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Estimating the Propagation and Extent of Cascading Line Outages from Utility Data with a Branching Process

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Estimating the Propagation and Extent of Cascading Line Outages from Utility Data with a Branching Process PREPRINT OF DOI 10.1109/TPWRS.2012.2190112, IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 27, NO 4, NOV 2012, 2146-2155 1 Estimating the propagation and extent of cascading line outages from utility data with a branching process Ian Dobson, Fellow IEEE Abstract—Large blackouts typically involve the cascading out- have approximately reproduced the distribution of blackout age of transmission lines. We estimate from observed utility sizes obtained from simulations of mechanisms of cascading data how much transmission line outages propagate, and obtain failure in blackouts [16], [17]. Branching processes are used parameters of a probabilistic branching process model of the cascading. The branching process model is then used to predict to analyze observed blackout data in [6], [18]. Moreover, the distribution of total number of outages for a given number of branching processes can approximate other high-level models initial outages. We study how the total number of lines outaged of cascading failure [5], [19], [20], [21]. depends on the propagation as the cascade proceeds. The analysis gives a new way to quantify the effect of cascading failure from Ren and Dobson [18] analyze propagation in a transmission standard utility line outage data. line outage data set that is smaller and has a different source Index Terms—Power transmission reliability, reliability mod- than the data set considered in this paper. They estimate an eling, reliability theory, risk analysis, failure analysis average value of propagation and use a branching process to predict the distribution of the number of line outages from the estimated propagation and the distribution of the initial I. INTRODUCTION line outages. One key difference between this paper and [18] Cascading failure is a series of dependent failures that is that this paper accounts for the varying propagation as the progressively weakens the system. Large electric power trans- cascade progresses whereas [18] assumes a constant value of mission systems occasionally have cascading failures that propagation throughout the cascade. cause widespread blackouts, with up to tens of millions of Chen and McCalley describe an accelerated propagation people affected [1], [2], [3]. The many mechanisms involved in model for the number of transmission line outages in [22]. cascading outages are complicated and varied, but all cascades For parameters based on combined data for North American include transmission line outages. In this paper we describe transmission line outages from [23], the accelerated propaga- and quantify the bulk statistical behavior of cascading trans- tion model applies to up to 7 outages. They examine the fit mission line outages from standard utility data that records of the accelerated propagation model, a generalized Poisson the times of line outages. Working with observed data to distribution, and a negative binomial distribution to the North quantify cascading failure is complementary to simulation of American transmission line outage data. Both the accelerated cascading failure and has the advantage of not requiring the propagation model and the generalized Poisson distribution are approximation of a selected subset of cascading mechanisms. consistent with the data. The generalized Poisson distribution is the distribution of the total number of outages produced II. PREVIOUS WORK by a Galton-Watson branching process with Poisson offspring Branching process models are bulk statistical models of distribution, whose mean number of offspring is constant cascading failure that are analytically and computationally except in the first generation. tractable. Branching processes have long been used to Simulations of cascading failure blackouts in power trans- study cascading processes in many other subjects, including mission systems are reviewed in [8], [24]. These simulations genealogy, cosmic rays, and epidemics [4], but their approximate some selection of cascading mechanisms and application to cascading failure is much more recent, first compute some possible cascading sequences. Greig [25] rep- appearing in [5], [6]. Evidence is accumulating that branching resents cascading failure in more general flow networks. process models can represent probability distributions of blackout size. Qualitatively, observed [7], [8], [9], [10] There is a large literature on cascading in graphs and its and simulated [8], [11], [12], [13], [14], [15] blackout relation to graph topology [26], [27] that is largely motivated statistics show features such as probability distributions of by cascading failures in the internet. Roy [28] considers blackout sizes with power law regions that can be shown by Markov reliability models on abstract influence graphs. Work branching processes [5]. Quantitatively, branching processes on network vulnerability that accounts for network loading includes Watts [29], Motter [30], Crucitti [31], and Lesieutre Ian Dobson is with ECpE department, Iowa State University, Ames IA [32]. Lindley [33] and Swift [34] represent cascading failure 50011 USA [email protected]. We gratefully acknowledge funding in part by California Energy Commission, Public Interest Energy Research in systems with a few components by increasing the failure Program, Power Systems Engineering Research Center (PSERC), and US rate of remaining components when a component fails. Sun DOE project “The Future Grid to Enable Sustainable Energy Systems,” an [35] applies accelerating failure to gradual degradation of a initiative of PSERC. This paper does not necessarily represent the views of the Energy Commission, its employees or the State of California. It has not mechanical system. been approved or disapproved by the Energy Commission nor has the Energy Commission passed upon the accuracy or adequacy of the information. Some initial results for this paper are in [36]. This paper c IEEE 2012 rewrites [36] and has more data and new statistical analysis. PREPRINT OF DOI 10.1109/TPWRS.2012.2190112, IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 27, NO 4, NOV 2012, 2146-2155 2 III. TRANSMISSION LINE OUTAGE DATA generation number sum over 0 1 2 3 ··· generations Transmission line outages are useful diagnostics in monitor- (1) (1) (1) (1) (1) ing the progress and extent of blackouts. One common feature cascade 1 Z0 Z1 Z2 Z3 ··· Z (2) (2) (2) (2) (2) of large blackouts is the successive outage of transmission cascade 2 Z0 Z1 Z2 Z3 ··· Z lines, and the number of lines outaged is a measure of the cascade 3 Z(3) Z(3) Z(3) Z(3) ··· Z(3) blackout extent. The number of transmission lines outaged :::::0 1 2 3 : ::::: : is not a measure directly impacting society such as energy ::::: : unserved or customers disconnected, but it is a measure of (J) (J) (J) (J) (J) cascade JZ0 Z1 Z2 Z3 ··· Z blackout size that is useful to utilities, and for which data is sum over Z Z Z Z ··· available. The line outages that do not lead to load shed can cascades 0 1 2 3 be regarded as precursor data for the line outages that do lead (j) to load shed and blackout. Summing over the generations gives Z , the total number of outages in cascade number j. Summing over the cascades Transmission owners in the USA are required to report gives Z , the total number of outages in generation number transmission line outage data to NERC for the Transmission k k. Table I shows Z , Z ; ··· ;Z . Availability Data System (TADS). The transmission line 0 1 10 The probability distribution of the total number of outages outage data used in this paper is 10 512 outages in TADS data in a cascade is given by recorded by a North American utility over a period of 12.4 years [37]. The TADS data for each transmission line outage pr = probability of r outages in a cascade; r = 1; 2; 3; ··· includes the outage time (to the nearest minute) as well as and is shown in Fig. 1 and Table III. p is estimated by other data. All the line outages are automatic trips. More than r counting the number of cascades with a given number r of 99% of the outages are of lines rated 69 kV or above and outages and dividing by the number of cascades J: 97% of the outages are of lines rated 115 kV or above. There J are several types of line outages in the data and a variety of 1 X h i p^ = I Z(j) = r ; r = 1; 2; 3; ··· (1) reasons for the outages. In processing the data, both voltage r J levels and all types of line outages are regarded as the same j=1 and the reasons for the line outages are neglected. For this (The notation I[event] is the indicator function that evaluates initial bulk statistical analysis, neglecting these distinctions to one when the event happens and evaluates to zero when the is a useful first step as we proceed. It is best to start new event does not happen.) Similarly, the probability distribution methods of analysis in the simplest way first. of the number of initial outages is given by The power system is carefully designed and operated so p = probability of r initial outages in a cascade; that most transmission line outages have only one or a few 0r outages occurring together. Most of these short cascades do r = 1; 2; 3; ··· not cause blackouts (no load is shed), but the longer cascades and is shown in Fig. 1. p0r is estimated as can lead to blackouts. In this paper we statistically analyze the J propagation of line outages in observed cascades of automatic 1 X h (j) i p^ = I Z = r ; r = 1; 2; 3; ··· (2) line outages, regardless of whether load is shed. 0r J 0 j=1 Since there are J = 6316 cascades, an event that occurs for only one of these cascades has an estimated probability 1/6316 IV. GROUPING OUTAGES INTO CASCADES, GENERATIONS = 0.00016. The cascades with a large number of outages may only occur one or a few times.
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