Kron Reduction of Graphs with Applications to Electrical Networks

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 w...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Florian Dörfler, Francesco Bullo
• Keywords: CORR 2011, Education, Kron Reduction, Kron Reduction Process, Laplacian Matrix