DC optimal power flow: Uniqueness and algorithms

The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network nodes, and is generally hard to solve. We study the direct current special case by leveraging recent developments on the zero duality gap of OPF. We study the uniqueness of the OPF solution using differential topology especially the Poincare-Hopf Index Theorem, and characterize its global uniqueness for simple network topologies, e.g., line and mesh networks. This serves as a starting point to design local algorithms with global behavior that have low complexity and are computationally fast for practical smart power grids.