  Convex relaxations of non-convex optimal power flow (OPF) problems have
recently attracted significant interest. While existing relaxations globally
solve many OPF problems, there are practical problems for which existing
relaxations fail to yield physically meaningful solutions. This paper applies
moment relaxations to solve many of these OPF problems. The moment relaxations
are developed from the Lasserre hierarchy for solving generalized moment
problems. Increasing the relaxation order in this hierarchy results in
"tighter" relaxations at the computational cost of larger semidefinite
programs. Low-order moment relaxations are capable of globally solving many
small OPF problems for which existing relaxations fail. By exploiting sparsity
and only applying the higher-order relaxation to specific buses, global
solutions to larger problems are computationally tractable through the use of
an iterative algorithm informed by a heuristic for choosing where to apply the
higher-order constraints. With standard semidefinite programming solvers, the
algorithm globally solves many test systems with up to 300 buses for which the
existing semidefinite relaxation fails to yield globally optimal solutions.
