《工程分析的模糊集和非精确概率方法》课程教学课件（讲稿）L19_20_Efficient Uncertainty Quantification for Complex Systems Analysis

EfficientUncertaintyQuantificationforComplexSystemsAnalysisMichaelBeerInstituteforRisk and Reliability,Leibniz UniversityHannover,GermanyInstituteforRiskandUncertainty,UniversityofLiverpool,UKInternational JointResearchCenterforEngineeringReliability andStochasticMechanics(ERSM),TongjiUniversity,China

Michael Beer 1 / 24 Efficient Uncertainty Quantification for Complex Systems Analysis Michael Beer Institute for Risk and Reliability, Leibniz University Hannover, Germany Institute for Risk and Uncertainty, University of Liverpool, UK International Joint Research Center for Engineering Reliability and Stochastic Mechanics (ERSM), Tongji University, China

ReliabilityAssessmentofComplexSystemsinUncertainEnvironmentsCOMPLEXNETWORKEDSYSTEMSDependenceoncomplexnetworksconstantlygrows.increasing demand for modern networks to be highly reliablequantifythedegreeto whichanetworkisabletoprovide itsserviceunderstandnetworkbehaviorforreliableandefficientpredictionspower western USfinancial globalgas distribution EuropeGrand challengesmodelinterdependencebetweendifferentnetworksefficient simulation tools and uncertainty quantification methodsdynamicnatureof real-worldnetworksprocessesonnetworks(e.g.epidemics,cascadingfailures)MichaelBeer2/24

Michael Beer 2 / 24 COMPLEX NETWORKED SYSTEMS Dependence on complex networks constantly grows quantify the degree to which a network is able to provide its service understand network behavior for reliable and efficient predictions increasing demand for modern networks to be highly reliable dynamic nature of real-world networks model interdependence between different networks efficient simulation tools and uncertainty quantification methods processes on networks (e.g. epidemics, cascading failures) Reliability Assessment of Complex Systems in Uncertain Environments • • • Grand challenges gas distribution Europe power western US financial global

ReliabilityAssessmentofComplexSystemsinUncertainEnvironmentsENGINEEREDSYSTEMS:SPECIFICCHALLENGESRapidgrowthinscale,complexityandinterconnectionuncertaintiesandrisksappearto agreaterextentthan everbeforeincreasingvulnerabilityandlackof resilience;seeFukushima,financial crisis,cybercrime,...low-probability-high-conseguenceeventsvery difficulttoidentifybuthighlycritical;dramaticconsequencesthroughcascadingfailuresissuesspanacrossdisciplinesApproachesofUncertaintyQuantificationAdvancedMonte CarloGeneralizedmodelsStochasticModelingSimulationmethodsforvagueand impreciseinformation, x(h,) d(o)X(t,w)4μ(x) 41.0eiSe]/htXeX=MichaelBeer3/24

Michael Beer 3 / 24 Rapid growth in scale, complexity and interconnection uncertainties and risks appear to a greater extent than ever before low-probability-high-consequence events very difficult to identify but highly critical; dramatic consequences through cascading failures increasing vulnerability and lack of resilience; see Fukushima, financial crisis, cyber crime, . issues span across disciplines ENGINEERED SYSTEMS: SPECIFIC CHALLENGES Approaches of Uncertainty Quantification Advanced Stochastic Modeling Monte Carlo Simulation methods Generalized models for vague and imprecise information Reliability Assessment of Complex Systems in Uncertain Environments

EfficientSystemsReliabilityAnalysisANALYSISOECOMPLEXSYSTEMSReliability,optimalmaintenanceandrepairmphatiguetestinMalorusChargvettonBanen2900DriveBeltbernanea.cmmo.comGasuneollTrteparatioMichael Beer4/24

Michael Beer 4 / 24 Reliability, optimal maintenance and repair ANALYSIS OF COMPLEX SYSTEMS Efficient Systems Reliability Analysis

EfficientSystemsReliabilityAnalysisANALYSISOECOMPLEXSYSTEMSModelingfor efficient and realisticreliability analysistraditionalapproachesTopevent》faulttreeanalysis》reliabilityblockdiagrams>》limitations in modeling》dependencies》common-causefailures》time-dependentbehavior》lackofinformation》complexnetworkstructurealternativemodelingtoaddresslimitationsimprovedreliabilityandavailabilityanalysisof systems》identifyweakcomponents》defineoptimalmaintenance strategiesMichaelBeer5/24

Michael Beer 5 / 24 ANALYSIS OF COMPLEX SYSTEMS Modeling for efficient and realistic reliability analysis Efficient Systems Reliability Analysis • traditional approaches » fault tree analysis alternative modeling to address limitations » reliability block diagrams » . limitations in modeling » dependencies » common-cause failures » time-dependent behavior » lack of information » complex network structure improved reliability and availability analysis of systems » identify weak components » define optimal maintenance strategies

EfficientSvstemsReliabilityAnalysisANALYSISOFCOMPLEXSYSTEMSSurvival signaturesystems availability interms of numberof working components》statevectorX = (X1..,Xm)Xi=1:componentiis workingX,=O:componentiisNoTworking》structurefunction((≤) = (X1*,X)p(x)=1 : system is workingp(×)=O : system is NOTworking》exampleβ(×1 =1,X2 = 0,× = 0)= ((1,0,0)= 0((× = 1, ×, = 1,×, = 0) = (P(1,1,0) = 1Michael Beer6/24

Michael Beer 6 / 24 ANALYSIS OF COMPLEX SYSTEMS Survival signature Efficient Systems Reliability Analysis • systems availability in terms of number of working components » state vector xx x = ( 1 m ,., ) xi = 1: component i is working xi = 0: component i is NOT working » structure function ϕ =ϕ (x xx ) ( 1 m ,., ) : system is working : system is NOT working ϕ = (x 1 ) ϕ = (x 0 ) » example ϕ = = = =ϕ = (x 1x 0x 0 100 0 12 3 , , ) ( , , ) ϕ = = = =ϕ = (x 1x 1x 0 110 1 123 , , ) ( , , )

EfficientSystemsReliabilityAnalysisANALYSISOFCOMPLEXSYSTEMSSurvival signaturecompressed representationof systemavailability》systemwithmcomponentsof the sametype》Ioutof mcomponentsareworkingD(Z(α)Φ(0) = 0, @(m) = 1XESprobability that system works whenI out of m components are working》examplem=3, 1=2:S, = ((1,1,0),(1,0,1), (0,1,1)(β(1,1,0) = 1, (1,0,1) = 1, β(0,1,1) = 0(区)=号(1+1+0)=号D(2and further: @(0) = 0, Φ(1) = 0, Φ(3) = 1MichaelBeer7/24

Michael Beer 7 / 24 ANALYSIS OF COMPLEX SYSTEMS Survival signature Efficient Systems Reliability Analysis • compressed representation of system availability ( ) ( ) − ∈   Φ= ⋅ϕ   ∑   l 1 x S m l x l » system with m components of the same type » l out of m components are working Φ= Φ = (0 0 m 1 ) , ( ) » example ϕ =ϕ =ϕ = (110 1 101 1 011 0 , , , , , , ) ( ) ( ) m = 3, l = 2: S 110 101 011 l = {( , , , , , , ) ( ) ( )} ( ) ( ) ( ) − ∈   Φ = ⋅ ϕ = ⋅ ++ =   ∑   l 1 x S 3 1 2 2 x 110 2 3 3 and further: Φ=Φ=Φ= (0 0 1 0 3 1 ) , , ( ) ( ) probability that system works when l out of m components are working

EfficientSystemsReliabilityAnalysisANALYSISOFCOMPLEXSYSTEMSSurvivalsignatureexpansiontosystemswithseveraltypesofcomponents》systemwithKcomponenttypes,mkcomponentsoftypek(k=1,,K)》lkoutofmkcomponentsareworking (k=1,..,K)0(...)=PxN》exampleK = 2, m1 = 3, m2 = 3:Φ(1,2Zp(μ)12XeS1①+(0+0 +0) +(0+0+0) =3329(μ)=1：1 (1+1+0)=号D(2,33U8/24MichaelBeer

Michael Beer 8 / 24 ANALYSIS OF COMPLEX SYSTEMS Survival signature Efficient Systems Reliability Analysis • expansion to systems with several types of components » system with K component types, mk components of type k (k=1, ., K) » lk out of mk components are working (k=1, ., K) » example K = 2, m1 = 3, m2 = 3: ( ) ( ) − = ∈     Φ = ⋅ϕ   ∏   ∑       l l 1 k 1 K k 1 k k 1 x S k m l l x l ,., ,., ( ) ( ) ( ) − − ∈   Φ = ⋅ ⋅ ϕ = ⋅⋅ ++ =   ∑   l 1 1 x S 3 3 1 1 2 2 3 x 1 1 0 2 3 3 1 3 , ( ) ( ) (( ) ( ) ( )) − − ∈   Φ = ⋅ ⋅ϕ   ∑   = ⋅⋅ ++ + ++ + ++ = l 1 1 x S 3 3 1 2 x 1 2 1 1 1 001 000 000 3 3 9

EfficientSvstemsReliabilityAnalvsisANALYSISOECOMPLEXSYSTEMSSurvival functiontimedependentsystem reliabilityCk=numberof componentsof typekstill workingattimetproductforindependentP(T. )=()(6(C-)failureofcomponentssystem structureprobability structureefficientsimulationontheseparatedprobabilitystructuresuperiorfeatures》applicabletocomplexsystemstructures》canworkwitharbitrary dependencystructures betweenfailureevents》captureschangingofcomponenttypes(e.g.,duetorepair)》canoperatewithimpreciseprobabilities(capturesdependabilityissuesMichael Beer9/24

Michael Beer 9 / 24 system structure probability structure ( ) ( ) ( { }) = = = >= Φ ⋅ = ∑ ∑  1 k 1 k m m K k S 1 k t k l0 l0 k 1 PT t . ,., l l P C l • time dependent system reliability = number of components of type k still working at time t k Ct efficient simulation on the separated probability structure ANALYSIS OF COMPLEX SYSTEMS Survival function Efficient Systems Reliability Analysis product for independent failure of components • superior features » applicable to complex system structures » can work with arbitrary dependency structures between failure events » captures changing of component types (e.g., due to repair) » can operate with imprecise probabilities (captures dependability issues)

EfficientSystemsReliabilityAnalysisANALYSISOFCOMPLEXSYSTEMSImprecise lifetimedistributionsforcomponentsboundsonsurvivalfunction》stochasticindependencebetween components[E (t)]*[1 - Ek (t)]P(T, >t) = ...(I.l)n0[Ft)-2...2o(..)Assessment of the influence of individual componentstimedependentrelativeimportanceindex(RI)RI (t) =P(T, >tT, >t)-P(T, >t/T, ≤t)Effectoffailureof componentionsystemreliabilitytargeted maintenance or repairof critical componentsatanypointintimeMichael Beer10/24

Michael Beer 10 / 24 Imprecise lifetime distributions for components • bounds on survival function » stochastic independence between components Assessment of the influence of individual components ( ) ( ) ( ) ( ) − = = =   >= Φ ⋅ ∑ ∑ ∏      −      1 k k k k 1 k m m K m l l k S 1 k k k l0 l0 k 1 k m PT t l l Ft 1Ft l . ,., ( ) ( ) ( ) ( ) − = = =   >= Φ ⋅ ∑ ∑ ∏      −      1 k k k k 1 k m m K m l l k k k S 1 k l0 l0 k 1 k m PT t l l Ft 1Ft l . ,., • time dependent relative importance index (RI) RI t P T t T t P T t T t i Si Si ( ) = > >− > ≤ ( ) ( ) Effect of failure of component i on system reliability targeted maintenance or repair of critical components at any point in time ANALYSIS OF COMPLEX SYSTEMS Efficient Systems Reliability Analysis