Convex Relaxations of Optimal Power Flow Problems: An Illustrative Example

TL;DR

This paper examines the limitations of semidefinite programming relaxations for optimal power flow problems, showing that physical system characteristics alone cannot explain relaxation success or failure, and explores tighter relaxations using polynomial optimization tools.

Contribution

It demonstrates that physical characteristics do not solely determine SDP relaxation success, and introduces advanced polynomial optimization relaxations to improve problem-solving capabilities.

Findings

SDP relaxation fails or succeeds depending on problem formulation, not physical system properties.

Physical characteristics alone cannot universally predict SDP relaxation outcomes.

Tighter relaxations from polynomial optimization hierarchies can address broader classes of OPF problems.

Abstract

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite programming (SDP) relaxation globally solves many OPF problems. However, there exist practical problems for which the SDP relaxation fails to yield the global solution. Conditions for the success or failure of the SDP relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the SDP relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The SDP relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the SDP relaxation must involve factors beyond just the network physics. The lack of universal physical conditions for success of the SDP relaxation motivates the development of tighter convex relaxations capable of solving a broader class of problems. Tools from polynomial optimization theory provide a means of developing tighter relaxations. We use the example OPF problem to illustrate relaxations from the Lasserre hierarchy for polynomial optimization and a related "mixed semidefinite/second-order cone programming" hierarchy.