Why the Swing Parameters Are Sloppy

Abstract

Evangelou, Stanković, Kevrekidis and Transtrum compared analytic and data-driven assessments of parameter identifiability for the infinite-bus synchronous generator. Fisher information with the manifold boundary approximation method (MBAM), and output diffusion maps with geometric harmonics, report the same six identifiable parameters on the late window $t \in [3, 5]\text{ s}$, and both place the inertia $H$ and the damping $D$ among the unidentifiable ones. We explain this result from the geometry of the dynamics rather than from that of the parameter-to-data map. The rotor swing is a fast fibre that has collapsed onto a two-dimensional normally attracting invariant manifold (NAIM) of the transient fluxes before the window opens. The zeroth-order NAIM is exactly independent of $H$, $D$, $T_{d0}''$ and $T_{q0}''$, four of the five MBAM limits. We prove, under stated smoothness and rank conditions, that each of the four limits is a boundary of the model manifold at a distance equal, to first order in the parameter, to its current profiled speed. The same analysis predicts how identifiability changes with the experiment: neither a larger initial speed deviation nor a load change restores $H$ or $D$ in the late window, whereas the early window $[0, 2]\text{ s}$ makes $H$ the best-determined parameter. Evaluated in a physical metric fixed by synchrophasor accuracy limits, the late window identifies five parameters rather than six and the early window nine. The spiked-covariance structure of the linearised trajectory map explains why the principal components of whitened, sampled trajectories reproduce the prior-scaled Fisher spectrum, and regression on those components attains the Fisher-predicted recovery error in 113 of 114 parameter--scenario pairs.