  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.
