Competitive percolation strategies for network recovery.

Restoring operation of critical infrastructure systems after catastrophic events is an important issue, inspiring work in multiple fields, including network science, civil engineering, and operations research. We consider the problem of finding the optimal order of repairing elements in power grids and similar infrastructure. Most existing methods either only consider system network structure, potentially ignoring important features, or incorporate component level details leading to complex optimization problems with limited scalability. We aim to narrow the gap between the two approaches. Analyzing realistic recovery strategies, we identify over- and undersupply penalties of commodities as primary contributions to reconstruction cost, and we demonstrate traditional network science methods, which maximize the largest connected component, are cost inefficient. We propose a novel competitive percolation recovery model accounting for node demand and supply, and network structure. Our model well approximates realistic recovery strategies, suppressing growth of the largest connected component through a process analogous to explosive percolation. Using synthetic power grids, we investigate the effect of network characteristics on recovery process efficiency. We learn that high structural redundancy enables reduced total cost and faster recovery, however, requires more information at each recovery step. We also confirm that decentralized supply in networks generally benefits recovery efforts.

Cascading failures in scale-free interdependent networks.

Large cascades are a common occurrence in many natural and engineered complex systems. In this paper we explore the propagation of cascades across networks using realistic network topologies, such as heterogeneous degree distributions, as well as intra- and interlayer degree correlations. We find that three properties, scale-free degree distribution, internal network assortativity, and cross-network hub-to-hub connections, are all necessary components to significantly reduce the size of large cascades in the Bak-Tang-Wiesenfeld sandpile model. We demonstrate that correlations present in the structure of the multilayer network influence the dynamical cascading process and can prevent failures from spreading across connected layers. These findings highlight the importance of internal and cross-network topology in optimizing robustness of interconnected systems.

Exotic states in a simple network of nanoelectromechanical oscillators.

Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.

Self-organization of dragon king failures.

The mechanisms underlying cascading failures are often modeled via the paradigm of self-organized criticality. Here we introduce a simple network model where nodes self-organize to be either weakly or strongly protected against failure in a manner that captures the trade-off between degradation and reinforcement of nodes inherent in many network systems. If strong nodes cannot fail, any failure is contained to a single, isolated cluster of weak nodes and the model produces power-law distributions of failure sizes. We classify the large, rare events that involve the failure of only a single cluster as "black swans." In contrast, if strong nodes fail once a sufficient fraction of their neighbors fail, then failure can cascade across multiple clusters of weak nodes. If over 99.9% of the nodes fail due to this cluster hopping mechanism, we classify this as a "dragon king," which are massive failures caused by mechanisms distinct from smaller failures. The dragon kings observed are self-organized, existing over a wide range of reinforcement rates and system sizes. We find that once an initial cluster of failing weak nodes is above a critical size, the dragon king mechanism kicks in, leading to piggybacking system-wide failures. We demonstrate that the size of the initial failed weak cluster predicts the likelihood of a dragon king event with high accuracy and we develop a simple control strategy that can dramatically reduce dragon kings and other large failures.

Escape routes, weak links, and desynchronization in fluctuation-driven networks.

Shifting our electricity generation from fossil fuel to renewable energy sources introduces large fluctuations to the power system. Here, we demonstrate how increased fluctuations, reduced damping, and reduced intertia may undermine the dynamical robustness of power grid networks. Focusing on fundamental noise models, we derive analytic insights into which factors limit the dynamic robustness and how fluctuations may induce a system escape from an operating state. Moreover, we identify weak links in the grid that make it particularly vulnerable to fluctuations. These results thereby not only contribute to a theoretical understanding of how fluctuations act on distributed network dynamics, they may also help designing future renewable energy systems to be more robust.

Network susceptibilities: Theory and applications.

We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.

Cascading failures in ac electricity grids.

Sudden failure of a single transmission element in a power grid can induce a domino effect of cascading failures, which can lead to the isolation of a large number of consumers or even to the failure of the entire grid. Here we present results of the simulation of cascading failures in power grids, using an alternating current (AC) model. We first apply this model to a regular square grid topology. For a random placement of consumers and generators on the grid, the probability to find more than a certain number of unsupplied consumers decays as a power law and obeys a scaling law with respect to system size. Varying the transmitted power threshold above which a transmission line fails does not seem to change the power-law exponent q≈1.6. Furthermore, we study the influence of the placement of generators and consumers on the number of affected consumers and demonstrate that large clusters of generators and consumers are especially vulnerable to cascading failures. As a real-world topology, we consider the German high-voltage transmission grid. Applying the dynamic AC model and considering a random placement of consumers, we find that the probability to disconnect more than a certain number of consumers depends strongly on the threshold. For large thresholds the decay is clearly exponential, while for small ones the decay is slow, indicating a power-law decay.

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.

Impact of network topology on synchrony of oscillatory power grids.

Replacing conventional power sources by renewable sources in current power grids drastically alters their structure and functionality. In particular, power generation in the resulting grid will be far more decentralized, with a distinctly different topology. Here, we analyze the impact of grid topologies on spontaneous synchronization, considering regular, random, and small-world topologies and focusing on the influence of decentralization. We model the consumers and sources of the power grid as second order oscillators. First, we analyze the global dynamics of the simplest non-trivial (two-node) network that exhibit a synchronous (normal operation) state, a limit cycle (power outage), and coexistence of both. Second, we estimate stability thresholds for the collective dynamics of small network motifs, in particular, star-like networks and regular grid motifs. For larger networks, we numerically investigate decentralization scenarios finding that decentralization itself may support power grids in exhibiting a stable state for lower transmission line capacities. Decentralization may thus be beneficial for power grids, regardless of the details of their resulting topology. Regular grids show a specific sharper transition not found for random or small-world grids.

Self-organized synchronization in decentralized power grids.

Robust synchronization (phase locking) of power plants and consumers centrally underlies the stable operation of electric power grids. Despite current attempts to control large-scale networks, even their uncontrolled collective dynamics is not fully understood. Here we analyze conditions enabling self-organized synchronization in oscillator networks that serve as coarse-scale models for power grids, focusing on decentralizing power sources. Intriguingly, we find that whereas more decentralized grids become more sensitive to dynamical perturbations, they simultaneously become more robust to topological failures. Decentralizing power sources may thus facilitate the onset of synchronization in modern power grids.