The uniqueness of the local minimum for power economic dispatch problems

The functions of the cost curve and transmission losses are nonlinear and simultaneously subject to equality and inequality constraints in the economic power dispatch problem. Mathematically, the solution of the nonlinear equations of the economic power dispatch problem may result in multiple solutions. Conventional methods may fail to find all the multiple solutions if the initial guess values are not appropriately located within the region of convergency. To overcome this difficulty, the homotopy method is suggested for the solution technique. Concurrently, the properties of the convex function are applied to analyze the intersection equation of the objective equation and the equality constraints equation. Hence we can demonstrate that only one local minimum exists in the augmented cost function, so that the local minimum is the optimal solution of the economic power dispatch problem.