# Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility

## TL;DR

This paper investigates the limitations of convex relaxations in AC optimal power flow problems, analyzing solution quality, proposing new metrics, and evaluating methods for improving AC feasibility in large-scale systems.

## Contribution

It provides a detailed analysis of inexact convex relaxations, introduces new metrics for solution quality, and assesses feasibility recovery methods across extensive test cases.

## Key findings

- Many solutions with low optimality gaps still lack AC feasibility.
- Penalization methods often fail to recover feasible solutions in large systems.
- Warm-starting non-convex solvers can significantly reduce computation time in most cases.

## Abstract

Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted significant interest as in several instances they provably yield the global optimum to the original non-convex problem. If, however, the relaxation is inexact, the obtained solution is not AC-feasible. The quality of the obtained solution is essential for several practical applications of AC-OPF, but detailed analyses are lacking in existing literature. This paper aims to cover this gap. We provide an in-depth investigation of the solution characteristics when convex relaxations are inexact, we assess the most promising AC feasibility recovery methods for large-scale systems, and we propose two new metrics that lead to a better understanding of the quality of the identified solutions. We perform a comprehensive assessment on 96 different test cases, ranging from 14 to 3120 buses, and we show the following: (i) Despite an optimality gap of less than 1%, several test cases still exhibit substantial distances to both AC feasibility and local optimality and the newly proposed metrics characterize these deviations. (ii) Penalization methods fail to recover an AC-feasible solution in 15 out of 45 cases, and using the proposed metrics, we show that most failed test instances exhibit substantial distances to both AC-feasibility and local optimality. For failed test instances with small distances, we show how our proposed metrics inform a fine-tuning of penalty weights to obtain AC-feasible solutions. (iii) The computational benefits of warm-starting non-convex solvers have significant variation, but a computational speedup exists in over 75% of the cases.