Obtaining statistics of cascading line outages spreading in an electric transmission network from standard utility data

We show how to use standard transmission line outage historical data to obtain the network topology in such a way that cascades of line outages can be easily located on the network. Then we obtain statistics quantifying how cascading outages typically spread on the network. Processing real outage data is fundamental for understanding cascading and for evaluating the validity of the many different models and simulations that have been proposed for cascading in power networks.


I. INTRODUCTION
Complicated series of cascading outages in the transmission network occasionally cause blackouts. These large cascading blackouts are rare, but of substantial risk since their impact is high [1], [2]. In cascading, initial outages propagate and progressively spread across the network, and, if there are many outages, load is shed and there is a blackout. The initial outages can be random failures due to many different causes, including weather, animals, equipment malfunction, earthquakes, operational errors and malicious attacks. The subsequent spreading of the outages beyond the initial outages in a cascade of dependent outages is complicated and includes many ways in which multiple previous outages or a common cause such as weather can weaken the transmission network to make further outages more likely.
To motivate our study, we first consider how the transmission line outages spread in the August 10 1996 Western interconnection blackout. The NERC blackout report [3] shows the initial spread of the cascading as reproduced in Fig. 1. The numbers show the order of the outages. It is clear that the outages propagate to other outages both near and far in the network and that the total extent of the cascade spreading can be large. In particular, the first 18 line outages of the blackout occur on the network formed in section II that is a subnetwork of the Western interconnection, so we located the 18 outages on this network. One way to measure the distance between two line outages counts the minimum number of buses in a path  in the network joining the two lines. For example, two lines with a common bus are a distance one apart. In the case of the first 18 line outages of the blackout, we find that the distance between successive line outages ranges from 2 to 6 buses and has a mean value of 3.2. The maximum extent of the first 18 cascading outages is 4 buses away from the initial outage.
This example of the August 10 1996 blackout shows the initial spread of one blackout and shows what can possibly happen. However, one cannot draw general conclusions about how cascades typically spread from one sample. Indeed, the August 10 1996 blackout is one of the more serious blackouts that has ever occurred in the Western interconnection, whereas the most common cascades, by careful design and operation of the power system, are that an initial outage occurs and either no outages or only a few outages follow. In order to account for successful mitigation as well as failures of mitigation, the assessment and mitigation of cascading risk must account for cascades of all sizes.
The detail of the spreading of cascading outages can be studied either by analyzing the complexities of particular blackouts after they occur [4], [5], or by simulating some subset of the mechanisms for cascading [6], [7], [8], [9]. These approaches are very useful both in understanding cascading blackouts and suggesting ways to mitigate particular mechanisms of cascading failure. The spatial correlation of Euclidean distance between outages is computed in [10] for the July 2 and August 10 1996 Western interconnection blackouts. However, cascading failure remains a hard problem requiring multiple approaches. In this paper we pursue another and complementary approach which is bulk statistical analysis of typical observed cascading data. One advantage of analyzing real data is that there are no modeling assumptions.
Bulk statistical analysis can describe the size and extent of cascading from real or simulated cascades so that blackout risk can be quantified. For example, cascading transmission line outages from utility data [11], [12] can be used to estimate the average propagation of transmission line outages during the cascade as well as the probability distribution of cascading size in terms of number of transmission line outages. Similar efforts for simulated cascades in [13], [14] characterize the distribution of blackout size in terms of load shed. These studies quantify the average growth in blackout size during the cascade and the probability of the cascade growing to a given blackout size. Observed WECC data for the causes and frequencies of common mode and dependent outages are analyzed in [15].
There has also been general progress in describing how cascades spread in simulated cascades, especially those assuming that cascades propagate by line outages causing static overloads. Overloaded line cascading outage interactions are analyzed using network resistance distance and line outage distribution factors in [16]. Line outage distribution factor calculations of the effect of double contingencies in overloading other lines are used to find all the critical N-2 contingencies in [17]. Progression of cascading through sets of lines is described in the simulation of [18]. Interactions between cascading line outages are described by line interactions graphs different than the transmission network topology in [19], [20].
This paper processes observed transmission line outage data to obtain the statistics of how cascades spread in a real power network. This is, to our knowledge, the first statistical study of typical cascade spreading and spatial extent based on real data. The statistics of the manner and extent of real cascade spreading is basic information that can support the analysis and mitigation of cascading. For example, the chance of a cascade spreading a certain amount can inform the design of area monitoring and control to mitigate cascading, and the fraction of cascading interactions at a given distance in the network is of interest in distinguishing the mechanisms of cascading that more frequently arise in practice.
Moreover, there is a large variety of many different simulation models of cascading that are claimed to represent cascading in power networks [21], [22], and the extent to which the statistics of cascade spreading match the observed statistics serves either to validate the simulation or to suggest ways to improve or disprove the simulation model [23]. It is especially important to make this comparison since the real data incorporates all the mechanisms of cascading whereas the current simulations only represent a limited and varying subset of the dozens of plausible cascading mechanisms [21], [23]. More generally, the objective of the validation of the models and simulations with real data is to determine which mechanisms need to be represented and in what detail in order to be able to do cascading failure risk analysis with confidence in the results. To achieve this objective, it is necessary to develop methods of data processing so that the statistics of typical real cascades can be obtained and compared with the statistics of simulated cascades.
Many countries, including the United States and Canada, collect useful transmission line outage data, and it would seem straightforward to sort this data into individual cascades and determine where on the network the successive line outages are located, and hence obtain the statistics of cascade spreading. However, this is difficult unless a network model consistent with the outage data is available. Indeed, an initial effort using observed North American transmission line outage data encountered substantial difficulties in automating the location of the line outages in a network model that was not consistent with the outage data [24]. For example, single buses representing a single substation in the observed line outage data can correspond to multiple buses of a detailed network model, and the details of corresponding bus names can differ. Single lines in the observed line outage data can correspond to several sections of lines in the detailed network model. Devices such as transformers in the detailed network model need to be accounted for, and the areas and voltage levels covered by the observed line outage data and the detailed network model need to be coordinated. Overall, an automated analysis is difficult since the observed line outage data corresponds to a particular reduction of a detailed network model, and it is not straightforward to perform that reduction in order to be able to relate the network implicit in the observed data with the detailed network model. These difficulties can be overcome to some extent by a sustained combination of automatic and hand processing; indeed [24] processes one year of line outage data for higher voltage levels, but it remains challenging and arduous to process enough data for statistically meaningful results. These difficulties are not surprising; they are an example of the general difficulty of coordinating different data bases containing different power system network descriptions. This paper describes a much better approach: we discovered that it is practical to form a satisfactory network directly from the line outage data as explained in section II.
The goal of this paper is to analyze observed cascading data to quantify how cascades typically spread. We describe a practical method to process standard utility data to locate outages on a network and obtain some bulk statistics of the spread, and illustrate the new method with real data that is publicly available. Similar data is produced by North American utilities for reporting to NERC, and also by some utilities worldwide, so that the method can be applied broadly to existing utility data.

II. FORMING THE NETWORK FROM UTILITY DATA
The required data is a list of recorded transmission line outages 1 , including the outage start time (to the nearest minute suffices) and the names of the buses at both ends of the line, and, for multiple circuits between the same two buses, the circuit number. The automatic line outages should be identified, since the cascade analysis should primarily address the automatic outages. For some purposes it is also useful to know the line length, nominal voltage rating and district. All this data is standard. For example, this data is reported by North American utilities in NERC's Transmission Availability Data System (TADS) [25], [26] and is also collected in several other countries.
The transmission line outage data used in this paper starts from 44 593 automatic and planned line outages 2 recorded by a North American utility over a period of 14 years starting in January 1999 [27]. The data requires some cleaning adjustments before the main processing. Outages in districts remote from the main network, outages of 9 lines rated below 68 kV, outages of 7 lines that did not have bus names for each end of the line and 10 rural lines that seemed disconnected from the main network were all deleted. About 20 bus names were adjusted to eliminate duplicate forms of the same bus name or to combine buses in the same or adjacent substation. This left 42 561 automatic and planned line outages from the main connected network of the utility, with each line outage having a sending end and receiving end bus identified from a list of unique bus names. 3 Then the network model was constructed simply by joining two buses with a transmission line if there was in the data an automatic or planned outage of the line joining those buses. This procedure produces a subset of the actual network, capturing only those branches that have experienced an outage within the time horizon of the data. Since it is not obvious how much outage data is needed to form a sufficiently complete network model in this way, we address and confirm the completeness of the network model formed in this way in section VII.
The network model obtained from the data is shown in Fig. 2. It is a connected network with 361 buses and 614 lines. An important practical advantage of forming the network model directly from the outage data is that there is then no difficulty establishing the correspondence between the network model and the outage data; the correspondence is immediate by construction. For example, an observed cascade obtained from the data set is located on the network as shown in Fig. 3. Fig. 3 changes or omits identifying details since it is bad practice to publish these when it is not absolutely necessary. 2 Lines that are normally out are ignored. 3 We do not process outages of sections of lines or taps of lines or feeders in forming the network and analyzing the outages.

GENERATIONS
Having formed the network model from both the automatic and planned line outages, the analysis of the cascade spreading proceeds with only the automatic outages. There are 10 942 automatic outages in the data. One motivation for analyzing only the automatic outages is that cascading focusses on uncontrolled outages; for example, NERC defines cascading as "the uncontrolled successive loss of system elements triggered by an incident at any location [28]." The structure of cascading is that each cascade starts with initial outages in the first generation followed by further outages grouped into generations 2, 3, 4, ... until the cascade stops. The first step in processing the line outages is to group the line outages into individual cascades, and then within each cascade to group the outages that occur in close succession into generations. The grouping of the outages into cascades and generations within each cascade is done based on the outage start times according to the method of [12]. We summarize the procedure here and refer to [12] for the details. The grouping is done by looking at the gaps in start time between successive outages. If successive outages have a gap of one hour or more, then the outage after the gap starts a new cascade (note that operator actions are usually completed within one hour). Within each cascade, if successive outages have a gap of more than one minute, then the outage after the gap starts a new generation of the cascade (note that fast transients and protection actions such as auto-reclosing are completed within one minute). Note that since the outage times are only known to the nearest minute, the order of outages within a generation often cannot be determined. This procedure applied to the 10 942 automatic outages yields 6687 cascades. 84% of these cascades have only one generation of outages and do not spread further.

IV. NETWORK DISTANCES
To quantify the spatial spreading of the cascading line outages, we specify two measures of distance in the network between two lines.
The network distance between lines L i and L j in terms of number of buses 45 is defined as d bus (L i , L j ) = minimum number of buses in a network path joining midpoint of L i to midpoint of L j .
For example, the distance d bus of line to itself is zero and the distance d bus of a line to a neighboring line with at least one bus in common is one. The network distance between lines L i and L j in terms of miles of transmission line 6 is defined as Cascading lines occur in generations and we define the network distance between two generations of lines. (Note that from the point of view of the processing that groups 4 It is common to define the network distance between buses as the minimum number of lines in a path between the buses, and d bus (L i , L j ) can be conveniently evaluated using this network distance between buses: Except for the case of the distance of a line to itself, d bus (L i , L j ) is one plus the minimum bus distance between either of the end buses in L i and either of the end buses in L j . This follows since a path between the midpoints of L i and L j with the minimum number of buses must include a path between the end buses of L i and the end buses of L j with the minimum number of buses. 5 The distance d bus (L i , L j ) is the same as the network distance between L i and L j in the line graph of the network. 6 d miles (L i , L j ) can also be conveniently evaluated using a network distance between buses: Except for the case of the distance of a line to itself, d miles (L i , L j ) is half the length of L i plus half the length of L j plus the minimum bus distance in miles between either of the end buses of L i and either of the end buses of L j . outages into generations, lines outaging in the same generation outage simultaneously and their outage times cannot be distinguished.) We write d for the network distance which can either be in terms of number of buses or in terms of miles. Then the mean network distance between generation of lines G i and generation of lines G j is defined as and the maximum network distance between generation G i and generation G j is defined as Probability distribution of network distance d bus mean (G i , G i+1 ) between successive generations of line outages. The error bars show a 95% confidence interval. There are 6687 cascades and 84% of these cascades have only one generation and so do not spread further. To analyze the spreading cascades, we exclude the cascades with only one generation, and only analyze the remaining 1098 cascades with more than one generation.
We are interested in the distance between successive generations of line outages d mean (G i , G i+1 ) where G i and G i+1 are successive generations in the same cascade. There are 2426 such pairs of successive generations in the data. (There are The statistics of the network distance for these successive generations in terms of number of buses is shown in Fig. 4 and Table I. The mean number of buses between successive generations is 2.9 and the median number of buses between successive generations is 2. The most frequent number of buses (23%) is zero; these are cases in which an outaged line is restored but outages again after more than one minute and less than one hour after the initial outage. Next most frequent (17%) is one bus, in which a neighboring line outages. But more than one bus, or equivalently not a repeat outage and not a neighbor, has frequency 60%. So, while neighboring lines do outage, this is less likely than the same line tripping again and much less likely than a non-neighboring line outaging.
The distribution of the distance in network miles of successive generations of the spreading cascades is shown in Fig. 5. Fig. 5 shows that half of the successive generations spread more than 100 miles, one third of the successive generations spread more than 200 miles, and one eighth of the successive generations spread more than 400 miles.
In the processing described so far, we have counted a repeated outage of the same line after more than one minute delay as one additional outage, and the distance spread in this case is zero. While this is reasonable since repeated outages have more impact than a single unrepeated outage, one could alternatively regard the repeat outages as the same as the original outage and count them only once. Then all the successive outages in a cascade move in the network a distance that is greater than zero. The effect of this alternative assumption on the spreading statistics is obtained by conditioning the probabilities on the spreading distance being greater than zero and the results for the bus network distance are also shown in Table I.
We are also interested in the maximum distance that a cascade spreads from the initial generation of outages G 0 : G 0 the initial generation in cascade C k and G i any generation in cascade C k } This maximum spreading distance is shown in Fig. 6 and Table I for number of buses and Fig. 7 for network miles. Most of the probability of zero spreading is due to cascades with only one line outage. If the cascades with only one line outage are excluded, the mean maximum distance spread is 3.8 buses.  All the spreading statistics show the effects of the finite size and edges of the network. In terms of network bus distance, the diameter of a network is the maximum possible distance d bus (L i , L j ) between any two lines L i and L j . The diameter of the network is 15, so this is an upper bound to the spreading results shown in Figs. 4, 6, and Table I. There are two types of cascades: cascades that are confined to the network lines for which the data includes all the outages, and cascades that involve lines outside the network that have missing data. The cascades that involve lines outside the network have spreads that can exceed the spread confined to the network and affect the results in Fig. 6 by tending to increase moderate spreads, tending to reduce larger observed spreads such as spreads of more than 10 buses, and eliminating cascades of more than 15 buses. This "network edge effect" is of interest for future work in trying to quantify how many cascades spread to or from a neighboring power system area.

VI. INDEPENDENT OUTAGES AND DEPENDENT OUTAGES
Cascading arises from a variety of types of dependent outages, and it is useful to check that independently occurring outages do not contribute much to the results. Independent outages occur randomly throughout the year. Most of these independent line outages are isolated in time from each other and from dependent outages, and do not contribute to the measures of cascading effects used in this paper. By chance, occasionally these independent line outages occur in close proximity to each other or to dependent outages in time and do contribute to the measures of cascading. This section quantifies the contribution of statistically independent outages towards the measures of cascading used in this paper.
Each cascade contains at least one initiating outage, and we assume that the first initiating outage 7 in each cascade is independent, and that a fraction R of the remaining outages in all the cascades are also independent outages. Then, since there are 6687 initiating outages and 4255 remaining outages, there are 6687 + 4255R independent outages in the 10 942 observed outages, and this corresponds to an independent outage rate of 0.055 + 0.035R per hour. We model the independent outages as a Poisson process [30] with this rate. In particular, the time differences between independent outages are exponentially distributed with rate parameter 0.055 + 0.035R.
It is convenient to call the outages that remain after the first initiating outage of each cascade is omitted "remaining outages." A remaining independent outage is processed as belonging to a cascade when it occurs after the initiating outage but no later than one hour after the last outage of a cascade. The average time between the initiating outage and last outage of a cascade is 7 minutes or 0.12 hour. Therefore, on average, a remaining independent outage is processed as belonging to a cascade when it occurs less than 1.12 hour after the initiating outage of a cascade. Therefore, assuming that the preceding independent outage is the initiating outage 8 , the fraction R of remaining independent outages that are processed as belonging to a cascade is Solving (1) numerically gives R = 0.06. That is, approximately 6% of the remaining outages, or 4% of all outages, are independent but classified as cascading outages. 7 If there are several initiating outages at the same time, then we arbitrarily choose one of these to be the first one. 8 In the much rarer case that the preceding independent outage during the cascade is not the initiating outage, the approximation (1) is not much different since in this case the remaining independent outage is processed as belonging to a cascade when it occurs on average less than a time T after the preceding outage, where 1.0 < T < 1.12.
One way to appreciate the strong effect of cascading dependence in the cascade spreading results is to remove the dependence by retaining the observed outages, but assigning them artificial and random outage times sampled from a Poisson process. Then, with the same processing described in section III, the number of cascades increases from 6687 to 9956 because there are more cascades with only one outage, but these initial outages propagate very weakly as shown in Table II; the average propagation of outages 9 reduces from 0.28 to 0.08. While it is useful for some purposes, such as classifying outages and their mechanisms, to find out how much independent outages contribute to the cascading results by being lumped together with dependent outages, it should also be emphasized that the power system operators have to deal with multiple outages closely spaced in time regardless of their independence or dependence.
One consequence of our analysis method of grouping the outages into generations is that it classifies automatic outages as initial outages (in the first generation), or as dependent outages (in second or higher generations). Of the 10 942 automatic outages there are 7911 initial outages (comprising 72%) and 3031 dependent outages (comprising 28%).
Since many of the mechanisms for initial outages differ from the mechanisms for dependent outages, it can be expected that there can be some differences between the initial and dependent outages, as observed in simulated cascades [29]. We examine the most frequently involved lines in initial outages and in dependent outages. One half of the 20 lines most frequently involved in initial outages differ from the 20 lines most frequently involved in dependent outages. And one third of the 100 lines most frequently involved in initial outages differ from the 100 lines most frequently involved in dependent outages.
It is also very useful to analyze the causes of the initial outages from utility data since some of these causes can be mitigated, and mitigating the initial outages is one way to reduce cascading outages [25], [26]. This paper does not address this useful aspect of cascade analysis and mitigation because this paper has the complementary objective of opening up possibilities for analysis and mitigation of the dependent cascading outages that follow the initial outages.

VII. COMPLETENESS OF THE NETWORK
As more outages are processed, outages of new lines are encountered, and the network formed from all the outages  processed so far becomes more complete, and, if sufficiently many outages from a fixed network are processed, the network formed from all the outages processed so far converges to the entire network. This section examines this convergence to show that enough outages were processed to form a good approximation of the network. This verifies building the network from the outage data.
In practice, the network slowly changes as new lines are added and old lines retire. Therefore the network formed from the outages includes both retired and new lines over the time period of the processed outages.
Both convergence towards a complete network and the effect of the network changing can be seen in Fig. 8. Fig. 8 initially shows the number of lines in the network converging as more outages are processed and then finally increasing slowly as new lines are added in the later outages. To confirm the convergence, we want to remove the effect of the network changing from Fig. 8. This is done by splitting the outage data into two halves at the midpoint, reversing the order of the data in the second half, and then interleaving the reversed second half with the first half. To give a small example, if 10 outages were originally in the order 1,2,3,4,5,6,7,8,9,10 then the reordering yields 1, 10,2,9,3,8,4,7,5,6. The problem of confirming convergence arises from new lines added in the later, converging portion of the data (new lines added in the earlier portion of the data only affect the transient before the convergence), and the reversal of the second half of data ensures that these new lines are likely to appear in the data before convergence. (Note that simply reversing all the data does not work: the new lines added during the converging portion of the data would now appear before convergence, solving the problem of the new lines, but now there would be a new problem of lines retiring at the beginning of the data appearing as added lines at the end of the reversed data.) The reordered outages that remove the effect of the changing network from the convergence are shown in Fig. 9 and the convergence is clear. The network constructed from the data would ideally include all the lines that have been present for a portion of or all of the time period, and the convergence analysis shows that the network constructed with the data converges to almost all such network lines.

VIII. SENSITIVITY TO CASCADE INTERVAL
We check the sensitivity of the results to the one-hour minimum interval between cascades assumed in processing the data in section III. Changing the minimum interval between cascades from one hour to 30 minutes causes the number of cascades to increase from 6687 to 7332 and changes the results in Table I to the results shown in Table III. The results are close; probabilities change by less than 0.01, with the exception of the lowest distance results in each column, which change by less than 0.04. OPA SIMULATION WITH THE OBSERVED DATA We make an initial comparison between the statistics of cascade spreading simulated by the OPA simulation and the statistics of the observed cascade spreading data from the previous sections. The intent is to show a specific example of using the paper results for improvement and validation of models, and to show how some technical issues in such a comparison may be addressed.
We start by briefly summarizing the OPA simulation, which includes a fast time scale for cascading transmission line outages and a slow time scale for the complex systems feedback shaping the reliability; for details see [9]. The slow time scale of the OPA simulation in which the power system evolves is of the order of days to years. In the slow timescale, the load power demand slowly increases and transmission lines involved in blackouts are upgraded as engineering responses to blackouts and maximum generator power is increased in response to the increasing demand. These slow opposing forces of load increase and network upgrade self organize the system to a complex system dynamic equilibrium that is close to the critical points of the system [9], [1], [33]. The results used here were obtained from OPA in this complex system dynamic equilibrium condition.
OPA was validated on a model of WECC with respect to historically observed statistics in [31], using a 1553-bus network model of WECC developed in a California Energy Commission project for analysis of extreme blackout events [32]. The OPA parameters used were derived from WECC data. The simulated and observed statistics compared were the distribution of blackout sizes, the number of line outages, the number of generations, and the average propagation of number of line outages between generations. The simulated and observed statistics agreed well, except for the average propagation of number of line outages in the later cascade stages. For the present paper, we extend the comparison to the statistics of cascade spatial spreading using the same 1553-bus network and OPA parameters as in [31].
16 788 cascades and 28 361 line outages across the entire WECC were simulated with OPA. Many of these cascades occur wholly or partially outside of the Northwest region 10 of WECC that covers the collection area for the observed data analyzed in the paper. To approximate the conditions of the observed data, we limited the analysis to the 6534 line outages that occurred inside the Northwest region. That is, the analyzed simulated outages correspond to the cascades or parts of cascades that occurred inside the Northwest region. This yielded 6534 outages which are organized into 2768 cascades and 5082 generations using the method of section III. These were then processed to obtain the distances of section IV between generations of line outages using the network distance on the 1553-bus network. Fig. 10 shows the probability distribution of distance in the 1553 network between successive generations of simulated outages as open circles. The mean distance in the  1553 network between the simulated successive generations of outages is 9.8. Since OPA (in common with other cascading failure simulations) does not simulate repeated outages of the same line, the simulated results should be compared with the probability distribution of nonzero distances in the formed network between the observed successive generations of outages. This is the data of the second column of Table I and it is plotted as the solid dots in Fig. 10. The mean distance in the formed network between the observed successive generations of outages is 3.8.
A problem with the comparison in Fig. 10 is that the network distances are measured in different network representations of a similar area of the power system. To correct this, we computed the mean network distance between 10 000 pairs of randomly chosen buses in each network, yielding a mean bus distance of 14.0 in the 1553 bus network and a mean bus distance of 6.4 in the network formed from the data. The ratio of the bus distances, 6.4/14.0 = 0.46, is applied as a scaling factor to the simulated distances to allow the comparison with the same distance scale in Fig. 11 (to allow direct comparison of the probabilities despite the distance scale change, Fig. 11 shows a probability density on the vertical scale). The mean distances expressed in terms of the distance for the formed network are now 4.5 for the simulation and, as before, 3.8 for the observed data. Fig. 11 shows agreement between the statistics of spreading between the simulated and observed data for long-range cas-cading interactions and disagreement for shorter-range interactions. In particular, the simulated data shows fewer interactions at distances 1 or 2 and more interactions at distances 4 or 5 than the observed data. Beyond showing which aspects of reality are well described by the simulation and for which purposes the predictions of the simulation are validated, a particular value of this comparison of spreading statistics is that it suggests the aspects of the simulation to be reconsidered to improve the match. In this case, the results indicate that the modeling of short range cascading interactions is not captured well enough by OPA. Obvious candidates for improvement in the short-range modeling would include protection system effects (such as hidden failures [6]) and representing parallel transmission lines in the network.

X. CONCLUSIONS
It is fundamental to the study of cascading blackouts in transmission networks to be able to process and characterize real cascading outage data. This paper gives new methods to process the spread of transmission line outages from standard utility data already collected by utilities and gives the first statistical characterization of how the cascading outages typically spread on the network. We also discuss the opportunities in cascading risk analysis opened up by these new methods, including the validation of simulations and models for cascading risk analysis that is needed to advance the field. The quantification of typical cascade spatial spreading in this paper complements and augments the quantification of cascading propagation in terms of number of line outages in [12].

A. Contributions to methods of outage data analysis
The paper contributes new methods of analyzing standard outage data: • We solved the problem of obtaining a network model at the same level of detail as and compatible with the outage data by using the outage data itself to form the network. This is shown to be an effective and practical solution to an otherwise messy problem coordinating different network descriptions. Then the recorded outages can be readily located on the network so that their spread can be observed in terms of network distance between generations of outages. • We demonstrated methods to verify that a substantially complete network is formed from the line outage data.
Even when the network changes over the period of observation, reordering the data can verify the network completion. • Cascading is initial outages followed by dependent outages. The processing methods include a small fraction of independent outages among the dependent outages, and we show how to estimate the fraction of independent outages. • We define metrics describing the average distances between generations of line outages on the network in terms of both average number of buses between the generations and network miles.

B. Contributions of the results of the data analysis
We present for the first time some basic statistics of real cascades spreading on a power transmission network. The spreading is quantified in terms of the minimum number of buses or network miles between generations of cascading line outages. In the case of the average minimum number of buses between generations, a generation of outages is followed by a repeated outage of the same line in about one quarter of the cases, and only one sixth are followed by a neighboring line outage. (Even if we ignore the repeated outages, less than one quarter are followed by a neighboring line outage.) A generation of cascading outages is followed by an outage of a non-neighboring line in over half of the cases. As detailed below, these statistics of typical cascade spreading can be used to help validate cascading models and simulations, understand the mechanisms of dependent cascading outages, design cascade mitigation schemes, and develop new cascading models.
The data shows that the lines most involved in initial outages and the subsequent propagation of dependent outages differ somewhat, as might be expected from the different mechanisms involved. Also, there are dramatic differences in the amount of propagation between realistic outages that have cascading dependencies and outages that occur at artificially randomized times.
We show that, after the initial automatic outage, the following cascades of mostly dependent outages contain about 6% independent outages. The distinction between independent and dependent outages is important in understanding and mitigating cascading, but it is also worth noting that in any case the power system operators have to cope with multiple outages regardless of their cause.
While the data used in the paper is from one large transmission operator, the methods can be applied much more broadly because the data is already routinely collected by some transmission operators internationally. In the USA, since TADS data is reported by all transmission operators, the approach can be applied by any transmission operator or by reliability organizations that aggregate the TADS data.
C. Using spatial spreading data to validate models and simulations for cascading risk analysis A large variety of cascading outage models and simulations have been proposed. For example, a 2008 survey paper references a sample of about 25 different models and simulations and many more have been subsequently proposed. However, there has been very little quantitative validation of these models with real data in the sense of reproducing observed cascade statistics so that they can be relied upon to quantify cascading risk. 11 There is a strong need for this validation so that the most important mechanisms of cascading risk analysis can be determined and represented at an appropriate level of detail. The statistics of cascading spread in this paper are a new contribution to the observed cascade statistics. This will enable the validation and improvement of cascading models and simulations that are close to the reality of power systems and the rejection of models that are unrealistic.
Two examples of this validation process are: • Section IX shows an example of this validation process by comparing the statistics of the next generation spreading of line outages in the OPA simulation of cascading with the observed statistics, after an appropriate normalization of the distances in the simulated network. We show how the comparison of spatial spreads distinguishes the matches obtained for long-range and short-range cascading mechanisms, giving insight into which mechanisms need to be differently modeled to get a better match. • Our data clearly shows a substantial fraction of nonlocal propagation of outages. Therefore cascading models from complex network theory that hypothesize nearestneighbor propagation on the topology of the electrical network are inconsistent with the data observed in this paper.

D. Opening up other possibilities in cascading risk analysis
Other possible research directions based on the cascading spread data include: • When analyzing outage data to better understand cascading, dependent outages can now be classified according to local or more global mechanisms according to how far they are from preceding outages in the cascade. The larger risk analysis context is that one can start from available outage data and then use the interaction distances and other attributes to classify the observations of dependent outages into groups of mechanisms. 12 • The observed statistical data on cascading spread could enable approximate high-level stochastic models of the effect of cascading. For example, given some initial damaged components outaging in an earthquake [34], [35], one could sample from a branching process model calibrated with the outage statistics [12] to determine the number of line outages in the next generation and then use the statistics generated in this paper to sample the position on the network of the line outages. This would give a Monte Carlo way to approximate the extent to which the blackout cascaded beyond the initial damage caused by the quake. While such a method is a rough approximation, it is grounded in the reality of the observed data, and may be a useful approximation in some contexts. For example, optimized transmission planning investments accounting for the risk of earthquakes is already highly computationally intensive [34], [35], and a fast, approximate assessment sampling the effect of cascading would be useful 13 . The problem of 12 This general "top-down" and data-driven approach is complementary to detailed modeling of a particular dependent outage mechanism and then seeking data for the detailed model. 13 Even when it is desirable in principle to model cascading more exactly, there are some computations and contexts in which the computational, modeling and data limitations require a simple stochastic approximation, and it is better to use an approximate model than to omit the effect of cascading entirely.
estimating the further spread of cascading blackout is particularly important for earthquakes (and other natural disasters or attacks [36]) because earthquakes typically cause much more death, destruction and economic losses than blackouts, but if the response to the earthquake is delayed by a widespread cascading blackout, the losses from the earthquake will be significantly increased.
• The statistics of how far cascades typically spread are a starting point for designing local and wide area schemes for mitigating cascading. In particular, for design one needs to know typical interaction distances for dependent outages (available from probability distributions such as Fig. 4) and the fraction of cascades that are confined to the design area of the scheme (available from probability distributions such as Fig. 6). That is, in addition to validating cascading models and simulations, there are several promising avenues of engineering risk analysis that open up given that real cascades can be readily tracked spatially on a network.