Geometry-Driven Islanding Detection and Fault Classification for Grid-Forming Inverters: A Normally Hyperbolic Invariant Manifold Framework with Physics-Derived Thresholds
This paper presents a geometry-driven detection and fault-classification framework for grid-forming (GFM) inverters based on normally hyperbolic invariant manifolds (NAIM) and stochastic hypothesis testing. The GFM droop manifold $\mathcal{M}_0$ is identified as a NAIM of the closed-loop dynamics. Transverse fluctuations under grid noise are modeled as an Ornstein--Uhlenbeck process, and the long-run covariance is obtained from the algebraic Lyapunov equation. The detection statistic $D_t=T_w\bar{\xi}_{\perp}^{\top}\Sigma_{\mathrm{long}}^{-1}\bar{\xi}_{\perp}$ converges to $\chi^2(2)$ under the null hypothesis, yielding the tuning-free threshold $D_{\alpha}=-2\ln\alpha$ and an asymptotically exact false-alarm rate $\alpha$. A factor-of-2 error in earlier formulations is corrected and validated using 8,000 Monte Carlo realizations over nine window lengths and three significance levels. The Berry--Esseen bound $d_{\mathrm{KS}}\leq1.6704/(\beta T_w)$ is confirmed empirically. The minimum window condition $T_w\geq10/\beta_{\min}\approx1.0$ s, where $\beta_{\min}=\min(\omega_f,\omega_v)$, satisfies the IEEE 1547-2018 two-second detection requirement. A co-design theorem shows that increasing $(\omega_f,\omega_v)$ simultaneously enlarges the Fenichel spectral gap, tightens the null covariance, and reduces the false-alarm rate. Modal decomposition separates frequency and voltage contributions, enabling classification of islanding and voltage faults without additional sensors. Case studies confirm correct acceptance of normal operation, rapid detection of soft islanding, and accurate identification of a 10\% voltage sag.