Strong SOCP Relaxations for the Optimal Power Flow Problem

TL;DR

This paper introduces three novel strong SOCP relaxations for the AC optimal power flow problem, achieving near-SDP solution quality, faster computation, and practical feasibility for real-time power system operations.

Contribution

The paper presents three new strong SOCP relaxations that are incomparable to each other and outperform existing convex relaxations in solution quality and computational efficiency.

Findings

Solutions are within 99.96% of SDP relaxations on average.

Relaxations can be used as warm starts for local solvers like IPOPT.

Achieved feasible solutions for large systems within seconds, close to global optimality.

Abstract

This paper proposes three strong second order cone programming (SOCP) relaxations for the AC optimal power flow (OPF) problem. These three relaxations are incomparable to each other and two of them are incomparable to the standard SDP relaxation of OPF. Extensive computational experiments show that these relaxations have numerous advantages over existing convex relaxations in the literature: (i) their solution quality is extremely close to that of the SDP relaxations (the best one is within 99.96% of the SDP relaxation on average for all the IEEE test cases) and consistently outperforms previously proposed convex quadratic relaxations of the OPF problem, (ii) the solutions from the strong SOCP relaxations can be directly used as a warm start in a local solver such as IPOPT to obtain a high quality feasible OPF solution, and (iii) in terms of computation times, the strong SOCP relaxations can be solved an order of magnitude faster than standard SDP relaxations. For example, one of the proposed SOCP relaxations together with IPOPT produces a feasible solution for the largest instance in the IEEE test cases (the 3375-bus system) and also certifies that this solution is within 0.13% of global optimality, all this computed within 157.20 seconds on a modest personal computer. Overall, the proposed strong SOCP relaxations provide a practical approach to obtain feasible OPF solutions with extremely good quality within a time framework that is compatible with the real-time operation in the current industry practice.