Post on 23-Apr-2018
transcript
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:mcreddy@yahoo.com# 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
emission minimization objective
64
6.1.4 Short-term hydrothermal scheduling : CO2
emission minimization objective
70
6.1.5 Multi-objective, short-term hydrothermal
scheduling : determining the best compromise
solution considering 24 sub-intervals
76
6.1.6 Multi-objective, short-term hydrothermal
scheduling : determining the best compromise
solution considering 72 sub-intervals
80
6.1.7 Multi-objective, short-term hydrothermal
scheduling : determining the best compromise
solution considering 168 sub-intervals
82
6.1.8 Multi-objective, short-term hydrothermal
scheduling with random generation of weight
vectors : trial -1
84
6.1.8.1 Generation of random weight vector of
four weights
84
6.1.9 Multi-objective, short-term hydrothermal
scheduling with random generation of weight
vectors : trial -2
88
6.1.10 Effectiveness of search space reduction
technique
ix
91
6.1.11 Multi-objective, short-term hydrothermal
scheduling with 161 user-fed weight vectors
93
6.1.12 Multi-objective, short-term hydrothermal
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
6.2.2 Multi-objective, short term hydrothermal
scheduling with genetically generated weight
vectors for 24 sub-intervals- trial 2
122
6.2.3 Multi-objective, short term hydrothermal
scheduling with genetically generated weight
vectors for 72 sub-intervals
129
6.2.4 Multi-objective, short term hydrothermal
scheduling : optimization interval –execution
time characteristic
139
6.2.5 Multi-objective, short term hydrothermal
scheduling : cost –total emission characteristic
141
6.2.6 Multi-objective, short term hydrothermal
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
6.2.6.2 Determining objective function values
by a genetic search
146
6.2.6.3 Determining objective function values
by a random search – trial 1
148
6.2.6.4 Determining objective function values
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
4.2 Search space reduction in Generation 2 42
4.3 Search space reduction in Generation 3 42
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
6.2a Objective function values in optimization sub-intervals 59
6.2b λ, power allocations and losses in optimization sub-intervals 61
6.3a Objective function values in optimization sub-intervals 65
6.3b λ, power allocations and losses in optimization sub-intervals 67
6.4a Objective function values in optimization sub-intervals 71
6.4b λ, power allocations and losses in optimization sub-intervals 73
6.5a Weight vectors and objective function values in the
optimization interval
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.
6.7a Weight vectors and objective function values in the
optimization interval
80
6.7b Membership function values and fitness 81
6.8 Optimum solution 81
6.9a Weight vectors and objective function values in the
optimization interval
82
6.9b Membership function values and fitness 83
6.10 Optimum solution 83
6.11a Weight vectors and objective function values in the
optimization interval
85
6.11b Membership function values and fitness 86
6.12 Optimum solution 86
6.13a Weight vectors and objective function values in the
optimization interval
89
6.13b Membership function values and fitness 89
6.14 Optimum solution 90
6.15 Effectiveness of search space reduction technique 91
6.16a Weight vectors and objective function values in the
optimization interval
94
6.16b Membership function values and fitness 102
6.17 Optimum solution 110
6.18a Optimum solution –trial 1 112
6.18b Optimum solution –trial 2 112
xiii
LIST OF TABLES CONTINUED
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
vectors given in Table 6.20
119
6.23 Power demand, power allocations , losses and λ in each sub-
interval corresponding to the highest fitness solution
120
6.24 Highest fitness in 10 generations 123
6.25 Set of weight vectors and fitness values in the last generation 124
6.26 Objective function values corresponding to the weight
vectors given in Table 6.25
125
6.27 Fuzzy membership functions corresponding to the weight
vectors given in Table 6.25
126
6.28 Power demand, power allocations , losses and λ in each sub-
interval corresponding to the highest fitness solution
127
6.29 Highest fitness in 10 generations 130
6.30 Set of weight vectors and fitness values in the last generation 131
6.31 Objective function values corresponding to the weight
vectors given in Table 6.30
132
xiv
LIST OF TABLES CONTINUED
Table
No.
Title of Table Page
no.
6.32 Fuzzy membership functions corresponding to the weight
vectors given in Table 6.30
133
6.33 Power demand, power allocations , losses and λ in each sub-
interval corresponding to the highest fitness solution
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
6.38a Weight vectors and fitness values 146
6.38b Objective function values 147
6.38c Fuzzy membership functions 147
6.39 No. of iterations, water withdrawals, coal-consumption
corresponding to each weight vector
147
6.40a Weight vectors and fitness values 148
6.40b Objective function values 148
6.40c Fuzzy membership functions 149
6.41 No. of iterations, water withdrawals, coal-consumption
corresponding to each weight vector
149
6.42a Weight vectors and fitness values 150
6.42b Objective function values 150
xv
LIST OF TABLES CONTINUED
Table
No.
Title of Table Page
no.
6.42c Fuzzy membership functions 151
6.43 No. of iterations, water withdrawals, coal-consumption
corresponding to each weight vector
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.
6.3 Variation of highest fitness value in ten generations 123
6.4 Power demand, total thermal and total hydro generations
corresponding to 24 sub-intervals
128
6.5 Variation of highest fitness value in ten generations 130
6.6 Power demand, total thermal and total hydro generations
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
Table 4.2 Search space reduction in Generation 2
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
Table 4.3 Search space reduction in Generation 3
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 water withdrawal (unit 4) = 110000.0 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
Weight vector [0 1 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
NOx emission minimization objective and hence the corresponding weight vector
is fed-in as [0 1 0 0].
The solution converged in 5 iterations and 0.407 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 6.2a. Values of λ, power allocations and
losses in sub-intervals are listed in Table 6.2b.
59
Table 6.2a Objective function values in optimization sub-intervals
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
Table 6.2a continued
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
Table 6.2b 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 3.83 166.35 342.82 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
Table 6.2b continued
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 NOx emission = 43369.68 kg
Total SO2 emission =103243.45 kg
Total CO2 emission = 691241.83 kg
Total water withdrawal (unit 3) = 100000.0 m3
Total water withdrawal (unit 4) = 110000.0 m3
Total coal consumption (unit 1) = 676000.0 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
MINIMIZATION OBJECTIVE
Weight vector [0 0 1 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
SO2 emission minimization objective and hence the corresponding weight vector
is fed-in as [0 0 1 0].
The solution converged in 5 iterations and 0.36 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 6.3a. Values of λ, power allocations and
losses in sub-intervals are listed in Table 6.3b.
65
Table 6.3a Objective function values in optimization sub-intervals
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
Table 6.3a continued
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
Table 6.3b 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 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
Table 6.3b continued
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
The total cost, emissions, water withdrawal and coal-consumption for
the 24 sub-intervals are as given below.
Total cost = R 75765.51
Total NOx emission = 44339.13 kg
Total SO2 emission = 102742.33 kg
Total CO2 emission = 702307.73 kg
Total water withdrawal (unit 3) = 100000.0 m3
Total water withdrawal (unit 4) = 110000.0 m3
Total coal consumption (unit 1) = 676000.0 kg
No. of iterations= 5
69
Time of execution= 0.36 s
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
MINIMIZATION OBJECTIVE
Weight vector [0 0 0 1]
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
CO2 emission minimization objective and hence the corresponding weight vector
is fed-in as [0 0 0 1].
The solution converged in 5 iterations and 0.422 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 6.4a. Values of λ, power allocations and
losses in sub-intervals are listed in Table 6.4b.
71
Table 6.4a Objective function values in optimization sub-intervals
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
Table 6.4b 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 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
Table 6.4b continued
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
The total cost, emissions, water withdrawal and coal-consumption for
the 24 sub-intervals are as given below.
Total cost = R 76212.06
Total NOx emission = 43372.02 kg
Total SO2 emission = 103193.89 kg
Total CO2 emission = 691183.05 kg
Total water withdrawal (unit 3) = 100000.0 m3
Total water withdrawal (unit 4) = 110000.0 m3
Total coal consumption (unit 1) = 676000.0 kg
No. of iterations = 5
75
Time of execution= 0.422 s
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
Method to determine objective function values N-R Method
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
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
3 0.72739 0.59941 1.00000 0.62405 0.73771
78
Time of execution = 1.016000 s
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
F2max = 45789.70 corresponding to sl. no. 1 in Table 6.5a
F2min = 43369.68 corresponding to sl. no. 2 in Table 6.5a
F3max = 103243.45 corresponding to sl. no. 2 in Table 6.5a
F3min = 102742.33 corresponding to sl. no. 3 in Table 6.5a
F4max = 720774.15 corresponding to sl. no. 1 in Table 6.5a
F4min = 691183.05 corresponding to sl. no. 4 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
6. 1.6 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING:
DETERMINING THE BEST COMPROMISE SOLUTION CONSIDERING 72 SUB-
INTERVALS
Weight vectors 4
Method to determine objective function values N-R Method
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.
Table 6.7 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 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
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 6.8 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 228752.35 130109.05 309730.35 2073725.48
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
3 0.72739 0.59941 1.00000 0.62405 0.73771
Time of execution = 5.125000 s
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
6. 1.7 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING:
DETERMINING THE BEST COMPROMISE SOLUTION CONSIDERING 168 SUB-
INTERVALS
Weight vectors 4
Method to determine objective function values N-R Method
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.
Table 6.9 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 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
Table 6.9 b 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 6.10 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 530358.58 310373.88 719196.30 4916154.08
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
3 0.72739 0.59941 1.00000 0.62405 0.73771
Time of execution = 45.641000 s
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
Method to determine objective function values N-R Method
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
Table 6.11 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
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
Table 6.11 b 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 0.99788 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
Table 6.12 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
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
3 0.72739 0.59941 1.00000 0.62397 0.73769
Time of execution = 0.766000 s
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
6.1.9 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH
RANDOM GENERATION OF WEIGHT VECTORS: TRIAL 2
Weight vectors 10 (4 fixed, 6random generated
Method to determine objective function values N-R Method
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. 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
Table 6.13 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
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
Table 6.13 b 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 0.99786 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
Table 6.14 Optimum solution
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
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
6.1.11 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH
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
Table 6.16 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
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
Table 6.16 a continued
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
Table 6.16 a continued
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
Table 6.16 a continued
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
Table 6.16 a continued
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
Table 6.16 a continued
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
Table 6.16 a continued
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
Table 6.16 b 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 0.99786 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
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
Table 6.16 b continued
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
Table 6.16 b continued
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
Table 6.16 b continued
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
Table 6.16 b continued
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
Table 6.16 b continued
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
Table 6.16 b continued
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
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
Table 6.17 Optimum solution
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
Sl.No. m(F1) m(F2) m(F3) m(F4) fitness
66 0.67789 0.76984 0.89854 0.79273 0.78475
Time of execution = 12.67 s
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
6.1.12 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH
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
Time of execution = 19.234000 s
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.
m(F1) m(F2) m(F3) m(F4) fitness
382 0.66768 0.77381 0.90154 0.79682 0.78496
Time of execution = 29.641000 s
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.
m(F1) m(F2) m(F3) m(F4) fitness
318 0.66841 0.76551 0.91660 0.78890 0.78486
Time of execution = 23.812000 s
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.
m(F1) m(F2) m(F3) m(F4) fitness
58 0.66033 0.78316 0.88835 0.80584 0.78442
Time of execution = 3.231000 s
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.
m(F1) m(F2) m(F3) m(F4) fitness
367 0.66315 0.77090 0.91116 0.79417 0.78484
Time of execution = 25.125 s
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
6.2.1 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH
GENETICALLY GENERATED WEIGHT VECTORS FOR 24 SUB-INTERVALS: TRIAL 1
Weight vectors Geneticallygenerated
Method to determine objective function values N-R Method
Test system 1
Sub-intervals 24
First four weight vectors are of the format shown below which is
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
6.2.2 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH
GENETICALLY GENERATED WEIGHT VECTORS FOR 24 SUB-INTERVALS: TRIAL 2
Weight vectors Geneticallygenerated
Method to determine objective function values N-R Method
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.
Table 6.25 lists the weight vectors and fitness values obtained in the
last generation. Table 6.26 lists the objective function values corresponding to the
weight vectors given in Table 6.25 and Table 6.27 lists the corresponding fuzzy
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
Table 6.24 Highest fitness in ten generations
Generations Highest fitness1 0.7496
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
Sl.no. w1 w2 w3 w4 fitness
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
Table 6.26 Objective function values corresponding to the weight vectors given in Table 6.25
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
Table 6.27 Fuzzy membership functions corresponding to the weight vectors given in Table 6.25
Sl.no. m(F1) m(F2) m(F3) m(F4)
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
6.2.3 MULTI-OBJECTIVE, SHORT-TERM HYDROTHERMAL SCHEDULING WITH
GENETICALLY GENERATED WEIGHT VECTORS FOR 72 SUB-INTERVALS
Weight vectors Geneticallygenerated
Method to determine objective function values N-R Method
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.
Table 6.30 lists the weight vectors and fitness values obtained in the
last generation. Table 6.31 lists the objective function values corresponding to the
weight vectors given in Table 6.30 and Table 6.32 lists the corresponding fuzzy
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
Table 6.29 Highest fitness in ten generations
Generations Highest fitness1 0.7411
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
Table 6.31 Objective function values corresponding to the weight vectors given in Table 6.30
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
Table 6.32 Fuzzy membership functions corresponding to the weight vectors given in Table 6.30
Sl.no. m(F1) m(F2) m(F3) m(F4)
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
Table 6.33 continued
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
Table 6.33 continued
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
Method to determine objective function values N-R Method
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
Method to determine objective function values N-R Method
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
Sl.no. w1 w2 w3 w4 fitness
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
Sl.no. m(F1) m(F2) m(F3) m(F4)
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)
Waterwithdrawal,Unit-4 (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
Sl.no. w1 w2 w3 w4 fitness
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
Table 6.38b Objective function values
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
Table 6.38c Fuzzy membership functions
Sl.no. m(F1) m(F2) m(F3) m(F4)
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
6.39 No. of iterations, water withdrawals and coal-consumption corresponding to each weight vector
Weight vector No. ofiterations
Waterwithdrawal,Unit-3 (m3)
Waterwithdrawal,Unit-4 (m3)
Coalconsumption,Unit-1 (kg)
[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
Table 6.40a lists weight vectors and fitness. Table 6.40b lists objective function
values and Table 6.40c lists the fuzzy membership functions of objective function
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
Sl.no. w1 w2 w3 w4 fitness
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
Table 6.40b Objective function values
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
Table 6.40c Fuzzy membership functions
Sl.no. m(F1) m(F2) m(F3) m(F4)
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
Weight vector No. ofiterations
Waterwithdrawal,Unit-3 (m3)
Waterwithdrawal,Unit-4 (m3)
Coalconsumption,Unit-1 (kg)
[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.
Table 6.42a lists weight vectors and fitness. Table 6.42b lists objective function
values and Table 6.42c lists the fuzzy membership functions of objective function
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
Sl.no. w1 w2 w3 w4 fitness
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
Table 6.42b Objective function values
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
Table 6.42c Fuzzy membership functions
Sl.no. m(F1) m(F2) m(F3) m(F4)
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
6.43 No. of iterations, water withdrawals and coal-consumption corresponding to each weight vector
Weight vector No. ofiterations
Waterwithdrawal,Unit-3 (m3)
Waterwithdrawal,Unit-4 (m3)
Coalconsumption,Unit-1 (kg)
[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 george.ab2008@gmail.com
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