图的Kron约简及其在电网中的应用
Kron Reduction of Graphs with Applications to Electrical Networks
AI总结：
本文提出一种图论框架用于Kron约简，通过Schur补将加权无向图（含自环）约简为子图，并应用于电路理论、电网监测等领域，提供拓扑、代数、谱、电阻和灵敏度分析。
摘要：
考虑一个加权无向图，可能包含自环，及其对应的拉普拉斯矩阵，该矩阵可能因自环而增加额外的对角元素。该图的Kron约简是一个新图，其拉普拉斯矩阵由原拉普拉斯矩阵关于某个节点子集的Schur补得到。Kron约简过程在经典电路理论及相关学科中普遍存在，例如电阻抗成像、智能电网监测、电力网络暂态稳定评估，以及感应电机和电力电子的分析与仿真。Kron约简的更广泛应用出现在稀疏矩阵算法、多重网格求解器、有限元分析和马尔可夫链中。拉普拉斯矩阵的Schur补及相关概念也在线性代数文献中以不同名称作为纯理论问题被研究。本文提出了一个通用的图论框架用于Kron约简，从而在数学和物理方面都带来了新颖而深刻的见解。我们展示了该框架在工程应用和计算中出现的各种实际问题设置中的适用性。此外，我们对Kron约简过程进行了全面而详细的图论分析，包括拓扑、代数、谱、电阻和灵敏度分析。在整个理论阐述中，我们特别强调了结果的实际适用性。
英文摘要：
Consider a weighted and undirected graph, possibly with self-loops, and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to the self-loops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment in power networks, or analysis and simulation of induction motors and power electronics. More general applications of Kron reduction occur in sparse matrix algorithms, multi-grid solvers, finite--element analysis, and Markov chains. The Schur complement of a Laplacian matrix and related concepts have also been studied under different names and as purely theoretic problems in the literature on linear algebra. In this paper we propose a general graph-theoretic framework for Kron reduction that leads to novel and deep insights both on the mathematical and the physical side. We show the applicability of our framework to various practical problem setups arising in engineering applications and computation. Furthermore, we provide a comprehensive and detailed graph-theoretic analysis of the Kron reduction process encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
