Improved differential evolution approach based on cultural algorithm and diversity measure applied to solve economic load dispatch problems

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 79 (2009) 3136–3147

Improved differential evolution approach based on culturalalgorithm and diversity measure applied to solve economic

load dispatch problems

Leandro dos Santos Coelho a,∗, Rodrigo Clemente Thom Souza a,Viviana Cocco Mariani b

a Industrial and Systems Engineering Graduate Program, Pontifical Catholic University of Paraná, Imaculada Conceicão,

1155, 80215­910, Curitiba, PR, Brazilb Department of Mechanical Engineering, Pontifical Catholic University of Paraná, Imaculada Conceicão,

1155, 80215­910, Curitiba, PR, Brazil

Received 17 June 2008; received in revised form 16 January 2009; accepted 10 March 2009

Available online 21 March 2009

Abstract

Evolutionary algorithms (EAs) are general-purpose stochastic searchmethods that use themetaphor of evolution as the key element

in the design and implementation of computer-based problems solving systems. During the past two decades, EAs have attracted

much attention and wide applications in a variety of fields, especially for optimization and design. EAs offer a number of advantages:

robust and reliable performance, global search capability, little or no information requirement, and others. Among various EAs,

differential evolution (DE), which characterized by the different mutation operator and competition strategy from the other EAs, has

shown great promise in many numerical benchmark problems and real-world optimization applications. The potentialities of DE are

its simple structure, easy use, convergence speed and robustness. To improve the global optimization property of DE, in this paper, a

DE approach based onmeasure of population’s diversity and cultural algorithm technique using normative and situational knowledge

sources is proposed as alternative method to solving the economic load dispatch problems of thermal generators. The traditional and

cultural DE approaches are validated for two test systems consisting of 13 and 40 thermal generators whose nonsmooth fuel cost

function takes into account the valve-point loading effects. Simulation results indicate that performance of the cultural DE present

best results when compared with previous optimization approaches in solving economic load dispatch problems.

© 2009 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Optimization; Evolutionary algorithms; Economic dispatch; Cultural algorithm; Differential evolution

1. Introduction

The economic dispatch problem (EDP) is one of the important problems in operation and control of modern power

systems. The objective of the EDP of electric power generation is to schedule the committed generating unit outputs

so as to meet the required load demand at minimum operating cost while satisfying all unit and system equality and

inequality constraints [30].

∗ Corresponding author. Tel.: +55 4132711345; fax: +55 4132711345.

E­mail addresses: [email protected] (L.D.S. Coelho), [email protected] (R.C.T. Souza), [email protected]

(V.C. Mariani).

0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.matcom.2009.03.005

L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147 3137

In traditional EDPs, the cost function of each generator is approximately represented by a simple quadratic function

and the valve-points effects [5,31] are ignored. These traditional EDPs are solved using mathematical programming

based on deterministic optimization techniques.

However, the EDPwith valve-point effects can be represented as a nonsmooth optimization problem having complex

and nonconvex features with heavy equality and inequality constraints [5]. Moreover, this kind of optimization problem

is hard, if not impossible, to solve using deterministic optimization algorithms. Recently, as an alternative to the

conventional optimization approaches, modern stochastic optimization techniques based on evolutionary algorithms

(EAs) [5,7,8,10,11,14,16,31,35,37] have been given much attention by many researchers due to their ability to find

potential solutions.

EAs, derived from biological adaptation paradigms, are stochastic population based methods that have proven to be

powerful and robust techniques to solve complex optimization problems. The advantages of EAs include global search

capability, effective constraints handling capacity, reliable performance and minimum information requirements, make

it a potential choice for solving EDPs.

In this paper, an alternative hybrid method based on EAs is proposed. The proposed hybrid method combines the

differential evolution (DE), an EA, with cultural algorithm (CA) based on normative and situational knowledge sources

to solve the EDPs associated with the valve-point effect.

DE as developed by Storn and Price [26] is one of the best EAs, and has proven to be a promising candidate to solve

real valued optimization problems [27]. The DE is a method based on stochastic searches, in which function parameters

are encoded as floating-point variables. TheDE algorithm presents also simple structure, convergence speed, versatility,

and robustness, with only a few parameters required to be set by a user. Nevertheless, this faster convergence of DE

results in a higher probability of searching toward a local optimum or getting premature convergence. The application

of CAs in DE is an alternative strategy to improve the convergence performance and local search.

CAs were proposed in Reynolds [18] as a complement to the metaphor adopted by EAs. The CA was introduced

as a vehicle for modeling social evolution and learning in agent based societies. CAs are classes of models based on

some theories proposed in sociology and archaeology to model cultural evolution, which extract information from the

domain of the problem during the evolutionary process itself. In this context, a CA can incorporate domain knowledge

to render a search process more efficient. Cultural algorithms have been successfully applied to global optimization of

unconstrained [1], constrained functions [21], and scheduling problems [19,36].

In this paper, a new cultural DE approach inspired in a measure of population’s diversity for crossover rate tuning

and selection of mutation operation is proposed. The EDPs with 13 and 40 thermal generators with nonsmooth fuel

cost functions [11,30] are employed in this paper to validate the efficiency of the proposed cultural DE approach.

Simulation results obtained with the traditional DE and cultural DE approaches were compared to those obtained using

other optimization methods presented in recent literature.

The remainder of this paper is organized as follows. Section 2 describes the formulation of the EDP, while Section

3 explains the concepts of optimization methods based on DE approaches. Simulations and comparisons are provided

in Section 4. Last, Section 5 outlines the conclusion with a brief summary of results and future research.

2. Description of economic dispatch problem

The objective of the EDP is to minimize the total fuel cost at thermal power plants subjected to the operating con-

straints of a power system. Therefore, it can be formulated mathematically as an optimization problem (minimization)

with an objective function and constraints. The equality and inequality constraints are represented by Eqs. (1) and (2)

given by:

n∑

i=1

Pi − PL − PD = 0 (1)

Pmini ≤ Pi ≤ Pmaxi (2)

In the power balance criterion, an equality constraint must be satisfied, as shown in Eq. (1). The generated power

should be the same as the total load demand plus total line losses. The generating power of each generator should lie

between maximum and minimum limits represented by Eq. (2), where Pi is the power of generator i (in MW); n is

3138 L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147

the number of generators in the system; PD is the system load demand (in MW); PL represents the total line losses (in

MW) and Pmini and Pmaxi are, respectively, the minimum and maximum power outputs of the i-th generating unit (in

MW). The total fuel cost function fc is formulated as follows [34]:

min fc =

n∑

i=1

Fi(Pi) (3)

where Fi is the total fuel cost for the generator unity i (in $/h), which is defined by equation:

Fi(Pi) = aiP2i + biPi + ci (4)

where ai, bi and ci are cost coefficients of generator i.

A cost function is obtained based on the ripple curve for more accurate modeling. This curve contains higher order

nonlinearity and discontinuity due to the valve point effect, and should be refined by a sinusoidal function. Therefore,

Eq. (4) can be modified, as:

Fi(Pi) = F (Pi)+ |ei sin(fi(Pmini − Pi))| (5)

or

Fi(Pi) = aiP2i + biPi + ci + |ei sin(fi(P

mini − Pi))| (6)

where ei and fi are constants of the valve point effect of generators. Hence, the total fuel cost that must be minimized,

according to Eq. (3), is modified to:

min fc =

n∑

i=1

Fi(Pi) (7)

where Fi is the cost function of generator i (in $/h) defined by Eq. (6). In the case study presented here, we disregarded

the transmission losses, PL; thus, PL = 0. The Eq. (7) represents the fitness function. We are minimizing the fitness

function in this paper.

3. Differential evolution approaches

In general, all EAs work as follows: a population of individuals is randomly initialized where each individual

represents a potential solution to the problem. The quality of each solution is evaluated using a fitness function. A

selection process is applied during each iteration of an EA in order to form a new population. The selection process is

biased toward the fitter individuals in order to increase their chances of being included in the newpopulation. Individuals

are altered using unary transformation (mutation) and higher-order transformation (crossover). This procedure is

repeated until a stopping criterion is met is reached. The best solution found is expected to be a near-optimum solution

[2].

Meanwhile, the DE combines simple arithmetic operators with the classical events of recombination (crossover),

mutation and selection to evolve from randomly generated initial population to final individual solution. The key idea

behind DE is a scheme for generating trial parameter vectors. Mutation and crossover are used to generate new vectors

(trial vectors), and selection then determines which of the vectors will survive the next generation.

The different variants of DE are classified using the following notation: DE/α/β/δ, where α indicates the method for

selecting the parent chromosome that will form the base of the mutated vector, β indicates the number of difference

vectors used to perturb the base chromosome, and δ indicates the recombination mechanism used to create the off-

spring population. The bin acronym indicates that the recombination is controlled by a series of independent binomial

experiments.

This section describes the evaluated DE approaches. First, a brief overview of the traditional DE is provided, and

then the proposed DE approach with normative and situational knowledge sources is detailed.

L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147 3139

3.1. The traditional DE algorithm

In DE, individuals are represented as floating-point (real-valued) vectors. In each step, the DE mutates vectors by

adding weighted, random vector differentials to them. If the fitness function of the trial vector is better than that of the

target, the target vector is replaced by the trial vector in the next generation. The variant implemented here was the

DE/rand/1/bin, which involved the following steps and procedures for a minimization problem:

Step 1: Design of the parameters setup: The user must choose the key parameters that control the DE, i.e., population

size (N), boundary constraints of optimization variables, mutation factor (fm), recombination or crossover rate

(CR), and the stopping criterion (tmax).

Step 2: Initialize the initial population of individuals: Initialize the generation’s counter, t= 1, and also initialize

a population of individuals (solution vectors) x(t) with random values generated according to a uniform

probability distribution in the n-dimensional problem space.

Step 3: Evaluate the fitness function value: For each individual, evaluate its fitness (objective function) value. We are

minimizing, rather than maximizing, the fitness function in this paper. The fitness function is given by Eq. (7)

considering the equality and inequality constraints given by Eqs. (1) and (2).

Step 4: Mutation operation (or differential operation): Mutate individuals in according to equation:

zi(t + 1) = xi,r1 (t)+ fm[xi,r2 (t)− xi,r3 (t)] (8)

In the above equations, i= 1, 2, . . ., N is the individual’s index of population; j= 1, 2, . . ., n is the decision

variable (component) in i-th individual; t is the generation (time-step); xi(t) = [xi1 (t), xi2 (t), ..., xin (t)]T

stands for the position of the i-th individual of population of N real-valued n-dimensional vectors; zi(t) =

[zi1 (t), zi2 (t), ..., zin (t)]T stands for the position of the i-th individual of a mutant vector; r1, r2 and r3 are

mutually different integers and also different from the running index, i, randomly selected with uniform

distribution from the set {1, 2, . . ., i−1, i+1, . . ., N}; fm > 0 is a real parameter, called mutation factor, which

controls the amplification of the difference between two individuals so as to avoid search stagnation and it is

usually taken form the range [0.1, 1].

Step 5: Crossover (recombination) operation: Following themutation operation, crossover is applied in the population.

For eachmutant vector, zi(t+ 1), an index rnbr(i)∈{1, 2, . . .,n} is randomly chosen using a uniformdistribution,and a trial vector, ui(t + 1) = [ui1 (t + 1), ui2 (t + 1), . . . , uin (t + 1)]

T , is generated with

uij (t + 1) =

zij (t + 1), if randb(j) ≤ CR or j = rnbr(i),

xij (t), if randb(j) > CR or j /= rnbr(i),(9)

where randb(j) is the j-th evaluation of a uniform random number generation with [0,1] and CR is a scalar

parameter of the DE, called the crossover rate.

To decide whether or not the vector ui(t+ 1) should be a member of the population comprising the next

generation, it is compared to the corresponding vector xi(t). Thus, if fc denotes the objective function under

minimization, then

xi(t + 1) =

{

ui(t + 1), if fc(u(t + 1)) < fc(xi(t)),

xi(t), otherwise.(10)

Step 6: Verification of the stopping criterion: Loop to Step 3 until a stopping criterion is met, usually a maximum

number of iterations (generations), tmax.

3.2. The cultural DE approach

CAs have been developed in order to model the evolution of the cultural component of an evolutionary computa-

tional system over time as it accumulates experience. As a result, CAs can provide an explicit mechanism for global

knowledge and a useful framework within which to model self-adaptation in an evolutionary or swarm intelligence

system [12,13,20,22]. CAs consist three components. First, there is a population component (or population space)

3140 L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147

that contains the population to be evolved and the mechanisms for its evaluation, reproduction and modification. The

population space consists of a set of possible solutions to the problem, and can be modeled using any population-based

optimization method, in this work, the individual’s population in DE.

Second, there is a belief space that represents the bias that has been acquired by the population during its problem-

solving process. In CAs, the information acquired by a member of the population can be shared with the entire

population. The third component is the communication protocol that is used to determine the interaction between the

population and the beliefs.

Becerra and Coello [4] were the first to propose the use of DE as the population space of a CA. The proposed

approach in [4] uses a CA with a DE population. In that approach, the belief space is divided in four knowledge

sources (situational, normative, topographical, and historical). In this work, in DE design, concepts of optimization

are presented based on normative and situational knowledge sources of proposed DE approach proposed in [4]. These

concepts are combined with a population space approach using a modified cultural DE approach.

Summarizing, CAs model two levels of evolution: social population level and belief space level. In addition to a

population space, CA has a belief space in which the beliefs (problem-solving knowledge) acquired from the pop-

ulation’s evolution can be stored and integrated. An acceptance function is used to generate beliefs by gleaning the

experience of individuals from the population space. In return, this problem-solving knowledge can bias the evolution

of the population component by means of the influence function. The belief space itself also evolves by the adjust

function [13].

The normative knowledge contains the intervals for decision variables (individuals) where good solutions have

been found, in order to move new solutions towards those intervals. The lj and uj are the lower and upper bounds,

respectively, for the j-th decision variable, and Lj and Uj are the values of the fitness function.

To initialize the normative knowledge, all the bounds are set to the intervals given as input data of the problem. Lj

and Uj are set to +∞, assuming a minimization problem. Updating the normative knowledge can reduce or expand

the intervals stored on it. An expansion takes place when the accepted individuals do not fit into the current interval,

while a reduction occurs when all the accepted individuals lie within the current interval, and the extreme values have

a better fitness and represent feasible solutions.

The number of individuals accepted, ηaccepted, for update of the belief space is choice randomly of a part of population

using the following expression:

ηaccepted = pN (11)

where p% is a parameter given by the user (in this work, 0.3 is adopted) and N is the population size. Summarizing,

the update of the belief space is based on the lower and upper bounds given by interval [lj; uj] for the j-th decision

variable using ηaccepted individuals.

In this case, the modification of Eq. (8) (traditional DE) proceeds as follows in the pseudo-code presented in Fig. 1

for the proposed cultural DE (CDE) using normative and situational knowledge sources.

The situational knowledge source consists of the best exemplar xi,best (t) found along the evolutionary process in

DE. It represents a leader for the other individuals to follow in DE/best/1/bin method employed in CDE.

3.3. The cultural DE approach using a measure of population’s diversity

In context of EAs, an attractive and repulsive approach was introduced by Ursem [28,29] in particle swarm opti-

mization algorithm to escape from the current local optimum. It uses a diversity measure to control the population. The

result is a powerful algorithm that alternates between phases of attraction and repulsion. The application of diversity

measure can be an alternative strategy to improve the convergence performance and local search in DE design.

The trade-off between the exploration (i.e. the global search) and the exploitation (i.e. the local search) of the

search space is critical to the success of a DE approach. The crossover rate CR is a key factor affecting the DE’s

convergence.

The utilization of improvements in CDE based on diversity measure can be useful to escape more easily from

local minima than with the traditional DE. However, there are different ways of introducing diversity and controlling

the degree of diversity introduced in DE [6]. In this context, a new improved CDE method based on measure of

population’s diversity (CDEMD) is proposed. The CDEMD uses information of population’s diversity for (i) tuning of

CR and (ii) selection of mutation operation based on normative and situational knowledge sources or re-initialization

L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147 3141

Fig. 1. Pseudo-code of CDE.

of j-th decision variable of the i-th individual if the diversity measure value of j-th decision variable is low). In this

work, the diversity measure is given by

diversity (j, t) =1

N(xmaxj − xminj )

N∑

i=1

√

√

√

√

n∑

j=1

(xij − xj)2 (12)

where xmaxj and xminj are minimum and maximum values (bounds given by Eq. (2)) of the i-th individual, N is the

population size, n is the dimensionality of the problem, xij is the j-th value of the i-th individual, and xj is the j-th value

of the midpoint x. The pseudo-code of CDEMD is presented in Fig. 2. A re-initialization mechanism is needed to avoid

the premature convergence of the CDEMD algorithm when the diversity is low. In this context, the re-initialization

with random decision variables makes the individuals explore the search region as they get very close to the global

best position found.

The advantage of CDEMD is two-fold. Due to its greediness in scouting local minima using CAs concepts, it finds

promising regions rapidly during the initial phase of the search, while due to re-initialization with random decision

variables, the individuals do not loose the global exploration capability and thus they will tend to find better optima as

the search continues.

4. Simulation results

In this section, we judge the performance of the DE approaches using two case studies of EDP with 13 and 40

thermal generators (units) are evaluated.

Each optimizationmethodwas implemented inMatlab (MathWorks) usingMicrosoftWindowsXP.All the programs

were run on a 3.2GHz Pentium IV processor with 2GB of random access memory. In each case study, 50 independent

runs were made for each of the optimization methods involving 50 different initial trial solutions for each optimization

method.

The individuals xi(t) = [xi1 (t), xi2 (t), ..., xin (t)]T of tested DE approaches represent the real power generation of

generator i of a possible solution. The population size N was 25 and the stopping criterion tmax was 1000 generations

3142 L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147

Fig. 2. Pseudo-code of CDEMD.

(25,000 evaluations of fitness function) for the tested DE approaches in two case studies. The crossover rate ofCR= 0.9

and fm = 0.5 was adopted for the traditional DE, CDE, and CDEMD approaches.

A key factor in the application of DE approaches is how the algorithm handles the constraints relating to the problem.

When DE approaches are used for constrained optimization problems, it is common to handle constraints using the

concept of penalty functions (which penalize unfeasible solutions). However, in this work is adopted a repair procedure

based on [17] instead of penalizing infeasible solutions to the equality constraints in two case studies.

4.1. Case study 1

This case study consisted of 13 thermal units of generation with the effects of valve-point loading, as given in

Table 1. The data shown in Table 1 is also available in Wong and Wong [33] and Gomes and Saavedra [11]. In this

case, the load demand expected to be determined was PD = 1800MW. This EDP has many local minima, and the global

minimum is difficult to determine.

The results obtained for this case study are given in Table 2, which shows that the CDEMD succeeded in finding the

best solution for the tested methods. The best results obtained for solution vector Pi, i= 1, . . ., 13 with CDEMD with

minimum cost of 17961.9440 is given in Table 3. Furthermore, the CDE and CDEMD approaches show performance

which is clearly better than that of traditional DE in terms of mean fitness value.

L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147 3143

Table 1

Data for the 13 thermal units.

Thermal unit Pmini Pmaxi a b c e f

1 0 680 0.00028 8.10 550 300 0.035

2 0 360 0.00056 8.10 309 200 0.042

3 0 360 0.00056 8.10 307 150 0.042

4 60 180 0.00324 7.74 240 150 0.063

5 60 180 0.00324 7.74 240 150 0.063

6 60 180 0.00324 7.74 240 150 0.063

7 60 180 0.00324 7.74 240 150 0.063

8 60 180 0.00324 7.74 240 150 0.063

9 60 180 0.00324 7.74 240 150 0.063

10 40 120 0.00284 8.60 126 100 0.084

11 40 120 0.00284 8.60 126 100 0.084

12 55 120 0.00284 8.60 126 100 0.084

13 55 120 0.00284 8.60 126 100 0.084

Table 2

Convergence results (50 runs) of DE approaches for the case study with 13 thermal units.

Optimization method CPU mean time (s) Minimum cost ($/h) Mean cost ($/h) Maximum cost ($/h) Standard deviation ($/h)

Traditional DE× 10.5 17968.3601 18002.9099 18133.4582 38.3352

CDE 12.1 17967.4000 17995.5850 18065.8044 27.0900

CDEMD 12.6 17961.9440 17974.6869 18061.4110 20.3066

Best result in bold font.

Table 4 compares the results obtained in this paper with those of other studies reported in the literature. Note that in

studied case, the best result reported here using CDEMD is comparatively lower than studies presented in the recent

literature.

4.2. Case study 2

This case study involved 40 thermal units with quadratic cost functions together with the effects of valve-point

loading, as shown in Table 5. The data of Table 5 are also available in [33]. In this case, the load demand expected

to be determined was PD = 10500MW. Table 6 shows the minimum, mean and standard deviation, and the maximum

fitness function achieved by the traditional DE, CDE, and CDEMD approaches.

As indicated in Table 6, theCDEMDwas the approach that obtained the best fuel cost for the EDPof 40 thermal units.

The best results obtained for solution vector Pi, i= 1, . . ., 40 with CDEMD with minimum cost of 121423.4013 $/h is

given in Table 7.

Table 8 compares the results obtained in this paper with those of other studies reported in the literature. Note that in

studied case, the best result reported here using CDEMD is comparatively lower than recent studies presented in the

recent literature.

Table 3

Best result (50 runs) obtained for the case study with 13 thermal units using CDEMD.

Power Generation (MW) Power Generation (MW)

P1 628.3287 P8 109.7869

P2 149.5060 P9 109.7691

P3 222.9850 P10 40.0000

P4 109.8517 P11 40.0000

P5 109.8752 P12 55.0000

P6 60.0000 P13 55.0228

P7 109.8746

13∑

i=1

Pi 1800.0000

3144 L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147

Table 4

Comparison of best results for fuel costs presented in recent literature for the case study with 13 thermal units.

Optimization technique Best objective function

Particle swarm optimization [30] 18030.72

Evolutionary programming [25] 17994.07

Hybrid evolutionary programming with SQP [30] 17991.03

Genetic algorithm [5] 17975.3437

Hybrid differential evolution [32] 17975.73

Hybrid particle swarm with SQP [30] 17969.93

Pattern search method [3] 17969.17

Differential evolution [17] 17963.83

Improved genetic algorithm with multiplier updating [5] 17963.9848

Quantum particle swarm optimization [9] 17963.95

Best result of this paper using CDEMD approach 17961.9440

Best result in bold font.

Table 5

Data for the 40 thermal units.

Thermal unit Pmini Pmaxi a b c e f

1 36 114 0.00690 6.73 94.705 100 0.084

2 36 114 0.00690 6.73 94.705 100 0.084

3 60 120 0.02028 7.07 309.54 100 0.084

4 80 190 0.00942 818 369.03 150 0.063

5 47 97 0.01140 5.35 148.89 120 0.077

6 68 140 0.01142 8.05 222.33 100 0.084

7 110 300 0.00357 8.03 278.71 200 0.042

8 135 300 0.00492 6.99 391.98 200 0.042

9 135 300 0.00573 6.60 455.76 200 0.042

10 130 300 0.00605 12.90 722.82 200 0.042

11 94 375 0.00515 12.90 635.20 200 0.042

12 94 375 0.00569 12.80 654.69 200 0.042

13 125 500 0.00421 12.50 913.40 300 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 7.10 801.32 300 0.035

26 254 550 0.00277 7.10 801.32 300 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 0.01140 5.35 148.89 120 0.077

31 60 190 0.00160 6.43 222.92 150 0.063

32 60 190 0.00160 6.43 222.92 150 0.063

33 60 190 0.00160 6.43 222.92 150 0.063

34 90 200 0.00010 8.95 107.87 200 0.042

35 90 200 0.00010 8.62 116.58 200 0.042

36 90 200 0.00010 8.62 116.58 200 0.042

37 25 110 0.01610 5.88 307.45 80 0.098

38 25 110 0.01610 5.88 307.45 80 0.098

39 25 110 0.01610 5.88 307.45 80 0.098

40 242 550 0.00313 7.97 647.83 300 0.035

L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147 3145

Table 6

Convergence results (50 runs) of DE approaches for the case study with 40 thermal units.

Optimization method CPU mean time (s) Minimum cost ($/h) Mean cost ($/h) Maximum cost ($/h) Standard deviation ($/h)

Traditional DE 29.3 121552.3516 121708.0739 122056.6991 105.0894

CDE 43.7 121551.3797 121671.8582 121859.6490 61.1927

CDEMD 44.3 121423.4013 121526.7330 121696.9868 54.8617

Best result in bold font.

Table 7

Best result (50 runs) obtained for the case study with 40 thermal units using CDEMD.

Power Generation (MW) Power Generation (MW)

P1 111.1110 P21 523.2958

P2 110.8299 P22 523.2849

P3 97.4122 P23 523.2856

P4 179.7443 P24 523.2979

P5 88.1510 P25 523.2799

P6 139.9959 P26 523.2910

P7 259.6065 P27 10.0064

P8 284.6045 P28 10.0018

P9 284.6149 P29 10.0000

P10 130.0002 P30 96.2132

P11 168.8029 P31 189.9996

P12 94.0000 P32 189.9998

P13 214.7591 P33 189.9981

P14 394.2716 P34 164.9126

P15 304.5206 P35 199.9941

P16 394.2834 P36 200.0000

P17 489.2912 P37 109.9988

P18 489.2877 P38 109.9994

P19 511.2977 P39 109.9974

P20 511.2791 P40 511.2800

40∑

i=1

Pi 10500.0000

Table 8

Comparison of best results for fuel costs presented in recent literature for the case study with 40 thermal units.

Optimization technique Best objective function

Particle swarm optimization [30] 122930.45

Evolutionary programming [25] 122624.35

Hybrid evolutionary programming with SQP [30] 122379.63

Improved genetic algorithm with multiplier updating [5] 121819.25

Anti-predatory particle swarm optimization [23] 121663.52

Quantum particle swarm optimization [9] 121501.14

Civilized swarm optimization [24] 121461.67

Best result of this paper using CDEMD approach 121423.4013

Best result in bold font.

5. Conclusion and further research

DE algorithm is a simple but powerful stochastic global optimizer. The crucial idea behind DE is a scheme for

generating trial parameter vectors. It has been proven a very good global optimizer for engineering design and opti-

mization. As is argued in [27], there are several advantages for this algorithm to outperform some other EAs [15], e.g.

DE is a very simple and straightforward strategy, and it is easy to use yet a very powerful algorithm.

3146 L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147

In this paper, traditional DE, CDE, and CDEMD approaches have been successfully introduced to solve two case

studies of EDP. In these case studies, tested DE approaches can provide accurate solutions for EDPs in reasonable

time.

In relation to procedure of solution of the EDP of electric energy with effect of valve point, the results with the

CDEMD for optimization were superior that the results presented in recent literature.

Future research is to investigate the effect of different knowledge sources incorporation into DE design for solving

the multiobjective EDPs in power systems.

Acknowledgments

This work was partially supported by the National Council of Scientific and Technologic Development of Brazil

— CNPq — under Grants 309646/2006-5/PQ, 302786/2008-2/PQ, 568221/2008-7, and 474408/2008-6. The authors

would like to thank anonymous reviewers and the editor for constructive comments and suggestions.

References

[1] J. Alami, A. El Imrani, A. Bouroumi, A multipopulation cultural algorithm using fuzzy clustering, Applied Soft Computing 7 (2) (2007)

506–519.

[2] F.S. Al-Anzi, A. Allahverdi, A self-adaptive differential evolution heuristic for two-stage assembly scheduling problem to minimize maximum

lateness with setup times, European Journal of Operational Research 182 (10) (2007) 80–94.

[3] J.S. Al-Sumait, A.K. Al-Othman, J.K. Sykulski, Application of pattern search method to power system valve-point economic load dispatch,

Electrical Power and Energy Systems 29 (10) (2007) 720–730.

[4] R.L. Becerra, C.A.C. Coello, Cultural differential evolution for constrained optimization, Computer Methods Applied in Mechanical Engi-

neering 195 (33–36) (2006) 4303–4322.

[5] C.L. Chiang, Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels, IEEE Transactions

on Power Systems 20 (4) (2005) 1690–1699.

[6] L.S. Coelho, P. Alotto, Electromagnetic optimization based on an improved diversity-guided differential evolution approach and adaptive

mutation factor, in: Proceedings of 10th InternationalWorkshop onOptimization and Inverse Problems in Electromagnetism, Ilmenau,Germany,

2008.

[7] L.S. Coelho, V.C. Mariani, Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with

valve-point effect, IEEE Transactions on Power Systems 21 (2) (2006) 989–996.

[8] L.S. Coelho, V.C. Mariani, Improved differential evolution algorithms for handling economic dispatch optimization with generator constraints,

Energy Conversion and Management 48 (5) (2007) 1631–1639.

[9] L.S. Coelho, V.C. Mariani, Particle swarm approach based on quantum mechanics and harmonic oscillator potential well for economic load

dispatch with valve-point effects, Energy Conversion and Management 49 (11) (2008) 3080–3085.

[10] J.G. Digalakis, K.G. Margaritis, A multipopulation cultural algorithm for the electrical generator scheduling problem, Mathematics and

Computers in Simulation 60 (3–5) (2002) 293–301.

[11] J.R. Gomes, O.R. Saavedra, A Cauchy-based evolution strategy for solving the reactive power dispatch problem, Electrical Power and Energy

Systems 24 (4) (2002) 277–283.

[12] R. Iacoban, R.G. Reynolds, J. Brewster, Cultural swarms: modelling the impact of culture on social interaction and problem solving, in:

Proceedings of the Swarm Intelligence Symposium, Indianapolis, IN, USA, 2003, pp. 205–211.

[13] X. Jin, R.G. Reynolds, Using knowledge-based evolutionary computation to solve nonlinear constraint optimization problems: a cultural

algorithm approach, in: Proceedings of Congress on Evolutionary Computation, Washington, DC, USA, 1999, pp. 1672–1678.

[14] I. Kitsios, T. Pimenides, H∞ controller design for a distillation column using genetic algorithms, Mathematics and Computers in Simulation

60 (3–5) (2002) 357–367.

[15] A. Leontitsis, D. Kontogiorgos, J. Pagge, Repel the swarm to the optimum, Applied Mathematics and Computation 173 (1) (2006) 265–272.

[16] H. Lin, K. Yamashita, Hybrid simplex genetic algorithm for blind equalization using RBF networks, Mathematics and Computers in Simulation

59 (4) (2002) 293–304.

[17] N. Noman, H. Iba, Differential evolution for economic load dispatch problems, Electric Power Systems Research 78 (3) (2008) 1322–1331.

[18] R.G.Reynolds, An introduction to cultural algorithms, in:A.V. Sebald, L.J. Fogel (Eds.), Proceedings of 3rdAnnual Conference onEvolutionary

Programming, World Scientific, River Edge, NJ, USA, 1994, pp. 131–139.

[19] R.G. Reynolds, M. Ali, Computing with the social fabric: the evolution of social intelligence within a cultural framework, IEEE Computational

Intelligence Magazine 3 (1) (2008) 18–30.

[20] R.G. Reynolds, C. Chung, The use of cultural algorithms to evolve multiagent cooperation, in: Proceedings of Micro-Robot World Cup Soccer

Tournament, Taejon, Korea, 1996, pp. 53–56.

[21] R.G. Reynolds, Z. Michalewicz, M. Cavaretta, Using cultural algorithms for constraint handling in GENOCOP, in: Proceedings of the 4th

Annual Conference on Evolutionary Programming, MIT Press, Cambridge, Massachusetts, 1995, pp. 298–305.

[22] R.G. Reynolds, S. Zhu, Knowledge-based function optimization using fuzzy cultural algorithms with evolutionary programming, IEEE

Transactions on Systems, Man, and Cybernetics – Part B: Cybernetics 31 (1) (2001) 1–18.

L.D.S. Coelho et al. / Mathematics and Computers in Simulation 79 (2009) 3136–3147 3147

[23] A.I. Selvakumar, K. Thanushkodi, Anti-predatory particle swarm optimization: solution to nonconvex economic dispatch problems, Electric

Power Systems Research 78 (1) (2008) 2–10.

[24] A.I. Selvakumar, K. Thanushkodi, Optimization using civilized swarm: solution to economic dispatch with multiple minima, Electric Power

Systems Research 79 (1) (2009) 8–16.

[25] N. Sinha, R. Chakrabarti, P.K. Chattopadhyay, Evolutionary programming techniques for economic load dispatch, IEEE Transactions on

Evolutionary Computation 7 (1) (2003) 83–94.

[26] R. Storn, K. Price, Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical

Report TR-95-012, International Computer Science Institute, Berkeley, USA, 1995.

[27] R. Storn, K. Price, Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global

Optimization 11 (4) (1997) 341–359.

[28] R.K. Ursem, Diversity-guided evolutionary algorithms, in: Proceedings of Parallel Problem Solving form Nature Conference-PPSN VII,

Granada, Spain, 2002, pp. 462–471.

[29] R.K.Ursem,Models for evolutionary algorithms and their applications in system identification and control optimization, PhD thesis, Department

of Computer Science, University of Aarhus, Aarhus, Denmark, 2003.

[30] T.A.A. Victoire, A.E. Jeyakumar, Hybrid PSO-SQP for economic dispatch with valve-point effect, Electric Power Systems Research 71 (1)

(2004) 51–59.

[31] D.C. Walter, G.B. Sheblé, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Transactions on Power Systems 8

(3) (1993) 1325–1332.

[32] S.-K. Wang, J.-P. Chiou, C.-W. Liu, Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm, IET

Generation, Transmission and Distribution 1 (5) (2007) 793–803.

[33] K.P. Wong, Y.W. Wong, Genetic and genetic/simulated-annealing approaches to economic dispatch, IEE Proceedings – Control, Generation

Transmission and Distribution 141 (5) (1994) 507–513.

[34] A.J. Wood, B.F. Wollenberg, Power Generation, Operation and Control, John Wiley & Sons, New York, USA, 1994.

[35] M. Xiangping, Z. Huaguang, T. Wanyu, A hybrid method of GA and BP for short-term economic dispatch of hydrothermal power systems,

Mathematics and Computers in Simulation 51 (3–4) (2000) 341–348.

[36] X. Yuan, Y. Yuan, Application of cultural algorithm to generation scheduling of hydrothermal systems, Energy Conversion and Management

47 (15–16) (2006) 2192–2201.

[37] X. Yuan, Y. Yuan, Y. Zhang, A hybrid chaotic genetic algorithm for short-term hydro system scheduling, Mathematics and Computers in

Simulation 59 (4) (2002) 319–327.