Analysis of Failures in Power Grids

Abstract

This paper focuses on line failures in the transmission system of power grids. Recent large-scale power outages demonstrated the limitations of percolation-and epidemic-based tools in modeling failures and cascades in power grids. Hence, we study failures and cascades by using computational tools and a linearized power flow model. We first obtain results regarding the Moore-Penrose pseudo-inverse of the power grid admittance matrix. Based on these results, we analytically study the impact of a single line failure on the flows on other lines and introduce metrics to evaluate the robustness of grids to failures. We also illustrate via simulation the impact of the distance and resistance distance on the flow increase following a failure, and discuss the difference from the epidemic models. We use the pseudoinverse of admittance matrix to develop an efficient algorithm to identify the cascading failure evolution, which can be a building block for cascade mitigation. Finally, we show that finding the lines whose removal results in the minimum yield (the fraction of demand satisfied after the cascade) is NP-Hard and present a simple heuristic for finding such a set . Overall, the results demonstrate that using the resistance distance and the pseudoinverse of admittance matrix provides important insights and can support the development of algorithms for designing robust power grids and controlling the evolution of a cascade upon failures.

On distributed submodular maximization with limited information

“…The classes of random graphs chosen here are Erdös-Rényi, Barabási-Albert preferential attachment, and WattsStrogatz model graphs, which are commonly used to model social networks [29], as well as physical networks [30]. Given a graph G, we compute the chromatic number χ(G) and the and clique number ω(G) using their standard integer programming formulations [31], and solve each integer program using Gurobi [32].…”

On Extension of Effective Resistance With Application to Graph Laplacian Definiteness and Power Network Stability

“…By (6c) and (6d), the first term of right-hand-side of (27) is zero, and the second term is r eff (L G , V a , V b ). Therefore, r eff (L G , V a , V b ) is the optimal value of problem (22) as well as the original problem in (9). Moreover, we have r eff (L G , V a , V b ) > 0 since the objective v T L G v is nonnegative and the only case that makes it zero (v = 1 n ) is excluded by (22b).…”

“…For instance, the effective resistance is closely connected to other graph theory concepts such as Kirchhoff Index, random walks and Foster's theorem [5], which have applications in, e.g., designing online algorithms [6] and describing molecular structure in chemistry [7]. The effective resistance has also been used in many fields of electric power networks, such as cascading failures [8,9], network partitioning [10] and power network stability [11][12][13]. In addition, since the effective resistance is defined by the graph Laplacian matrix [1], it is useful in quantifying the performance of network control problems where the graph Laplacian plays an important role.…”

Risk Assessment of Multi-Timescale Cascading Outages Based on Markovian Tree Search

“…According to [37], cut branches can be identified with complexity of () OE , where E is the number of connected branches. Denote the admittance matrix of the grid as Y , then its Penrose-Moore pseudo-inverse uniquely exists, denoted as Z ZY…”

“…However the inverse of (26) requires that {} k i is not a cut set. If {} k i is a cut set, then the update of Z has to be realized with SVD of Y , which has complexity of 3 () OV [37]. Theoretically, the SVD is computationally expensive, and an alternative that searches islands and simulate events on each island respectively consumes less computational resources.…”