The $N-k$ Problem in Power Grids: New Models, Formulations, and Numerical Experiments
SIAM Journal on Optimization
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This paper focuses on cascading line failures in the transmission system of the power grid. Such a cascade may have a devastating effect not only on the power grid but also on the interconnected communication networks. Recent large-scale power outages demonstrated the limitations of epidemic- and percolation-based tools in modeling the cascade evolution. Hence, based on a linearized power flow model (that substantially differs from the classical packet flow models), we obtain results regarding the various properties of a cascade. Specifically, we consider performance metrics such as the the distance between failures, the length of the cascade, and the fraction of demand (load) satisfied after the cascade. We show, for example, that due to the unique properties of the model: (i) the distance between subsequent failures can be arbitrarily large and the cascade may be arbitrarily long, (ii) a large set of initial line failures may have a smaller effect than a failure of one of the lines in the set, and (iii) minor changes to the network parameters may have a significant impact. Moreover, we show that finding the set of lines whose removal has the most significant impact (under various metrics) is NP-Hard. Moreover, we develop a fast algorithm to recompute the flows at each step of the cascade. The results can provide insight into the design of smart grid measurement and control algorithms that can mitigate a cascade.