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

An attempt is made in This work to present critical literature review and an up-to-date and exhaustive bibliography on the AGC of power systems. Various control aspects concerning the AGC problem have been highlighted. AGC schemes based on parameters, such as linear and nonlinear power system models, classical and optimal control, and centralized, decentralized, and multilevel control, are discussed. AGC strategies based on digital, self-tuning control, adaptive, VSS systems, and intelligent/soft computing control have been included. Finally, the investigations on AGC systems incorporating BES/SMES, wind turbines, FACTS devices, and PV systems have also been discussed.

Recent philosophies of automatic generation control strategies in power systems

We consider the problem of security constrained optimal control for discrete-time, linear dynamical systems in which control and measurement packets are transmitted over a communication network. The packets may be jammed or compromised by a malicious adversary. For a class of denial-of-service (DoS) attack models, the goal is to find an (optimal) causal feedback controller that minimizes a given objective function subject to safety and power constraints. We present a semi-definite programming based solution for solving this problem. Our analysis also presents insights on the effect of attack models on solution of the optimal control problem.

Safe and Secure Networked Control Systems under Denial-of-Service Attacks

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

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Mixed-integer nonlinear optimization

We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

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Recent Advances in Graph Partitioning

In this paper, the traditional automatic generation control (AGC) two-area system is modified to take into account the effect of bilateral contracts on the dynamics. The concept of distribution companies (DISCO) participation matrix to simulate these bilateral contracts is introduced and reflected in the two-area block diagram. Trajectory sensitivities are used to obtain optimal parameters of the system using a gradient Newton algorithm.

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Simulation and Optimization in an AGC System after Deregulation

Identifying small groups of lines, whose removal would cause a severe blackout, is critical for the secure operation of the electric power grid. We show how power grid vulnerability analysis can be studied as a mixed integer nonlinear programming (minlp) problem. Our analysis reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem. The key new observation behind our analysis is the correspondence between the Jacobian matrix (a representation of the feasibility boundary of the equations that describe the flow of power in the network) and the Laplacian matrix in spectral graph theory (a representation of the graph of the power grid). The reduced combinatorial problem is known as the network inhibition problem, for which we present a mixed integer linear programming formulation. Our experiments on benchmark power grids show that the reduced combinatorial model provides an accurate approximation, to enable vulnerability analyses of real-sized problems with more than 10,000 power lines.

Optimization strategies for the vulnerability analysis of the electric power grid.

We propose a computationally efficient approach to detect severe multiple contingencies. We pose a contingency analysis problem using a nonlinear optimization framework, which enables us to detect the fewest possible transmission line outages resulting in a system failure of specified severity, and to identify the most severe system failure caused by removing a specified number of transmission lines from service. Illustrations using a three-bus system and the IEEE 30-bus system aim to exhibit the effectiveness of the proposed approach.

Severe Multiple Contingency Screening in Electric Power Systems

Identifying small groups of lines, whose removal would cause a severe blackout, is critical for the secure operation of the electric power grid. We show how power grid vulnerability analysis can be studied as a bilevel mixed integer nonlinear programming problem. Our analysis reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem. The key new observation behind our analysis is the correspondence between the Jacobian matrix (a representation of the feasibility boundary of the equations that describe the flow of power in the network) and the Laplacian matrix in spectral graph theory (a representation of the graph of the power grid). The reduced combinatorial problem is known as the network inhibition problem, for which we present a mixed integer linear programming formulation. Our experiments on benchmark power grids show that the reduced combinatorial model provides an accurate approximation, to enable vulnerability analyses of real-sized problems with more than 16,520 power lines.

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Optimization Strategies for the Vulnerability Analysis of the Electric Power Grid

In this paper we propose a two-stage screening and analysis process for identifying multiple contingencies that may result in very severe disturbances and blackouts. In a screening stage we form an optimization problem to find the minimum change in the network to move the power flow feasibility boundary to the present operating point and that will cause the system to separate with a user-specified power imbalance. The lines identified by the optimization program are used in a subsequent analysis stage to find combinations that may lead to a blackout. This approach is applied to a 30-bus system with encouraging results.

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Vaibhav Donde

Papers

Simulation and Optimization in an AGC System after Deregulation

Optimization strategies for the vulnerability analysis of the electric power grid.

Severe Multiple Contingency Screening in Electric Power Systems

Optimization Strategies for the Vulnerability Analysis of the Electric Power Grid

Identification of severe multiple contingencies in electric power networks