Critical Links and Nonlocal Rerouting in Complex Supply Networks

Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network’s capability to maintain a stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link’s redundant capacity and a renormalized linear response theory we derive. These criteria outperform the critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing countermeasures and support network planning and design.

Figure 1

Limits to predicting critical links by local measures. (a) Stationary loads in a coarse-grained model of the British power grid. For three links $(a,b)$ the loads $L_{ab}=F_{ab}/K_{ab}$ are indicated. (b) The load does not necessarily indicate whether a local failure causes a desynchronization and thus a major malfunction. (b1) A highly loaded link [labeled “link 1” in panel (a)] induces desynchronization and is thus critical. (b2) Moderately loaded link “2” still induces desynchronization and thus is also critical. (b3) In the same network, even a more heavily loaded link “3” does not yield desynchronization and leaves the network fully functional. Accordingly, link 3 is highly loaded but noncritical (stable). We analyze critical links for test networks based on the topology of the British transmission grid [10, 28]: Transmission lines have a capacity $K_0=15{\mathrm{s}}^{-2}$, except for lines connecting to generator nodes which have doubled capacity. Ten nodes are randomly chosen as generators ($\square ,P_j=+11P_0$) and the others as consumers ($\circ ,P_j=-P_0$), with $P_0=1{\mathrm{s}}^{-2}$, modeling a heavily loaded grid. Other test grids exhibit qualitatively the same phenomena [31].

Figure 2

Redundant capacity indicates flow rerouting options. (a) Ratio between flows and redundant capacities (color coded), with the three links from Fig. 1 indicated by arrows. (b) Dominant rerouting paths. (b1),(b2) For the two links (1, heavily loaded) and (2, moderately loaded), the flows to be rerouted are larger than the redundant capacity. The bottlenecks on the rerouting path are indicated by dashed red arrows. Thus, $|F_{ab}/K_{ab}^{\mathrm{red}}|>1$ and these links are critical. (b3) For link 3, whereas more heavily loaded than link 2, the flow to be rerouted is smaller than the total available redundant capacity. Thus, $|F_{ab}/K_{ab}^{\mathrm{red}}|<1$ and the link is stable.

Figure 3

The renormalized linear response theory predicts critical links. (a) The renormalized linear response not only predicts which links are critical (red squares) and which are stable (green disks), it also accurately tells the maximum loads for stable links (solid line, no fit parameter). (b) The response as a function of the load $L_{ab}$ of the failing link does grow proportionally to the load but also depends on other factors. (c) The response grows proportionally to the overlap $|\mathit{q}·{\mathit{v}}_2|$ with the Fiedler vector. The solid straight line with slope $1/{\lambda }_2$ gives a strict lower bound; see Eq. (10). Networks and parameters are as in Fig. 2.

Figure 4

Improved quality of the critical link prediction. The receiver operating characteristic curves indicate a multifold increase in the prediction quality using the classifiers (4) and (9) in comparison to the load or flow. (Statistics for 66 000 links from 400 random grid realizations; model, topologies, and parameters are as in Fig. 1.)