A normalization method for solving the combined economic and emission dispatch problem with meta-heuristic algorithms

Highlights

• A normalized objective function is proposed to avoid unit and scale difference.

• Eight metaheuristic algorithms are employed to solve the ED, MED, and CEED problems.

• Four benchmark instances with different number of generating units are tested to verify algorithm performance.

• VOA performs the best considering both solution quality and computational efficiency.

Abstract

Solving the economic and emission dispatch (ED/MED) problems separately becomes more complex when the combined version (CEED) of the two aforementioned cases is considered. The basic idea is to achieve the lowest possible cost with the smallest amount of pollutant and this problem is known as the combined economic–emission dispatch (CEED). A new approach for solving the CEED is presented in this paper. The idea consists of normalizing the two conflicting objective functions, ED and MED, using the mean and standard deviation of the members contained in the population-based meta-heuristic algorithms implemented in this study thus preventing units and scale differences when optimizing the CEED problem. The mathematical model for each problem (ED, MED, and CEED) presented in this study is optimized implementing a nonlinear optimization package named TOMLAB available for MATLAB, which helps us to determine the best possible solution for the tested instances. A novel meta-heuristic named Virus Optimization Algorithm (VOA) is implemented along with seven well-known algorithmic tools, which are the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Harmony Search (HS), Differential Evolution (DE), FireFly algorithm (FF), Gravitational Search Algorithm (GSA), and Seeker Optimization Algorithm (SOA). A comprehensive statistical study is performed to determine the quality of the solutions delivered by the algorithmic tools when compared with TOMLAB. From the test instances, it was observed that the proposed normalization method does not only show to be feasible, but also helps the algorithms to achieve similar results from that coming when solving the ED and MED separately. Furthermore, among the eight meta-heuristic tools (VOA, GA, PSO, HS, DE, FF, GSA, and SOA), VOA showed outstanding performance in both solution quality and computational efficiency.

Introduction

Finding the best possible allocation of power among the committed generating thermal units in order to satisfy a given demand is the basic idea of the economic dispatch (ED) problem [1], [2], [3], [4]. The minimization of the operating cost is the main objective of the ED problem, where each generating unit is subject to physical and technological constraints. However, given that thermal power generation is a major cause of atmospheric pollution, the minimization of the operating cost is no longer the only concern when satisfying power demand. Consequently, the emission of pollutants such as nitrogen and sulfur oxides (NOx and SOx) has to be minimized, where this problem is known as the minimum emission dispatch (MED) [5], [6], [7].

Due to the nature of the ED and MED problems, finding a high-quality solution that not only provides a reasonable operating cost in ($/h), but also reduces the emission of pollutants (NOx and SOx) in (ton/h) is a very difficult task. Therefore, implementation of exact optimization methods [8], [9] is impractical, since they required enormous computational effort and sometimes are unable to find a solution to the problem under study. The aforementioned problem is known as the combined economic–emission dispatch (CEED) problem, where several approaches have been proposed for solving the CEED problem [10], [11], [12], [13] whereas, in recent years meta-heuristic algorithms have gained huge popularity [14], [15], [16], [17], [18], [19], [20].

The objectives of this paper are to solve the ED and MED problems independently, to later solve and compare the results with that from the combined economic–emission dispatch problem. Implementing and comparing a nonlinear optimization package for MATLAB along with eight meta-heuristic algorithms will help us to optimize the mathematical model proposed in this study. The algorithmic tools are Virus Optimization Algorithm (VOA) [21], Genetic Algorithm (GA) [22], Particle Swarm Optimization (PSO) [23], Harmony Search (HS) [24], Differential Evo3ution (DE) [25], FireFly algorithm (FF) [26], Gravitational Search Algorithm (GSA) [27], and Seeker Optimization Algorithm (SOA) [28]. The second goal and main contribution of this paper, is to further investigate a newly proposed approach for the objective function evaluation when solving the CEED problem using population-based algorithms as in [16], throughout a more comprehensive and detailed testing process. Therefore, not only showing the feasibility of the proposed approach but also its robustness when dealing with a variety of instances implemented on different optimization algorithms to reduce the computational effort.

The approach as in [29] avoids differences in units and scale when combining the ED and MED; therefore, penalty factors and/or multi-objective optimization [30], [31], [6] are not required. The major difference between this paper and the one presented in [29] is that, the robustness of the proposed method (normalization) is extensively studied with a variety of instances and population based algorithmic tools. The aforementioned, gives more insides in the behavior and power of the proposed normalization method.

Section 2 of this paper introduces the necessary background for the ED, MED, and CEED problems, respectively. Section 3 details the approach proposed as in [29], where the normalized objective function to be optimized by the VOA, GA, PSO, HS, DE, FF, GSA, and SOA is introduced as well. The computational results for the test instances are shown in Section 4; where the ED, MED and CEED are analyzed. Lastly, conclusions and future research directions are addressed in Section 5.