Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow

TL;DR

This paper analyzes the inexactness of SDP relaxations in optimal power flow problems with generation lower bounds, providing a complete characterization, a library of challenging instances, and improved solution techniques.

Contribution

It offers a complete characterization of SDP relaxation outcomes with generation bounds, analytical gap expressions, and enhanced solution methods for radial networks.

Findings

SDP relaxation can be exact, inexact, or feasible while the OPF is infeasible.

A complete characterization of approximation outcomes is provided.

Valid inequalities and bound tightening improve solver performance.

Abstract

It has been recently proven that the semidefinite programming (SDP) relaxation of the optimal power flow problem over radial networks is exact under technical conditions such as not including generation lower bounds or allowing load over-satisfaction. In this paper, we investigate the situation where generation lower bounds are present. We show that even for a two-bus one-generator system, the SDP relaxation can have all possible approximation outcomes, that is (1) SDP relaxation may be exact or (2) SDP relaxation may be inexact or (3) SDP relaxation may be feasible while the OPF instance may be infeasible. We provide a complete characterization of when these three approximation outcomes occur and an analytical expression of the resulting optimality gap for this two-bus system. In order to facilitate further research, we design a library of instances over radial networks in which the SDP relaxation has positive optimality gap. Finally, we propose valid inequalities and variable bound tightening techniques that significantly improve the computational performance of a global optimization solver. Our work demonstrates the need of developing efficient global optimization methods for the solution of OPF even in the simple but fundamental case of radial networks.