Optimal power flow in direct current networks

The optimal power flow (OPF) problem seeks to control power generation/demand to optimize certain objectives such as minimizing the generation cost or power loss. Direct current (DC) networks (e.g., DC-microgrids) are promising to incorporate distributed generation. This paper focuses on the OPF problem in DC networks. The OPF problem is nonconvex, and we study solving it via a second-order cone programming (SOCP) relaxation. In particular, we prove that the SOCP relaxation is exact if there are no voltage upper bounds, and that the SOCP relaxation has at most one solution if it is exact.