  This paper presents a graph-structured Gaussian process (GP) model for
data-driven risk assessment of critical voltage constraints. The proposed GP is
based on a novel kernel, named the vertex-degree kernel (VDK), that decomposes
the voltage-load relationship based on the network graph. To estimate the GP
efficiently, we propose a novel active learning scheme that leverages the
additive structure of VDK. Further, we prove a probabilistic bound on the error
in risk estimation using VDK-GP model that demonstrates that it is
statistically comparable to using standard AC power flow (AC-PF), but does not
require computing a large number of ACPF solutions. Simulations demonstrate
that the proposed VDK-GP achieves more than two fold sample complexity
reduction, compared to a generic GP on medium scale 500-Bus and large scale
1354-Bus power systems. Moreover, active learning achieves an impressive
reduction of over 15 times in comparison to the time complexity of Monte-Carlo
simulations (MCS), and have risk estimation error of order 1E-4 for both
500-Bus and 1354-Bus system, demonstrating its superior efficiency in risk
estimation.
