Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials

TL;DR

This paper enhances convex relaxations for the AC optimal power flow problem by introducing voltage difference variables and improved envelopes for trilinear monomials, resulting in tighter bounds and smaller optimality gaps.

Contribution

It proposes two novel methods—voltage difference constraints and Meyer-Floudas envelopes—to improve QC relaxations of AC-OPF problems.

Findings

Smaller optimality gaps compared to existing QC methods.

Significant bound tightening on voltage difference variables.

Enhanced relaxation tightness leads to better solution quality.

Abstract

AC optimal power flow (AC~OPF) is a challenging non-convex optimization problem that plays a crucial role in power system operation and control. Recently developed convex relaxation techniques provide new insights regarding the global optimality of AC~OPF solutions. The quadratic convex (QC) relaxation is one promising approach that constructs convex envelopes around the trigonometric and product terms in the polar representation of the power flow equations. This paper proposes two methods for tightening the QC relaxation. The first method introduces new variables that represent the voltage magnitude differences between connected buses. Using "bound tightening" techniques, the bounds on the voltage magnitude difference variables can be significantly smaller than the bounds on the voltage magnitudes themselves, so constraints based on voltage magnitude differences can tighten the relaxation. Second, rather than a potentially weaker "nested McCormick" formulation, this paper applies "Meyer and Floudas" envelopes that yield the convex hull of the trilinear monomials formed by the product of the voltage magnitudes and trignometric terms in the polar form of the power flow equations. Comparison to a state-of-the-art QC implementation demonstrates the advantages of these improvements via smaller optimality gaps.