Minimal Sparsity for Second-Order Moment-SOS Relaxations of the AC-OPF Problem
Abstract
AC-OPF (Alternative Current Optimal Power Flow)
aims at minimizing the operating costs of a power grid
under physical constraints on voltages and power injections.
Its mathematical formulation results in a nonconvex polynomial optimization
problem which is hard to solve in general,
but that can be tackled
by a sequence of SDP
(Semidefinite Programming) relaxations
corresponding 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 relaxations
exploiting the correlative sparsity pattern of AC-OPF
are hardly numerically tractable for large
instances --- with thousands of power buses.
Our contribution lies in a new sparsity
framework, termed minimal sparsity, inspired
from the specific structure of power flow
equations.
Despite its heuristic nature,
numerical examples show that minimal sparsity allows the computation of
highly accurate second-order moment-SOS relaxations
of 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 cases
with about 6000 power buses, which we believe
to be unprecedented.
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