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

电网现代化仍然是一项重大的国家和国际挑战。气候变化对电力基础设施的破坏性影响，以及当这种基础设施只受到轻微破坏，更不用说造成重大破坏时，可能发生的社会后果，加剧了这一现代化的紧迫性。数学和统计学将在这种现代化中发挥特殊的作用，因为电力系统是由复杂的数学方程系统建模的。该项目将开发新的快速和准确的计算技术，用于求解建模网格的核心数学方程。这些新技术的关键将基于电网的网络结构，它暴露了电网中的电力流动。这些技术中一个常见且重要的问题是确定电网中最相关的发电机、负荷和变电站。了解这些组件将使电网工程师能够确定电网的哪些部分最容易受到自然灾害和网络攻击的干扰，从而有助于设计出更具弹性的电网。该项目将为研究生提供培训机会。该项目旨在开发快速、准确的算法来分析大规模、真实世界的电力系统。这些算法的一个共同特点是利用系统的关联加权图拉普拉斯来揭示网络中的电力扩散。这项研究的基本组成部分是确定网络总线的相关性和相关性，这是通过最先进的加权图拉普拉斯和近似扩散距离度量的代数多重网格技术来完成的。这种相关性和相关性将被用来开发(1)用于高度事故分析的快速准确的筛选技术；(2)用于电网分析中回归的多尺度图神经网络(GNN)；(3)动态电力系统的最优并行时间积分器。此外，为了避免随机梯度法中的梯度消失问题，并允许自然地在GNN设置中包含不确定性，将使用集合卡尔曼滤波法来训练GNN权重。通过将从多尺度GNN获得的替代模型与传统电力系统模型适当地结合起来，将构建更稳健的模型，在不确定性方面稳健。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.