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

Abstract

Evolutionary algorithms (EAs) are general-purpose stochastic search methods that use the metaphor 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 on measure 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.

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].

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 EDP with 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. The DE 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.