ML120540604

Project, Units 1 and 2 - Meeting Materials for 3/1/12 Conference Call (TAC Nos. ME7735 and ME7736)

ML120540604
Person / Time
Site:	South Texas
Issue date:	02/22/2012
From:	Galenko A, Popova E University of Texas at Austin
To:	Plant Licensing Branch IV
Singal, B K, NRR/DORL, 301-415-301
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ML120540492	List: ML120540509 ML120540557 ML120540570 ML120540604 ML120540610 ML120540667 ML120540694 ML120540727 ML120540758 ML120540784 ML120540816
References
TAC ME7735, GSI-191, TAC ME7736
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UncertaintymodelingofLOCAfrequenciesandbreaksizedistributionsfortheSTPGSI-191resolutionAlexanderGalenko,ElmiraPopovaagalenko@gmail.com,elmira@mail.utexas.eduTheUniversityofTexasatAustin1IntroductionIntheinitialquan(Crenshaw,2012),Flemingetal.(2011)per-formedasubstantialstudydesignedtobuildupontheestablishedEPRIrisk-informedinserviceinspectionprogram(EPRI,1999).Thismethod-ologywasusedasprimarybasistodevelopthesizeandlocation-sprupturefrequencies(eventsperyear)fortheinitialquanAlthoughtheoverallmethodologyappearstobesoundbasedonpeerreview(Mosleh,2011)andreasonablenessofthevaluesobtained,NRCfeedbackinthePilotProjectreviewshasresultedinfurtherreviewoftheapproach.InthispaperweproposeanewapproachtoassignlocationspLOCAfrequenciesthatarederivedfromtheoverallfrequencies,asinTregoningetal.(2008).WerefertothissourcereferenceasNUREG-1829.TheNUREG-1829annualfrequenciesarenotplantsporplant-locationspc.YettheyareusedthroughoutthenuclearindustryasanimportantinputtoPRAanalyses,andtherefore,theyneedtobepreserved.ConservationoftheNUREG-1829breakfrequenciesisourguidingprinciple.Inthisdocumentwewillworkwiththesixbreaksizecategories,inTable1,NUREG-1829Volume1,pagexxiasivebreaksize"forthe1 PWRplants.Table1showsthemappingbetweenNUREG-1829notationandours.Inaddition,theuseofthetermdistributionwillbeequivalenttoadistributionfunction(eithercumulativedistributionorprobabilitydensityfunction)ofarandomvariableusedtomodelaspuncertainty.Table1:LOCAcategoriesnotationmapebreaksize(inch)forPWRNotation12cat1158cat23cat37cat414cat531cat6WeshouldpointoutthattheSouthTexasProjectPRAanalysisusesonly3LOCAcategories,small,medium,andlarge.Ourproposedmethodologycanbeappliedtoanyenumberofcategories.Inthisdocumentwewillusethetermlocationtorepresentweldloca-tions.Overalltherearetwowaystocomeupwithlocation-spLOCAfrequencies:bottom-upandtop-down.Thebottom-upapproachrequireslocationspfailuredatatoestimatethecorrespondingprobabilityofaweldfailure.IfweassumethereareMjtlocationsintheplantwherebreaksofsizecatjcanoccur,location1;:::;locationMj,thenusingthelawoftotalprobabilitywecanwrite:P[catj]=MjXi=1P[catjjlocationi]P[locationi];j=1;2;:::;6whereP[catj]isthetotalprobabilityofacatjLOCA,P[catjjlocationi]isthe2 conditionalprobabilityofacatjLOCAgivenlocationihasbeenchosen,andP[locationi]istheprobabilitythatlocationiischosen.Inourapplicationwewillassumethatalllocationsareequallylikelytobeselected,i.e.P[locationi]=1Mj;i=1;:::;MjInthebottom-upapproachwemustdetermineP[catjjlocationi](us-ingestimationorexpertelicitation).Ifthathappens,thenwecanmultiplytheseprobabilitiesby1=Mj,sumthemup,andwewillobtainthetotalprob-abilityofacatjLOCA.Ifthebottom-upapproachisfollowedweclaimthattheresultingtotalcatjLOCAprobabilitywillNOTequalthenumberpro-videdinNUREG-1829(oratleastitisveryunlikelytogetthatnumber).TheapproachtakenbyFlemingetal.(2011)isaninherentlybottom-upapproach.TopreservetheNUREG-1829frequenciesFlemingetal.(2011)developedanapproximationscheme.Intheirreview,theNRCtechnicalteamraisedseveralquestionsaboutthebottom-upapproach,whichhasleadustoproposearentmethodology.Themethodthatweproposeisrootedinthetop-downapproach:startwiththeNUREG-1829frequenciesanddevelopanintuitivewaytodistributethecumulativefrequenciesamongerentweldlocations.WewilluseagainthecatjLOCAasanillustrativeexample.AssumethatwehavecomputedthetotalprobabilityofacatjLOCAfromtheNUREG-1829frequencytableusingtheformulaP[catj]=Frequency[catj]P6i=1Frequency[cati]AgainweassumethereareMjtlocationsintheplantwherebreaksofsizecatjcanoccur,(location1;:::;locationMj),andtheyareequallylikely,3 i.e.P[locationi]=1Mj;i=1;:::;Mj:NotethatthedenominatoristhetotalfrequencyofallLOCAsizes.WewillassumethatallP[catjjlocationi]=P[catj];i=1;:::;Mj.ThenapplyingthelawoftotalprobabilityweseethattheresultingprobabilityofacatjLOCAequalsexactlytheNUREG-1829probabilityandP[catjatlocationi]=P[catj]=Mj.TheabovemethodologydistributesequallytheLOCAfrequenciesasde-inNUREG-1829Table1betweenalllocationsthatcanexperiencebreaksfromoneormoreofthesixsizecategories.Thesixbreaksizecate-gories(columnsinNUREG-1829Table1)arerangesboundedbysixdiscretepoints.Foraparticularweldweneedtobeabletosamplefromthecon-tinuousrangeofbreaksizevalues.Inaddition,wewouldliketobeabletosamplefromthedistributionofthefrequencies.TherowsinTable1fromNUREG-1829representsthedistributionofthefrequenciesbyreportingthemean,median,5theand95thepercenttiles.Wewillusethisinformationtosixcontinuousdistributionsforeachbreaksizecategory.2Proposedmethodology2.1FittingdistributionstotheLOCAfrequenciesWewilldescribethedistributiontothefrequenciesforeachbreaksizecategory.Intheory,therearenumberofdistributionsthatonecantotheLOCAfrequenciesrepresentedinNUREG-1829:twosplitLognormaldistributionsareusedinNUREG-1829andGammadistributionsareusedinNUREG/CR6928.4 WechosetotheboundedJohnsondistribution,Johnson(1949)forthefollowingreasons:Ithasfourparametersthatwillallowustomatchcloselythefourdistri-butionalcharacteristicsprovidedbyNUREG-1829.InordertogettheparametersoftheJohnsondistributionwesolveanoptimizationprob-lemwithconstraintsbythefourdistributionalcharacteristics:the5thpercentile,themean,themedian,andthe95thpercentile.Itcanhavevarietyofshapes.Inparticular,skewed,symmetric,bi-modal,orunimodalshapescanbeobtained.Thecumulativedistributionfunction(CDF)oftheboundedJohnsonis:F[x]=f+f[(x)]g;wherex]-CDFofastandardNormal(0,1)randomvariable,andareshapeparameters,isalocationparameter,isascaleparameter,andf(z)=log[z=(1z)].TheparametersoftheJohnsondistributionforeachofthesixcategoriesaregiveninTable2.ThecomparisonbetweentheNUREG-1829distributionalcharacteristicsoftheLOCAfrequenciesandtheonesarepresentedinTable3.Thelargesterrorinourestimationis3.78%whichweconsidertobesmallenough.Figures1and2showtheCDFsandprobabilitydensityfunctions(PDFs)oftheJohnsondistributionforeachcategory.5 Table2:FittedJohnsonParametersJohnsonParametersCat10.72882460.38933260.000634490.02449228Cat26.95E-012.40E-017.41E-062.44E-03Cat37.24E-012.44E-012.06E-076.24E-05Cat47.14E-012.39E-011.36E-086.19E-06Cat54.73E-012.69E-011.87E-105.93E-07Cat64.75E-012.73E-011.77E-148.52E-08Table3:NUREG-1829andJohnsonmean,5thand95thpercentilevaluesNUREG-1829FittedJohnsonError5%Mean95%5%Mean95%5%Mean95%Cat16.90E-047.30E-032.30E-026.89E-047.30E-032.30E-020.08%0.01%0.00%Cat27.60E-066.40E-042.40E-037.56E-066.42E-042.40E-030.59%0.38%0.04%Cat32.10E-071.60E-056.10E-052.09E-071.60E-056.12E-050.40%0.23%0.38%Cat41.40E-081.60E-066.10E-061.40E-081.59E-066.08E-060.26%0.65%0.39%Cat54.10E-102.00E-075.80E-074.14E-101.98E-075.86E-070.94%0.94%0.98%Cat63.50E-112.90E-088.10E-083.59E-112.84E-088.41E-082.60%2.03%3.78%6 (a)Cat1(b)Cat2(c)Cat3(d)Cat4(e)Cat5(f)Cat6Figure1:JohnsonCDFsforeachcategory7 (a)Cat1(b)Cat2(c)Cat3(d)Cat4(e)Cat5(f)Cat6Figure2:JohnsonPDFsforeachcategory8 Oncethebestisfound,wesampletheLOCAfrequenciesforeachcategorytogetFrequency[catj]-realizationofLOCAfrequencyforcategoryj.2.2DistributionofLOCAfrequenciestotweldlocationsWeconvertLOCAfrequenciestoprobabilitiesusingP[catj]=Frequency[catj]=Xl2JFrequency[catl];whereP[catj]-probabilityofobservingabreakthatfallsintocategoryjgivenabreakwasobservedFrequency[catj]-frequencyoffailureoftypejwherej2JJ=fcat1;cat2;cat3;:::;catBg-setofpossiblebreaktypes(categories)GivenP[catj]thenextstepistodistributethatprobabilityamongallweldsthatcanexperienceabreakfromthesamecategory.WecomputeP[catjatlocationi]-theprobabilitythatweldiwillexperiencebreakoftypejusingP[catjatlocationi]=P[catj]=Mj,whereMj=jIjj-numberofweldsthatcanexperiencecategoryjbreaks,i2Ij;Ij-setofweldsthatcanexperiencebreakcategoryj.Hereweassumethateveryweldthatcanexperienceabreakofcategoryjhasequalprobabilityofactuallyexperiencingit.9 2.3SamplingofthebreaksizeThestepistosampletheactualbreaksizeconditionedonthebreakcategory.Hereweassumethatthebreaksizehasauniformdistributionwithinagivencategory,formallybreakSizeijU[minBreakij;maxBreakij];j2J;i2Ij;whereU[minBreakij;maxBreakij]istheUniformdistributionboundedbyminBreakijandmaxBreakijminBreakij=catminBreakjmaxBreakij=min[catmaxBreakj;weldsizei]catminBreakj-minimumbreaksizethatwouldputitintocategoryjcatmaxBreakj-maximumbreaksizethatwouldputitintocategoryjweldsizei-actualweldsize(itcannotexperiencebreaksizelargerthanit'sdiameter).2.4MethodologysummaryThismethodologywillrequiretwosamplingloopsinoursimulatorCASAGrande,Letellier(2011).WeneedonesamplingloopforthebreaksizewithineachcategoryandasecondloopthatsamplesLOCAfrequenciesfromtheirdistributions.Belowisastep-by-stepdescriptionofthatprocedure:1.SetN-numberofLOCAfrequencysamplesandS-numberofbreaksizesamplestogenerate10 2.SampleLOCAfrequenciesFrequency[catj]fromtheJohnsondis-tributions,seeSection2.13.DistributefrequencyacrossplantspweldsasdescribedinSection2.24.Sampleactualbreaksizeforeachpossibleweld/breakcategorycom-binationasdescribedinSection2.35.Estimateperformancemeasures,storethem6.IfweranSbreaksizessamplesgotothenextstep,otherwisegotostep47.Computetheperformancemeasuressummary,storethem8.IfweranNLOCAfrequenciessamplesgotothenextstep,otherwise,gotostep29.Makeaggregatedperformancemeasuressummary3IllustrativeexampleWeillustrateourapproachdescribedinthefourstepsfromSection2.4usingthefollowingexample,seeFigure3.Assumewehaveatotalofsixweldsandthesearetheonlylocationswhereabreakcanoccur.Threeofthem(welds1,2and3)aresmallandhaveasizeof2.5inchesandhencecanexperienceonlysmallbreaks(category1andcategory2).Twoofthosesix(welds4and5)areofmediumsizeandhaveasizeof4inchesandthuscanhavesmallandmediumbreaks(category1,category2andcategory3only,11 theycan'texperiencecategory4break).Thelastweld(weld6)islargeandhasasizeof35inchesandcanhaveallthreetypesofbreaks-small,mediumandlarge(category1,category2,category3,category4,category5andcategory6).Figure3:IllustrativeExample1.AssumeS=1;N=1.12 2.ThesampledLOCAfrequencies(usingthedJohnsondistributions)aregiveninTable4Table4:SampledLOCAfrequenciesandcorrespondingprobabilitiesFailureTypeCategoryBreakSizeFrequencyProbabilitysmall10.5-1.6253.9E-039.64E-01small21.625-31.4E-043.46E-02medium33-73.4E-068.41E-04medium47-143.1E-077.67E-05large514-311.2E-082.97E-06large631-411.2E-092.97E-073.WehaveJ=fcat1;cat2;cat3;cat4;cat5;cat6g,Icat1=fweld1;weld2;weld3;weld4;weld5;weld6g,Icat2=fweld1;weld2;weld3;weld4;weld5;weld6g,Icat3=fweld4;weld5;weld6g,Icat4=fweld6g,Icat5=fweld6g,Icat6=fweld6g.Mcat1=6,Mcat2=6,Mcat3=3,Mcat4=1,Mcat5=1,Mcat6=1.BreakSizeweld1cat1U[0:5;1:625]BreakSizeweld2cat1U[0:5;1:625]BreakSizeweld3cat1U[0:5;1:625]BreakSizeweld4cat1U[0:5;1:625]13 BreakSizeweld5cat1U[0:5;1:625]BreakSizeweld6cat1U[0:5;1:625]BreakSizeweld1cat2U[1:625;2:5]BreakSizeweld2cat2U[1:625;2:5]BreakSizeweld3cat2U[1:625;2:5]BreakSizeweld4cat2U[1:625;3]BreakSizeweld5cat2U[1:625;3]BreakSizeweld6cat2U[1:625;3]BreakSizeweld4cat3U[3;4]BreakSizeweld5cat3U[3;4]BreakSizeweld6cat3U[3;7]BreakSizeweld6cat4U[7;14]BreakSizeweld6cat5U[14;31]BreakSizeweld6cat6U[31;35]UsingtheformulaforP[catjatlocationi]wecomputeprobabilitiesforeachweld.TheresultsaregiveninTable5.Weseethatthesumofthedistributedprobabilitiesandthetargetedprobabilitiesarethesame.4.Wesimulatebreaksizesforeachweldwithineachcategoryusinguni-14 formdistributionwiththespaboveparameters.ThesampleisshowninTable6.Weitisworthmentioningthatourassumptionsleadtoapiece-wiselinearCDFfunctionofthebreaksizedistributionforagivenweld.Forexample,considerweld6.TheCDFoftheBreaksizeforthatweldwillhave6breakpointswiththeslopesdeterminedbytheP[catjatlocationweld6]valuesandbreakpointsatcatmaxBreakjvalues(1.625,3,7,14,31),seeFigure4.Table5:DistributedLOCAprobabilitiesamongallweldsWeld123456ActualTargetCat11.61E-011.61E-011.61E-011.61E-011.61E-011.61E-019.64E-019.64E-01Cat25.77E-035.77E-035.77E-035.77E-035.77E-035.77E-033.46E-023.46E-02Cat32.80E-042.80E-042.80E-048.41E-048.41E-04Cat47.67E-057.67E-057.67E-05Cat52.97E-062.97E-062.97E-06Cat62.97E-072.97E-072.97E-07Table6:SampledbreaksizesforallweldswithineachbreakcategoryWeld123456Cat11.10.60.871.340.791.23Cat22.41.92.12.91.752.36Cat34.566.545.97Cat49.67Cat525.68Cat632.6715 Figure4:CDFfunctionofbreaksizedistributionforweld6ConclusionInthisdocumentwearepresentingsolutionstothreeproblems:1.HowtodistributetheNUREG1829LOCAfrequenciestotlocations(welds)inanuclearplant.Wemakethesimpleassumptionthatsmallbreaksareequallylikelytooccurinsmall,medium,orlargewelds;mediumbreaksareequallylikelytooccurinmediumandlargewelds;largebreakscanoccuronlyinlargewelds.ThisallowsustopreservetheNUREG1829LOCAfrequencies.2.Thesixbreaksizecategories(columnsinNUREG-1829Table1)arerangesboundedbysixdiscretepoints.Foraparticularweldweneedtobeabletosamplefromthecontinuousrangeofbreaksizevalues.We16 proposetousethelinearinterpolationwhichisequivalenttoassigningequallylikelyprobabilitieswithineachbreaksizecategory.3.HowtomodelthedistributionoftheLOCAfrequencies-weproposeandtheJohnsondistributions.Webelievethattheproblemisthemostimportantthatweneedtoagreeonitssolution.Theothertwocanbemodeledwithtdistribu-tions.WehavealreadyimplementedtheGammadistributionsfromtheNUREG/CR6928andworkingonasetofBetadistributions.Thiswillgiveaportfolioofoptionstoapply.Inthisdocumentwedonotdiscussthetsamplingtechniquesneeded.PopovaandGalenko(2011)describeallthesamplingmethodologiesthatweimplement.17 ReferencesCrenshaw,J.W.(2012,January).SouthTexasProjectUnits1and2DocketNos.STN50-499,SummaryoftheSouthTexasProjectRisk-InformedApproachtoResolveGenericSafetyIssue(GSI-191).LetterfromJohnW.CrenshawtoUSNRC.EPRI(1999).RevisedRisk-InformedIn-ServiceInspectionProcedure.TR112657RevisionB-A,ElectricPowerResearchInstitute,PaloAlto,CA.Fleming,K.N.,B.O.Lydell,andD.Chrun(2011,July).DevelopmentofLOCAInitiatingEventFrequenciesforSouthTexasProjectGSI-191.Technicalreport,KnFConsultingServices,LLC,Spokance,WA.Johnson,N.(1949).Systemsoffrequencycurvesgeneratedbymethodsoftranslations.Biometrika36,149{176.Letellier,B.(2011).Risk-InformedResolutionofGSI-191atSouthTexasProject.TechnicalReportRevision0,SouthTexasProject,Wadsworth,TX.Mosleh,A.(2011,October).TechnicalReviewofSTPLOCAFrequencyEs-timationMethodology.LetterReportRevision0,UniversityofMaryland,CollegePark,MA.Popova,E.andA.Galenko(2011,Deecember).UncertaintyQuan(UQ)Methods,Strategies,andIllustrativeExamplesUsedforResolvingtheGSI-191ProblematSouthTexasProject.TechnicalReportRevision0,TheUniversityofTexasatAustin,Austin,TX.18 Tregoning,R.,P.Scott,andA.Csontos(2008,April).EstimatingLoss-of-CoolantAccident(LOCA)FrequenciesThroughtheElicitationProcess:MainReport(NUREG-1829).NUREG1829,NRC,Washington,DC.19