A Time-to-Boundary Margin for Transient Stability: Unifying Critical Clearing Time and Operating-Point Drift

Abstract

The loading margin to voltage collapse -- the distance in parameter space to the closest saddle-node bifurcation -- is a standard proximity index for voltage stability. This paper develops its transient-stability counterpart: a margin M that measures the time to the synchronism boundary rather than a distance, and that unifies two limits usually treated separately. The critical clearing time (CCT) is the fast, fixed-parameter limit; the slow drift of the operating point toward a static loadability limit is the other. M is defined as the first-passage time of the joint state-parameter motion to the survival boundary. We prove and verify that M equals the CCT exactly on the one-machine-infinite-bus reduction (deviation <= 0.01% across loadings on a published benchmark), establishing a certified single-machine pillar. Under operating-point drift, M yields an operational lead time before faults become unclearable; we take the 28 April 2025 Iberian blackout timeline as an illustrative time scale for the drift rate. On the New England 39-bus system, an independent benchmark, the single-machine-equivalent reduction reproduces the CCT within 1.8-6.0% (conservatively), and a critical slowing-down signature flags proximity to the boundary. For the multimachine case we characterize the limits explicitly: the transfer-conductance work is tightly boundable, while the controlling unstable equilibrium is the binding obstruction to a certified margin.