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Top1. Introduction
The electrical load demand is increasing due to the continuous growth of industries at the global level (Bhattacharjee 2018). Therefore, the power system economics become a major and urgent problem to supply continues changing load (Jatin M. Soni et al. 2020; J.M. Soni and Pandya 2018). Moreover, a huge amount of fuel gases like nitrogen oxides (NOX), sulfur oxides (SOX), etc. have been emitted into the atmosphere by traditional thermal power plants (Bhattacharjee, Bhattacharya, and nee Dey 2013). Therefore, the emission of pollutants must be added to the economic dispatch optimization problem to minimize fuel cost as well as the emission of fuel gases. Hence, the Dynamic Economic Emission Dispatch (DEED) problem has been considered as a multi-objective optimization problem to reduce fuel cost and pollutant emission by satisfying various operating constraints like generator operating constraints, ramp-rate limit, and power balance constraints (Bhattacharjee, Shah, and Soni 2022). The electrical load is changing with respect to time in the DEED problem. Therefore, the main aim of the DEED is to allocate electrical load among the thermal generators during the dispatch period (Bhattacharjee, Bhattacharya, and nee Dey 2014). The DEED helps to enhance power system economics and eco-friendliness. The entire time period has been divided into 24 h cycles (Bhattacharjee and Patel 2018). Therefore, the output power of generators changes with load demand in the dispatching period (Bhattacharjee , Bhattacharya, and Nee Dey 2014).
The objective function of the DEED problem becomes highly complex because the emission and fuel cost functions are non-linear, non-convex, and non-smooth (Wang et al. 2021). The output curve of the generating units becomes non-smooth due to the presence of multiple valves in the steam turbine. The quadratic objective function changes to the summation of a sine function and quadratic function due to the Valve Point Loading Effect (VPLE) (Verma et al. 1AD). The complexity of the objective function increases with increasing number of connected thermal generators in the power system. The mathematical model requires initial values to get nearly optimum solution (Bhattacharjee, Shah, and Soni 2022). Therefore, intelligent optimization methods have been developed to solve the DEED problem (Jatin M. Soni and Pandya 2018). The intelligent methods have not been affected by the non-differentiability and the dimension of the problem. Classical optimization methods like Linear Programming (LP) (Basak, Bhattacharyya, and Dey 2022), Particle Swarm Optimization (PSO) (Hamedi 2013), Genetic Algorithm (GA) (Sumit Banerjee, Maity, and Chanda 2015), and Simulated Annealing (SA) (Kumarappan and Suresh 2015) have been used to solve DEED problem. The classical methods have reduced fuel cost which leads increasing emission of pollutants. Hence, the optimum solution cannot be obtained by classical optimization methods (Dasgupta, Roy, and Mukherjee 2021). The SA method has better convergence ability in iteration process. The PSO method has given satisfactory solution in solving the DEED problem. However, the classical methods trap into local solution in solving non-smooth and non-convex DEED problem (Basak, Bhattacharyya, and Dey 2022).