Nonsmooth Economic Power Dispatch through an Enhanced Differential Evolution Approach

Samir Sayah * , Abdellatif Hamouda QUERE Laboratory, Electrical Engineering Department,

Ferhat Abbas University, Setif 19000, Algeria

samir_pg04@yahoo.fr , a_hamouda1@yahoo.fr

Abstract—Economic power dispatch (EPD) is an important tool for optimal operation and planning of modern power systems. To solve effectively EPD problems, most of the conventional calculus methods rely on the assumption that the fuel cost characteristic of a generating unit is a continuous and convex function, resulting in inaccurate dispatch. This paper presents the design and application of an enhanced differential evolution (EDE) algorithm for the solution of the economic power dispatch problem, where the nonsmooth and nonconvex characteristics of the generators, such as valve-point effects and multi-fuel options of the practical generator operation are considered. The 10-generator benchmark test system with valve-point loading effects and multi-fuel options was used for testing and validation purposes. The results obtained demonstrate the effectiveness of the proposed method for solving the nonsmooth economic dispatch problem.

Keywords-nonsmooth economic dispatch; differential evolution; valve-point effect; multi-fuel options.

I. INTRODUCTION The conventional economic power dispatch (EPD) problem of power generation involves allocation of power generation to different thermal units to minimize the operating cost subject to diverse equality and inequality constraints of the power system. This makes the EPD problem a large-scale highly nonlinear constrained optimization problem. The EPD problem has been solved via many traditional techniques, such us linear programming, nonlinear programming, quadratic programming, Newton-based techniques and interior point methods. Usually, these methods rely on the assumption that the fuel cost characteristic of a generating unit is a smooth, convex function. However, there are situations where it is not possible, or appropriate, to represent the unit’s fuel cost characteristic as a convex function. For example, this situation arises when valve-points, unit prohibited operating zones, or multiple fuels are present. Hence, the true global optimum of the problem could not be reached easily. New numerical methods are then needed to cope with these difficulties, specially, those with high speed search to the optimal and not being trapped in local minima. In recent years, new optimization techniques based on the principles of natural evolution, and with the ability to solve

extremely complex optimization problems, have been developed. These techniques, also known as evolutionary algorithms, search for the solution of optimization problems, using a simplified model of the evolution process found in nature [1]. Differential Evolution (DE) is one of these recently developed evolutionary computation techniques [2, 3]. Differential evolution improves a population of candidate solutions over several generations using the mutation, crossover and selection operators in order to reach an optimal solution. Differential evolution presents great convergence characteristics and requires few control parameters, which remain fixed throughout the optimization process and need minimum tuning [4]. In DE, the fitness of an offspring is one-to-one competed with that of the corresponding parent, making convergence speed of DE faster than other evolutionary algorithms. Nevertheless, this faster convergence property yields in a higher probability of searching toward a local optimum or getting premature convergence.

In this paper, an enhanced differential evolution (EDE) based technique is presented and used to solve the economic dispatch problem, with nonconvex, noncontinuous, and nonlinear cost functions. Modifications in mutation rule are suggested to the original DE algorithm that explores efficiently the solution space, for a better solution quality. An application was performed on the 10-generator benchmark test system with valve-point loading effects and multi-fuel options. The results obtained through EDE and those of the previous methods are compared. The outcome of the comparisons confirms the effectiveness of the proposed EDE approach.

II. PROBLEM DESCRIPTION The main objective of EPD is to minimize the total generation cost of the power system within a defined interval (typically 1 h). The basic EPD considers the power balance constraint apart from the generating capacity limits. However, a practical EPD must take a variety of practical operating conditions into consideration, such as valve-point effects and multi-fuel options, to provide the completeness for the EPD problem formulation.

978-1-4673-4766-2/12/$31.00 ©2012 IEEE

2.1. Objective function The objective function for the EPD reflects the cost

associated with generating power in the system. The objective function F for the entire power system can then be written as the sum of the fuel cost model for each generator:

��

�ng

iiFF

1 (1)

where iF is the fuel cost function of the ith generator in ($/h) and ng is the number of online generating units to be dispatched. Traditionally, the fuel cost curve is approximated using a simple smooth quadratic function [5] given by:

iiiiiii cPbPaPF ��� 2)( (2)

where iP is the power of generator i. ia , ib and ic are the cost coefficients of generator i.

However, it is more practical to consider the valve-point loading effects for fossil-fuel-based plants. These effects, which occur as each steam admission valve in a turbine starts to open, produce a rippling effect on the unit’s cost curve [6]. Usually, valve-point effect is modelled by adding a recurring rectified sinusoid to the basic quadratic cost curve [6]. Therefore, (2) can be modified as:

� �)(sin)( min2iiiiiiiiiii PPfecPbPaPF ����� (3)

where ie and if are the coefficients of generator i reflecting

valve point effects and miniP is the minimum generation limit

of unit i. Moreover, there are many thermal generating units that can

be supplied by multiple fuel sources [7]. In those cases, it is more appropriate to represent the unit’s fuel cost characteristic as a piecewise function, reflecting the effects of fuel type changes. To obtain an accurate and practical EPD solution, valve-point effects and multi-fuel options should be included in the cost function which is formulated as follows [7]:

� �)(sin

)(min

2

iiijij

ijiijiijii

PPfe

cPbPaPF

��

���

if max,

min, jiiji PPP �� , j = 1, …, nf

(4)

where i and j denote the index of unit and index of fuel type, respectively; ija , ijb , ijc , ije and ijf are the cost coefficients

of the unit i for fuel type j; min, jiP and max

, jiP are the minimum and maximum power output of unit i with fuel option j, respectively, and nf is the number of fuel types for each unit.

2.2. Problem constraints

2.2.1. Power balance constraint

This constraint is based on the principle of equilibrium between total system generation and total system loads DP and transmission losses LP [5]. That is

LD

ng

ii PPP ���

�1 (5)

LP can be obtained using the B matrix loss formula [5], given by:

� ��� ��

���ng

i

ng

iiiiij

ng

jiL BBPPBPP

1 1000

1 (6)

where ijB , 0iB and 00B are the loss coefficients.

2.2.2. Generator capacity constraints For stable operation, the real power generation of each

generator should be restricted between its lower and upper limits of generation. The generator capacity constraints are expressed as a pair of inequality constraints, as follows:

maxminiii PPP �� (7)

where miniP and

maxiP are the minimum and maximum

generation of unit i, respectively.

III. OVERVIEW OF DIFFERENTIAL EVOLUTION ALGORITHM The differential evolution algorithm (DE) is a population

based algorithm like genetic algorithm using the similar operators; crossover, mutation and selection. In DE, each decision variable is represented in the chromosome by a real number. As in any other evolutionary algorithm, the initial population of DE is randomly generated, and then evaluated. After that, the selection process takes place. During the selection stage, three parents are chosen and they generate a single offspring which competes with a parent to determine which one passes to the following generation. DE generates a single offspring (instead of two like in the genetic algorithm) by adding the weighted difference vector between two parents to a third parent. If the resulting vector yields a lower objective function value than a predetermined population member, the newly generated vector replaces the vector to which it was compared.

An optimization task consisting of D parameters can be presented by a D-dimensional vector. In DE, a population of NP solution vectors is randomly created at the start. This population is successfully improved over G generations by applying mutation, crossover and selection operators to reach an optimal solution [3, 4]. The main steps of the DE algorithm are given bellow:

Initialization Evaluation RRepeat

Mutation Crossover Evaluation Selection

Until (Termination criteria are met)

Initialization Typically, each decision parameter in every vector of the initial population is assigned a randomly chosen value from within its corresponding feasible bounds:

� �minmaxmin)0(, jjjjij xxxx ��� , i = 1,…, NP, j = 1, …,

D (8)

where j denotes a uniformly distributed random number within the range [0, 1], generated anew for each value of j.

maxjx and min

jx are the upper and lower bounds of the jth decision parameter, respectively.

Mutation The mutation operator creates mutant vectors ix by perturbing a randomly selected vector ax with the difference of two other randomly selected vectors bx and cx , according to the following equation [8]:

)( )()()()( Gc

Gb

Ga

Gi xxxx ��� � , i = 1, …, NP (9)

where ax , bx and cx are randomly chosen vectors among the NP population, and a ≠ b ≠ c. The scaling constant� is an algorithm control parameter used to adjust the perturbation size in the mutation operator and improve algorithm convergence.

Crossover The crossover operation generates trial vectors ix by mixing

the parameters of the mutant vectors ix with its target or parent vectors ix , based on a series selected probability distribution of the following form [8]:

��

���

� ��

�

otherwise

or if

)(,

)(,

)(,

Gij

RjGij

Gij

x

qjCx

x

�

,

i = 1, …, NP, j = 1,…, D

(10)

where j� denotes a uniformly distributed random number within [0, 1], generated anew for each value of j. q is a randomly chosen index � �PN,...,1 � that guarantees that the trial vector gets at least one parameter from the mutant vector. The crossover constant CR is an algorithm parameter that controls the diversity of the population and aids the algorithm to escape from local minima.

Selection The selection operator forms the population by choosing between the trial vectors and their predecessors (parent vectors) those individuals that present a better fitness or are more optimal according to (11) [8].

�

��

� ���

otherwise ,

)()( if ,

)(

)()()(

)1(

Gi

Gi

Gi

Gi

Gi

x

xfxfxx ,

i=1, …, NP.

(11)

This optimization process is repeated for several generations, allowing individuals to improve their fitness as they explore the solution space in search of optimal values. In this work, the search procedure will terminate whenever the predetermined maximum number of generations maxG is reached, or whenever the global best solution does not improve over a predetermined number of iterations.

DE has three essential control parameters which are: the scaling factor� , the crossover constant CR and the population size NP. The scaling factor is a value in the range [0, 2] that controls the amount of perturbation in the mutation process. The crossover constant is a value in the range [0, 1] that controls the diversity of the population. The population size determines the number of individuals in the population and provides the algorithm enough diversity to search the solution space.

IV. ENHANCED DIFFERENTIAL EVOLUTION ALGORITHM In order to improve the global search capability, a new

scheme of differential evolution algorithm has been proposed by Kaelo and Ali [9]. In the original DE three vectors are chosen at random for mutation and the base vector is then chosen at random within the three. This has an exploratory effect but it slows down the convergence of DE. Also the original DE uses a fixed positive value for the scaling factor � in mutation. This has an effect of restricting the exploration. The first modification to DE is to replace the random base vector )(G

ax in the mutation rule (9) with the

tournament best )(Gtbx . From the three random vectors the best

is used as the base vector and the remaining two are used to find the differential vector in (9). This process explores the region around each )(G

tbx for each mutated point. This maintains the exploratory feature and at the same time expedites the convergence [8]. Also, instead of using a fixed � throughout a run of DE, we use a random � in � � � �1 ,4.04.0 ,1 ��� for each mutated point [13]. This random localization feature gradually transforms itself into the search intensification feature for rapid convergence when the points in the solution space form a cluster around the global minimizer. This version of DE is referred to as the differential evolution algorithm with random localization (DERL) [9].

V. CONSTRAINTS HANDLING A penalty function approach is used to handle the power

balance constraint and inequality constraints [10]. The extended objective function FT (or fitness function) is defined by:

����

����

����

�����

nc

jj

ng

iLDiT PFPPPKFF

1

2

1 (12)

where K denotes the penalty factor of the equality constraint; nc represents the number of inequality constraints and jPF is

the penalty function for the jth inequality constraint; given as:

�

��� �

� otherwise 0

violatedif )( 2limjjj

jUUKPF (13)

Kj is the penalty factor for the jth inequality constraint; limjU is

the limit value of the variable jU .

VI. NUMERICAL RESULTS AND ANALYSIS In order to evaluate the performance of the proposed EDE

approach, experiments are performed on 10-generator system used in [7] with valve-point loading effects and multi-fuel options. Note that the unit curves of this system have nondifferential points according to valve-point loading and multiple fuel changes. The first generator of the system has two fuel options and the remaining generators have three fuel options each. The total system demand is 2700 MW and no transmission losses are considered. The input data and related constraints of the test system are described in [7].

Due to the stochastic nature of the proposed approach, this system was repeatedly solved twenty times by the EDE method, using the control parameters listed in Table 1. Simulations were carried out using MATLAB computational environment, on an Intel Core 2 Duo 2.10 GHz Laptop computer with 3 GB total memory.

The best solution produced in the 20 trials was 623.83 $/h. The worst solution obtained was 623.88 $/h with an average of 623.85 $/h. The average execution time required for one complete solution was 2.90 s, which is very tolerable for EPD solutions. Also, it is important to point out that for all the trial runs, the convergence was reached without any violation of the generator capacity constraints. The global optimal dispatch solution of EDE is summarized in Table 2, which converged after 360 generations and 2.83 s. The convergence characteristic of the EDE is depicted in Fig. 1. The best solution found by DE is shown in Table 3, with the following control parameter settings: 5.0�� , CR = 0.9, NP = 50 and Gmax

= 500. The comparison of the convergence characteristics is depicted in Fig. 1. It is quite clear that EDE appears to be superior to DE with regard to the solution quality and convergence rapidity.

The comparison results of the EDE with several recently published methods are presented in Table 3. It is interesting to note that the exact fuel costs computed from the best schedules given by BBO and DE/BBO [15] are

628.20 $/h and 628.82 $/h, respectively. It is clear that these costs are higher than those published in Bhattacharya [15].

Consequently, we can say that EDE gives better and accurate optimal generation cost than other algorithms. It can be concluded that the proposed method is reliable and can provide high quality solutions in the presence of generating units with nonsmooth cost curves.

The statistical results of 20 runs by EDE with 20 different initial trial solutions are depicted in Fig. 2. From this figure it is observed that the EDE algorithm consistently produces solutions at or very near to the global optimum, indicating a good convergence characteristic. It can be concluded that EDE algorithm is robust and effective in solving the EPD problem considering practical generator operation constraints, such us valve-point effects and multiple fuel changes.

TABLE I. CONTROL PARAMETER SETTINGS OF EDE

Parameter Setting

Population size (Np) 50

Crossover constant (CR) 0.9

Penalty factor of the equality constraint (K) 3600

Maximum number of generations (Gmax) 500

TABLE II. BEST RESULTS OBTAINED BY THE POPOSED EDE

Unit miniP max

iP Fuel type

Generation (MW)

1 100 250 2 218.105 2 50 230 1 211.660 3 200 500 1 280.657 4 99 265 3 239.283 5 190 490 1 279.935 6 85 265 3 240.064 7 200 500 1 290.099 8 99 265 3 239.283 9 130 440 3 425.046 10 200 490 1 275.869 Total power generation (MW) 2700.000 Total fuel cost ($/h) 623.83

Figure 1. Convergence of EDE and DE algorithms

TABLE III. COMPARISON RESULTS

Method Minimum cost ($/h) Maximum cost ($/h) Mean cost ($/h) CGA-MU [7] 624.72 633.87 627.61 IGA-MU [7] 624.52 630.87 625.87 NPSO-LRS [11] 624.13 627.00 625.00 APSO1 [12] 624.01 627.30 624.82 APSO2 [13] 623.91 NA 624.51 CBPSO-RVM [14] 623.96 624.29 624.08 BBO [15] 605.54, (628.20) a 605.91 605.86 DE/BBO [15] 605.62, (628.82) a 605.62 605.63 DE 623.89 623.97 623.93

Proposed EDE 623.83 623.88 623.85 NA: Not Available. a Denotes the exact fuel cost computed from the reported schedule. 1 Anti-predatory particle swarm optimization. 2 Adaptive particle swarm optimization.

Figure 2. Distribution of fuel cost obtained by EDE

VII. CONCLUSION In this study, an efficient approach based on enhanced

differential evolution (EDE) algorithm was developed and successfully applied to solve the economic load dispatch problem with taking into account nonlinear generator features such as valve-point loading effects and multiple fuel options. An improvement of the original DE algorithm was accomplished with a modification in mutation rule that enhances its rate of convergence without compromising solution quality.

The feasibility of the proposed EDE was demonstrated on the 10-unit benchmark test system. Simulation results have demonstrated that the proposed improvement was a powerful strategy to prevent premature convergence to local minima, providing high quality solutions. Also, EDE algorithm has shown superior features such as accurate solution, stable convergence characteristic, and good computation efficiency, compared to the original DE algorithm and other heuristics reported in the literature recently.

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