  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 in the network. It is becoming increasingly important for distribution
networks, which are tree networks, due to the emergence of distributed
generation and controllable loads. In this paper, we study the OPF problem in
tree networks. The OPF problem is nonconvex. We prove that after a "small"
modification to the OPF problem, its global optimum can be recovered via a
second-order cone programming (SOCP) relaxation, under a "mild" condition that
can be checked apriori. Empirical studies justify that the modification to OPF
is "small" and that the "mild" condition holds for the IEEE 13-bus distribution
network and two real-world networks with high penetration of distributed
generation.
