Adaptive particle swarm optimization approach for static and dynamic economic load dispatch

Abstract

This paper presents a novel heuristic optimization approach to constrained economic load dispatch (ELD) problems using the adaptive–variable population – PSO technique. The proposed methodology easily takes care of different constraints like transmission losses, dynamic operation constraints (ramp rate limits) and prohibited operating zones and also accounts for non-smoothness of cost functions arising due to the use of multiple fuels. Simulations were performed over various systems with different numbers of generating units, and comparisons are performed with other existing relevant approaches. The findings affirmed the robustness, fast convergence and proficiency of the proposed methodology over other existing techniques.

Introduction

Among the different issues in power system operation, the economic load dispatch (ELD) problem lies at the kernel [1], [2]. Essentially, the ELD problem is a constrained optimization problem in power systems that have the objective of dividing the total power demand among the on-line participating generators economically while satisfying the various constraints. Over the years, many efforts have been made to solve the problem, incorporating different kinds of constraints or multiple objectives, through various mathematical programming and optimization techniques. The conventional methods include the lambda iteration method [3], [4], base point and participation factors method [3], [4], gradient method [3], [5], etc. Among these methods, the lambda iteration is the most common one, and owing to its ease of implementation, it has been applied through various software packages to solve ELD problems. However, for effective implementation of this method, the formulation needs to be continuous. The basic ELD considers the power balance constraint apart from the generating capacity limits. However, a practical ELD must take ramp rate limits, prohibited operating zones, valve point loading effects, and multi-fuel options [6] into consideration to provide completeness for the ELD problem formulation. The resulting ELD is a non-convex optimization problem, which is a challenging one and cannot be solved by the traditional methods. An ELD problem with valve point loading has also been solved by dynamic programming (DP) [7], [8]. Though promising results are obtained in small sized power systems while solving it with DP, it unnecessarily raises the length of the solution procedure, resulting in its vulnerability to solve large size ELD problems in stipulated time frames.

Moreover, evolutionary and behavioral random search algorithms such as genetic algorithm (GA) [9], [10], [11], particle swarm optimization (PSO) [12], [13], etc. have previously been implemented on the ELD problem at hand. In addition, an integrated parallel GA incorporating ideas from simulated annealing (SA) and tabu search (TS) techniques was also proposed in Ref. [14] utilizing the generator’s output power as the encoded parameter. Yalcinoz et al. have used a real coded representation technique along with arithmetic genetic operators and elitist selection to yield a quality solution [15]. GAs have been deployed to solve the ELD problem with various modifications over the years. In a similar attempt, a unit independent encoding scheme has also been proposed based on an equal incremental cost criterion [16]. In spite of its successful implementation, GAs do possess some weaknesses leading to longer computation time and less guaranteed convergence, particularly in the case of an epistatic objective function containing highly correlated parameters [17], [18]. Moreover, premature convergence of GAs is accompanied by a very high probability of entrapment in local optima [19]. Some other hybrid approaches, like GA combined with simulated annealing (SA) [20], evolutionary programming (EP) [21], improved tabu search (ITS) [22], improved fast EP (IFEP) [23], and evolutionary strategy optimization (ESO) [24] have been successfully applied to solve the ELD problem. Besides these soft computing methodologies, some other promising techniques, like Hopfield neural networks [25], [26] and two phase neural network [27], have been successfully applied to solve the constrained ELD.

This paper proposes a new optimization approach, to solve the ELD problem using an adaptive particle swarm optimization (APSO) technique. As mentioned earlier, to solve the proposed problem of economic load dispatch, many researchers have suggested different modified versions of the PSO, and the results are reported in some of the papers [33], [34], [35], [36]. In this paper, an attempt has been made to solve both the static and dynamic economic load dispatch problem using APSO methodology. In order to establish the capability of APSO to optimize smooth as well as non-smooth cost functions, this paper considers two alternative models, one being a constrained quadratic cost function with generator constraints, power loss and ramp rate limits with the other being an unconstrained non-smooth cost function with multiple fuels. The APSO is tested on three generator, six generator and fifteen generator test systems in the case of smooth cost curves, and for 3 generator and 10 generator systems in the case of multiple fuels. The results obtained are compared with those of GA, PSO and other promising methods. The proposed methodology was also applied to solve a dynamic load dispatch problem as reported in Ref. [37]. The proposed methodology emerges as a robust optimization technique for solving the ELD problem for various cost curve natures and different size power systems.