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

• 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.

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.