Analytical Solutions for Economic Dispatch via Wolfe’s Duality and Exponential Convex Model

Optimization plays a fundamental role in applied sciences. With the advancement of computational tools, numerical (non-analytical) methods for optimization have progressed considerably. However, such methods often face challenges related to computational accuracy, convergence time, and reliance on sophisticated hardware. These issues are particularly relevant in electrical energy production optimization. In power system economic dispatch (ED), the objective is to minimize total generation cost by coordinating multiple electrical generators—a problem that is inherently mathematical. Ensuring fast convergence and reduced computational time is essential. Traditionally, the production cost is modeled as a quadratic function of generator output, assuming a linearly increasing cost rate. However, for some generators, as efficiency declines, the cost rate increases non-linearly. Consequently, Quadratic Models yield sub-optimal solutions. In this study, we propose a new cost model based on an exponential function to better capture the nonlinear cost behavior. For the first time, an analytical solution using Wolfe’s duality is derived for ED with exponential cost functions. This approach enhances accuracy and reduces computational overhead. Simulation results show that our method matches the accuracy of the exponential Lambda Iteration method while reducing execution time by 26.6%. It is especially suitable for real-time applications such as Optimal Power Flow (OPF) in power distribution systems.