Exploiting the Weighted Graph Laplacian for Power Systems: High-Degree Contingency, Machine Learning, Data Assimilation, and Parallel-in-Time Integration

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Modernizing the power grid continues to be a major national and international challenge. The urgency of this modernization has been accentuated by the devastating effects of climate change on the power infrastructure and the societal consequences that can occur when only minor, let alone major, damages are made to this infrastructure. Mathematics and statistics will play a special role in this modernization since the power system is modeled by complex systems of mathematical equations. This project will develop new fast and accurate computational techniques for solving the core mathematical equations modeling the grid. The crux of these new techniques will be based on the grid's network structure, which exposes the flow of electricity in the power grid. A common and significant problem in these techniques is determining the most relevant generators, loads, and substations in the grid. Knowing these components will allow power grid engineers to determine which parts of the grid are most vulnerable to disruptions coming from natural disasters and cyber attacks, and hence, help design a more resilient grid. The project will provide training opportunities to graduate students. This project aims to develop fast and accurate algorithms for analyzing large-scale, real-world power systems. A common feature of these algorithms is exploiting the system's associated weighted graph Laplacian to expose the diffusion of electricity in the network. The fundamental component of this research is determining the dependency and relevancy of the network buses, which is accomplished through state-of-the-art algebraic multigrid techniques for weighted graph Laplacians and approximate diffusion distance measures. The dependency and relevancy will be used to develop (1) fast and accurate screening techniques for high-degree contingency analysis; (2) multi-scale graph neural networks (GNNs) for regression in power grid analysis; and (3) optimal parallel-in-time integrators for dynamical power systems. Moreover, to avoid the vanishing gradient problem in the stochastic gradient method and to permit the natural incorporation of uncertainties in the GNN setting, an ensemble Kalman filter method will be used to train the GNN weights. More robust models, robust with respect to uncertainties, will be constructed by appropriately combining the surrogates obtained from the multi-scale GNNs with traditional power system models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.