Economic Load Dispatch for Single-Area and Multi-Area systems using
Cuckoo Search Algorithm
Thesis
Submitted in Partial Fulfillment of the Requirement for the Degree of
Master of Power Engineering
2013
By
ANIRBAN CHOWDHURY
Registration No. 117214 0f 2011-2012
University Roll No. 001111502001
Examination Roll No. M4POW13-01
Under the Supervision of
Dr. MOUSUMI BASU
DEPARTMENT OF POWER ENGINEERING
JADAVPUR UNIVERSITY, SALT LAKE CAMPUS
KOLKATA-700098, INDIA
FACULTY OF ENGINEERING AND TECHNOLOGY
JADAVPUR UNIVERSITY
KOLKATA-700032, INDIA
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Department of Power Engineering
Faculty of Engineering & Technology
Jadavpur University, Kolkata, India
Certificate of Recommendation
This is to certify that the thesis entitled Economic Load Dispatch for Single-
Area and Multi-Area systems using Cuckoo Search Algorithm being submitted
by Sri. Anirban Chowdhury to Jadavpur University for the award of the
degree of Master of Power Engineering is a record of his bonafide project work
carried out under my supervision & guidance.
This work, in my opinion has reached the standard of fulfilling the requirement
for the award of degree of Master of Engineering.
Date: _____________________________
Countersigned: Dr. Mousumi Basu
(Thesis Advisor)
ASSOCIATE PROFESSOR
Dept. of Power Engineering
Jadavpur University, Salt Lake Campus
Kolkata, India
_____________________________ _____________________________
Dr. Mousumi Basu
HEAD DEAN
Dept. of Power Engineering Faculty of Engineering & Technology
Jadavpur University, Salt Lake Campus Jadavpur University
Kolkata, India Kolkata, India
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Department of Power Engineering
Faculty of Engineering & Technology
Jadavpur University, Kolkata, India
Certificate of Approval
The foregoing thesis is hereby approved as a creditable study in the area
of Power Engineering, carried out and presented in a manner
satisfactorily by Sri. Anirban Chowdhury to warrant its acceptance as a
prerequisite for the award of the degree of Master of Power Engineering
from Jadavpur University, Kolkata, India. It is understood that by this
approval the undersigned do not necessarily endorse or approve any
statement made, opinion expressed or conclusion drawn therein, but
approve the thesis only for the purpose of which it is submitted.
Board of Thesis Examiners:
_____________________________ _____________________
_____________________________ _____________________
_____________________________ _____________________
Signature Date
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Acknowledgement
In the beginning, I must convey my honest gratitude towards Dr. Mousumi Basu,
Head of the Department, Power Engineering, Jadavpur University, for giving me
this opportunity to carry out this project under her supervision.
Thereafter, I must express my sincere gratefulness to my project supervisor, Dr.
Mousumi Basu, for providing me immense support, guidelines along with
indispensable documents required to conduct the project work.
Consequently, I would like to express my truthful gratitude towards all the
respected faculty members of Department of Power Engineering for providing
their continuous endorsement to make this learning procedure, a great experience.
Afterward, I must convey my earnest thankfulness towards the Librarian for giving
their extended support.
Subsequently, I would like to recognize my seniors & batch mates for providing
their moral support with lot of encouragement.
Last, but not the least, I would definitely name my parents, Sri. Abhijit Chowdhury
& Smt. Sheela Chowdhury, for their blessings & continuous moral support to
conclude this one year of hard work as a Final Year Thesis.
Date:
Place: Kolkata, India Anirban Chowdhury
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Abstract
This thesis presents cuckoo search algorithm (CSA) for solving convex and non-
convex economic dispatch (ED) problems of fossil fuel fired generators in single &
multiple areas considering transmission losses, multiple fuels, valve-point loading,
prohibited operating zones & tie line power flow(in case of multiple areas). CSA is
a new meta-heuristic algorithm. It is a nature-based searching technique which is
inspired from the obligate brood parasitism of some cuckoo species by laying their
eggs in the nests of other host birds of other species. The effectiveness of the
proposed algorithm has been verified on four different single area based test
systems & three different multiple area based test systems, both small and large,
involving varying degree of complexity. Compared to the other existing
techniques, considering the quality of the solution obtained, the proposed
algorithm seems to be a promising alternative approach for solving the ED
problems in practical power system.
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Contents Acknowledgement iii
Abstract iv
List of Tables ix
List of Figures x
Chapter-1 LITERATURE REVIEW, MOTIVATION
BEHIND THE WORK & OVERVIEW 1-5
1.1 Introduction 1
1.2 Literature Review 1
1.3 Motivation behind the work 4
1.4 Overview 5
Chapter-2 CUCKOO SEARCH via LEVY FLIGHTS 6-14
2.1 Introduction 6
2.2 Cuckoo Breeding Behavior 7
2.3 Levy Flight 8
2.4 Assumptions 9
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2.5 Cuckoo Search Algorithm 10
2.6 Flowchart of CSA via Lvy flights 13
Chapter-3 ECONOMIC LOAD DISPATCH 15-36
3.1 Introduction 15
3.2 Economic Load Dispatch 15
3.2.1 Economic Load Dispatch without Losses 17
3.2.2 Economic Load Dispatch with Losses 20
3.3 Practical situations that should be taken into account during
operation 24
3.3.1 Valve Point Loading 24
3.3.2 Multiple Fuels 26
3.3.3 Prohibited Operating Zones 27
3.4 Types of ED Problems 27
3.4.1 Economic Dispatch with Quadratic Cost Function and
Transmission Loss (EDQCTL) 27
3.4.2 Economic Dispatch with Quadratic Cost Function, Prohibited
Operating Zones and Transmission Loss (EDQCPOZTL) 27
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3.4.3 Economic Dispatch with Valve-point Loading Effect
and without Transmission Loss (EDVPL) 28
3.4.4 Economic Dispatch with Valve-point Loading Effect
and Multi-fuel Options (EDVPLMF) 28
3.5 Results 28
Chapter-4 MULTI AREA ECONOMIC DISPATCH 37-51
4.1 Introduction 37
4.2 Operational Constraints in MAED 38
4.2.1 Real Power Balance constraint 38
4.2.2 Tie-Line Capacity constraint 38
4.2.3 Real power generation constraint 38
4.3 Types of MAED problems 38
4.3.1 Multi-Area Economic dispatch with Quadratic cost function,
Prohibited Operating Zones & Transmission Losses
(MAEDQCPOZTL) 39
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4.3.2 Multi-Area Economic Dispatch with Valve Point Loading
(MAEDVPL) 39
4.3.3 Multi-Area Economic Dispatch with Valve Point Loading,
Multiple Fuels Sources & Transmission Losses
(MAEDVPLMFTL) 40
4.4 Determination of generation level of the slack generator 41
4.5 Results 42
Chapter-5 CONCLUSION & SCOPE OF FUTURE WORK 52-53
5.1 Conclusion 52
5.2 Scope of future work 53
REFERENCES 54
APPENDIX A 59
APPENDIX B 63
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List of Tables
Table Number Table Name Page No
Table 3.1 Simulation results for 6-generator system 29
Table 3.2 Simulation results for 10-generator system 31
Table 3.3 Simulation results for 20-generator system 33
Table 3.4 Simulation results for 40-generator system 35
Table 4.1 Simulation results for 2-Area System 44
Table 4.2 Simulation results for 3-Area System 47
Table 4.3 Simulation results for 4-Area System 50
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List of Figures
Figure Figure Page
Number Name No.
Fig. 2.1 Flowchart of Cuckoo Search Algorithm via Lvy flights 14
Fig. 3.1 Schematic diagram of a set of generators connected to a load 16
Fig. 3.2 Simple Model of a Fossil Plant 16
Fig. 3.3 Variation of Operating Cost of a fossil-fired generator with
active power generation 17
Fig. 3.4 Ripples in the Cost-Function due to Valve Point Loading 25
Fig. 3.5 Resultant Fuel Cost vs. Power Output due to Multiple Fuels 26
Fig. 3.6 Cost convergence characteristic of 6-generator system 30
Fig. 3.7 Cost convergence characteristic of 10-generator system 32
Fig. 3.8 Cost convergence characteristic of 20-generator system 34
Fig. 3.9 Cost convergence characteristic of 40-generator system 36
Fig. 4.1 Four-Area Generation System connected via. Tie-lines 37
Fig. 4.2 Cost convergence characteristic of 2-Area System 44
Fig. 4.3 Cost convergence characteristic of 3-Area System 48
Fig. 4.4 Cost convergence characteristic of 4-Area System 51
Chapter 1
LITERATURE REVIEW, MOTIVATION BEHIND THE WORK & OVERVIEW
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Chapter1: LITERATURE REVIEW, MOTIVATION BEHIND THE WORK &
OVERVIEW
1.1 Introduction
Modern power systems consist of number of areas, each area containing a number of generators,
all the areas are inter-connected via tie-lines. Now, one of the major challenges faced by the
network operators is to transfer the generated electrical power to the load end, economically &
safely, satisfying several constraints.Now, as the number of areas & generators increases & so do
the constraints, the complexity & dimensionality of Economic Dispatch(ED) problem increases.
Classical Methods of solving ED problems fail to perform in such cases. Thus, the necessity of
developing meta-heuristic techniques to handle such problems has been felt by the researchers.
Meta-heuristic search techniques can handle such problems with efficient utilization of the
search space & satisfactory computation time. Cuckoo Search Algorithm (CSA) is a newly
developed nature-inspired, meta-heuristic search algorithm by Yang & Deb [48]. This thesis
aims at solving single area & multi-area, convex & non-convex ED problems using CSA &
compares the obtained results with some popular algorithms.
1.2 Literature Review
ED [1] is one of the most important optimization problems in power system operation and
planning. ED allocates the load demand among the committed generators most economically
while satisfying the physical and operational constraints. Since the cost of power generation in
fossil fuel fired plants is exorbitant, an optimum dispatch saves a considerable amount of money.
Classical methods such as lambda iteration, base point participation factor, gradient method,
Newtons method and Lagrange multiplier method can solve economic dispatch problem under
the assumption that the incremental cost curves of the generating units are monotonically
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increasing piecewise-linear functions. However, in reality, large steam turbines have a number of
steam admission valves which contribute non convexity in the fuel cost function of the
generating units. Classical calculus-based techniques fail to address these types of problems
satisfactorily and lead to sub optimal solutions producing huge revenue loss over time. Dynamic
programming (DP) can solve ELD problem with inherently nonlinear and discontinuous cost
curves. But it suffers from the curse of dimensionality or local optimality.
In this respect, stochastic search algorithms such as simulated annealing (SA) [2], genetic
algorithm (GA) [3-4], evolutionary programming (EP) [5], artificial neural networks (ANN) [6],
ant colony optimization (ACO) [7], particle swarm optimization (PSO) [8], artificial immune
system (AIS) [9], differential evolution (DE) [10], bacterial foraging algorithm (BFA)
[11],biogeography-based optimization (BBO) [12], etc., have been applied successfully to solve
complex ED problem without any restriction in the shape of the cost curves.
Recently, different hybridization and modification of EP, GA, PSO, DE methods like PSO-SQP
[13], IFEP [14], IGA [15],DEC-SQP [16], NPSO-LRS [17], SOH-PSO [18], ICA-PSO [19],
hybrid differential evolution with biogeography-based optimization [20] etc., have been
proposed for solving ED problem in search of better quality solution. However, the hybrid
methods contain many controllable parameters which may not be properly selected.
In practical cases, generators are distributed in several generation areas, interconnected via tie-
lines. Multi-area economic dispatch (MAED) is an extension of economic dispatch. MAED
determines the level of power generation & exchange of power between the areas such the total
fuel cost in all areas is minimized while satisfying power balance constraints, generation limits
constraint & tie-line power capacity constraints.
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The power transmission losses are often not considered while solving ED problems. However,
some researchers did take transmission capacity constraints into consideration. Shoults et al. [21]
considered the import & export constraints between the areas. This study provides a complete
formulation of multi-area generation scheduling, and a framework for multi-area studies.
Romano et al. [22] presented the Dantzig-Wolfe decomposition principle to the constrained ED
of multi-area systems. Doty and McEntire [23] used spatial dynamic programming to solve
MAED problem & the result obtained was a global optimum. An application of linear
programming to transmission constrained production cost analysis was proposed in Ref. [24]
MAED with Area Control Error was solved by Helmick et al. [25]. Ouyang et al [26] proposed
heuristic multi-area unit comment with ED. Wang and Shahidehpour [27] proposed a
decomposition approach for solving multi-area generation scheduling with tie-line constraints
using expert systems. Streiffert [28] proposed network flow models for solving the MAED
problem with transmission constraints. An algorithm for MAED and calculation of short range
margin cost based prices has been presented by Wernerus and Soder [29], where MAED problem
was solved via. Newton-Raphsons method. Yalcinoz and Short [30] solved MAED problems by
using Hopfield neural network approach. Jayabarathi et al. [31] solved MAED problems with tie
line constraints using evolutionary programming. The direct search method for solving ED
problem considering transmission capacity constraints was presented in Ref. [32]. Manoharan et
al. [33] explored the performance of the various evolutionary algorithms such as the Real-coded
Genetic Algorithm (RCGA), Particle Swarm Optimization (PSO), Differential Evolution (DE)
and Covariance Matrix Adapted Evolution Strategy (CMAES) are considered. Multi-area
economic environmental dispatch (MAEED) problem proposed in [34]. In this case, MAEED
problem is handled by an improved multi-objective particle swarm optimization (MOPSO)
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algorithm for searching out the Pareto-optimal solutions. Sharma et al. [35] have presented a
close comparison of classic PSO & DE strategies and their variants for solving the reserve
constrained MAED problem with power balance constraint, upper/lower generation limits, ramp
rate limits, transmission constraints and other practical constraints. A discussion of Reserve
constrained multi-area economic dispatch employing differential evolution with time-varying
mutation has been presented in [36]. Naturally inspired algorithms have also been applied to
solve MAED problems. Swarm Intelligence [37]-[39], a branch of naturally inspired algorithms,
focuses on the behavior of insect in order to develop meta-heuristic algorithms. MAED using
Artificial Bee Colony Optimization (ABCO) has been presented by M.Basu in [40].
1.3 Motivation behind the work
With the incorporation of large-scale ED problems where there are anumber of areas, each area
is having a number of generators, numerous operational constraints such as multiple fuel, valve
point loading, transmission losses, tie line power capacity etc., ED problems become more &
more complex & multidimensional, as a result, more & more challenging day by day. This needs
the development of more & more efficient algorithms, which can handle these non-convex
multidimensional ED problems efficiently & in less computation time. CSA is a newly
developed nature inspired meta-heuristic algorithm which is indeed a promising search algorithm
when applied to ED problems.
This thesis aims at application of CSA in convex & non-convex, single area & multi-area based
ED problems & compares the obtained results with those obtained by some popularly existing
algorithms. The cost of operation & the computation time are the two figures of merit of any ED
algorithm. The main objective of this thesis is to find out that how well CSA performs in
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handling ED problems, subjected to several constraints with respect to some popularly existing
algorithms.
1.4 Overview
Chapter 1 gives a brief survey on the works that has been done on single area economic dispatch
as well as MAED by various authors & the motivation behind this thesis work. Chapter 2
explains & describes Cuckoo Search, the Levy Flight concept, Cuckoo Search Algorithm & the
Flowchart. Chapter 3 explains ELD on single area, the operational constraints, the results & cost
convergence characteristics obtained on application of CSA on four different test systems & also
a tabulated comparison of the obtained results with those from some popularly existing
algorithms. Chapter 4 explains Multi-Area ELD, application of CSA on 3 different multi-area
test systems, the results & cost convergence characteristics & also a tabulated comparison with
some popularly existing algorithms. Chapter 5 highlights some of the conclusions drawn & the
scope of future work with CSA.
Chapter 2
CUCKOO SEARCH via LEVY FLIGHTS
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Chapter 2: CUCKOO SEARCH via LEVY FLIGHTS
2.1 Introduction
Conventional Numerical Methods had computational drawbacks in solving complex
optimization problems. For this reason, researchers had to rely on meta-heuristic algorithms.
Two main characteristics of meta-heuristic algorithms are: Intensification and Diversification, or
exploitation and exploration (Blum and Roli, 2003) [41]. Intensification means to focus the
search in a local region knowing that a current good solution is found in this region.
Diversification means to generate diverse solutions so as to explore the search space on a global
scale. A good balance between intensification & diversification is to be found during the
selection of the best solutions to improve the rate of algorithm convergence. A good combination
of these two major components will usually ensure that global optimality is achievable. Cuckoo
Search is a nature-based searching technique which is inspired from the obligate brood
parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other
species.Cuckoo Search Algorithm is a meta-heuristic algorithm developed in recent times. It can
act as a very efficient tool for selecting the proper combinationof generatorsfor practical non-
convex economic load dispatch subjected to several constraints, especially for large scale
systems.In addition, this algorithm is enhanced by the so-called Lvy flights, rather than by
simple isotropic random walks (Pavlyukevich 2007). Test results have revealed that the proposed
method can obtain less expensive solutions than many other methods reported in the literature.
This thesis aims at description, formulation & seeking the best solutions of Cuckoo Search
Algorithm & implementation of the Lvy flight searching technique in finding out the best
quality solutions.Economic Dispatch involves distribution of the consumer load demand amongst
multiple generating units in such a manner, so that that the cost of generation & transmission is
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minimized. The classical solution says that the costs are minimal when the incremental costs at
all the generating units are equal. But this, however, does not occur in practice. It is due to many
practically faced physical constraints, such as, the throttling losses due to opening of steam
admission valves, which causes non-linearity in the fuel cost curve, multiple-fuels, etc. which
will be discussed later. Another constraint is multiple fuel sources which lead to different
calorific value for same mass of fuel. Likewise, there are several other constraints which increase
the dimensionality & nonlinearity of Economic Load Dispatch Problem. So, it is very difficult to
solve ED problems for large scale systems until simplifications & assumptions are made while
handling the above said constraints. Meta-heuristic algorithms like Cuckoo Search Algorithm
can handle the above complexities& lead to good quality of solutions.
2.2 Cuckoo Breeding Behavior
Cuckoos are fascinating birds, not only because of beautiful sounds they make, but also because
of their aggressive reproduction strategy. Some species like the ani & Guira cuckoos lay their
eggsin communal nests, though they may remove others eggs so that the probability of hatching
oftheir own eggs increases (Payne et al 2005) [42]. Quite a number of species engage the
obligate brood parasitism by laying their eggs in the nests of other host birds (often other
species). There are threebasic types of brood parasitism: intraspecific brood
parasitism,cooperative breeding, and nest takeover. If a host bird finds eggs in its nest which are
not itsown, it can do two things-
a) Throw away the alien eggs out of the nest.
b) Abandon the nest & build a new one elsewhere.
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Some cuckoos like New World Brood Parasitic Tapera lays eggs of such color & patternwhich
matches with some specific host birds. This reduces the probability of its eggs beingabandoned
& increases the chances of its reproduction. The time of laying eggs of somecuckoo species is
quite amazing. Parasitic cuckoos often choose a nest where the host bird justlaid its own eggs
.After the cuckoo chick hatches, its first instinct will be evicting the host eggs byblindly
propelling out the eggs off the nest. This increases its chance of getting more share offood.
2.3 Levy Flight
Levy Flight is a random walk of step lengths having direction of the steps as isotropic &
random. This concept has been propounded Paul Pierre Lvy (1886-1971). It is very useful in
stochastic measurements & simulations of random & pseudo-random phenomena. In practical
world, we see that when sharks & other predators cannot procure food, they abandon Brownian
motion, the random motion seen in swirling gas molecules & adopt Lvy flight ([43], [44], [45],
[46])a mix of long trajectories and short, random movements. Birds &other animals, such as
insects, also seem to follow Lvy flights when searching for food. A recent study by Reynolds
and Frye shows that fruit flies or Drosophila melanogaster; explore their landscape using a series
of straight flight paths punctuated by a sudden 90 degrees turn, leading to a Levy-flight-style
intermittent scale free search pattern. Studies on human behavior such as the Ju/hoansi hunter-
gatherer foraging patterns also show the typical feature of Levy flights. Even light can be related
to Levy flights [47].
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2.4 Assumptions
Before understanding Cuckoo Search Algorithm as described by Yang & Deb [48], we have to
keep in mind three idealized rules.
1) Each Cuckoo lays one egg at a time & dumps it to a randomly chosen nest.
2) The best nests with high quality of eggs will carry over to the next generations.
3) The number ofavailable host nests is fixed& and the egg laid by a cuckoo is discovered by
the host bird with a probability pa [0, 1].
In that case, the host bird can do two things; it may either throw the egg away from its nest or
abandon the nest & build a new nest elsewhere. In the last assumption as stated above, it can be
simplified to the fact that a fraction pa of n nests is replaced by new nests (with new random
solutions).
For a problem to be maximized, it is evident that the quality or fitness of a solution is directly
proportional to the objective function. In a very simple manner, we can represent each egg in the
nest as a solution & a cuckoo egg as a new solution. Our objective is to use the new &
potentially better solutions (cuckoos) to replace relatively less fit solutions in the nest. This
algorithm also has the scope of extending it towards complicated cases where multiple eggs
representing a set of solutions exist.
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2.5 Cuckoo Search Algorithm
Step1: Initialization
Let us consider a population of Np host nests which are represented by X=[X1, X2,.,XNp]T
,where each nest Xd =[Pd1,Pd2,., PdN] (d=1,2,.,Np) represents the power output of the
generating units except the slack unit is initialized by:
(2.1)
rand1 is an uniformly distributed random number between 0 & 1 for each population of host
nests.The initial solution is further checked for POZ violation. If POZ violation is found, then the
corrective action must be taken.
Each POZ is divided into two subzones, the midpoint of which is given by
if
if
(2.2)
Step2: Evaluation of the fitness function
The fitness function to be minimized is given by
(
)
(2.3)
Ks & Kr are penalty factors for the slack unit corresponding to the nest d in the population &
spinning reserve of the ith
unit corresponding to the nest d in the population.
The slack unit has the limits given by
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if
if
otherwise (2.4)
The initial population of host nests is set to be the best value of each nest Xbestd
(d=1,2,.,Nd) The nest corresponding to the best fitness function is given by Gbest among all
the nests in the population.
Step3: Generation of new solutions via Lvy flights
The new solution for each nest is calculated as follows:
Where > 0 is the updated step size & rand2 is a normally distributed stochastic number.
is calculated below.
(2.5)
(2.6)
randx & randy are two normally distributed stochastic variables.
Where x() & y() are their standard deviations given by :
(
)
{
}
(2.7)
(2.8)
Where is the distribution factor ranging between 0.2 1.99
must be checked weather it satisfies the units operating limits & POZ violation. Repairing
strategy must be undertaken to satisfy the above conditions if necessary.
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Step4: Alien egg discover & randomization
When there is a probability pa of discovery of alien eggs by a host bird in its nest, the new
solution can be found out in the following way.
(2.9)
K is the updated coefficient based on the probability
if rand3< pa
otherwise
(2.10)
rand3 & rand4 are distributed random numbers lying between 0 & 1.
& are the random perturbation positions of nests in .
Like Lvy flights, the above obtained solution must be checked weather it satisfies the units
operating limits & POZ violation. Repairing strategy must be undertaken to satisfy the above
conditions if necessary. The newly best value for each nest Xbestd & the best value of all the
nests Gbest are determined after comparing the calculated fitness function FTd from the new
solution & the previously stored one.
Step5: Stopping Criteria
The above algorithm must be stopped when the number of iterations reaches the predefined
value, of course if the computer storage permits.
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2.6 Flowchart of CSA via Lvy flights
Now, based on the algorithm for CSA, the flowchart can be easily drawn. The flowchart of CSA
will definitely help in better understanding of the cuckoo search technique & give a clear view of
it. The flowchart describing CSA via Lvy flights is shown Fig.2.1.
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Fig. 2.1 Flowchart of Cuckoo Search Algorithm via Lvy flights
START
Initialize: Population of NP host nests represented by
X, each nest represented by Xd& the power output of
all the units except the slack unit is denoted by Xdi
Check for Generating Limits
Constraint & POZ Violation
Violated?
Y
Evaluate the fitness function FTd
N
Set the best value of each nest Xbestd& the
nest corresponding to the best fitness
function is given by Gbest
Generate new solutions
Xdnew via Levy Flight
technique
Check whether the newly generated solutions
satisfy the generating limits & violating POZ
Violated?
Y
N
Update the best value of each nest
Xbestd& the nest corresponding to the best
fitness function is given by Gbest
Re-evaluate the fitness function FTd
Iteration starts
Evaluate new solutions Xddis due
to alien egg discovery by the host
bird
Again, check whether the newly
generated solutions satisfy the
generating limits & violating POZ
Violated?
Y
N
Re-evaluate the fitness function FTd
Update the best value of each nest
Xbestd& the nest corresponding to the
best fitness function is given by Gbest
Maximum
number of
iterations
reached?
Y
STOP
N
Chapter 3
ECONOMIC LOAD DISPATCH
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Chapter 3: ECONOMIC LOAD DISPATCH
3.1 Introduction
The most important concern in the planning & operation of electric power generation system is
the effective scheduling of all generators in a system to meet the required demand. Economic
Load Dispatch (ELD) is a phenomenon where an optimal combination of power generating units
are selected so as to minimize the total fuel cost while satisfying the load demand & several
operational constraints [49]. In a deregulated electricity market, the optimization of economic
dispatch is of utmost economic importance to the network operator. The main objective of ELD
problem is to minimize the operation cost by satisfying the various operational constraints in
order met the load demand. Many traditional algorithms (Wood & Wollenberg,1996) like
Lambda-Iteration, Gradient search, Newton Method are applied to optimize ELD problems
however in these methods it is assumed that the incremental cost curves of the units are
monotonically increasing piecewise linear functions, but the practical systems are nonlinear.
3.2 Economic Load Dispatch
An electrical generation system consists of one or more generators connected to the load. Fig. 3.1
shows N number of generators connected to a load. Now, the objective is to dispatch the
generated power to the connected load safely & economically, satisfying all the operational
constraints. Economical load dispatch problem can be classified into two types convex & non-
convex. Convex/Smooth ED problems neglect transmission losses & other constraints while non-
convex or non-smooth ED problems deviates from idealities & takes them into consideration.
Thus the complexity or dimensionality of a non-convex ED problem increases & so does the
computation time. In this chapter, both convex & non-convex ED problems have been
considered.
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Fig.3.1 Schematic diagram of a set of generators connected to a load
Fig. 3.2 shows a simple schematic diagram of a thermal power station. Fossil fuel (i.e. coal) is
supplied to the boiler in which superheated steam is generated, which hits the turbine blades. The
turbine starts rotating whose shaft is coupled to that of the alternator, which in turn rotates
generating electric power at its output terminals. Now, the main concern is to make this
generation economic, subject to several constraints which will be discussed later. The operating
cost of a generating unit includes the fuel cost, cost of labor, supply & maintenance. Cost of
labor, supplies & maintenance are generally fixed with respect to incoming fuel costs. A
schematic diagram of a fossil-fired power plant is shown in Fig 3.2.
Fig 3.2 Simple Model of a Fossil Plant
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3.2.1 Economic Load Dispatch without Losses
Economic Load Dispatch attains its simplest form when the transmission losses are neglected.
So, the total load demand PD is equal to the sum of power generated by the units. A cost function
is assumed to be a known parameter for each plant. The variation of the cost function with
respect to active power generation is shown in Fig. 3.3. Ideally, it is a monotonically increasing
quadratic function given by the equation (3.1).
( ) (3.1)
Fig 3.3 Variation of Operating Cost of a fossil-fired generator with respect to active power
generation
Now, the problem is to find the real power generation Pgi for which the operating cost becomes
minimum & the generating limits are satisfied &
[50]. Let there be a generating station
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with NG generators committed & the given active load demand be PD. The real power generation
Pgi for each generator has to be allocated so as to minimize the total cost. The optimization
problem can be written as:
{ ( )}
(3.2)
Equation (3.2) is subjected to
(a)The equality constraint equation is the energy balance equation.
(3.3)
(b)The inequality constraint equation is the generating limits.
(3.4)
The above constrained optimization problem can be converted into an unconstrained
optimization problem. Lagrange Multiplier is used in which a function is minimized (or
maximized) with side conditions in the form of equality constraints. Using this method, the
augmented function becomes:
19 | P a g e
( ) ( )
(3.5)
A necessary condition for the function, ( ) subject to energy balance constraint to have a
relative minimum at point is that the partial derivative of the Lagrange function defined by
( )with respect to each of its arguments must be zero. So, the necessary conditions for
the optimization problem are
( )
( )
(3.6)
From (3.3) & (3.6), we get
( )
(3.7)
Also, from equation (3.6), we get
( )
(3.8)
When incremental costs of the generators in operation are equal, they are said to be optimally
loaded. We can get NG equations from equation (3.8) & they are called co-ordination equations.
Differentiating equation (3.1) with respect to , we get
20 | P a g e
( )
(3.9)
Combining equations (3.8) & (3.9), we get
(3.10)
(3.11)
Substituting the value of in equation (3.7), we get,
(3.12)
3.2.2 Economic Load Dispatch with Losses
Transmission losses are neglected when they are small in magnitude but when distances of
transmission are large in case of large network; transmission losses become accountable &
cannot be neglected. They affect the process of economic load dispatch. The economic load
dispatch problem, considering the transmission power loss PL, for the objective function, is same
as, equation (3.1) & equation (3.2). But what change are the constraints to which the equations
are subjected to.
21 | P a g e
(a)The equality constraint, i.e., Energy Balance Equation gets modified to,
(3.13)
(b) The inequality constraints, i.e., the generator limits, although, remain the same.
The general loss formula using B-coefficients is given by
(3.14)
Using Lagrange multiplier , augmented function can be written as follows.
( ) ( )
(3.15)
For minimization of the augmented function,
( )
(3.16)
( )
(3.17)
( )
( )
(3.18)
22 | P a g e
From equations (3.15) & (3.16), we get
( )
( )
(3.19)
From equations (3.16) & (3.19), we get
( )
(3.20)
( )
(3.21)
( )
, the incremental fuel cost, is denoted by (IC)i &
,the incremental transmission loss, is
denoted by (ITL)i.
Rearranging equation (17), we get
( )
(3.22)
( ( )
) (3.23)
, the penalty factor, is denoted by .
( ( )
) (3.24)
23 | P a g e
Equation (3.24) shows that the minimum cost is obtained when penalty factor, multiplied with
the incremental fuel cost is same for all the plants.
Equation (3.20) can also be re-written as given in [51],
(3.25)
The above equation is referred to as the co-ordination equation for ELD considering losses & we
get a set of i equations. Computation of ITL is necessary for each plant in order to solve ELD
problems & therefore, functional dependence of transmission loss on real powers of a generating
plant must be determined. One of the most important, simple, yet approximate methods of
determining ITL is expressing it with the help of general loss formula using B-coefficients given
in equation (3.14) .
Simplifying Equation (3.14) & assuming we get
(3.26)
Substituting the value of
&
( )
in Equation (3.25), we get
(3.27)
24 | P a g e
Collecting all the terms for & solving for we get,
(3.28)
(3.29)
The above equation can be solved for any values of , iteratively, by assuming initial values of
Pgi. Iterations are stopped when Pgi converge with the specified accuracy.
3.3 Practical situations that should be taken into account during operation
The practical situations that are encountered during a real life ED problem formulation are
described below.
3.3.1 Valve Point Loading
Loading effects at which a new steam admission valve is opened are called valve points. These
lead to discontinuities in the cost curves & in the incremental heat rate curves due to steep
increase in throttle losses. As the valve is opened gradually, the losses decrease until the valve is
completely opened. This produces a rippling effect on the units input-output curve as shown in
wire drawings. In most of the optimization techniques the input-output characteristics of a
generating unit is approximated by a smooth quadratic functions which lead to inaccuracy of the
25 | P a g e
obtained resulting dispatch. A units input-output curve considering valve point loading is shown
in Fig 3.4.
Fig 3.4 Ripples in the Cost-Function due to Valve Point Loading
In the above figure the dotted line shows, the generation cost without valve point loading. The
effect of valve point loading on the fuel cost is shown on the above diagram. As the number of
valves in the system increases, more & more nonlinearity is introduced in the cost curve. It is
evident from the above figure that the cost function consists of two components, one quadratic
function & a rectified sinusoidal function superimposed on it. The second component is due to
valve point loading effect.
Now, the resultant function due to valve point loading becomes
(3.30)
26 | P a g e
3.3.2 Multiple Fuels
Since a single type of fuel can be procured from different sources, they have got different
calorific values. Also, different types of fuels can be used in a power generating unit. Multiple
fuels for different generating units are represented with the help of piecewise quadratic function.
The resultant cost curve due to operation of a unit with multiple fuels is shown in Fig 3.5.
. .
(3.31.1)
for fuel type j and j F ,...,2,1 (3.31.2)
Fig 3.5 Resultant Fuel Cost vs. Power Output due to Multiple Fuels
27 | P a g e
3.3.3 Prohibited Operating Zones
The prohibited operating zones are the range of power output of a generator where the operation
causes undue vibration of the turbine shaft bearing caused by opening or closing of the steam
valve. This undue vibration might cause damage to the shaft and bearings. Normally operation is
avoided in such regions. The feasible operating zones of unit can be described as follows:
j=2,,
(3.32)
j represents the number of prohibited operating zones of i the generator. is the upper limit
of jth
prohibited operating zone of ith
generator.
is the lower limit of jth
prohibited operating
zone of ith
generator. Total number of prohibited operating zone of ith
generator is .
3.4 Types of ED Problems
The ED may be formulated as a nonlinear constrained optimization problem. Four different types of ED
problems have considered.
3.4.1 Economic Dispatch with Quadratic Cost Function and Transmission Loss (EDQCTL)
In this problem, the cost function is assumed to be quadratic in nature as given by (3.1). The
transmission losses are considered in this case. Equation (3.1) is subjected to the constraints as
given by (3.13) & (3.4). The transmission loss is given by equation (3.14).
3.4.2 Economic Dispatch with Quadratic Cost Function, Prohibited Operating Zones and
Transmission Loss (EDQCPOZTL)
28 | P a g e
In this problem, the cost function is also assumed to be quadratic in nature as given by (3.1). The
transmission losses are considered in this case. The generating units have prohibited operating
zones (POZ).Equation (3.1) is subjected to the constraints as given by (3.13), (3.32) & (3.4). The
transmission loss is given by equation (3.14).
3.4.3 Economic Dispatch with Valve-point Loading Effect and without Transmission Loss
(EDVPL)
In this type of ED problem, the cost function is quadratic with a rectified sine component
superimposed on it, due to valve point loading effect. This is given by equation (3.30). The
transmission losses are not considered in this problem. Equation (3.30) is subjected to the
constraints as given by (3.3) and (3.4).
3.4.4 Economic Dispatch with Valve-point Loading Effect and Multi-fuel Options
(EDVPLMF)
This type of ED problem considers the effect of valve point loading & has Multiple Fuel options
for the units. The transmission losses are neglected in this problem. Equations, as given in
(3.31.1), have rectified sine components superimposed with the quadratic components. The
resultant equations are subjected to the constraints given by (3.31.2) & (3.3).
3.5 Results
The proposed cuckoo search algorithm has been applied to solve ED problems in four different
test systems for verifying its feasibility. The software has been written in MATLAB 7 on a PC
(Pentium IV, 80 GB, 3.0 GHZ).
29 | P a g e
Test System 1: A six generator system with prohibited operating zone is considered here. The
generator data and B-coefficients have been taken from [8]. The load demand is 1263 MW. For
this test system, the population size ( ), maximum number of iterations and the value of
probability pa have been selected 50, 300 and 0.7 respectively. Results obtained from proposed
CSA, BBO [12], SOH-PSO [18], NPSO-LRS [17] and PSO [8] have been presented in Table 3.1.
The cost convergence characteristic of six generator system obtained from CSA is shown in Fig.
3.6.
Table 3.1: Simulation results for 6-generator system
Unit Power
Output
(MW)
CSA BBO [12] SOH-PSO
[18]
NPSO-LRS
[17]
PSO [8]
1 447.4768
447.3997 438.21 446.96 447.50
2 173.2234
173.2392 172.58 173.3944 173.32
3 263.3787
263.3163 257.42 262.3436 263.47
4 138.9524
138.0006 141.09 139.5120 139.06
5 165.4120
165.4104 179.37 164.7089 165.48
6 87.0024
87.07979 86.88 89.0162 87.13
Total Power
Output
(MW)
1275.447 1275.446 1275.55 1275.94 1276.01
Ploss (MW) 12.447 12.446 12.55 12.936 12.958
Total cost
($/h)
15443.08 15443.096 15446.02 15450 15450
30 | P a g e
Fig.3.6. Cost convergence characteristic of 6-generator system
Test System 2: This system consists of ten generators with valve-point loading and multi-fuel
sources. The generator data has been adopted from [15]. The load demand is 2700 MW.
Transmission loss has not been considered here. For this test system, the population size ( ),
maximum number of iterations and the value of probability pa have been selected 50, 500 and 0.7
respectively. Results obtained from proposed CSA, BBO [12], NPSO-LRS [17], NPSO [17] and
IGA [15] have been summarized in Table 3.2. The cost convergence characteristic of this test
system obtained from CSA is shown in Fig. 3.7.
0 50 100 150 200 250 3001.5442
1.5444
1.5446
1.5448
1.545
1.5452
1.5454
1.5456
1.5458x 10
4
iteration
cost(
$/h
our)
31 | P a g e
Table 3.2: Simulation results for 10-generator system
Unit
Power
Output
(MW)
CSA BBO [12] NPSO-LRS
[17]
NPSO [17] IGA [15]
F
u
e
l
F
u
e
l
F
u
e
l
F
u
e
l
F
u
e
l
1 236.4387
2 212.96 2 223.33 2 220.657 2 219.126 2
2 230.0000
1 209.43 1 212.19 1 211.785 1 211.164 1
3 417.3113
2 332.02 3 276.21 1 280.402 1 280.657 1
4 135.9952
1 238.34 3 239.41 3 238.601 3 238.477 3
5 328.6017
1 269.25 1 274.64 1 277.562 1 276.417 1
6 197.6450
1 237.64 3 239.79 3 239.120 3 240.467 3
7 257.0953
1 280.61 1 285.53 1 292.139 1 287.739 1
8 228.2969
3 238.47 3 240.63 3 239.153 3 240.761 3
9 411.4391
3 414.85 3 429.26 3 426.114 3 429.337 3
10 257.1768
1 266.38 1 278.95 1 274.463 1 275.851 1
Total cost
($/h)
598.0243
605.6387 624.1273 624.1624 624.5178
32 | P a g e
Fig.3.7: Cost convergence characteristic of 10-generator system
Test System 3: A twenty generator system with quadratic cost function is considered here. The
generator data and B-coefficients have been taken from [6]. The load demand is 2500 MW. For
this test system, the population size ( ), maximum number of iterations and the value of
probability pa have been selected 50, 500 and 0.7 respectively. Results obtained from proposed
CSA, BBO [12], Hopfield Model [15], and Lambda Iteration [15] have been shown in Table 3.3.
The cost convergence characteristic of twenty generator system obtained from CSA is shown in
Fig.3.8.
0 50 100 150 200 250 300 350 400 450 500590
600
610
620
630
640
650
660
iteration
cost(
$/h
our)
33 | P a g e
Table 3.3: Simulation results for 20-generator system
Unit Power Output (MW) CSA BBO [12] Hopfield
Model [15]
Lambda
Iteration [15]
1 512.8467
513.0892 512.7804 512.7805
2 168.8534
173.3533 169.1035 169.1033
3 126.8549
126.9231 126.8897 126.8898
4 102.8784
103.3292 102.8656 102.8657
5 113.6863
113.7741 113.6836 113.6386
6 73.5482
73.06694 73.5709 73.5710
7 115.4766
114.9843 115.2876 115.2878
8 116.4497
116.4238 116.3994 116.3994
9 100.7505
100.6948 100.4063 100.4062
10 106.1438
99.99979 106.0267 106.0267
11 150.2221
148.977 150.2395 150.2394
12 292.7736
294.0207 292.7647 292.7648
13 118.9029
119.5754 119.1155 119.1154
14 30.8736
30.54786 30.8342 30.8340
15 115.7864
116.4546 115.8056 115.8057
16 36.2102
36.22787 36.2545 36.2545
17 66.8828
66.85943 66.8590 66.8590
18 87.8848
88.54701 87.9720 87.9720
19 100.7805
100.9802 100.8033 100.8033
20 54.1771
54.2725 54.3050 54.3050
Total Power Output (MW) 2555.80 2592.1011 2591.9670 2591.9670
Ploss (MW) 55.80 92.1011 91.5670 91.9670
Total cost ($/h) 62456.63
62456.7926 62456.63 62456.63
34 | P a g e
Fig.3.8. Cost convergence characteristic of 20-generator system
Test System 4: This system consists of forty generators with valve-point loading. The generator
data has been adopted from [6]. The load demand is 10500 MW. Transmission loss has not been
considered here. For this test system, the population size ( ), maximum number of iterations
and the value of probability pa have been selected 50, 500 and 0.7 respectively. Results obtained
from proposed CSA, BBO [12], NPSO-LRS [17], and SOH-PSO [18] have been depicted in
Table 3.4. The cost convergence characteristic of this test system obtained from CSA is shown in
Fig. 3.9.
0 50 100 150 200 250 300 350 400 450 5006.245
6.25
6.255
6.26x 10
4
iteration
cost(
$/h
our)
35 | P a g e
Table 3.4: Simulation results for 40-generator system
Output
(MW) CSA BBO
[12]
NPSO-
LRS [17]
SOH-
PSO
[18]
Output
(MW) CSA BBO
[53]
NPSO-
LRS
[17]
SOH-
PSO
[18]
1 112.0518 111.0465 113.9761 110.80
21 523.3012 523.417 523.2916 523.28
2 111.4948 111.5915 113.9986 110.80
22 523.2928 523.2795 523.2853 523.28
3 97.5626 97.6077 97.4141 97.40
23 523.2892 523.3793 523.2797 523.28
4 179.8000 179.7095 179.7327 179.73
24 523.4340 523.3225 523.2994 523.28
5 88.9834 88.3060 89.6511 87.80
25 523.2839 523.3661 523.2865 523.28
6 140.0000 139.9992 105.4044 140.00
26 523.2810 523.4362 523.2936 523.28
7 299.9993 259.6313 259.7502 259.60
27 10.0000 10.05316 10.0000 10.00
8 284.9506 284.7366 288.4534 284.60
28 10.0009 10.01135 10.0001 10.00
9 284.9653 284.7801 284.6460 284.60
29 10.0014 10.00302 10.0000 10.00
10 130.0006 130.2484 204.8120 130.00
30 92.0666 88.47754 89.0139 97.00
11 94.0000 168.8461 168.8311 94.00
31 190.0000 189.9983 190.0000 190.00
12 94.0000 168.8239 94.00 94.00
32 190.0000 189.9881 190.0000 190.00
13 214.7621 214.7038 214.7663 304.52
33 190.0000 189.9663 190.0000 190.00
14 304.5194 304.5894 394.2852 304.52
34 199.9998 164.8054 199.9998 185.20
15 394.2799 394.2761 304.5187 394.28
35 199.9999 165.1267 165.1397 164.80
16 394.2793 394.2409 394.2811 394.28
36 200.0000 165.7695 172.0275 200.00
17 489.2802 489.2919 489.2807 489.28
37 110.0000 109.9059 110.0000 110.00
18 489.2776 489.4188 489.2832 489.28
38 110.0000 109.9971 110.0000 110.00
19 511.2797 511.2997 511.2845 511.28
39 110.0000 109.9695 93.0962 110.00
20 511.2799 511.3073 511.3049 511.27
40 511.2824 511.2794 511.2996 511.28
Total cost ($/h)
121425.61
121426.95 121664.43 121501.14
36 | P a g e
Fig. 3.9.Cost convergence characteristic of 40-generator system
0 50 100 150 200 250 300 350 400 450 5001.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.3
1.31x 10
5
iteration
cost(
$/h
our)
Chapter 4
MULTI AREA ECONOMIC DISPATCH
37 | P a g e
Chapter 4: MULTI AREA ECONOMIC DISPATCH
4.1 Introduction
Economic Dispatch allocates the load demand among all the committed generators most
economically while satisfying the physical & operational constraints in a single area. Generally,
the generators are divided into several generation areas which are inter-connected by tie-lines.
Multi-Area Economic Dispatch (MAED) is an extension of Economic Dispatch as described in
the previous chapter. MAED determines the level of generation & the exchanged power between
the areas such that the total fuel cost in all the areas get minimized while satisfying all the
constraints such as ; power balance, generating limits, tie-line capacity etc.. Fig. 4.1 shows a
Multi-Area generation system connected via tie-lines.
Fig.4.1 Four-Area Generation System connected via. Tie-lines
The objective of MAED is to minimize the total production cost of supplying loads to the areas
while satisfying all the above said constraints.
38 | P a g e
4.2 Operational Constraints in MAED
4.2.1 Real Power Balance constraint
It says that the total generated power by the committed generators in an area should be equal to
the summation of the load demand in that particular area, the transmission loss & the tie line
power flows from that area to the other areas.
iN (4.1)
With the help of B-coefficients, the transmission loss, can be expressed as
(4.2)
4.2.2 Tie-Line Capacity constraint
The tie line real power transfer from area i to area k & it should not exceed the tie line transfer
capacity for security consideration.
iN & j (4.3)
4.2.3 Real power generation constraint
The generator output in a particular area must lie within its limits.
(4.4)
4.3 Types of MAED problems
Three different types of MAED problems have been taken into account. [40]
39 | P a g e
4.3.1 Multi-Area Economic dispatch with quadratic cost function, prohibited operating
zones & transmission losses (MAEDQCPOZTL)
The prohibited operating zones are the range of power output of a generator where its operation
causes undue vibration of the shaft bearing caused by opening or closing of the steam valve. This
undue vibration may cause damage to the shaft & bearings. Operation of a generator is generally
avoided in such regions. The feasible operating zones of a generating unit can be described as
follows:
(4.5)
The objective function Ft, total cost of committed generators of all areas, can be written as
(4.6)
, the cost function of the jth
generator in the area i, is expressed as a quadratic polynomial.
Equation (4.6) is subjected to the constraints as given by equations (4.1) to (4.5).
4.3.2 Multi-Area Economic Dispatch with Valve Point Loading (MAEDVPL)
The generator cost function is obtained from data points taken during heat-run tests, when
input & output data are measured as the unit is slowly varied through its operating region. Wire
drawing effects, occurring as each steam admission valve in a turbine starts to open, produce a
40 | P a g e
rippling effect on the unit curve. To model the effect of valve-points, a recurring rectified
sinusoid contribution is added to the quadratic function [3]. The fuel cost function considering
valve-point loading of the generator is given as
{ (
)}
(4.7)
The objective of MAEDVPL is to minimize subject to the constraints given in equations (4.1),
(4.3) & (4.4). The transmission loss ( ) is not considered here.
4.3.3 Multi-Area Economic Dispatch with Valve Point Loading, Multiple Fuel Sources &
Transmission Losses (MAEDVPLMFTL)
Since generators are supplied with multiple fuel sources [15] in practical, each generator should
be represented with several piecewise quadratic functions superimposed sine terms reflecting the
effect of fuel type m changes & the generator must identify the most economical fuel to burn.
The fuel cost function of the ith
generator with NF fuel types considering valve point loading is
expressed as
( )
(4.8)
(4.9)
41 | P a g e
The objective function Ft is given by
(4.10)
Equation (4.10) can be expanded as shown in equation (4.11)
(4.11)
The objective function is to be minimized subject to the constraints given in equations (4.1),
(4.3) & (4.4).
4.4 Determination of generation level of the slack generator
Mi committed generators in area i deliver their output power subject to the power balance
constraint (4.1), tie line capacity constraints (4.3) &the respective generation limit constraint
(4.4). Assuming the power loading of the first (Mi-1) generators are known, the power level of
the Mith
generator (i.e. the slack generator) is given by
(
)
(4.12)
42 | P a g e
Expanding & rearranging, equation (4.12), we get
(
)
(
)
(4.13)
4.5 Results
The proposed cuckoo search algorithm has also been applied to solve ED problems in multi area
systems interconnected via. Tie-lines in two different test systems for verifying its feasibility.
The software has been written in MATLAB 7 on a PC (Pentium IV, 80 GB, 3.0 GHZ).Test
Systems 1, 2 & 3.
Test System 1: This system consists of two areas. Each area consists of three generators with
prohibited operating zones. Transmission loss is considered here. The generator data has been
modified from [8]. The generator data & B-coefficients are given in Appendix 2. The percentage
of load demand in area 1 is 60% & 40% in area 2. The total load demand is 1263 MW and the
power flow limit of the system is 100 MW.
The problem is solved by CSA via Levy flight. Here, the parameters are selected as follows.
Number of iterations (nit) =100
Number of population (np) = 10
Probability of getting an alien egg discovered = 70%
43 | P a g e
Updated coefficient based on probability (F) =1
Distribution factor () while incorporating Levy flight = 1.67
To validate the proposed algorithm, CSA via Levy flight, the same test system is solved by
using Artificial Bee Colony Optimization (ABCO) [40], Differential Evolution (DE),
Evolutionary Programming (EP), & Real Coded Genetic Algorithm (RCGA). In case of ABCO
algorithm, the parameters are selected as ns =50, m =30, nb =10, mulG =0.1 mulT =0.01 &
Nmax=100 for this test system under consideration. In case of DE, the population size, scaling
factor & crossover constant has been selected as 200, 1.0 and 1.0 respectively. In case of EP, the
population size & scaling factor have been selected as 100 & 0.1 respectively. In RCGA, the
population size, crossover & mutation probabilities have been selected as 100, 0.9, & 0.2
respectively. Maximum number of generations has been selected 100, for ABCO, DE, EP &
RCGA. Results obtained from the proposed CSA via Levy flight, ABCO, DE, EP, & RCGA
have been summarized in Table 4.1. The cost convergence characteristic of this test system
obtained from CSA via Levy flight is shown in Fig 4.2.
44 | P a g e
Table 4.1: Simulation results for 2-Area System
Fig. 4.2 Cost convergence characteristic of 2-Area System
0 10 20 30 40 50 60 70 80 90 1001.2137
1.2137
1.2137
1.2137
1.2138
1.2138
1.2138
1.2138
1.2138
1.2139x 10
4
Generation
Cost(
$/h
)
CSA ABCO DE EP RCGA
P1,1 (MW) 492.6194 500.0000 500.0000 500.0000 500.0000
P1,2 (MW) 200.0000 200.0000 200.0000 200.0000 200.0000
P1,3 (MW) 149.8511 149.9997 150.0000 149.9919 149.6328
P2,1 (MW) 203.5850 204.3358 204.3341 206.4493 205.9398
P2,2 (MW) 171.6067 154.9954 154.7048 154.8892 155.8322
P2,3 (MW) 57.3578 67.2915 67.5770 65.2717 65.2209
T1,2 (MW) 82.7000 82.7728 82.7731 82.7652 82.4135
PL1 (MW) 7.0200 9.4269 9.4269 9.4267 9.4193
PL2 (MW) 5.0000 4.1955 4.1890 4.1754 4.2064
Cost($/h) 12137.35 12255.39 12255.42 12255.43 12256.23
CPU
time(seconds)
8.2993 10.9844 11.9219 16.8906 19.6094
45 | P a g e
Test System 2: This system comprises of ten generators with valve-point loading and multiple
fuel sources having three fuel options. Transmission loss is considered here. The generator data
has been taken from [6]. The total load demand is 2700 MW. The ten generators are divided into
three areas. Area 1 consists of the first four units; area 2 consists of the next three units and area
3 consists of the last three units. The load demand in area 1 is assumed as 50 % of the total
demand. The load demand in area 2 is assumed to be 25% of the total demand and in area 3 is
taken as 25 % of the total demand. The power flow limit from area 1 to area 2 or vice-versa is
100MW. The power flow limit from area 1 to area 3 or vice-versa is 100MW. The power flow
limit from area 2 to area 3 or vice-versa is 100MW. The B-coefficients are given in Appendix 2.
The problem is solved by CSA via Levy flight. Here, the parameters are selected as follows.
Number of iterations (nit) =100
Number of population (np) = 50
Probability of getting an alien egg discovered = 70%
Updated coefficient based on probability (F) =1
Distribution factor () while incorporating Levy flight = 1.99
To validate the proposed algorithm, CSA via Levy flight, the same test system is solved by
using Artificial Bee Colony Optimization (ABCO), Differential Evolution (DE), Evolutionary
Programming (EP), & Real Coded Genetic Algorithm (RCGA). In case of ABCO algorithm, the
parameters are selected as ns =50, m =30, nb =10, mulG =0.1 mulT =0.01 & Nmax=300 for this test
system under consideration. In case of DE, the population size, scaling factor & crossover
46 | P a g e
constant has been selected as 200, 1.0 and 1.0 respectively. In case of EP, the population size &
scaling factor have been selected as 100 & 0.1 respectively. In RCGA, the population size,
crossover & mutation probabilities have been selected as 100, 0.9, & 0.2 respectively. Maximum
number of generations has been selected 300, for ABCO, DE, EP & RCGA. Results obtained
from the proposed CSA via Levy flight, ABCO, DE, EP, & RCGA have been summarized in
Table 4.2. The cost convergence characteristic of this test system obtained from CSA via Levy
flight is shown in Fig 4.3.
47 | P a g e
Table 4.2: Simulation results for 3-Area System
CSA ABCO DE EP RCGA
Fuel Fuel Fuel Fuel Fuel
P1,1 (MW) 225.5422 2 225.9431 2 225.4448 2 223.8491 2 239.0958 2
P1,2 (MW) 211.0131 1 211.9514 1 210.1667 1 209.5759 1 216.1166 1
P1,3 (MW) 489.4703 2 489.9216 2 491.2844 2 496.0680 2 484.1506 2
P1,4 (MW) 243.0732 3 240.6232 3 240.8956 3 237.9954 3 240.6228 3
P2,1 (MW) 238.6128 1 254.0397 1 251.0049 1 259.4299 1 259.6639 1
P2,2 (MW) 201.2374 3 235.4927 3 238.8603 3 228.9422 3 219.9107 3
P2,3 (MW) 294.5629 1 263.8837 1 264.0906 1 264.1133 1 254.5140 1
P3,1 (MW) 249.5153 3 237.0006 3 236.9982 3 238.2280 3 231.3565 3
P3,2 (MW) 223.5721 1 328.7373 1 326.5394 1 331.2982 1 341.9624 1
P3,3 (MW) 361.0306 1 248.8607 1 250.3339 1 246.6025 1 248.2782 1
T2,1(MW) 95.0078 99.8288 99.4680 100 93.1700
T3,1(MW) 97.7667 99.7334 100 100 93.8739
T3,2(MW) 35.8230 31.2615 30.2810 32.5231 43.7824
PL1 (MW) 11.9365 17.2095 17.2680 17.4884 17.0297
PL2 (MW) 0.1973 9.8488 9.7688 10.0085 9.7010
PL3 (MW) 25.5015 8.6037 8.5905 8.6056 8.9408
Cost($/h) 646.9233 653.9995 654.0184 655.1716 657.3325
CPU
time(seconds)
85.3622
90.4381
95.0351
108.0625
133.8438
48 | P a g e
Fig. 4.3 Cost convergence characteristic of 3-Area System
Test System 3: This system comprises of 40 generators with valve point loading. The generator
data has been taken from [14]. The total load demand is 10500 MW. The forty generators are
divided into four areas. Area 1 includes 1st ten units and 15% of the total load demand. Area 2
has 2nd
ten generators and 40% of the total load demand. Area 3 includes 3rd
ten generators and
30% of the total load demand. Area 4 includes last 10 generators and 15% of the total load
demand. The power flow limit from area 1 to area 2 or vice-versa is 200MW. The power flow
limit from area 1 to area 3 or vice-versa is 200MW. The power flow limit from area 2 to area 3
or vice-versa is 200MW. The power flow limit from area 1 to area 4 or vice-versa is 100MW.
49 | P a g e
The power flow limit from area 2 to area 4 or vice-versa is 100MW. The power flow limit from
area 3 to area 4 or vice-versa is 100MW. Transmission losses are neglected here.
The problem is solved by CSA via Levy flight. Here, the parameters are selected as follows.
Number of iterations (nit) =500
Number of population (np) = 25
Probability of getting an alien egg discovered = 70%
Updated coefficient based on probability (F) =1
Distribution factor () while incorporating Levy flight = 0.99
To validate the proposed algorithm, CSA via Levy flight, the same test system is solved by
using Artificial Bee Colony Optimization (ABCO)i, Differential Evolution (DE), Evolutionary
Programming (EP), & Real Coded Genetic Algorithm (RCGA). In case of ABCO algorithm, the
parameters are selected as ns =100, m =50, nb =20, mulG =0.1 mulT =0.01 & Nmax=500 for this test
system under consideration. In case of DE, the population size, scaling factor & crossover
constant has been selected as 400, 1.0 and 1.0 respectively. In case of EP, the population size &
scaling factor have been selected as 200 & 0.1 respectively. In RCGA, the population size,
crossover & mutation probabilities have been selected as 200, 0.9, & 0.2 respectively. Maximum
number of generations has been selected 500, for ABCO, DE, EP & RCGA. Results obtained
from the proposed CSA via Levy flight, ABCO, DE, EP, & RCGA have been summarized in
Table 4.3. The cost convergence characteristic of this test system obtained from CSA via Levy
flight is shown in Fig 4.4.
50 | P a g e
Table 4.3: Simulation results for 4-Area System
POWER
(MW)
CSA
ABCO
DE
EP
RCGA
POWER
(MW)
CSA
ABCO
DE
EP
RCGA
P1,1 112.9320
111.1020
93.0826
114.0000
94.0855 P3,4
527.1708
542.3424
545.9437
531.7377
524.9246
P1,2 113.7681
109.9774
109.0592
114.0000
47.7313
P3,5 527.4900
520.2448
523.6608
526.7530
495.4096
P1,3 101.0535
100.9238
89.7493
63.7726
85.4353 P3,6
541.0723
533.6389
527.3677
550.0000
442.8850
P1,4 80.6090
190.0000
116.9489
138.8847
131.2807
P3,7 12.6768
10.0000
10.0000
10.0000
51.7060
P1,5 96.7969
96.9390
97.0000
75.3245
79.1711 P3,8
10.1249
10.0000
15.7851
10.0000
42.4448
P1,6 140.0000
96.9675
140.0000
106.4216
131.4026
P3,9 10.0000
10.0000
10.0000
10.0000
47.9812
P1,7 262.0304
259.6950
263.7266
300.0000
176.5484 P3,10
87.3564
96.7699
93.0253
89.7589
95.5812
P1,8 300.0000
276.8725
286.2646
300.0000
232.6707
P4,1 162.0848
190.0000
190.0000
173.5365
149.1883
P1,9 296.9276
300.0000
284.9088
284.9513
292.1746 P4,2
190.0000
168.6841
157.8968
190.0000
159.4065
P1,10 131.2435
130.6977
131.6349
136.7335
130.1531
P4,3 162.1322
173.6165
190.0000
116.4310
161.6999
P2,1 102.3412
245.1007
169.8738
175.3639
340.9307 P4,4
155.6870
186.3740
200.0000
180.6554
167.5135
P2,2 94.0436
94.0000
110.9708
94.000
185.7976
P4,5 166.4450
200.0000
90.0000
162.0916
172.4220
P2,3 125.0000
125.0000
229.8845
263.8126
462.1471 P4,6
164.7566
164.9570
149.4540
173.0920
179.2210
P2,4 499.4270
434.8062
387.4742
331.0545
391.6765 P4,7
110.0000
92.5627
110.0000
109.4254
91.9333
P2,5 489.8462
390.6743
427.7543
394.2191
376.9261
P4,8 110.0000
96.9911
88.1630
74.3342
92.5453
P2,6 396.3797
395.0043
478.2780
413.0955
484.3564 P4,9
81.2047
109.8153
25.0000
99.6914
89.0354
P2,7 499.9881
500.0000
490.1819
499.6763
481.2045
P4,10 512.0916
431.4011
538.4695
541.9711
458.8239
P2,8 492.8359
500.0000
490.9476
500.0000
421.9451 T1,2
160.0000
191.7078
200.0000
200.0000
-118.7357
P2,9 550.0000
530.7889
511.9152
533.8328
469.0019
T3,1 12.7800
6.6740
91.5412
94.6831
-25.9549
P2,10 511.8146
514.4090
511.8241
508.9305
511.2801 T3,2
183.0000
183.1852
147.8992
186.0147
174.0405
P3,1 527.1287
527.1989
547.6323
520.6865
513.0630
T4,1 86.8590
86.8590
51.0838
46.2286
81.5599
P3,2 511.5380
502.0795
523.4937
531.7618
513.8375 T4,2
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