Certified Loadability Brackets for AC Power Networks

The loadability margin measures how far loads can grow along a direction while some operating point, with full redispatch, satisfies the power-flow equations and every limit. Local solvers estimate it without proof; convex relaxations bound it with unknown looseness. We compute an interval that provably contains it. Its upper end comes from the dual of the semidefinite relaxation's maximum-loadability problem, in one conic solve on all but the 2000-bus grid. Where the relaxation is loose, certified bound tightening proves single loads infeasible. Interval Newton tests prove its lower end along a path of interpolated optimal dispatches or along the optimal point continued with binding limits held. Exact rational arithmetic re-verifies every upper end. We bracket the margin along 130 directions of 17 systems with up to 2000 buses. Over all 130 directions the bracket is 1.00% wide in the median, and 0.01% along the 35 where the relaxation is within 0.1% of the local maximum. Bound tightening, run along 54 of the 82 directions where the relaxation is over 1% loose, proves the margin at most 0.1% above the local maximum along 32, on systems with up to 300 buses.