DIMACS Center, CoRE Building, Rutgers University

**Organizers:****Daniel Bienstock**, Columbia University, dano at columbia dot edu**Michael Chertkov**, Los Alamos National Lab, chertkov at lanl.gov

Title: Lyapunov Functions to Characterize Cascading Failure Vulnerability in Dynamic Power Grid Models

In an electric power network experiencing a transient following some initial disturbance, relay actions interrupting transmission lines or disconnecting generators may create further transient over-current in other lines or over-frequency in other generators. Relay actions that disconnect equipment in response to such overloads may induce cascading equipment disconnection, and possibly system wide outage. Full time domain simulations to credibly examine the wide range of possible switching actions that can lead to cascading failure events under transient conditions is extremely costly. To address these problems, this work extends prior results on bi-stable models of transmission line over-current relays, to now also include generator disconnection under the action of over/under-frequency relays. By incorporating such relay action in a smooth dynamic model, the problem becomes amenable to analytic stability analysis, with new insights and computational efficiencies to be gained. In particular, partially degraded network configurations appear as multiple equilibria in this augmented transient stability model, and vulnerability to transition to a degraded configuration may be quantified via a closed form "energy-like" Lyapunov function. This work develops the model, and demonstrates the construction of the associated Lyapunov function, confirming this function to be locally positive definite about stable equilibria and non-increasing along system trajectories (including trajectories that capture the relay action). This work is intended as a first step to demonstrate the promise for Lyapunov methods to contribute to cascading failure analysis. As one application, we suggest that short duration simulation be augmented with a computationally inexpensive Lyapunov function threshold test, to yield a sufficient condition for a post-disturbance state to be "captured" in the potential well about a partially degraded stable equilibrium, thereby ensuring no further line or generator outages can occur along the remaining portion of the trajectory.

Title: Sampling and Speed for Cascading Outage Simulations

It is challenging to estimate large blackout risk from simulations of cascading outages in electric power transmission systems. In order to estimate risk, proper sampling is needed to represent the joint uncertainties of the initial conditions, initial outages, and evolution of the cascade. I will also discuss new ways to speed up the simulation using the splitting method and efficiently obtain the probabilities of large blackouts from the simulation output using high-level branching process models.

Title: Estimating Cascading Failure Risk with Random Chemistry

Estimating cascading failure risk is hard for two primary reasons. First, the process of cascading in real power systems is very complicated, involving many different mechanisms. Some of these mechanisms can be modeled with tractable models (ie, cascading overloads using DC or AC power flow models), but others (such as operator errors) are very difficult to model. Second, cascades are rare events typically triggered by unanticipated combinations of disturbances or errors that interact to produce an unanticipated outcome. Identifying the potentially dangerous combinations in advance is very difficult because the search space grows exponentially with the size of the network. This talk will present a way to deal with the combinatorial problem using a new Random Chemistry algorithm that can find large collections of dangerous combinations (malignancies) of disturbances, given a model of cascading failure. This talk will describe the algorithm, results from its application on a system with 2896 possible single outages, and describe preliminary thoughts on how this method can be used to obtain measures of cascading failure risk.

Title: Modeling and prevention of cascading outages

I will presents the results of our recent studies of cascade propagation mechanisms in large scale power systems. We have developed a simple model of cascading outage propagation based on classical DC power flow description. Within the framework of this model we first study the effect of random load and generation related to intermittent renewable sources. We show that the system experiences a sharp second order phase transition at some critical fluctuation level, where the average damage grows dramatically. Second, we analyze the statistical distribution of individual cascade outcomes and quantify the effect of operator intervention on the final outcome. The distribution of the outcomes is very broad and suggests that optimized remedial action can have dramatic effect on the overall blackout risks. Finally, we present novel techniques for selection of dangerous cascade initiating contingencies and discuss the mechanisms for cascade propagation. The most dangerous contingency configurations correspond to weak cuts of the system. We propose new ways of remedial action selection and alternative explanations of the blackout distribution size, relating the the observed power laws to hierarchical structure of real power grids.

Title: Network Resilience to Correlated and Cascading Failures: A Percolation View

Critical infrastructure such as communication networks and power grids are susceptible to correlated node and link failures, as well as cascading failures resulting from virus epidemics and power blackouts. In this talk, we discuss a framework for understanding network resilience based on the theory of percolation. We present analytical conditions for the existence and non-existence, respectively, of a large connected component of operational nodes and links after degree-dependent failures in large-scale networks with geometric constraints. Combining these conditions with a simple but descriptive model for the spread of cascades, we derive analytical conditions for the occurrence and non-occurrence of cascading node and link failures, respectively, in large-scale networks with geometric constraints.

Title: Power Grid Vulnerability to Geographically Correlated Failures Analysis and Control Implications

We consider power line outages in the transmission network of the power grid, and specifically those caused by a natural disaster or a large scale physical attack. In the transmission network, an outage of a line may lead to overload on other lines, thereby eventually leading to their outage. While such cascading failures have been studied before, our focus is on cascading failures that follow an outage of several lines in the same geographical area. Such an event may have a devastating effect not only on the power grid but also on the interconnected communication networks. We study an analytical model of such failures and show that it differs from other models used to analyze cascades in networks (e.g., epidemic/percolation-based models). Inspired by methods developed for computer networks survivability analysis, we then show how to identify the most vulnerable locations in the transmission network. We perform extensive numerical experiments with real grid data to estimate the various effects of geographically correlated outages. The results provide insights into the relationships between various parameters and performance metrics, such as the size of the original event, the final number of connected components, and the fraction of demand (load) satisfied after the cascade. Finally, we consider the sensitivity of the model to various assumptions and briefly discuss cascade mitigation methods. The developed techniques can indicate where grid monitoring efforts should take place, and hence, will have impact on the deployment of the smart grid networking infrastructure. Joint work with Andrey Bernstein (EPFL), Daniel Bienstock (Columbia University), David Hay (Hebrew University) and Meric Uzunoglu (Qualcomm)

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Document last modified on January 31, 2013.