Approximating the ACOPF problem with a hierarchy of SOCP problems

Semidefinite programming (SDP) relaxations for the Alternating Current Optimal Power Flow (ACOPF) problem have been shown to be tight for well studied problem instances. However, due to the computational demands of SDP, it becomes difficult to use SDP relaxations to approximate large-scale instances of the ACOPF problem. Recently, computationally cheaper second-order cone relaxations have been proposed for the ACOPF problem that are tight for networks with a simple topology. In this paper, we exploit recent results in polynomial optimization to construct a hierarchy of second-order cone relaxations that provide increasingly better approximations for ACOPF problems. We show that in comparison with proposed related SDP hierarchies, the second-order cone hierarchies provide good approximations to the ACOPF problems for larger scale networks. We illustrate this with numerical examples on well studied instances of the ACOPF problem.