Reducing Cascading Failure Risk by Increasing Infrastructure Network Interdependence

Increased interconnection between critical infrastructure networks, such as electric power and communications systems, has important implications for infrastructure reliability and security. Others have shown that increased coupling between networks that are vulnerable to internetwork cascading failures can increase vulnerability. However, the mechanisms of cascading in these models differ from those in real systems and such models disregard new functions enabled by coupling, such as intelligent control during a cascade. This paper compares the robustness of simple topological network models to models that more accurately reflect the dynamics of cascading in a particular case of coupled infrastructures. First, we compare a topological contagion model to a power grid model. Second, we compare a percolation model of internetwork cascading to three models of interdependent power-communication systems. In both comparisons, the more detailed models suggest substantially different conclusions, relative to the simpler topological models. In all but the most extreme case, our model of a “smart” power network coupled to a communication system suggests that increased power-communication coupling decreases vulnerability, in contrast to the percolation model. Together, these results suggest that robustness can be enhanced by interconnecting networks with complementary capabilities if modes of internetwork failure propagation are constrained.

where x * indicates the complex conjugate of x. With some manipulation of equations (S1) and (S2), we can find the active (P) and reactive (Q) power flowing from k to m as follows: P km = V 2 k g km −V k V m (g km cos θ km + b km sin θ km ) (S3) where θ km = θ k − θ m is the phase angle difference between k and m. If we assume that the voltage amplitudes V k and V m are at their nominal levels, that we have normalized y km such that this nominal level is 1.0 (common practice), and that the resistance r km is small (nearly zero) relative to the reactance x km (a reasonable assumption for bulk power systems), then g km ∼ = 0, and P km becomes: If we assume that θ km is small, then sin θ km ∼ = θ km and we get: If we furthermore assume that Q km = 0 (not a particularly good assumption), then the current magnitude and the power are equal, |I km | = P km , and we can use equation (S6) to roughly simulate power flows in a power system. In order to solve for the flows P km in simulation, we put equation (S6) into matrix form as follows. Let A denote the line-to-node incidence matrix with 1 and −1 in each row indicating the endpoints of each line, θ be the vector of voltage phase angles, X be a diagonal matrix of line reactances, and P flow be a vector of active power flows along transmission lines. Then, we can solve for the vector of power flows P flow given that we know the vector of voltage phase angles θ as shown in the following: In order to solve for θ , we use information about the sources (generators) and sinks (loads) to build a vector of net injected powers (generation minus load), P. Given P, we can solve the following to find θ : The matrix B is known as the bus susceptance matrix, and has the properties of a weighted graph Laplacian matrix describing the network of transmission lines, where the link weights are the susceptances b km = 1/x km .
Comparing P N/2 to P ∞ In this paper, we measured the impact of disturbances of various sizes, f , on the probability of at least half of the network remaining within the "giant component" (GC) after the resulting cascade had subsided: P N/2 ; or the probability of half of the load still being served after the cascade completed: P D/2 . An alternative way to measure the impact of the disturbances is to measure the average cascade size (sometimes known as the yield), rather than the probability of a cascade in a given size range. Indeed, after random removal of f N nodes from the network, N 0 = (1 − f )N nodes remain in the GC, and the cascade of failures results in gradual fragmentation of the network, leading to a reduction in the number of nodes in the GC until a post-cascade steady state is reached. At this point, we calculated the ratio of the number of nodes remaining within the end-state GC, N ∞ , to the size of the network after the initial removal of nodes, N 0 , and measured the average cascade size, N ∞ /N 0 , across a set of samples. This measure would be more analogous to the P ∞ -metric that is commonly used in the literature on phase transitions in percolation systems. We chose not to use P ∞ as our primary measure of network robustness since the modeling assumptions described in the above discussion of "DC power flow" become particularly inaccurate for very large cascades. Essentially, P ∞ would, in many cases, average over small numbers that were not particularly accurate. However, the results that one obtains by measuring the average cascade impact do not lead one to substantially different conclusions than those reported in the paper (aside from the fact that the transitions are much more gradual).
Supplementary Figure S1 compares the response of various networks to random failures using the P ∞ -and P N/2measures for the topological contagion and power grid models. For the power grid model, the relative robustness of the five network structures is unchanged. The lattice is the most vulnerable and the scale-free network is the most robust. In the topological model, the P ∞ -measure indicates that the power grid, random graph, random regular, and scale-free networks have similar levels of robustness, for f < 0.15. The lattice remains to be the most vulnerable of the five network structures.
Supplementary Figure S2 compares the response of various coupled models to random failures with different levels of coupling between the power and communications network. In this case, we compare the original metrics used in the paper (i.e., P N/2 and P D/2 ) to P ∞ . Our analogous measure of robustness for the three smart grid models is D ∞ /D : the average ratio of the amount of demand (load) served at the end of the cascade to the original load, which amounts to 24.5725 GW. The results for the three different smart grid models are not substantially changed. We still see that increased coupling increases robustness in both the Ideal and the Intermediate Smart Grid models, whereas coupling is detrimental (though only slightly) in the Vulnerable Smart Grid model. For the Coupled Topological model, coupling is detrimental to robustness; indeed, by measuring the results using both P ∞ and P N/2 , the decrease in performance with q is monotonic.

50% coupling results
To better understand the impact of the level of coupling, we recomputed the results shown in Supplementary Fig. S4 using 50% coupling, i.e., q = 0.5.

Network vulnerability indices
One way to compare the various topological configurations and models described in this paper is to convert the sigmoidal results shown in Supplementary Figs S3 and S4 into a single metric of robustness (or conversely, vulnerability). To quantify the effects of topology, physics, and coupling among different synthetic networks, we define the network vulnerability index (β ) as follows: where f is the initiating failure size; L is the total number of f values simulated; and P GC = P N/2 is the probability of observing a GC whose size is more than half the number of grid nodes. The β -values corresponding to five network structures and six models of cascading studied in this paper are aggregately displayed in a bar chart in Supplementary  Fig. S5.

Sensitivity analysis on λ
Sensitivity of the three smart grid models with respect to the weight vector λ using various levels of coupling ranging from q = 0 to q = 1 is displayed in Supplementary Fig. S6. This analysis is carried out using our load-based P ∞ -measure, in which the ratios of the amount of post-cascade power, D ∞ , to the precascade load amount, D, are averaged across the same 1,000 initial outage sets of failure size f = 0.05, as in other cases. It is clearly seen that the choice of the weight vector, λ , of unavoidable overloads has only negligible impact on the values of the chosen robustness metric, D ∞ /D .  Figure S6. Sensitivity of the three Smart Grid models to the overload weight vector, λ , for overloads that cannot be eliminated through changes to generators and loads for the Polish power grid. The D ∞ /D -results shown herein are produced using three different uniform weight vectors (i.e., with λ i = 50, 100, 200), varying levels of coupling from q = 0 to q = 1, and a failure size of f = 0.05.