Kron Reduction of Graphs With Applications to Electrical Networks

By Florian Dörfler, Francesco Bullo. 2011

870 citations recorded by the source corpus.

Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to 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 specified 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, and analysis of power electronics. Kron reduction is also relevant in other physical domains, in computational applications, and in the reduction of Markov chains. Related concepts have also been studied as purely theoretic problems in the literature on linear algebra. In this paper we analyze the Kron reduction process from the viewpoint of algebraic graph theory. Specifically, we provide a comprehensive and detailed graph-theoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results to various problem setups arising in engineering, computation, and linear algebra. Our analysis of Kron reduction leads to novel insights both on the mathematical and the physical side.

Subfields: Computational Theory and Mathematics, Electrical and Electronic Engineering, Materials Chemistry

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Citations in the corpus

Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs (2017, background): “This is followed by Kron reduction [13], which also deﬁnes the new edge labeling, enhanced with spectral sparsiﬁcation of edges [37].”
Synchronization in complex networks of phase oscillators: A survey (2014, background): “We remark that different loads such as constant power/current/susceptance loads and synchronous motors can be modeled by the same set of equations (8)- (10), see (Sastry and Varaiya, 1980; Chiang et al., 1995; Sauer and Pai, 1998; Dörfler and Bullo, 2013a).”
Voltage Stability and Reactive Power Sharing in Inverter-Based Microgrids With Consensus-Based Distributed Voltage Control (2016, background): “The admittance loads in the Kron-reduced network are a conglomeration of the individual loads in the original network [10], [41].”
Graph Learning From Data Under Laplacian and Structural Constraints (2016, methodology): “In [26], a Kron reduction procedure is developed based on GGLs for simplified modeling of electrical networks.”
Analysis of Power Sharing and Voltage Deviations in Droop-Controlled DC Grids (2013, methodology): “Using the Kron reduction technique, which is discussed in greater detail in [26], the nodes without current injections can be considered as internal nodes that can be eliminated.”
The Small-Signal Stability Analysis of the Droop-Controlled Converter in Electromagnetic Timescale (2019, methodology): “State-space-based approach was a global stability analysis method to assess the stability of microgrids regardless of the location of the instability source [15]–[17], [23].”
Synchronization and power sharing for droop-controlled inverters in islanded microgrids (2012, neutral): “Solving the set of |V L | algebraic equations and substituting into the dynamics for ∆ θ I , we obtain d(∆ θ I is also a Laplacian matrix, and therefore shares the same properties as L ( θ ∗ ) [13].”
A Multiscale Pyramid Transform for Graph Signals (2013, background): “A positive (negative) strong nodal domain of f on G is a maximally connected subgraph such that f(i) > 0 (f(i) 0) for all vertices i in the subgraph [18, Chapter 3].”