On the Existence of and Lower Bounds for the Number of Optimal Power Flow Solutions

An optimal power flow (OPF) problem can have one or multiple locally optimal solutions. It can also admit no solution. Few studies have addressed the existence of OPF solutions and derived a (tight) lower bound for the number of OPF solutions. The present paper is devoted to the analytical aspects of OPF solutions. Specifically, a necessary and sufficient condition for the existence of an OPF solution is presented and a tight lower bound for the number of OPF solutions is derived; in other words, an existence theorem is developed for the OPF solution and a computable lower bound for the number of solutions is derived by investigating a dynamical system called the quotient gradient system (QGS), which is built from the set of constraints. It is shown that an OPF solution exists if and only if the QGS has a regular stable equilibrium manifold (SEM) and the number of OPF solutions is not less than the number of regular SEMs for the QGS. Two test systems with 9 and 118 buses, respectively, are evaluated, and the numerical results are summarized to validate the presented analytical results and illustrate that multiple OPF solutions can lie in the same SEM.