Pareto-based Multi-objective Firefly Algorithm with Hierarchical Clustering for Environmental-Economic Power Dispatch Problem

This paper proposes a multi-objective firefly algorithm (MOFFA) to solve the environmental-economic power dispatch (EEPD) problem. Also, it presents a modified firefly algorithm (FFA) to solve the economic power dispatch (EPD) and the emission dispatch (ED) problems as single goals. The modifications proposed on the traditional FFA aim to improve its exploration and ensure the feasibility of the obtained solutions. The proposed MOFFA uses an external Pareto set to keep the non-dominated solutions, where a hierarchical clustering algorithm is used to get a representative and controlled set of Pareto-optimal solutions. The best compromise solution is also extracted from the Pareto set using an approach based on the fuzzy set theory. The constraints of EEPD, EPD, and ED problems are power balance constraint, generation limits constraint, and transmission power losses. To verify the effectiveness of the proposed algorithms, two methodologies are adopted and tested on the IEEE 30-bus test and the 10-unit test system with valve-point loading. In methodology 1, the EPD problem and the ED problem are solved separately using the modified FFA. In methodology 2, the EEPD is solved as a true multi-objective optimization problem using the proposed MOFFA. The simulation results and the statistical analysis ensure the high-quality solutions of the proposed algorithms and prove the ability of the MOFFA to produce well-distributed Pareto-optimal solutions.