Algorithms for mathematical programs with equilibrium constraints with applications to deregulated electricity markets

Mathematical programs with equilibrium constraints (MPECs) are optimization problems with some constraints defined in terms of complementarity systems. Important applications of these problems arise in engineering design problems of mechanical structures, economic models, and option pricing. We have developed a new algorithm for MPECs, which we apply to solve novel economic models of deregulated electricity markets. It can be shown that constraint qualifications typically assumed to prove convergence of standard algorithms fail to hold for MPECs. As a result, applying standard algorithms is problematic. To circumvent these problems, various reformulations of MPECs have been proposed. One of these approaches involves the use of smoothing functions with favorable properties to substitute for the complementarity constraints. We investigate a new sequential quadratic programming algorithm for equilibrium-constrained optimization (ECOPT) based on such a smooth reformulation. The algorithm employs a specialized termination criterion as well as update rules for the Lagrangian Hessian. Numerical tests on standard test problems show its performance is superior to that of state of the art nonlinear optimization algorithms as well as some other algorithms specifically designed to solve MPEC problems. We also present a new mathematical model of electricity forward markets. The lack of working forward markets in electricity has been identified as one of the main obstacles to current deregulation efforts. Our new model incorporates a Cournot equilibrium for the spot market and considers actions by producers in the forward market. The mathematical model is an instance of an MPEC. Using ECOPT to solve the producer's problem, one can find Nash equilibria in the forward market. The application of the model to a six-node network with two competing producers reveals a fundamental relationship between transmission capacity and forward markets. We also demonstrate how to apply the model to gain a better understanding of transmission investment decisions in deregulated electricity markets.