A Lagrange-Based Multi-Objective Framework for Wind–Thermal Economic Emission Dispatch

Economic dispatch using wind power plants plays a role in reducing the price of electricity production by dispatching power among different generating units for thermal and wind power plants, and supplying load demand while meeting the power system equality and inequality constraints. Adding wind power plants to the economic dispatch model can significantly reduce electricity production costs and reduce carbon dioxide emissions. In this paper, fuel cost and emission minimization are considered as the objective function of the economic dispatch problem, taking into account transmission loss using the B matrix. The quadratic model of the fuel cost and emission criterion functions is modeled without considering a valve-point loading effect. The real power generation limits for both wind and conventional generating units are considered. In addition, a closed-form expression based on the incomplete gamma function is provided to define the impact of wind power, which includes the cost of wind energy, including overestimation and underestimation of available wind power using a Weibull-based probability density function. In this research work, Lagrange&rsquo:s algorithm is proposed to solve the Wind&ndash:Thermal Economic Emission Dispatch (WTEED) problem. The developed Lagrange classical optimization algorithm for the WTEED problem is validated using the IEEE test systems with 6-, 10-, and 40-generation unit systems. The proposed Lagrange optimization method for WTEED problem solutions demonstrates a notable improvement in both economic and environmental performance compared to other heuristic optimization methods reported in the literature. Specifically, the fuel cost was reduced by an average of 4.27% in the IEEE 6-unit system, indicating more economical power dispatch. Additionally, the emission cost was lowered by an average 22% in the IEEE 40-unit system, reflecting better environmental compliance and sustainability. These results highlight the effectiveness of the proposed approach in achieving a balanced trade-off between cost minimization and emission reduction, outperforming several existing heuristic techniques such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE) under similar test conditions. The research findings report that the proposed Lagrange classical method is efficient and accurate for the convex wind&ndash:thermal economic emission dispatch problem.