We give an overview of a complex systems approach to large blackouts of electric power transmission systems caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics and dynamics of series of blackouts with approximate global models. Blackout data from several countries suggest that the frequency of large blackouts is governed by a power law. The power law makes the risk of large blackouts consequential and is consistent with the power system being a complex system designed and operated near a critical point. Power system overall loading or stress relative to operating limits is a key factor affecting the risk of cascading failure. Power system blackout models and abstract models of cascading failure show critical points with power law behavior as load is increased. To explain why the power system is operated near these critical points and inspired by concepts from self-organized criticality, we suggest that power system operating margins evolve slowly to near a critical point and confirm this idea using a power system model. The slow evolution of the power system is driven by a steady increase in electric loading, economic pressures to maximize the use of the grid, and the engineering responses to blackouts that upgrade the system. Mitigation of blackout risk should account for dynamical effects in complex self-organized critical systems. For example, some methods of suppressing small blackouts could ultimately increase the risk of large blackouts.

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Cascading failures in large-scale electric power transmission systems are an important cause of blackouts. Analysis of North American blackout data has revealed power law ͑algebraic͒ tails in the blackout size probability distribution which suggests a dynamical origin. With this observation as motivation, we examine cascading failure in a simplified transmission system model as load power demand is increased. The model represents generators, loads, the transmission line network, and the operating limits on these components. Two types of critical points are identified and are characterized by transmission line flow limits and generator capability limits, respectively. Results are obtained for tree networks of a regular form and a more realistic 118-node network. It is found that operation near critical points can produce power law tails in the blackout size probability distribution similar to those observed. The complex nature of the solution space due to the interaction of the two critical points is examined. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1505810͔From the analysis of a 15-year time series of North American electric power transmission system blackouts, we have found that the frequency distribution of the blackout sizes does not decrease exponentially with the size of the blackout, but rather has a power law tail. The existence of a power tail suggests that the North American power system has been operated near a critical point. To see if this is possible, here we explore the critical points of a simple blackout model that incorporates circuit equations and a process through which outages of lines may happen. In spite of the simplifications, this is a complex problem. Understanding the different transition points and the characteristic properties of the distribution function of the blackouts near these points offers a first step in devising a dynamical model for the power transmission systems.

Transport of tracer particles is studied in a model of three-dimensional, resistive, pressure-gradient-driven plasma turbulence. It is shown that in this system transport is anomalous and cannot be described in the context of the standard diffusion paradigm. In particular, the probability density function (pdf) of the radial displacements of tracers is strongly non-Gaussian with algebraic decaying tails, and the moments of the tracer displacements exhibit superdiffusive scaling. To model these results we present a transport model with fractional derivatives in space and time. The model incorporates in a unified way nonlocal effects in space (i.e., non-Fickian transport), memory effects (i.e., non-Markovian transport), and non-Gaussian scaling. There is quantitative agreement between the turbulence transport calculations and the fractional diffusion model. In particular, the model reproduces the shape and space-time scaling of the pdf, and the superdiffusive scaling of moments.

A model for plasma transport near marginal stability is presented. The model is based on subcritical resistive pressure-gradient-driven turbulence. Three-dimensional nonlinear calculations based on this model show effective transport for subcritical mean profiles. This model exhibits some of the characteristic properties of self-organized criticality. Perturbative transport techniques are used to elucidate the transport properties. Propagation of positive and negative pulses is studied. The observed results suggest a possible explanation of the apparent nonlocal effects observed with perturbative experiments in tokamaks.

Abstract

A recently introduced tool for the analysis of turbulence, wavelet bicoherence [B. Ph. van Milligen, C. Hidalgo and E. Sánchez, Phys. Rev. Lett. 16 (1995) 395], is investigated. It is capable of detecting phase coupling-nonlinear interactions of the lowest (quadratic) orderwith time resolution. To demonstrate its potential, it is applied to numerical models of chaos and turbulence and to real measurements. It detected the coupling interaction between two coupled van der Pol oscillators. When applied to a model of drift wave turbulence relevant to plasma physics, it detected a highly localized coherent structure. Analyzing reflectometry measurements made in fusion plasmas, it detected temporal intermittency and a strong increase in nonlinear phase coupling coinciding with the L/H (Low-to-High confinement mode) transition.

We define a model for the evolution of a long series of electric power transmission system blackouts. The model describes opposing forces which have been conjectured to cause self-organized criticality in power system blackouts. There is a slow time scale representing the opposing forces of load growth and growth in system capacity and a fast time scale representing cascading line overloads and outages. The time scales are coupled: load growth leads to outages and outages lead to increased system capacity. The opposing forces result in a dynamic equilibrium in which blackouts of all sizes occur. The model is a means to study the complex dynamics of this dynamic equilibrium. The Markov property of the model is briefly discussed. The model dynamic equilibrium is illustrated using initial results from the 73 bus IEEE reliability test system.