Certified Safety Radii in Forecast-Error Space for Wasserstein Distributionally Robust Small Signal Stability-Constrained AC Optimal Power Flow via Lifted Spectrahedral Containment
2026-08-31 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address the problem of ensuring small-signal stability in AC optimal power flow under uncertainty, which is difficult because the boundary between stable and unstable conditions is complex and changes with decisions. They use a special mathematical transformation to make the stability condition easier to handle by turning it into a convex constraint in a lifted variable space. Instead of trying to directly find the unstable boundary, they focus on certifying safe regions around sample points where stability is guaranteed. Their method provides provable guarantees on stability margins and works well when combined with robust optimization techniques, as shown in numerical tests.
AC Optimal Power FlowSmall-Signal StabilityConvex OptimizationPositive Semidefinite ConstraintPerron CertificateJacobian RegularityRobust OptimizationWasserstein Distributionally Robust Chance ConstraintAdjoint EliminationPower Flow
Authors
Ziqi Zhang, Xi Chen
Abstract
Directly robustifying small-signal stability in AC optimal power flow is challenging since the stability boundary in the original uncertainty space is implicit, highly nonconvex, and changes with the operating decision. This paper exploits an alternative geometry. For a fixed model-specific stability certificate admitting suitable physical lifts, the small-signal stability requirement becomes an affine positive semidefinite constraint in the lifted variables, thereby defining a convex certified safe region. Instead of approximating the nonlinear instability boundary itself, we optimize a sample-wise safe radius in the original uncertainty space and certify, in the lifted space, that the entire power-flow image of the corresponding uncertainty ball is contained in the convex stability region. To this end, a componentwise Perron certificate guarantees existence, uniqueness, and Jacobian regularity of the target AC power-flow branch throughout each ball. An adjoint elimination then provides an exact affine-quadratic representation of the stability-relevant quantities, while rigorous matrix remainder bounds convert their nonlinear variation into finite robust PSD constraints. The resulting radii are certified lower bounds on the distances from empirical samples to failure and can therefore be coupled directly to the distance-based reformulation of a Wasserstein distributionally robust chance constraint, without directly approximating the instability boundary. Numerical studies demonstrate the effectiveness of the proposed framework.