Cascading failures in power systems propagate non-locally, making the control of outages extremely difficult. In this work, we propose a new framework that offers strong analytical guarantees on both the localization and mitigation of cascading failures in power systems. The key component of this framework leverages the concept of tree partition, which characterizes regions of a power network inside which line failures are automatically localized. In Part I of this paper we establish a mathematical theory that underlies all the performance guarantees of tree partition, as well as its failure localization properties. This theory consists of a set of tools developed using the Laplacian matrix of the transmission network and reveals a novel perspective that precisely captures the Kirchhoff's Law in terms of topological structures. Our results show that the distribution of different families of subtrees of the transmission network plays a critical role on the patterns of power redistribution, and motivates tree partitioning of the network as a strategy to eliminate long-distance propagation of disturbances. These results are used in Parts II and III of this paper to design strategies to localize and mitigate line failures.

中文翻译：

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电力系统级联故障的定位和缓解，第 I 部分：频谱表示和树分区

电力系统中的级联故障会在非本地传播，这使得对停电的控制变得极其困难。在这项工作中，我们提出了一个新框架，该框架为电力系统级联故障的定位和缓解提供了强有力的分析保证。该框架的关键组件利用了树分区的概念，该概念表征了电力网络的区域，在该区域内自动定位线路故障。在本文的第一部分中，我们建立了一个数学理论，该理论是树分区的所有性能保证及其故障定位特性的基础。该理论由一组使用传输网络的拉普拉斯矩阵开发的工具组成，并揭示了一种新颖的视角，可以根据拓扑结构精确捕捉基尔霍夫定律。我们的结果表明，输电网络不同族子树的分布对功率再分配的模式起着关键作用，并促使网络的树分区作为消除干扰长距离传播的策略。本文的第二部分和第三部分将使用这些结果来设计定位和减轻线路故障的策略。