An estimator of propagation of cascading failure

Transcription

1 Thirty-ninth Hawaii International Conerence on System Sciences, January 2006, auai, Hawaii. c 2006 IEEE An estimator o propagation o cascading ailure Ian Dobson, evin R. Wierzbicki, Benjamin A. Carreras, Vickie E. Lynch, David E. Newman Abstract We suggest a statistical estimator to measure the extent to which ailures propagate in cascading ailures such as large blackouts. The estimator is tested on a saturating branching process model o cascading ailure and on ailure data generated by the OPA simulation o cascading blackouts. The estimator is a standard estimator or the branching process parameter modiied so as to avoid saturation eects. We discuss the statistical eectiveness o the estimator and show how estimating the ailure propagation and the initial ailures leads to estimates o the distribution o the numbers o cascading ailures. The estimator is derived rom a simple and high-level branching process model o cascading ailure. We discuss how branching process approximations and in particular the propagation parameter may arise in several dierent models, including models o interacting inrastructures. I. INTRODUCTION Cascading ailure is the main way that blackouts become widespread. For example, the August 2003 blackout spread to a sizable region o Northeastern America by cascading [28]. A traditional way to address cascading ailure is to limit the ailures that initiate cascades. To pursue the complementary approach o limiting the propagation o ailures ater they are initiated, we irst need to be able to quantiy how much ailures propagate rom data. This paper suggests such an estimator and tests it on a branching process model o cascading ailure and on the OPA model [4] o cascading ailure blackouts. The estimator may be applicable to cascading ailure o other large, interconnected inrastructures. We model the growth o blackout ailures using a branching process and then estimate the branching process parameter λ that measures the extent to which ailures propagate. Branching process models are an obvious choice o stochastic model to capture the gross eatures o cascading blackouts because they have been developed and applied to other cascading processes such as genealogy, epidemics and cosmic rays [2]. The irst suggestion to apply branching processes to blackouts appears to be in [5] and subsequent applications to blackouts appear in [4], [6]. I. Dobson and.r. Wierzbicki are with the ECE department, University o Wisconsin, Madison WI USA; dobson@engr.wisc.edu. B.A. Carreras and V.E. Lynch are with Oak Ridge National Laboratory, Oak Ridge TN 3783 USA; carrerasba@ornl.gov. D.E. Newman is with the Physics department, University o Alaska, Fairbanks A USA; den@ua.edu. We grateully acknowledge support in part rom NSF grants ECS and ECS This paper is an account o work sponsored in part by the Power Systems Engineering Research Center (PSERC). Part o this research has been carried out at Oak Ridge National Laboratory, managed by UT-Battelle, LLC, or the U.S. Department o Energy under contract number DE-AC05-00OR Thirty-ninth Hawaii International Conerence on System Sciences, January 2006, auai, Hawaii. c 2006 IEEE. We now explain how branching processes can be useul approximations to some o the gross eatures o cascading blackouts. Our idealized probabilistic model o cascading ailure [7] describes a general cascading process in which component ailures weaken and urther load the system so that subsequent ailures are more likely. We have shown that this cascade model and variants o it can be well approximated by a Galton-Watson branching process with each ailure giving rise to a Poisson distribution o ailures in the next stage [5], [3]. Moreover, some eatures o this cascade model are consistent with results rom cascading ailure simulations [6], [4], [25]. All o these models can show criticality and power law regions in the distribution o ailure sizes or blackout sizes consistent with NERC data [8]. The distribution o the number o high voltage transmission lines lost in North American contingencies rom 965 to 985 [] also has a heavy tail distribution [9]. Initial work itting branching process models to observed blackout data is in [6]. The Galton-Watson branching process model [2], [2] gives a way to quantiy the propagation o cascading ailures with a parameter λ. In the Galton-Watson branching process the ailures are produced in stages. The process starts with Z 0 ailures at stage zero to represent the initial disturbance. The ailures in each stage independently produce urther ailures in the next stage according to an probability distribution with mean λ called the ospring distribution. That is, each ailure in each stage produces an average o λ ailures in the next stage. The eventual behavior o the branching process is governed by the parameter λ. In the subcritical case o λ <, the ailures will die out (that is, reach and remain at zero ailures at some stage) and the mean number o ailures in each stage decreases exponentially. In the supercritical case o λ>, although it possible or the process to die out, oten the ailures increase exponentially until the system size or saturation eects are encountered. Saturation is thought to be a signiicant eect because many observed cascading blackouts do not proceed to the entire interconnection blacking out. The reasons or this may well include inhibition eects such as load shedding relieving system stress, or successul islanding, that apply in addition to the stochastic variation that will limit some cascading sequences. Understanding and modeling these inhibition or saturation eects is important, but in this paper we avoid describing the saturation process by only estimating the propagation o ailures beore saturation is encountered. At the critical case o λ =, the branching process has a power law distribution o number o ailures with exponent

2 .5. A corresponding power law region can be observed in the distributions o number o ailures in the cascading ailure model [7] and in the distribution o blackout size in blackout models [4], [0], [25] when the system has a particular loading called the critical loading. The implications or blackout risk o the power law region are that the risk o large blackouts is approximately the same or even exceeding the risk o small blackouts [5]. This observation justiies the study o large blackouts; an exponential tail in the distribution o blackout size would imply that large blackouts have negligible risk and that a risk-based analysis would ignore large blackouts. Moreover, at criticality the mean blackout size starts to increase more rapidly and above criticality there is an increasing risk o large blackouts. The terminology o criticality comes rom statistical physics and does not, at this stage o knowledge o overall blackout risk, necessarily imply improper power system operation. Indeed there is some evidence that power systems may organize themselves to near critical loading in response to strong societal orces balancing economic use o the transmission system and reliability [7], [8]. One requirement on the ailure data in order to estimate λ is that the ailures be grouped into stages. The estimator we propose depends on the number o ailures in the stages and particularly on the total number o ailures and the number o ailures in the initial and inal stages. Many cascading ailure simulations naturally produce ailures in stages as the simulation iterates. However, i the method is applied to real data, the problem o grouping the data into stages must be addressed (see [6] or initial work grouping ailures together according to closeness in time). One direct way to estimate the probability distribution o number o line ailures at a given level o system loading is simply to run the simulation or record real blackout data until suicient data is accumulated to estimate the probability distribution o blackout sizes. This is straightorward but requires a large number o simulations or an impractically long observation time. I the distribution o line ailures is near criticality and it has a power law character, the probability distribution requires many observations to determine its orm or the larger blackouts. For example, it can take o the order o 000 to 0000 real or simulated blackouts to accurately estimate the probability o the larger number o line ailures in the near critical case. The near critical case is pertinent because there is some evidence and explanation indicating that the North American power transmission system is designed and operated near criticality [8], [7]. However, [8], [7] mainly apply to the distribution o energy unserved and the exact relationship between an external, societal impact measure o blackout size such as energy unserved and an internal measure o blackout size such as number o lines ailed remains unclear. On the other hand, empirical data or North American line outages [] has a heavy tail that is airly close to a power law [9]. I one assumes a branching process model, one can predict the probability distribution o the number o ailures rom the initial distribution o ailures and the estimate o λ using an analytic ormula (an example is shown in section III- D). We test the estimate or λ on line ailure data rom the OPA cascading ailure simulation by predicting the probability distribution o the number o line ailures using the estimate o λ and comparing this distribution to the probability distribution o line ailures observed in OPA. This approach o estimating the distribution o line ailures by irst estimating λ has the potential to be much aster because λ is the mean o the ospring distribution that generates the branching process and the ospring distribution does not have heavy tails. II. BRANCHING PROCESS WITH SATURATION This section describes the details o the main branching process model used in this paper. Suppose that there are N identical components and all components are initially unailed. The process saturates when S N components ail. Component ailures occur in stages with Z n the number o ailures in stage n and Y n the total number o ailures up to and including stage n. Y n = Z 0 + Z + Z Z n () There are a deterministic number Z 0 o initial ailures. Each o the Z n ailures in stage n independently causes a urther number o ailures in stage n + according to a Poisson distribution with mean λ, except that i the total number o ailures exceeds S, then the total number o ailures is limited to S. That is, the jth ailure in stage n causes Z [j] n+ ailures in stage n + according to the Poisson distribution and the total number o ailures in stage n + is { } Z n+ = min Z [] n+ + Z[2] n+ + + Z[Zn] n+,s Y n, (2) where Z [] n+,z[2] n+,,z[zn] n+ are independent. (A dierent orm o saturation is described in [5], [4].) The total number o ailures Y is distributed according to a saturating Borel- Tanner distribution: Z 0 λ(rλ) r Z0 e rλ (r Z 0 )! ; Z 0 r<s P [Y = r] = S Z 0 λ(sλ) s Z0 e sλ (s Z 0 )! ; r = S s=z 0 (3) Some simulations such as OPA may produce a random number Z 0 o initial ailures. I the initial ailures are independent and o equal probability p 0, then they are distributed according to a binomial distribution. I the number o components is large and p 0 is small, the distribution o initial ailures is approximately Poisson with mean θ = Np 0 and in this case the total number o ailures is distributed according to a saturating generalized Poisson distribution: r e rλ θ θ(rλ + θ) r! ; 0 r<s S P [Y = r] = s e sλ θ θ(sλ + θ) ; r = S s! s=0 We are interested in the total number o ailures conditioned on there being a nonzero number o ailures and this is (4)

3 distributed according to θ(rλ + θ) r e rλ θ r!( e θ ) ; r<s S P [Y = r] = θ(sλ + θ) s e sλ θ s!( e θ ) ; r = S s= (5) I there is an arbitrary distribution o nonzero initial ailures P [Z 0 = z 0 ] or z 0 =, 2, 3,..., then the total number o ailures is distributed according to a combination o the saturating Borel-Tanner distributions: r P [Z 0 =z 0 ]z 0 λ(rλ) r z0 e rλ (r z z 0 )! ; r<s P [Y =r] = 0= S P [Y =s]; r = S s= III. ESTIMATORS OF PROPAGATION λ s AND MEAN INITIAL FAILURES θ A. Deinition o λ s We suppose that the cascading ailure simulation produces Z 0 > 0 initial ailures in stage 0 and then iterates to produce urther numbers o ailures Z, Z 2, Z 3,... in stages,2,3,... respectively. The assumption o Z 0 > 0 implies that all statistics are conditioned on the start o a cascade. The simulation is run times to produce independent realizations o the cascade. The ailures in the kth run are written as Z (k) 0, Z(k), Z(k) 2,,.... The simulation results can be tabulated as ollows: Z (k) 3 stage 0 stage stage 2 stage 3 run Z () 0 Z () Z () 2 Z () 3 run 2 Z (2) 0 Z (2) Z (2) 2 Z (2) 3 run 3 Z (3) 0 Z (3) Z (3) 2 Z (3) run Z () 0 Z () Z () 2 Z () 3 Deine the cumulative number o ailures in run k up to and including stage n as Y (k) n (6) (7) = Z (k) 0 + Z (k) + Z (k) Z (k) n (8) Each run has a stage at which the number o ailures is zero and remains zero or all subsequent stages, either because the cascade dies out, or all N o the components in the system have ailed. The number o ailures at which saturation occurs is S N. Deine s(k) = max{ n Y n (k) <S and Z (k) n > 0 } (9) Then s(k) is either the irst stage at which there are zero ailures or the last stage beore saturation. We deine the estimator o λ as λ s = = (Z (k) + Z (k) Z (k) s(k) (Z (k) 0 + Z (k) Z (k) ) (k) s(k) Z(k) 0 Y (k) ) ) (0) () The appendix derives ormulas or the mean and variance o λ s. B. The standard estimator λ n The estimator λ s is a variant o the well known estimator: ) n (k) Z (k) 0 λ n = Y (2) (k) n The dierence is that λ n uses a ixed number o stages n or each run whereas λ s only uses inormation rom stages beore saturation. λ n is a maximum likelihood estimator [], [24], [9]. Moreover, or ixed n and as, (i) I λ<, λ n λ almost surely and E λ n λ; that is, λ n is an strongly consistent and asymptotically unbiased estimate o λ; (ii) i the ospring distribution variance σ 2 <, then λ n has an asymptotically normal distribution ] n ( ) N λ, σ [ 2 λ i = N λ, σ2 (λ ) (λ n ) i=0 Note that the standard deviation o λ n /. C. Estimator o mean initial ailures θ samples o the initial ailures are given by Z () 0, Z(2) 0,..., Z () 0. We it the initial ailures with a Poisson distribution with parameter θ conditioned on nonzero number o ailures: P [Z 0 = r] = e θ θ r e θ, r =, 2, 3,... (3) r! Let the sample mean o the initial ailures be Z 0 = Z (k) 0. (4) Then both maximum likelihood and method o moments estimation o θ in (3) yields an estimate θ satisying θ Z 0 =. (5) e θ D. Estimating distribution o number o ailures rom λ I we assume that the cascading process is approximated by a branching process with a Poisson ospring distribution, then we can obtain the probability distribution o the number o ailures rom λ. For example, assuming one initial ailure, Figure shows the probability distributions obtained or S = N = 000 and three values o λ. For subcritical λ =0.6 well below, the probability o large number o ailures o size near S is exponentially small. The probability o exactly S ailures is also very small. As λ increases in the subcritical range λ<, the mechanism by which there develops a signiicant probability o large number o ailures near S is that the power law region extends towards S ailures [4]. For near critical λ, there is a power law region extending to S ailures. For supercritical λ =.2, there is an exponential tail. This again implies that the probability o large number o ailures <Sis exponentially small. However there is a signiicant probability o exactly S ailures that increases with λ.

4 probability number o components ailed Fig.. Log-log plot o PDF o total number o ailures in branching process model or three values o λ. λ =0.6 is indicated by the diamonds. λ =.0 (criticality) is indicated by the boxes. λ =.2 is indicated by the triangles ˆ 2 λ s Fig. 2. λs plotted against λ rom 20 runs o saturating branching process with saturation S =20. 5 computations o λs are shown or each value o λ to indicate the scatter o the results around the line o slope one. λ IV. RESULTS We irst consider the statistical perormance o the estimator λ s on an ideal saturating branching process and then test the use o analytic branching process ormulas including the estimator λ s in predicting the distribution o line ailures rom the OPA simulation. A. Testing on saturating branching process The estimator λ s was tested on the saturating branching process with Poisson ospring distribution and Z 0 =or various speciied values o λ and saturation S. Figures 2 and 3showhowwell λ s matches λ or saturations S =20and S = 00. The necessity or λ s taking account o saturation is shown by the corresponding results or λ n in Figure 4 (compare to Figure 2). For subcritical λ well below λ =, λ n and λ s give the same perormance because it is likely that cascades die out beore reaching saturation. We determined the bias and variance o λ s numerically by computing λ s 000 times and computing the sample mean and standard deviation o λ s. The standard deviation σ( λ s ) was ound to decrease / as increases. For saturation S =20, 0 <λ<2, and number o runs 0 000, λ s underestimated λ with a bias less than 0.: 0. <µ( λ s ) λ 0 and σ( λ s ) ˆ λ s Fig. 3. λs plotted against λ rom 20 runs o saturating branching process with saturation S = computations o λs are shown or each value o λ to indicate the scatter o the results around the line o slope one..5 ˆ λ n 2 λ For example, =20runs gives σ( λ s ) 0.3. For saturation S = 00, 0 <λ<2, and number o runs 0 50, λ s underestimated λ with a bias less than 0.07: 0.07 <µ( λ s ) λ λ and σ( λ s ) 0.5. For example, =20runs gives σ( λ s ) 0.. Fig. 4. Standard estimator λn plotted against λ rom 20 runs o saturating branching process with saturation S =20. 5 computations o λn are shown or each value o λ to indicate the scatter o the results around the line o slope one.

5 B. Computing λ s rom OPA results and predicting distribution o line ailures The OPA model produces cascading line ailures in stages resulting rom a random initial set o line ailures. (Here the intent o the term line ailure includes line outages in which the line is intact but temporarily unavailable to transmit power as well as line outages in which the line is damaged.) The power transmission system is modelled using DC load low and LP generator dispatch and cascading line overloads and ailures are represented [4]. The power system is assumed to be ixed with no transmission line upgrade process. For each case considered, OPA was run so as to produce 5000 cascading ailures with a nonzero number o line ailures. These 5000 runs yield line ailure data in the orm (7). All our statistics are conditioned on a nonzero number o line ailures. Then λ s and θ were obtained using equations () and (5). The empirical distribution o the initial ailures Z 0 was also estimated. The number o runs and the resulting data set are large enough that the standard deviation o λ s is negligible. The objective is to test an accurate λ s by evaluating its ability to predict the probability distribution o line outages. (The inluence on statistical accuracy o the small number o runs desirable in practice is evaluated in section IV-A.) For each case, the probability distribution o line ailures was predicted using two methods and compared to the empirical probability distribution o line ailures. The irst method used the estimates λ s and θ in the saturating generalized Poisson distribution ormula (5) to predict the distribution. This tests the validity o the branching process model o initial ailures and propagation underlying (5) and the estimates λ s and θ. The second method used the empirical distribution o initial ailures and the estimate λ s in the ormula (6) to predict the distribution. This tests the validity o the branching process propagation o ailures and the estimate λ s. The irst three cases used the IEEE 8 bus system at average load levels o 0.9,.0, and.3 times the base case loading. (The OPA parameters (explained in [4]) are γ =.67, p 0 =0.000 and p =and () is evaluated with saturation S =5lines.) The results are shown in Figures 5-7 and Table I. The matches in Figures 5 and 6 are very good. Note that the OPA simulation o cascading ailure producing the empirical probability distributions o line ailures is substantially more complicated than the simple analytic probability distribution ormulas (5) and (6) used to predict the same distributions. Figure 7 shows a case with a large initial disturbance (the average number o lines initially ailed is θ 2). In this case the match or the generalized Poisson distribution is good and the match or the ormula (6) using the empirical distribution o initial ailures is poorer, especially or the smaller number o ailures. We suspect that the poorer match is due to some o the cascades reaching saturation because o the large initial disturbance. The last two cases used the 90 bus tree-like test system [4] at average load levels o.0 and.2 times the base case loading. (The OPA parameters are γ =.94, p 0 =0.005 and p =0.5, and () is evaluated with saturation S =5lines.) The results are shown in Figures 8 and 9 and Table I. The results in Figure 9 show saturation eects not well captured in the branching process model that make the prediction o the distribution poorer. (One could set S = 5 in (5) and (6) to model all cascades stopping when 5 lines ail, but this very crude model o saturation gives a spike at 5 lines that is also a poor it to the empirical distribution since quite a ew o the cascades proceed beyond 5 lines ailed.) The position o the peak in the empirical distribution o Figure 9 indicates a concentration o number o line ailures near 5, and we associate this with a tendency or the cascades to saturate at approximately 5 lines ailed. There also seems to be a secondary peak and saturation at approximately 30 lines ailed. At smaller loadings such as in Figure 8, ewer cascades encounter the saturation and, although the saturation eect is perhaps discernible, it is small. We expect larger saturation values in larger test systems. It is interesting to note that the July and August 996 Western area blackouts saturated at over 30 high voltage line trips and the August 2003 Eastern interconnect blackout saturated at several hundred high voltage line trips [6]. The results suggest that good predictions o the probability distributions o the number o line ailures can be obtained as long as saturation eects are not signiicant. We do not understand how to accurately model the saturation eects at present. The ability to predict the probability distribution o the number o line ailures in non saturating cases supports the applicability o branching models to cascading ailure beore saturation is reached and the useulness o the estimate λ s o ailure propagation. λ s can be obtained much more eiciently than empirical probability distributions o line ailures obtained by brute orce. TABLE I ESTIMATORS FOR OPA CASES power system loading actor θ λs IEEE 8 bus IEEE 8 bus IEEE 8 bus tree-like 90 bus tree-like 90 bus V. GENERALITY AND LIMITATIONS OF λ CONCEPT This paper approximates cascading ailure by a single-type Galton-Watson branching process with random or deterministic initial ailures. The most immediate motivation is that this branching process approximates a loading dependent model o cascading ailure [7] and captures some, but not all, qualitative eatures o cascading ailure in blackout simulations [4], [0], [25]. Moreover, the single-type Galton Watson branching process also approximates other cascading processes, such as avalanches in idealized sandpiles [23], [27], initial spread o epidemics, growth o populations etc. Thus estimation o λ is already established in cascading ailure in other ields [9]. This section shows two examples o more elaborate models that involve cascading ailure to start to explore how the parameter λ may apply more generally.

6 probability probability number o lines ailed Fig. 5. IEEE 8 bus system with loading actor 0.9. PDF estimated with initial ailure distribution and λ s (solid line) and θ and λs (dashed line) compared with OPA empirical PDF (dots). Note the log-log scales. probability number o lines ailed Fig. 8. Tree-like 90 bus system with loading actor.0. PDF estimated with initial ailure distribution and λ s (solid line) and θ and λs (dashed line) compared with OPA PDF (dots). probability number o lines ailed Fig. 6. IEEE 8 bus system with loading actor.0. PDF estimated with initial ailure distribution and λ s (solid line) and θ and λs (dashed line) compared with OPA empirical PDF (dots). probability number o lines ailed Fig. 7. IEEE 8 bus system with loading actor.3. PDF estimated with initial ailure distribution and λ s (solid line) and θ and λs (dashed line) compared with OPA empirical PDF (dots) number o lines ailed Fig. 9. Tree-like 90 bus system with loading actor.. PDF estimated with initial ailure distribution and λ s (solid line) and θ and λs (dashed line) compared with OPA PDF (dots). A. Demon model o two coupled inrastructures The Demon model [26] abstractly models cascading inrastructure ailure as a dynamic complex system. The Demon model is based on the orest ire model o Bak, Chen and Tang [3] with modiications by Drossel and Schwabl [8]. For a single system, the model is deined on a 2-d network. Network nodes represent components o the inrastructure system and lines represent couplings between components. The components can be operating, ailed or ailing. The rules o the model are that or each time step: ) Any operating component ails with a probability P. P is the spontaneous ailure rate independent o other components. 2) An operating component ails with a probability P n i at least one o the nearest components is ailing. P n describes the extent to which ailures propagate to neighboring components. 3) A ailing component becomes a ailed one. 4) A ailed component is repaired and becomes operational with probability P r.

7 We can generalize this single system to two coupled inrastructure systems by taking two o these 2-d networks, making a raction g o the nodes in each system correspond to the component in the corresponding 2-d position in the other system, and adding another rule: 5) An operating component in system ails i the corresponding component in system 2 is ailed or ailing. An operating component in system 2 ails i the corresponding component in system is ailed or ailing. In this paper, we will assume that the two coupled systems have identical networks and identical parameters except that the probability P () o spontaneous ailures in system may dier rom the the probability P (2) o spontaneous ailures in system 2. We assume that the systems are coupled symmetrically with coupling coeicient c and that the raction o components coupled to other system is g. These rules allow both analytic mean ield and numerical calculations and we start with a mean ield calculation that generalizes the calculation in [8]. The number o components in each system is N. Let O (i) (t) be the raction o operating components in system i at time t; that is, the number o operating components at time t divided by N. In the same way, we can deine the raction o ailed components, F (i) (t), and the raction o ailing components, B (i) (t). The mean ield equations or the coupled system are: B () (t+) = P () O () (t)+p n O () (t)b () (t) +cgo () (t)(b (2) (t)+f (2) (t)) F () (t+)=( P r )F () (t)+b () (t) O () (t+)=( P () )O () (t)+p r F () (t) P n O () (t)b () (t) cgo () (t)(b (2) (t)+f (2) (t)) B (2) (t+) = P (2) O (2) (t)+p n O (2) (t)b (2) (t) +cgo (2) (t)(b () (t)+f () (t)) F (2) (t+)=( P r )F (2) (t)+b (2) (t) O (2) (t+)=( P (2) )O (2) (t)+p r F (2) (t) P n O (2) (t)b (2) (t) cgo (2) (t)(b () (t)+f () (t)) (6) In (6), is the average number o neighboring nodes within a system that a ailing node can propagate ailure to. During ailure propagation, since at least one o the nodes neighboring a ailing node has ailed, k, where the network degree k o each system is the average number o nodes within each system that are joined to a given node. Equations (6) are consistent with the condition O (i) (t)+b (i) (t)+f (i) (t) =. In the limit with no spontaneous ailures, P () = P (2) =0, and or a steady state solution, (6) can be reduced to two coupled equations, ( ( Pn O ()) P r ( O () )=cg( + P r )( O (2) )O (), Pn O (2)) P r ( O (2) )=cg( + P r )( O () )O (2). (7) I c 0, then O () = implies O (2) = ; that is, the systems are eectively decoupled. Thereore, to have truly coupled systems, system must be in a supercritical state with O () <. Here criticality occurs in passing rom an average state with almost no ailures to an average state with some ailures. For this mean ield computation we have assumed identical system symmetrically coupled. In particular P () = P (2) =0. Then O () = O (2), and (7) lead to the ollowing identical solutions or the two systems in steady state: O (i) = F (i) = B (i) =, ĝ ĝ, ĝ> (8) 0, ĝ ĝ ĝ( + P r ), ĝ> (9) 0, ĝ ĝ ĝ( + P r ) P r, ĝ> Here, ĝ is the control parameter given by (20) ĝ = P n + cg( + P r) (2) P r The critical point occurs at ĝ =. Observe rom (2) that i the systems are uncoupled so that c =0, then ĝ = P n both measures the average propagation o ailures in each system and detects criticality when ĝ =. Thus or uncoupled systems (or a system considered in isolation) ĝ has similar characteristics to λ in the branching process model. However, the situation is more complicated when the systems are coupled and then (2) shows that the criticality depends also on the coupling and the probability o repair P r. We have tested the results rom the mean ield computation by comparing them with numerical results or the identical coupled systems ormed rom the systems listed in Table II. The results or the average number o operating components are shown in Figure 0. These results were obtained or ixed P r =0.00, c =0.0005, P () =0.0000, andp (2) =0 and we have varied the propagation parameter P n. The results show very good agreement with the mean ield theory results as the network degree k increases. For k =2, the system are practically -d and the mean ield theory is not applicable. TABLE II SYSTEMS TO BE COUPLED system k number o nodes Open 3-branch Tree Closed 3-branch Tree Square Hexagon The density o operating components is practically the same in both systems. This is expected because the systems are nearly identical with the only symmetry breaking eature being the probability o spontaneous ailures P (2) that is zero in the second system. Perhaps the most important point or the purposes o this paper is that the critical point moves as the coupling cg increases. Figure shows the critical point as a unction o

8 critical point discussed above, the critical value o λ d at the critical point is typically less then (up to 20% less or the cases examined) and it depends on the coupling and the details o the system. We are continuing to investigate measures o propagation and their interpretation and possible applicability to the coupled system. Fig. 0. Variation o raction o operating components O () in system as the propagation parameter P n is varied. B. Two-type branching process model To show a more elaborate branching process model where the generalization o λ with all the characteristics o the single type case is not yet apparent, consider a two-type Galton-Watson branching process [2]. One motivation or the two types in blackout models is that type could represent component ailures within the network and type 2 could represent the amount o load shed or energy unserved. Another motivation is that the two types could represent ailures in two coupled inrastructures [26]. One way to generalize λ is the matrix ( ) λ λ Λ= 2 λ 2 λ 22 where λ ij is the average number o type j ailures caused by each type i ailure in the previous stage. The coupled cascading processes still show criticality but the condition or criticality becomes λ max =where λ max is the largest eigenvalue o Λ [26]. λ max is a quadratic root unction o the entries o Λ. The criticality property remains a unction o Λ, but the unction is not trivial as in the single type case. Suppose that the two-type branching process represents two coupled inrastructures. Then, i the irst inrastructure is incorrectly modeled in isolation, it will be a single-type branching process with parameter λ, and its proximity to criticality will be computed as λ. On the other hand, the model o the two coupled inrastructures will yield a proximity to criticality o λ max. This illustrates the importance o mutual couplings with other inrastructures in determining the propensity or cascading ailure. Fig.. Fraction o operating components O (i) at critical point as a unction o coupling cg or system and system 2. The triangles with dots inside are results obtained with g =(all components coupled between the systems) and the open triangles are results obtained with g =0.and a tenold increase in c. the coupling cg. A actor o our variation in the critical point is seen. Figure shows the critical points o both system and system 2 calculated rom the change in the unctional orm o the operating node density (8). The two systems are virtually indistinguishable rom each other. Following [8], we computed a measure o propagation o ailing components B(4) λ d = (22) B(3) where denotes averaging over all the cascades occurring in the model runs. While there is critical value or λ d at the VI. DISCUSSION AND CONCLUSION In this paper, we approximate cascading ailure by a single type Galton-Watson branching process with saturation in order to propose a method o quantiying the propagation o ailures λ and hence estimate the probability distribution o the number o ailures. The proposed estimator λ s requires observations o cascades with the initial and inal ailures grouped in stages. Although λ s is not asymptotically unbiased, it does work in the presence o saturation eects that are thought to occur in blackouts. Testing on a saturating branching process suggests that an order o =0observations can be suicient to determine λ s with worst-case bias and standard deviation o order 0.. The standard deviation o λ s decreases like / as increases and it may be possible to correct or the bias in uture work. This compares avorably with the much larger number o observations needed to estimate the oten heavytailed distribution o number o ailures directly. The ability to estimate the propagation o ailures and the distribution o ailures with a modest number o observations would expand

9 the opportunities or using cascading ailure simulations to study the eect o transmission system upgrades on cascading ailure and is crucial or the practicality o monitoring ailures in the power grid to assess cascading ailure. Initial testing o the estimator λ s by predicting the empirical probability distributions o line ailures rom OPA using branching process models shows a good it or cases in which saturation eects are not present. Similarly good its or the non-saturating case are obtained by approximating the initial ailures as a Poisson distribution and estimating the mean θ o the Poisson distribution. These results are also promising in showing the applicability o simple branching process models to cascading processes in blackouts beore saturation. Further progress on saturating cases is likely to require a better understanding o the saturation eects. We also examine quantities sharing some properties o λ in an abstract complex systems dynamical model o two interacting inrastructures. While there remains a useul parameter ĝ governing the criticality o the model, the value o ĝ at criticality varies with the amount o coupling between the two inrastructures. When the inrastructures are not coupled, ĝ is analogous to λ in that it describes the average propagation o ailures in each system and the system becomes critical when ĝ =. However, the situation is more complicated when the inrastructures become coupled. A two-type generalization o the branching process is a dierent model o coupled cascading systems and in this model λ can be generalized to a more complicated unction o the model parameters that measures the dominant propagation and becomes at criticality. The complex systems dynamical model and the two-type branching model are very dierent but both are simple models o coupled cascading processes that have the potential to capture some overall eatures o ailures cascading in and between inrastructures. Both models show that the propensity or cascading ailure is strongly inluenced by the mutual couplings between the inrastructures. In particular, consideration o the inrastructures in isolation could give a misleading assessment o the propensity or cascading ailure. As more is learned about the universal or applicationspeciic properties o cascading ailure, the high level models o cascading ailure may be elaborated. Each elaborated branching process has urther parameters to it to describe the cascade. Further work is needed to determine the best compromise between simplicity and elaboration, but this paper makes progress in estimating λ or a single-type Galton- Watson branching process with saturation. This branching process model or its elaborations could well be applicable to assessing cascading ailure risk in large networked inrastructures and in interacting inrastructures. REFERENCES [] R. Adler, S. Daniel, C. Heising, M. Lauby, R. Ludor, T. White, An IEEE survey o US and Canadian overhead transmission outages at 230 kv and above, IEEE Transactions on Power Delivery, vol. 9, no., Jan. 994, pp [2].B. Athreya, P.E. Ney, Branching Processes, Dover NY 2004 (reprint o Springer-verlag Berlin 972). [3] P. Bak,. Chen, C. Tang, A orest-ire model and some thoughts on turbulence, Physics Letters A, vol. 47, July 990, pp [4] B.A. Carreras, V.E. Lynch, I. Dobson, D.E. Newman, Critical points and transitions in an electric power transmission model or cascading ailure blackouts, Chaos, vol. 2, no. 4, December 2002, pp Note: all the authors papers are available at dobson/home.html [5] B.A. Carreras, V.E. Lynch, D.E. Newman, I. 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10 [27] F. Slanina, Sel-organized branching process or a one-dimensional ricepile, European Physical Journal B, , [28] U.S.-Canada Power System Outage Task Force, Final Report on the August 4th blackout in the United States and Canada. United States Department o Energy and National Resources Canada, April APPENDIX This appendix derives ormulas or mean and variance o λ s similar to the ormulas or λ n in [9, pp ]. Deine the event { } A = s(k) =s k,y (k) s k 2 = y k,z (k) s k = z k This uses VarX =EX 2 (EX) 2 =EE(X 2 Y ) E(E(X Y )) 2 + E(E(X Y )) 2 (EE(X Y )) 2 =EVar(X Y ) + VarE(X Y ) E λ s =E = s k,y k,z k E = s k,y k,z k E = s k,y k,z k =+λe Let (k) s(k) Z(k) 0 ) Y (k) (k) s(k) Z(k) 0 ) Y A P (A) (k) ( ) (y k + z k + Z s (k) k Z (k) 0 ) (y A P (A) k + z k ) ( ) + (λz k Z (k) 0 ) (y P (A) k + z k ) Z(k) Var λ s =Var =Var + =Var =EVar = σ 2 E = σ2 Y (k) B(k) = E Z(k) 0 Y (k) ( ) s(k),y (k) s(k) 2,Z(k) (k) s(k) Z(k) 0 ) Y (k) (Z(k) s(k) Z(k) 0 ) (k) s(k) 2 + Z(k) ) (Z(k) s(k) Z(k) 0 ) (k) s(k) 2 + Z(k) ) (Z(k) s(k) Z(k) 0 ) (k) s(k) 2 + Z(k) ) (Z(k) s(k) Z(k) 0 ) +Var E Z(k) ( Y (k) E ( (k) s(k) 2 + Z(k) ) ) 2 +Var Z(k) ) 2 Y (k) + Var (λz(k) B(k) B(k) (23) (λz(k) Z(k) 0 ) Y (k) Z(k) 0 ) Y (k)