Semidefinite relaxations of equivalent optimal power flow problems: An illustrative example

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite relaxation globally solves many OPF problems. However, there exist practical problems for which the semidefinite relaxation fails to yield the global solution. Conditions for the success or failure of the semidefinite 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 semidefinite 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 semidefinite 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 semidefinite relaxation must involve factors beyond just the network physics.