A Computational Algebraic Geometry Based Global Optimization Technique to Address Economic Dispatch

In algebraic geometry, the concept of Grobner basis allows a systematic study of the solution of a system of polynomial equations. This concept can be applied to find the global (and all local optima) optimum of a nonlinear, not necessarily convex function, the only restriction being that the objective function be polynomial. The method is based on computing a lexicographic (lex) ordered Grobner basis for the ideal generated by the first order necessary conditions defined by the Lagrangian. Computing the optimal solution is then equivalent to computing the variety corresponding to this ideal. By virtue of the (lex) ordering, the system is transformed in to set of polynomials which can be solved successively to obtain the solutions. Here, we illustrate the application of the method on a non-convex function and identify the global optimum from the set of fifteen stationary points (6 local minima, 2 local maxima and 7 saddles). Then we apply the method to solve the classical economic dispatch problem including a combined cycle heat plant (CCHP) whose piecewise linear cost function is approximated by a smooth tenth order polynomial. Interestingly, the the method yields two possible solutions from which the least cost solution can be picked. While the work reported here is only preliminary, we find the results encouraging and hope that the method will find applicability in identifying the global optimum of non-convex power systems optimization problems.