Kron Reduction of Graphs with Applications to Electrical Networks

TL;DR

This paper introduces a comprehensive graph-theoretic framework for Kron reduction, providing new insights and analysis applicable to electrical networks and various engineering problems, with a focus on practical applications.

Contribution

It develops a general framework for Kron reduction that offers novel mathematical and physical insights, along with detailed analyses relevant to engineering and computational applications.

Findings

Unified graph-theoretic perspective on Kron reduction

Deep topological, algebraic, and spectral analysis of the process

Applicability to practical engineering problems and computation

Abstract

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.