Economic dispatch using particle swarm optimization with bacterial foraging effect Ahmed Yousuf Saber Research and Development Department, Operation Technology Inc., ETAP, Irvine, CA 92618, USA article info Article history: Received 7 June 2008 Received in revised form 22 August 2011 Accepted 5 September 2011 Available online 1 November 2011 Keywords: Particle swarm optimization Bacterial foraging technique Random walk Swimming Dynamic economic dispatch Local minima abstract This paper presents a novel modified particle swarm optimization (MPSO), which includes advantages of bacterial foraging (BF) and PSO for constrained dynamic economic dispatch (ED) problem. The proposed modified PSO consists of problem dependent four promising values in velocity vector to incorporate repellent advantage of bacterial foraging in PSO for the complex dynamic ED problem. It reliably and accurately tracks a continuously changing solution of the complex cost functions. As there is no differen- tiation operation in this method, all cost functions can easily be handled. The modified PSO has better balance between local and global search abilities and it can avoid local minima quickly. Finally, a bench- mark data set and existing methods are used to show the effectiveness of the proposed method. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Economic dispatch (ED) is one of the important optimization problems in power system that distributes instant and/or dynamic energy demand among online available energy resources econom- ically while satisfying various system constraints. An optimal ED should meet load demand, generation limit, ramp rate, prohibited operating zone, etc. considering network losses at every time inter- val such that the total cost is minimum. The problem is more com- plex when dynamically changing load is considered in dynamic ED problems. It is the main complex computational intensive part of unit commitment problems [1]. Therefore, dynamic ED is one of the most important problems in power system optimization area. Various numerical optimization techniques have been em- ployed to approach the ED problem. ED is solved traditionally using mathematical programming based on optimization tech- niques such as lambda iteration, gradient method, dynamic pro- gramming (DP) and so on [2–6]. Generator units usually have several non-linear characteristics, such as discontinuous opera- tional zones, ramp rate limits, and non-smooth or convex cost functions which render exact mathematical methods infeasible for complex dynamic ED problems. Since the global optima of ED problems are usually very difficult to achieve mathematically, a large number of meta-heuristic meth- ods have been developed in order to solve them. Among these methods, some of them are genetic algorithm (GA) [7,8], evolution- ary programming (EP) [9,10], tabu search [11], hybrid EP [12], neu- ral network (NN) [13], adaptive Hopfield neural network (AHNN) [14], particle swarm optimization (PSO) [15–23], bacterial foraging (BF) [24–27], etc. For calculation simplicity, existing methods use second order fuel cost function, which involves approximation; and constraints are handled separately. The author proposes higher order cost functions for (a) better curve fitting of fuel costs, (b) less approximation, (c) more practical, accurate and reliable results, and a modified particle swarm optimization with bacterial foraging is introduced to solve the dynamic dispatch of the complex cost functions with practical constraints. The method is generic and fuel cost functions does not affect its execution complexity. Typically lambda iteration, gradient method [2–4] can solve simple ED calculations, which are not sufficient for real world com- plex applications. However, they are fast. Intelligent optimization methods [7–27] are general purpose and thus they have random- ness/blindness for a particular problem. For a specific problem, like ED, the intelligent methods should be modified accordingly, which are suitable to solve efficiently dynamic economic dispatch prob- lems with complex cost functions and practical constraints. Some general optimization software packages are also available for modeling the ED [28–30]. However, software companies urge state-of-the-art technologies to continuously update their prod- ucts in competitive market. Swarm optimization is very popular in recent days because it has information sharing and conveying mechanisms. Among swarm optimization methods, bacterial foraging and particle swarm optimization are very promising and recently they are ap- plied in ED [31–35]. Each method has different set of advantages and disadvantages regarding local minima, randomness, direction of movement, attraction/repulsion, etc. The proposed method 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.09.003 E-mail address: aysaber@ieee.org Electrical Power and Energy Systems 34 (2012) 38–46 Contents lists available at SciVerse ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes