Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective

Physics-informed line graph neural network for power flow calculation

Power flow calculation plays a significant role in the operation and planning of modern power systems. Traditional numerical calculation methods have good interpretability but high time complexity. They are unable to cope with increasing amounts of data in power systems; therefore, many machine learning based methods have been proposed for more efficient power flow calculation. Despite the good performance of these methods in terms of computation speed, they often overlook the importance of transmission lines and do not fully consider the physical mechanisms in the power systems, thereby weakening the prediction accuracy of power flow. Given the importance of the transmission lines as well as to comprehensively consider their mutual influence, we shift our focus from bus adjacency relationships to transmission line adjacency relationships and propose a physics-informed line graph neural network framework. This framework propagates information between buses and transmission lines by introducing the concepts of the incidence matrix and the line graph matrix. Based on the mechanics of the power flow equations, we further design a loss function by integrating physical information to ensure that the output results of the model satisfy the laws of physics and have better interpretability. Experimental results on different power grid datasets and different scenarios demonstrate the accuracy of our proposed model.

No free lunch for avoiding clustering vulnerabilities in distributed systems

Emergent design failures are ubiquitous in complex systems, and often arise when system elements cluster. Approaches to systematically reduce clustering could improve a design's resilience, but reducing clustering is difficult if it is driven by collective interactions among design elements. Here, we use techniques from statistical physics to identify mechanisms by which spatial clusters of design elements emerge in complex systems modelled by heterogeneous networks. We find that, in addition to naive, attraction-driven clustering, heterogeneous networks can exhibit emergent, repulsion-driven clustering. We draw quantitative connections between our results on a model system in naval engineering to entropy-driven phenomena in nanoscale self-assembly, and give a general argument that the clustering phenomena we observe should arise in many distributed systems. We identify circumstances under which generic design problems will exhibit trade-offs between clustering and uncertainty in design objectives, and we present a framework to identify and quantify trade-offs to manage clustering vulnerabilities.

Predicting dynamic stability from static features in power grid models using machine learning

A reliable supply with electric power is vital for our society. Transmission line failures are among the biggest threats for power grid stability as they may lead to a splitting of the grid into mutual asynchronous fragments. New conceptual methods are needed to assess system stability that complement existing simulation models. In this article, we propose a combination of network science metrics and machine learning models to predict the risk of desynchronization events. Network science provides metrics for essential properties of transmission lines such as their redundancy or centrality. Machine learning models perform inherent feature selection and, thus, reveal key factors that determine network robustness and vulnerability. As a case study, we train and test such models on simulated data from several synthetic test grids. We find that the integrated models are capable of predicting desynchronization events after line failures with an average precision greater than 0.996 when averaging over all datasets. Learning transfer between different datasets is generally possible, at a slight loss of prediction performance. Our results suggest that power grid desynchronization is essentially governed by only a few network metrics that quantify the networks' ability to reroute the flow without creating exceedingly high static line loadings.

Large fluctuations in locational marginal prices

This paper investigates large fluctuations of locational marginal prices (LMPs) in wholesale energy markets caused by volatile renewable generation profiles. Specifically, we study events of the form [Formula: see text] where LMP is the vector of LMPs at the n power grid nodes, and α^-, [Formula: see text] are vectors of price thresholds specifying undesirable price occurrences. By exploiting the structure of the supply-demand matching mechanism in power grids, we look at LMPs as deterministic piecewise affine, possibly discontinuous functions of the stochastic input process, modelling uncontrollable renewable generation. We use techniques from large deviations theory to identify the most likely ways for extreme price spikes to happen, and to rank the nodes of the power grid in terms of their likelihood of experiencing a price spike. Our results are derived in the case of Gaussian fluctuations, and are validated numerically on the IEEE 14-bus test case. This article is part of the theme issue 'The mathematics of energy systems'.

A rare-event study of frequency regulation and contingency services from grid-scale batteries

We perform a rare-event study on a simulated power system in which grid-scale batteries provide both regulation and emergency frequency control ancillary services. Using a model of random power disturbances at each bus, we employ the skipping sampler, a Markov Chain Monte Carlo algorithm for rare-event sampling, to build conditional distributions of the power disturbances leading to two kinds of instability: frequency excursions outside the normal operating band, and load shedding. Potential saturation in the benefits, and competition between the two services, are explored as the battery maximum power output increases. This article is part of the theme issue 'The mathematics of energy systems'.

Open database analysis of scaling and spatio-temporal properties of power grid frequencies

The electrical energy system has attracted much attention from an increasingly diverse research community. Many theoretical predictions have been made, from scaling laws of fluctuations to propagation velocities of disturbances. However, to validate any theory, empirical data from large-scale power systems are necessary but are rarely shared openly. Here, we analyse an open database of measurements of electric power grid frequencies across 17 locations in 12 synchronous areas on three continents. The power grid frequency is of particular interest, as it indicates the balance of supply and demand and carries information on deterministic, stochastic, and control influences. We perform a broad analysis of the recorded data, compare different synchronous areas and validate a previously conjectured scaling law. Furthermore, we show how fluctuations change from local independent oscillations to a homogeneous bulk behaviour. Overall, the presented open database and analyses constitute a step towards more shared, collaborative energy research.

Power grid frequencies mirror the state of the grid. Here, Rydin Gorjão et al. analyse measurements of power grid frequencies across areas and continents and uncover scaling laws of their fluctuations and spatio-temporal dynamics, which could aid the design, operation and control of power systems.

Emergence of Scale-Free Blackout Sizes in Power Grids

We model power grids as graphs with heavy-tailed sinks, which represent demand from cities, and study cascading failures on such graphs. Our analysis links the scale-free nature of blackout sizes to the scale-free nature of city sizes, contrasting previous studies suggesting that this nature is governed by self-organized criticality. Our results are based on a new mathematical framework combining the physics of power flow with rare event analysis for heavy-tailed distributions, and are validated using various synthetic networks and the German transmission grid.

Quieting a noisy antenna reproduces photosynthetic light-harvesting spectra

Photosynthesis achieves near unity light-harvesting quantum efficiency yet it remains unknown whether there exists a fundamental organizing principle giving rise to robust light harvesting in the presence of dynamic light conditions and noisy physiological environments. Here, we present a noise-canceling network model that relates noisy physiological conditions, power conversion efficiency, and the resulting absorption spectra of photosynthetic organisms. Using light conditions in full solar exposure, light filtered by oxygenic phototrophs, and light filtered under seawater, we derived optimal absorption characteristics for efficient solar power conversion. We show how light-harvesting antennae can be tuned to maximize power conversion efficiency by minimizing excitation noise, thus providing a unified theoretical basis for the observed wavelength dependence of absorption in green plants, purple bacteria, and green sulfur bacteria.

Network desynchronization by non-Gaussian fluctuations

Many networks must maintain synchrony despite the fact that they operate in noisy environments. Important examples are stochastic inertial oscillators, which are known to exhibit fluctuations with broad tails in many applications, including electric power networks with renewable energy sources. Such non-Gaussian fluctuations can result in rare network desynchronization. Here we build a general theory for inertial oscillator network desynchronization by non-Gaussian noise. We compute the rate of desynchronization and show that higher moments of noise enter at specific powers of coupling: either speeding up or slowing down the rate exponentially depending on how noise statistics match the statistics of a network's slowest mode. Finally, we use our theory to introduce a technique that drastically reduces the effective description of network desynchronization. Most interestingly, when instability is associated with a single edge, the reduction is to one stochastic oscillator.

On structural and dynamical factors determining the integrated basin instability of power-grid nodes

In electric power systems delivering alternating current, it is essential to maintain its synchrony of the phase with the rated frequency. The synchronization stability that quantifies how well the power-grid system recovers its synchrony against perturbation depends on various factors. As an intrinsic factor that we can design and control, the transmission capacity of the power grid affects the synchronization stability. Therefore, the transition pattern of the synchronization stability with the different levels of transmission capacity against external perturbation provides the stereoscopic perspective to understand the synchronization behavior of power grids. In this study, we extensively investigate the factors affecting the synchronization stability transition by using the concept of basin stability as a function of the transmission capacity. For a systematic approach, we introduce the integrated basin instability, which literally adds up the instability values as the transmission capacity increases. We first take simple 5-node motifs as a case study of building blocks of power grids, and a more realistic IEEE 24-bus model to highlight the complexity of decisive factors. We find that both structural properties such as gate keepers in network topology and dynamical properties such as large power input/output at nodes cause synchronization instability. The results suggest that evenly distributed power generation and avoidance of bottlenecks can improve the overall synchronization stability of power-grid systems.

Degree Dispersion Increases the Rate of Rare Events in Population Networks

There is great interest in predicting rare and extreme events in complex systems, and in particular, understanding the role of network topology in facilitating such events. In this Letter, we show that degree dispersion-the fact that the number of local connections in networks varies broadly-increases the probability of large, rare fluctuations in population networks generically. We perform explicit calculations for two canonical and distinct classes of rare events: network extinction and switching. When the distance to threshold is held constant, and hence stochastic effects are fairly compared among networks, we show that there is a universal, exponential increase in the rate of rare events proportional to the variance of a network's degree distribution over its mean squared.

Optimal Noise-Canceling Networks

Natural and artificial networks, from the cerebral cortex to large-scale power grids, face the challenge of converting noisy inputs into robust signals. The input fluctuations often exhibit complex yet statistically reproducible correlations that reflect underlying internal or environmental processes such as synaptic noise or atmospheric turbulence. This raises the practically and biophysically relevant question of whether and how noise filtering can be hard wired directly into a network's architecture. By considering generic phase oscillator arrays under cost constraints, we explore here analytically and numerically the design, efficiency, and topology of noise-canceling networks. Specifically, we find that when the input fluctuations become more correlated in space or time, optimal network architectures become sparser and more hierarchically organized, resembling the vasculature in plants or animals. More broadly, our results provide concrete guiding principles for designing more robust and efficient power grids and sensor networks.

Rare slips in fluctuating synchronized oscillator networks

We study rare phase slips due to noise in synchronized Kuramoto oscillator networks. In the small-noise limit, we demonstrate that slips occur via large fluctuations to saddle phase-locked states. For tree topologies, slips appear between subgraphs that become disconnected at a saddle-node bifurcation, where phase-locked states lose stability generically. This pattern is demonstrated for sparse networks with several examples. Scaling laws are derived and compared for different tree topologies. On the other hand, for dense networks slips occur between oscillators on the edges of the frequency distribution. If the distribution is discrete, the probability-exponent for large fluctuations to occur scales linearly with the system size. However, if the distribution is continuous, the probability is a constant in the large network limit, as individual oscillators fluctuate to saddles while all others remain fixed. In the latter case, the network's coherence is approximately preserved.