Penalized Parabolic Relaxation for Optimal Power Flow Problem

TL;DR

This paper introduces a novel parabolic relaxation method for the optimal power flow problem, transforming it into convex QCQPs and using penalization to recover feasible, near-optimal solutions efficiently, outperforming traditional relaxations.

Contribution

It proposes a new parabolic relaxation technique and a compatible penalization method for OPF, providing an effective alternative to SDP and SOCP relaxations.

Findings

Successfully solved large-scale benchmark systems with up to 13659 buses.

Obtained solutions within 0.32% of the best-known solutions.

Demonstrated the effectiveness of the penalized relaxation approach through extensive experiments.

Abstract

This paper is concerned with optimal power flow (OPF), which is the problem of optimizing the transmission of electricity in power systems. Our main contributions are as follows: (i) we propose a novel parabolic relaxation, which transforms non-convex OPF problems into convex quadratically-constrained quadratic programs (QCQPs) and can serve as an alternative to the common practice semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, (ii) we propose a penalization technique which is compatible with the SDP, SOCP, and parabolic relaxations and guarantees the recovery of feasible solutions for OPF, under certain assumptions. The proposed penalized convex relaxation can be used sequentially to find feasible and near-globally optimal solutions for challenging instances of OPF. Extensive numerical experiments on small and large-scale benchmark systems corroborate the efficacy of the proposed approach. By solving a few rounds of penalized convex relaxation, fully feasible solutions are obtained for benchmark test cases from [1]-[3] with as many as 13659 buses. In all cases, the solutions obtained are not more than 0.32% worse than the best-known solutions.