Non-local impact of link failures in linear flow networks

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.

Collective effects of link failures in linear flow networks

“…We then apply the predictor to different test grids and reveal its performance in forecasting collective effects for multiple link failures quantitatively, outperforming also distance measures proven to be good predictors in the case of single link failures. Finally, we extend on previous work [22] that successfully established an analogy between flow rerouting after single link failures and the fields of electromagnetic dipoles in regular grids by demonstrating that flows after multiple link failures may be treated as a superposition of multiple individual dipole fields in such grids in the continuum limit.…”

“…Additionally, one can define the geodesic distance between edges as the smallest possible distance between the nodes incident to the corresponding edges plus half of each edge's length, Here the subscript 'ge' denotes the geodesic distance while (r, s) and (m, n) are the respective edges given by the nodes they are incident in. As we demonstrated in a recent publication [22], this distance measure does not capture essential aspects of the flow rerouting after a link failure. Instead, we proposed the rerouting distance given by the length of the shortest cycle crossing both edges (r, s) and (m, n).…”

“…If no such cycle exists, the rerouting distance is defined to be ¥. This distance measure is strongly correlated with the magnitude of the LODFs as shown in [22]. Figure 6 shows the scaling of ξ with distance between the failing links for a failure of two links.…”

“…Here, [ ] B k is the Laplacian of the network after removal of link k and ( ) F k 0 is the pre-outage flow on link k. This equation was studied in the past in different settings [35,22,36,33]. The failure of single links is thus comparably well understood [22,37,38], whereas the simultaneous failure of multiple links was not yet studied to the same extend on a theoretical level.…”

“…Basic mathematical tools have been developed extending the concept of Line Outage Distribution Factors (LODFs) originally used for single link contingencies [11] to include multiple link failures thereby allowing for a mathematical description of these contingencies [19][20][21]. These tools demonstrate that the nature of multiple outages may be fundamentally different from the outage of single lines, thus making a direct transfer of understanding and intuition developed for single link failures [22] difficult. In particular, multiple outages can enhance or attenuate each other in a counterintuitive manner.…”