The QC Relaxation: Theoretical and Computational Results on Optimal Power Flow

TL;DR

This paper systematically studies the QC relaxation for AC Optimal Power Flow, demonstrating its theoretical strength over SOC relaxation and its practical advantages in accuracy and computational efficiency compared to SDP relaxation.

Contribution

It provides the first comprehensive theoretical comparison of QC relaxation with SOC and SDP relaxations, and evaluates its computational performance on realistic power networks.

Findings

QC relaxation is stronger than SOC relaxation.

QC relaxation improves accuracy over SOC, especially with tight phase angle bounds.

QC and SOC relaxations are faster and more reliable than SDP in practice.

Abstract

Convex relaxations of the power flow equations and, in particular, the Semi-Definite Programming (SDP) and Second-Order Cone (SOC) relaxations, have attracted significant interest in recent years. The Quadratic Convex (QC) relaxation is a departure from these relaxations in the sense that it imposes constraints to preserve stronger links between the voltage variables through convex envelopes of the polar representation. This paper is a systematic study of the QC relaxation for AC Optimal Power Flow with realistic side constraints. The main theoretical result shows that the QC relaxation is stronger than the SOC relaxation and neither dominates nor is dominated by the SDP relaxation. In addition, comprehensive computational results show that the QC relaxation may produce significant improvements in accuracy over the SOC relaxation at a reasonable computational cost, especially for networks with tight bounds on phase angle differences. The QC and SOC relaxations are also shown to be significantly faster and reliable compared to the SDP relaxation given the current state of the respective solvers.