Graph-Structured Kernel Design for Power Flow Learning using Gaussian Processes

This paper presents a physics-inspired graph-structured kernel designed for power flow learning using Gaussian Process (GP). The kernel, named the vertex-degree kernel (VDK), relies on latent decomposition of voltage-injection relationship based on the network graph or topology. Notably, VDK design avoids the need to solve optimization problems for kernel search. To enhance efficiency, we also explore a graph-reduction approach to obtain a VDK representation with lesser terms. Additionally, we propose a novel network-swipe active learning scheme, which intelligently selects sequential training inputs to accelerate the learning of VDK. Leveraging the additive structure of VDK, the active learning algorithm performs a block-descent type procedure on GP's predictive variance, serving as a proxy for information gain. Simulations demonstrate that the proposed VDK-GP achieves more than two fold sample complexity reduction, compared to full GP on medium scale 500-Bus and large scale 1354-Bus power systems. The network-swipe algorithm outperforms mean performance of 500 random trials on test predictions by two fold for medium-sized 500-Bus systems and best performance of 25 random trials for large-scale 1354-Bus systems by 10%. Moreover, we demonstrate that the proposed method's performance for uncertainty quantification applications with distributionally shifted testing data sets.