The Idaho National Laboratory (INL) has been engaged in a significant multiyear effort to modernize the computational reactor physics tools and validation procedures used to support operations of the Advanced Test Reactor (ATR) and its companion critical facility.
Trang 1Nuclear Engineering and Design 295 (2015) 615–624
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j o ur na l h o me pa g e :w w w e l s e v i e r c o m / l o c a t e / n u c e n g d e s
Idaho National Laboratory, 1955 N., Fremont Avenue, PO Box 1625, Idaho Falls, ID 83402, USA
a r t i c l e i n f o
Article history:
Received 1 July 2015
Accepted 30 July 2015
Available online 30 October 2015
a b s t r a c t
TheIdahoNationalLaboratory(INL)hasbeenengagedinasignificantmultiyearefforttomodernize thecomputationalreactorphysicstoolsandvalidationproceduresusedtosupportoperationsofthe AdvancedTestReactor(ATR)anditscompanioncriticalfacility(ATRC).Severalnewprotocolsfor vali-dationofcomputedneutronfluxdistributionsandspectraaswellasforvalidationofcomputedfission powerdistributions,basedonnewexperimentsandwell-recognizedleast-squaresstatisticalanalysis techniques,havebeenunderdevelopment.Inthecaseofpowerdistributions,estimatesoftheapriori ATR-specificfuelelement-to-elementfissionpowercorrelationandcovariancematricesarerequired forvalidationanalysis.Apracticalmethodforgeneratingthesematricesusingtheelement-to-element fissionmatrixispresented,alongwithahigh-orderschemeforestimatingtheunderlyingfissionmatrix itself.TheproposedmethodologyisillustratedusingtheMCNP5neutrontransportcodefortherequired neutronicscalculations.Thegeneralapproachisreadilyadaptableforimplementationusingany multi-dimensionalstochasticordeterministictransportcodethatofferstherequiredlevelofspatial,angular, andenergyresolutioninthecomputedsolutionfortheneutronfluxandfissionsource
©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND
license(http://creativecommons.org/licenses/by-nc-nd/4.0/)
1 Introduction
TheIdahoNationalLaboratory(INL)hasinitiatedafocusedeffort
toupgradelegacycomputationalreactorphysicssoftwaretoolsand
protocolsusedforsupportofAdvancedTestReactor(ATR)corefuel
management,experimentmanagement,andsafetyanalysis.This
isbeingaccomplishedthroughtheintroductionofmodern
high-fidelitycomputationalsoftware andprotocols, withappropriate
verificationandvalidation(V&V)accordingtoapplicablenational
standards.Asuiteofwell-recognizedstochasticanddeterministic
transporttheorybasedreactorphysicscodesandtheirsupporting
nucleardatalibraries(HELIOS(StudsvikScandpower,2008),NEWT
(DeHart,2006),ATTILA(McGheeetal.,2006),KENO6(Hollenbach
etal.,1996)andMCNP5(Goorleyetal.,2004))isinplaceatthe
INLforthispurpose,andcorrespondingbaselinemodelsoftheATR
anditscompanioncriticalfacility(ATRC)areoperational
Further-more,acapabilityforrigoroussensitivityanalysisanduncertainty
quantification based on the TSUNAMI (Broadhead et al., 2004)
systemhasbeenimplementedandinitialcomputationalresults
havebeenobtained.Finally,wearealsoincorporatingtheMC21
∗ Corresponding author Tel.: +1 208 526 4257.
E-mail address: joseph.nielsen@inl.gov (J.W Nielsen).
(Suttonetal.,2007)andSERPENT(Leppänen,2012)stochastic sim-ulationanddepletioncodesintothenewsuiteasadditionaltools forV&Vintheneartermandpossiblyasadvancedplatformsfor full3-dimensionalMonteCarlobasedfuelcycleanalysisandfuel managementinthelongerterm
Ontheexperimentalsideoftheeffort,severalnew benchmark-qualitycodevalidationmeasurementsbasedonneutronactivation spectrometryhavebeenconductedattheATRC.Resultsforthefirst three experiments,focusedondetailed neutronspectrum mea-surements within the Northwest Large In-Pile Tube (NW LIPT) wererecentlyreported(Niggetal.,2012a)asweresomeselected resultsforthefourthexperiment,featuringneutronfluxspectra withinthecorefuelelementssurroundingtheNWLIPTandthe diametricallyopposite SoutheastIPT(Niggetal., 2012b).In the currentpaperwefocusoncomputationandvalidationofthefuel element-to-elementpowerdistributionintheATRC(andby exten-siontheATR)usingdatafromanadditional,recentlycompleted, ATRCexperiment Inparticularwepresentamethoddeveloped for estimating thecovariance matrix for thefission power dis-tribution using the corresponding fission matrix computed for theexperimentalconfigurationofinterest.Thiscovariancematrix
is a key inputparameter that is required for theleast-squares adjustmentvalidation methodologyemployedforassessmentof the bias and uncertainty of the various modeling codes and techniques
http://dx.doi.org/10.1016/j.nucengdes.2015.07.049
0029-5493/© 2015 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.
Trang 2616 J.W Nielsen et al / Nuclear Engineering and Design 295 (2015) 615–624
Fig 1.Core and reflector geometry of the Advanced Test Reactor References to core lobes and in-pile tubes are with respect to reactor north, at the top of the figure.
2 Facility description
TheATR(Fig.1)isalight-waterandberylliummoderated,
beryl-liumreflected,light-watercooledsystemwith40fully-enriched
(93wt%235U/UTotal)plate-typefuelelements,eachwith19curved
fuelplates separated bywater channels Thefuel elementsare
arrangedinaserpentinepatternasshown,creatingfiveseparate
8-element“lobes”.Grossreactivityandpowerdistributioncontrol
duringoperationareachievedthrough theuseofrotating
con-troldrums withhafnium neutronabsorber plates on one side
TheATRcanoperateatpowers ashighas250MWwith
corre-spondingthermalneutronfluxesinthefluxtrapsthatapproach
5.0×1014N/cm2s.Typicaloperatingcyclelengthsareintherange
of45–60days
TheATRCisanearly-identicalopen-poolnuclearmockupofthe
ATRthattypicallyoperatesatpowersintherangeofseveral
hun-dredwatts.Itismostoftenusedwithprototypeexperimentsto
characterizetheexpectedchangesincorereactivityandpower
dis-tributionforthesameexperimentsintheATRitself.Usefulphysics
datacanalsobeobtainedforevaluatingtheworthandcalibrationof
controlelementsaswellasthermalandfastneutrondistributions
3 Computational methods and models
Computational reactorphysics modeling is used extensively
to support ATR experiment design, operations and fuel cycle
management,coreandexperimentsafetyanalysis,andmanyother applications.ExperimentdesignandanalysisfortheATRhasbeen supportedforanumberofyearsbyverydetailedandsophisticated three-dimensionalMonteCarloanalysis,typicallyusingtheMCNP5 code,coupledtoextensivefuelisotopebuildupanddepletion anal-ysis where appropriate On the other hand, the computational reactorphysicssoftwaretoolsandprotocolscurrentlyusedforATR corefuelcycleanalysisandoperationalsupportarelargelybasedon four-groupdiffusiontheoryinCartesiangeometry(Pfeifer,1971) withheavyrelianceon“tuned”nuclearparameterinputdata.The latterapproachisnolongerconsistentwiththestateofmodern nuclearengineeringpractice,havingbeensupersededinthe gen-eralreactorphysicscommunitybyhigh-fidelitymultidimensional transport-theory-basedmethods.Furthermore,someaspectsofthe legacyATRcoreanalysisprocessarehighlyempiricalin nature, withmany“correctionfactors”andapproximationsthatrequire veryspecializedexperiencetoapply.Butthestaffknowledgefrom the1960sand1970sthatisessentialforthesuccessful applica-tionofthesevariousapproximationsandoutdatedcomputational processesisrapidlybeingdepletedduetopersonnelturnoverand retirements
Fig.2showsthesuiteofnewtoolsmentionedearlier,howthey generallyrelatetooneanother,andhowtheywillbeappliedto ATR.Thisillustrationisnotacomputationalflowchartor proce-dureperse.Specificcomputationalprotocolsusingthetoolsshown
inFig.2forroutineATRsupportapplicationswillbepromulgated
Trang 3J.W Nielsen et al / Nuclear Engineering and Design 295 (2015) 615–624 617
Fig 2.Advanced computational tool suite for the ATR and ATRC, with supporting verification, validation and administrative infrastructure.
Fig 3.ATR Fuel element geometry, showing standard fission wire positions used for intra-element power distribution measurements.
inapprovedproceduresandotheroperationaldocumentation.The
mostrecentreleaseoftheEvaluatedNuclearDataFiles(ENDF/B
Version7) is generally used to provide thebasic cross section
dataandothernuclearparametersrequiredforallofthe
model-ingcodes.TheENDFphysicalnucleardatafilesareprocessedinto
computationally-usefulformatsusingtheNJOYorAMPX(Radiation
SafetyInformationComputationalCenter,2010)codesas
applica-bletoaparticularmodule,asshownatthetopofFig.2
4 Validation measurements
Inthenewvalidationexperimentofinterest here,activation
measurementsthat canberelated tothetotalfission powerof
eachofthe40ATRCfuelelementsweremadewithfissionwires
composedof10%byweight235Uinaluminum Thewireswere
1mmindiameterandapproximately0.635cm (0.25 inlength andwereplacedinvariouslocationswithinthecoolingchannelsof eachfuelelementasshowninFig.3,atthecoreaxialmidplane.The totalmeasuredfissionpowersforthefuelelementsareestimated usingappropriately-weightedsumsofthemeasuredfissionrates
intheU/Alwireslocatedineachelement(DurneyandKaufman,
1967)
Fig.4 shows the computeda priori(MCNP5)fission powers forthe40ATRCfuelelements,alongwiththemeasuredelement powers basedonthefissionwire measurements.The top num-ber(black)inthecenterofeachelementistheapriorielement power(W)calculatedbyMCNP5.Thebottomnumber(red)isthe measurement.Totalmeasuredpowerwas875.5W.Uncertainties
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Fig 4.Calculated (black) and measured (red) fuel element powers (W) for ATRC Depressurized Run Support Test 12-5 The fuel element numbers are in bold type.
associatedwiththemeasuredelementpowersareapproximately
5%(1).Thepowersforthefive8-elementATRcore“lobes”arealso
keyoperatingparametersandareformedbysummingthepowers
ofElements2–9fortheNortheastLobe,Elements12–19forthe
SoutheastLobe,Elements22–29fortheSouthwestLobe,Elements
32–39fortheNorthwestLobeandElements1,10,11,20,21,30,
31,and40fortheCenterLobe.Thesignificanceofthelobepowers
willbediscussedinmoredetaillater
5 Power distribution adjustment protocol
Analysisof the computed and measuredpower distribution
forcodevalidationpurposesisaccomplishedbyanadaptationof
standardleast-squaresadjustmenttechniquesthatarewidelyused
inthereactorphysicscommunity(ASTM,2008).Theleast-square
methodologyisquitegeneral,andcanbeusedtoadjustanyvector
ofaprioricomputedquantitiesagainstavectorofcorresponding
measureddatapointsthatcanberelatedtothequantitiesof
inter-estthroughamatrixtransform.Thisproducesa“bestestimate”of
thequantitiesofinterestandtheiruncertainties,whichcanthen
beusedtoestimatethebias,ifany,andtheuncertainty ofthe
computationalmodel,and asatoolfor improvingthemodelas
appropriate
Inthefollowingdescriptionoftheadjustmentequationsused
inthiswork,matrixandvectorquantitieswillgenerallybe
indi-catedbyboldtypeface.Insomecases,matricesandvectorswill
beenclosedinsquarebracketsforclarity.Thesuperscripts,“−1”
and“T”,respectively,indicatematrixinversionandtransposition, respectively
Webeginthemathematicaldevelopmentbyconstructingthe followingoverdeterminedsetoflinearequations:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
a 11 a 12 a 13 ··· ··· a1,NE
a 21 a 22 a 23 ··· ··· a2 ,NE
a NM,1 aNM,2 aNM,3 ··· ··· aNM,NE
1 0 0 ··· ··· 0
0 1 0 ··· ··· 0
0 0 1 ··· ··· 0
0 0 0 ··· ··· 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
•
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P 1
P 2
P 3
P NE
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Pm 1
Pm 2
Pm NM
P 01
P 02
P 03
P0NE
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
andthesupportingdefinition
[Cov (Z)]=
[Cov (Pm)] [0]
[0] [Cov (P0)]
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J.W Nielsen et al / Nuclear Engineering and Design 295 (2015) 615–624 619
whereNEisthetotalnumberoffuelelements(i.e.40forATR)and
NMisthenumberoftheseelementsforwhichelementpower
mea-surementshavebeenmade.NMistypicallyanumberbetween1
andNEalthoughmultiplepowermeasurementsforthesamefuel
elementsmayoptionallybeincludedifavailable,possibly
caus-ingNMtobegreaterthan NE.ThevectorPis thedesiredbest
least-squaresestimateforthepowersofall40fuelelements,the
vectorPm(thefirstNMentriesin[Z])containstheNMmeasured
powersandthevectorP 0 (thelast40entriesin[Z])containsthe
40aprioriestimates,P0ifortheelementpowers,extractedfrom
thecomputationalmodelofthevalidationexperiment
configura-tion.ThetopNMrowsofthematrixAeachcontainentriesai,jthat
areequaltozeroexceptforthecolumncorrespondingtothe
ele-mentforwhichthemeasurementontheright-handsideinthat
rowwas made,where theentry would be1.0 The bottom 40
rowsofthematrixAcorrespondtotherowsofa40×40identity
matrix
NotethattheformulationdescribedbyEq.(1)variesfromthat
ofseveralotherleast-squaresadjustmentalgorithmsusedin
reac-torphysicsinthesensethattheparametersinthematrixonthe
left-handsideareallconstants.Thisisasimplificationinthatthere
arenoadjustableparameters(e.g.nuclearcrosssections)onthe
lefthandsidethatcanbemanipulatedwithintheiruncertainties
toproducestatisticalconsistencyinaleast-squaressensebetween
thecomputedaprioripowervectorandthemeasuredpowervector
BasicallyEq.(1)maybethoughtofasamethodologyforadjusting
theaprioripowervectorandthemeasurementvector(withintheir
respectiveuncertainties)directlytothesamebest-estimatefuel
elementpowervectorP,whichtherebycontainsallofthe
avail-ableinformationabouttheaprioriandmeasuredpowervectors
and theircorresponding covariance matrices The methodology
alsoenablesamathematicallyvalidadjustmentoftheentirea
pri-orielementpowervectorandcomputationofassociatedreduced
uncertaintyforallofthefuelelementsevenifmeasurementsare
notavailableforsomeofthefuelelements.Thisasaresultofthe
waythattheaprioricovariancematrix(describedfurtherbelow)
canserveasaninterpolatingfunctionaswellasastatistical
weight-ingfunctionintheadjustment(Williams,2012)
Eq.(2)includestheNM×NMandNE×NEcovariancematrices
forthemeasuredpowervectorandfortheaprioripowervector,
respectively.Thenumericalentriesfor[Cov(Pm)]arebasedonthe
reporteduncertaintiesoftheexperimentaldataintheusual
man-ner.Thecovariancematrix[Cov(P 0 )]fortheaprioripowervectoris
fundamentaltothesimplifiedadjustmentmethodologydescribed
here.Itmaybecomputedexplicitly(atleastthediagonalelements)
bypropagatingallofthecomputationalmodeluncertainties(i.e
uncertaintiesassociatedwiththenucleardata,component
dimen-sions, materialcompositions and densities, etc.) throughtothe
computedpowervectorusingvariousestablishedtechniques.On
theotherhand,andwithmanysimplifyingassumptionsthatmayor
maynotbeappropriate,[Cov(P 0)]canalsobeapproximatedbased
ontheassumptionofanelement-to-elementfissionpower
correla-tionfunctionthatdecreasesexponentiallywithdistancebetween
anytwoelements, normalizedtotheestimatedvariancesofthe
computedpowersbasedonhistoricalexperienceandengineering
judgment
However,itmaynotalwaysbepracticaltocomputethefulla
prioricovariancematrixexplicitlybypropagatingalloftheinput
uncertaintiesbut,atthesametime,asimpleexponential
approx-imationfor theoff-diagonal entriesmaynot bewellsuited for
computingthefuelelementpowercorrelationmatrixneededto
construct[Cov(P 0)]inEq.(2).Foranyofseveralphysicalreasons
thefuelelementpowercorrelationmatrixforaparticularfacility
mayhaveamorecomplexstructurethanthesimple
diagonally-dominant arrangement that an exponential formula provides
Nonetheless, the availability of an accurate, realistic power
correlationmatrixisacrucialprerequisiteforthesuccessful appli-cationoftheleast-squaresmethodology(Williams,2012)
Toaddressthisissue,weintroduceanintermediate methodol-ogyforobtaining[Cov(P 0)]forATRapplicationsbasedonthefission matrixconcept,furtherdescribedbelow.Themethodfeaturesthe abilitytoincorporateexplicitcalculations ortouseengineering estimatesforthediagonalentriesof[Cov(P 0)]whilestill represent-ingtheoff-diagonalentriesrealistically,butsignificantlyreducing thecomputationaleffortrequired,offeringthepossibilityof effi-cientreal-timeonlinevalidationdataassimilation.Thisapproach was required for ATR because of the complex serpentine core arrangement
5.1 CalculationoftheATR/ATRCfissionmatrix Eachentry,fi,j,oftheso-called“FissionMatrix”,Ffora criti-calsystemcomposedofaspecifiednumberofdiscretefissioning regionsisdefinedasthenumberoffirst-generationfission neu-tronsborninregioniduetoaparentfissionneutronborninregion
j(CarterandMcCormick,1969).Theindexicorrespondstoarow
ofthefissionmatrixandtheindexjcorrespondstoacolumn.In thecaseoftheATRandtheATRCapplicationofinterestherethe fissioningregionsaredefinedtocorrespondtothefuelelements,
sothefissionmatrixhasdimensionsof40×40
Assumenowthattheexactspace,angularandenergy distribu-tionoftheparentfissionsourceneutronswithineachfuelelement
isknownfroma detailedhigh-fidelitytransport calculationand thatthisinformationisincorporatedintotheformationofF.Then constructthefollowingeigenvalueequation:
S= 1 k
whereSisthesuitably-normalized40-elementfundamentalmode vectoroftotalfissionsourceneutronsproducedineachofthe40 fuelelementsandkisthefundamentalmodemultiplicationfactor UndertheseconditionsthesolutiontoEq.(3)willbethesameasis obtainedbyperformingthecorrespondinghigh-fidelitytransport calculationforthesameconfigurationandintegratingtheresulting fissionsourceovereachfuelelement.Ofcourse,ifonealreadyhas thesolutionforthedetailedhigh-fidelitytransportmodelthenEq (3)doesnotprovideanynewinformation,butthefissionmatrix conceptcanstillbeveryusefulandinstructive.Inparticular,there hasbeenagreatdealofeffortovertheyearsfocusedonacceleration
ofMonteCarlocalculationsusingfissionmatrixbasedtechniques, withcertainassumptionstosimplifytheestimationofthefission matrixelementsasthecalculationproceeds,withoutfully solv-ing thehigh-fidelity problemexplicitly beforehand (Carter and McCormick,1969;KitadaandTakeda,2001;DufekandGudowski, 2009;WennerandHaghighat,2011;Carneyetal.,2012)
In theATR application presented here we employ a fission matrixbasedapproachtodeterminethefuelelementtoelement fissionpowercorrelationmatrixandtherebytheassociated covari-ancematrix[Cov(P 0)]thatisrequiredinEq.(2).Theexampleuses theMCNP5codefortherequiredcomputations,butinprincipal theideashouldbeamenabletoimplementationusingany multidi-mensionaldeterministicorstochastictransportsolutionmethod, providedthatasufficientlevelofspatial,angular,andenergy res-olutioncanbeachievedinthedetailedtransportsolutionneeded foranaccuratecalculationofthefissionmatrix
Inthecase oftheATRand ATRC,thefuelelementgeometry (Fig.3)isrepresentedessentiallyexactlyinMCNP5.Eachfuelplate hasaseparateregionforthehomogeneousuranium–aluminum fis-silesubregionandtheadjacentaluminumcladdingsubregionson eachsideofthefueledlayer.Burnableboronpoisonisalsoexplicitly representedinthefuelplateswhereitispresent.Coolantchannels betweentheplatesareexplicitlyrepresented,asarethealuminum
Trang 6620 J.W Nielsen et al / Nuclear Engineering and Design 295 (2015) 615–624
sideplatestructures.Theactivefuelheightis1.2192m(48 and
theelementshaveessentiallythesametransversegeometric
struc-tureatallaxiallevelswithintheactiveheight.Eachfuelelement
contains1075gof235U
High-fidelitycomputation of thefission matrixwithMCNP5
(orwithanyotherMonteCarlocodethatfeaturessimilar
capa-bilities) for this particular application is accomplished in two
easily-automatedstepsasfollows:
First,runawell-convergedfundamental-modeeigenvalue
(“K-Code” in MCNP5 parlance) calculation for the ATR or ATRC
configurationofinterest.Savethedetailedvolumetricfission
neu-tronsourceinformationthatincludesallfissionneutronsstarting
fromwithineachfuelelement.Theabsolutespatial,angular,and
energydistributionofthefissionneutronsbornineachfuelelement
mustbefullyspecifiedinthesourcefiledataforthatelement
Second,usingthefissionneutronsourcefileinformation
cre-atedasdescribedabove,runasetof40correspondingfixed-source
MCNP5calculationsfor thesamereactorconfigurationof
inter-est,oneseparatewell-convergedcalculationforeachfuelelement
fissionneutronsourceseparately.Thesecalculationsarerunwith
fission neutron production turned off using the “NONU” input
parameter.Fissionsinducedbytheoriginalfissionsourceneutrons
sampledfromthesourcefilearetherebytreatedascaptureinthe
sensethatnoadditionalfissionneutronsareproducedtobe
fol-lowedinsubsequenthistories.The“fission”ratethatistalliedin
thismannerforeachfuelelementinagivenMCNPfixed-source
cal-culationthusincludesonlythefirst-generationfissionsinducedin
thatelementbytheoriginalsourceneutronsemittedbythesource
fuelelementthatwasactiveforthatcalculation.Multiplyingthis
quantityforeachfuelelementinagivenMCNPcalculationbythe
averagenumberofneutronsperfissionandthendividingtheresult
bytheabsolutemagnitudeoftheoriginalfissionneutronsource
associatedwiththeactivefuelelementthenyieldsthecolumnof
thefissionmatrixcorrespondingtothatsourcefuelelement
Substitutionofthefissionmatrixfromtheaboveprocessinto
Eq.(3)shouldreproduce(withintheapplicablestatistical
uncer-tainties)theeigenvalueand thefuelelement-to-elementfission
neutronproductiondistributionoftheoriginalMCNPK-Code
cal-culation.Oncethisisverified,thefissionmatrixisreadyforusein
generatingtherequiredfuelelementfissioncorrelationmatrixas
describedbelow
5.2 Constructionofthefissioncovariancematrix
Tobeginthefissioncovariancematrixdevelopment,wemake
akeyfacilitatingassumptionthattheaveragenumberofneutrons
producedperfissionisthesameforallofthefissioningregionsin
themodel.ThisisreasonablefortheATRCexperimentofinterest
herebecauseall40fuelelementswereidenticalandunirradiated
Furthermore,MCNPcalculationsshowthattheneutronspectrum
doesnotvaryfromoneATRCfuelelementtothenextinamanner
thatsignificantlyaffectstheratioof238Ufissionsto235Ufissions
Thereforeinthiscaseeachentry,fi,j,ofthefissionmatrixalsocan
beinterpretedasthenumberoffirst-generationdaughterfissions
induced(orcorrespondingfissionenergyreleased)ineachregioni
duetoaparentfissionoccurringinregionj
Turningnowtotheactualcomputation ofthefissionpower
covariancematrixneededinEq.(2),itisimportanttonotethat
the40-elementfundamentalmodevectoroffissionpowers(or
fis-sionneutronsources)foreachofthe40ATRorATRCfuelelements
maybeviewedasavectorofrandomvariablesthatarecorrelated
becausefissionneutronsborninonefuelelementcaninducenew
fissionsnotonlyinthesameelement,butinanyotherfuelelement
aswell,althoughtheprobabilitythataneutronborninoneelement
willinduceafissioninanotherelementgenerallydecreaseswith
physicalseparationofthetwofuelelements
ReferringtoEq.(3),itcanbeseenthatifthefundamentalmode fission source(orpower) vector is premultipliedby thefission matrixtheresultingvector is,bydefinition,simplytheoriginal vectorwithall entriesmultipliedbyk-effective Furthermoreif thefundamentalmodesourceorpowervectorisarbitrarily per-turbedinsomemanner,thenpremultiplicationoftheperturbed vectorbythefissionmatrixwillforceitbacktowardtheoriginal fundamentalmodeshape,althoughanumberofiterationsmaybe requiredtoconvergebacktotheoriginalvectorinapplicationssuch
asATR,wherethedominanceratioisfairlylarge.Theabove obser-vationssuggestthefollowingstochasticestimationprocedurefor constructingtherequiredfissioncorrelationmatrix:
(1)Generateavectorof40normally-distributedrandomnumbers whosemeanis1.0andwhosestandarddeviationissome nom-inalsmallfractionofthemean,e.g.10%.Thefractionspecified forthestandarddeviationisarbitrary,butitshouldbesmall enoughsuchthatessentiallynonegativerandomnumbersare everproducedandatthesametimeitshouldbelargeenough
toavoidround-offerrorsintheprocessdescribedbelow (2)Multiplyeachofthe40elementsofthefundamentalmode fis-sionpowervectorbythecorrespondingelementoftherandom numbervectorfromStep1.Ontheaverage,halfofthefission powerentriesthatarerandomlyperturbedinthismannerwill increaseandhalfwilldecrease
(3)Premultiplytheperturbed fundamental-mode fissionpower vectorfromStep2bythefissionmatrixandstoretheresulting perturbed“first-generation”fissionpowervector
(4)RepeatSteps1–3astatisticallyappropriatenumberoftimes,N (e.g.N=1000),toproduceabatchofN40-elementperturbed
“first-generation”fissionpowervectors
(5)Compute the40×40 covariance matrix for theelementsof theN40-elementperturbed “first-generation”fissionpower vectorsusingthefundamentaldefinition ofcovariance.This completesan“inneriteration”,producingastatisticalestimate
ofthefissionpowercovariancematrix
(6)Repeat Steps1–5many times,tallyinga runningaverageof thecovariance matrices that areproduced untilsatisfactory convergenceisobtained.Thencomputethecorrelationmatrix associatedwiththeconvergedcovariancematrix
(7)Constructthecovariancematrixfortheaprioripowers com-puted by the modeling code by combining the correlation matrixfromStep6withavectorofassumedapriori uncer-tainties that are to be associated with the a priori power vector At this point one could also manually add a fully-correlated componenttothecovariancematrixtorepresent potentialsystematicuncertainties(e.g.uncertaintyinthetotal powernormalizationoftheapriorimodel)inadditiontothe partially-correlated uncertainties that are estimated by the aboveprocedure
Inmathematicaltermsthisprocesscanbeprogrammedas fol-lows:
First,define
[PD]=
⎡
⎢
⎢
⎢
⎢
P01
P02
P03
..
P0,NE
⎤
⎥
⎥
⎥
⎥
(4)
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wherethediagonalelementsof[PD]correspondtotheapriori
com-putedfuelelementfissionpowersandallotherentriesarezero
Nowdefinethematrixofrandomnumbers
[R] =
⎡
⎢
⎢
⎢
⎢
⎣
r11 r12 ··· ··· r1 ,N
rNE ,1 rNE ,2 rNE ,N
⎤
⎥
⎥
⎥
⎥
⎦
(5)
where Nislargeand each rij is a randomnumber drawnfrom
anormallydistributedpopulationwhosemeanis1.0andwhose
standarddeviationisasmallfractionofthemean(e.g.10%).Then
formthematrixproduct:
[PP]= [PD][R] =
⎡
⎢
⎢
⎢
⎢
⎣
pp11 pp12 ··· ··· pp1 ,N
ppNE,1 ppNE,2 ppNE ,N
⎤
⎥
⎥
⎥
⎥
⎦
(6)
whereeach columnof[PP]isa vectorofapriorielement
pow-ersperturbedbythecorrespondingrandomnumbersinthesame
columnof[R]
Nowpremultiply[PP]bythefissionmatrix[F]toobtainamatrix
[FPP] of N“first-generation”fuel element powervectors
corre-spondingtoeachoriginalperturbedpowervector:
Theelementsoftherandomly-perturbedpowervectors
com-prising [PP] are uncorrelated, but the elements of each of the
correspondingfirst-generationpowervectorscomprising[FPP]will
bepositivelycorrelatedbyvirtueofthefactthatafissionoccurring
inonefuelelementcancauseanext-generationfissionnotonly
inthatelementbutalsoinanyotherelement,asquantifiedbythe
fissionmatrix
Now,recognizingthattheNcolumnsof[FPP]arerandom
sam-ples of an “average” first-generation fission power vector [P1]
(whosespatialshapecanincidentallybeshowntobestatistically
identicaltothatoftheoriginalpowervector[P0]),thecovariance
oftheelementsof[P1]maybecomputedas:
where[DM]isthedifferencematrix:
and[U]isanNE-row,N-columnmatrixwhoseentriesareall1.0
Repeattheprocessdescribedaboveanumberoftimes,
tally-ingarunningaverageof[Cov(P1)]untilsatisfactoryconvergence
isobtained.Thencomputethecorrelationmatrixcorresponding
totheconvergedcovariancematrix[Cov(P1)]usingthestandard
definition This is the desired fuel element-to-element power
correlationmatrix.Finally, usethis powercorrelationmatrixto
constructamatrix[Cov(P0)]thatcorrespondstotheactualabsolute
uncertaintiesassociatedwiththeelementsof[P0]ratherthanthe
arbitraryuniformperturbationusedtoobtain[Cov[P1]],andthen
addafully-correlatedcomponentto[Cov(P0)]ifdesired
5.3 Solutionoftheadjustmentequations Withthefissionpowercovariancematrixnowavailable,Eqs (1)and(2)canbecombinedintheusualmannertoconstructthe covariance-weighted“NormalEquations”(e.g.Meyer,1975)forthe system,yielding:
with
Eq.(10)canbesolvedbyanysuitablenumericaloranalytical methodtoyieldtheadjustedelementpowervectorP.The differ-encebetweentheadjustedpowervectorandtheaprioripower vectorthengivesanestimateofthebiasofthemodel,ifany,relative
tothebest-estimatepowervector
Also,sincethesolutiontoEq.(10)is:
thecovariancematrixfortheadjustedpowersmaybecomputed
bythestandarduncertaintypropagationformula:
where
Thediagonalelementsofthecovariancematrixfortheadjusted powerscanthenalsobeusedtoestimatetheuncertaintyinthe differencebetweentheaprioriandtheadjustedpowervectors.It mayalsobenotedinpassingthatthecovariance matrixforthe adjustedpowervectorisalsosimplytheinverseofB.
6 Results and discussion
The a priori and measured power distributions from Fig 4 areplottedinFig.5,alongwiththeadjustedpowerdistribution correspondingtothe measuredpowers ofall 40elements The covariancematrixfortheaprioripowervectorwascomputedas describedaboveandnormalized toanestimatedapriori uncer-tainty of 10%(1) for the diagonalentries, based onhistorical experience.Thecovariancematrixforthemeasuredpowerswas assumedtohavediagonalentriesof5%(1)basedonhistorical experienceandnooff-diagonalentriesforthisexample.Itisa sim-plemattertoincludeappropriateoff-diagonalelementsinthelatter matrixtoaccountforcorrelations,forexamplefromacommon cal-ibrationofthedetectorusedtomeasuretheactivityofthefission wires,ifdesired.Thereduceduncertaintiesfortheadjusted ele-mentpowersinFig.5,computedusingEq.(7),rangedfrom3.1%
Fig 5.Fuel element power distributions for ATRC Depressurized Run Support Test 12-5 The adjusted power is computed using the measured powers of all 40 fuel
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Fig 6.Fission power correlation matrix for the ATRC The axis numbering
corre-sponds to the fuel element numbers shown in Fig 4
Fig 7. Fission matrix for the ATRC The axis numbering corresponds to the fuel
element numbers shown in Fig 4
to3.7%.Thecorrelationmatrixassociatedwiththefissionpower
covariancematrixusedtocomputetheadjustedpowervectoris
shownasacontourplotinFig.6.Keyoff-diagonalstructural
fea-tures,suchasthecorrelationsbetweennearby,butnon-adjacent,
Elements1and10,orElements11and20,etc.arereadilyapparent
TheunderlyingfissionmatrixforthisexampleisshowninFig.7
Thesamegeneralstructureisapparent.Notealsothatthefission
matrixisnotnecessarilysymmetric,whilethefissioncorrelation
matrixissymmetricbydefinition
Fig.8showstheresultofanadjustmentoftheMCNPapriori
fluxwhereonlythepowersoftheodd-numberedfuelelementsin
Test12-5wereincludedintheanalysis.Thissimulatesthe
rela-tivelycommonATRpracticewhereonlytheodd-numberedfuel
elementpowersareactually measured,and thepowerforeach
even-numberedelementisassumedtobeequaltothemeasured
powerintheodd-numberedelementontheoppositesideofthe
samelobe.Forexample,thepowerinElement2isassumedequal
tothepowerinElement9,thepowerinElement4isassumed
equaltothepowerin Element7,and soforth aroundthecore
Theoften-questionablevalidityofthisassumptiondependsonthe
overallsymmetryof thereactorconfiguration.Inthefuturethe
assumptionofsymmetrywillbereplacedbythemorerigorous
least-squareadjustmentproceduredescribedheretoestimatethe
powersintheeven-numberedelements.Thereduced
uncertain-tiesfortheadjustedelementpowersinFig.8rangedfrom3.9%
to4.3% forthe odd-numbered elementsand from4.0% to5.2%
fortheeven-numberedelements,demonstratinghowsignificant
uncertaintyreductioncanoccurintheadjustedpowersevenfor
Fig 8. Fuel element power distributions for ATRC Depressurized Run Support Test 12-5 The adjusted power is computed using the measured powers of only the 20 odd-numbered fuel elements.
elementsforwhichnomeasurementisincluded.Thisisaresultof theweightedinterpolationeffectprovidedbytheelementpower covariancematrix
Economizing onthenumber of measurements even further, Fig.9showsanadjustmentwhereonlythemeasuredpowersfor Elements 8,18, 28, and 38 wereincluded in theanalysis.This arrangementsimulates anotherATRprotocolthat is sometimes usedbecausetheseelementsare representativeof the highest-poweredelementsin each outer lobe Inthis case thereduced uncertaintiesfortheadjustedelementpowersrangedfrom4.4%
to4.5%forElements8,18,28and38,from6.6%to7%forthe imme-diatelyadjacentelementsandupto9.9%fortheelementsthatwere themostdistantfromtheelementsforwhichmeasurementswere made.It isnotableherethatsomeuncertaintyreductionoccurs evenforthemostremotefuelelements
Fig.10illustratesanotherpossibleuseofthetechniques devel-opedinthiswork.TheATRhasanonlinelobepowermeasurement systembutitdoesnothaveanonlinesystemformeasurementof individualfuelelementpowers.Measurementsofindividual ele-ment powerscurrently canonlybe donebytherather tedious fissionwiretechniquedescribedearlier.Theleast-squares method-ology outlined here also offers a simple, but mathematically rigorous,approachforestimatingthefissionpowersofall40fuel ATRfuelelementsand theiruncertainties usingtheonlinelobe powermeasurementsasfollows:
InthecaseofFig.10theonlinelobepowermeasurementsare simulatedbythefissionwiremeasurementsusedfortheprevious examples.Thefirstfiverowsofthematrixontheleft-handside
ofEq.(1)describethefivesimulatedonlinelobepower measure-ments.Theserowseachcontainentriesof0.125ontheleft-hand sidefortheeight(8)elementsincludedinthelobecorresponding
tothatrowandentriesofzeroelsewhere.Therighthandsideof eachofthesefirstfiverowscontainstheaverageofthemeasured powersfromthefissionwiresfortheloberepresentedbythatrow Forexamplethefirstrow(Lobe1)contains entriesof0.125for elements2through9,andtheaverageofthemeasuredpowersfor elements2through9appearsontherighthandside,andsoforthfor theotherlobes.Thereduceduncertaintiesfortheadjustedpowers showninFig.10forthe40elementsrangefrom6.4%to8.3% TheresultsshowninFig.10thusillustrateapractical applica-tionwherethepowersforeachATRlobethataremeasuredonline couldbeenteredintoEq.(1)eachtimetheyareupdated(everyfew seconds),andacorrespondingestimateforalloftheindividual ele-mentpowerscouldbeimmediatelyproduced.Ofcoursetheapriori powervectorwouldneedtoberecalculatedregularlyasthecore depletes,controldrumsrotate,andneckshimsarepulledduring
acycle.Thiscouldhoweverbeautomatedtoalargeextent,andit
Trang 9J.W Nielsen et al / Nuclear Engineering and Design 295 (2015) 615–624 623
Fig 9. Fuel element power distributions for ATRC Depressurized Run Support Test 12-5 The adjusted power is computed using the measured powers of elements 8, 18, 28 and 38 only.
Fig 10.Fuel element power distributions for ATRC Depressurized Run Support Test
12-5 The adjusted power is computed using the measured powers of the five core
lobes.
Fig 11.Comparison of a priori element powers (MCNP5), the adjusted element
powers based on the measured lobe powers formed from the original detailed fuel
element power measurements, and the actual detailed element power
measure-ments.
shouldultimatelybequitepractical,forexample,toupdatethea prioripowervectorfromthemodelatleastdailyandperhapseven hourly
Finally,Fig.11showsacomparisonoftheapriorielement pow-ers andtheadjusted elementpowers based onthelobepower measurements(Fig.10)withtheoriginaldetailed40-element mea-suredpowerdata.Recallthattheadjustedpowersinthisfigureare basedonlyonthemeasuredlobepowersthatwerepre-computed
byaveragingthedetailedelementpowermeasurementsforeach lobe.Itisinterestingtonotethattheadjustedpowerdistribution curvestillrecapturesasignificantamountofthedetailedshape changerelativetotheaprioripowerdistribution,eventhoughthe detailsinthemeasuredpowerdistributionwerelargelyaveraged outwhencomputingthesimulatedmeasuredlobepowersused fortheadjustment.Thecovariancematrixplaysakeyroleinthis process
7 Conclusions
Insummary,thispaperpresentsarelativelysimplebut effec-tivefission-matrix-basedmethodforgeneratingtherequiredfuel elementcovarianceinformationneededfordetailedstatistical vali-dation and best-estimate adjustment analysis of fission power distributionsproducedbycomputationalreactorphysicsmodelsof theATR(orforthatmatter,anyothertypeofreactor).Themethod hasbeendemonstratedusingtheMCNP5neutronicscodebutitcan
beusedwithanyotherMonteCarloneutronicssimulationcodeas wellaswithanydeterministicneutrontransportcodethat pro-videsasufficientlevelofspatial,angular,andenergyresolution withineachfissioningregionofinterest.Analysesofthistypeare usefulnotonlyforquantifyingthebiasanduncertaintyof com-putationalmodelsforaspecificmeasuredreactorconfigurationof interest,buttheyalsocanserveasguidesformodelimprovement andforestimation ofapriorimodelinguncertaintiesforrelated reactorconfigurationsforwhichnomeasurementsareavailable
Acknowledgements
Thiswork wassupportedby theU.S Department of Energy (DOE), via the ATR Life Extension Program under BattelleEn-ergy Alliance, LLC Contract no.DE-AC07-05ID14517 with DOE The authorsalsowish togratefully acknowledgeseveral useful
Trang 10624 J.W Nielsen et al / Nuclear Engineering and Design 295 (2015) 615–624
discussionswithDr.JohnG.Williams,UniversityofArizona,onthe
generalsubjectofcovariancematricesandtheirroleinthistypeof
analysis
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