  AC-OPF (Alternative Current Optimal Power Flow)aims at minimizing the
operating costs of a power gridunder physical constraints on voltages and power
injections.Its mathematical formulation results in a nonconvex polynomial
optimizationproblem which is hard to solve in general,but that can be tackled
by a sequence of SDP(Semidefinite Programming) relaxationscorresponding to the
steps of the moment-SOS (Sums-Of-Squares) hierarchy.Unfortunately, the size of
these SDPs grows drastically in the hierarchy,so that even second-order
relaxationsexploiting the correlative sparsity pattern of AC-OPFare hardly
numerically tractable for largeinstances -- with thousands of power buses.Our
contribution lies in a new sparsityframework, termed minimal sparsity,
inspiredfrom the specific structure of power flowequations.Despite its
heuristic nature, numerical examples show that minimal sparsity allows the
computation ofhighly accurate second-order moment-SOS relaxationsof AC-OPF,
while requiring far less computing time and memory resources than the standard
correlative sparsity pattern. Thus, we manage to compute second-order
relaxations on test caseswith about 6000 power buses, which we believe to be
unprecedented.
