Dynamic Relationship Between KKT Saddle Solutions and Optimal Solutions in AC OPF Problems

Abstract

Many KKT condition-based methods for solving AC OPF models can compute KKT-saddle solutions that are feasible solutions but not local optimal solutions. This paper explores a dynamic relationship between KKT-saddle solutions and optimal solutions in AC OPF problems and presents a “buy one, get two free” theorem by showing that if one KKT saddle is computed, then this KKT saddle is dynamically connected to two local optimal solutions. In addition, it is shown that by following the unstable manifold of the computed KKT saddle, one can easily compute two local optimal solutions. This “buy one, get two free” property occasionally degenerates into a “buy one, get one free” property when the unstable equilibrium point (UEP) associated with the computed KKT saddle does not lie on a quasi-stability boundary. Numerical demonstration of these two properties is conducted on several test systems including the IEEE 118-bus system. The robustness of these two properties under the perturbations of AC OPF models is analytically and numerically shown.