Equivalency of Continuation and Optimization Methods to Determine Saddle-Node and Limit-Induced Bifurcations in Power Systems

This paper presents a comprehensive and detailed study of an optimization-based approach to identify and analyze saddle-node bifurcations (SNBs) and limit-induced bifurcations (LIBs) of a power system model, which are known to be directly associated with voltage stability problems in these systems. Theoretical studies are presented, formally demonstrating that solution points obtained from an optimization model, which is based on complementarity constraints used to properly represent generators' voltage controls, correspond to either SNB or LIB points of this model. These studies are accomplished by proving that optimality conditions of these solution points yield the transversality conditions of the corresponding bifurcation points. A simple but realistic test system is used to numerically illustrate the theoretical discussions.