Robustness of power systems under a democratic-fiber-bundle-like model

Abstract

We consider a power system with N transmission lines whose initial loads (i.e., power flows) L1, . . . , LN are independent and identically distributed with PL(x) = P [L ≤ x]. The capacity Ci defines the maximum flow allowed on line i, and is assumed to be given by Ci = (1 + α)Li, with α > 0. We study the robustness of this power system against random attacks (or, failures) that target a p-fraction of the lines, under a democratic fiber bundle-like model. Namely, when a line fails, the load it was carrying is redistributed equally among the remaining lines. Our contributions are as follows: i) we show analytically that the final breakdown of the system always takes place through a first-order transition at the critical attack size p, where E [·] is the expectation operator; ii) we derive conditions on the distribution PL(x) for which the first order break down of the system occurs abruptly without any preceding diverging rate of failure; iii) we provide a detailed analysis of the robustness of the system under three specific load distributions: Uniform, Pareto, and Weibull, showing that with the minimum load Lmin and mean load E [L] fixed, Pareto distribution is the worst (in terms of robustness) among the three, whereas Weibull distribution is the best with shape parameter selected relatively large; iv) we provide numerical results that confirm our mean-field analysis; and v) we show that p ⋆ is maximized when the load distribution is a Dirac delta function centered at E [L], i.e., when all lines carry the same load; we also show that optimal p ⋆ equals α α+1. This last finding is particularly surprising given that heterogeneity is known to lead to high robustness against random failures in many other systems.PACS numbers: 64.60. Ht,

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“…Also, there is a body of works on modeling the load and capacity of the transmission lines in the power grid literature, the most relevant of which aim to model the cascading failures caused by load fluctuation on the transmission lines and propose effective network designs, e.g., [29], [30], [31]. In this literature, the network consists of multiple transmission lines; upon failure, the load of the failed lines will be redistributed among all the remaining ones.…”

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“…This leads to an increase in load carried by all remaining nodes, which in turn may lead to further failures of overloaded nodes, and so on, potentially leading to a cascade of failures. The equal load redistribution rule takes its roots from the democratic fiber bundle model [2], [12], and has been recently used by Pahwa et al [25] in the context of power systems; see also [36], [39]. The relevance of the equal load-redistribution model for power systems stems from its ability to capture the long-range nature of the Kirchhoff's law, at least in the meanfield sense, as opposed to the topological models where failed load is redistributed only locally among neighboring lines [11], [34].…”

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