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The optimal power flow (OPF) problem seeks to control power generation/demandto optimize certain objectives such as minimizing the generation cost or powerloss in the network. It is becoming increasingly important for distributionnetworks, which are tree networks, due to the emergence of distributedgeneration and controllable loads. In this paper, we study the OPF problem intree networks. The OPF problem is nonconvex. We prove that after a "small"modification to the OPF problem, its global optimum can be recovered via asecond-order cone programming (SOCP) relaxation, under a "mild" condition thatcan be checked apriori. Empirical studies justify that the modification to OPFis "small" and that the "mild" condition holds for the IEEE 13-bus distributionnetwork and two real-world networks with high penetration of distributedgeneration.

Exact Convex Relaxation of Optimal Power Flow in Tree Networks