Improved differential evolution approach based on cultural algorithm and diversity measure applied to solve 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.
Section snippets
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 constraints 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:In the power balance criterion, an equality constraint must be satisfied, as shown in Eq. (1). The generated
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 new population. Individuals are altered using unary
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 optimization method was implemented in Matlab (MathWorks) using Microsoft Windows XP. All the programs were run on a 3.2 GHz Pentium IV processor with 2 GB 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
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 optimization. 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.
In this paper, traditional DE, CDE, and CDEMD
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 (37)
- et al.
A multipopulation cultural algorithm using fuzzy clustering
Applied Soft Computing
(2007) - et al.
A self-adaptive differential evolution heuristic for two-stage assembly scheduling problem to minimize maximum lateness with setup times
European Journal of Operational Research
(2007) - et al.
Application of pattern search method to power system valve-point economic load dispatch
Electrical Power and Energy Systems
(2007) - et al.
Improved differential evolution algorithms for handling economic dispatch optimization with generator constraints
Energy Conversion and Management
(2007) - et al.
A multipopulation cultural algorithm for the electrical generator scheduling problem
Mathematics and Computers in Simulation
(2002) - et al.
A Cauchy-based evolution strategy for solving the reactive power dispatch problem
Electrical Power and Energy Systems
(2002) - et al.
H∞ controller design for a distillation column using genetic algorithms
Mathematics and Computers in Simulation
(2002) - et al.
Repel the swarm to the optimum
Applied Mathematics and Computation
(2006) - et al.
Hybrid simplex genetic algorithm for blind equalization using RBF networks
Mathematics and Computers in Simulation
(2002) - et al.
Differential evolution for economic load dispatch problems
Electric Power Systems Research
(2008)