A Semidefinite Optimization-based Branch-and-Bound Algorithm for Several Reactive Optimal Power Flow Problems

The Reactive Optimal Power Flow (ROPF) problem consists in computing an optimal power generation dispatch for an alternating current transmission network that respects power flow equations and operational constraints. Some means of action on the voltage are modelled in the ROPF problem such as the possible activation of shunts, which implies discrete variables. The ROPF problem belongs to the class of nonconvex MINLPs (Mixed-Integer Nonlinear Problems), which are NP-hard problems. In this paper, we solve three new variants of the ROPF problem by using a semidefinite optimization-based Branch-and-Bound algorithm. We present results on MATPOWER instances and we show that this method can solve to global optimality most instances. On the instances not solved to optimality, our algorithm is able to find solutions with a value better than the ones obtained by a rounding algorithm. We also demonstrate that applying an appropriate clique merging algorithm can significantly speed up the resolution of semidefinite relaxations of ROPF large instances.