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C-Grasp Application to the Economic Dispatch Problem

Permanent Link: http://ufdc.ufl.edu/UFE0041893/00001

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Title: C-Grasp Application to the Economic Dispatch Problem
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Radziukyniene, Ingrida
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: dispatch, economic, ga, grasp, sa
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Economic dispatch plays an important role in power system operations, which is a complicated nonlinear constrained optimization problem. It has non-smooth and non-convex characteristic when generation unit valve-point effects are taken into account. This work adopts the C-GRASP algorithm to solve differently formulated economic dispatch problems. The comparison of the feasibility and effectiveness of the C-GRASP, SA and GA is given as well.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ingrida Radziukyniene.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Pardalos, Panagote M.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041893:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041893/00001

Material Information

Title: C-Grasp Application to the Economic Dispatch Problem
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Radziukyniene, Ingrida
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: dispatch, economic, ga, grasp, sa
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Economic dispatch plays an important role in power system operations, which is a complicated nonlinear constrained optimization problem. It has non-smooth and non-convex characteristic when generation unit valve-point effects are taken into account. This work adopts the C-GRASP algorithm to solve differently formulated economic dispatch problems. The comparison of the feasibility and effectiveness of the C-GRASP, SA and GA is given as well.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ingrida Radziukyniene.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Pardalos, Panagote M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041893:00001


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C-GRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM


By

INGRIDA RADZIUKYNIENE


















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2010































@ 2010 Ingrida Radziukyniene

































I dedicate this to my wonderful son, Matas









ACKNOWLEDGMENTS

I am grateful to many people for supporting me throughout my graduate study in

United States. First of all, I would like to express my earnest gratitude to my advisor,

Dr. Panos M. Pardalos, for directing this study and reading previous drafts of this work.

Without his guidance, inspiration, and support throughout the course of my research,

this work would not be complete. Many thanks to Arturas who has been there for me,

listening to me and supporting me. I am also thankful to my friends at the Center for

Applied Optimization who mentally supported and made my student life more colorful.









TABLE OF CONTENTS


ACKNOW LEDGMENTS ............................

LIST O F TABLES . .

LIST O F FIG URES . .

ABSTRACT ..................................

CHAPTER

1 INTRO DUCTIO N .. .. .. .. .. .. ..

1.1 M otivation . .
1.2 Literature Overview ........................

2 ECONOMIC DISPATCH (ED) PROBLEM ...............

2.1 ED C onstraints . .
2.1.1 Load-Generation Balance .................
2.1.2 Generation Capacity Constraint ..............
2.1.3 Generating Unit Ramp Rate Limits ............
2.1.4 Reserve Contribution ................ ...
2.1.5 System Spinning Reserve Requirement .........
2.1.6 Tie-line Limits .. ............... .....
2.1.7 Prohibited Zone ................ .....
2.2 Objective Functions .. .....................
2.2.1 Smooth Cost Function .. ................
2.2.2 Non-smooth Cost Functions with Valve-point Effects ..
2.2.3 Non-smooth Cost Functions with Multiple Fuels .....
2.2.4 Non-smooth Cost Functions with Valve-Point Effects and
Fuels ...........................
2.2.5 Em mission Function .. ..................


page
. 4


Multiple


3 SOLUTION METHODS .. ...........................

3.1 Continuous Greedy Randomized Adaptive Search Procedure (C-GRASP)
3.2 Genetic Algorithms (GA) ..........................
3.3 Simulated Annealing (SA) ........................
3.4 Constraints Handling ..............................
3.4.1 Penalty-Based Approach .. ....................
3.4.2 Heuristic Strategy .........................

4 EXPERIMENTS AND RESULTS .. ......................

4.1 Experim ents . . .
4.1.1 System 1 .









4.1.2 System 2 .................... ........... 33
4.1.3 System 3 .................... ........... 35
4.1.4 System 4 .................... ........... 35
4.1.5 System 5 .................... ........... 37
4.2 Results ........................ ............. 37
4.2.1 Case 1 ................... ............. 38
4.2.2 Case 2 ................... ............ 38
4.2.3 Case 3 ................... ............ 40
4.2.4 Case 4 .................. ............ 41

5 CO NCLUSIO N .. .. .. . .. .. 43

REFERENCES ...................................... 44

BIOGRAPHICAL SKETCH .................... ........... 50










LIST OF TABLES


Table

4-1

4-2

4-3

4-4

4-5

4-6

4-7

4-8

4-9

4-10

4-11

4-12

4-13

4-14

4-15


Generation costs for 13-unit system with demand 1800 MW

Generation costs for 40-unit system with demand 10500 MV

Best solution for case 4 ....................


Generating units characteristics of five-unit system .

Load demand ................ .........

Generating units characteristics of six-unit system .

Rump-up limits and prohibited zones of six-unit system .

Generating units characteristics of 13-unit system .

Generating units characteristics of 40-unit system .

Generating units characteristics of 10-unit system .

Load demand for 24 hours . .

Generation costs for case 1 ..................

Best solution for case 1 . .

Best solutions for case 2 ... ...

Best results, when demand is1263 MW .


&


page

33

34

34

35

35

3 6

37

37

38

38

39

39

4 0

4 0
V ........... 41
41









LIST OF FIGURES
Figure page

2-1 Example of cost function with two prohibited operating zones ... 19

2-2 Cost function with valve-point effects ..... .... .. 21

2-3 Cost function with multiple fuels ... 22









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

C-GRASP APPLICATION TO THE ECONOMIC DISPATCH PROBLEM

By

Ingrida Radziukyniene

August 2010

Chair: Panos M. Pardalos
Major: Industrial and Systems Engineering

Economic dispatch plays an important role in power system operations, which

is a complicated nonlinear constrained optimization problem. It has non-smooth and

non-convex characteristic when generation unit valve-point effects are taken into

account. This work adopts the C-GRASP algorithm to solve differently formulated

economic dispatch problems. The comparison of the feasibility and effectiveness of the

C-GRASP, SA and GA is given as well.









CHAPTER 1
INTRODUCTION

1.1 Motivation

The economic dispatch (ED) optimization problem is one of the fundamental

issues in power systems to obtain optimal benefits with the stability, reliability and

security [52]. Essentially, the ED problem is a constrained optimization problem in

power systems that have the objective of dividing the total power demand among the

on-line participating generators economically while satisfying the various constraints.

ED problem have complex and nonlinear nonconvex characteristics with equality and

inequality constraints. Therefore, good solutions of the ED problem would result in great

economical benefits.

Over the years, many efforts have been made to solve this problem, incorporating

different kinds of constraints or multiple objectives, through various mathematical

programming and optimization techniques [42]. In the conventional methods such

as the lambda-iteration method, the base point and participation factors, and the

gradient methods, an essential assumption is that the incremental cost curves of

the units are monotonically increasing piece wise linear functions, but the practical

systems are nonlinear [52]. Hence, global optimization techniques, such as the genetic

algorithms (GAs), simulated annealing (SA), and particle swarm optimization (PSO)

have been studied in the past decade and have been successfully used to solve the ED.

However, the references with continuous greedy randomized adaptive search procedure

(C-GRASP) application to such type of problems hadn't appear yet.

The aim of this work is to apply the C-GRASP to the ED problem and compare its

effectiveness and produced solution feasibility with ones of other heuristic methods like

the GAs and SA.









1.2 Literature Overview


Since Carpentier introduced a network constrained economic dispatch problem in

1962 [9] and the first paper in the area of dynamic dispatching was published by Bechert

and Kwatny in 1972 [6], a lot of researches have employed various mathematical

programming optimization methods for solving ED problems [30]. These optimization

techniques can be classified into three main categories.

The first category contains deterministic methods that include the linear programming

algorithm [26, 57, 69], quadratic programming algorithm [18, 37], non-linear programming

algorithm [39], etc. The LP method application to the power-system rescheduling

problem with security-constrained economic dispatch/control for multiple-valved-turbine

units was given by Stott and Marinho [57]. Rosehart et al. [48] discovered that for the

economic dispatch problem, SLP appears to be a better tool than SQP. An approach

based on efficient SLP techniques to solve the multi-objective environmental/economic

load dispatch problem was presented by Zehar and Sayah [69]. Granelli et al. [18]

solved a security constrained economic dispatch problem using modified SQP

techniques. A dual feasible starting point is found by relaxing transmission limits and

then constraint violations are enforced applying the dual quadratic algorithm. In [59]

and [35], a security constrained economic dispatch problem was solved by SLP and the

interior point dual-affine scaling algorithm. Momoh et al. [37] proposed an IPM for ED

problem formulated as linear and convex QP. However each of traditional methods has

some defects: it would generate large errors to use the linear programming algorithm

to linearize the ED model; for the quadratic programming and nonlinear programming

algorithms, the objective function should be continuous and differentiable [30].

The second category contains the methods based on artificial intelligence. Artificial

intelligence technology has been successfully used to solve the ED problem. A chaos

optimization algorithm (CAO) has been proposed by Jiang et al. [29] to deal with the

economic dispatch problem of a hydro power plant. Zhijiang et al. [71] also applied a









COA and the simulation results verified that the proposed approach is effective and

precise. A mutative scale COA was applied by Xu et al. [65] to the economic operation

of power plants. However, the results showed that the method is time-consuming. An

improved mutative scale COA hes been developed by Han and Lu [19]. According to

the authors, their algorithm is highly efficient and can be applied not only to ED but to

many power system problems, such as economic operation, OPF, system identification

and optimal control, as well. In [36], Mahdad et al. proposed an efficient decomposed

parallel GA to solve the multi-objective environmental/economic dispatch problem.

In the first stage, the original network is decomposed into multi sub-systems and

the problem is transformed to optimize the active power demand associated with

each partitioned network. In the second stage, an active power dispatch strategy

is proposed to enhance the final solution of the optimal power flow of the original

network. The proposed approach was tested on the Algerian 59-bus test system.

The computational results showed the convergence at the near solution and obtain a

competitive solution at a reduced time. GAs with fuzzy logic controllers to adjust its

crossover and mutation probabilities was applied by Song et al. [56] to solve a combined

environmental economic dispatch problem. SA techniques were used by Roa-Sepulveda

and Pavez-Lazo [47], however, long computational time to obtain an optimal solution

was reported. Tabu search was applied by Altun and Yalcinoz [2]. Simulation results

on power systems consisting of 6 and 20 generating units exhibited good performance.

In [38], an application of TS for solving security constrained ED problem was given by

Muthuselvan and Somasundaram. Base case and contingency case line flow constraints

were considered. Tests on 66-bus and 191-bus Indian utility systems revealed the

reliability, efficiency and suitability of the proposed algorithm for practical applications.

The third category consists the hybrid methods, which combine two or more

techniques in order to get best features in each algorithm. Typically, significant

improvement with hybrid methods can be achieved over each of the individual methods.









Hybrid methods gained increasing popularity in the last 10 years. For the ED problem,

Wong and Wong [63] combined an incremental GA with SA techniques. Coelho and

Mariani [12] proposed a method combining a DE algorithm with self-adaptive mutation

factor in the global search stage and chaotic local search techniques in the local search

to solve an ED problem associated with the valve-point effect. The same authors report

another successful application of chaotic PSO in combination with an implicit filtering

local search method to solve economic dispatch problems [13]. The chaotic PSO

approach is used to produce good potential solutions, while the implicit filtering is used

to fine-tune the final solution of the PSO. The hybrid methodology is validated for a

test system consisting of 13 thermal units whose incremental fuel cost function takes

into account the valve-point loading effects. In [11], Coelho and Lee improved PSO

approaches for solving an ED problem taking into account non-linear generator features

such as ramp-rate limits. Prohibited operating zones in the power system operation are

developed as well. Their algorithm combines the PSO, Gaussian probability distribution

functions and/or chaotic sequences. The PSO and its variants are validated for two test

systems consisting of 15 and 20 thermal generation units, respectively. A combination of

chaotic and self-organization behavior of ants in the foraging process was presented by

Cai et al. [8]. This algorithm was applied to ED problems with thermal generators.

The thesis is organized as follows: In Section 2, we briefly discuss a general

ED problem formulation. The methods applied to solve ED are shortly discussed in

Section 3. Section 4 describes experimental cases and presents calculation results. We

conclude with Section 5.









CHAPTER 2
ECONOMIC DISPATCH (ED) PROBLEM

ED is one of the important optimization problems in power system operations, which

is used to determine the optimal combination of power outputs of all generating units to

minimize the total fuel cost while satisfying various constraints over the entire dispatch

periods [67].

The traditional or static ED problem assumes constant power to be supplied by a

given set of units for a given time interval and attempts to minimize the cost of supplying

this energy subject to constraints on the static behavior of the generating units like

system load demand. Shortly, static ED determines the loads of generators in a system

that will meet a power demand during a single scheduling period for the least cost.

Therefore, it might fail to capture large variations of the load demand due to the ramp

rate limits of the generators. Due to large variation of the customers load demand and

the dynamic nature of the power systems, it became necessary to schedule the load

beforehand so that the system can anticipate sudden changes in demand in the near

future.

Dynamic ED is an extension of static ED to determine the generation schedule of

the committed units so that to meet the predicted load demand over the entire dispatch

periods at minimum operating cost under ramp rate and other constraints [64]. The

ramp rate constraint is a dynamic constraint which used to maintain the life of the

generators, i.e. plant operators, to avoid shortening the life of the generator, try to keep

thermal stress within the turbines safe limits [20]. Since the violations of the ramp rate

constraints are assessed by examining the generators output over a given time interval,

this problem cannot be solved for a single value of MW generation [20]. The objective

function of dynamic ED is formulated as follows
T N
minC(P) =_ C,(Pf) (2-1)
t=l i=1









where N is the set of committed units; Pi is the generation of unit i; C,(Pi) is the cost of

producing Pi from unit i; T is the number of intervals in the study period. The fuel cost

functions C,(.) is derived from the fuel consumption function that can be measured and

are discussed in Section 2.2.

The dynamic ED is not only the most accurate formulation of the economic dispatch

problem but also the most difficult to solve because of its large dimensionality [3]. The

DED problem is normally solved by discretization of the entire dispatch period into a

number of small time intervals, over which the load demand is assumed to be constant

and the system is considered to be in a temporal steady state. Over each time interval

a static ED problem is solved under static constraints and the ramp rate constraints are

enforced between the consecutive intervals [34]. In the DED problem the optimization is

done with respect to the dispatchable powers of the units.

Some researchers have considered the ramp rate constraints by solving SED

problem interval by interval and enforcing the ramp rate constraints from one interval

to the next. However, this approach can lead to suboptimal solutions [23]; moreover, it

does not have the look-ahead capability.

Since dynamic ED was introduced, variuos methods have been used to solve this

problem. However, all of those methods may not be able to provide an optimal solution

and usually getting stuck at a local optimal.

2.1 ED Constraints

The constrained ED problem is subjected to a variety of constraints depending

upon assumptions and practical implications. Usually, formulation of ED problem

includes such constraints as load generation balance, minimum and maximum capacity

constraints. To maintain system reliability and security, spinning reserve constraints

and security constraints can be added to the dynamic ED problem. The inclusion of the

prohibited zones, ramp-rate limits and other practical constraints results in nonconvex

ED of generating units. All these constraints are discussed bellow.









2.1.1 Load-Generation Balance

The generated power from all the running units must satisfy the load demand and

the system losses given by (2-2)
N
Pf = D' + Losst, t=1,2,..., T (2-2)
i= 1

where Dt is the demand and Losst is the system transmission loss. Their sum

represents the effective load to be satisfied at the tth interval. The transmission line

losses can be expressed in terms of the unit outputs:
N N N
Losst = PfBPj + P BoPf + Boo
i=1 j=1 i=1

where B, is the ijth element of the loss coefficient square matrix, Bio is the ith element

of the loss coefficient, and Boo is the constant loss coefficient. Sometimes the last two

terms are omitted.

In a competitive environment, the load-generation balance constraint is relaxed

and each generating company schedules its production to maximize its profits given a

forecast of electricity prices for the scheduling period. As a first approximation, each

generating unit could be optimized separately in this problem because of the decoupling

made possible by the availability of prices at each period. Dynamic constraints (such

as ramp rates and minimum up and down time constraints) complicate the problem

because a generating company that owns a portfolio of units must then decide whether

to buy "flexibility" on the market or meet the dynamic constraints with its own resources

[21].

2.1.2 Generation Capacity Constraint

For normal system operations, real power output of each generator is restricted by

lower and upper bounds as follows:


P + Sf < P ax = 1,2,... N, t= 1,2,..., T (2-3)









min < p i= 1,2,...N, t = 1,2,..., T (2-4)

where Pmin and pmax are the minimum and maximum power produced by generator i, Sf

is the reserve contribution of unit during time interval t.

2.1.3 Generating Unit Ramp Rate Limits

One of unpractical assumption that prevailed for simplifying the problem in many of

the earlier research is that the adjustments of the power output are instantaneous [43].

Therefore, the power output of a practical generator cannot be adjusted instantaneously

without limits. The operating range of all online units is restricted by their ramp-rate limits

during each dispatch period. So, the subsequent dispatch output of a generator should

be limited between the constraints of up and down ramp-rates [66] as follows

pit+_ P < URi. At (2-5)

Pf Pf+ < DR; At i = 12 ...N, t= 1, 2..., T 1 (2-6)

where URi and DR, are the maximum ramp up/down rates for unit i and At is the

duration of the time intervals into which the study period is divided. The inclusion of

ramp rate limits modifies the generator operation constraints (2-3, 2-4) as follows


max(P"i, Pf- DR,) < Pi < min(Pmax, P,-1 + UR,) (2-7)

2.1.4 Reserve Contribution

The maximum reserve contribution has to satisfy following constraints:

0 < < S ax i = 1, 2, ... N, t = 1,2,..., T (2-8)

where S"ax is the maximum contribution of unit i to the reserve capacity.

Maximum-ramp spinning reserve contribution is defined as in (2-9)

0< Sf < UR At = 1,2,..., N, t= 1,2,..., T (2-9)

where Sf is the spinning reserve of unit i.









2.1.5 System Spinning Reserve Requirement

Sufficient spinning reserve is required from all running units to maximize and

maintain system reliability [14]. There are many ways to determine the system spinning

reserve requirement. It can be calculated as the size of the largest unit in operation or

as a percentage of forecast load demand or even as a function of the probability of not

having sufficient generation to meet the load [64]. The spinning rezerve can be defined

by (2-10)
N
St > SRt t= 1,2,..., T (2-10)
i=1
where SRt is the system spinning reserve requirement for time interval t. Also, the

system spinning reserve requirement for interval t can sometimes be given by the

following equation [20]:


SRt = odDt +g -g *max(Pmax scheduled at time t, = 1,2,...N) (2-11)

where ad and ag are constants which depend on the system required reliability level

[55]. Besides the determination of the system spinning reserve requirement, the issue of

allocation the spinning reserve among the committed units is very important; however, it

has received very little attention in the dynamic ED literature.

2.1.6 Tie-line Limits

The economic dispatch problem can be extended by importing additional constraint

like transmission line capacity limit given by (2-12)


PTjk,rn < PTk + Sjk < PTkmax (2-12)

where PrTk,mn and PTjkmax specify the tie-line trasnmission capability, i.e. the transfer

from area to area k should not exceed the tie-line transfer capacities for security

consideration [28]. Each area has own special load and its spinning reserve [68].









2.1.7 Prohibited Zone

The generating units may have certain ranges where operation is restricted on the

grounds of physical limitations of machine components or instability, e.g. due to steam

valve or vibration in shaft bearings. So, there is a quest to avoid operation in these

zones in order to economize the production [43]. These ranges are prohibited from

operation and a generator with prohibited regions (zones) has discontinuous fuel-cost

characteristics (Fig. 2.1.7) [53]. The acceptable operating zones of a generating unit can

be formulated as follows

pmin < Pt < P' (2-13)

P:j_ < P < Pj, ie O, j =2,3,..., ni, t = 1, 2,..., T (2-14)

PiUn, < < < pmax (2-15)

where ni is the number of the prohibited zones in unit i, 0 is the set of units that have

prohibited zones, P/i, P P are the lower and upper bounds of thejth prohibited zone.




PZ: Prohibited Zone

0
IPZ1I IPZ2, I
I I J I
II I I

.-- I I I I
I I I I I -
Min Max
Power output (MW)

Figure 2-1. Example of cost function with two prohibited operating zones


2.2 Objective Functions

The dynamic ED problem has been solved with many different forms of the cost

function, such as the smooth quadratic cost function (2-16) or the nonsmooth cost









function due to the valve-point effects (2-17). Also, a linear cost function [20] and

piecewise linear cost function [27, 41] have been employed. For smooth cost function it

is usually assumed that its incremental cost function. In some power systems combined

cycle units are used to supply the base load. For these units the cost function can be

given as linear, piecewise or quadratic with decreasing incremental cost function [41].

For units with prohibited zones, the fuel cost function is discontinuous and nonconvex.

An interesting departure from this standard formulation is the approach proposed by

Wang and Shahidehpour [61] who include in the objective function a term representing

the reduction in the life of the turbine caused by excessive ramping rates. This flexible

technique makes possible a tradeoff between the system operating cost and the life

cycle cost of the generating units [21].

2.2.1 Smooth Cost Function

The most simplified cost function of each generator can be represented as a

quadratic function as given in (2-16) whose solution can be obtained by the conventional

mathematical methods

Ci(Pf,) = ai + bPf + c,(Pf)2 (2-16)

where ai, bi,c, are cost coefficients of generator i.

2.2.2 Non-smooth Cost Functions with Valve-point Effects

The generating units with multi-valve steam turbines exhibit a greater variation in

the fuel cost functions because in order to meet the increased demand a generator

with multi-valve steam turbines increase its output and various steam valves are

to be opened [67]. This valve-opening process produces ripple like effect in the

heat-rate curve of the generator. The inclusion of valve-point loading effects makes

the modeling of the incremental fuel cost function of the generators more practical [60].

Therefore, in reality, the objective function of ED problem has non-differentiable property.

Consequently, the objective function should be composed of a set of non-smooth cost

functions. Considering non-smooth cost functions of generation units with valve-point










$'MlWh E

D-

/.
C







A: Primary Vah B: Secondary Valve MW
C : Tertiary Valv D : Quateramry Vahl
E: Qukmary Valv


Figure 2-2. Cost function with valve-point effects


effects, the objective function is generally described as the superposition of sinusoidal

functions and quadratic functions [52]


Ci(Pf) = ai + bPf + ci(Pf)2 + leisin(hi(P" Pft)) (2-17)

where ei and hi are the coefficients of generator i reflecting valvepoint effects. As shown

in Fig. 2.2.2, this increases the non-linearity of curve as well as number of local optima

in the solution space [60] compared with the smooth cost function due to the valvepoint

effects. Also the solution procedure can easily trap in the local optima in the vicinity of

optimal value.

2.2.3 Non-smooth Cost Functions with Multiple Fuels

Since the dispatching units are practically supplied with multi-fuel sources [49],

each unit should be represented with several piecewise quadratic functions reflecting

the effects of fuel type changes, and the generator must identify the most economic fuel

to burn. The resulting cost function is called a "hybrid cost function." Each segment of

the hybrid cost function implies some information about the fuel being burned or the





















PowerlMWI
Min PI P2 Max

Figure 2-3. Cost function with multiple fuels

units operation. Thus, generally, the fuel cost function is a piecewise quadratic function

described as follows

ail + biPf + cil(Pf)2 if P/ mi < Pf < p

ai2 + bi2Pf + Ci2(Pf)2 if Pf < Pf < pt 2
ci(P,) (2-18)


ain + binPf + cn(Pft)2 if Pt-1 < f
where are a,p, bp, cp the cost coefficients of generator for the pth power level. The

incremental cost functions are illustrated in Fig. (2.2.3)

2.2.4 Non-smooth Cost Functions with Valve-Point Effects and Multiple Fuels

To obtain an accurate and practical economic dispatch solution, the realistic

operation of the ED problem should consider both valve-point effects and multiple

fuels. The reference [10] proposed an incorporated cost model, which combines the

valve-point loadings and the fuel changes into one frame. Therefore, the cost function

should combine (2-17) with (2-18), and can be realistically represented as shown in









(2-19)


ail + bilPf + cil(Pt)2 + lei,1sin(hi,l(Pm n P- ))

ai2 + bi2Pf + i2(Pit)2 + ei,2sin(hi,2(Pin Pt,2))
ci(P,) =


ain + binPf + Cin(Pf)2 + lei,nsin(hi,n(Pirn Pitn))|


if Pt i < Pt < pt
,,mln -- I i,1

f pt < ptf< pt
1,1 i,2



if i < pit < Pimax
(2-19)


2.2.5 Emission Function

Due to increasing concern over the environmental considerations, society demands

adequate and secure electricity, i.e. not only at the cheapest possible price, but also at

minimum level of pollution. In this case, two conflicting objectives, i.e., operational costs

and pollutant emissions, should be minimized simultaneously [4, 5, 7, 62].

The atmospheric pollutants such as sulphur oxides (SO) and nitrogen oxides

(NOx) caused by fossil-fueled generating units can be modeled separately or as the total

emission of them which is the sum of a quadratic [4] and an exponential function and

can be expressed as


(2-20)


T N
Z ai + p,Pf + 7i(Pf)2 + iexp(6iPft)
t=l i=1


where a, iP,








CHAPTER 3
SOLUTION METHODS

3.1 Continuous Greedy Randomized Adaptive Search Procedure (C-GRASP)

Continuous-GRASP (C-GRASP) extends the greedy randomized adaptive search

procedure (GRASP) that was introduced by Feo and Resende [16, 17] from the

domain of discrete optimization to that of continuous global optimization in [24, 25].

It is described as a multi-start local search procedure, where each C-GRASP iteration

consists of two phases, namely, a construction phase and a local search phase [24].

Construction combines greediness and randomization to produce a diverse set of

good-quality starting solutions for local search. The local search phase attempts to

improve the solutions found by construction. The best solution over all iterations is kept

as the initial solution. The advantages of this method is simplicity to implement and no

requirement for derivative information

Pseudo-code for C-GRASP is shown in (3.1). C-GRASP works by discretizing

the domain into a uniform grid. Both the construction (see the high level pseudo-code

3.2) and local improvement phases (see the high level pseudo-code 3.3) move along

points on the grid. As the algorithm progresses, the grid adaptively becomes more

dense. The main difference between GRASP and C-GRASP is that an iteration of

C-GRASP does not consist of a single greedy randomized construction followed by

local improvement, but rather a series of construction-local improvement cycles with

the output of construction serving as the input of the local improvement, as in GRASP,

but unlike GRASP, the output of the local improvement serves as the input of the

construction procedure [25].

Since C-GRASP is essentially an unconstrained optimization algorithm, the

constraints handling strategy needs to be incorporated into it in order to deal with

the constrained ED problem. Approaches to manage these constraints are discussed in

section 3.4.










pseudo-code 3.1 C-GRASP (n, /, u, f(.),Maxlters, MaxNumlterNolmprov,
NumTimesToRun, MaxDirToTry,a)
1: f* -- c0
2: forj 1,..., NumTimesToRun do
3: x UnifRand(/, u); h 1; NumlterNolmprov 0;
4: for Iter 1,..., Maxlters do
5: x +- ConstructGreedyRandomized(x, f(.), n, h, I, u, a);
6: x LocalSearch(x, f(.), n, h, I, u, MaxDirToTry);
7: if f(x) < f* then
8: x* +- x; f* f(x); NumlterNolmprov 0;
9: else
10: NumlterNolmprovw- NumlterNolmprov+1
11: end if
12: if NumlterNolmprov> MaxNumlterNolmprov then
13: h h/2; NumlterNolmprov 0; {/}*make grid more dense*/
14: end if
15: end for
16: end for
17: return x*



pseudo-code 3.2 ConstructGreedyRandomizedSolution (Problem Instance)
1: Solution 0;
2: while Solution construction not done do
3: MakeRCL(RCL);
4: S +- SelectRandomElement(RCL);
5: Solutions- Solution U S;
6: AdaptGreedyFunction(S);
7: end while
8: return (Solution);



pseudo-code 3.3 LocalSearch(Solution,Neighborhood)
1: Solution* Solution
2: while Solution* not locally optimal do
3: Solution*- SelectRandomElement(Neighborhood(Solution*));
4: if Solution better than Solution* then
5: Solution*- Solution;
6: end if
7: end while
8: return (Solution*)









3.2 Genetic Algorithms (GA)

This section engages into the concept of genetic algorithms that reflects the

nature of chromosomes in genetic engineering. GAs are a class of stochastic search

algorithms that start with the generation of an initial population or set of random

solutions for the problem at hand. Each individual solution in the population called a

chromosome or string represents a feasible solution. The objective function is then

evaluated for these individuals. If the best string (or strings) satisfies the termination

criteria, the process terminates, assuming that this best string is the solution of the

problem. If the termination criteria are not met, the creation of new generation starts,

pairs, or individuals are selected randomly and subjected to crossover and mutation

operations. The resulting individuals are selected according to their fitness for the

production of the new offspring. Genetic algorithms combine the elements of directed

and stochastic search while exploiting and exploring the search space [31]. More details

about GA can be found in [22, 46, 58].

pseudocode 3.4 Genetic algorithm
1: initialize population(
2: while not converge do
3: assign population fitness(
4: for 1,..., npopsiz do
5: select parents(p1,p2)
6: reproduction(p ,p2,child)
7: end for
8: select next generation(
9: end while


The advantages of GA over other traditional optimization techniques can be

summarized as follows:

*GA searches from a population of points, not a single point. The population can
move over hills and across valleys. GA can therefore discover a globally optimal
point, because the computation for each individual in the population is independent
of others. GA has inherent parallel computation ability.









* GA uses payoff (fitness or objective functions) information directly for the search
direction, not derivatives or other auxiliary knowledge. GA therefore can deal
with non-smooth, non-continuous and non-differentiable functions that are the
real-life optimization problems. This property also relieves GA of the approximate
assumptions for a lot of practical optimization problems, which are quite often
required in traditional optimization methods.

GA uses probabilistic transition rules to select generations, not deterministic rules.
They can search a complicated and uncertain area to find the global optimum. GA
is more flexible and robust than the conventional methods [33].

The first attempt of the application of genetic algorithms in power systems is in the

load flow problem [70]. It has been found that the simple genetic algorithm quickly finds

the normal load flow solution for small-size networks by specifying an additional term in

the objective function. A number of approaches to improving convergence and global

performance of GAs have been investigated [70].

3.3 Simulated Annealing (SA)

The SA is a generic probabilistic meta-heuristic for the global optimization problem

that was proposed by Kirkpatric et al. [32]. In the SA method, each point s of the search

space is analogous to a state of some physical system, and the function E(s) to be

minimized is analogous to the internal energy of the system in that state. The goal is

to bring the system, from an arbitrary initial state, to a state with the minimum possible

energy. In each step of the SA algorithm the current solution is replaced by a random

"nearby" solution, chosen with a probability that depends on the difference between the

corresponding function values and on a global parameter T (called the temperature),

that is gradually decreased during the process. The dependency is such that the current

solution changes almost randomly when T is large, but increasingly "downhill" as T

goes to zero. The allowance for "uphill" moves saves the method from becoming stuck

at local minima which are the bane of greedier methods. For certain problems, SA may

be more effective than exhaustive enumeration. It has been shown that this technique

converges asymptotically to the global optimal solution with probability one [1].









SA is an effective global optimization algorithm because of the following advantages

[50]:

* suitability to problem in wide area,

* no restriction on the form of cost function,

* high probability to find global optimization,

* easy implementation by programming.

The pseudocode implementing SA is given bellow. It starts from state sO and

continue for kmax of steps or until a state with energy emax or less is found. The call

neighbour (s) should generate a randomly chosen neighbour of a given state s; the call

random() should return a random value in the range [0,1]. The annealing schedule is

defined by the temp(r), which should yield the temperature to use, given the fraction r of

the time budget that has been expended so far.

pseudocode 3.5 Simmulated Annealing
1: s so; e E(s)
2: Sbest +- S; ebest <- e;
3: k 0;
4: while k < kmax and e > emax do
5: Snew neighbour(s)
6: enew E(snew)
7: if enew < best then
8: best Snew; best enew
9: end if
10: if P(e, enew, temp(k/max)) > random() then
11: S Snew; e enew
12: k- k + 1
13: end if
14: end while
15: return Sbest


Actually, the "pure" SA algorithm does not keep track of the best solution found so

far: it does not use the variables Sbest and ebest, it lacks the first if inside the loop, and,

at the end, it returns the current state s instead of Sbest. While saving the best state is a









standard optimization, that can be used in any metaheuristic, it breaks the analogy with

physical annealing since a physical system can "store" a single state only.

In strict mathematical terms, saving the best state is not necessarily an improvement,

since one may have to specify a smaller kmax in order to compensate for the higher cost

per iteration. However, the step Sbest -- Snew happens only on a small fraction of the

moves. Therefore, the optimization is usually worthwhile, even when state-copying is an

expensive operation.

SA has the ability to avoid getting local solutions; then it can generate global or near

global optimal solutions for optimization problems without any restriction on the shape

of the objective functions [44]. SA is not memory intensive [45]. However, the setting of

control parameters of the SA algorithm is a difficult task and the computation time is high

[3]. The computational burden can be reduced by means of parallel processing [44].

3.4 Constraints Handling

Constraints lie at the hear to fall constrained engineering optimization applications.

Practical constraints, which are often nonlinear and non-trivial,confine the feasible

solutions to a small subset of the entire search space. There are several approaches

which can be applied to handle constraints in heuristic approaches. These methods can

be grouped into four categories: methods that preserve the feasibility of solutions,

penalty-based methods, methods that clearly distinguish between feasible and

unfeasible solutions, and hybrid methods [15, 62].

3.4.1 Penalty-Based Approach

The penalty function method is frequently applied to manage constraints in

evolutionary algorithms. Such a technique converts the primal constrained problem

into an unconstrained problem by penalizing constraint violations. The penalty function

method is simple in concept and implementation. However, its primal limitation is

the degree to which each constraint is penalized. These penalty terms have certain

weaknesses that become fatal when penalty parameters are large. Such a penalty









function tends to be ill conditioned near the boundary of the feasible domain where the

optimum point is usually located [10]. The penalized fuel cost function in ED problem

was employed in [51].

In [40] the ED problem was transformed into an unconstrained one by constructing

an augmented objective function incorporating penalty factors for any value violating the

constraints:
Neq Nueq
H(X) = J(X) + k, Z(hj(X))2 + k2 max[0, -g(X)]2 (3-1)
j=1 j=1
where J(X) is the objective function value of the ED problem. Neq and Nueq are the

number of equality and inequality constraints, respectively; hj(X) and gj(X) are the

equality and inequality constraints, respectively; kl and k2 are the penalty factors. Since

the constraints should be met, the value of thek, and k2 parameters were chosen to

have high value of 10,000. This approach was epmpoyed when applying SA method.

The heuristic startegy that is discussed in nex section was used to get a feasible solution

while applying C-GRASP method.

3.4.2 Heuristic Strategy

When the C-GRASP is applied to solve ED problem, a key problem is how to

handle constraints with efficiency. In this section we mainly focus on handling the real

power limits and generators ramp-up constraints. Other than penalty based way to

satisfy the real power balance equality constraints (2-2), is to specify the output of

(N 1) generating units and to find the Nth from the equality constraint like in [4, 67]. In

reference [67], authors employed a dependent generation power pt of randomly selected

unit /.

The heuristic strategy applied in LocalSearchO procedure in C-GRASP algorithm

can be formulated in a following way:

Step 1. Set the dispatch period index t = 1 and iteration i = 1.









Step 2. Calculate the violation of power balance constraint Pr,, at dispatch time t is

calculated from 3-2 as follows
N
Pt = Dt + Losst Pf (3-2)
i=1

If Prr = 0, then go to Step 5, otherwise to Step 3.

Step 3. Randomly generate / the index of generating unit and calculate the real

power of selected dependent generating unit pt from (3-3).

N
Pf =Dt Pft t = 1,2,..., T (3-3)
i=1
i#/

However, considering transmission losses (2.1.1), these equality constraints become

nonlinear and the output of dependent generating unit for every dispatch period t can be

found from by solving a following equation

N N N N N
BII(Pf)2+(2Z BiiPf+B10-1)Pf+(Dt+ PfBiiPj +Boo+ BioPf-, P) = 0 (3-4)
i=1 i=1j=1 i=1 i=1
i-/ i / jil i / i-/

If it doesn't violate the generator operating limits and ramp-up constraints (if they are

present), go to Step 5. Otherwise, the value has to be modified according to 3-5

P max if Pt > pmax

pmin if Pf < pmin

If ED incorporates ramp-up limits and dispatch period t > 1, then dependent unit output

has to be calculated as 3-6

max(Pmin, P-1 DR,) if Pf > max(Pmin, Pf-1 DR) (3-6)
P {(f= <(3-6)
min(Pmax, Pf + URi) if Pf < min(P/ax, Pt- + URI)

After adjustment, go to Step 4.









Step 4. Increase the iteration number by 1, i.e. I = / + 1. If I < /max go to Step 2,

otherwise go to Step 5.

Step 5. Increase the period number by 1, i.e. t = t + 1. If t < T go to Step 2,

otherwise stop.

The applied strategy for constraints handling will produce solutions satisfying

real power limits constraint and generating unit ramp rate limits constraint, however

not always the the real power balance constraint will be satisfied in dynamic ED

due to ramp-up limits. The situation can be that in one dispatch period demand will

meet generation, however in the next period the demand can be not because due to

generating unit power increase or reduction limitation. In order not to consider such

infeasible solution a large penalty is added to objective function value.









CHAPTER 4
EXPERIMENTS AND RESULTS

4.1 Experiments

In order to verify the feasibility and effectiveness of adopted C-GRASP capabilities

for solving ED problems, different ED problem formulations, i.e. static and dynamic

ED and different systems were used. The C-GRASP algotihm with heuristic strategy

to deal with constraints was implemented in Matlab 7.5. For GA and SA algorithms,

the standard Matlab functions form Genetic Algorithm and Direct Search Toolbox

were employed. In standard GA function ga(), it is possible to include both linear and

nonlinear equality and inequality constraints. However, SA function simulannealbnd(

incorporates only lower and upper bound constraints, other constraints as a penalty

function is added to objective function. Next, we will provide descriptions of systems

used for our experiments.

4.1.1 System 1

The system consists of five generating units, whose the maximum total output is

925 MW. On this system dynamic ED problem was solved with the dispatch horizon one

day with 12 intervals of one hour each. The demand of the system and generating unit

data are given in Tables (4-1) and (4-2), respectively.

Table 4-1. Generating units characteristics of five-unit system
ai, $ /h bi, $ /MWh ci, $ /(MW)2h pin, MW pmax, MW
Unit 1 25.000 2.000 0.008 10.000 75.000
Unit 2 60.000 1.800 0.003 20.000 125.000
Unit 3 100.000 2.100 0.0012 30.000 175.000
Unit 4 120.000 2.000 0.001 40.000 250.000
Unit 5 40.000 1.800 0.0015 50.000 300.000


4.1.2 System 2

The system contains six thermal generating units. The total maximum output

of generating units is 1470 MW. This system was used to solve static ED problem

where load demand on the system is 1263 MW. Parameters of all the thermal units are









Table 4-2. Load demand


Time, h


Load, MW


Time, h


Load, MW


1 410 7 626
2 435 8 654
3 475 9 690
4 530 10 704
5 558 11 720
6 608 12 740


reported in [30] and are given in Tables 4-3 and 4-4. In normal operation of the system,

the loss coefficients B are as follows:


0.0017 0.0012 0.0007 -0.0001 -0.0005 -0.0002

0.0012 0.0014 0.0009 0.0001 -0.0006 -0.0001

0.0007 0.0009 0.0031 0.0001 -0.001 -0.0006


0.0001 0.0001


0 0.0024 -0.0006 -0.0008


x 10-2


Table


-0.0005 -0.0006 -0.001 -0.0006 0.0129 -0.0002

-0.0002 -0.0001 -0.0006 -0.0008 -0.0002 0.015

Boi = [-0.3908 0.12970.70470.05910.2161 0.6635] x 10-3

Boo = 0.056

4-3. Generating units characteristics of six-unit system


Unit Pmin, MW Pmax, MW a, $ /h bi, $/MWh c,,$/(MW)2h Po, MW
1 100 500 0.007 7 240 440
2 50 200 0.0095 10 200 170
3 80 300 0.009 8.5 220 200
4 50 150 0.009 11 200 150
5 50 200 0.008 10.5 220 190
6 50 120 0.0075 12 190 110


Bi -









4-4. Rump-up limits and prohibited zones of six-unit system


Unit URi,MW DR,,MW Prohibited zone
1 80 120 [210 240] [350 380]
2 50 90 [90 110] [140 160]
3 65 100 [150 170] [210 240]
4 50 90 [80 90] [110 120]
5 50 90 [90 110] [140 150]
6 50 90 [75 85] [100 105]


4.1.3 System 3

This system consists of 13 generating units with valve-point loading as given in

Table (4-5). The parameters of this system showed is taken from [54]. The expected

demand is 1800 MW and 2520 MW.

Table 4-5. Generating units characteristics of 13-unit system


pmin, MW pmax, MW
i i


680
360
360
180
180
180
180
180
180
120
120
120
120


ai bi ci ei f


0.00028
0.00056
0.00056
0.00324
0.00324
0.00324
0.00324
0.00324
0.00324
0.00284
0.00284
0.00284
0.00284


8.1
8.1
8.1
7.74
7.74
7.74
7.74
7.74
7.74
8.6
8.6
8.6
8.6


550
309
307
240
240
240
240
240
240
126
126
126
126


300
200
200
150
150
150
150
150
150
100
100
100
100


0.035
0.042
0.042
0.063
0.063
0.063
0.063
0.063
0.063
0.084
0.084
0.084
0.084


4.1.4 System 4

This system is composed of 40 generating units with valve-point loading effects

supplying a total demand of 10500 MW. Therefore, this system has nonconvex solution

spaces and there are many local minima due to valve-point effects and the global

minimum is very difficult to determine. The parameters of this system showed in the

Table (4-6) are available in [54] as well.


Unit


Table










Table 4-6. Generating units characteristics of 40-unit system
Unit pmin, MW pmax, MW ai bi ci ei
1 36 114 0.0069 6.73 94.705 100 0.084
2 36 114 0.0069 6.73 94.705 100 0.084
3 60 120 0.02028 7.07 309.54 100 0.084
4 80 190 0.00942 8.18 369.03 150 0.063
5 47 97 0.0114 5.35 148.89 120 0.077
6 68 140 0.01142 8.05 222.33 100 0.084
7 110 300 0.00357 8.03 287.71 200 0.042
8 135 300 0.00492 6.99 391.98 200 0.042
9 135 300 0.00573 6.6 455.76 200 0.042
10 130 300 0.00605 12.9 722.82 200 0.042
11 94 375 0.00515 12.9 635.2 200 0.042
12 94 375 0.00569 12.8 654.69 200 0.042
13 125 500 0.00421 12.5 913.4 300 0.035
14 125 500 0.00752 8.84 1760.4 300 0.035
15 125 500 0.00708 9.15 1728.3 300 0.035
16 125 500 0.00708 9.15 1728.3 300 0.035
17 220 500 0.00313 7.97 647.85 300 0.035
18 220 500 0.00313 7.95 649.69 300 0.035
19 242 550 0.00313 7.97 647.83 300 0.035
20 242 550 0.00313 7.97 647.81 300 0.035
21 254 550 0.00298 6.63 785.96 300 0.035
22 254 550 0.00298 6.63 785.96 300 0.035
23 254 550 0.00284 6.66 794.53 300 0.035
24 254 550 0.00284 6.66 794.53 300 0.035
25 254 550 0.00277 7.1 801.32 300 0.035
26 254 550 0.00277 7.1 801.32 300 0.035
27 10 150 0.52124 3.33 1055.1 120 0.077
28 10 150 0.52124 3.33 1055.1 120 0.077
29 10 150 0.52124 3.33 1055.1 120 0.077
30 47 97 0.0114 5.35 148.89 120 0.077
31 60 190 0.0016 6.43 222.92 150 0.063
32 60 190 0.0016 6.43 222.92 150 0.063
33 60 190 0.0016 6.43 222.92 150 0.063
34 90 200 0.0001 8.95 107.87 200 0.042
35 90 200 0.0001 8.62 116.58 200 0.042
36 90 200 0.0001 8.62 116.58 200 0.042
37 25 110 0.0161 5.88 307.45 80 0.098
38 25 110 0.0161 5.88 307.45 80 0.098
39 25 110 0.0161 5.88 307.45 80 0.098
40 242 550 0.00313 7.97 647.83 300 0.035









4.1.5 System 5

This system has 10 generating units with valve-point loading effects. Therefore,

this system has nonconvex solution spaces and there are many local minima due to

valve-point effects. The parameters of this system are given in the Table 4-10 and are

available in [4] as well. The forecasted demand with the dispatch horizon one day with

24 intervals of one hour each is shown in Table 4-8.

Table 4-7. Generating units characteristics of 10-unit system
Unit pin", MW pmax, MW ai bi ci ei f URi URi
1 150 470 786.7988 38.5397 0.1524 450 0.041 80 80
2 135 470 451.3251 46.1591 0.1058 600 0.036 80 80
3 73 340 1049.9977 40.3965 0.028 320 0.028 80 80
4 60 300 1243.5311 38.3055 0.0354 260 0.052 50 50
5 73 243 1658.5696 36.3278 0.0211 280 0.063 50 50
6 57 160 1356.6592 38.2704 0.0179 310 0.048 50 50
7 20 130 1450.7045 36.5104 0.0121 300 0.086 30 30
8 47 120 1450.7045 36.5104 0.0121 340 0.082 30 30
9 20 80 1455.6056 39.5804 0.109 270 0.098 30 30
10 10 55 1469.4026 40.5407 0.1295 380 0.094 30 30


Table 4-8. Load demand for 24 hours
Time, h Load, MW Time, h Load, MW Time, h Load, MW
1 1036 9 1924 17 1480
2 1110 10 2022 18 1628
3 1258 11 2106 19 1776
4 1406 12 2150 20 1972
5 1480 13 2072 21 1924
6 1628 14 1924 22 1628
7 1702 15 1776 23 1332
8 1776 16 1554 24 1184


4.2 Results

One of the features that the heuristic algorithms possess is randomness. Therefore,

their performances cannot be judged by the result of a single run and many trials

with different initializations should be made to reach a valid conclusion about the

performance of the algorithms. An algorithm is robust, if it can guarantee an acceptable









performance level under different conditions. In this paper, 50 different runs of C-GRASP

have been carried out.

4.2.1 Case 1

In this case, the dynamic ED problem on system 1 is solved. It can be seen from

Table (4-9) that the C-GRASP provided the best solution compared to SA and GA.

Table 4-9. Generation costs for case 1


Min
19645.87118
19817.50206
19675.35508


Avg
19725.6329
19819.44773
19777.37761


Max
19837.55210
19817.50206
19855.63880


The smallest total production cost is obtained by SA and it is $19645.87. Morever,

we can notice that on the average C-GRASP algorithm performs better than SA and

GA. The lowest maximum value is provided by C-GRASP as well, while the highest

maximum value was produced by SA. shows that SA solutions are very sensitive to

starting points and are more volatile. The best found solution satisfying demand and

power limits is given in Table 4-10.

Table 4-10. Best solution for case 1


Unit 1
10.00006975
10.00003632
26.00796462
26.75734182
49.04650601
19.63298415
36.5415225
13.70517701
41.8141424
21.88623928
28.88128653
25.81847079


Unit 2
20.00005644
52.60222864
87.4541663
95.20329384
56.01226378
103.7721597
92.86540909
92.67194012
102.4755387
102.6285523
58.20434065
113.8283152


Unit 3
80.00001477
30.00004416
80.84053733
170.8232815
91.33351254
51.19984471
61.43881398
138.6821921
123.5565313
129.2017437
161.8266865
174.999958


Unit 4
105.6819645
151.8901212
45.99580774
68.03647655
110.4114641
196.8271002
157.4967338
178.1269249
244.850131
249.9999302
222.1801306
147.8022832


Unit 5
194.3178945
190.5075697
234.701524
169.1796063
251.1962536
236.5679112
277.6575207
230.8137659
177.3036566
200.2835345
248.9075557
277.5509728


4.2.2 Case 2

Here, the static ED problem includes the nonlinear generation-demand equality

constraints due to included transmission losses. The ramp up limits and prohibited


Method
C-GRASP
GA
SA


St.Dev.
26.8471
0.93349
36.1006


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









zones of generators are incorporated as well. The efficient of C-GRASP is tested on

six-unit systems that is described in Section 4.1.2. The same problem has been solved

in [30] and their best solution and applied methods are presented in Table 4-11. The

losses and total generation cost are given in Table 4-12. The best solutions among

all solutions have been illustrated in the bold prints. From these data we can see that

their provided objective function values are smaller that one obtained by C-GRASP, but

it should be noted that solutions gained by CPSO 1 and CPSO 2 violate the ramp-up

limits of generator 3. When the solution of PSO has been pluged, it has been found the

violation of generation-demand balance equality by 0.4661 MW, because according to

given solution, the generation is equal to 1275.9571 MW and loss is 12.4910 MW.

The minimum generation cost found by C-GRASP is $15456.54469, while the

average cost is $15507.10954 with standard deviation of value $28.10037477.

According to these facts, it can be stated that C-GRASP approach with applied heuristic

strategy can produced feasible and good solutions. The results produced by SA and

GA were not feasible or reasonably close to results presneted here, so they are not

presneted here.

Table 4-11. Best solutions for case 2
Pi P2 P3 P4 P5 P6
PSO 447.4970 173.3221 263.4745 139.0594 165.4761 87.1280
CPSO 1 434.4236 173.4385 274.2247 128.0183 179.7042 85.9082
CPSO 2 434.4295 173.3231 274.4735 128.0598 179.4759 85.9281
C-GRASP 447.8181 200 253.5570 149.9999 150.3202 73.5022


Table 4-12. Best results, when demand is1263 MW
Total output Loss Total generation cost
PSO 1276.0 12.9584 15451
CPSO 1 1276.0 12.9583 15447
CPSO 2 1276.0 12.9582 15446
C-GRASP 1275.1974 12.1974 15456.54









4.2.3 Case 3

In this case, the static ED problem with nonsmooth cost function due to the

valve-point effects is considered to check the ability of C-GRASP to solve such type

problems and its competitiveness with both GA and SA approaches. The experiment is

performed on two different systems, namely, system 3 and system 4. The final fuel costs

obtained using applied approaches are summarized in Table 4-13. It shows the minimum,

average and maximum cost and standard deviation achieved by applied methods for

75 runs. From the computational results, the minimum cost achieved by C-GRASP was

the best, followed by SA and GA. The minimum cost, maximum cost and the mean

cost values obtained by C-GRASP are 18394.07 $/h, 18699.339 $/h, and 18550.105

$/h, respectively, which are lower than those obtained by SA and GA. The worst results

are obtained by GA. It can be noticed that results produced by GA vary the least, this

can be confirmed by the low standard deviation that is $12.562. In literature [54], the

lowest reported generation cost for 1800 MW is $17994.07, however the solution is not

presented.

Table 4-13. Generation costs for 13-unit system with demand 1800 MW
Method Min Avg Max St. Dev.
C-GRASP 18394.070 18550.105 18699.339 65.729
GA 19384.229 19417.964 19438.914 12.562
SA 18950.174 19393.114 19782.516 181.920


The results on 40-units are presented in table 4-14. C-GRASP, SA and GA

algorithms were run for 75 times and the minimum, maximum and average value of

objective function are reported.

Table 4-14. Generation costs for 40-unit system with demand 10500 MW
Method Min Avg Max St.Dev.
C-GRASP 128883.1965 130268.9796 132839.2181 972.757
GA 163401.9977 163534.9817 163623.3423 64.0606
SA 138975.7844 150757.5002 162578.6271 6118.65









4.2.4 Case 4


The last problem solved by C-GRASP is dynamic ED problem including ramp-up

limits, that makes this problem more difficult than in a case 1. For simplicity, the

transmission losses are neglected. The minimum cost obtained by the C-GRASP

coupled with heuristic strategy is found to be $1,735,176.10, the best solution that

satisfies demand-balance constraints as well as generators operation constraints

including ones of ramp-up is given in Table 4-15. GA ans SA applied in this work

couldn't produce the feasible solution.

Table 4-15. Best solution for case 4


Hour
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24


P1
197.5775
199.3353
279.3352
357.6864
341.5554
364.4919
291.8407
275.9255
355.9255
373.6142
453.6141
469.9999
405.7058
325.7058
308.0081
228.0081
252.7930
214.1486
253.8865
333.8865
334.2637
254.2637
174.2637
150.0001


P2
181.2115
142.0086
222.0086
196.9907
276.9907
356.9907
436.9906
452.7053
382.1891
395.9434
447.3342
469.9999
425.0176
367.8957
320.2882
369.5967
361.9796
441.9796
470.0000
448.3762
375.3046
295.3046
215.3046
171.8146


P3
163.4175
243.3161
163.3161
83.3162
104.5803
92.7176
162.9068
238.0856
318.0856
310.8613
339.9999
327.0836
281.3943
339.9999
284.4522
235.5079
165.9723
160.0032
240.0032
320.0032
329.5451
249.5451
218.8263
144.0519


P4
62.3581
112.3580
162.3580
212.3580
226.3765
205.6510
249.2158
230.3986
233.8956
235.4000
285.3999
235.4000
279.5788
229.5788
279.5788
230.6255
237.6272
187.6272
137.6272
149.1631
195.2469
245.2469
256.1493
259.4389


P5
127.0141
115.8294
80.9696
130.9696
180.9696
202.5198
210.4313
196.6733
197.8604
242.9999
197.8604
229.2473
226.4663
240.9742
190.9742
174.2616
169.1177
211.3444
242.9999
211.3444
242.9999
195.4727
161.5679
132.1630









Best solutions for case 4 continued


Hour
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24


P6
84.1156
87.3169
60.2349
107.8558
128.7563
140.9288
137.0179
89.7335
139.7335
159.9999
112.7650
112.5501
88.9431
65.2257
70.0661
108.1979
62.7680
112.7679
122.1729
159.9999
153.7373
127.9852
122.7771
72.7771


P7
77.5922
63.8052
93.8052
111.1225
81.1226
76.3086
46.3086
76.3086
106.3086
128.7672
98.7671
128.7671
129.9999
127.7126
97.7126
67.7126
70.2996
99.9863
97.8341
126.0825
100.4907
70.4907
40.4908
70.4908


Ps
83.0176
90.0000
119.9999
108.2973
78.2974
98.3418
68.3418
84.7578
93.1458
84.5977
114.5977
84.5977
114.5977
95.1739
119.9999
90.2077
97.6767
119.9999
112.1255
105.6771
119.9999
96.0201
66.0202
73.5756


P9
20.0001
46.0304
46.7507
59.4590
38.9946
56.9712
71.6510
80.0000
62.0940
73.0386
43.0386
73.0386
77.8105
76.7334
62.4338
32.4338
32.8367
48.3857
55.9241
78.8549
48.8549
70.0305
56.0837
60.6486


Pl0
39.6960
10.0001
29.2219
37.9445
22.3567
33.0785
27.2956
51.4117
34.7621
16.7777
12.6230
19.3156
42.4861
54.9999
42.4861
17.4482
28.9293
31.7572
43.4265
38.6122
23.5569
23.6405
20.5164
49.0393









CHAPTER 5
CONCLUSION

* Economic disatch problem can be formulated in very different ways: as a simple
linear programing problem to nonlinear nonconvex problem.

* In this work, four different cases were analysed and three heuristic methods:
C-GRASP, GA and SA were applied to solve ED problem.

* Since C-GRASP is able to cope with optimization problem having box constraints,
the heuristic strategy to deal with equality and inequality constraints for ED
problem was incorporated.

* The experimental results revealed, that C-GRASP adopted to ED problem is able
to provide good results and can outperfom SA and GA.









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BIOGRAPHICAL SKETCH

Ingrida Radziukyniene got Bachelor of Science and Master of Science in computer

science at Vytautas Magnus University, Lithuania in 2003 and 2005, respectively. In

addition, she got a certificate of business management from Department of Business

at Vytautas Magnus University. In 2010, she earned the Master of Science in industrial

engineering from University of Florida. More information about her research interest can

be found in her webpage http://plaza.ufl.edu/ingridar/.





PAGE 1

C-GRASPAPPLICATIONTOTHEECONOMICDISPATCHPROBLEMByINGRIDARADZIUKYNIENEATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2010

PAGE 2

c2010IngridaRadziukyniene 2

PAGE 3

Idedicatethistomywonderfulson,Matas 3

PAGE 4

ACKNOWLEDGMENTS IamgratefultomanypeopleforsupportingmethroughoutmygraduatestudyinUnitedStates.Firstofall,Iwouldliketoexpressmyearnestgratitudetomyadvisor,Dr.PanosM.Pardalos,fordirectingthisstudyandreadingpreviousdraftsofthiswork.Withouthisguidance,inspiration,andsupportthroughoutthecourseofmyresearch,thisworkwouldnotbecomplete.ManythankstoArturaswhohasbeenthereforme,listeningtomeandsupportingme.IamalsothankfultomyfriendsattheCenterforAppliedOptimizationwhomentallysupportedandmademystudentlifemorecolorful. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Motivation .................................... 10 1.2LiteratureOverview .............................. 11 2ECONOMICDISPATCH(ED)PROBLEM ..................... 14 2.1EDConstraints ................................. 15 2.1.1Load-GenerationBalance ....................... 16 2.1.2GenerationCapacityConstraint .................... 16 2.1.3GeneratingUnitRampRateLimits .................. 17 2.1.4ReserveContribution .......................... 17 2.1.5SystemSpinningReserveRequirement ............... 18 2.1.6Tie-lineLimits .............................. 18 2.1.7ProhibitedZone ............................. 19 2.2ObjectiveFunctions .............................. 19 2.2.1SmoothCostFunction ......................... 20 2.2.2Non-smoothCostFunctionswithValve-pointEffects ........ 20 2.2.3Non-smoothCostFunctionswithMultipleFuels ........... 21 2.2.4Non-smoothCostFunctionswithValve-PointEffectsandMultipleFuels ................................... 22 2.2.5EmissionFunction ........................... 23 3SOLUTIONMETHODS ............................... 24 3.1ContinuousGreedyRandomizedAdaptiveSearchProcedure(C-GRASP) 24 3.2GeneticAlgorithms(GA) ............................ 26 3.3SimulatedAnnealing(SA) ........................... 27 3.4ConstraintsHandling .............................. 29 3.4.1Penalty-BasedApproach ........................ 29 3.4.2HeuristicStrategy ............................ 30 4EXPERIMENTSANDRESULTS .......................... 33 4.1Experiments .................................. 33 4.1.1System1 ................................ 33 5

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4.1.2System2 ................................ 33 4.1.3System3 ................................ 35 4.1.4System4 ................................ 35 4.1.5System5 ................................ 37 4.2Results ..................................... 37 4.2.1Case1 .................................. 38 4.2.2Case2 .................................. 38 4.2.3Case3 .................................. 40 4.2.4Case4 .................................. 41 5CONCLUSION .................................... 43 REFERENCES ....................................... 44 BIOGRAPHICALSKETCH ................................ 50 6

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LISTOFTABLES Table page 4-1Generatingunitscharacteristicsofve-unitsystem ................ 33 4-2Loaddemand ..................................... 34 4-3Generatingunitscharacteristicsofsix-unitsystem ................ 34 4-4Rump-uplimitsandprohibitedzonesofsix-unitsystem ............. 35 4-5Generatingunitscharacteristicsof13-unitsystem ................ 35 4-6Generatingunitscharacteristicsof40-unitsystem ................ 36 4-7Generatingunitscharacteristicsof10-unitsystem ................ 37 4-8Loaddemandfor24hours .............................. 37 4-9Generationcostsforcase1 ............................. 38 4-10Bestsolutionforcase1 ............................... 38 4-11Bestsolutionsforcase2 ............................... 39 4-12Bestresults,whendemandis1263MW ...................... 39 4-13Generationcostsfor13-unitsystemwithdemand1800MW ........... 40 4-14Generationcostsfor40-unitsystemwithdemand10500MW .......... 40 4-15Bestsolutionforcase4 ............................... 41 7

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LISTOFFIGURES Figure page 2-1Exampleofcostfunctionwithtwoprohibitedoperatingzones .......... 19 2-2Costfunctionwithvalve-pointeffects ........................ 21 2-3Costfunctionwithmultiplefuels ........................... 22 8

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceC-GRASPAPPLICATIONTOTHEECONOMICDISPATCHPROBLEMByIngridaRadziukynieneAugust2010Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineeringEconomicdispatchplaysanimportantroleinpowersystemoperations,whichisacomplicatednonlinearconstrainedoptimizationproblem.Ithasnon-smoothandnon-convexcharacteristicwhengenerationunitvalve-pointeffectsaretakenintoaccount.ThisworkadoptstheC-GRASPalgorithmtosolvedifferentlyformulatedeconomicdispatchproblems.ThecomparisonofthefeasibilityandeffectivenessoftheC-GRASP,SAandGAisgivenaswell. 9

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CHAPTER1INTRODUCTION 1.1MotivationTheeconomicdispatch(ED)optimizationproblemisoneofthefundamentalissuesinpowersystemstoobtainoptimalbenetswiththestability,reliabilityandsecurity[ 52 ].Essentially,theEDproblemisaconstrainedoptimizationprobleminpowersystemsthathavetheobjectiveofdividingthetotalpowerdemandamongtheon-lineparticipatinggeneratorseconomicallywhilesatisfyingthevariousconstraints.EDproblemhavecomplexandnonlinearnonconvexcharacteristicswithequalityandinequalityconstraints.Therefore,goodsolutionsoftheEDproblemwouldresultingreateconomicalbenets.Overtheyears,manyeffortshavebeenmadetosolvethisproblem,incorporatingdifferentkindsofconstraintsormultipleobjectives,throughvariousmathematicalprogrammingandoptimizationtechniques[ 42 ].Intheconventionalmethodssuchasthelambda-iterationmethod,thebasepointandparticipationfactors,andthegradientmethods,anessentialassumptionisthattheincrementalcostcurvesoftheunitsaremonotonicallyincreasingpiecewiselinearfunctions,butthepracticalsystemsarenonlinear[ 52 ].Hence,globaloptimizationtechniques,suchasthegeneticalgorithms(GAs),simulatedannealing(SA),andparticleswarmoptimization(PSO)havebeenstudiedinthepastdecadeandhavebeensuccessfullyusedtosolvetheED.However,thereferenceswithcontinuousgreedyrandomizedadaptivesearchprocedure(C-GRASP)applicationtosuchtypeofproblemshadn'tappearyet.TheaimofthisworkistoapplytheC-GRASPtotheEDproblemandcompareitseffectivenessandproducedsolutionfeasibilitywithonesofotherheuristicmethodsliketheGAsandSA. 10

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1.2LiteratureOverviewSinceCarpentierintroducedanetworkconstrainedeconomicdispatchproblemin1962[ 9 ]andtherstpaperintheareaofdynamicdispatchingwaspublishedbyBechertandKwatnyin1972[ 6 ],alotofresearcheshaveemployedvariousmathematicalprogrammingoptimizationmethodsforsolvingEDproblems[ 30 ].Theseoptimizationtechniquescanbeclassiedintothreemaincategories.Therstcategorycontainsdeterministicmethodsthatincludethelinearprogrammingalgorithm[ 26 57 69 ],quadraticprogrammingalgorithm[ 18 37 ],non-linearprogrammingalgorithm[ 39 ],etc.TheLPmethodapplicationtothepower-systemreschedulingproblemwithsecurity-constrainedeconomicdispatch/controlformultiple-valved-turbineunitswasgivenbyStottandMarinho[ 57 ].Rosehartetal.[ 48 ]discoveredthatfortheeconomicdispatchproblem,SLPappearstobeabettertoolthanSQP.AnapproachbasedonefcientSLPtechniquestosolvethemulti-objectiveenvironmental/economicloaddispatchproblemwaspresentedbyZeharandSayah[ 69 ].Granellietal.[ 18 ]solvedasecurityconstrainedeconomicdispatchproblemusingmodiedSQPtechniques.Adualfeasiblestartingpointisfoundbyrelaxingtransmissionlimitsandthenconstraintviolationsareenforcedapplyingthedualquadraticalgorithm.In[ 59 ]and[ 35 ],asecurityconstrainedeconomicdispatchproblemwassolvedbySLPandtheinteriorpointdual-afnescalingalgorithm.Momohetal.[ 37 ]proposedanIPMforEDproblemformulatedaslinearandconvexQP.Howevereachoftraditionalmethodshassomedefects:itwouldgeneratelargeerrorstousethelinearprogrammingalgorithmtolinearizetheEDmodel;forthequadraticprogrammingandnonlinearprogrammingalgorithms,theobjectivefunctionshouldbecontinuousanddifferentiable[ 30 ].Thesecondcategorycontainsthemethodsbasedonarticialintelligence.ArticialintelligencetechnologyhasbeensuccessfullyusedtosolvetheEDproblem.Achaosoptimizationalgorithm(CAO)hasbeenproposedbyJiangetal.[ 29 ]todealwiththeeconomicdispatchproblemofahydropowerplant.Zhijiangetal.[ 71 ]alsoapplieda 11

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COAandthesimulationresultsveriedthattheproposedapproachiseffectiveandprecise.AmutativescaleCOAwasappliedbyXuetal.[ 65 ]totheeconomicoperationofpowerplants.However,theresultsshowedthatthemethodistime-consuming.AnimprovedmutativescaleCOAhesbeendevelopedbyHanandLu[ 19 ].Accordingtotheauthors,theiralgorithmishighlyefcientandcanbeappliednotonlytoEDbuttomanypowersystemproblems,suchaseconomicoperation,OPF,systemidenticationandoptimalcontro,aswelll.In[ 36 ],Mahdadetal.proposedanefcientdecomposedparallelGAtosolvethemulti-objectiveenvironmental/economicdispatchproblem.Intherststage,theoriginalnetworkisdecomposedintomultisub-systemsandtheproblemistransformedtooptimizetheactivepowerdemandassociatedwitheachpartitionednetwork.Inthesecondstage,anactivepowerdispatchstrategyisproposedtoenhancethenalsolutionoftheoptimalpowerowoftheoriginalnetwork.TheproposedapproachwastestedontheAlgerian59-bustestsystem.Thecomputationalresultsshowedtheconvergenceatthenearsolutionandobtainacompetitivesolutionatareducedtime.GAswithfuzzylogiccontrollerstoadjustitscrossoverandmutationprobabilitieswasappliedbySongetal.[ 56 ]tosolveacombinedenvironmentaleconomicdispatchproblem.SAtechniqueswereusedbyRoa-SepulvedaandPavez-Lazo[ 47 ],however,longcomputationaltimetoobtainanoptimalsolutionwasreported.TabusearchwasappliedbyAltunandYalcinoz[ 2 ].Simulationresultsonpowersystemsconsistingof6and20generatingunitsexhibitedgoodperformance.In[ 38 ],anapplicationofTSforsolvingsecurityconstrainedEDproblemwasgivenbyMuthuselvanandSomasundaram.Basecaseandcontingencycaselineowconstraintswereconsidered.Testson66-busand191-busIndianutilitysystemsrevealedthereliability,efciencyandsuitabilityoftheproposedalgorithmforpracticalapplications.Thethirdcategoryconsiststhehybridmethods,whichcombinetwoormoretechniquesinordertogetbestfeaturesineachalgorithm.Typically,signifcantimprovementwithhybridmethodscanbeachievedovereachoftheindividualmethods. 12

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Hybridmethodsgainedincreasingpopularityinthelast10years.FortheEDproblem,WongandWong[ 63 ]combinedanincrementalGAwithSAtechniques.CoelhoandMariani[ 12 ]proposedamethodcombiningaDEalgorithmwithself-adaptivemutationfactorintheglobalsearchstageandchaoticlocalsearchtechniquesinthelocalsearchtosolveanEDproblemassociatedwiththevalve-pointeffect.ThesameauthorsreportanothersuccessfulapplicationofchaoticPSOincombinationwithanimplicitlteringlocalsearchmethodtosolveeconomicdispatchproblems[ 13 ].ThechaoticPSOapproachisusedtoproducegoodpotentialsolutions,whiletheimplicitlteringisusedtone-tunethenalsolutionofthePSO.Thehybridmethodologyisvalidatedforatestsystemconsistingof13thermalunitswhoseincrementalfuelcostfunctiontakesintoaccountthevalve-pointloadingeffects.In[ 11 ],CoelhoandLeeimprovedPSOapproachesforsolvinganEDproblemtakingintoaccountnon-lineargeneratorfeaturessuchasramp-ratelimits.Prohibitedoperatingzonesinthepowersystemoperationaredevelopedaswell.TheiralgorithmcombinesthePSO,Gaussianprobabilitydistributionfunctionsand/orchaoticsequences.ThePSOanditsvariantsarevalidatedfortwotestsystemsconsistingof15and20thermalgenerationunits,respectively.Acombinationofchaoticandself-organizationbehaviorofantsintheforagingprocesswaspresentedbyCaietal.[ 8 ].ThisalgorithmwasappliedtoEDproblemswiththermalgenerators.Thethesisisorganizedasfollows:InSection 2 ,webrieydiscussageneralEDproblemformulation.ThemethodsappliedtosolveEDareshortlydiscussedinSection 3 .Section 4 describesexperimentalcasesandpresentscalculationresults.WeconcludewithSection 5 13

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CHAPTER2ECONOMICDISPATCH(ED)PROBLEMEDisoneoftheimportantoptimizationproblemsinpowersystemoperations,whichisusedtodeterminetheoptimalcombinationofpoweroutputsofallgeneratingunitstominimizethetotalfuelcostwhilesatisfyingvariousconstraintsovertheentiredispatchperiods[ 67 ].ThetraditionalorstaticEDproblemassumesconstantpowertobesuppliedbyagivensetofunitsforagiventimeintervalandattemptstominimizethecostofsupplyingthisenergysubjecttoconstraintsonthestaticbehaviorofthegeneratingunitslikesystemloaddemand.Shortly,staticEDdeterminestheloadsofgeneratorsinasystemthatwillmeetapowerdemandduringasingleschedulingperiodfortheleastcost.Therefore,itmightfailtocapturelargevariationsoftheloaddemandduetotherampratelimitsofthegenerators.Duetolargevariationofthecustomersloaddemandandthedynamicnatureofthepowersystems,itbecamenecessarytoscheduletheloadbeforehandsothatthesystemcananticipatesuddenchangesindemandinthenearfuture.DynamicEDisanextensionofstaticEDtodeterminethegenerationscheduleofthecommittedunitssothattomeetthepredictedloaddemandovertheentiredispatchperiodsatminimumoperatingcostunderramprateandotherconstraints[ 64 ].Theramprateconstraintisadynamicconstraintwhichusedtomaintainthelifeofthegenerators,i.e.plantoperators,toavoidshorteningthelifeofthegenerator,trytokeepthermalstresswithintheturbinessafelimits[ 20 ].Sincetheviolationsoftheramprateconstraintsareassessedbyexaminingthegeneratorsoutputoveragiventimeinterval,thisproblemcannotbesolvedforasinglevalueofMWgeneration[ 20 ].TheobjectivefunctionofdynamicEDisformulatedasfollows minC(P)=TXt=1NXi=1Ci(Pti)(2) 14

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whereNisthesetofcommittedunits;Piisthegenerationofuniti;Ci(Pi)isthecostofproducingPifromuniti;Tisthenumberofintervalsinthestudyperiod.ThefuelcostfunctionsCi()isderivedfromthefuelconsumptionfunctionthatcanbemeasuredandarediscussedinSection 2.2 .ThedynamicEDisnotonlythemostaccurateformulationoftheeconomicdispatchproblembutalsothemostdifculttosolvebecauseofitslargedimensionality[ 3 ].TheDEDproblemisnormallysolvedbydiscretizationoftheentiredispatchperiodintoanumberofsmalltimeintervals,overwhichtheloaddemandisassumedtobeconstantandthesystemisconsideredtobeinatemporalsteadystate.OvereachtimeintervalastaticEDproblemissolvedunderstaticconstraintsandtheramprateconstraintsareenforcedbetweentheconsecutiveintervals[ 34 ].IntheDEDproblemtheoptimizationisdonewithrespecttothedispatchablepowersoftheunits.SomeresearchershaveconsideredtheramprateconstraintsbysolvingSEDproblemintervalbyintervalandenforcingtheramprateconstraintsfromoneintervaltothenext.However,thisapproachcanleadtosuboptimalsolutions[ 23 ];moreover,itdoesnothavethelook-aheadcapability.SincedynamicEDwasintroduced,variuosmethodshavebeenusedtosolvethisproblem.However,allofthosemethodsmaynotbeabletoprovideanoptimalsolutionandusuallygettingstuckatalocaloptimal. 2.1EDConstraintsTheconstrainedEDproblemissubjectedtoavarietyofconstraintsdependinguponassumptionsandpracticalimplications.Usually,formulationofEDproblemincludessuchconstraintsasloadgenerationbalance,minimumandmaximumcapacityconstraints.Tomaintainsystemreliabilityandsecurity,spinningreserveconstraintsandsecurityconstraintscanbeaddedtothedynamicEDproblem.Theinclusionoftheprohibitedzones,ramp-ratelimitsandotherpracticalconstraintsresultsinnonconvexEDofgeneratingunits.Alltheseconstraintsarediscussedbellow. 15

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2.1.1Load-GenerationBalanceThegeneratedpowerfromalltherunningunitsmustsatisfytheloaddemandandthesystemlossesgivenby( 2 ) NXi=1Pti=Dt+Losst,t=1,2,...,T(2)whereDtisthedemandandLosstisthesystemtransmissionloss.Theirsumrepresentstheeffectiveloadtobesatisedatthetthinterval.Thetransmissionlinelossescanbeexpressedintermsoftheunitoutputs:Losst=NXi=1NXj=1PtiBijPtj+NXi=1Bi0Pti+B00whereBijistheijthelementofthelosscoefcientsquarematrix,Bi0istheithelementofthelosscoefcient,andB00istheconstantlosscoefcient.Sometimesthelasttwotermsareomitted.Inacompetitiveenvironment,theload-generationbalanceconstraintisrelaxedandeachgeneratingcompanyschedulesitsproductiontomaximizeitsprotsgivenaforecastofelectricitypricesfortheschedulingperiod.Asarstapproximation,eachgeneratingunitcouldbeoptimizedseparatelyinthisproblembecauseofthedecouplingmadepossiblebytheavailabilityofpricesateachperiod.Dynamicconstraints(suchasrampratesandminimumupanddowntimeconstraints)complicatetheproblembecauseageneratingcompanythatownsaportfolioofunitsmustthendecidewhethertobuyexibilityonthemarketormeetthedynamicconstraintswithitsownresources[ 21 ]. 2.1.2GenerationCapacityConstraintFornormalsystemoperations,realpoweroutputofeachgeneratorisrestrictedbylowerandupperboundsasfollows: Pti+StiPmaxii=1,2,...,N,t=1,2,...,T(2) 16

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PminiPtii=1,2,...,N,t=1,2,...,T(2)wherePminiandPmaxiaretheminimumandmaximumpowerproducedbygeneratori,Stiisthereservecontributionofunitduringtimeintervalt. 2.1.3GeneratingUnitRampRateLimitsOneofunpracticalassumptionthatprevailedforsimplifyingtheprobleminmanyoftheearlierresearchisthattheadjustmentsofthepoweroutputareinstantaneous[ 43 ].Therefore,thepoweroutputofapracticalgeneratorcannotbeadjustedinstantaneouslywithoutlimits.Theoperatingrangeofallonlineunitsisrestrictedbytheirramp-ratelimitsduringeachdispatchperiod.So,thesubsequentdispatchoutputofageneratorshouldbelimitedbetweentheconstraintsofupanddownramp-rates[ 66 ]asfollowsPt+1i)]TJ /F4 11.955 Tf 11.96 0 Td[(PtiURit (2)Pti)]TJ /F4 11.955 Tf 11.96 0 Td[(Pt+1iDRiti=1,2,...,N,t=1,2,...,T)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (2)whereURiandDRiarethemaximumrampup/downratesforunitiandtisthedurationofthetimeintervalsintowhichthestudyperiodisdivided.Theinclusionoframpratelimitsmodiesthegeneratoroperationconstraints( 2 2 )asfollows max(Pmini,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1i)]TJ /F4 11.955 Tf 11.95 0 Td[(DRi)Pimin(Pmaxi,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1i+URi)(2) 2.1.4ReserveContributionThemaximumreservecontributionhastosatisfyfollowingconstraints: 0StiSmaxii=1,2,...,N,t=1,2,...,T(2)whereSmaxiisthemaximumcontributionofunititothereservecapacity.Maximum-rampspinningreservecontributionisdenedasin( 2 ) 0StiURiti=1,2,...,N,t=1,2,...,T(2)whereStiisthespinningreserveofuniti. 17

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2.1.5SystemSpinningReserveRequirementSufcientspinningreserveisrequiredfromallrunningunitstomaximizeandmaintainsystemreliability[ 14 ].Therearemanywaystodeterminethesystemspinningreserverequirement.Itcanbecalculatedasthesizeofthelargestunitinoperationorasapercentageofforecastloaddemandorevenasafunctionoftheprobabilityofnothavingsufcientgenerationtomeettheload[ 64 ].Thespinningrezervecanbedenedby( 2 ) NXi=1StiSRtt=1,2,...,T(2)whereSRtisthesystemspinningreserverequirementfortimeintervalt.Also,thesystemspinningreserverequirementforintervaltcansometimesbegivenbythefollowingequation[ 20 ]: SRt=dDt+gmax(Pmaxischeduledattimet,i=1,2,...N)(2)wheredandgareconstantswhichdependonthesystemrequiredreliabilitylevel[ 55 ].Besidesthedeterminationofthesystemspinningreserverequirement,theissueofallocationthespinningreserveamongthecommittedunitsisveryimportant;however,ithasreceivedverylittleattentioninthedynamicEDliterature. 2.1.6Tie-lineLimitsTheeconomicdispatchproblemcanbeextendedbyimportingadditionalconstraintliketransmissionlinecapacitylimitgivenby( 2 ) PTjk,minPTjk+SjkPTjk,max(2)wherePTjk,minandPTjk,maxspecifythetie-linetrasnmissioncapability,i.e.thetransferfromareajtoareakshouldnotexceedthetie-linetransfercapacitiesforsecurityconsideration[ 28 ].Eachareahasownspecialloadanditsspinningreserve[ 68 ]. 18

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2.1.7ProhibitedZoneThegeneratingunitsmayhavecertainrangeswhereoperationisrestrictedonthegroundsofphysicallimitationsofmachinecomponentsorinstability,e.g.duetosteamvalveorvibrationinshaftbearings.So,thereisaquesttoavoidoperationinthesezonesinordertoeconomizetheproduction[ 43 ].Theserangesareprohibitedfromoperationandageneratorwithprohibitedregions(zones)hasdiscontinuousfuel-costcharacteristics(Fig. 2.1.7 )[ 53 ].TheacceptableoperatingzonesofageneratingunitcanbeformulatedasfollowsPminiPtiPli,1 (2)Pui,j)]TJ /F8 7.97 Tf 6.59 0 Td[(1PtiPli,j,i2,j=2,3,...,ni,t=1,2,...,T (2)Pui,niPtiPmaxi (2)whereniisthenumberoftheprohibitedzonesinuniti,isthesetofunitsthathaveprohibitedzones,Pli,j,Pui,jarethelowerandupperboundsofthejthprohibitedzone. Figure2-1. Exampleofcostfunctionwithtwoprohibitedoperatingzones 2.2ObjectiveFunctionsThedynamicEDproblemhasbeensolvedwithmanydifferentformsofthecostfunction,suchasthesmoothquadraticcostfunction( 2 )orthenonsmoothcost 19

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functionduetothevalve-pointeffects( 2 ).Also,alinearcostfunction[ 20 ]andpiecewiselinearcostfunction[ 27 41 ]havebeenemployed.Forsmoothcostfunctionitisusuallyassumedthatitsincrementalcostfunction.Insomepowersystemscombinedcycleunitsareusedtosupplythebaseload.Fortheseunitsthecostfunctioncanbegivenaslinear,piecewiseorquadraticwithdecreasingincrementalcostfunction[ 41 ].Forunitswithprohibitedzones,thefuelcostfunctionisdiscontinuousandnonconvex.AninterestingdeparturefromthisstandardformulationistheapproachproposedbyWangandShahidehpour[ 61 ]whoincludeintheobjectivefunctionatermrepresentingthereductioninthelifeoftheturbinecausedbyexcessiverampingrates.Thisexibletechniquemakespossibleatradeoffbetweenthesystemoperatingcostandthelifecyclecostofthegeneratingunits[ 21 ]. 2.2.1SmoothCostFunctionThemostsimpliedcostfunctionofeachgeneratorcanberepresentedasaquadraticfunctionasgivenin( 2 )whosesolutioncanbeobtainedbytheconventionalmathematicalmethods Ci(Pti)=ai+biPti+ci(Pti)2(2)whereai,bi,ciarecostcoefcientsofgeneratori. 2.2.2Non-smoothCostFunctionswithValve-pointEffectsThegeneratingunitswithmulti-valvesteamturbinesexhibitagreatervariationinthefuelcostfunctionsbecauseinordertomeettheincreaseddemandageneratorwithmulti-valvesteamturbinesincreaseitsoutputandvarioussteamvalvesaretobeopened[ 67 ].Thisvalve-openingprocessproducesripplelikeeffectintheheat-ratecurveofthegenerator.Theinclusionofvalve-pointloadingeffectsmakesthemodelingoftheincrementalfuelcostfunctionofthegeneratorsmorepractical[ 60 ].Therefore,inreality,theobjectivefunctionofEDproblemhasnon-differentiableproperty.Consequently,theobjectivefunctionshouldbecomposedofasetofnon-smoothcostfunctions.Consideringnon-smoothcostfunctionsofgenerationunitswithvalve-point 20

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Figure2-2. Costfunctionwithvalve-pointeffects effects,theobjectivefunctionisgenerallydescribedasthesuperpositionofsinusoidalfunctionsandquadraticfunctions[ 52 ] Ci(Pti)=ai+biPti+ci(Pti)2+jeisin(hi(Pmini)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti))j(2)whereeiandhiarethecoefcientsofgeneratorireectingvalvepointeffects.AsshowninFig. 2.2.2 ,thisincreasesthenon-linearityofcurveaswellasnumberoflocaloptimainthesolutionspace[ 60 ]comparedwiththesmoothcostfunctionduetothevalvepointeffects.Alsothesolutionprocedurecaneasilytrapinthelocaloptimainthevicinityofoptimalvalue. 2.2.3Non-smoothCostFunctionswithMultipleFuelsSincethedispatchingunitsarepracticallysuppliedwithmulti-fuelsources[ 49 ],eachunitshouldberepresentedwithseveralpiecewisequadraticfunctionsreectingtheeffectsoffueltypechanges,andthegeneratormustidentifythemosteconomicfueltoburn.Theresultingcostfunctioniscalledahybridcostfunction.Eachsegmentofthehybridcostfunctionimpliessomeinformationaboutthefuelbeingburnedorthe 21

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Figure2-3. Costfunctionwithmultiplefuels unitsoperation.Thus,generally,thefuelcostfunctionisapiecewisequadraticfunctiondescribedasfollows ci(Pi)=8>>>>>>>>>><>>>>>>>>>>:ai1+bi1Pti+ci1(Pti)2ifPti,minPtiPti,1ai2+bi2Pti+ci2(Pti)2ifPti,1PtiPti,2......ain+binPti+cin(Pti)2ifPti,n)]TJ /F8 7.97 Tf 6.59 0 Td[(1PtiPti,max(2)whereareaip,bip,cipthecostcoefcientsofgeneratorforthepthpowerlevel.TheincrementalcostfunctionsareillustratedinFig.( 2.2.3 ) 2.2.4Non-smoothCostFunctionswithValve-PointEffectsandMultipleFuelsToobtainanaccurateandpracticaleconomicdispatchsolution,therealisticoperationoftheEDproblemshouldconsiderbothvalve-pointeffectsandmultiplefuels.Thereference[ 10 ]proposedanincorporatedcostmodel,whichcombinesthevalve-pointloadingsandthefuelchangesintooneframe.Therefore,thecostfunctionshouldcombine( 2 )with( 2 ),andcanberealisticallyrepresentedasshownin 22

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( 2 ) ci(Pi)=8>>>>>>>>>><>>>>>>>>>>:ai1+bi1Pti+ci1(Pti)2+jei,1sin(hi,1(Pmini,1)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti,1))jifPti,minPtiPti,1ai2+bi2Pti+ci2(Pti)2+jei,2sin(hi,2(Pmini,2)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti,2))jifPti,1PtiPti,2......ain+binPti+cin(Pti)2+jei,nsin(hi,n(Pmini,n)]TJ /F4 11.955 Tf 11.96 0 Td[(Pti,n))jifPti,n)]TJ /F8 7.97 Tf 6.59 0 Td[(1PtiPti,max(2) 2.2.5EmissionFunctionDuetoincreasingconcernovertheenvironmentalconsiderations,societydemandsadequateandsecureelectricity,i.e.notonlyatthecheapestpossibleprice,butalsoatminimumlevelofpollution.Inthiscase,twoconictingobjectives,i.e.,operationalcostsandpollutantemissions,shouldbeminimizedsimultaneously[ 4 5 7 62 ].Theatmosphericpollutantssuchassulphuroxides(SOx)andnitrogenoxides(NOx)causedbyfossil-fueledgeneratingunitscanbemodeledseparatelyorasthetotalemissionofthemwhichisthesumofaquadratic[ 4 ]andanexponentialfunctionandcanbeexpressedas TXt=1NXi=1i+iPti+i(Pti)2+iexp(iPti)(2)wherei,i,i,i,andiareemissioncoefcientsofithgeneratingunit. 23

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CHAPTER3SOLUTIONMETHODS 3.1ContinuousGreedyRandomizedAdaptiveSearchProcedure(C-GRASP)Continuous-GRASP(C-GRASP)extendsthegreedyrandomizedadaptivesearchprocedure(GRASP)thatwasintroducedbyFeoandResende[ 16 17 ]fromthedomainofdiscreteoptimizationtothatofcontinuousglobaloptimizationin[ 24 25 ].Itisdescribedasamulti-startlocalsearchprocedure,whereeachC-GRASPiterationconsistsoftwophases,namely,aconstructionphaseandalocalsearchphase[ 24 ].Constructioncombinesgreedinessandrandomizationtoproduceadiversesetofgood-qualitystartingsolutionsforlocalsearch.Thelocalsearchphaseattemptstoimprovethesolutionsfoundbyconstruction.Thebestsolutionoveralliterationsiskeptastheinitialsolution.TheadvantagesofthismethodissimplicitytoimplementandnorequirementforderivativeinformationPseudo-codeforC-GRASPisshownin( 3.1 ).C-GRASPworksbydiscretizingthedomainintoauniformgrid.Boththeconstruction(seethehighlevelpseudo-code 3.2 )andlocalimprovementphases(seethehighlevelpseudo-code 3.3 )movealongpointsonthegrid.Asthealgorithmprogresses,thegridadaptivelybecomesmoredense.ThemaindifferencebetweenGRASPandC-GRASPisthataniterationofC-GRASPdoesnotconsistofasinglegreedyrandomizedconstructionfollowedbylocalimprovement,butratheraseriesofconstruction-localimprovementcycleswiththeoutputofconstructionservingastheinputofthelocalimprovement,asinGRASP,butunlikeGRASP,theoutputofthelocalimprovementservesastheinputoftheconstructionprocedure[ 25 ].SinceC-GRASPisessentiallyanunconstrainedoptimizationalgorithm,theconstraintshandlingstrategyneedstobeincorporatedintoitinordertodealwiththeconstrainedEDproblem.Approachestomanagetheseconstraintsarediscussedinsection 3.4 24

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pseudo-code3.1C-GRASP(n,l,u,f(),MaxIters,MaxNumIterNoImprov,NumTimesToRun,MaxDirToTry,) 1: f 1 2: forj 1,...,NumTimesToRundo 3: x UnifRand(l,u);h 1;NumIterNoImprov 0; 4: forIter 1,...,MaxItersdo 5: x ConstructGreedyRandomized(x,f(),n,h,l,u,); 6: x LocalSearch(x,f(),n,h,l,u,MaxDirToTry); 7: iff(x)
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3.2GeneticAlgorithms(GA)Thissectionengagesintotheconceptofgeneticalgorithmsthatreectsthenatureofchromosomesingeneticengineering.GAsareaclassofstochasticsearchalgorithmsthatstartwiththegenerationofaninitialpopulationorsetofrandomsolutionsfortheproblemathand.Eachindividualsolutioninthepopulationcalledachromosomeorstringrepresentsafeasiblesolution.Theobjectivefunctionisthenevaluatedfortheseindividuals.Ifthebeststring(orstrings)satisestheterminationcriteria,theprocessterminates,assumingthatthisbeststringisthesolutionoftheproblem.Iftheterminationcriteriaarenotmet,thecreationofnewgenerationstarts,pairs,orindividualsareselectedrandomlyandsubjectedtocrossoverandmutationoperations.Theresultingindividualsareselectedaccordingtotheirtnessfortheproductionofthenewoffspring.Geneticalgorithmscombinetheelementsofdirectedandstochasticsearchwhileexploitingandexploringthesearchspace[ 31 ].MoredetailsaboutGAcanbefoundin[ 22 46 58 ]. pseudocode3.4Geneticalgorithm 1: initializepopulation() 2: whilenotconvergedo 3: assignpopulationtness() 4: for1,...,npopsizdo 5: selectparents(p1,p2) 6: reproduction(p1,p2,child) 7: endfor 8: selectnextgeneration() 9: endwhile TheadvantagesofGAoverothertraditionaloptimizationtechniquescanbesummarizedasfollows: GAsearchesfromapopulationofpoints,notasinglepoint.Thepopulationcanmoveoverhillsandacrossvalleys.GAcanthereforediscoveragloballyoptimalpoint,becausethecomputationforeachindividualinthepopulationisindependentofothers.GAhasinherentparallelcomputationability. 26

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GAusespayoff(tnessorobjectivefunctions)informationdirectlyforthesearchdirection,notderivativesorotherauxiliaryknowledge.GAthereforecandealwithnon-smooth,non-continuousandnon-differentiablefunctionsthatarethereal-lifeoptimizationproblems.ThispropertyalsorelievesGAoftheapproximateassumptionsforalotofpracticaloptimizationproblems,whicharequiteoftenrequiredintraditionaloptimizationmethods. GAusesprobabilistictransitionrulestoselectgenerations,notdeterministicrules.Theycansearchacomplicatedanduncertainareatondtheglobaloptimum.GAismoreexibleandrobustthantheconventionalmethods[ 33 ].Therstattemptoftheapplicationofgeneticalgorithmsinpowersystemsisintheloadowproblem[ 70 ].Ithasbeenfoundthatthesimplegeneticalgorithmquicklyndsthenormalloadowsolutionforsmall-sizenetworksbyspecifyinganadditionaltermintheobjectivefunction.AnumberofapproachestoimprovingconvergenceandglobalperformanceofGAshavebeeninvestigated[ 70 ]. 3.3SimulatedAnnealing(SA)TheSAisagenericprobabilisticmeta-heuristicfortheglobaloptimizationproblemthatwasproposedbyKirkpatricetal.[ 32 ].IntheSAmethod,eachpointsofthesearchspaceisanalogoustoastateofsomephysicalsystem,andthefunctionE(s)tobeminimizedisanalogoustotheinternalenergyofthesysteminthatstate.Thegoalistobringthesystem,fromanarbitraryinitialstate,toastatewiththeminimumpossibleenergy.IneachstepoftheSAalgorithmthecurrentsolutionisreplacedbyarandomnearbysolution,chosenwithaprobabilitythatdependsonthedifferencebetweenthecorrespondingfunctionvaluesandonaglobalparameterT(calledthetemperature),thatisgraduallydecreasedduringtheprocess.ThedependencyissuchthatthecurrentsolutionchangesalmostrandomlywhenTislarge,butincreasinglydownhillasTgoestozero.Theallowanceforuphillmovessavesthemethodfrombecomingstuckatlocalminimawhicharethebaneofgreediermethods.Forcertainproblems,SAmaybemoreeffectivethanexhaustiveenumeration.Ithasbeenshownthatthistechniqueconvergesasymptoticallytotheglobaloptimalsolutionwithprobabilityone[ 1 ]. 27

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SAisaneffectiveglobaloptimizationalgorithmbecauseofthefollowingadvantages[ 50 ]: suitabilitytoprobleminwidearea, norestrictionontheformofcostfunction, highprobabilitytondglobaloptimization, easyimplementationbyprogramming.ThepseudocodeimplementingSAisgivenbellow.Itstartsfromstates0andcontinueforkmaxofstepsoruntilastatewithenergyemaxorlessisfound.Thecallneighbour(s)shouldgeneratearandomlychosenneighbourofagivenstates;thecallrandom()shouldreturnarandomvalueintherange[0,1].Theannealingscheduleisdenedbythetemp(r),whichshouldyieldthetemperaturetouse,giventhefractionrofthetimebudgetthathasbeenexpendedsofar. pseudocode3.5SimmulatedAnnealing 1: s s0;e E(s) 2: sbest s;ebest e; 3: k 0; 4: whilekemaxdo 5: snew neighbour(s) 6: enew E(snew) 7: ifenewebestthen 8: sbest snew;ebest enew 9: endif 10: ifP(e,enew,temp(k=max))>random()then 11: s snew;e enew 12: k k+1 13: endif 14: endwhile 15: returnsbest Actually,thepureSAalgorithmdoesnotkeeptrackofthebestsolutionfoundsofar:itdoesnotusethevariablessbestandebest,itlackstherstifinsidetheloop,and,attheend,itreturnsthecurrentstatesinsteadofsbest.Whilesavingthebeststateisa 28

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standardoptimization,thatcanbeusedinanymetaheuristic,itbreakstheanalogywithphysicalannealingsinceaphysicalsystemcanstoreasinglestateonly.Instrictmathematicalterms,savingthebeststateisnotnecessarilyanimprovement,sinceonemayhavetospecifyasmallerkmaxinordertocompensateforthehighercostperiteration.However,thestepsbest snewhappensonlyonasmallfractionofthemoves.Therefore,theoptimizationisusuallyworthwhile,evenwhenstate-copyingisanexpensiveoperation.SAhastheabilitytoavoidgettinglocalsolutions;thenitcangenerateglobalornearglobaloptimalsolutionsforoptimizationproblemswithoutanyrestrictionontheshapeoftheobjectivefunctions[ 44 ].SAisnotmemoryintensive[ 45 ].However,thesettingofcontrolparametersoftheSAalgorithmisadifculttaskandthecomputationtimeishigh[ 3 ].Thecomputationalburdencanbereducedbymeansofparallelprocessing[ 44 ]. 3.4ConstraintsHandlingConstraintslieattheheartofallconstrainedengineeringoptimizationapplications.Practicalconstraints,whichareoftennonlinearandnon-trivial,connethefeasiblesolutionstoasmallsubsetoftheentiresearchspace.Thereareseveralapproacheswhichcanbeappliedtohandleconstraintsinheuristicapproaches.Thesemethodscanbegroupedintofourcategories:methodsthatpreservethefeasibilityofsolutions,penalty-basedmethods,methodsthatclearlydistinguishbetweenfeasibleandunfeasiblesolutions,andhybridmethods[ 15 62 ]. 3.4.1Penalty-BasedApproachThepenaltyfunctionmethodisfrequentlyappliedtomanageconstraintsinevolutionaryalgorithms.Suchatechniqueconvertstheprimalconstrainedproblemintoanunconstrainedproblembypenalizingconstraintviolations.Thepenaltyfunctionmethodissimpleinconceptandimplementation.However,itsprimallimitationisthedegreetowhicheachconstraintispenalized.Thesepenaltytermshavecertainweaknessesthatbecomefatalwhenpenaltyparametersarelarge.Suchapenalty 29

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functiontendstobeillconditionedneartheboundaryofthefeasibledomainwheretheoptimumpointisusuallylocated[ 10 ].ThepenalizedfuelcostfunctioninEDproblemwasemployedin[ 51 ].In[ 40 ]theEDproblemwastransformedintoanunconstrainedonebyconstructinganaugmentedobjectivefunctionincorporatingpenaltyfactorsforanyvalueviolatingtheconstraints: H(X)=J(X)+k1NeqXj=1(hj(X))2+k2NueqXj=1max[0,)]TJ /F4 11.955 Tf 9.3 0 Td[(gj(X)]2(3)whereJ(X)istheobjectivefunctionvalueoftheEDproblem.NeqandNueqarethenumberofequalityandinequalityconstraints,respectivel;hj(X)andgj(X)aretheequalityandinequalityconstraints,respectively;k1andk2arethepenaltyfactors.Sincetheconstraintsshouldbemet,thevalueofthek1andk2parameterswerechosentohavehighvalueof10,000.ThisapproachwasepmpoyedwhenapplyingSAmethod.TheheuristicstartegythatisdiscussedinnexsectionwasusedtogetafeasiblesolutionwhileapplyingC-GRASPmethod. 3.4.2HeuristicStrategyWhentheC-GRASPisappliedtosolveEDproblem,akeyproblemishowtohandleconstraintswithefciency.Inthissectionwemainlyfocusonhandlingtherealpowerlimitsandgeneratorsramp-upconstraints.Otherthanpenaltybasedwaytosatisfytherealpowerbalanceequalityconstraints( 2 ),istospecifytheoutputof(N)]TJ /F5 11.955 Tf 12.05 0 Td[(1)generatingunitsandtondtheNthfromtheequalityconstraintlikein[ 4 67 ].Inreference[ 67 ],authorsemployedadependentgenerationpowerptlofrandomlyselectedunitl.TheheuristicstrategyappliedinLocalSearch()procedureinC-GRASPalgorithmcanbeformulatedinafollowingway:Step1.Setthedispatchperiodindext=1anditerationi=1. 30

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Step2.CalculatetheviolationofpowerblanceconstraintPterratdispatchtimetiscalculatedfrom 3 asfollows Pterr=Dt+Losst)]TJ /F6 7.97 Tf 17.3 14.95 Td[(NXi=1Pti(3)IfPterr=0,thengotoStep5,otherwisetoStep3.Step3.Randomlygenerateltheindexofgeneratingunitandcalculatetherealpowerofselecteddependentgeneratingunitptlfrom( 3 ). Ptl=Dt)]TJ /F6 7.97 Tf 17.3 14.94 Td[(NXi=1i6=lPtit=1,2,...,T(3)However,consideringtransmissionlosses( 2.1.1 ),theseequalityconstraintsbecomenonlinearandtheoutputofdependentgeneratingunitforeverydispatchperiodtcanbefoundfrombysolvingafollowingequation Bll(Ptl)2+(2NXi=1i6=lBliPti+Bl0)]TJ /F5 11.955 Tf 9.48 0 Td[(1)Ptl+(Dt+NXi=1i6=lNXj=1j6=lPtiBijPtj+B00+NXi=1i6=lBi0Pti)]TJ /F6 7.97 Tf 14.83 14.94 Td[(NXi=1i6=lPti)=0(3)Ifitdoesn'tviolatethegeneratoroperatinglimitsandramp-upconstraints(iftheyarepresent),gotoStep5.Otherwise,thevaluehastobemodiedaccordingto 3 Ptl=8>><>>:PmaxlifPtl>PmaxlPminlifPtl1,thendependentunitoutputhastobecalculatedas 3 Ptl=8>><>>:max(Pminl,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1l)]TJ /F4 11.955 Tf 11.95 0 Td[(DRl)ifPtl>max(Pminl,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1l)]TJ /F4 11.955 Tf 11.95 0 Td[(DRl)min(Pmaxl,Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1i+URl)ifPtl
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Step4.Increasetheiterationnumberby1,i.e.l=l+1.Ifl
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CHAPTER4EXPERIMENTSANDRESULTS 4.1ExperimentsInordertoverifythefeasibilityandeffectivenessofadoptedC-GRASPcapabilitiesforsolvingEDproblems,differentEDproblemformulations,i.e.staticanddynamicEDanddifferentsystemswereused.TheC-GRASPalgotihmwithheuristicstrategytodealwithconstraintswasimplementedinMatlab7.5.ForGAandSAalgorithms,thestandardMatlabfunctionsformGeneticAlgorithmandDirectSearchToolboxwereemployed.InstandardGAfunctionga(),itispossibletoincludebothlinearandnonlinearequalityandinequalityconstraints.However,SAfunctionsimulannealbnd()incorporatesonlylowerandupperboundconstraints,otherconstraintsasapenaltyfunctionisaddedtoobjectivefunction.Next,wewillprovidedescriptionsofsystemsusedforourexperiments. 4.1.1System1Thesystemconsistsofvegeneratingunits,whosethemaximumtotaloutputis925MW.OnthissystemdynamicEDproblemwassolvedwiththedispatchhorizononedaywith12intervalsofonehoureach.ThedemandofthesystemandgeneratingunitdataaregiveninTables( 4-1 )and( 4-2 ),respectively. Table4-1. Generatingunitscharacteristicsofve-unitsystem ai,$/hbi,$/MWhci,$/(MW)2hPmini,MWPmaxi,MW Unit1 25.0002.0000.00810.00075.000Unit2 60.0001.8000.00320.000125.000Unit3 100.0002.1000.001230.000175.000Unit4 120.0002.0000.00140.000250.000Unit5 40.0001.8000.001550.000300.000 4.1.2System2Thesystemcontainssixthermalgeneratingunits.Thetotalmaximumoutputofgeneratingunitsis1470MW.ThissystemwasusedtosolvestaticEDproblemwhereloaddemandonthesystemis1263MW.Parametersofallthethermalunitsare 33

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Table4-2. LoaddemandTime,h Load,MW Time,h Load,MW 1 410 7 6262 435 8 6543 475 9 6904 530 10 7045 558 11 7206 608 12 740 reportedin[ 30 ]andaregiveninTables 4-3 and 4-4 .Innormaloperationofthesystem,thelosscoefcientsBareasfollows:Bij=26666666666666640.00170.00120.0007)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0005)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00020.00120.00140.00090.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00010.00070.00090.00310.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00010.000100.0024)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0008)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0005)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00060.0129)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0002)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0002)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0001)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0006)]TJ /F5 11.955 Tf 9.3 0 Td[(0.0008)]TJ /F5 11.955 Tf 9.3 0 Td[(0.00020.015377777777777777510)]TJ /F8 7.97 Tf 6.59 0 Td[(2B0i=[)]TJ /F5 11.955 Tf 9.29 0 Td[(0.3908)]TJ /F5 11.955 Tf 11.95 0 Td[(0.12970.70470.05910.2161)]TJ /F5 11.955 Tf 11.95 0 Td[(0.6635]10)]TJ /F8 7.97 Tf 6.58 0 Td[(3B00=0.056 Table4-3. Generatingunitscharacteristicsofsix-unitsystemUnit Pmini,MWPmaxi,MWai,$/hbi,$/MWhci,$/(MW)2hP0i,MW 1 1005000.00772404402 502000.0095102001703 803000.0098.52202004 501500.009112001505 502000.00810.52201906 501200.007512190110 34

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Table4-4. Rump-uplimitsandprohibitedzonesofsix-unitsystemUnit URi,MWDRi,MWProhibitedzone 1 80120[210240][350380]2 5090[90110][140160]3 65100[150170][210240]4 5090[8090][110120]5 5090[90110][140150]6 5090[7585][100105] 4.1.3System3Thissystemconsistsof13generatingunitswithvalve-pointloadingasgiveninTable( 4-5 ).Theparametersofthissystemshowedistakenfrom[ 54 ].Theexpecteddemandis1800MWand2520MW. Table4-5. Generatingunitscharacteristicsof13-unitsystemUnit Pmini,MWPmaxi,MWaibicieifi 1 06800.000288.15503000.0352 03600.000568.13092000.0423 03600.000568.13072000.0424 601800.003247.742401500.0635 601800.003247.742401500.0636 601800.003247.742401500.0637 601800.003247.742401500.0638 601800.003247.742401500.0639 601800.003247.742401500.06310 401200.002848.61261000.08411 401200.002848.61261000.08412 551200.002848.61261000.08413 551200.002848.61261000.084 4.1.4System4Thissystemiscomposedof40generatingunitswithvalve-pointloadingeffectssupplyingatotaldemandof10500MW.Therefore,thissystemhasnonconvexsolutionspacesandtherearemanylocalminimaduetovalve-pointeffectsandtheglobalminimumisverydifculttodetermine.TheparametersofthissystemshowedintheTable( 4-6 )areavailablein[ 54 ]aswell. 35

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Table4-6. Generatingunitscharacteristicsof40-unitsystemUnit Pmini,MWPmaxi,MWaibicieifi 1 361140.00696.7394.7051000.0842 361140.00696.7394.7051000.0843 601200.020287.07309.541000.0844 801900.009428.18369.031500.0635 47970.01145.35148.891200.0776 681400.011428.05222.331000.0847 1103000.003578.03287.712000.0428 1353000.004926.99391.982000.0429 1353000.005736.6455.762000.04210 1303000.0060512.9722.822000.04211 943750.0051512.9635.22000.04212 943750.0056912.8654.692000.04213 1255000.0042112.5913.43000.03514 1255000.007528.841760.43000.03515 1255000.007089.151728.33000.03516 1255000.007089.151728.33000.03517 2205000.003137.97647.853000.03518 2205000.003137.95649.693000.03519 2425500.003137.97647.833000.03520 2425500.003137.97647.813000.03521 2545500.002986.63785.963000.03522 2545500.002986.63785.963000.03523 2545500.002846.66794.533000.03524 2545500.002846.66794.533000.03525 2545500.002777.1801.323000.03526 2545500.002777.1801.323000.03527 101500.521243.331055.11200.07728 101500.521243.331055.11200.07729 101500.521243.331055.11200.07730 47970.01145.35148.891200.07731 601900.00166.43222.921500.06332 601900.00166.43222.921500.06333 601900.00166.43222.921500.06334 902000.00018.95107.872000.04235 902000.00018.62116.582000.04236 902000.00018.62116.582000.04237 251100.01615.88307.45800.09838 251100.01615.88307.45800.09839 251100.01615.88307.45800.09840 2425500.003137.97647.833000.035 36

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4.1.5System5Thissystemhas10generatingunitswithvalve-pointloadingeffects.Therefore,thissystemhasnonconvexsolutionspacesandtherearemanylocalminimaduetovalve-pointeffects.TheparametersofthissystemaregivenintheTable 4-10 andareavailablein[ 4 ]aswell.Theforecasteddemandwiththedispatchhorizononedaywith24intervalsofonehoureachisshowninTable 4-8 Table4-7. Generatingunitscharacteristicsof10-unitsystemUnit Pmini,MWPmaxi,MWaibicieifiURiURi 1 150470786.798838.53970.15244500.04180802 135470451.325146.15910.10586000.03680803 733401049.997740.39650.0283200.02880804 603001243.531138.30550.03542600.05250505 732431658.569636.32780.02112800.06350506 571601356.659238.27040.01793100.04850507 201301450.704536.51040.01213000.08630308 471201450.704536.51040.01213400.08230309 20801455.605639.58040.1092700.098303010 10551469.402640.54070.12953800.0943030 Table4-8. Loaddemandfor24hoursTime,h Load,MW Time,h Load,MW Time,h Load,MW 1 1036 9 1924 17 14802 1110 10 2022 18 16283 1258 11 2106 19 17764 1406 12 2150 20 19725 1480 13 2072 21 19246 1628 14 1924 22 16287 1702 15 1776 23 13328 1776 16 1554 24 1184 4.2ResultsOneofthefeaturesthattheheuristicalgorithmspossessisrandomness.Therefore,theirperformancescannotbejudgedbytheresultofasinglerunandmanytrialswithdifferentinitializationsshouldbemadetoreachavalidconclusionabouttheperformanceofthealgorithms.Analgorithmisrobust,ifitcanguaranteeanacceptable 37

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performancelevelunderdifferentconditions.Inthispaper,50differentrunsofC-GRASPhavebeencarriedout. 4.2.1Case1Inthiscase,thedynamicEDproblemonsystem1issolved.ItcanbeseenfromTable( 4-9 )thattheC-GRASPprovidedthebestsolutioncomparedtoSAandGA. Table4-9. Generationcostsforcase1 MethodMinAvgMaxSt.Dev. C-GRASP19645.8711819725.632919837.5521026.8471GA19817.5020619819.4477319817.502060.93349SA19675.3550819777.3776119855.6388036.1006 ThesmallesttotalproductioncostisobtainedbySAanditis$19645.87.Morever,wecannoticethatontheaverageC-GRASPalgorithmperfomsbetterthanSAandGA.ThelowestmaximumvalueisprovidedbyC-GRASPaswell,whilethehighestmaximumvaluewasproducedbySA.showsthatSAsolutionsareverysensitivetostartingpointsandaremorevolatile.ThebestfoundsolutionsatisfyingdemandandpowerlimitsisgiveninTable 4-10 Table4-10. Bestsolutionforcase1 HourUnit1Unit2Unit3Unit4Unit5 110.0000697520.0000564480.00001477105.6819645194.3178945210.0000363252.6022286430.00004416151.8901212190.5075697326.0079646287.454166380.8405373345.99580774234.701524426.7573418295.20329384170.823281568.03647655169.1796063549.0465060156.0122637891.33351254110.4114641251.1962536619.63298415103.772159751.19984471196.8271002236.5679112736.541522592.8654090961.43881398157.4967338277.6575207813.7051770192.67194012138.6821921178.1269249230.8137659941.8141424102.4755387123.5565313244.850131177.30365661021.88623928102.6285523129.2017437249.9999302200.28353451128.8812865358.20434065161.8266865222.1801306248.90755571225.81847079113.8283152174.999958147.8022832277.5509728 4.2.2Case2Here,thestaticEDproblemincludesthenonlineargeneration-demandequalityconstraintsduetoincludedtransmissionlosses.Therampuplimitsandprohibited 38

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zonesofgeneratorsareincorporatedaswell.TheefcientofC-GRASPistestedonsix-unitsystemsthatisdiscribedinSection 4.1.2 .Thesameproblemhasbeensolvedin[ 30 ]andtheirbestsolutionandappliedmethodsarepresentedinTable 4-11 .ThelossesandtotalgenerationcostaregiveninTable 4-12 .Thebestsolutionsamongallsolutionshavebeenillustratedintheboldprints.FromthesedatawecanseethattheirprovidedobjectivefunctionvaluesaresmallerthatoneobtainedbyC-GRASP,butitshouldbenotedthatsolutionsgainedbyCPSO1andCPSO2violatetheramp-uplimitsofgenerator3.WhenthesolutionofPSOhasbeenpluged,ithasbeenfoundtheviolationofgeneration-demandbalanceequalityby0.4661MW,becauseaccordingtogivensolution,thegenerationisequalto1275.9571MWandlossis12.4910MW.TheminimumgenerationcostfoundbyC-GRASPis$15456.54469,whiletheaveragecostis$15507.10954withstandarddeviationofvalue$28.10037477.Accordingtothesefacts,itcanbestatedthatC-GRASPapprochwithappliedheuristicstrategycanproducedfeasibleandgoodsolutions.TheresultsproducedbySAandGAwerenotfeasibleorreasonablyclosetoresultspresnetedhere,sotheyarenotpresnetedhere. Table4-11. Bestsolutionsforcase2 P1P2P3P4P5P6 PSO447.4970173.3221263.4745139.0594165.476187.1280CPSO1434.4236173.4385274.2247128.0183179.704285.9082CPSO2434.4295173.3231274.4735128.0598179.475985.9281C-GRASP447.8181200253.5570149.9999150.320273.5022 Table4-12. Bestresults,whendemandis1263MW TotaloutputLossTotalgenerationcost PSO1276.012.958415451CPSO11276.012.958315447CPSO21276.012.958215446C-GRASP1275.197412.197415456.54 39

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4.2.3Case3Inthiscase,thestaticEDproblemwithnonsmoothcostfunctionduetothevalve-pointeffectsisconsideredtochecktheabilityofC-GRASPtosolvesuchtypeproblemsanditscompetitivenesswithbothGAandSAapproches.Theexperimentisperformedontwodifferentsystems,namely,system3andsystem4.ThenalfuelcostsobtainedusingappliedapprochesaresummarizedinTable 4-13 .Itshowstheminimum,averageandmaximumcostandstandarddeviationachievedbyappliedmethodsfor75runs.Fromthecomputationalresults,theminimumcostachievedbyC-GRASPwasthebest,followedbySAandGA.Theminimumcost,maximumcostandthemeancostvaluesobtainedbyC-GRASPare18394.07$/h,18699.339$/h,and18550.105$/h,respectively,whicharelowerthanthoseobtainedbySAandGA.TheworstresultsareobtainedbyGA.ItcanbenoticedthatresultsproducedbyGAvarytheleast,thiscanbeconrmedbythelowstandarddeviationthatis$12.562.Inliterature[ 54 ],thelowestreportedgenerationcostfor1800MWis$17994.07,howeverthesolutionisnotpresented. Table4-13. Generationcostsfor13-unitsystemwithdemand1800MW MethodMinAvgMaxSt.Dev. C-GRASP18394.07018550.10518699.33965.729GA19384.22919417.96419438.91412.562SA18950.17419393.11419782.516181.920 Theresultson40-unitsarepresentedintable 4-14 .C-GRASP,SAandGAalgorithmswererunfor75timesandtheminimum,maximumandaveragevalueofobjectivefunctionarereported. Table4-14. Generationcostsfor40-unitsystemwithdemand10500MW MethodMinAvgMaxSt.Dev. C-GRASP128883.1965130268.9796132839.2181972.757GA163401.9977163534.9817163623.342364.0606SA138975.7844150757.5002162578.62716118.65 40

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4.2.4Case4ThelastproblemsolvedbyC-GRASPisdynamicEDproblemincludingramp-uplimits,thatmakesthisproblemmoredifcultthaninacase1.Forsimplicity,thetransmissionlossesareneglected.TheminimumcostobtainedbytheC-GRASPcoupledwithheuristicstrategyisfoundtobe$1,735,176.10,thebestsolutionthatsatisesdemand-balanceconstraintsaswellasgeneratorsoperationconstraintsincludingonesoframp-upisgiveninTable 4-15 .GAansSAappliedinthisworkcouldn'tproducethefeasiblesolution. Table4-15. Bestsolutionforcase4 HourP1P2P3P4P5 1197.5775181.2115163.417562.3581127.01412199.3353142.0086243.3161112.3580115.82943279.3352222.0086163.3161162.358080.96964357.6864196.990783.3162212.3580130.96965341.5554276.9907104.5803226.3765180.96966364.4919356.990792.7176205.6510202.51987291.8407436.9906162.9068249.2158210.43138275.9255452.7053238.0856230.3986196.67339355.9255382.1891318.0856233.8956197.860410373.6142395.9434310.8613235.4000242.999911453.6141447.3342339.9999285.3999197.860412469.9999469.9999327.0836235.4000229.247313405.7058425.0176281.3943279.5788226.466314325.7058367.8957339.9999229.5788240.974215308.0081320.2882284.4522279.5788190.974216228.0081369.5967235.5079230.6255174.261617252.7930361.9796165.9723237.6272169.117718214.1486441.9796160.0032187.6272211.344419253.8865470.0000240.0032137.6272242.999920333.8865448.3762320.0032149.1631211.344421334.2637375.3046329.5451195.2469242.999922254.2637295.3046249.5451245.2469195.472723174.2637215.3046218.8263256.1493161.567924150.0001171.8146144.0519259.4389132.1630 41

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Bestsolutionsforcase4continued HourP6P7P8P9P10 184.115677.592283.017620.000139.6960287.316963.805290.000046.030410.0001360.234993.8052119.999946.750729.22194107.8558111.1225108.297359.459037.94455128.756381.122678.297438.994622.35676140.928876.308698.341856.971233.07857137.017946.308668.341871.651027.2956889.733576.308684.757880.000051.41179139.7335106.308693.145862.094034.762110159.9999128.767284.597773.038616.777711112.765098.7671114.597743.038612.623012112.5501128.767184.597773.038619.31561388.9431129.9999114.597777.810542.48611465.2257127.712695.173976.733454.99991570.066197.7126119.999962.433842.486116108.197967.712690.207732.433817.44821762.768070.299697.676732.836728.929318112.767999.9863119.999948.385731.757219122.172997.8341112.125555.924143.426520159.9999126.0825105.677178.854938.612221153.7373100.4907119.999948.854923.556922127.985270.490796.020170.030523.640523122.777140.490866.020256.083720.51642472.777170.490873.575660.648649.0393 42

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CHAPTER5CONCLUSION Economicdisatchproblemcanbeformulatedinverydifferentways:asasimplelinearprogramingproblemtononlinearnonconvexproblem. Inthiswork,fourdifferentcaseswereanalysedandthreeheuristicmethods:C-GRASP,GAandSAwereappliedtosolveEDproblem. SinceC-GRASPisabletocopewithoptimizationproblemhavingboxconstraints,theheuristicstrategytodealwithequalityandinequalityconstraintsforEDproblemwasincorporated. Theexperimentalresultsrevealed,thatC-GRASPadoptedtoEDproblemisabletoprovidegoodresultsandcanoutperfomSAandGA. 43

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BIOGRAPHICALSKETCH IngridaRadziukynienegotBachelorofScienceandMasterofScienceincomputerscienceatVytautasMagnusUniversity,Lithuaniain2003and2005,respectively.Inaddition,shegotacerticateofbussinessmanagementfromDepartmentofBusinessatVytautasMagnusUniversity.In2010,sheearnedtheMasterofScienceinindustrialengineeringfromUniversityofFlorida.Moreinformationaboutherresearchinterestcanbefoundinherwebpagehttp://plaza.u.edu/ingridar/. 50