International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Dynamic restricted equilibrium model to determine statistically the resilience of a traffic network to… Nogal, Maria; Martinez-Pastor, Beatriz; O’Connor, Alan; Caulfield, Brian Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Dynamic Restricted Equilibrium Model to Determine Statisticallythe Resilience of a Traffic Network to Extreme Weather EventsMaria NogalPost-Doc Researcher, Dept. of Civil, Struct. & Env. Eng., Trinity College Dublin, IrelandBeatriz Martinez-PastorPh.D. Student, Dept. of Civil, Struct. & Env. Eng., Trinity College Dublin, IrelandAlan O’ConnorAssociate Professor, Dept. of Civil, Struc. & Env. Eng., Trinity College Dublin, IrelandBrian CaulfieldAssociate Professor, Dept. of Civil, Struct. & Env. Eng., Trinity College Dublin, IrelandABSTRACT: Extreme weather events lead transportation systems to critical situations, which imply highsocial, economical and environmental costs. Developing a tool to quantify the damage suffered by a trafficnetwork and its capacity of response to these phenomena is essential to reduce the damage of this hazardand to improve the system. With this aim, a statistical analysis of the resilience of a traffic network underextreme climatological events is presented. The resilience of a traffic network is determined by meansof a dynamic restricted equilibrium model together with a travel cost function that includes the effect ofweather on a traffic network. The cost function parameters related to the hazard effect are assumed asrandom, following Generalized Beta distributions. Then, the fragility curves of the target traffic networkare defined using the Monte Carlo method and Latin Hypercube sampling. Fragility curves are a usefultool to analyse of the vulnerability of a traffic network, assisting in the decision-making for the preventionand response to the extreme weather events.1. INTRODUCTIONThe loss of serviceability of a traffic network occursseveral times during its service life. Nevertheless,this problem should be limited to a certain level ofacceptable risk and should be easily recoverable.Extreme weather events lead the transportation sys-tems to anomalous situations and in some cases,even to critical ones, which imply a high social,economical and environmental cost. The quantifi-cation of the damage suffered by a traffic networkand its capacity of response to these phenomena isof great importance in order to identify and enhanceany network weaknesses, generating more efficientdesigns.In recent years a holistic concept has been usedto define the capacity of a system potentially ex-posed to hazards, to adapt by resisting or chang-ing in order to reach and maintain an acceptablelevel of functioning (UN/ISDR (2004)), i.e. the re-silience. Quantifying the resilience of a system isnot a straightforward task, because it includes as-pects such as robustness, redundancy, resourceful-ness, adaptability, ability to recover quickly, amongothers (Bruneau et al. (2003); Murray-Tuite (2006);Park et al. (2013)). Despite the complexity of thistask, some numerical models exist which quan-tify the resilience of traffic systems (Ip and Wang(2009); Vugrin et al. (2010); Henry and Ramirez-Marquez (2012); Chen and Miller-Hooks (2012);Nogal et al. (2014b)). Nevertheless, the disrup-tive events considered in most of the existing lit-erature are not related to climatological events butmodification of the traffic network characteristics.With this purpose, some authors such as Lam et al.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015(2008) or Nogal et al. (2014a), propose some travelcost functions to capture the effects of the weatheron the traffic network.On the other hand, systems have some capac-ity of adaptation to a certain degree of perturba-tion, however, when a disruption occurs in a sud-den and intense manner, critical states are gener-ated. When the perturbation is produced by clima-tological phenomena, the extreme weather eventswill cause these critical states. According to thelatest Intergovernmental Panel On Climate Changereport (IPCC (2014)), confidence has increased thatsome extremes will become more frequent, morewidespread and/or more intense during the 21stcentury. In this present-day context, tools andguidelines to assess the resilience of transportationsystems under extreme weather events becomesnecessary.Due to the uncertainties inherent to this problem,a statistical approach guarantees a more realisticand valuable estimation of the response of the sys-tems. Therefore, the aim of this paper is to presenta statistical analysis of the resilience of a traffic net-work under extreme climatological events. The re-silience of a traffic network is determined by meansof a dynamic restricted equilibrium model togetherwith a travel cost function that includes the weathereffect on a traffic network. As a result of the uncer-tainties of the cost function parameters related tothe hazard effect, their values are assumed as ran-dom. The Latin Hypercube is used to sample differ-ent network models to be analysed under real andsynthetic climatological extreme events. Finally,by means of the Monte Carlo Method the fragilitycurves of the traffic system are obtained.This is the first time fragility curves have beenused to analyse the resilience of a traffic systemunder extreme weather events, since this statisticaltool has been traditionally used in the field of struc-tures (Mander and Basöz (1999); Shinozuka et al.(2000); Ghosh and Padgett (2010)). It is noted thatthe response of a traffic system is highly non-linearand its analysis becomes tough, especially when theextreme weather is the source of the traffic networkdisruption. For that reason, the fragility curves be-come a useful tool to assist in the decision-makingfor the prevention and response to the hazards.The paper is organized as follows; Section 2 ex-plains the main parameters required to estimate theresilience of the system by means of the fragilitycurves. The process to obtain these parameters aredescribed in detail in the next sections. Section 3deals with the concept of resilience as well as thedynamic model required to assess the resilience in-dex. Section 4 explains how to determine the haz-ard intensity and Section 5 gives an example to il-lustrate the performance of the proposed method.Finally, in Section 6 some conclusions and futureresearch paths are drawn.2. FRAGILITY CURVESIn the context of transportation systems, thefragility curves are a representation of the proba-bility that a specific traffic network exceeds a givendamage state, as a function of the hazard degree.The fragility function can be expressed as follows:FDSi(HD) = P[DS > DSi|HD), (1)where HD is a parameter indicating the hazard de-gree and DS, a parameter indicating the damagestate.Therefore, to obtain these curves, the followingvariables have to been defined, (a) the hazard de-gree; (b) the discrete Damage States DSi; and (c)a variable that allows the quantification of the re-sponse of the traffic network and, at the same timethat can be related to the damage state, i.e. the re-silience. Consequently, the discrete damage statesare associated with different resilience levels.Moreover, the parameter related to the hazarddegree has to integrate the main aspects affectingthe system resilience and, should be easily com-puted. As indicated, the response of a traffic systemmainly depends on the rapidity with a hazard occursand its intensity. Considering these two aspects, theslope of the cumulative curve of the hazard inten-sity function has been successfully used to evaluateHD.By means of the Monte Carlo Method (MCM),the cumulative distribution functions (CDF) of theresilience associated with different values of HDare obtained. Finally, the fragility curves are de-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015fined using the CDF for the resilience levels relatedto the DSi.In the following sections, obtaining the resilienceand the hazard intensity functions are explained indepth.3. RESILIENCE. A DYNAMIC RE-STRICTED EQUILIBRIUM MODELTo estimate the response of a traffic network un-der extreme weather events it is necessary to con-sider a parameter that represents the global systemresponse. With this aim the resilience has beenselected, which can be defined as the capacity ofa transportation network (a) to absorb disruptiveevents, maintaining its level of service, and (b) toreturn to a level of service equal to or greater thanthe pre-disruption level of service within a reason-able time frame. In this paper only the resilience inthe perturbation stage is analysed.The assessment of the traffic network resiliencerequires a dynamic approach. With this aim, Nogalet al. (2014b) propose a “Dynamic Restricted Equi-librium Assignment Model”, which allows the sim-ulation of the network behaviour when a disruptiveevent occurs.Considering a connected traffic network definedby a set of nodes and a set of links, for certainorigin-destination (OD) pairs of nodes, there aregiven positive demands which give rise to a linkflow pattern when distributed through the network.Then, a dynamic “equilibrium-restricted” state canbe obtained when, for each OD pair, the actual routetravel cost experienced by travelers entering duringthe same time interval tends to be equal and min-imal. Nevertheless the system could be unable toreach this state in such a time interval. The rea-son that the system does not reach an optimal (orminimum cost) state in a given time interval is be-cause the traffic network behaviour is restricted bya system impedance to alter its previous state. Thisimpedance is due to the actual capacity of adapta-tion to the changes, the lack of knowledge of thenew situation and the lack of knowledge of the be-haviour of other users.It is noted that when a disturbance takes place,the travel costs increase. Nevertheless, these costscan remain in low values if the users modify theirroute choices, increasing the stress level of the sys-tem. As this mechanism of response cost-stress islimited, the larger the disruption, the lower the re-maining response capacity. Therefore, the trafficnetwork behaviour when suffering a disruption canbe assessed by means of its exhaustion level (por-tion of used resources), and its evolution with thetime by the exhaustion curve.The proposed approach permits the inclusion ofboth, the stress level of the system and the extracost generated by the hazard, when assessing theexhaustion level of the network.Following the concept of resilience as the capac-ity of the network to absorb a shock, the perturba-tion resilience evaluates how far the system is fromcomplete exhaustion. For that reason, Nogal et al.(2014b) evaluate the perturbation resilience, χκ , asthe normalized area over the exhaustion curve, asindicated:χκ =∫ tp1tp0 (1−ψκ(t))dttp1− tp0100, (2)where tp0 and tp1 denote the initial and the finaltime of the disruptive event, respectively; and ψκ ,the exhaustion level associated with a given state ofperturbation κ . The perturbation resilience is de-fined between [0,100]. Moreover, a cost thresholdis included to assume the system break-down. Thisvalue restricts the perturbation resilience and allowsthe comparison of different resilience indices.Furthermore, a travel cost function that includesthe climatological events has to be considered.More precisely, Nogal et al. (2014a) propose thefollowing expressionτa(t) = τ0a1+ma exp−βaXa(t)Xmaxa+ pah(t)1−h(t)−γa, (3)where τa and τ0a are the actual travel time and thefree travel time, respectively; ma, βa and γa areshape parameters; Xa(t) and Xmaxa are the actualflow and the link capacity to provide a certain ser-vice level; h(t) is the hazard intensity whose rangeis [0,1), and pa is the specific sensitivity of eachlink to this type of hazard. For instance, in the case312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015of pluvial flooding, pa depends on the catchmentarea, slope of the road, type of pavement, existenceof element of protection, etc. Subindex a impliesassociation with link a.It is noted that when any hazard does not exist,h(t) = 0, Eq. 3 becomesτa(t) = τ0a{1+ma exp[−(βaXa(t)Xmaxa)−γ]},(4)with only three parameters related to traffic fea-tures.We indicate that the knowledge of the OD de-mands of a given network and their associated routeflows allows the calibration of the model, i.e. thedefinition of these three parameters for the steadystate.The parameter pa, which includes the local vul-nerability of the network, can be defined based onthe previous experiences and expert opinion. Dueto the uncertainty involved, the proposed approachassumes that pa is a random variable. More pre-cisely, they are assumed to follow a GeneralizedBeta Distribution pa ∼ GBeta(αa,θa; p0a, p f a),where αa and θa are the shape parameters, and p0aand p f a are the range of the parameter pa. Thisdistribution allows the definition of the randomvariable on the interval [p0a, p f a] (Castillo et al.(2012)). After that, by means of the Latin Hyper-cube Sampling (LHS), different networks are mod-elled and analysed under real and synthetic clima-tological extreme events.4. EXTREME WEATHER EVENTSThe definition of extreme event depends on the re-sponse capacity of the system to this event. For in-stance, regarding the rainfall, one of the criteria fordefining “extreme weather” is associated with therainfall level which causes floods; and this level de-pends on the drainage systems, among others. Itis highlighted that this extreme rainfall is not nec-essarily related to its return period. It has beendemonstrated that the diary precipitation with thelargest return period does not always produce theworst flood, as constant, mild rainfall can be evenmore damaging.In this section it is explained how to determinethe hazard intensity for the case of the pluvial flooddue to long periods of rain as well as intense show-ers. With this aim, the data of the daily precipitationof Valencia (Spain) has been selected because of theannual weather phenomenon called “cold drop”1,which causes important floods.The daily precipitation provided by the Euro-pean Climate Assessment Dataset (ECA (2014))has been used to compute the cumulative precipita-tion and the cumulative drainage in Valencia from1961 to 2010. The cumulative drainage has beenobtained by assuming an average drainage capacity,450 mm/day, and 148.5 mm/day when the soil be-comes saturated. When the capacity of drainage isexceeded by the rainfall amount, the flood occurs.For example, Figure 5, which illustrates the dailyprecipitation, cumulative precipitation and cumula-tive drainage in Valencia from 2007 to 2010, showsfour floods in this period of time, two in 2007, onein 2008 and other in 2009. It is noted that thedrainage capacity has been estimated by comparingthe time and duration of the obtained hazards withthe real historical events registered.Time (days)0 200 400 600 800 1000 1200 1400Cumulative precipitation (mm)x10 400.511.522.5Year 2008 Year 2009 Year 2010 Daily precipitation (mm)02 4 6 8 Cumulative Capacity DrainageCumulative Precipitationx10 3Rain flooding registered in October 2007 in ValenciaFigure 1: Daily precipitation, cumulative precipitationand cumulative drainage in Valencia.In order to correctly define the fragility curves,it is necessary to have a great amount of data ofhazards, nevertheless, due to the extreme nature of1This phenomenon is associated with extremely violentdownpours and storms, but not always accompanied by sig-nificant rainfall. This phenomenon usually lasts a very shorttime, from a few hours to a maximum of four days.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015these events, this requirement is fulfilled with dif-ficulty. To overcome this obstacle, synthetic haz-ards have been proposed. This synthetic responsehas been generated using real data, and taking intoaccount aspects as the intensity of the hazard, dura-tion, and the rapidity at which the peaks occur.Figure 2 shows in continuous red colour the 77pluvial flooding cases observed since 1961 to 2010;and the dashed blue curves represent the 100 syn-thetic curves considered in this paper.Time (days)0 2 4 6 8 10 12 14 16 18 20 22Hazard h(t) curvesSynthetic curvesFigure 2: Real and synthetic curves of the pluvialfloods obtained from Valencia historical data.5. ILLUSTRATIVE EXAMPLETo facilitate the understanding of the proposedmethod, we consider an illustrative traffic networkwhich consists of 5 nodes and 16 links as indicatedin Figure 3. Within the set of possible combina-tions, only 8 routes and 5 OD pairs have been con-sidered in this simple example.Table 1 shows the average OD demand for differ-ent OD pairs, the routes defined by their links andthe route flow associated with the equilibrium state(without any hazard).Initially the system is in equilibrium, i.e. allusers have selected those routes that actually mini-mize their route travel time. The route flow associ-ated with this equilibrium state is given in Table 1.When the hazard occurs, the system tries to reach anew equilibrium state. However, due to the systemimpedance this equilibrium is not reached immedi-ately.12345678910111213 14151612 3 45NetworkFigure 3: Illustrative network showing nodes and links.Table 1: OD demand, routes and equilibrium routeflow.OD Average Route Links EquilibriumDemand Route Flow1 - 4 100 1 1 13 14 55.4162 9 10 3 44.5842 - 1 50 3 2 11 12 50.0002 - 5 50 4 13 2 50.0005 - 1 100 5 3 4 0 50.4646 7 8 0 49.5361 - 2 100 7 5 15 16 45.7278 5 6 7 54.273The parameters of the cost function given by theequation 3 are indicated in Table 2.With these values, and using the MCM, differenttraffic networks have been exposed to the real andthe synthetic hazards shown in Figure 2.In each simulation, the link travel cost functionsare computed using the link parameters obtained byLHS. Figure 4 shows the link travel cost functionsof a selected set of links for different levels of haz-ard. The lower green line represents the link traveltime associated with the state without hazard2, theupper blue curve provides the link travel time func-tion associated with the maximum level of hazard.The red dashed line marks the Xmax,a value and thegrey band indicates the actual range of link flow2This curve roughly matches with the well known BRPfunction for ratios xaXmax,a < 1.2)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Link flow0 20 40 60 80Link travel cost0.511.5 h=0.00h=0.43 Link 1  (p1=0.32)Link flow0 20 40 60 80Link travel cost0.511.5 h=0.00h=0.43 Link 3  (p3=0.24)Link flow0 20 40 60 80Link travel cost0.511.5 h=0.00h=0.43 Link 9  (p9=0.49)Link flow0 20 40 60 80Link travel cost0.511.5 h=0.00h=0.43 Link 10 (p10=0.15)Link flow0 20 40 60 80Link travel cost0.511.5 h=0.00h=0.43 Link 14 (p14=0.21)Link flow0 20 40 60 80Link travel cost0.511.5 h=0.00h=0.43 Link 16 (p16=0.20)XmaxActual range of link flowFigure 4: Link travel cost functions for different levels of hazard.during the process stability-hazard-stability. For in-stance, it can be appreciated that in link 1, whoseparameter p = 0.32 in this simulation, the flow ratevaries from 50.056 users to 46.44 users as a con-sequence of the hazard. The link travel cost beforethe hazard is 0.12 hours and when the hazard levelis h = 0.43, the link travel cost becomes 0.74 hours.Subsequently, the resilience associated with eachhazard is assessed using Eq. (2) by means of thestress level and the cost function, as shown Fig-ure 5.After applying the MCM, the CDF of the re-silience index for different hazard degrees are com-puted, as shown by Figure 6.For this traffic network, four discrete damagestates have been considered, viz., negligible, light,light to moderate and moderate, associated with theresilience levels 95%, 85%, 75% and 65%, respec-tively. Finally, making use of these resilience lev-Time0 10 20 30 40 50 60Stress Level00.51Perturbation start Perturbation end Stress LevelTime0 10 20 30 40 50 60Cost Level00.51Perturbation start Perturbation end Cost LevelTime0 10 20 30 40 50 60Exhaustion Level00.51Perturbation start Perturbation end Exhaustion LevelResilienceFigure 5: Example of resilience assessment.els, the fragility curves of the traffic network are ob-tained, as Figure 7 illustrates. Based on the analysisof the fragility curves it can be stated, for instance,612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 2: Parameters of the cost function. ma = 7; βa =0.9 and γ = 5.5.Free flow pLink speed Capacity Generalized Beta(km/h) (users) αa θa p0a p0 f1 120 90 3 2 0.2 0.62 120 90 10 10 0.15 0.253 120 90 2 3.5 0.1 0.44 120 90 3 2 0.2 0.65 120 90 3 2 0.2 0.66 120 90 2 3.5 0.1 0.47 120 90 10 10 0.15 0.258 120 90 3 2 0.2 0.69 90 60 3 2 0.2 0.610 90 60 2 3.5 0.1 0.411 90 60 10 10 0.15 0.2512 90 60 3 2 0.2 0.613 90 90 10 10 0.15 0.2514 90 60 2 3.5 0.1 0.415 90 60 2 3.5 0.1 0.416 90 60 10 10 0.15 0.25Resilience (R)40 50 60 70 80 90 100p(R <= Rk | HD) = 0.038HD = 0.073HD = 0.108HD = 0.142HD = 0.177HD = 0.212HD = 0.247HD = 0.281HD = 0.316HD = 0.351HD = 0.385HD = 0.420Figure 6: CDF of the resilience for different hazarddegrees.that a hazard of HD = 0.22, there is a 98% proba-bility that the damage is worse than light, and 8%probability that the damage is worse than light-to-moderate.6. CONCLUSIONSIn this paper the fragility curves have been used todetermine the traffic network response to a extremeweather event. This representation of the probabil-ity that a specific traffic network exceeds a givenFigure 7: Fragility curves of the traffic network.damage state, is given as a function of the haz-ard degree. The resilience has been used to definethe damage states and the hazard degree has beencomputed as the normalized slope of the cumula-tive curve of the hazard intensity function.Additionally, the following conclusions can bedrawn from this paper:1. A Dynamic Restricted Equilibrium Assign-ment Model has been used to compute the re-silience of the system. This model takes intoaccount important and complex features of thetraffic network such as the stress level, thevulnerability and the capacity of adaptation.Moreover, this model allows a dynamic anal-ysis, which is a key aspect when consideringclimatological events.2. By means of a travel cost function which cap-tures the consequences of the extreme weatheron the traffic network, time-varying hazardshave been introduced coupled with the effectof these hazards on each specific link.3. The cost function parameter pa, which in-cludes the local vulnerability of the network,is assumed as a random variable following aGeneralized Beta Distribution. This assump-tion permits the definition of the random vari-able on the interval [p0a, p f a].4. The joined probability of the involved param-eters is computed by using the Monte CarloMethod together with the Latin Hypercube712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Sampling.5. The uncertainties of the involved parametersmake the deterministic approaches inadequateto evaluate the response of a traffic system tothis type of disruption. Therefore the prob-abilistic methods are required to provide amore realistic analysis. More precisely, thefragility curves are a useful tool to evaluate thevulnerability of a traffic network, assisting inthe decision-making for the prevention and re-sponse to the hazards.Finally, in following publications, this method-ology shall be applied to real networks. However,due to the social sensitivity of the results on realnetworks, this task has to be addressed carefully.On the other hand, the analysis of fragility curvesconsidering recovery resilience has yet to be car-ried out. This study will imply a new definition ofthe HD parameter, since the recovery of the trafficnetwork depends on different aspects.ACKNOWLEDGEMENTSThis project has received funding from the Euro-pean Union’s Seventh Framework Programme forresearch, technological development and demon-stration under grant agreement no 608166.7. 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A global review ofdisaster reduction initiatives.” Report No. 1, UnitedNations.Vugrin, E., Warren, D., Ehlen, M., and Camphouse, R.(2010). “A framework for assessing the resilienceof infrastructure and economic systems.” Sustain-able and Resilient Critical Infrastructure Systems,Springer, 77–116.8


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