SelectingRepresentativeScenariosforContingencyAnalysisof ... · 2020. 2. 5. · Hernández-Fajardo and Dueñas-Osorio,2013). If a more expensive model is available and preferable,

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13Seoul, South Korea, May 26-30, 2019

Selecting Representative Scenarios for Contingency Analysis ofInfrastructure Systems with Dependent Component Failures

Hugo Rosero-VelásquezPhD Student, Engineering Risk Analysis Group, Technische Universität München,Germany

Daniel StraubProfessor, Engineering Risk Analysis Group, Technische Universität München, Germany

ABSTRACT:

Classical contingency analysis assesses the robustness of infrastructure systems by removing one compo-nent at the time (sometimes up to two components) and evaluating the effect on the system performanceY . In systems with dependent component failures, this approach might not identify critical system failurescenarios, e.g. if failures are caused by natural hazards or other common causes. In this contribution,we develop an approach to identify representative scenarios ST of component failures that are associatedwith damages or performance losses of a specific return period T . In practice, such scenarios are mostlydefined based on historical data and expert knowledge, which often reflect past events but might not berepresentative of future events. Our approach is based on an initial Monte Carlo analysis of the system,resulting in an annual exceedance probability function of the system performance Y . Samples that ap-proximately correspond to the value of Y associated with the return period of interest T are selected.The representative scenario for T is then identified by means of a clustering algorithm applied to thesesamples. The approach is demonstrated on a numerical example.

1. INTRODUCTION

Reliability of infrastructure networks, such aspower grids, is often evaluated through contingencyanalyses, in which one or two components are as-sumed to fail. For a system with N componentsthat might fail, there are at most N(N−1)

2 out of 2N

cases considered. An application for analyzing apower distribution system considering N − 1 con-tingencies can be found in Acosta et al. (2018),while N − 2 contingencies are compared with aninfluence-graph model in Hines et al. (2017). Theyare also studied for defining a risk-based ranking inKengkla and Hoonchareon (2018). Additionally, aso-called N−1−1 contingency has been proposed,which considers a sequence of two failures (Mitraet al., 2016). This requires the analysis of N(N−1)out of N! cases.

As highlighted by Kengkla and Hoonchareon(2018), for complex reliable systems with large Nand high redundancy, contingencies with more thantwo failed components should be considered forobserving extreme black outs. N −M contingen-cies have already been studied by Srivastava et al.(2018) in the context of attack scenarios, combininggraph theory and DC power flow measures. In ad-dition, Pescaroli and Alexander (2018) provides aframework for analyzing the impact of natural haz-ards to lifeline systems, while Scherb et al. (2017)develops a generic hazard model for being coupledwith a system and ranking single components basedon a network efficiency measure.

In a large system, there is a very large numberof possible scenarios leading to some performanceloss. For risk assessment and management pur-

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poses, it is often convenient or even required to con-sider only a few representative loss scenarios. Typ-ically, these are scenarios associated to a specificreturn period. Such representative scenarios can in-form the contingencies to be analyzed in detail.

In this paper, we propose a methodology for se-lecting representative scenarios of component fail-ures in an infrastructure system, based on the re-liability of the system components, a performancemeasure and a return period of interest.

2. GENERIC INFRASTRUCTURE SYS-TEM MODEL

We model a system with N components as aweighted graph with ND nodes and NE edges. Acomponent can either be a node (e.g. a substation ora bus in a transmission grid) or an edge (e.g. trans-mission lines), thus N = ND +NE . Each edge has aweight, which for example represents the resistancein a transmission line or the pressure gradient in apipeline. Moreover, the system shall have at leastone source node and one terminal node.

Each component has a binary state Xi. Duringa hazard event, the reliability of the component ispi = Pr(Xi = 1). The complete system is describedby the component vector state X = (X1,X2, ...,XN),which takes 2N possible values. The probabil-ity associated to each state is a function of themarginal component reliabilities and the depen-dence between component states.

The component vector state X is mapped to thesystem performance measure Y (e.g. power loss) bymeans of a system model g (see Straub, 2018) :

Y = g(X) (1)

In the following, Y represents a loss of performancerelative to the fully functioning system.

The system model g can be deterministic orstochastic. In most systems, it should consider cas-cading effects, caused by a redistribution of loadsin the system following initial component failures.Cascading failure models are described in the liter-ature (e.g., Crucitti et al., 2004; Hernández-Fajardoand Dueñas-Osorio, 2013; Scherb et al., 2017).

The system loss Y is a random variable,with exceedance probability (EP) function

HY (y) = Pr(Y > y). Since Y is the loss of thesystem following a hazard event, this EP functionis related to the annual exceedance rate AERthrough

AER(y) = λHHY (y) (2)

with λH representing the annual frequency of thehazard occurrence. It follows that the loss with re-turn period T is

YT = H−1Y

(1

λHT

)(3)

Based on the above, a scenario S is defined asthe tuple (x,y), where x,y are realizations of therandom variables X,Y . A representative scenarioST associated to a return period T is a scenariowhich in some sense is representative of all scenar-ios within a range 1

T (1±∆).

3. METHODOLOGYIn this Section we introduce a methodology foridentifying the representative scenarios ST . Ithinges upon the availability of a computationallysufficiently cheap system model g, such that a crudeMonte Carlo simulation (MCS) with around 10T λHsamples can be performed. Many network systemmodels used in the literature fullfill this criterion,(e.g., Shinozuka et al., 1999; Poljansek et al., 2011;Hernández-Fajardo and Dueñas-Osorio, 2013). If amore expensive model is available and preferable,the methodology might nevertheless be applied on acheaper (surrogate) model. The identified scenariosST can then be evaluated with the expensive modelto evaluate the risk.

The first step is to perform a MCS, where ns sam-ples of X are generated. The minimum number ofsamples ns should be selected in function of themaximum return period of interest Tmax, such thatns > 10TmaxλH .

Samples of the component vector statex1,x2, ...,xns are generated and the model g(xk) isevaluated for each sample k, resulting in samplesof system performance y1, ...,yns . The EP functionHY (y) is then constructed empirically as the com-plement of the sample CDF. Based on this function,one can associate to each y1, ...,yns a correspondingreturn period T1, ..., Tns by means of Equation (3).

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The second step is to associate sample tu-ples (xk,yk) to a return period of interestT . To this end, one defines an intervalΩT,∆ =

[ 1T (1−∆), 1

T (1+∆)), with the parameter

∆ > 0 specifying the width of the interval. All sam-ple tuples (xk,yk) whose corresponding 1

Tkfalls into

this interval are collected. Note that the interval issymmetric around T in terms of exceedance prob-ability. This ensures that the expected exeedanceprobability of all samples in the interval is 1

T .The choice of ∆ must balance the accuracy and

robustness of the selected scenarios. A small ∆

leads to a high accuracy in the scenario return pe-riod, but implies that a smaller number of samplesfalls into ΩT,∆. Hence the identified scenario maybe less robust. In contrast, a larger ∆ causes sce-narios with return periods different than T to be in-cluded in the interval. As a consequence, the se-lected scenario ST can be associated with a returnperiod that differs more from T .

Since the number of samples that fall into the in-terval decrease with increasing T for a fixed ∆, itcan be optimal to vary ∆ with T . For example, onecan select ∆ by fixing the expected number of sam-ples nv in ΩT,∆, which approximates the probabilityof one sample to fall into that interval:

Pr(

1λH Ti

∈ΩT,∆

)≈ nv

ns(4)

Solving for ∆ results in:

∆≈ λHT nv

2ns(5)

The third step is to identify a representative xamong the collected samples in ΩT,∆. A cluster-ing algorithm is applied to these samples. If thereis a cluster whose size is significantly larger, thenthe tuple (x,y) corresponding to the mode of thatcluster is selected as the representative scenario ST .

In most cases, multiple return periodsT1,T2, ...,Tq = Tmax are of interest. In thiscase, steps 2 and 3 are repeated for each T = Tj.

3.1. Clustering algorithmAmong the available clustering algorithms, we uti-lize DBSCAN (see Ester et al., 1996) for identify-ing the representative scenarios. DBSCAN deter-mines the clusters based on the parameter ε , the

maximum distance between points in a cluster, andthe parameter k, the minimum number of pointsfor a subset to become a cluster. These two pa-rameters determine whether two points are densityreachable, i.e. whether they are part of non-disjointneighborhoods of radius ε , each of them containingat least k points. Finally, clusters are determined asthe non-empty maximal subsets of points that aredensity connected with respect to ε and k. Somesamples do not belong to any cluster, these are la-beled as noise. Importantly, DBSCAN does not re-quire one to predefine the number of clusters.

The choice of k determines the minimum size ofany cluster. In cases where there are few repeatedsamples, there is a higher chance that the largestcluster has a size close to k. If k is small, then thealgorithm is likely to result in multiple small clus-ters, while a large k increases the fraction of sam-ples labeled as noise.

Ester et al. (1996) suggest to choose ε and k ac-cording to the EP function of the distances of eachpoint to their k-nearest neighbor. That function pro-vides an indication on the expected percentage ofsamples labelled as noise for a given combinationof ε and k. In the context of the considered ap-plication, we suggest to fix k and then determineε based on this EP function. By increasing ε , thesmall clusters are merged and some samples, for-merly classified as noise, might now be added tothe new clusters. However, an ε close to

√N clas-

sifies the nv samples into one single cluster.

3.2. Example with a small systemFor illustrating the methodology, we consider asmall system with N = 10 components, whose per-formance is measured by the following weightedsum:

g(x) =2

N(N +1)

N

∑i=1

i(1− xi) (6)

That is 0 ≤ g(x) ≤ 1. The performance functionis evaluated through a MCS with sample size ofns = 2×104.

The component failure probability follows a betadistribution with mean pi = p = 0.99 and standarddeviation σp = 0.05. Thus the number of avail-able components Na follows a beta-binomial distri-

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bution, whose probability mass function is:

Pr(Na = n) =(

Nn

)B(n+ r,N−n+ s)

B(r,s)(7)

The beta-binomial distribution introduces a depen-dence among the components, which is defined byσp. If σp = 0, the components fail independently,while σp =

√p(1− p) implies that either none or

all components fail, the latter event with probability1− p. Therefore, it is convenient to describe the de-pendence between the components by the normal-ized measure γp =

σp√p(1−p)

(Straub, 2018).

Figure 1 shows the EP function obtained from thens samples. The dashed lines bound the intervalsΩT,∆. We consider two return periods of interest(T1 = 102 and T2 = 103 years) and λH = 1.0yr−1.Based on Equation (5), for the return period of102 years, we found that ∆ = 0.1 results in an ex-pected number of 200 samples, while for 103 yearswe expect 100 samples with ∆ = 0.5. The resultingsample size for each interval for the specific MCSis 241 samples for 102 years and 105 for 103 years.

Figure 1: System EP function, with the intervalsΩT,∆ for return periods of T = 102 years (red) andT = 103 years (blue).

The samples of the MCS are displayed in Fig-ure 2, located according to the vertices of the unit10-hypercube graph. The hypercube projection ar-ranges the vertices in such way that each columngroups the vertices with the same number of failedcomponents. Since every edge in an unit hypercubehas length one, such a projection has the propertythat the edges only connect nodes located in differ-ent but adjacent columns.

Figure 2: Sample of vector states located on the pro-jection of a 10-hypercube, with vertices arranged bynumber of failed components. Sample sets for returnperiods of T = 102 years (red) and T = 103 years (blue)are highlighted, and the marker size is proportional tothe frequency of each state in the samples.

As one can see from Figure 1, the state withoutfailed components is the most frequent one, repre-senting more than 90% of the samples. Accord-ingly, it is the largest vertex of the hypercube inFigure 2, which also shows that the frequency ofeach state decreases with the number of failed com-ponents.

The result of applying the DBSCAN algorithmto the highlighted sample subsets is shown in Fig-ure 3, for selected values of ε and k. With ε ≥

√2,

all samples are assigned to the same cluster, whichis consequence of the small number of componentsin this example. Therefore, we recommend to fixε = 1 for systems with few components and highcomponent reliability. In contrast, in systems withmany components, a larger ε can be a reasonablechoice. In such case, the EP function of the k-distances gets smoother, and the expected percent-age of noise can be controlled by following the pro-cedure of Ester et al. (1996).

From Figure 3, one can also observe that the in-crease of parameter k increases the percentage ofsamples labelled as noise. This becomes critical forlarge return periods, where ∆ is larger and the sam-ples to be clustered are more sparse in the hyper-cube. For this set of samples, increasing k from 4 to6 increases the the percentage of noise from 9% to23% for a return period of 102 years, and from 27%to 67% for a return period of 103 years. However,the sample size is sufficiently large that the mode

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Return period: 102 years Return period: 103 years

Figure 3: Clustered vector states for return periods of 102 and 103 years and the labels of their representativescenarios. The dots correspond to states classified as noise. The number label indicates the representative sce-nario for each interval, whose binary representation tells which are the components that fail, e.g. state 2 j−1 cor-responds to the single failure of the j-th component, for j = 1, ...,N.

of the largest cluster does not change with variationof ε and k, which indicates the robustness of therepresentative scenario.

4. APPLICATION TO AN ELECTRIC GRID

We apply the methodology to the IEEE39 bus sys-tem, whose topology is displayed in Figure 4. Itconsists of 39 nodes and 43 edges. The edgeweights correspond to the line reactance values;the line capacities are here modeled as being pro-portional to the number of shortest paths passingthrough them. This example was previously stud-ied by Scherb et al. (2017) for quantifying the net-work reliability considering cascading effects andexposure to spatially distributed hazards.

The system performance is evaluated by the effi-ciency loss:

g(x) = 1− E(x)E(1)

(8)

E(x) =1

|SN||T N| ∑s∈SNt∈T Ns 6=t

effst (9)

where effst is the efficiency of the most efficientpath from source node s to terminal node t, SNis the set of source nodes, T N the set of terminalnodes. E(x) is the network efficiency associated tothe component vector state x and E(1) is the effi-ciency of the intact system. The efficiency of a path

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Edge Capacities Edge Reactances

Figure 4: IEEE39 bus system, with edge thicknesses proportional to their estimated capacities (left) and reac-tances (right).

is equal to the sum of the reactance values alongthat path. Equation (9) is a modified version of thenetwork efficiency defined by Latora and Marchiori(2001), restricted to paths that connect nodes fromSN to nodes in T N. Equation (8) is the complimentof the network efficiency defined by Scherb et al.(2017).

In this example, two types of dependencies areconsidered: Initial failures at nodes and line failuresby cascading effects. The first one is modelled bythe beta-binomial distribution with different valuesof γp following Section 3.2. The cascading effectsare evaluated with the efficiency-based model pro-posed by Crucitti et al. (2004) and later extended byHernández-Fajardo and Dueñas-Osorio (2013) andScherb et al. (2017). This model of cascading ef-fects is deterministic, i.e. for a given set of initiallyfailed nodes, there is only one possible final state ofthe system.

For the performance loss, the return period ofinterest is 102 years. The generic hazard has fre-quency λH = 1.0yr−1. The marginal componentprobability of failure during a hazard event is 0.01.A MCS is performed with sample size ns = 5×105.

4.1. EP function for different γp

Figure 5 shows the EP functions for dependencymeasure γp = 0.0, 0.2, 0.5 and 0.7. One can ob-serve the transition from the case of independentinitial failures, where scenarios of high losses are

rare and initial failures of few nodes are mostlikely, to the limit case of fully dependent com-ponents (γp = 1.0), in which the system per-formance following a hazard event correspondsto a Bernoulli random variable with probabilityPr(Y = 0|Hazard) = 0.01.

0 0.2 0.4 0.6 0.8 1

Efficiency loss

10-4

10-3

10-2

10-1

100

Exceedance P

robabili

ty

p=0.0 (Independent)

p=0.2

p=0.5

p=0.7

p=1.0 (Fully dependent)

Figure 5: EP functions for different dependence mea-sures γp and mean component reliability of p = 0.99

4.2. Clustering and representative scenariosWe find the representative scenario for theT = 100 year return period, by requiring a mini-mum number of samples nv = 103 whose EP fallsin ΩT,∆. According to Equation (5), this requires∆ = 0.1. Since we apply the clustering algorithm to

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γp = 0.0 γp = 0.2 γp = 0.5

Cluster modes Rel. freq.22 0.4718 0.4035 0.13

Cluster modes Rel. freq.11,22 0.3322,38 0.173,13 0.17

28,35 0.13

Cluster modes Rel. freq.14,22 0.235,10 0.175,32 0.175,22 0.13

Figure 6: Representative scenarios for return period of T = 100 years and different dependence measures γp

based on 30 repeated analyses. The size of the markers is proportional to the relative frequency that failure ofthe node is part of a representative scenario among the 30 repetitions. The table summarizes the most frequentrepresentative scenarios, together with their relative frequencies in the 30 repetitions.

the final states of the edges, the samples are verticesof the 43-hypercube graph.

For assessing the robustness of the scenario se-lection, we repeat the MCS and the subsequentanalysis 30 times for each dependence measure γp.The results are summarized in Figure 6.

As evident from Figure 6, the identified repre-sentative scenarios can differ when repeating theanalysis with different MCS samples. This effectis more severe with increasing dependence amongcomponent failures. Among 30 repetitions, in thecase of γp = 0 we find three different representativescenarios, in the case of γp = 0.2 we find eight dif-ferent representative scenarios, in case of γp = 0.5we find nine different representative scenarios, andfor γp = 0.7 we find 28 different representative sce-narios. This effect might be related to the fact thatthe EP functions become flatter around EP = 10−2

with increasing γp (see Figure 5). This indicatesthat the choice of the mode in the cluster as the rep-resentative scenario might not generally lead to ro-bust choices.

5. CONCLUSIONSWe introduce a methodology for selecting represen-tative scenarios of component failures in infrastruc-ture systems, which can be utilized in contingencyanalyses and risk assessments. The scenarios arerepresentative for a fixed return period of perfor-mance losses. The procedure is based on perform-ing an initial Monte Carlo simulation, assuming asufficiently cheap computational model of the sys-tem performance is available, and a cluster anal-ysis to identify the representative combination ofcomponent failures. The application on two exam-ples demonstrates the feasibility of the method, buthighlights that selecting the representative scenarioas the mode of the identified clusters may not al-ways be a robust approach.

6. ACKNOWLEDGEMENTSThis work has been sponsored by the researchand development project RIESGOS (Grant No.03G0876), funded by the German Federal Ministryof Education and Research (BMBF) as part of thefunding programme "BMBF CLIENT II – Interna-tional Partnerships for Sustainable Innovations".

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7. REFERENCESAcosta, J. S., López, J. C., and Rider, M. J. (2018).

“Optimal multi-scenario, multi-objective allocationof fault indicators in electrical distribution systemsusing a mixed-integer linear programming model.”IEEE Transactions on Smart Grid (early access), 1–1.

Crucitti, P., Latora, V., and Marchiori, M. (2004).“Model for cascading failures in complex networks.”Phys. Rev. E, 69, 045104.

Ester, M., Kriegel, H.-P., Sander, J., and Xu, X. (1996).“A density-based algorithm for discovering clustersa density-based algorithm for discovering clusters inlarge spatial databases with noise.” KDD’96 Confer-ence Proceedings, AAAI Press, 226–231.

Hernández-Fajardo, I. and Dueñas-Osorio, L. (2013).“Probabilistic study of cascading failures in com-plex interdependent lifeline systems.” Reliability En-gineering System Safety, 111, 260–272.

Hines, P., Dobson, I., and Rezaei, P. (2017). “Cascadingpower outages propagate locally in an influence graphthat is not the actual grid topology.” IEEE Transac-tions on Power Systems, 32(2), 958–967.

Kengkla, N. and Hoonchareon, N. (2018). “Risk-basedn-2 contingency ranking in transmission system usingoperational condition.” ICEAST Conference Proceed-ings.

Latora, V. and Marchiori, M. (2001). “Efficient behav-ior of small-world networks.” Phys. Rev. Lett., 87,198701–198701–4.

Mitra, P., Vittal, V., Keel, B., and Mistry, J. (2016). “Asystematic approach to n-1-1 analysis for power sys-tem security assessment.” IEEE Power and EnergyTechnology Systems Journal, 3(2), 71–80.

Pescaroli, G. and Alexander, D. (2018). “Understandingcompound, interconnected, interacting, and cascad-ing risks: A holistic framework.” Risk Anal., 38(11),2245–2257.

Poljansek, K., Bono, F., and Gutiérrez, E. (2011). “Seis-mic risk assessment of interdependent critical infras-tructure systems: The case of european gas and elec-tricity networks.” Earthquake Engineering and Struc-tural Dynamics, 41(1), 61–79.

Scherb, A., Garrè, L., and Straub, D. (2017). “Relia-bility and component importance in networks subjectto spatially distributed hazards followed by cascadingfailures.” ASME J. Risk Uncertain. B, 3(2), 021007.

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SelectingRepresentativeScenariosforContingencyAnalysisof ... · 2020. 2. 5. · Hernández-Fajardo and Dueñas-Osorio,2013). If a more expensive model is available and preferable,