An Adaptive Metamodel-Based Subset Importance Sampling approach for the assessment of the functional failure probability of a thermal-hydraulic passive system

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An Adaptive Metamodel-Based Subset Importance Sampling approach for the assessment of the functional failure probability of a thermal-hydraulic passive system

Nicola Pedroni, Enrico Zio

To cite this version:

Nicola Pedroni, Enrico Zio. An Adaptive Metamodel-Based Subset Importance Sampling approach

for the assessment of the functional failure probability of a thermal-hydraulic passive system. Applied

Mathematical Modelling, Elsevier, 2017, 48, pp.269-288. �10.1016/j.apm.2017.04.003�. �hal-01652225�

Applied Mathematical Modelling 48 (2017) 269–288

ContentslistsavailableatScienceDirect

Applied Mathematical Modelling

journalhomepage:www.elsevier.com/locate/apm

An Adaptive Metamodel-Based Subset Importance Sampling approach for the assessment of the functional failure

probability of a thermal-hydraulic passive system

Nicola Pedroni

^a,∗

, Enrico Zio

^a,b

aChair on Systems Science and the Energetic Challenge, Fondation Electricité de France (EdF), Laboratoire Genie Industriel (LGI), Université Paris-Saclay-Ecole CentraleSupelec, Grande Voie des Vignes, 92290 Chatenay-Malabry, France

bEnergy Department , Politecnico di Milano , Via Ponzio 34/3 , 20133 Milano , Italy

a r t i c l e i n f o

Article history:

Received 9 June 2015 Revised 8 March 2017 Accepted 3 April 2017 Available online 7 April 2017 Keywords:

Nuclear passive safety system Small failure probability Efficient Monte Carlo Subset Simulation

Non-parametric importance sampling Adaptive metamodeling

a b s t r a c t

AnAdaptiveMetamodel-BasedSubsetImportanceSampling(AM-SIS)approach,previously developedbytheauthors,ishereemployedtoassessthe(small)functionalfailureprob- abilityofathermal-hydraulic(T-H)nuclearpassivesafetysystem.Theapproachrelieson aniterativeImportanceSampling(IS)schemethatefficientlycouplesthepowerfulcharac- teristicsofSubsetSimulation(SS)andfast-runningArtificialNeuralNetworks(ANNs).In particular,SSandANNsareintelligentlyemployedtobuildandprogressivelyrefineafully nonparametricestimatoroftheideal,zero-varianceImportanceSamplingDensity(ISD),in orderto:(i)increasetherobustnessofthefailureprobabilityestimatesand(ii)decrease thenumberofexpensiveT-Hsimulationsneeded(togetherwiththerelatedcomputational burden).

The performance ofthe approach is thoroughlycompared tothat of otherefficient MonteCarlo(MC)techniquesonacasestudyinvolvinganadvancednuclearreactorcooled byhelium.

© 2017 Elsevier Inc. All rights reserved.

1. Introduction

Thenuclearindustryischaracterizedbylevelsofriskthatareamongthelowestwithrespecttoanyengineeringappli- cation,asrequestedbythestrongpublicperception.Inthepastandcurrentgenerationsofreactordesigns,theselowlevels ofriskaretypically achievedby resortingmostlytoactive safetysystems.Ontheother hand,themostinnovative reactor concepts,suchasthoseofGenerationIV,aimatrelyingalmostentirelyonpassivesafetysystems,thusbecomingasmuch independentaspossible fromany kindofexternal input, especiallyenergy [1]. Moreover, thesimplicityof thedesign of thesesystemsissuchthatmechanicalfailuresbecomeveryunlikelytohappen,incontrasttoactivesystems[2–7].

Yet, uncertainties in the operation of these systems (due to not fully understood physical phenomenaand scarce, if notnull,operatingexperience[8,9])maygiverisetofunctionalfailures.Apassivesystemfailstoperformitsfunction(e.g., decayheatremovalbynaturalcirculation)duetodeviationsfromitsexpectedbehavior,whichleadtheloadimposedonthe systemtoovercomeitscapacity[10–24].Aclassicalapproachtoestimatingsuchfunctionalfailureprobabilitystemsfromthe deterministicmodelingofthe Thermo-Hydraulics(T-H)ofthepassivesystemby a(possiblylong-running)computer code

∗ Corresponding author.

E-mail addresses: [email protected] , [email protected] (N. Pedroni), [email protected] , [email protected] (E. Zio).

http://dx.doi.org/10.1016/j.apm.2017.04.003 0307-904X/© 2017 Elsevier Inc. All rights reserved.

andthesubsequentMonteCarlo(MC)propagationoftheuncertainties,describedbyproperlyidentifiedProbabilityDensity Functions (PDFs), tosome modeloutputs withrespecttowhichthe failureeventis defined.The propagationisbasedon repeatedcomputerruns(orsimulations)oftheT-Hcodeincorrespondenceofdifferentsetsoftheuncertaininputvalues sampledfromtheirPDFs[25–36].

Inpractice,apassivesafetysystemisobviouslydesignedsothatitsfailure isarareevent.As aconsequence,its prob- ability offailure isquite small(e.g.,about10⁻⁵ orless) andits estimationmayrequirea large amountof T-Hcomputer coderuns[37].Sinceeachsimulationofthedetailed,mechanisticT-Hmodelmaytakeseveralhoursinsomecases[33],the standardMCsimulation-basedproceduretypicallyrequiressignificantcomputationalefforts.Moreefficienttechniquesmust thenbesought[38,39].

Twofamiliesofmethodsaretypicallyemployed toaddresstheseissues.Onone side,fast-runningsurrogateregression models(alsocalledresponsesurfacesormetamodels)canbeproperlybuilttomimictheresponseoftheoriginalT-Hmodel code:sincethemetamodelis,ingeneral,muchfastertobeevaluated,theproblemoflongcomputingtimesiscircumvented.

Severalkindsofsurrogatemetamodelshavebeenrecentlyapplied tosafety-relatednuclear, structuralandhydrogeological problems,includingpolynomialResponseSurfaces(RSs)[40–43],polynomialchaos expansions[44–47],stochastic colloca- tions[48,49],ArtificialNeuralNetworks(ANNs)[50–53],SupportVectorMachines(SVMs)[54,55]andkriging[56–61].

Ontheotherside,advancedMonteCarloSimulationmethodscanbeadoptedtorealizerobustestimationsbymeansof alimitedamountofrandomsamples[62–64].ThisclassofmethodsincludesStratifiedSampling[65,66],SubsetSimulation (SS) [67–74],the Response Conditioning Method (RCM) [75,76]for structuralreliability analysis (also developed from SS andemployingapproximatesolutions), LineSampling(LS) [77–84]andsplittingmethods[85–89].However, theadvanced MCSmethodsthatisadoptedmorefrequentlyisperhapsImportanceSampling(IS),wheretheoriginalPDFoftheuncertain inputsis replaced by aproper Importance SamplingDensity (ISD):this favorsthe MC samplesto lie closeto the failure region, thusartificiallyincreasing the frequencyofthe (rare)failure event[90–92].Inthis respect,itis possibleto show thatthereisanoptimal(ideal)ISDthatmakesthestandarddeviationoftheMCestimatorequaltozero.Ontheotherhand, thisoptimalISDcannot be used inpractice,because itsmathematical expressioncontainsthe (aprioriunknown) failure probabilityitself.Withinthiscontext,severaltechniquesinvariousfieldsofresearchhavebeenproposedtoapproximatethe

“ideal” optimalISD:see,e.g.,theAdaptive Kernel(AK)[93,94],theCross-Entropy(CE)[95–97],theVariance Minimization (VM)[98]andtheMarkovChainMonteCarlo-ImportanceSampling(MCMC-IS)[99]methods.

Inthepresentpaper,theAdaptiveMetamodel-basedSubsetImportanceSampling(AM-SIS)method,originallydeveloped bytheauthors[100],isadopted.Intheapproach,thecharacteristicsofseveraladvancedcomputationalmethodsofliterature areefficientlycombined.Inparticular,SSandANNmetamodelsarecoupledwithinan adaptiveISschemeinthefollowing twostages[99,100]:

1. arandomsampleapproximatelydistributedastheideal,zero-varianceISDiscreatedbyresortingtotheSubsetSimulation (SS)technique.NoticethatatthisstepwesubstitutetheexpensiveT-HcodewithanANNmetamodel(adaptivelyrefined intheareasclosetothefailureregionbymeansofthesamplesiterativelyproducedbySS),withtheaimofreducingthe relatedcomputationalburden.Then,weadoptafullynonparametricPDF(proposedinRef.[99]andrelyingontheGibbs Sampler[101])to“fit” thesamplestherebygeneratedandobtainanestimatorfortheoptimalISD;

2. weemploythequasi-optimalISDestimatorbuiltatstep(1)above toperformImportanceSampling(IS)andassessthe functionalfailureprobabilityoftheT-Hpassivesystem.

Noticethattheideaof“interlacing” metamodelswithefficientMCSschemeshasbeenalreadyproposedintheliterature fordecreasingthecomputationalburdenassociatedtothequantificationofsmallprobabilities:see,e.g.,Refs.[102–113].In theseworks,themetamodelisbuiltandprogressivelyrefined(bymeansofsamplesintelligentlygeneratedbytheadvanced MCtechnique)untiladesiredlevelofaccuracyandprecisioninthefailureprobabilityestimateisobtained.

ThemaincontributionsofthepresentworkwithrespecttothereferencepaperbyRef.[99]arethefollowing:

• weadoptSubsetSimulation(SS)forthegenerationofsamplesapproximatelydistributedaccordingtothezero-variance ISD;

• weemployANNstodecreasethecomputationaleffortrelatedtotheapproximationofthezero-varianceISD:inpartic- ular,wetrainthesemetamodelsthroughanadaptivealgorithmthat makesasmartuseofthesamplesproducedby SS withtheobjectiveofprogressivelyamelioratingtheANNaccuracyaroundthefailureregion;

• weassesstheefficiencyofAM-SISintheestimationofsmallfunctionalfailureprobabilitieswithaverysmallnumberof callstotheT-Hcode(e.g.,oftheorderoffewtensorhundreds):thisisrelevantforthosepracticalcaseswhereasingle simulationneedsseveralhourstorun;

• weextensivelyandsystematicallycomparetheperformanceofAM-SIStothatofnumerousotheradvancedprobabilistic simulationmethodsofliterature.

Inaddition,thenovelcontentsofthispaperinrelationtothepreviousarticlebytheauthors[100]arethefollowing:

• inRef. [100], thefocus ofthe comparisonsbetween AM-SISand other MC schemes (inparticular, standard MCS, LHS [65],IS[90–94]andLS[77–84])isexclusivelyputontheefficiencyofstage(2)ofthealgorithm,i.e.,ofrandomimpor- tancesamplingforfailureprobabilityestimation.Inthiswork,instead,comparisonsaremadealsotoothermethodsthat combineSSandmetamodelsandthatareveryclosetostage(1)ofourapproach.Inparticular,thefollowingtechniques

N. Pedroni, E. Zio / Applied Mathematical Modelling 48 (2017) 269–288 271

are considered: (i) the ²SMART algorithm [105],which combines SSandSVMs inan original active learningscheme, wherefew trainingpointsare placedsequentiallyinthe uncertainvariablespace,inorderto builda sequenceofSVM metamodels asaccurate surrogatesoftheSSintermediate thresholds;(ii)theNeuralNetwork-basedSubset Simulation (SS-NN) method [113], inwhich informationis transferred from theMCMC algorithm to theANN trainingprocess in ordertoefficientlyconstructthemetamodelinasequenceof“movingwindows”;

• also the performance of stage (1) of the AM-SIS method is assessed more thoroughly by carrying out a larger set of comparisons(e.g.,byconsideringdifferentsamplesizesforfailureprobabilityestimation);

• we have modifiedandimproved the techniqueadoptedto identify, selectand“distribute” in the entireuncertain in- put space the (training) samples used to adaptivelyrefine the ANN metamodelthrough the iterations of theAM-SIS algorithm.

AlltheanalysesareperformedwithregardstothesamecasestudyconsideredinRef.[100],whichconcernsaGas-cooled FastReactor(GFR)cooledbynaturallycirculatingheliumafteraLossOfCoolantAccident(LOCA)(modifiedfromRef.[26]).

Theremainderofthepaperisorganizedasfollows.InSection2,theconceptsoffunctionalfailureanalysisforT-Hpassive systemsaresyntheticallysummarized.Section3,aquitedetaileddescriptionoftheAM-SIStechniqueproposedinthiswork isgiven.Section4presentsthecasestudyconcerningthepassivecoolingofaGFR,andthecorrespondingresultsareshown anddiscussed inSection5.Someconclusionsaredrawn inthelastSection. Finally,forthesake ofpaperself-consistency, theAppendixreportsthoroughmathematicaldetails concerningtheconstruction ofthefullynonparametricISDproposed inRef.[99]andbasedonthewell-knownGibbsSampler[101].

2. ThequalitativeandquantitativestepsoffunctionalfailureanalysisofT-Hpassivesystems

Thebasicsteps ofthe qualitative andquantitativephase ofthe functionalfailure analysisofa T-Hpassivesystem are [25]:

1.Characterizetheaccidentscenarioinwhichthepassivesystemunderconsiderationoperates;

2.Definethefunctionthatthepassivesystemisexpectedtoperform,e.g. decayheatremoval,vesselcooling anddepres- surization,etc.

3.Identify thedesignparameters values(e.g.,power level,systempressure,heatexchangerinitial walltemperature, etc.) relatedtothereferenceconfigurationofthepassivesystemandthecorrespondingnominalfunction.

4.Identifythepossiblefailuremodesofthepassivesystemfortheaccidentscenariounderconsideration.

5.Build a mechanistic, best-estimate (typicallylong-running) T-H modelto simulatethe passive systembehavior in the accidentscenarioofinterestandperformbestestimatecalculations.

6.IdentifythefactorsofuncertaintyintheresultsofthebestestimateT-Hcalculations.Inthepresentwork, weconsider onlyepistemicuncertainties(duetothelimitedknowledgeonsomephenomenaandprocesses– e.g.,models,parameters andcorrelationsusedintheT-Hanalysis– andtothepaucityoftherelatedoperationalandexperimentaldataavailable), asdonealsointhepreviousarticlebytheauthors[100].

7.Quantifytheuncertaintiesidentifiedatthepreviousstepbyproperprobabilitydistributions.

8.Propagatethe(epistemic)uncertaintiesthroughthedeterministic T-Hmodelinorderto assesstheprobabilityoffunc- tionalfailureofthepassivesystem.Formally,let:x={x₁,x₂,…,x_j,…,xn_i}bethevectoroftherelevantuncertain(input) model parameters/variables;

μ

^{y(x) be a (possiblynonlinear)} deterministic function representing thecomplex, compu- tationally expensiveT-Hcode;y={y₁,y₂,…,y_l,…,yn_o}=

μ

^{y(x) bethe vectorofthe} (uncertain)T-H modeloutputsof interest;Y(x)beasingle-valuedindicatoroftheperformanceofthepassivesystem(e.g.,thefuelpeakcladdingtemper- ature); and

α

Yalimitvaluethat representsthesystemfailure criterion.¹Forillustratingpurposes,wesupposethatthe passive systemfunctionscorrectlyaslongasY(x) <

α

Y.The probability P(F) ofsystemfunctionalfailure can,then, be expressedbythemultidimensionalintegral:

P

(

^F

)

^{= ... IF}

(

^x

)

^q

(

^x

)

^dx, (1)

whereq(·) is thejoint ProbabilityDensity Function (PDF)describing the uncertaintyinthe parameters x,Fis thefailure region(where Y(x) ≥

α

Y) andI_F(·) isan indicator function such that I_F(x) = 1,if x ∈F(i.e.,ifY(x) ≥

α

Y) and I_F(x) = 0, otherwise(i.e.,ifY(x)<

α

Y).

TheMCSprocedure forestimating(1)entailsthat a typicallylarge numberN_Tofsamples ofthevalues ofthesystem parameters xbe drawn fromthe corresponding probability distributionsand usedto evaluate(y(x) and) Y(x) by running theT-Hcode.AnestimatePˆ(^F)^{NT ofthe}probabilityoffailureP(F)can,then,becomputedbydividingthenumberoftimes thatY(x) >

α

Y by thetotalnumberofsamplesN_T.The costrelatedtothisprocess canbe impractical,whentherunning time foreach T-Hmodel simulation (i.e.,foreach evaluation ofY(x))takes severalhours (which is oftenthe caseforT- Hpassivesystems). Inordertotackletheseissues,we adopttheAdaptive Metamodel-basedSubsetImportance Sampling

1Note that the choice of a single-valued performance function does not reduce the generality of the approach, because any multidimensional vector of physical quantities (i.e., the vector y of the outputs of the T-H code in this case) can be conveniently re-expressed as a scalar parameter by resorting to suitable min-max transformations: see Refs. [67,68,72] for details.

(AM-SIS)method(proposedbytheauthorsinRef.[100]),whichcombinesthecharacteristicsoftwoadvancedMCSschemes (i.e.,ImportanceSampling-ISandSubsetSimulation-SS)andametamodel(i.e.,anArtificialNeuralNetwork-ANN)(Section3).

3. TheAdaptiveMetamodel-BasedSubsetImportanceSampling(AM-SIS)method

TheAdaptiveMetamodel-basedSubsetImportanceSampling(AM-SIS)method,proposedinRef.[100]andhereemployed, basicallyrelies onan iterative Importance Sampling(IS) scheme,whereby theImportance SamplingDensity (ISD) iscon- structedandadaptivelyrefinedsoastofavortheMCsamplestobenearthefailureregion,thusincreasingthefrequencyof occurrenceofthe(rare)failureevent[90–92].Inmoredetail,itconsistsoftwomainsteps[98,99,118]:

1. Webuildan estimatorgˆ^∗(^x)^{oftheso-called“optimalISD”g∗}(^x)= ^IF(_P^x₍⁾_F^q₎^(x),i.e.,theISDthat minimizesthevariance of theMCestimatorPˆ(^F)^NT [90–92,107](Section3.1);

2. WeperformImportanceSampling(IS),inwhichtheestimatorgˆ^∗(^x)^{builtatstep1aboveisemployedasanISDinorder} toassessthefunctionalfailureprobabilityP(F)(1)oftheT-Hpassivesystem(Section3.2).

3.1. Adaptiveapproximationoftheideal,zero-varianceISDbymeansofSubsetSimulationandmetamodels

Manyapproacheshavebeenproposed to approximatethe optimalISDg^∗(x) ortoatleast identifyoneproviding small standarddeviationoftheMCestimator:see,e.g.,Refs.[90,91,94–99]fordetails.Theestimatorgˆ^∗(^x)^is,here,constructedin twostages:

a. weemploytheSubset Simulation(SS)method[67,68,70,73]to createapopulation

{

^zmF : m=1,2,...,M

}

^ofMsamples

approximatelydistributedaccordingtotheoptimalISDg^∗(·)(4),i.e.,

{

^zmF : m=1,2,...,M

}

∼g^∗(·)=I_F(·)q(·)/P(F).²No- ticethatherewesubstitutetheexpensiveT-Hcodey(x)=

μ

y(x)withaquick-runningANNregressionmodel(adaptively refinedbymeansofthesamplesiterativelyproducedbySS),withtheaimofreducingthecomputationalburdenrelated tothisstep(Section3.1.1);

b. we “fit” the samples

{

^zmF :m=1,2,...,M

}

^{by means of a proper PDF, in order to obtain an estimator gˆ∗}(^x) ^for g^∗(x). To this aim, we adopt the fully nonparametric PDF proposed in Ref. [99] and relying on the Gibbs Sampler [101](Section3.1.2).

3.1.1. SubsetSimulationandadaptivelytrainedANNmetamodelsforproducingsamplesdistributedaccordingtotheoptimalISD As highlighted above, in order to build an estimator gˆ^∗(·) ^{for g∗(}·)=I_F( · )q( · )/P(F) we need a population

{

^zmF : m=1,2,...,M

}

^{ofM samples} approximately distributedaccording to the optimal ISD, i.e.,

{

^zmF : m=1,2,...,M

}

∼ g^∗(·).TheSubsetSimulation (SS)methodishereadoptedto thisaim.Thestrength ofSSisthatitconvertsthesimulation of a singleevent (in thiscase, the failure eventF) into a sequence ofn_F simulations ofintermediate conditional events correspondingtosubsetsF_i,i=1,2,…,n_F,oftheuncertainparameterspace.Duringsimulation,MarkovChainMonteCarlo (MCMC)basedontheMetropolis–Hastings(MH)algorithm[67]isemployedtoproducetheconditionalsamples(belonging to theintermediate subregionsF_i):in thisway,thesuccessive intermediate subsetsF_iare progressivelypopulatedby the conditionalsamplesuptothetarget(failure)regionF[67,68].Itcanbeshownthatfollowingthisprocedure,thestationary densityofthe(failure)pointslyinginFisq(·|F)=I_F(·)q(·)/P(F),whichisexactlytheideal,zero-varianceISDg^∗(·)[67,73]. Ontheother hand,itis worthnotingthat asinglerun oftheSSmethodmayrequirea significant numberNc ofevalua- tionsofthelong-runningmodel,dependingonthemagnitudeofP(F) (i.e.,approximatelyNc=M·logp0[P(F)],withM≥100 forproducingreliable estimates,whichmayimplyeventhousandsofcalls to theT-Hcode).Thus, inorderto reducethe associatedcomputational burden,we substitutethe T-Hcodey(x)=

μ

^{y(x) bya regressionmodelf(x,w∗),dependingona}

vectorofadjustableparametersw^∗tobeproperlycalibratedbymeansofafinite(andpossiblyreduced)setofinput/output dataexamples (i.e.,patterns),alsocalledDesignOfExperiments(DOE)[114].In thiswork, three-layered feed-forwardAr- tificialNeuralNetworks (ANNs)trained bythe errorback-propagation algorithm areconsidered [52,53,115–117]. Theyare structuredinthreelayers(namely,theinput,hiddenandoutputlayers),whichcontainrespectivelyn_i,n_handnocomputing elements(named neuronsornodesandmathematicallydescribedbysigmoidalbasisfunctions).Thesecomputingunitsare linkedbyweighedconnections(calledsynapses).DetailsaboutANNregressionmodelsarenotreportedhereforbrevity:the interestedreadermayrefertothecitedreferencesandthecopiousliteratureinthefield.

The coreof the method (proposedin Ref. [100] andadopted inthe present manuscript) isthat the ANN metamodel is builtandadaptivelyrefined bymeans ofthe samplesiterativelyproduced by SSitself,accordingto thefollowing steps [105,119](Fig.1):

1. ConstructandtrainaninitialANNregressionmodelf(z,w⁰)bymeansofaDOED⁰_TR=

{

(^z0p,y⁰_p),p=1,2,...,N_T⁰_R

}

^consti-

tutedbyN_TR°elementsselectedbyplainrandomsampling,LatinHypercubeSampling(LHS),partial/fullfactorialdesigns orother more sophisticated experimental design methods. Byso doing, we create a “first trial” surrogate ofthe T-H

2Notice that the change of name of the ‘dummy input vector’ from x to z has no ‘conceptual reason’ and is done only for notational simplicity and coherence with the description of the method reported in the following Sections.

N. Pedroni, E. Zio / Applied Mathematical Modelling 48 (2017) 269–288 273

Fig. 1. Sketch of the procedure used for generating samples distributed as the ideal, zero-variance ISD by means of Subset Simulation and adaptively trained ANN regression models within the AM-SIS algorithm ( Section 3.1.1 ).

model coderepresentedby theunknown nonlineardeterministic function y=

μ

^{y(z).Notice that theANNstructure (in}

practice,thenumbern_h⁰ ofhiddenneurons)hastobechosensothatthetotalnumbern⁰_p,ANN ofadjustableparameters w⁰islowerthanthesizeoftheDOE(i.e.,n⁰_p,ANN <N_TR⁰);

2.Run SS employing the ANN regression model f(z, w⁰) (instead of the original T-H code) to generate M samples

{

^zmnF^,0:m=1,2,...,M

}

approximatelydistributedasg^∗(·)=q(· |F)=I_F(·)q(·)/P(F).Noticethatthissimulationisalmost costlessfromthecomputationalviewpoint,sinceitrequirescallingonlythefast-runningANNmodelf(z,w⁰)insteadof thelong-runningT-Hcode

μ

y(z):thus,thenumberMofconditionalsamplesiterativelyproducedbytheSSalgorithmin eachintermediateconditionalregionF_icanbechosentobequitelarge,inordertoincreasetheaccuracyandrobustness oftheprocedure(e.g.,M=50,000inthispaper);

3.ProduceanestimatePˆ(^F)^NANN^T,0ofthefailureprobabilityP(F)byusingf(z,w⁰)(theestimateprovidedbySSattheprevious step2canbestraightforwardlyused).Asforstep2above,thisoperationisalmostcomputationallycostless;

4.Sett=0and

ε

P(F)=100%,wheretistheiterationindexand

ε

P(F)standsfortherelativevariationbetweentwoconsec- utiveANN-basedestimatesofP(F)producedduringtheadaptiveanditerativeprocedureofmetamodelrefinement;

5.while

ε

P(F) >

ε

^thP(^F)(i.e.,thechangebetweentheANN-basedfailureprobabilityestimatesattwoconsecutiveiterationsis largerthanaproperthreshold

ε

^thP(F))andNTRt<NTRmax(i.e.,thesizeoftheDOE– inotherwords,theoverallnumberof coderuns– remainssmallerthanamaximalpredefinedvalueN_TR^max),repeatthefollowingoperationsinordertorefine theANNregressionmodelinproximityofthefailureregionFandtoproduceasampleapproximatelydistributedasthe

“ideal” ISDg^∗(·)=I_F(·)q(·)/P(F):

(a) t=t+1;

(b) selectN_add points{z^add_p : p= 1,2,…, N_add}amongall those producedby SSatiteration (t – 1):a highfractionof thesepointsisselected“around” thelimitstateinordertobetterdescribetheresponseoftheT-Hpassivesystemin theareasclosetothefailureregionF;theremainingelementsare“distributed” inproximityoftheboundariesofthe otherintermediateconditionalsubsetsF_iidentifiedbySS.Bysodoing,areliable“anchoring” oftheANNmetamodel is possible also in the relevant areasof the uncertain input space “visited” by the SS algorithm (in order not to interrupttheflowofthepresentation,furthertechnicaldetailsaboutthissteparegivenattheendofthisSection);

(c)join the N_add new elements to the current DOE after N_add new evaluations of the system perfor- mance function by means of the original T-H model

μ

^{y(z): thus, Dt}TR=D^t−1_TR ∪D^add_TR , where D^add_TR =

{

(^zaddp ,y^add_p =

μ

^y(^zaddp )),p=1,2,...,N_add

}

^{; also, the size Nt}TR of D^t_TR is N_T^t_R=N^t_T⁻_R¹+N_add. It is important to note thatnowthecurrentDOEisobviouslyenrichedintheareasclosetothefailureregion;

(d)employtheupdatedDOED^t_TR toconstructanupdated ANNregressionmodelf(x,w^t).SincethesizeoftheDOEhas increased, alsothe numberofthe ANNadjustable parameters(in practice, thenumber ofhiddenneurons) can be increaseduntiln^t_p,ANN <N_T^t_R.Also,noticethatthenewANNmetamodelf(x,w^t)isnowexpectedtobemoreaccurate indescribingthebehavioroftheoriginalT-HcodearoundtheregionFofinterest;

(e) generateMsamples

{

^zm,tnF : m=1,2,...,M

}

approximatelydistributedasg^∗(·)byrunningSSwiththeANNregression modelf(x,w^t);

(f)produceanANN-basedestimatePˆ(^F)^N_ANN^{T,t ofP(F)andcompute}

ε

P(F)as

|

^Pˆ(^F)^N_ANN^T,t−Pˆ(^F)^N_ANN^T,t−1

|

/Pˆ(^F)^N_ANN^{T,t−1.NoticethatSS} hastoberunateachiterationinordertoobtainboththesamples

{

^zmnF^,t:m=1,2,...,M

}

^{andthefailure}probability estimates Pˆ(^F)^N_ANN^{T,t used forthe} convergencecheck. In order to “remove” thevariability that stems from different random numbers (in subsequent simulations), the random generator is reinitialized using the same seed at each iteration;

6. thefailureregionFdoesnotchangeexcessivelyfromoneiterationtoanother,i.e.,itisreliably“located”,andtheANNis sufficientlyaccurateinitsproximity [119].Thus,thevectors

{

^zm,t_nF ^:m=1,2,...,M

}

^{producedatthelast iterationtare}

retainedasifthey werethe“optimalones”

{

^zmF : m=1,2,...,M

}

^,distributedastheideal,zero-varianceISDg^∗(·)=I_F(· )q(·)/P(F).

At the end of the procedure, the total number Nc,aux of T-H code runs carried out to refine the metamodel is Nc,aux=N_TR^t=N_TR⁰+t·N_add.

Someconsiderationsareinorderwithrespecttothealgorithmabove.Asalreadysaid,atstep4.bN_addpointsareselected (amongallthosegeneratedbySS)tobeaddedtothecurrentDOED^t−1_TR .ArelevantfractionKoftheseN_addpointsischosen

“across” thelimit state,inproximity ofthefailure regionF_nF ≡F.Thisispractically implementedasfollows: (i)consider all the SSsamplesbelongingto thefailure region FnF ≡ F={z: Y(z) >

α

Y} andto theprecedingintermediate conditional regionF_nF−1={z:Y(z)>y_nF-1}asa‘pool’ofpossiblecandidatestobeincludedintotheDOE(noticethattheseSSsamples

‘cover’ thelimitstate);(ii)identifythecandidateforwhichtheminimaldistancefromtheelementsofthecurrentDOEis thelargestwithrespecttotheother candidatesofthe‘pool’,andaddittothecurrentDOE(thisallowsfillingthe“empty”

areasoftheinputspace);(iii)removetheselectedvectorfromthe‘pool’ofcandidates;(vi)repeatsteps(ii)–(iii)untilK·N_add newelementsareincludedintotheDOE.

Bymeansofthesameprocedure,theremaining(1–K)·N_addvectorsareequally“distributed” inproximityofthebound- ariesoftheotherSSintermediateconditionalregions:bywayofexample,thesetofcandidatesconsideredforrefiningthe ANNinproximityofregion F_i={z:Y(z) >y_i}isobviouslyrepresentedby allthe SSsamples lyingacross thresholdy_i,i.e., {z:Y(z) > y_i-1 ∩ Y(z) <y_i+1}.Thiswayofproceedingensures an acceptablequality ofthe ANNmetamodelalsoinother

“key areas” ofthe uncertain input spacebesides the failure region F itself: actually, in order to produce a set of points

{

^zmF : m=1,2,...,M

}

^thataredistributedasg^∗(·)=I_F( ·)q(·)/P(F),theSSalgorithmneedstograduallyandaccuratelyvisit theentireuncertain inputspaceandits intermediateconditional regions.Differently fromRef.[100] (whereK wassetto 0.5bytrial-and-errorandtheremaining(1–K)N_addpointswererandomlydistributedacrosstheuncertaininputspace),in thispaperthe fractionK mentionedaboveiscomputedas2•N_add/n_F:inotherwords,thenumberofpointsusedtorefine the metamodelaroundthe limit state{z: Y(z)=

α

Y}is twicethe numberofthose usedto improvetheANNinproximity ofthe other intermediate SSthresholds.Thischoice is motivatedasfollows:(i) higherimportance is giventothe failure regionwithrespecttootherareasoftheuncertaininputspace;(ii)thevalueofKchangesdependingonthenumbern_F of intermediate regions,i.e.,accordingtothemagnitudeofthefailureprobability P(F);(iii)thisvalue isempiricallyfoundto performsatisfactorilyonthecasestudyanalysed.

Onlyforillustrationpurposes,Fig.2pictoriallyshowsthekeyconceptsunderlyingtheprocedureforenrichingtheDOE.

Therectangles representtheelementsofthe currentDOEinregions F_nF≡ FandF_nF−1,whereasthecircles representthe

‘pool’ofpossiblecandidatesgeneratedbySS.Wesupposethatthreeamongthesecandidateshavetobeselectedandadded tothe DOE.At eachstep,we identifythecandidatewhoseminimaldistancefromtheelementsofthecurrentDOEisthe largest withrespect to the other candidates ofthe ‘pool’ (see thearrows inFig. 2). Thiscandidate is addedto the DOE, whichisthusdynamicallyupdated.Theprocedure isrepeated(illustrative stepsa–dinFig.2) untilthedesirednumberof points(i.e.,threeinthecaseofFig.2)isincluded.

Also, noticethatsteps 2and4.fofthealgorithmabove requirethecomputationofan ANN-basedestimate Pˆ(^F)^N_ANN^{T of} P(F).However, approximatingtheoriginal systemmodeloutputby anANNempiricalregressionmodel(trained bymeans ofafiniteset ofdata) bearstwoeffects: (i)thecorresponding estimatesmaybesubjecttoabias (whichmakes themnot accurate); (ii) the associated estimates are subject to additional uncertainty (which makes them less precise). To address issue(i), we employa Bootstrap-basedBias Correction(BBC) procedurerelying on an ensembleofB(e.g., B= 100)ANN regressionmodelsconstructedonB‘fictitious’datasetsrandomlysampledwithreplacementfromtheoriginalone:seeRefs.

[52,53,120]fordetails.Ontheotherhand,wedonottackleissue(ii).Actually,contrarytootherworksofliteratureinvolving thecombinationofmetamodelingandadvancedMCS,e.g., Refs.[103–111],inthispaperwe donotusethemetamodelto

N. Pedroni, E. Zio / Applied Mathematical Modelling 48 (2017) 269–288 275

Fig. 2. Conceptual steps underlying the procedure for enriching the DOE in proximity of the failure region F (step 4.b of the iterative algorithm of ANN refinement). Rectangles: current DOE; circles: ‘pool’ of candidates generated by SS; arrows: candidate added to the DOE. a) identification of the first candidate to be added; b) inclusion of the first candidate into the DOE and selection of the second; c) inclusion of the second candidate into the DOE and selection of the third; and d) inclusion of the third candidate into the DOE.

providethefinalestimatePˆ(^F)^{NT ofthefailure}probability: instead,we employitonlyasa“support tool” forconstructing thequasi-optimalISD(infacts, intheISphaseofthepresentmethodtheoriginalsystemmodelisused,seethefollowing Section3.2). Inother words,theonlyobjectiveoftheANN-based iterativealgorithmdescribedabove isthatofaccurately

“locating” the failure region F of interest without any bias: in this view,we perform only the amount of system model evaluationsthat are strictly necessaryto achieve thisobjective,whereas no furthercode runsare devotedto increase the precision(or,conversely,toreducetheuncertainty)ofthemetamodelestimates.

Inaddition,discussioniswarrantedontheconvergencecriterionadopted(step4.fabove).Actually,(i)theultimategoal ofthisfirststage ofAM-SISistodrawsamplesthatare asatisfactoryrepresentationofthequasi-optimalISDand(ii)the ANNtrainedhereisnot usedinthefollowing failureprobability estimationstep(Section3.2). Thismayleadtoconclude thatthefailure probability estimatedthrough theiterationsofSSandANN(steps3 and4.f) isnota“proper” measure of convergenceforthealgorithmofFig.1.Ontheother hand,itisimportanttonoticethattheaccuracyandunbiasednessof theANNdiscussed above(i.e.,itscapabilityofcorrectly“classifying” and“distinguishing” safeandfailure configurations)is absolutelyfundamentalforobtainingsamplesdistributedasg^∗(·)=I_F(·)q(·)/P(F)bymeansofSS.Inthisrespect,alsonote thatthe correctidentificationoffailure/safe pointscombinedwiththe mechanismofMCMC(which cangenerate random samplesdistributedaccordingto anyPDF)should beenoughto guaranteethatthesystemconfigurations producedatthe endofstep6ofthealgorithmarerepresentativeoftheoptimalISDg^∗(·).

Obviously,theseconsiderations do not implythat theconvergence criterion selected isthe only possible(or the best overall). Forexample, one could alsolook atthe difference (say,through relative entropy) [121] betweentheestablished ISDsatsubsequentiterations.Ontheotherhand,convergenceofanentireprobabilitydistributionoverthewholeuncertain parameterspacemaybemuchmoredemanding(intermsofrequiredcoderuns)thanasimpleclassificationoffailure/safe configurations.

Finally,aconsiderationisinorderwithrespecttotheimpactofthedimensionalityoftheproblem(i.e.,ofthedimension ofx)ontheANNqualityand,thus,onthedensityapproximation.Increasingthedimensionoftheuncertainvariablespace actuallybrings ratherchallengingissues,includingtheso-calledcurseofdimensionality.In practice,weare tryingto train a metamodelon a learningdataset whose size isnot increased sufficiently withthe dimension ofthe space, giving rise totheso-calledempty spacephenomenon[105].Thisissueseems unavoidableandlimitsthe applicationofanyalgorithm belongingto themetamodelfamily toengineering problemswitha smallnumber ofinput parameters (say tens), unless some dimensionality reduction strategy is adopted: e.g., principal component analysis, feature selection orextraction and sensitivityanalysismethods[122–124].

3.1.2. ConstructinganestimatoroftheoptimalISDbymeansofafullynonparametricProbabilityDensityFunction(PDF) TheiterativeANN-basedalgorithmpresentedabove producesapopulationofsamplesapproximatelydistributedasthe ideal,zero-variance ISD: however,these samplescannot be used directlyto define an unbiased IS estimator,because the corresponding sampling densityg^∗(·) isnot knownexplicitly. Moreover, thesampled states

{

^zmF :m=1,2,...,M

}

^{are not}

independent.A first(naive) ideamight beto justusetheempirical distributionof

{

^zmF :m=1,2,...,M

}

^{forIS sampling.}

Butthis distributiondoesnot satisfy therequirementof havinga nonzerodensitywherever I_F(x)q(x) > 0.Tosatisfy this requirement, asecond ideaistofita kerneldensityestimatetothissample, basedonakernelwhosedensityisnonzero wherever I_F(x)q(x) > 0 [93,94]. However, kernel densityestimation by standard methods is difficult in highdimensions.

Finally, inRef. [99] the authors propose toadopt the one step ahead transitiondensity

κ

^{(x|z) ofa Markov chain,when}

the chain isin state z^m_F,andaverage over the Mstates

{

^zmF : m=1,2,...,M

}

^{.In thisway, the optimal ISDg∗(x) can be}

approximatedbygˆ^∗(^x)^(2):

gˆ^∗

(

^x

)

= 1 M

M

m=1

κ (

^x

|

^zmF

)

. (2)

FollowingthesuggestionofRef.[99],inthispaperweusethetransitiondensity

κ

(^x

|

^zmF)^{adoptedbytheGibbsSampler} [101](seetheAppendixforthoroughtechnicaldetails).

The advantageoftheISD(2)is thatit isnotlimitedby parametric models(e.g.,normal,exponential,lognormal, etc.), asithappensinsteadfortheCrossEntropy (CE)[95] orVarianceMinimization(VM)[125]techniques.Ontheother hand, forISD(2)thepositivitycondition(i.e.,thatgˆ^∗(^x)>0wheneverI_F(x)q(x)>0)isnotguaranteedintherare-eventsetting.

Thus,gˆ^∗(^x)^{isheremodifiedtoassurethatitisnonzeroovertheentiresupportofg∗(x).Tothisaim,weconsidera“weighed} average” of ˆg^∗(^x)with the original densityq(x), i.e.:

gˆ^∗_w

(

^x

)

=w·q

(

^x

)

+

(

^1−w

)

^·gˆ∗

(

^x

)

, (3)

wherewisaweightarbitrarilyselectedbetween0and1(inthiswork,w=0.01).

3.2. ImportancesamplingfromtheestimatoroftheoptimalISD

Thewell-known‘compositionmethod’isemployedtodrawN_Tsamplesfromgˆ^∗_w(^x)^{(3).Inextremesynthesis,fork}=1,2,

…,N_T,auniformrandomnumberuisgeneratedin[0,1):ifu<w,thenthevectorx^kissampledfromtheoriginalPDFq(x);

otherwise,fromgˆ^∗(^x)=_M^{1 M}

m=1

κ

(^x

|

^zmF)^{(2)(forthesakeofbrevity,wedonotreportheredetailsaboutthestepsnecessary} todrawsamplesfromthemixturegˆ^∗(^x)^{(2)oftransitiondensities:theinterestedreaderisreferredtotheAppendix).}

SincethefailureprobabilityP(F)=Eq[I_F(x)q(x)](1)canbeobviouslyexpressedalsoasE_gˆ∗ w[^IF_g⁽_ˆ^x_∗^)q(x)

w(^x) ],the AM-SISestimator Pˆ(^F)^NT of the failure probabilityP(F) is then[90–99]:

Pˆ

(

^F

)

^{NT =}_N¹

T NT

k=1

IF

x^k

q

x^k

ˆ g^∗_w

x^k

= 1 NT

NT

k=1

IF

x^k

q

x^k

w·q

x^k

+

(

^1−w

)

^·gˆ∗

x^k

. (4)

The sample standard deviation

σ

ˆ^[Pˆ(^F)^{NT] of (4) can be also} straightforwardly computed since the realizations I_F(^xk)^q(^xk)/gˆ^∗_w(^xk)^,k=1,2,…,N_T,areindependent.Withrespecttothat,itisalsoworthnotingthattheestimatorsPˆ(^F)^NT and

σ

ˆ^[Pˆ(^F)^{NT] are unbiased forthemeanandthe standarddeviation, even if(i)the samples}

{

^zmF : m=1,2,...,M

}

^itera-

tively generatedby SSare byconstruction not independentandare notexactly distributedasg^∗(x) (see Section3.1.1) and (ii)thequalityoftheISDapproximationinthefirststepofthealgorithmislow(seeSection3.1.2).Ontheotherhand,the performanceoftheAM-SISalgorithmobviouslydependsonthequalityofthisapproximation.

Noticeagainthattheoriginalsystemmodel(not theANNmetamodelemployedattheprevious step)isusedinthisIS- basedfailureprobabilityestimationphase.Actually,asexplainedinSection3.1.1,oneoftheeffectsoftheapproximationof theoriginalsystemmodeloutputbyanempiricalregressionmodel(trainedbymeansofafinitesetofdata)isthattheas- sociatedestimatesaresubjecttoadditionaluncertainty(whichmakesthemlessprecise).EmployinginsteadtheANNonlyas a“supporttool” forconstructingthequasi-optimalISD(i.e.,foraccurately“locating” thefailureregionF)allowsperforming onlytheamountofsystemmodelevaluationsthat arestrictly necessarytoachieve thisobjective: thus,noadditionalcode runsaredevotedtoincreasetheprecision(or,conversely,toreducetheuncertainty)oftheANNmetamodelestimates,with obviousbenefitsfromthepointofviewofthereducedcomputationalcost.

4. Casestudy:functionalfailureanalysisofaT-Hpassivesystemofliterature

Asinthepreviouspaperbytheauthors[100],thecasestudyinvolvesaGas-cooledFastReactor(GFR)cooledbynatural convectionaftera LossOfCoolantAccident(LOCA) [26].The reactorisa600-MWGFRcooled byheliumflowingthrough separatechannelsinasiliconcarbidematrixcore[126].

Incase ofaLOCA, long-termheatremovalis ensured bynaturalcirculation ina givennumberN_loops of identicaland parallelheatremovalloops.However,inordertoachieveasufficientheatremovalratebynaturalcirculation,itisnecessary

N. Pedroni, E. Zio / Applied Mathematical Modelling 48 (2017) 269–288 277

Fig. 3. One loop of passive decay heat removal system of the 600-MW GFR [26] .

tomaintainan elevatedpressureevenaftertheLOCA.Thisisaccomplishedby aguardcontainment,whichsurroundsthe reactorvesselandpowerconversionunit,andholdsthepressureatalevelthatisreachedafterthedepressurizationofthe system[26].

TheconfigurationofthepassivesystemfortheremovalofdecayheatisshownschematicallyinFig.3,whereonlyone loopoutofN_loopsisreportedforclaritysake:theflowpathofthecoolingheliumgasissuggestedbytheblackarrows.As showninFig.3,theloophasbeendividedintoN_sections =18sections;technicaldetailsaboutthegeometricalandstructural propertiesofthesesectionsarenotreportedhereforbrevity:theinterestedreadermayreferto[26].

It isworth notingthat as inRef. [26] the subjectof thepresent analysisis the quasi-steady-state naturalcirculation coolingthat takesplaceafter theLOCAtransienthasoccurred. Thus,the analyses hereafterreportedrefer tothissteady- state periodandare conditional tothe successfulinceptionof naturalcirculation.As aconsequence, theanalyses donot considerthe probability ofnot startingnaturalconvectionnor theprobability ofnot buildingup andmaintaininga high pressurelevelintheguardcontainment.

4.1. ThedeterministicT-Hmodel

Tosimulatethesteady-statebehaviorofthesystem,aone-dimensionalthermal-hydraulicMATLABcodehasbeenimple- mented.Thecodetreatsall theN_loopsmultipleloopsasidenticalandeachloopisdividedinN_sections =18sections(Fig.3).

Further,thesectionscorresponding tothe heater(i.e.,the reactorcore,item 4inFig.3) andthecooler (i.e.,theheatex- changer, item 12 in Fig. 3) are divided ina proper numberN_nodes of axial nodes to compute the temperatureand flow gradientswithsufficientdetail(40nodesarechosen forthepresentanalysis).Boththeaverageandhotchannelsaremod- eledinthecoresothattheincrease intemperatureinthe hotchannelduetotheradial peakingfactorcanbe calculated [26].

Fromastrictly mathematicalpointofview,obtaining asteady-statesolutionamountstodropping thetimedependent termsintheenergyandmomentumconservationequations[127];inpractice,thecodebalancesthepressurelossesaround theloopssothatfrictionandformlossesarecompensatedbythebuoyancyterm, whileatthesametimemaintainingthe heatbalanceintheheater(i.e.,thereactorcore,item4inFig.3)andcooler(i.e.,theheatexchanger,item12inFig.3).

Table 1

Equations governing the heat transfer process in each node l = 1, 2, …, N nodesof both the heater and the cooler of the 600-MW GFR passive decay heat removal system of Fig. 3 , together with a description of the physical parameters and their units of measure.

Heat transfer equations

Q ˙ l= m ˙ c _p,l(T _out,l−T _in,l), l = 1 , 2 , . . . , N nodes (5)

˙

m c _p,l(T _out,l−T _in,l)= S lh l(T wall,l−T _bulk,l), l = 1 , 2 , . . . , N nodes (6)

Parameter Description Unit of measure

Q ˙ l Heat flux in the l th node kW

˙

m l Mass flow rate in the l th node kg/s

c p,l Specific heat at constant pressure in the l th node kJ/kg K

T out,l Temperature measured at the outlet of the l th node K

T in,l Temperature measured at the inlet of the l th node K

T wall,l Temperature measured at the wall channel of the l th node K

T bulk,l Temperature measured at the bulk of the l th node K

S l Heat-exchanging surface in the l th node m ² h l Heat transfer coefficient in the l th node kW/m ²K

Table 2

Equation stating the balance between buoyancy and pressure losses along the closed loop of the 600-MW GFR passive decay heat removal system of Fig. 3 , together with a description of the physical parameters and their units of measure.

Mass flow rate equation

Nsections

s=1 [ ρsg H s+ f sLs Ds

˙ m² 2ρsA²s + K s m˙²

2ρsA²s] = 0 (7)

Parameter Description Unit of measure

ρs Coolant density in the s th section of the loop kg/m ³ H s Height of the s th section of the loop m f s Friction factor in the s th section of the loop / L s Length of the s th section of the loop m D s Hydraulic diameter of the s th section of the loop m

˙

m Mass flow rate in the s th section of the loop kg/s A s Flow area of the s th section of the loop m ² K s Form loss coefficient of the s th section of the loop –

Eqs.(5)and(6)inTable1governtheheattransferprocessineachnode l=1,2,…,N_nodesofboththeheaterandthe cooler.

Eq.(5)inTable1statestheequalitybetweentheenthalpy increasebetweentheflow attheinletandtheflow atthe outletinanynode,whereas Eq.(6)inTable 1regulatestheheat exchangebetweenthechannel wallandthebulkofthe coolant.

Themassflow rateisdetermined byabalance betweenbuoyancyandpressurelossesalongtheclosedloop according toEq.(7)inTable2.

Eq.(7)inTable2statesthatthesumofbuoyancy(firstterm),frictionlosses(secondterm)andformlosses(thirdterm) shouldbeequaltozeroalongtheclosedloop[26].

Noticethattheheattransfercoefficientsh_l,l=1,2,…,N_nodes,in(6)inTable1arefunctionsofthefluidcharacteristics andthegeometry,andarecalculatedthroughpropercorrelationscoveringforced-,mixed-andfree-convectionregimes,in bothturbulentandlaminarflow;further,differentNusseltnumbercorrelationsare usedinthedifferentregimestoobtain valuesfortheheattransfercoefficients.Alsothefrictionfactorsfs,s=1,2,…,N_sections,in(7)inTable2arefunctionsofthe fluid characteristicsandgeometryandarecalculated usingappropriatecorrelations[26].Aniterativealgorithmisusedto findasolutionthatsatisfiessimultaneouslytheheatbalanceandpressurelossequations.

4.2. UncertaintiesinthedeterministicT-Hmodel

There is incomplete knowledge onthe properties ofthe system andthe conditions inwhich the passive phenomena develop.Thiskindofuncertainty,oftentermedepistemic,affectsthemodelrepresentationofthepassivesystembehavior [9,29,128].

Theninemodelparametersx={x_j: j=1, 2, ..., 9}affectedbyepistemicuncertaintyaresummarizedinTable3[26]:the meanandstandarddeviationsofthecorrespondingGaussianPDFsarealsoreported.

Thesequentialstepsundertakentopropagatetheepistemic(i.e.,modelandparameter)uncertaintiesthroughthedeter- ministicT-HequationsdescribedinthepreviousSection4.1are:

i Drawsamplesoftheuncertaininput parametersx₁ (i.e.,power),x₂ (i.e.,pressure)andx₃ (i.e.,coolerwalltemperature) fromtherespectiveepistemicprobabilitydistributions(Table3).