Abstract: The dynamical behavior of power systems under stress frequently deviates from the predictions of deterministic models. Model-free methods for detecting signs of excessive stress before instability occurs would therefore be valuable. The mathematical frameworks of fast-slow systems and critical slowing down can describe the statistical behavior of dynamical systems that are subjected to random perturbations as they approach points of instability. This paper builds from existing literature on fast-slow systems to provide evidence that time series data alone can be useful to estimate the temporal distance of a power system to a critical transition, such as voltage collapse. Our method is based on identifying evidence of critical slowing down in a single stream of synchronized phasor measurements. Results from a single machine, stochastic infinite bus model, a three machine/nine bus system and the Western North American disturbance of 10 August 1996 illustrate the utility of the proposed method.
Abstract: Power grids are almost universally agreed to be complex systems, which means that it is not possible to fully understand the grid by just looking at its parts. Power grids, which we define here to include all of the physical infrastructure and human individuals and organizations that jointly work to produce, distribute and consume electricity, have many properties that are common to other complex systems. Like the international financial system, power grids are operated by many millions of physical (hardware/software) and human agents. Like the Internet, power systems are frequently subjected to both random failure and malicious attack. Like the weather systems interacting to form hurricanes, there are strong, non-linear connections among the components, and between the components and society at large. And power systems occasionally exhibit spectacularly large, and costly, failures. This essay attempts to help us to understand these failures by highlighting key mathematical properties of cascading failures in complex systems in general, and in power grids in particular. We focus particularly on the mathematical challenges of measuring cascading failure risk in large power grids, and discuss some techniques that may provide better information to power grid operators regarding cascading failure risk.
Abstract: Critical slowing down (CSD) is the phenomenon in which a system recovers more slowly from small perturbations. CSD, as evidenced by increasing signal variance and autocorrela- tion, has been observed in many dynamical systems approaching a critical transition, and thus can be a useful signal of proximity to transition. In this paper, we derive autocorrelation functions for the state variables of a stochastic single machine infinite bus system (SMIB). The results show that both autocorrelation and variance increase as this system approaches a saddle-node bifurcation. The autocorrelation functions help to explain why CSD can be used as an indicator of proximity to criticality in power systems revealing, for example, how nonlinearity in the SMIB system causes these signs to appear.
Abstract: This paper describes two new approaches to cascading failure analysis in power systems that can combine large amounts of data about cascading blackouts to produce information about the ways that cascades may propagate. In the first, we evaluate methods for representing cascading failure information in the form of a graph. We refer to these graphs as dual graphs because the vertices are the transmission lines (the physical links), rather than the more conventional approach of representing power system buses as vertices. Examples of these ideas using the IEEE 30 bus system indicate that the dual graph methods can provide useful insight into how cascades propagate. In the second part of the paper we describe a random chemistry algorithm that can search through the enormous space of possible combinations of potential component outages to efficiently find large collections of the most dangerous combinations. This method was applied to a power grid with 2896 transmission branches, and provides insight into component outages that are notably more likely than others to trigger a cascading failure. In the conclusions we discuss potential uses of these methods for power systems planning and operations.
Abstract: The topological (graph) structure of complex networks often provides valuable information about the performance and vulnerability of the network. However, there are multiple ways to represent a given network as a graph. Electric power transmission and distribution networks have a topological structure that is straightforward to represent and analyze as a graph. However, simple graph models neglect the comprehensive connections between components that result from Ohm's and Kirchhoff's laws. This paper describes the structure of the three North American electric power interconnections, from the perspective of both topological and electrical connectivity. We compare the simple topology of these networks with that of random (Erdos and Renyi, 1959), preferential-attachment (Barabasi and Albert, 1999) and small-world (Watts and Strogatz, 1998) networks of equivalent sizes and find that power grids differ substantially from these abstract models in degree distribution, clustering, diameter and assortativity, and thus conclude that these topological forms may be misleading as models of power systems. To study the electrical connectivity of power systems, we propose a new method for representing electrical structure using electrical distances rather than geographic connections. Comparisons of these two representations of the North American power networks reveal notable differences between the electrical and topological structure of electric power networks.
Abstract: In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern US power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack vectors that appear to cause the most damage depend on the measure chosen. While our topological and power grid model results show some trends that are similar, there is only a mild correlation between the vulnerability measures for individual simulations. We conclude that evaluating vulnerability in power networks using purely topological metrics can be misleading.
Abstract: Power grids are complex dynamical systems, and because of this complexity it is unlikely that we will completely eliminate blackouts. However, there are things that can be done to reduce the average size and cost of these blackouts. In this article we described two strategies that hold substantial promise for reducing the size and cost of blackouts. Both reciprocal altruism and survivability respect the necessarily decentralized nature of power grids. Both strategies can be implemented within the context of the existing physical infrastructure of the power grids,which is important because dramatic changes to the physical infrastructure are prohibitively expensive. However, additional engineering and innovation will be needed to bring strategies such as these to implementation and to create power grids with smaller, less costly blackouts.