Abstract
Link failures in supply networks can have catastrophic consequences that can lead to a complete collapse of the network. Strategies to prevent failure spreading are thus heavily sought after. Here, we make use of a spanning tree formulation of link failures in linear flow networks to analyse topological structures that prevent failures spreading. In particular, we exploit a result obtained for resistor networks based on the \textit{Matrix tree theorem} to analyse failure spreading after link failures in power grids. Using a spanning tree formulation of link failures, we analyse three strategies based on the network topology that allow to reduce the impact of single link failures. All our strategies do not reduce the grid's ability to transport flow or do in fact improve it - in contrast to traditional containment strategies based on lowering network connectivity. Our results also explain why certain connectivity features completely suppress any failure spreading as reported in recent publications.
Highlights
The theory of linear flow networks provides a powerful framework, allowing one to study systems ranging from water supply networks [1,2] and biological networks, such as leaf venation networks [3,4,5,6], to resistor networks [7,8,9], or ac power grids [10,11]
This connection dates back to work by Kirchhoff [8], who analyzed resistor networks and introduced several major tools that are the basis of the theory of complex networks, such as the matrix tree theorem [7,8,20]. These tools can serve as a basis for the analysis of failure spreading in ac power grids, which can be modeled as linear flow networks based on the dc approximation [11]
III, we demonstrate the analogy between such networks and ac power grids in the dc approximation and relate the spanning trees (STs) formulation to line outages studied in power system security analysis
Summary
The theory of linear flow networks provides a powerful framework, allowing one to study systems ranging from water supply networks [1,2] and biological networks, such as leaf venation networks [3,4,5,6], to resistor networks [7,8,9], or ac power grids [10,11]. The study of linear flow networks is intimately related to graph theory since most phenomena can be analyzed on purely topological grounds [7] This connection dates back to work by Kirchhoff [8], who analyzed resistor networks and introduced several major tools that are the basis of the theory of complex networks, such as the matrix tree theorem [7,8,20]. These tools can serve as a basis for the analysis of failure spreading in ac power grids, which can be modeled as linear flow networks based on the dc approximation [11]. IV we show how this formulation may be used to understand why certain connectivity features inhibit failure spreading extending on recent results [19]
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