Robustness of power systems under a democratic-fiber-bundle-like model

Scaling laws of failure dynamics on complex networks

The topology of the network of load transmitting connections plays an essential role in the cascading failure dynamics of complex systems driven by the redistribution of load after local breakdown events. In particular, as the network structure is gradually tuned from regular to completely random a transition occurs from the localized to mean field behavior of failure spreading. Based on finite size scaling in the fiber bundle model of failure phenomena, here we demonstrate that outside the localized regime, the load bearing capacity and damage tolerance on the macro-scale, and the statistics of clusters of failed nodes on the micro-scale obey scaling laws with exponents which depend on the topology of the load transmission network and on the degree of disorder of the strength of nodes. Most notably, we show that the spatial structure of damage governs the emergence of the localized to mean field transition: as the network gets gradually randomized failed clusters formed on locally regular patches merge through long range links generating a percolation like transition which reduces the load concentration on the network. The results may help to design network structures with an improved robustness against cascading failure.

Temporal evolution of failure avalanches of the fiber bundle model on complex networks

We investigate how the interplay of the topology of the network of load transmitting connections and the amount of disorder of the strength of the connected elements determines the temporal evolution of failure cascades driven by the redistribution of load following local failure events. We use the fiber bundle model of materials' breakdown assigning fibers to the sites of a square lattice, which is then randomly rewired using the Watts-Strogatz technique. Gradually increasing the rewiring probability, we demonstrate that the bundle undergoes a transition from the localized to the mean field universality class of breakdown phenomena. Computer simulations revealed that both the size and the duration of failure cascades are power law distributed on all network topologies with a crossover between two regimes of different exponents. The temporal evolution of cascades is described by a parabolic profile with a right handed asymmetry, which implies that cascades start slowly, then accelerate, and eventually stop suddenly. The degree of asymmetry proved to be characteristic of the network topology gradually decreasing with increasing rewiring probability. Reducing the variance of fibers' strength, the exponents of the size and the duration distribution of cascades increase in the localized regime of the failure process, while the localized to mean field transition becomes more abrupt. The consistency of the results is supported by a scaling analysis relating the characteristic exponents of the statistics and dynamics of cascades.

Modelling cascading failures in networks with the harmonic closeness

Many studies on cascading failures adopt the degree or the betweenness of a node to define its load. From a novel perspective, we propose an approach to obtain initial loads considering the harmonic closeness and the impact of neighboring nodes. Based on simulation results for different adjustable parameter θ, local parameter δ and proportion of attacked nodes f, it is found that in scale-free networks (SF networks), small-world networks (SW networks) and Erdos-Renyi networks (ER networks), there exists a negative correlation between optimal θ and δ. By the removal of the low load node, cascading failures are more likely to occur in some cases. In addition, we find a valuable result that our method yields better performance compared with other methods in SF networks with an arbitrary f, SW and ER networks with large f. Moreover, the method concerning the harmonic closeness makes these three model networks more robust for different average degrees. Finally, we perform the simulations on twenty real networks, whose results verify that our method is also effective to distribute the initial load in different real networks.

Mapping heterogeneities through avalanche statistics

Avalanche statistics of various threshold-activated dynamical systems are known to depend on the magnitude of the drive, or stress, on the system. Such dependences exist for earthquake size distributions, in sheared granular avalanches, laboratory-scale fracture and also in the outage statistics of power grids. In this work, we model threshold-activated avalanche dynamics and investigate the time required to detect local variations in the ability of model elements to bear stress. We show that the detection time follows a scaling law where the scaling exponents depend on whether the feature that is sought is either weaker, or stronger, than its surroundings. We then look at earthquake data from Sumatra and California, demonstrate the trade-off between the spatial resolution of a map of earthquake exponents (i.e. the b-values of the Gutenberg-Richter Law) and the accuracy of those exponents, and suggest a means to maximize both.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Cascading failures in interdependent systems under a flow redistribution model

Robustness and cascading failures in interdependent systems has been an active research field in the past decade. However, most existing works use percolation-based models where only the largest component of each network remains functional throughout the cascade. Although suitable for communication networks, this assumption fails to capture the dependencies in systems carrying a flow (e.g., power systems, road transportation networks), where cascading failures are often triggered by redistribution of flows leading to overloading of lines. Here, we consider a model consisting of systems A and B with initial line loads and capacities given by {L_{A,i},C_{A,i}}_{i=1}^{n} and {L_{B,i},C_{B,i}}_{i=1}^{n}, respectively. When a line fails in system A, a fraction of its load is redistributed to alive lines in B, while remaining (1-a) fraction is redistributed equally among all functional lines in A; a line failure in B is treated similarly with b giving the fraction to be redistributed to A. We give a thorough analysis of cascading failures of this model initiated by a random attack targeting p_{1} fraction of lines in A and p_{2} fraction in B. We show that (i) the model captures the real-world phenomenon of unexpected large scale cascades and exhibits interesting transition behavior: the final collapse is always first order, but it can be preceded by a sequence of first- and second-order transitions; (ii) network robustness tightly depends on the coupling coefficients a and b, and robustness is maximized at non-trivial a,b values in general; (iii) unlike most existing models, interdependence has a multifaceted impact on system robustness in that interdependency can lead to an improved robustness for each individual network.

Limits of Predictability of Cascading Overload Failures in Spatially-Embedded Networks with Distributed Flows

Cascading failures are a critical vulnerability of complex information or infrastructure networks. Here we investigate the properties of load-based cascading failures in real and synthetic spatially-embedded network structures, and propose mitigation strategies to reduce the severity of damages caused by such failures. We introduce a stochastic method for optimal heterogeneous distribution of resources (node capacities) subject to a fixed total cost. Additionally, we design and compare the performance of networks with N-stable and (N-1)-stable network-capacity allocations by triggering cascades using various real-world node-attack and node-failure scenarios. We show that failure mitigation through increased node protection can be effectively achieved against single-node failures. However, mitigating against multiple node failures is much more difficult due to the combinatorial increase in possible sets of initially failing nodes. We analyze the robustness of the system with increasing protection, and find that a critical tolerance exists at which the system undergoes a phase transition, and above which the network almost completely survives an attack. Moreover, we show that cascade-size distributions measured in this region exhibit a power-law decay. Finally, we find a strong correlation between cascade sizes induced by individual nodes and sets of nodes. We also show that network topology alone is a weak predictor in determining the progression of cascading failures.

Optimizing the robustness of electrical power systems against cascading failures

Electrical power systems are one of the most important infrastructures that support our society. However, their vulnerabilities have raised great concern recently due to several large-scale blackouts around the world. In this paper, we investigate the robustness of power systems against cascading failures initiated by a random attack. This is done under a simple yet useful model based on global and equal redistribution of load upon failures. We provide a comprehensive understanding of system robustness under this model by (i) deriving an expression for the final system size as a function of the size of initial attacks; (ii) deriving the critical attack size after which system breaks down completely; (iii) showing that complete system breakdown takes place through a first-order (i.e., discontinuous) transition in terms of the attack size; and (iv) establishing the optimal load-capacity distribution that maximizes robustness. In particular, we show that robustness is maximized when the difference between the capacity and initial load is the same for all lines; i.e., when all lines have the same redundant space regardless of their initial load. This is in contrast with the intuitive and commonly used setting where capacity of a line is a fixed factor of its initial load.