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Alternating current optimal power flow (AC OPF) is one of the mostfundamental optimization problems in electrical power systems. It can beformulated as a semidefinite program (SDP) with rank constraints. Solving ACOPF, that is, obtaining near optimal primal solutions as well as high qualitydual bounds for this non-convex program, presents a major computationalchallenge to today's power industry for the real-time operation of large-scalepower grids. In this paper, we propose a new technique for reformulation of therank constraints using both principal and non-principal 2-by-2 minors of theinvolved Hermitian matrix variable and characterize all such minors into threetypes. We show the equivalence of these minor constraints to the physicalconstraints of voltage angle differences summing to zero over three- andfour-cycles in the power network. We study second-order conic programming(SOCP) relaxations of this minor reformulation and propose strong cuttingplanes, convex envelopes, and bound tightening techniques to strengthen theresulting SOCP relaxations. We then propose an SOCP-based spatialbranch-and-cut method to obtain the global optimum of AC OPF. Extensivecomputational experiments show that the proposed algorithm significantlyoutperforms the state-of-the-art SDP-based OPF solver and on a simple personalcomputer is able to obtain on average a 0.71% optimality gap in no more than720 seconds for the most challenging power system instances in the literature.

Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem