Towards Estimating the Statistics of Simulated Cascades of Outages With Branching Processes

3410 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013

Towards Estimating the Statistics of SimulatedCascades of Outages With Branching ProcessesJunjian Qi, Student Member, IEEE, Ian Dobson, Fellow, IEEE, and Shengwei Mei, Senior Member, IEEE

Abstract—Branching processes can be applied to simulated cas-cading data to describe the statistics of the cascades and quicklypredict the distribution of blackout sizes. We improve the pro-cedures for discretizing load shed data so that a Galton-Watsonbranching process may be applied. The branching process param-eters such as average propagation are estimated from simulatedcascades and the branching process is then used to estimate the dis-tribution of blackout size. We test the estimated distributions withline outage and load shed data generated by the improved OPAand AC OPA cascading simulations on the IEEE 118-bus systemand the Northeast Power Grid of China.

Index Terms—Branching process, cascading failure, powertransmission reliability, simulation.

I. INTRODUCTION

C ASCADING blackouts of transmission systems are com-plicated sequences of dependent outages that lead to load

shed. Of particular concern are the larger cascading blackouts;these blackouts are rare and high impact events that have sub-stantial risk, and they pose many challenges in simulation, anal-ysis, and mitigation [1].Simulations of cascading outage produce samples of cascades,

and it is useful to be able to statistically describe these simulatedcascades with high-level probabilistic models such as branchingprocesses. Branching processes have descriptive parameters thatcharacterize the system resilience to cascading. For example, thepropagationofoutages isakeyparameter forbranchingprocessesthat quantifies the average extent to which outages cause furtheroutages. Moreover, the branching process can be used to predictthe probability distribution of blackout size (that is, the frequen-ciesofblackoutsasa functionofblackout size) frommuchshortersimulation runs. It is much quicker [2] to estimate the parametersof abranchingprocess fromashorter simulation runand thenpre-dict the distribution of blackout size using the branching processthan it is to run the simulation for very long time to accumulate

Manuscript received September 17, 2012; revised December 22, 2012; ac-cepted January 23, 2013. Date of publication March 07, 2013; date of currentversion July 18, 2013. This work was supported in part by the Tsinghua Schol-arship for Overseas Graduate Studies, National High Technology Research andDevelopment Program of China (863 Program, 2011AA05A118), NSFC grant-50525721, Special Funding for National Excellent Doctoral Dissertation Au-thor-201153, DOE grant DE-SC0002283, and NSF grant CPS-1135825. Paperno. TPWRS-01049-2012.J. Qi and S. Mei are with the State Key Laboratory of Power Systems, Elec-

trical Engineering Department, Tsinghua University, Beijing 100084, China(e-mail: [email protected]; [email protected]). J.Qi visited the Electrical and Computer Engineering Department at Iowa StateUniversity, Ames, IA 50011 USA.I. Dobson is with the Electrical and Computer Engineering Department, Iowa

State University, Ames, IA 50011 USA (e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRS.2013.2243479

enough cascades (especially the rare large cascades) to empiri-cally estimate the distribution of blackout size.However, these advantages of branching processes can only

be realized when the branching process is validated, and thisneeds to be done for a range of different simulations and testcases, and for real data. The progress so far in establishingbranching process models for estimating cascade propagationand distributions of blackout size is:1) Branching processes can match the distribution of numberof line outages [2] and load shed [3] simulated by the OPAsimulation on the IEEE 118- and 300-bus systems. Thereis also a match for the distribution of load shed for theTRELSS simulation in one case of an industrial system ofabout 6250 buses [3].

2) Branching processes can match the propagation and distri-bution of the number of cascading line outages in real data[4], [5].

3) Branching processes can approximate CASCADE, anotherhigh-level probabilistic model of cascading [6].

In matching load shed in [3] for item 1), two approaches andtypes of branching processes were tried to accommodate thecontinuously varying load shed data. (Note that line outages areeasier than load shed since number of line outages is a nonneg-ative integer whereas load shed varies continuously.) The ap-proach that worked best in [3] discretized the load shed dataso that it became integer multiples of the chosen discretizationunit. Then the discretized load shed data was processed with aGalton-Watson branching process, which works with nonneg-ative integers. However in [3], the choice of the discretizationunit was ad hoc and no systematic approach was given. Ana-lyzing the load shed data with the branching process is importantbecause load shed is measure of blackout size that is of great sig-nificance to both utilities and society, whereas line outages areof direct interest only to utilities.Instead of applying a Galton-Watson branching process to

discretized load shed, it is also possible to directly analyze loadshed using a continuous state branching process model [3]. Eachgeneration is a continuously varying amount of load shed thatpropagates according to a continuous offspring probability dis-tribution. One challenge with the continuous state branchingprocess is determining the form to assume for the continuousoffspring distribution. Reference [3] assumes a Gamma distri-bution for simplicity and computational convenience, but theproblem of a justified choice of continuous offspring distribu-tion for cascading load shed remains open. The calculations forcontinuous state branching processes are analogous to the calcu-lations for Galton-Watson branching processes but more tech-nically difficult.

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QI et al.: TOWARDS ESTIMATING THE STATISTICS OF SIMULATED CASCADES OF OUTAGES WITH BRANCHING PROCESSES 3411

In this paper, we advance the application of branching pro-cesses to cascading blackouts by improving the processing ofload shed data and validating the branching process on othersimulations and test systems. In particular:1) We describe a new and systematic procedure to discretizeload shed data so that it can be analyzed as a Galton-Watson branching process with a Poisson offspring distri-bution. This new procedure can be applied to any continu-ously varying measure used to track cascading.

2) We show how branching process models canmatch the dis-tribution of number of cascading line outages and load shedsimulated by two enhanced versions of the OPA simulationon the IEEE 118-bus system and the Northeast Power Gridof China (NPGC).

The rest of this paper is organized as follows. Section II brieflyintroduces Galton-Watson branching processes. Section IIIexplains estimating branching process parameters and the dis-tribution of outages. Section IV explains how to discretize theload shed data so that it can be analyzed with a Galton-Watsonbranching process. Section V introduces the improved OPA andAC OPA cascading simulations. Section VI tests the proposedmethod with line outage data and load shed data generated bythe two simulations on IEEE 118-bus system and the NPGCsystem. Finally the conclusion is drawn in Section VII.

II. BRANCHING PROCESSES

Branching processes have long been used in a variety of ap-plications to model cascading processes [7], [8], but have beenapplied to the risk of cascading outage only recently [2]–[6],[9]. This section gives an informal overview of branching pro-cesses. For more detail, see [5], [7], and [8].The Galton-Watson branching process gives a high-level

probabilistic model of how the number of outages in a blackoutpropagate. The initial outages propagate randomly to producesubsequent outages in generations. Each outage in each gen-eration (a “parent” outage) independently produces a randomnumber 0, 1, 2, 3, … of outages (“children” outages) in the nextgeneration. The distribution of the number of children from oneparent is called the offspring distribution. The children outagesthen become parents to produce another generation, and so on.If the number of outages in a generation becomes zero, thenthe cascade stops.The mean of the offspring distribution is the parameter .is the average family size (the average number of children

outages for each parent outage). quantifies the tendency forthe cascade to propagate, since large average family sizes willtend to cause the outages to grow faster. In this paper, we have

, and the outages will always eventually die out. Thebranching process model does not directly represent any of thephysics or mechanisms of the outage propagation, but, after itis validated, it can be used to predict the total number of out-ages. The parameters of a branching process model can be es-timated from a much smaller data set, and then predictions ofthe total number of outages can be made based on the estimatedparameters. While it is sometimes possible to observe or pro-duce large amounts of data to make an empirical estimate of thetotal number of outages (indeed this is the way the branchingprocess prediction is validated), the ability to do this via the

branching process model with much less data is a significant ad-vantage that enables practical applications. The simplicity of thebranching process model also allows a high-level understandingof the cascading process without getting entangled in the var-ious and complicated mechanisms of cascading. The branchingprocess should be seen as complementary to detailed modelingof cascading outage mechanisms.The intent of the branching process modeling is not that each

parent outage in some sense “causes” its children outages; thebranching process simply produces random numbers of outagesin each generation that can statistically match the outcome ofthe cascading. For example, when used to track the numberof transmission line outages, the branching process does notspecify which lines outage, or where they are, or explain whythey outaged. The branching process describes the statistics ofthe number of lines outaging in each generation, and the statis-tics of the total number of lines outaged.Similarly, when used to track load shed, the branching

process does not specify which load is shed, or where, orexplain why load is shed. It is worth noting that the underlyingcascading processes and interactions that we are tracking witha branching process are complicated and varied. For example,there are situations in which load shed tends to inhibit thechance of further cascading and there are other situations inwhich load shed can tend to increase the chance of furthercascading. It is not obvious that branching processes can sum-marize this complexity, and it is a goal of this paper to showevidence that this can be done. In particular, we show that thebranching process describes the statistics of the load shed ineach generation for the purpose of summarizing the cascadepropagation so that the distribution of the total load shed can bestatistically estimated. The branching process analysis is a bulkstatistical analysis that should be regarded as complementaryto detailed causal analysis.

III. ESTIMATING BRANCHING PROCESS PARAMETERS

This section explains how the branching process parametersare estimated from the simulated data and used to estimate thedistribution of blackout size, in particular the distribution of thenumber of line outages or the distribution of load shed.The simulation naturally produces the line outages or load

shed in generations or stages; each iteration of the “main loop”of the simulation produces another generation. In the case ofline outages, the number of line outages in each generation arecounted. In the case of load shed, the continuously varying loadshed amounts in each generation are processed and discretizedas described in Section IV to produce integer multiples of thechosen discretization unit. In either case, cascades1 are sim-ulated to produce nonnegative integer data that can be arrangedas

generation 0 generation 1 generation 2

cascade 1

cascade 2...

......

......

cascade

1Note that a single outage shedding load followed by no further load shed isregarded as a cascade with only one generation.


where is the number of outages produced by the simula-tion in generation of cascade number . Since the averagepropagation , each cascade eventually terminates with afinite number of generations when the number of outages in ageneration becomes zero. Each cascade has a nonzero number ofoutages in generation 0. The shortest cascades stop in generation1 by having no outages in generation 1 and higher generations,but some of the cascades will continue for several or occasion-ally many generations before terminating. The assumption of apositive number of outages in generation 0 implies that all sta-tistics assume (or, in statistical terminology, are conditioned on)a cascade starting.Note that all the outages are parent outages, and all the out-

ages in generations 1 and higher are children outages. It followsthat the average propagation (the average family size) can beestimated as the total number of children divided by the totalnumber of parents:

(1)

This is the standard Harris estimator of propagation [8], [10].We will also estimate the variance of the offspring distributionas explained in Appendix A and use this variance in Section IVto choose the load shed discretization.The empirical probability distribution of the number of initial

outages can be obtained as

(2)

where the notation is the indicator function that evalu-ates to one when the event happens and evaluates to zero whenthe event does not happen.The average number of initial outages is estimated as

(3)

We assume that the offspring distribution is a Poisson dis-tribution with mean . There are general arguments suggestingthat the choice of a Poisson offspring distribution is appropriate[3], based on the offspring outages being selected from a largenumber of possible outages that have small probability and areapproximately independent.We are most interested in statistics of the total number of

outages produced by the cascades, since indicates eitherthe total number of line outages or the total load shed. Given theprobability distribution of and the average propagation , theformula for calculating the probability distribution of is thefollowing mixture of Borel-Tanner distributions [4]:

IV. PROCESSING AND DISCRETIZING LOAD SHED

This section discusses the processing and discretization ofthe load shed data to produce integer counts of the discretiza-tion unit of load shed. Our Galton-Watson branching processassumes a Poisson offspring distribution, which always has itsvariance equal to its mean. The key idea is to choose the dis-cretization so that the offspring distribution also has its varianceequal to its mean and is therefore consistent with the Poissondistribution.

A. Initial Processing

Very small load shed amounts (less than 0.5% of total load)are considered negligible and rounded to zero. The cascadeswith no load shed are discarded. For those cascades that have noload shed in initial generations and non-negligible load shed insubsequent generations, the initial generations with no load shedare discarded to guarantee that generation zero always startswith a positive amount of load shed. Moreover, when any in-termediate generation with zero load shed is encountered, thecurrent cascade will be ended and a new cascade started at thenext generation with nonzero load shed. There are cascadesin total and denotes the load shed at generation of cas-cade .

B. Discretization

To apply a Galton-Watson branching process to load sheddata, we need to discretize the continuously varying load sheddata in MW to integer multiples of a unit of discretization .In particular, we use the following discretization to convert theload shed MW to integer multiples of MW:

(4)

where is the integer part of . We add 0.5 before takingthe integer part to ensure that the average values of and

are equal. The discretization (4) is straightforward ex-cept that the choice of the discretization unit matters. Here wegive a justifiable way to choose . A previous paper [3] madea heuristic and subjective choice of .We first discuss how the choice of the discretization unitaffects the mean and variance of the offspring distribution.

Consider a second choice of discretization unit and corre-sponding discretized data so that

Neglecting the effects of rounding, so thatand , yields

(5)

The mean of the offspring distribution is a dimensionless ratioof load shed that does not depend on the scaling or units of theload shed [see (1)]. To understand this, we recall that the off-spring distribution is defined as the distribution of the numberof units of load shed in generation assuming one unit ofload shed in generation . When the discretization is changed


TABLE IESTIMATED PARAMETERS FOR LINE OUTAGE DATA

from to , both and are multiplied by , sothat the mean of the offspring distribution does not change. Thevariance of is multiplied by , but since isalso multiplied by , the offspring distribution variance ,which is the variance in generation arising from one unitof load shed in generation , is only multiplied by . Toexplain these scalings with an example, suppose that in the firstcascade there are 2 units of load shed in generation with dis-cretization so that . Then, according to the principlesof branching processes, the distribution of load shed ingeneration is the sum of two independent copies of the off-spring distribution for discretization , and therefore has meanand variance . Now change the discretization to

so that and . Then the dis-

tribution of is the offspring distribution for discretization

, which has mean , and

variance. Thus changing the discretization unit strongly affects

, and in particular increasing decreases proportionally.When we consider the influence of rounding on these scaling

approximations, the mean of the offspring distribution is onlyslightly affected by while the variance of the offspring distri-bution has an overall strong tendency of decreasing propor-tionally with the increase of (strict monotonicity for smallchanges in is not guaranteed). We notate these strong andweak dependencies by writing and respectivelyand we can get andfrom the above analysis.Our calculations assume a Poisson offspring distribution, and

it is well known that the variance of a Poisson distribution isequal to its mean. Therefore, to be consistent with the Poissondistribution, we need to choose a discretization so that the vari-ance of the offspring distribution is equal to its mean. That is,we need to choose so that .Specifically, we discretize the data for

and estimate and and then the that satisfiesis approximately . Calculated

values of are shown in Table II. This procedure requiresto be estimated and this calculation is explained in the

Appendix. We have found that the distribution of load shed isnot very sensitive to the exact value of , so that the estimateof and the chosen need not be very accurate.

C. Processing After Discretization

After discretization the initial load shed in some cascadesmay become zero. These cascades are discarded.

TABLE IIESTIMATED PARAMETERS FOR LOAD SHED DATA

V. IMPROVED OPA AND AC OPA MODEL

This section briefly summarizes the main features of the im-proved OPA and AC OPA cascading outage simulations. Sincethese are enhanced versions of the OPA model, we first summa-rize the OPA model. For detailed descriptions, see [11]–[13] forOPA, [14] and [15] for improved OPA, and [15]–[17] for ACOPA. This paper uses the forms of the improved OPA and ACOPA simulations in which the power grid is fixed and does notevolve or upgrade.

A. OPA

TheOPAmodel represents transmission lines, loads, and gen-erators and computes the network power flows with a DC loadflow. Each simulation run starts from a solved base case solutionfor the power flows, generation, and loads that satisfy circuitlaws and constraints. To obtain diversity in the runs, the systemloads at the start of each run are varied randomly around theirmean values by multiplying by a factor uniformly distributed in

. Initial line outages are generated randomly by as-suming that each line can fail independently with probability. This crudely models initial line outages due to a variety of

causes including lightning, wildfires, bad weather, and opera-tional errors. Whenever a line fails, the generation and load isredispatched to satisfy the transmission line and generation con-straints using standard linear programming methods. The opti-mization cost function is weighted to ensure that load sheddingis avoided where possible. If any lines were overloaded duringthe optimization, then these lines are those that are likely to haveexperienced high stress, and each of these lines fails indepen-dently with probability . The process of redispatch and testingfor line outages is iterated until there are no more outages. OPAhas been validated against WECC data [18].

B. Improved OPA

Compared with OPA, the improved OPA considers in the cas-cading process the misoperation of protective relays and thefailure of EMS or communications in the dispatching center.The maloperation of relays is simulated by tripping lines thatare not overloaded with probability . is thebase probability of unwanted operation of relays andis the load ratio of the transmission line. The standard linear pro-gramming problem in OPA is calculated with probability and

is the failure rate of the dispatching center, which can becaused by breakdown of the EMS or communications.


C. AC OPA

Both OPA and improved OPA use DC power flow and onlyconsider active power. Bus voltages are considered constant. Incontrast, the AC OPA model uses AC power flow and thus canconsider reactive power and voltage. The operation mode of thesystem is first determined by AC OPF and load shedding andwill be readjusted by AC OPF until there is no further outageor failure once outages happen. The AC OPF takes into accountboth reactive power and voltage constraints.

VI. RESULTS

This section presents results giving the branching process pa-rameters computed from simulated cascades, the distributionsof outages predicted from these parameters, and the compar-ison with the simulated distributions of outages. The cascadingoutage data is produced by the improved OPA simulation [14]on the NPGC system and the AC OPA simulation [16], [17] onthe IEEE 118-bus test system.The NPGC system consists of Heilongjiang, Jilin, Liaoning,

and the northern part of Inner Mongolia. It has about 500 trans-mission lines, more than 300 transformers, 250 substations, andabout 200 generators. We consider 500-kV and 220-kV trans-mission lines and substations which correspond to a 568-bussystem. The NPGC system data includes the line limits.The IEEE 118-bus system data is standard, except that line

flow limits are determined by running the fast dynamics of theimproved OPA and the slow dynamics of OPA that selectivelyupgrades lines in response to their participation in blackouts[12], starting from an initial guess of the line limits. This proce-dure results in a coordinated set of line limits.For the improved OPA model, we use the same parameters as

those in [14] and , , , and .For the NPGC system , which is the same as [14],and for the IEEE 118-bus system , which is thesame as [2]. For both models the load variability asin [2].For testing the branching process model, the simulation is run

so as to produce 5000 cascading outages with a nonzero numberof line outages or non-negligible load shed.

A. Line Outage Results

We test the branching process model for estimating thedistribution of total line outages on cascading data from the im-proved OPA simulations on the NPGC system under differentload levels. The estimated branching process parameters andare shown in Table I. The distributions of line outages are

shown in Figs. 1 and 2. The distributions of total line outages(dots) and initial line outages (squares) are shown, as well asa solid line indicating the total line outages predicted by thebranching process. The branching process data is also discrete,but is shown as a line for ease of comparison. The branchingprocess prediction of the distribution of total line outagesmatches the empirical distribution quite well.The branching process model is also tested with cascading

data generated by the AC OPA model on the IEEE 118-bussystem. The results are shown in Figs. 3–5. The branching

Fig. 1. Probability distributions of number of line outages by improved OPAon NPGC system at load level 1.3 times the base case. Dots indicate total num-bers of line outages and squares indicate initial line outages; both distributionsare empirically obtained from the simulated cascades. The solid line indicatesthe distribution of total numbers of line outages predicted with the branchingprocess.

Fig. 2. Probability distributions of number of line outages by improved OPAon NPGC system at load level 1.6 times the base case.

Fig. 3. Probability distributions of number of line outages by ACOPA on IEEE118-bus system at base case load level.

process prediction of the distribution of total line outagesmatches the empirical distribution well.


Fig. 4. Probability distributions of number of line outages by ACOPA on IEEE118-bus system at load level 1.2 times the base case.

Fig. 5. Probability distributions of number of line outages by ACOPA on IEEE118-bus system at load level 1.4 times the base case.

B. Load Shed Results

We also test the branching process model for estimating thedistribution of total load shed on cascading data from the im-proved OPA and AC OPA simulations on the NPGC and IEEE118-bus systems. The estimated branching process parametersand are listed in Table II. In contrast with the line outages,

the average propagation does not always increase with loadlevel. When the load level increases from 1.3 times base caseto 1.6 times base case for improved OPA simulation on NPGCsystem, decreases from 0.29 to 0.22.The results comparing the distributions of load shed for im-

proved OPA simulations are shown in Figs. 6–8 and those forAC OPA are shown in Figs. 9–11. The branching process pre-dictions of the distribution of load shed match the empirical dis-tributions well, with a less good match for the high stress caseof Fig. 11.A reasonable objection to the comparison in Sections VI-A

and VI-B is that the same data is used both to estimate the distri-bution and to obtain the empirical distribution. To address thisobjection we divided the data into separate fitting and valida-tion sets. Specifically, we estimate the distribution from the oddnumbered cascades and compare with the empirical distribution

Fig. 6. Probability distribution of total load shed by improved OPA on NPGCsystem at load level 1.15 times the base case.



for the even numbered cascades, and vice verse. The resultingmatches, which are shown in Appendix B, are also satisfactory.

C. Efficiency

For testing and validating the branching process model, weuse 5000 cascades, but the analysis and results of [2] and [3]


Fig. 9. Probability distribution of total load shed by AC OPA on IEEE 118-bussystem at base case load level.

Fig. 10. Probability distribution of total load shed byACOPA on IEEE 118-bussystem at load level 1.2 times the base case.


suggest that, once validated, the approach can be applied formuch fewer cascades. Tables I and II show the propagationestimated using the first 500 cascades. is close to in allcases except for load shed in NPGC at loading level 1.3.Previous work has demonstrated the efficiency gained by es-

timating the distribution of discretized load shed by first es-timating the propagation and then using a branching process

TABLE IIIEFFICIENCY GAIN FOR LOAD SHED DATA

TABLE IVEFFICIENCY GAIN FOR LINE OUTAGE DATA

model. According to [3], “if the initial load shed distributionis known accurately, then accurately estimating the distribu-tion of the total amount of load shed via discretization and theGalton-Watson branching process requires substantially fewercascades.”In particular, let be the probability of shedding total

load MW, computed via estimating from simu-lated cascades with non-negligible load shed and then using thebranching process model. is conditioned on a non-neg-ligible amount of load shed. Let be the probability ofshedding total load MW, computed empirically by simu-lating cascades with non-negligible load shed. If werequire the same standard deviation for both methods, then [3,Section IV] derives the following approximation of the ratioof the required number of simulated cascades as

(6)is a ratio describing the gain in efficiency when using

branching process rather than empirical methods. To obtainnumerically a rough estimate of , we evaluate (6) for al-most largest total load shed MW for all six load shedcases. Here we sort the total load shed of cascadesin descending order and choose to be the th one, where

. is estimated by nu-merical differencing. The results in Table III show thatexceeds by an order of magnitude or more. That is,if the initial load shed distribution is known accurately, thenaccurately estimating the distribution of the total amount ofload shed via discretization and the Galton-Watson branchingprocess requires substantially fewer cascades.Similarly, we can also confirm the efficiency gains for line

outage data by evaluating (6) with for almost largesttotal line outages , which can be determined in a similar wayas choosing . The results in Table IV show thatexceeds by an order of magnitude or more. Similar orbetter efficiency gains for estimating the distribution of numberof lines outages have also been observed in [2].


The efficiency gains are not surprising because estimating theparameters of a distribution is generally expected to be moreefficient than estimating the distribution empirically.

VII. CONCLUSION

In this paper, we advance the application of branchingprocesses to cascades of power grid outages by improving theprocessing of load shed data and testing the capability of thebranching process to estimate distributions of blackout size.Load shed data needs to be discretized before applying a

Galton-Watson branching process with a Poisson offspring dis-tribution. We describe and justify a new way to select the unit ofdiscretization so that it is compatible with the Poisson offspringdistribution. The capability to describe load shed cascading datais useful, and the method should also apply to any continuouslyvarying quantity that is used to track the progression and impactof cascading outages.We used enhanced versions of the OPA simulation on the

IEEE 118-bus system and NPGC systems to produce cascadingoutage data for the transmission lines outaged and the load shed.After discretizing the load shed data, we estimated the averagepropagation and the average size of the initial outage that arethe parameters of a branching process model. We then used thebranching process model to estimate the distributions of linesoutaged and load shed. The estimated distributions are close tothe empirical distributions, suggesting that the Galton-Watsonbranching process model with an average propagation can cap-ture some overall statistical aspects of the cascading of line out-ages and load shed. While previous work on other simulationsand test cases has reached similar conclusions [3], [5], it is nec-essary to test branching processes with many different simula-tions and test cases to establish the use of branching processesfor power system cascading outages. In addition to providinga useful summary describing cascade statistics, the branchingprocess enables the average propagation and the distribution ofblackout size to be estimated with much fewer simulated cas-cades [2]. This is a significant advantage since simulation timeis a limiting factor when studying cascading blackouts.

APPENDIX AESTIMATING VARIANCE OF THE OFFSPRING DISTRIBUTION

This appendix explains estimating the variance of the off-spring distribution that is used to determine the discretizationof the load shed data in Section IV-B. Useful background is in[7] or [8].Let be the total outages

up to and including generation of cascade , and letbe the total outages in all generations of cascade . Group thecascades into groups so that each group has at least initialoutages. The choice of and is discussed below. Letbe the sum of the outages in group and let be the Harrisestimator (1) evaluated for the outages in group .Given , which is the Harris estimator (1) of the mean of

the offspring distribution for all the cascades, we estimate thevariance of the offspring distribution as

(7)

We discuss the choice of and . It is desirable to have largeand large , but there is a tradeoff between and . If there

are a total of initial outages in the simulated cascades, then. To determine values of and , we simulated 5000

realizations of an ideal Galton-Watson branching process withPoisson offspring distribution of knownmean and variance for arange of values of . We found that for small and large , (7)underestimates the variance whereas for large and small ,(7) is a noisy estimate. We chose andbased on this testing.2

There are other estimators of offspring variance in the liter-ature [10], [19], and while (7) seemed to perform well enoughfor our purposes here, the question of the most effective vari-ance estimator remains open, especially for our cascading datain which the offspring mean and variance may vary somewhatwith generation.Now we give some justification for (7), mainly by following

Yanev [20]. Consider a branching processwith and offspring mean and offspring variance .We assume and . (A branching process with

is equivalent to independent branching process cas-cades whose initial outages are all 1. Thus cor-responds to one of the groups of cascades discussed above.)Let denote the total outages up toand including generation . Letbe the Harris estimator for the offspring mean computed from

. Write for thebranching process starting from the th initial outage only. Let

.From [20, equation (8)]:

(8)

We have

[20] states that and .Then the strong law of large numbers implies

Moreover, , anonzero constant, as . Hence from (8)

Now letting , we get as

, and as .

2When load shed data is discretized, the number of initial outages is in-versely proportional to the discretization unit . Choosing proportional toensures that the grouping of the cascades into groups will remain unchangedwhen the unit of discretization changes.


Write to show the dependence of the Harris estimatorfor all the cascades on the total number of initial outages .

Define

By Cauchy-Schwartz

(9)

Since ,from [20, equation (6) and Lemma 1], we obtain

From [20, Theorem 2(ii)]

Choose for some . Then from (9)

and as .

For , and as

That is, . Moreover

, and henceand as . Since we have alreadyshown , we conclude that .Hence for

Since as ,

as .

APPENDIX BCOMPUTING ESTIMATED AND EMPIRICALDISTRIBUTIONS FROM SEPARATE DATA

This Appendix presents results by dividing the data into sep-arate fitting and validation sets. Typical results for both lineoutage data and load shed data are given.In Figs. 12 and 13, we estimate the distribution of the total

outages using the branching process from the odd numbered

Fig. 12. Probability distribution of total line outages by improved OPA onNPGC system at load level 1.3 times the base case.


Fig. 14. Probability distribution of total line outages by AC OPA on IEEE118-bus system at load level 1.2 times the base case.

cascades (solid line) and compare with the empirical distribu-tion from the even numbered cascades (dots). The squares areempirically obtained distribution of initial outages from theeven numbered cascades.In Figs. 14 and 15, we estimate the distribution using the

branching process from the even numbered cascades (solidline) and compare with the empirical distribution from theodd numbered cascades (dots). The squares are empirically



obtained distribution of initial outages from the odd numberedcascades.

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Junjian Qi (S’12) received the B.E. degree in elec-trical engineering from Shandong University, Jinan,China, in 2008 and is currently pursuing the Ph.D.degree in the Department of Electrical Engineeringof Tsinghua University, Beijing, China.His research interests include blackout model, cas-

cading failure, and power system state estimation.

Ian Dobson (F’06) received the B.A. degree inmathematics from Cambridge University, Cam-bridge, U.K., and the Ph.D. degree in electricalengineering from Cornell University, Ithaca, NY,USA.He is currently Sandbulte Professor of Engineering

at Iowa State University, Ames, IA, USA. His in-terests include cascading failure risk, blackouts, andsynchrophasors.

Shengwei Mei (SM’05) received the B.E. degreein mathematics from Xinjiang University, Urumqi,China, the M.S. degree in operations researchfrom Tsinghua University, Beijing, China, and thePh.D degree in automatic control from the ChineseAcademy of Sciences, Beijing, in 1984, 1989, and1996, respectively.He is currently a Professor at Tsinghua University.

His research interests include power system analysisand control.

Towards Estimating the Statistics of Simulated Cascades of Outages With Branching Processes

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