Ian Dobson

Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization

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.

Critical points and transitions in an electric power transmission model for cascading failure blackouts

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 (algebraic) 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.(c) 2002 American Institute of Physics.

Towards a theory of voltage collapse in electric power systems

Abstract Several recent major power system blackouts are characterised by a progressive decline in voltage magnitude at the system buses. These events are termed ‘voltage collapses’. The mechanisms of voltage collapse are not well defined and the dynamics of the process are not well understood. In this paper, we described the loss of stability when a stable equilibrium point disappears in a saddle node bifurcation and present a simple model of the system dynamics after the bifurcation. The results apply generally to any generic one parameter dynamical system. Then we use these results to propose a model for voltage collapse in power systems. The model gives an explicit mechanism for the dynamics of voltage collapse. We illustrate the model by constructing a simple power system model and simulating a voltage collapse.

Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters

Power system loading margin is a fundamental measure of a systems proximity to voltage collapse. Linear and quadratic estimates to the variation of the loading margin with respect to any power system parameter or control are derived. Tests with a 118-bus system indicate that the estimates accurately predict the quantitative effect on the loading margin of altering the system loading, reactive power support, wheeling, load model parameters, line susceptance and generator dispatch. The accuracy of the estimates over a useful range and the ease of obtaining the linear estimate suggest that this method will be of practical value in avoiding power system voltage collapse.

Evidence for self-organized criticality in a time series of electric power system blackouts

We analyze a 15-year time series of North American electric power transmission system blackouts for evidence of self-organized criticality (SOC). The probability distribution functions of various measures of blackout size have a power tail and rescaled range analysis of the time series shows moderate long-time correlations. Moreover, the same analysis applied to a time series from a sandpile model known to be self-organized critical gives results of the same form. Thus, the blackout data seem consistent with SOC. A qualitative explanation of the complex dynamics observed in electric power system blackouts is suggested.

New methods for computing a closest saddle node bifurcation and worst case load power margin for voltage collapse

Voltage collapse and blackout can occur in an electric power system when load powers vary so that the system loses stability in a saddle node bifurcation. The authors propose new iterative and direct methods to compute load powers at which bifurcation occurs and which are locally closest to the current operating load powers. The distance in load power parameter space to this locally closest bifurcation is an index of voltage collapse. The pattern of load power increase need not be predicted; instead the index is a worst case load power margin. The computations are illustrated in the six-dimensional load power parameter space of a five bus power system. The normal vector and curvature of a hypersurface of critical load powers at which bifurcation occurs are also computed. The sensitivity of the index to parameters and controls is easily obtained from the normal vector. >

On voltage collapse in electric power systems

Several voltage collapses have had a period of slowly decreasing voltage followed by an accelerating collapse in voltage. The authors clarify the use of static and dynamic models to explain this type of voltage collapse, where the static model is used before a saddle-node bifurcation and the dynamic model is used after the bifurcation. Before the bifurcation, a static model can be used to explain the slow voltage decrease. The closeness of the system to bifurcation can be interpreted physically in terms of the ability of transmission systems to transmit reactive power to load buses. Simulation results show how this ability varies with system parameters. It is suggested that voltage collapse could be avoided by manipulating system parameters so that the bifurcation point is outside the normal operating region. After the bifurcation, the system dynamics are modeled by the center manifold voltage collapse model. The essence of this model is that the system dynamics after bifurcation are captured by the center manifold trajectory. The behavior predicted by the model is found simply by numerically integrating the system differential equations to obtain this trajectory.<<ETX>>

An initial model fo complex dynamics in electric power system blackouts

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.

Observations on the geometry of saddle node bifurcation and voltage collapse in electrical power systems

Saddle node bifurcation is a generic instability of parameterized differential equation models. The bifurcation geometry and some implications for the study of voltage collapse in electric power systems is described. The initial direction in state space of dynamic voltage collapse can be calculated from a right eigenvector of a static power system model. The normal vector to the bifurcation set in parameter space is a simple function of a left eigenvector and is expected to be useful in emergency control near bifurcation and in computing the minimum distance to bifurcation in parameter space. >

Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered

When a generator of a heavily loaded electric power system reaches a reactive power limit, the system can become immediately unstable and a dynamic voltage collapse leading to blackout may follow. The statics and dynamics of this mechanism for voltage collapse are studied by example and by the generic theory of saddle node and transcritical bifurcations. It is shown that load power margin calculations can be misleading if the immediate instability phenomenon is neglected. >