Computing Rare Probabilities of Voltage Collapse

Abstract

This paper introduces a framework based on Large Deviation Theory (LDT) to accurately and efficiently compute the rare probabilities of voltage collapse. We formulate the problem as finding the most probable failure point (the instanton) on the stability boundary and derive both first-order and second-order approximations for the collapse probability. The second-order method incorporates the local curvature of the stability boundary, yielding higher accuracy. This LDT framework generalizes methods based on Mahalanobis distance and is extensible to non-Gaussian uncertainties. We validate our approach on test systems, demonstrating that the LDT estimates converge to Monte Carlo results in the rare-event regime where direct sampling becomes computationally prohibitive.

Figures (6)

• Figure 1: Local geometry at the bifurcation boundary for the 2-bus system. True boundary (blue), LDT-1 tangent (orange, dashed), and LDT-2 quadratic (green, dash-dot) at the instanton $\lambda^*$ (red triangle). Gray dashed ellipses are level sets of $I(\lambda)$ for $N(\mu,\Sigma)$. The separation highlights how anisotropy/rotation in $\Sigma$ changes the most probable failure direction and why LDT-2 improves LDT-1 by accounting for boundary curvature.
• Figure 2: Two-bus probability sweep for Gaussian uncertainty. The reference curve is computed by direct Monte Carlo for the larger covariance scales and by importance sampling in the rare-event regime.
• Figure 3: Local geometry at the GMM instanton in the 2-bus system.
• Figure 4: Two-bus probability sweep for the Gaussian-mixture extension. The reference curve is computed by Monte Carlo or importance sampling, while GMM-LDT1 and GMM-LDT2 correspond to \ref{['eq:gmm_1st_order_ldt']} and \ref{['eq:gmm_2nd_order_ldt']}.
• Figure 5: Five-bus probability sweep for Gaussian uncertainty. The reference curve combines direct Monte Carlo with importance sampling, while LDT1 and LDT2 denote the first-order and second-order approximations.