Modified differential evolution algorithm for optimal power flow with non-smooth cost functions

Abstract

Differential evolution (DE) is a simple but powerful evolutionary optimization algorithm with continually outperforming many of the already existing stochastic and direct search global optimization techniques. DE algorithm is a new optimization method that can handle non-differentiable, non-linear, and multimodal objective functions. This paper presents an efficient modified differential evolution (MDE) algorithm for solving optimal power flow (OPF) with non-smooth and non-convex generator fuel cost curves. Modifications in mutation rule are suggested to the original DE algorithm, that enhance its rate of convergence with a better solution quality. A six-bus and the IEEE 30 bus test systems with three different types of generator cost curves are used for testing and validation purposes. Simulation results demonstrate that MDE algorithm provides very remarkable results compared to those reported recently in the literature.

Introduction

Optimal power flow (OPF) is one of the main tools for optimal operation and planning of modern power systems. The OPF is, hence, the basic tool that allows electric utilities to determine secure and economic operating conditions for an electric power system. An OPF adjusts the controllable quantities in the system to optimize an objective function, while satisfying a set of physical and operational constraints. This makes the OPF problem a large-scale highly non-linear constrained optimization problem.

The OPF problem has been solved via many traditional optimization methods such as linear programming, non-linear programming, quadratic programming, Newton-based techniques and interior point methods. A comprehensive review of various optimization techniques available in the literature is reported in Refs. [1], [2]. Usually, these methods rely on the assumption that the fuel cost characteristic of a generating unit is a smooth, convex function. However, there are situations where it is not possible, or appropriate, to represent the unit’s fuel cost characteristic as a convex function. For example, this situation arises when valve-points, unit prohibited operating zones, or multiple fuels are present. Hence, the true global optimum of the problem could not be reached easily. New numerical methods are then needed to cope with these difficulties, specially, those with high speed search to the optimal and not being trapped in local minima.

In recent years, many heuristic algorithms such as genetic algorithms (GA) [3], [4], evolutionary programming (EP) [5], [6], tabu search (TS) [7], particle swarm optimization (PSO) [8] and simulated annealing (SA) [9], have been proposed to solve the OPF problem, without any restrictions on the shape of the cost curves. The results reported were promising and encouraging for further research in this direction.

Recently, a new evolutionary computation technique, called differential evolution (DE), has been developed and introduced by Storn and Price [10]. DE algorithm is a stochastic population-based search method successfully applied in global optimization problems. DE combines simple arithmetic operators with the classical operators of crossover, mutation and selection to evolve from a randomly generated starting population to a final solution [11], [12].

This paper presents an efficient modified differential evolution (MDE) algorithm for solving optimal power flow (OPF) with non-smooth cost functions. Modifications in mutation rule are suggested to the original DE algorithm that explores the solution space with a random localisation, enhancing its rate of convergence for a better solution quality. In order to demonstrate the suitability of the proposed approach, MDE algorithm was applied to the six-bus and IEEE 30 bus test systems with three different types of generator cost curves. Simulation results demonstrate that MDE algorithm is superior to the original DE and appears to be fast providing very remarkable results compared to those reported in the literature recently.