This work presents a new approach to economic dispatch (ED) problems with nonsmooth cost functions using a particle swarm optimization (PSO) technique. The practical ED problems have nonsmooth cost functions with equality and inequality constraints that make the problem of finding the global optimum difficult using any mathematical approaches. A modified PSO (MPSO) mechanism is suggested to deal with the equality and inequality constraints in the ED problems. A constraint treatment mechanism is devised in such a way that the dynamic process inherent in the conventional PSO is preserved. Moreover, a dynamic search-space reduction strategy is devised to accelerate the optimization process. To show its efficiency and effectiveness, the proposed MPSO is applied to test ED problems, one with smooth cost functions and others with nonsmooth cost functions considering valve-point effects and multi-fuel problems. The results of the MPSO are compared with the results of conventional numerical methods, Tabu search method, evolutionary programming approaches, genetic algorithm, and modified Hopfield neural network approaches.

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A particle swarm optimization for economic dispatch with nonsmooth cost functions

This paper proposes a new version of the classical particle swarm optimization (PSO), namely, new PSO (NPSO), to solve nonconvex economic dispatch problems. In the classical PSO, the movement of a particle is governed by three behaviors, namely, inertial, cognitive, and social. The cognitive behavior helps the particle to remember its previously visited best position. This paper proposes a split-up in the cognitive behavior. That is, the particle is made to remember its worst position also. This modification helps to explore the search space very effectively. In order to well exploit the promising solution region, a simple local random search (LRS) procedure is integrated with NPSO. The resultant NPSO-LRS algorithm is very effective in solving the nonconvex economic dispatch problems. To validate the proposed NPSO-LRS method, it is applied to three test systems having nonconvex solution spaces, and better results are obtained when compared with previous approaches

A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems

This paper presents an improved genetic algorithm with multiplier updating (IGA/spl I.bar/MU) to solve power economic dispatch (PED) problems of units with valve-point effects and multiple fuels. The proposed IGA/spl I.bar/MU integrates the improved genetic algorithm (IGA) and the multiplier updating (MU). The IGA equipped with an improved evolutionary direction operator and a migration operation can efficiently search and actively explore solutions, and the MU is employed to handle the equality and inequality constraints of the PED problem. Few PED problem-related studies have seldom addressed both valve-point loadings and change fuels. To show the advantages of the proposed algorithm, which was applied to test PED problems with one example considering valve-point effects, one example considering multiple fuels, and one example addressing both valve-point effects and multiple fuels. Additionally, the proposed algorithm was compared with previous methods and the conventional genetic algorithm (CGA) with the MU (CGA/spl I.bar/MU), revealing that the proposed IGA/spl I.bar/MU is more effective than previous approaches, and applies the realistic PED problem more efficiently than does the CGA/spl I.bar/MU. Especially, the proposed algorithm is highly promising for the large-scale system of the actual PED operation.

Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels

Evolutionary algorithms are heuristic methods that have yielded promising results for solving nonlinear, nondifferentiable, and multi-modal optimization problems in the power systems area. The differential evolution (DE) algorithm is an evolutionary algorithm that uses a rather greedy and less stochastic approach to problem solving than do classical evolutionary algorithms, such as genetic algorithms, evolutionary programming, and evolution strategies. DE also incorporates an efficient way of self-adapting mutation using small populations. The potentialities of DE are its simple structure, easy use, convergence property, quality of solution, and robustness. This paper proposes a new approach for solving economic load dispatch problems with valve-point effect. The proposed method combines the DE algorithm with the generator of chaos sequences and sequential quadratic programming (SQP) technique to optimize the performance of economic dispatch problems. The DE with chaos sequences is the global optimizer, and the SQP is used to fine-tune the DE run in a sequential manner. The combined methodology and its variants are validated for two test systems consisting of 13 and 40 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. The proposed combined method outperforms other state-of-the-art algorithms in solving load dispatch problems with the valve-point effect.

Correction to "Combining of Chaotic Differential Evolution and Quadratic Programming for Economic Dispatch Optimization with Valve-Point Effect"

The growing costs of fuel and operation of power generating units warrant improvement of optimization methodologies for economic dispatch (ED) problems. The practical ED problems have non-convex objective functions with equality and inequality constraints that make it much harder to find the global optimum using any mathematical algorithms. Modern optimization algorithms are often meta-heuristic, and they are very promising in solving nonlinear programming problems. This paper presents a novel approach to determining the feasible optimal solution of the ED problems using the recently developed Firefly Algorithm (FA). Many nonlinear characteristics of power generators, and their operational constraints, such as generation limitations, prohibited operating zones, ramp rate limits, transmission loss, and nonlinear cost functions, were all contemplated for practical operation. To demonstrate the efficiency and applicability of the proposed method, we study four ED test systems having non-convex solution spaces and compared with some of the most recently published ED solution methods. The results of this study show that the proposed FA is able to find more economical loads than those determined by other methods. This algorithm is considered to be a promising alternative algorithm for solving the ED problems in practical power systems.

Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

Evolutionary programming has emerged as a useful optimization tool for handling nonlinear programming problems. Various modifications to the basic method have been proposed with a view to enhance speed and robustness and these have been applied successfully on some benchmark mathematical problems. But few applications have been reported on real-world problems such as economic load dispatch (ELD). The performance of evolutionary programs on ELD problems is examined and presented in this paper in two parts. In Part I, modifications to the basic technique are proposed, where adaptation is based on scaled cost. In Part II, evolutionary programs are developed with adaptation based on an empirical learning rate. Absolute, as well as relative, performance of the algorithms are investigated on ELD problems of different size and complexity having nonconvex cost curves where conventional gradient-based methods are inapplicable.

R. Chakrabarti

Papers

Evolutionary programming techniques for economic load dispatch

Fast evolutionary programming techniques for short-term hydrothermal scheduling