lists the highest fitness values obtained in the ten

generations and Figure 6.5 gives the corresponding graph.




membership function values. Table 6.33 lists the power demand, power

allocations, losses and λ in each sub-interval corresponding to the highest fitness

solution given against sl. no.1 in Table 6.30. Figure 6.6 shows the variations in

power demand, total thermal generation and total hydro generations over the 72

sub-intervals.

130



2 0.7730

3 0.7840

4 0.7845

5 0.7846

6 0.7846

7 0.7846

8 0.7846

9 0.7847

10 0.7847

Figure 6.5 Variation of highest fitness value in ten generations

131

Table 6.30 Set of weight vectors and fitness values in the last generation

Sl.no. w1 w2 w3 w4 fitness String

no.

1 0.0485 0.0373 0.8955 0.0187 0.7847 1

2 0.0478 0.0074 0.9228 0.0221 0.7846 2

3 0.0669 0.0282 0.8838 0.0211 0.7845 3

4 0.0036 0.0547 0.9270 0.0146 0.7840 4

5 0.0601 0.0318 0.8869 0.0212 0.7840 11

6 0 0.0554 0.9299 0.0148 0.7838 5

7 0.1058 0.0273 0.8567 0.0102 0.7838 6

8 0.0118 0 0.9725 0.0157 0.7835 7

9 0 0.0379 0.9432 0.0189 0.7831 8

10 0.0547 0.0073 0.9124 0.0255 0.7826 10

11 0.0111 0.0517 0.9299 0.0074 0.7805 13

12 0.1051 0.0373 0.8508 0.0068 0.7799 14

13 0.0111 0.0221 0.9410 0.0258 0.7797 15

14 0.0449 0.0300 0.8989 0.0262 0.7794 9

15 0 0.0407 0.9333 0.0259 0.7772 12

16 0 0.0304 0.9696 0 0.7543 16

132


Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

Stringno.

1 226234.69 129565.27 306408.62 2061150.62 1

2 226238.73 129554.36 306411.57 2061021.29 2

3 226249.81 129504.90 306425.81 2060436.70 3

4 226218.09 129659.46 306383.97 2062268.32 4

5 226257.99 129485.22 306431.45 2060204.16 11

6 226222.38 129648.30 306386.73 2062135.57 5

7 226077.90 130080.72 306304.17 2067320.68 6

8 226155.81 129881.40 306335.82 2064919.31 7

9 226254.07 129543.75 306414.21 2060895.15 8

10 226282.21 129419.35 306451.50 2059427.45 10

11 226109.55 130077.51 306302.67 2067278.39 13

12 226038.90 130255.08 306281.44 2069430.09 14

13 226317.60 129347.75 306474.70 2058585.82 15

14 226325.05 129307.02 306489.11 2058109.05 9

15 226347.40 129274.25 306500.74 2057725.64 12

16 225986.92 130804.26 306240.14 2076127.57 16

133



Stringno.

1 0.61002661 0.80400159 0.89598343 0.82894890 1

2 0.60810815 0.80550917 0.89420347 0.83041373 2

3 0.60285291 0.81234135 0.88560680 0.83703456 3

4 0.61790015 0.79099016 0.91085549 0.81629021 4

5 0.59897486 0.81505854 0.88220420 0.83966824 11

6 0.61586632 0.79253208 0.90919464 0.81779370 5

7 0.68439625 0.73280013 0.95901478 0.75906896 6

8 0.64744043 0.76033336 0.93991228 0.78626603 7

9 0.60083100 0.80697478 0.89261101 0.83184235 8

10 0.58748359 0.82415821 0.87010712 0.84846496 10

11 0.66938551 0.73324428 0.95991949 0.75954792 13

12 0.70289586 0.70871511 0.97272784 0.73517858 14

13 0.57069893 0.83404776 0.85610781 0.85799699 15

14 0.56716235 0.83967518 0.84740978 0.86339669 9

15 0.55656169 0.84420139 0.84039489 0.86773911 12

16 0.72755470 0.63285459 0.99765110 0.65932513 16

134

Table 6.33 Power demand, Power allocations, losses and λ in each sub-intervalcorresponding to the highest fitness solution given against sl. no. 1 in Table 6.30

Sl.No.

PowerDemand

MW

P1MW

P2MW

P3MW

P4MW

LossesMW

λR/MWh

1 600 153.63 309.94 75.76 73.96 13.28 6.23

2 750 202.72 375.60 97.14 95.60 21.07 6.84

3 590 150.36 305.60 74.34 72.53 12.83 6.19

4 675 178.17 342.64 86.39 84.73 16.93 6.53

5 700 186.35 353.60 89.96 88.35 18.26 6.63

6 525 129.10 277.50 65.22 63.28 10.09 5.93

7 550 137.27 288.28 68.72 66.83 11.10 6.03

8 900 251.88 442.32 118.94 117.65 30.79 7.47

9 1230 360.20 593.02 168.46 167.63 59.32 8.95

10 1250 366.78 602.34 171.53 170.73 61.38 9.05

11 1350 399.65 649.21 187.03 186.34 72.23 9.52

12 1400 416.10 672.85 194.86 194.22 78.02 9.77

13 1200 350.35 579.10 163.86 163.00 56.31 8.81

14 1250 366.78 602.34 171.53 170.73 61.38 9.05

15 1250 366.78 602.34 171.53 170.73 61.38 9.05

16 1270 373.35 611.67 174.61 173.83 63.47 9.14

17 1350 399.65 649.21 187.03 186.34 72.23 9.52

18 1470 439.13 706.17 205.91 205.33 86.55 10.11

19 1330 393.08 639.80 183.91 183.20 69.98 9.43

20 1250 366.78 602.34 171.53 170.73 61.38 9.05

135

Table 6.33 continued

Sl. No. PowerDemand

MW

P1MW

P2MW

P3MW

P4MW

LossesMW

λR/MWh

21 1170 340.49 565.21 159.29 158.39 53.38 8.67

22 1050 301.09 510.14 141.17 140.11 42.52 8.13

23 900 251.88 442.32 118.94 117.65 30.79 7.47

24 600 153.63 309.94 75.76 73.96 13.28 6.23

25 600 153.63 309.94 75.76 73.96 13.28 6.23

26 750 202.72 375.60 97.14 95.60 21.07 6.84

27 590 150.36 305.60 74.34 72.53 12.83 6.19

28 675 178.17 342.64 86.39 84.73 16.93 6.53

29 700 186.35 353.60 89.96 88.35 18.26 6.63

30 525 129.10 277.50 65.22 63.28 10.09 5.93

31 550 137.27 288.28 68.72 66.83 11.10 6.03

32 900 251.88 442.32 118.94 117.65 30.79 7.47

33 1230 360.20 593.02 168.46 167.63 59.32 8.95

34 1250 366.78 602.34 171.53 170.73 61.38 9.05

35 1350 399.65 649.21 187.03 186.34 72.23 9.52

36 1400 416.10 672.85 194.86 194.22 78.02 9.77

37 1200 350.35 579.10 163.86 163.00 56.31 8.81

38 1250 366.78 602.34 171.53 170.73 61.38 9.05

39 1250 366.78 602.34 171.53 170.73 61.38 9.05

40 1270 373.35 611.67 174.61 173.83 63.47 9.14

136


Sl. No. PowerDemand

MW

P1MW

P2MW

P3MW

P4MW

LossesMW

λR/MWh

41 1350 399.65 649.21 187.03 186.34 72.23 9.52

42 1470 439.13 706.17 205.91 205.33 86.55 10.11

43 1330 393.08 639.80 183.91 183.20 69.98 9.43

44 1250 366.78 602.34 171.53 170.73 61.38 9.05

45 1170 340.49 565.21 159.29 158.39 53.38 8.67

46 1050 301.09 510.14 141.17 140.11 42.52 8.13

47 900 251.88 442.32 118.94 117.65 30.79 7.47

48 600 153.63 309.94 75.76 73.96 13.28 6.23

49 650 169.99 331.71 82.84 81.13 15.66 6.43

50 700 186.35 353.60 89.96 88.35 18.26 6.63

51 600 153.63 309.94 75.76 73.96 13.28 6.23

52 700 186.35 353.60 89.96 88.35 18.26 6.63

53 750 202.72 375.60 97.14 95.60 21.07 6.84

54 500 120.93 266.74 61.73 59.74 9.13 5.84

55 525 129.10 277.50 65.22 63.28 10.09 5.93

56 800 219.10 397.72 104.36 102.91 24.09 7.05

57 1270 373.35 611.67 174.61 173.83 63.47 9.14

58 1280 376.64 616.34 176.16 175.39 64.53 9.19

59 1300 383.21 625.71 179.25 178.51 66.68 9.28

60 1450 432.55 696.63 202.74 202.15 84.06 10.01

137


61 1250 366.78 602.34 171.53 170.73 61.38 9.05

62 1300 383.21 625.71 179.25 178.51 66.68 9.28

63 1200 350.35 579.10 163.86 163.00 56.31 8.81

64 1250 366.78 602.34 171.53 170.73 61.38 9.05

65 1300 383.21 625.71 179.25 178.51 66.68 9.28

66 1400 416.10 672.85 194.86 194.22 78.02 9.77

67 1300 383.21 625.71 179.25 178.51 66.68 9.28

68 1200 350.35 579.10 163.86 163.00 56.31 8.81

69 1100 317.50 533.00 148.69 147.69 46.88 8.35

70 1000 284.68 487.41 133.71 132.58 38.38 7.91

71 900 251.88 442.32 118.94 117.65 30.79 7.47

72 600 153.63 309.94 75.76 73.96 13.28 6.23

138

Figure 6.6 Power demand, total thermal and total hydro generations over72 sub-intervals

Discussion

In this case the highest fitness value obtained after ten generations is 0.7847,

almost equal to the fitness obtained in the case of 161 weight vectors fed in. The

fitness value reached almost saturation in five generations after which the

improvement was marginal.

139

6.2.4 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING:

OPTIMIZATION INTERVAL-EXECUTION TIME CHARACTERISTIC

Weight vectorsGeneticallygenerated


Test system 1

Sub-intervals 24,48,72,96,

120,144,168

Seven computations have been performed with sub-intervals 24, 48, 72,

96, 120, 144 and 168 and the execution time in each case is given in Table 6.34.

Figure 6.7 shows the corresponding graph.

Table 6.34 Optimization interval - execution time

Sl. no. Optimizationinterval (h) Execution time (s)

1 24 5.594

2 48 35.985

3 72 149.531

4 96 335.078

5 120 651.422

6 144 1383.406

7 168 1807.586

140

Figure 6.7 Optimization interval – execution time characteristic

Discussion

The execution time has varied from 5.594 s for 24 sub-intervals to approximately

30 minutes for 168 sub-intervals.

The trend line of the characteristic as a second degree polynomial is

t = 0.11 h2 - 8.32 h + 155.5 s

where ‘t’ is the execution time in seconds and ‘h’ is the optimization interval in

hours.

141

6.2.5 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING: COST-

TOTAL EMISSION CHARACTERISTIC

Weight vectorsGeneticallygenerated


Test system 1

Sub-intervals 72

Table 6.31 given in sub-section 6.2.3, is arranged in the ascending

order of cost and listed in Table 6.35.

A plot, Figure 6.8, is made between cost and total emission using Table

6.35. Cost represents the total cost in the72 sub-intervals and total emission

represents the total of CO2, SO2 and NOx emissions in the72 sub intervals. The

plot relates the values given in columns 2 and 6 of Table 6.35 for the 16 weight

vectors obtained in the last generation.

142

Table 6.35 Table 6.31 arranged in the ascending order of cost

Sl.no. F1

(R)F2

(kg)F3

(kg)F4

(kg)

Total Emission(F2+F3+F4)

kg

1 225986.92 130804.26 306240.14 2076127.57 2513172

2 226038.90 130255.08 306281.44 2069430.09 2505967

3 226077.90 130080.72 306304.17 2067320.68 2503706

4 226109.55 130077.51 306302.67 2067278.39 2503659

5 226155.81 129881.40 306335.82 2064919.31 2501137

6 226218.09 129659.46 306383.97 2062268.32 2498312

7 226222.38 129648.30 306386.73 2062135.57 2498171

8 226234.69 129565.27 306408.62 2061150.62 2497125

9 226238.73 129554.36 306411.57 2061021.29 2496987

10 226249.81 129504.90 306425.81 2060436.70 2496367

11 226254.07 129543.75 306414.21 2060895.15 2496853

12 226257.99 129485.22 306431.45 2060204.16 2496121

13 226282.21 129419.35 306451.50 2059427.45 2495298

14 226317.60 129347.75 306474.70 2058585.82 2494408

15 226325.05 129307.02 306489.11 2058109.05 2493905

16 226347.40 129274.25 306500.74 2057725.64 2493501

143

Figure 6.8 Cost-total emission characteristic

Discussion

In real life optimization problems the objectives are generally conflicting in

nature. In multi-objective hydrothermal scheduling problems cost and emission are

generally found conflicting.

The plot shows conflicting nature of the objectives, cost and total emission.

144

6.2.6 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING:

OBJECTIVE FUNCTION VALUES DETERMINED BY THREE DIFFERENT METHODS

Weight vectors 4

Method to determine objective function values N-R method, genetic search,

random search

Test system 1

Sub-intervals 24

System 1 is tested for four weight vectors with objective function values

determined by

1. N-R Method

2. a method using a genetic search

3. a method using a random search

6.2.6.1 Determining objective function values by N-R method

Table 6.36a lists weight vectors and fitness. Table 6.36b lists objective function

values and Table 6.36c lists the fuzzy membership functions of objective function

values. Table 6.37 lists number of iterations water withdrawals by hydro units and

coal consumption by coal-constrained thermal units for each weight vector.

145

Table 6.36a Weight vectors and fitness values


1 1.0 0.0 0.0 0.0 0.33535

2 0.0 1.0 0.0 0.0 0.49950

3 0.0 0.0 1.0 0.0 0.73771

4 0.0 0.0 0.0 1.0 0.53900

Table 6.36b Objective function values

Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

1 75583.64 45789.70 103072.37 720774.15

2 76250.78 43369.68 103243.45 691241.83

3 75765.51 44339.13 102742.33 702307.73

4 76212.06 43372.02 103193.89 691183.05

Table 6.36c Fuzzy membership functions


1 1.00000 0.00000 0.34140 0.00000

2 0.00000 1.00000 0.00000 0.99801

3 0.72739 0.59941 1.00000 0.62405

4 0.05804 0.99903 0.09891 1.00000

146

6.37 No. of iterations, water withdrawals and coal-consumption corresponding to each weight vector

Weight vector No. ofiterations

Waterwithdrawal,Unit-3 (m3)


Coalconsumption,Unit-1 (kg)

[1 0 0 0] 5 100000.00 110000.00 676000.00

[0 1 0 0] 5 100000.00 110000.00 676000.00

[0 0 1 0] 5 100000.00 110000.00 676000.00

[0 0 0 1] 5 100000.00 110000.00 676000.00

6.2.6.2 Determining objective function values by a genetic search

Table 6.38a lists weight vectors and fitness values. Table 6.38b lists objective

function values and Table 6.38c lists the fuzzy membership functions of objective

function values. Table 6.39 lists number of iterations, water withdrawals by hydro

units and coal consumption by coal-constrained thermal units corresponding to

each weight vector. A convergence tolerance of 10 (m3 of water) is specified for

hydro units and 50 (kg of coal) is specified for the coal-constrained thermal unit.

Table 6.38a Weight vectors and fitness values


1 1.0 0.0 0.0 0.0 0.33056

2 0.0 1.0 0.0 0.0 0.49899

3 0.0 0.0 1.0 0.0 0.73953

4 0.0 0.0 0.0 1.0 0.54485

147


Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

1 75585.67 45793.12 103075.68 720822.25

2 43369.27 43369.27 103236.01 691223.56

3 75763.36 44333.95 102738.46 702237.12

4 76201.92 43368.46 103179.69 691114.04



1 1.00000 0.00000 0.32224 0.00000

2 0.00000 0.99967 0.00000 0.99631

3 0.73075 0.60180 1.00000 0.62559

4 0.06619 1.00000 0.11319 1.00000






[1 0 0 0] 75 100003.20 110000.62 675981.22

[0 1 0 0] 92 100003.16 110003.68 676043.61

[0 0 1 0] 85 100000.39 109993.95 676048.38

[0 0 0 1] 79 100008.64 109990.58 676025.14

148

6.2.6.3 Determining objective function values by a random search-trial 1



values. Table 6.41 lists number of iterations, water withdrawals by hydro units and

coal consumption by coal-constrained thermal units corresponding to each weight

vector. Convergence tolerances used in this case is same as in the case of genetic

search method.

Table 6.40a weight vectors and fitness values


1 1.0 0.0 0.0 0.0 0.33232

2 0.0 1.0 0.0 0.0 0.49953

3 0.0 0.0 1.0 0.0 0.73773

4 0.0 0.0 0.0 1.0 0.54163


Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

1 75581.59 45788.31 103069.92 720752.77

2 76243.20 43366.27 103232.78 691180.90

3 75762.78 44334.75 102738.15 702246.36

4 76202.20 43369.21 103180.46 691124.64

149



1 1.00000 0.00000 0.32926 0.00000

2 0.00000 1.00000 0.00000 0.99810

3 0.72614 0.60014 1.00000 0.62462

4 0.06197 0.99878 0.10577 1.00000

Table 6.41 No. of iterations, water withdrawals and coal-consumption corresponding to each weight vector





[1 0 0 0] 51 99997.654 109994.816 675968.168

[0 1 0 0] 67 100001.101 110008.682 676022.376

[0 0 1 0] 54 100000.770 110008.385 676026.476

[0 0 0 1] 64 99997.847 110000.707 675998.752

150

6.2.6.4 Determining objective function values by a random search-trial 2.



values. Table 6.43 lists number of iterations, water withdrawals by hydro units and

coal consumption by coal-constrained thermal units corresponding to each weight

vector. A convergence tolerance of 5 (m3 of water) is specified for hydro units and

5 (kg of coal) is specified for coal-constrained thermal units.

Table 6.42a weight vectors and fitness values


1 1.0 0.0 0.0 0.0 0.33210

2 0.0 1.0 0.0 0.0 0.49950

3 0.0 0.0 1.0 0.0 0.73632

4 0.0 0.0 0.0 1.0 0.54180


Sl.no.

F1(R)

F2(kg)

F3(kg)

F4(kg)

1 75584.28 45790.66 103073.30 720787.69

2 76244.01 43368.77 103234.35 691214.41

3 75766.56 44340.70 102743.97 702330.03

4 76203.25 43371.49 103182.10 691155.68

151



1 1.00000 0.00000 0.32841 0.00000

2 0.00000 1.00000 0.00000 0.99802

3 0.72370 0.59869 1.00000 0.62289

4 0.06177 0.99887 0.10655 1.00000






[1 0 0 0] 93 100000.52 110001.24 675999.30

[0 1 0 0] 83 100004.40 109998.39 675998.11

[0 0 1 0] 78 99999.79 110003.18 675998.72

[0 0 0 1] 98 100003.75 110001.21 675997.89

6.2.6.5 Comparison of genetic and random search methods

Tables 6.39 and 6.41 list the number of iterations, water withdrawals by hydro

units and coal-consumption by coal-constrained thermal units corresponding to the

four weight vectors tested using the same convergence tolerances. Results are

almost the same in both the cases. Number of iterations mainly depends on

152

convergence tolerance values. Number of iterations and hence time of execution

would have been much more if fuzzy multipliers were not used.

6.2.6.6 Comparison of execution time

Table 6.44 lists execution time in the three different cases. N-R method is found

very fast, genetic search method is found very slow and random search method is

found moderately fast.

Table 6.44 Execution time for the three methods

Method Execution time(s)

N-R Method 1.75

Genetic search 2145.48

Random searchTrial-1 43.031

Random searchTrial-2 60.89

Discussion

N-R method of determining objective function values is found very fast and

accurate. In genetic search method high accuracy level is very difficult as low

convergence tolerances result in very large number of attempts. Also execution

time is found very high due to numerous operations to be performed on binary

strings. In random search method good accuracy is possible. Execution time is

also moderate.

153

In N-R method the time of execution does not vary much when simulation is

repeated. In genetic search method the variation is marginal and in random search

method the variation is considerable.

154

CHAPTER 7

CONCLUSION AND SCOPE FOR FUTURE WORK

7.1 CONCLUSION

The major strength of the main algorithm is the possibility of generating a large

number of non-inferior solutions and identifying the best among them whereas in

conventional algorithm only a limited number of non-inferior solutions are

generated. Also, a set of weight combinations with high fitness values can be

obtained, normally the weight combinations obtained in the last generation. This

provides an opportunity for one to select a weight combination depending on the

priority to be given to various objectives.

Further, a comparison is performed among the three methods used for determining

objective function values in the optimization interval as the main algorithm is the

same in all three cases. The Newton-Raphson method solves the problem with

quite good accuracy in minimum time. The major drawback with this method is

the numerous mathematical operations involved in forming the Hessian and

Jacobian. The genetic search method eliminates all laborious mathematical steps

and the problem becomes very simple. The number of generations to locate

optimum λ in sub-intervals is reduced by applying search space reduction

technique. Number of iterations to get hydro and coal constraints satisfied is

reduced by using Fuzzy multipliers. But, still the method is slow due to numerous

operations to be performed on binary strings, which is its major drawback.

Random search method overcomes the major drawbacks of other two methods to a

great extend. This method is much faster than genetic search method and the

simplest of all.

155

7.2 SCOPE FOR FUTURE WORK

As the search space is extremely vast or almost infinite, global optimum cannot be

guaranteed for the best compromise solution. Modifications can be done to the

main algorithm to increase the genetic diversity by fitness sharing or any other

method to explore global optimum. But this will definitely result in a much higher

execution time. Numerous simulations performed so far indicate that, the fitness

value corresponding to the global optimum cannot be much higher than the highest

value obtained so far. But still an attempt can be made to reach the global

optimum.

REFERENCES

1. Abdulla Konak, David W. Coit, Alice E. Smith (2006) “Multi-objectiveOptimization using genetic algorithms: A tutorial”, Reliability Engineeringand System Safety, Elsevier 91, pp 992-1007.

2. Abido M.A (2003) “Environmental/Economic Power Dispatch using Multi-objective Evolutionary Algorithm”, IEEE Transactions on Power Systems,Vol. 18, No.4, pp 1529-1537.

3. Allen J Wood, Bruce F Wollenberg (2005) “Power Generation, Operationand Control”, John Wiley & Sons, New Delhi.

4. Basu M, Chakrabarti R.N , Chattopadhyay P.K, Ghoshal T.K (2006)“Economic Emission Load Dispatch of Fixed Head hydrothermal PowerSystems through Interactive Fuzzy satisfying Method and SimulatedAnnealing Technique”, Journal of Institution of Engineers India, vol. 86,pp 275-281.

5. Catalao J.P.S, Mariano S.J.P.S, Mendes V.M.F, Ferreira L.A.F.M (2008)“Nonlinear optimization method for short-term hydro schedulingconsidering head-depencency”, European Transaction on Electrical Power.DOI: 10.1002/etep.301

6. Catalao J.P.S, Mariano S.J.P.S, Mendes V.M.F, Ferreira L.A.F.M (2009)“Scheduling of Head Sensitive Cascaded Hydro Systems: A NonlinearApproach”, IEEE Transactions on Power System, Vol. 24, No.1, pp 337-346.

7. Clodomiro Unsihuay, Marangon Lima, Zambroni D’Souza (2007) “Shrot-Term Operation Planning of Integrated Hydrothermal and Natural GasSystems”, 2007 IEEE PES Power Tech. Conference, July 1-5, Switzerland.

8. David E. Goldberg (2004) “Genetic Algorithms”, PHI, New Delhi.

9. Esteban Gil, Julian Bustos, Hugh Rudnick (2003) “Short-TermHydrothermal Generation Scheduling Model using Genetic Algorithm”,IEEE Transactions on Power Systems, vol. 18, No.4, pp 1256-1264.

10. Jarnail S. Dhillon, Dhillon J.S , Kothari D.P (2007) “Multi-objective ShortTerm Hydrothermal Scheduling based on Heuristic Search Technique”,Asian Journal of Information Technology, 6(4), pp 447-454.

11. Jaydeep Chakravorty, Sandeep Chakravorty, Samarjit Ghosh (2009) “AFuzzy Based Efficient Load Flow Analysis”, International Journal ofComputer and Electrical Engineering , Vol. 1, No.4.

12. Kothari D.P, Dhillon J.S (2004) “Power System Optimization”, PrenticeHall of India Pvt. Ltd., New Delhi.

13. Marco Dorigo, Thomas Stutzle (2004), “Ant Colony Optimization”, MITPress, London.

14. Maurice Clerc (2007) , “Particle Swarm Optimization”, ISTE Ltd., London.

15. Nallasivan C, Suman D.S, Joseph Henry , Ravichandran S (2006), “Anovel approach for short-term hydrothermal scheduling using hybridtechnique”, 2006 IEEE Power India Conference, 5pp , DOI :10.1109/POWERI.2006.1632593 .

16. Po-Hung Chen (2008), “Pumped-Storage Scheduling Using EvolutionaryParticle Swarm Optimization”, IEEE Transactions on Energy Conversion,vol. 23, Issue 1, pp 294-301.

17. Ruey-Hsun Liang , Ming-Huei Ke , Yie-Tone Chen (2009), “Co-evolutionary Algorithm Based on Lagrangian Method for HydrothermalGeneration Scheduling”, IEEE Transactions on Power Systems, Vol 24,Issue 2, pp 499-507.

18. Singeresu S. Rao (2003) “Engineering Optimization”, New AgeInternational (P) Ltd. New Delhi.

LIST OF PAPERS PUBLISHED BASED ON THIS THESIS

I. NATIONAL CONFERENCE

Nil.

II. INTERNATIONAL CONFERENCE

1. Abraham George, ‘Emission constrained thermal dispatch and

hydrothermal scheduling based on genetic algorithm: search space

reduction technique’, 18th Annual Symposium, IEEE Bangalore Section, 29

August 2009.

III. INTERNATIONAL JOURNALS

1. Abraham George, Channa Reddy M., Sivaramakrishnan A.Y., ‘Short term

Hydrothermal Scheduling based on Multi-objective Genetic Algorithm’,

International Journal of Electrical Engineering, ISSN 0974-2158, Volume 3,

Number 1, 2010, 13-26, Research India Publications, New Delhi.

SNIP : 0.076 SJR : 0.034

2. Abraham George, Channa Reddy M., Sivaramakrishnan A.Y. ‘Multi-

objective, short-term hydrothermal scheduling based on two novel search

techniques’, International Journal of Engineering Science and Technology,

ISSN: 0975-5462, Vol. 2(12), 2010, 7021- 7034.

Indexed in DOAJ, EBSCO, SCOPUS and PROQUEST

CURRICULAM VITAE

1. Personal background

1. Name Abraham George

2. Date of Birth 25 May 1957

3. Address No.9, Heerachand Road, Cox Town, Bangalore-560005.

4. Contact No. 080- 25488259, 919986769176 (Mobile)

5. Marital status Married

6. E-mail [email protected]

2. Educational Qualifications

Degree/Diploma

Subject Year ofcomp-letion

InstitutionStudied

Board/University

Class Percen-tage

SSLC 1972 Boys’ High School,Adoor, Kerala.

BSE,Kerala

I 69.8

Pre-Degree

PCM 1974 NSS College,Pandalam, Kerala

KeralaUniversity

I 64.8

B.Sc.(Engg.)

ElectricalEngg.

1979 NSS Engg. College,Palghat, Kerala

CalicutUniversity

I 63

M.Sc.(Engg.)

PowerSystem

1983 Govt. Engg. College,Trichur, Kerala

CalicutUniversity

II 57

3. Work Experience

InstitutionWorked

Period Duration Position TotalTeachingExperience

R V College of Engg.,Bangalore.

Sep. 83 -Jan 86

2 Yrs. ,4 months

Lecturer

27 years

UniversityVisveswaraya Collegeof Engg., Bangalore

Feb 86 –Aug 89

2 Yrs,6 months

Lecturer

Islamiah Institute ofTechnology, Bangalore

Sep 89 -Mar 02

12 Yrs,6 months

Asst. Professor

New Horizon College ofEngg. , Bangalore

Aug 02 -March 07

4 yrs,7 months

Asst. Professor

M.S EngineeringCollege,Bangalore.

April 07-date

4 years,3 months

Professor

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