Acc. No. DC 1385.pdf

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 4.1 Simulation results for 2-Area System 44



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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. 4.1 Four-Area Generation System connected via. Tie-lines 37

Fig. 4.2 Cost convergence characteristic of 2-Area System 44



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:

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( ) ( )

(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


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


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


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.








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


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.








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


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

Documents

Acc. No. DC 1385.pdf