Abstract
The failure of a single link can degrade the operation of a supply network up to the point of complete collapse. Yet, the interplay between network topology and locality of the response to such damage is poorly understood. Here, we study how topology affects the redistribution of flow after the failure of a single link in linear flow networks with a special focus on power grids. In particular, we analyze the decay of flow changes with distance after a link failure and map it to the field of an electrical dipole for lattice-like networks. The corresponding inverse-square law is shown to hold for all regular tilings. For sparse networks, a long-range response is found instead. In the case of more realistic topologies, we introduce a rerouting distance, which captures the decay of flow changes better than the traditional geodesic distance. Finally, we are able to derive rigorous bounds on the strength of the decay for arbitrary topologies that we verify through extensive numerical simulations. Our results show that it is possible to forecast flow rerouting after link failures to a large extent based on purely topological measures and that these effects generally decay with distance from the failing link. They might be used to predict links prone to failure in supply networks such as power grids and thus help to construct grids providing a more robust and reliable power supply.
Highlights
The robust operation of supply networks is essential for the function of complex systems across scales and disciplines
We focus on how the network topology determines the overall network response as well as the spatial flow rerouting
Adopting the language of power system security analysis [37, 38], we call the factor of proportionality the line outage distribution factor (LODF)
Summary
The robust operation of supply networks is essential for the function of complex systems across scales and disciplines. Huge amounts of money and assets are exchanged through a complex financial network [6] Structural damages to such networks can have catastrophic consequences such as a stroke, a power outage or a major economic crisis. Large scale outages are typically triggered by the failure of a single transmission or generation element [7,8,9,10,11]. The power flows were rerouted, causing secondary overloads and eventually a cascade of failures In these three examples, the cascades propagated mostly locally – overloads took place in the proximity of previous failures. The cascades propagated mostly locally – overloads took place in the proximity of previous failures This is not necessarily the case during power outages The linearity allows to obtain several rigorous bounds for flow rerouting in general network topologies and to solve special cases fully analytically
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