Applications of Lifted Nonlinear Cuts to Convex Relaxations of the AC Power Flow Equations

TL;DR

This paper introduces lifted nonlinear cuts (LNC) to improve convex relaxations of the AC power flow equations, significantly reducing optimality gaps and enhancing computational efficiency in power system optimization.

Contribution

The study demonstrates the effectiveness of LNC in tightening SOC, CDF, and NF relaxations of the AC-OPF problem, with extensive experimental validation on standard test cases.

Findings

LNC strengthen relaxations in 100% of cases for maximization.

Average optimality gap reduction of 23.1% with maximum 93.5%.

CDF relaxation outperforms SOC in runtime and iterations.

Abstract

We demonstrate that valid inequalities, or lifted nonlinear cuts (LNC), can be projected to tighten the Second Order Cone (SOC), Convex DistFlow (CDF), and Network Flow (NF) relaxations of the AC Optimal Power Flow (AC-OPF) problem. We conduct experiments on 36 cases from the PGLib-OPF library for two objective functions, (1) power generation maximization and (2) generation cost minimization. Significant optimality gap improvements are shown for the maximization problem, where the LNC strengthen the SOC and CDF relaxations in 100% of the test cases, with average and maximum differences in the optimality gaps of 23.1% and 93.5% respectively. The NF relaxation is strengthened in 79.2% of test cases, with average and maximum differences in the optimality gaps of 3.45% and 21.2% respectively. We also study the trade-off between relaxation quality and solve time, demonstrating that the strengthened CDF relaxation outperforms the strengthened SOC formulation in terms of runtime and number of iterations needed, while the strengthened NF formulation is the most scalable with the lowest relaxation quality provided by these LNC.