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

## TL;DR

This paper introduces a novel reformulation of the AC optimal power flow problem using matrix minors and develops an SOCP-based branch-and-cut method that improves solution quality and computational efficiency.

## Contribution

It proposes a new minor-based reformulation of AC OPF constraints and an SOCP relaxation with cutting planes, enhancing global optimization capabilities.

## Key findings

- Outperforms state-of-the-art SDP-based solvers in computational tests.
- Achieves an average 0.71% optimality gap within 720 seconds on challenging instances.
- Provides a scalable approach for real-time large-scale power system optimization.

## Abstract

Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today's power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a 0.71% optimality gap in no more than 720 seconds for the most challenging power system instances in the literature.