Data-Driven Estimation of Failure Probabilities in Correlated Structure-Preserving Stochastic Power System Models

TL;DR

This work addresses the challenge of efficiently quantifying tail-event probabilities in correlated, stochastic power systems. By deriving a reduced-order PDF PDE for selected quantities of interest and learning closure terms via regression, the authors propagate uncertainty with far fewer samples than conventional Monte Carlo or KDE approaches, while capturing joint dependencies. The method demonstrates accurate marginal and joint tail probabilities for line energies in standard test networks and reveals meaningful correlations that independent models miss. The resulting framework enables scalable uncertainty quantification for cascading-like events in structure-preserving stochastic power-system models, with practical implications for reliability assessment and risk mitigation.

Abstract

We propose a data-driven approach for propagating uncertainty in stochastic power grid simulations and apply it to the estimation of transmission line failure probabilities. A reduced-order equation governing the evolution of the observed line energy probability density function is derived from the Fokker--Planck equation of the full-order continuous Markov process. Our method consists of estimates produced by numerically integrating this reduced equation. Numerical experiments for scalar- and vector-valued energy functions are conducted using the classical multimachine model under spatiotemporally correlated noise perturbation. The method demonstrates a more sample-efficient approach for computing probabilities of tail events when compared with kernel density estimation. Moreover, it produces vastly more accurate estimates of joint event occurrence when compared with independent models.

Figures (5)

• Figure 1: (Top to bottom) Predicted PDF of line energies for WSCC case 9, IEEE case 30, and IEEE case 57, plotted at respective peak times, with exceedance thresholds zoomed in and marked in red. The 1D reduced-order marginal PDF equations (\ref{['eqn:coord-projection-qoi']}) were solved with $m_{\text{R}} = 5,000$ samples, compared with a $m_{\text{KDE}} = 10,000$ KDE benchmark.
• Figure 2: Predicted reduced-order joint PDFs (\ref{['eqn:joint-ropdf-equation']}) with $m_{\text{R}} = 5,000$ samples, at selected PDE solution times. Particularly, IEEE Case 30 and 57 show non-trivial evolution and correlation structure over time.
• Figure 3: (Left) All test cases reported at energy peak time, tail probabilities for individual exceedance (\ref{['eqn:individual-event']}), and joint exceedance (\ref{['eqn:tail2d-event']}). The values given by reduced-order PDF equations are computed with $m_{\text{R}} = 5,000$ samples and compared with those obtained from $m_{\text{KDE}} = 10,000$ KDE benchmark. The tail probabilities estimated by using independence assumption (\ref{['eqn:independence-assumption']}) are marked in grey. (Right) Ratios between predicted probability $\widehat{\mathbb{P}}$ and benchmark probability $\mathbb{P}$, as described in Section \ref{['sec:1dropdf']}. The grey markers indicate the computation when the line failures are assumed to be independent, which is significantly farther away from 1 (i.e. away from matching the empirical observation of the frequency of two simultaneous events).
• Figure 4: (Top) Convergence of $L^1$-error from benchmark (\ref{['eqn:l1-error-measure']}), averaged by the number of lines for each case. (Bottom) Aggregate number of samples required for KDE or reduced-order PDF method to reach within $\gamma=1\%$$L^1$-error from the benchmark.
• Figure 5: Estimated mutual information (\ref{['eqn:mutual-info']}) of WSCC case 9 and IEEE cases 30 and 57 over PDE simulation time up to $T=10$.