AREVA NP GmbH. AREVA NP GmbH, an AREVA and Siemens company

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1 1, an REV and Siemens company

2 Evaluation of Citicality cceptance Citeia Using Monte Calo Methods Jens Chistian Neube and xel Hoefe Gemany 2

3 J.C. Neube,. Hoefe, NEE-G kˆ (x) + λ Intoduction: Citicality cceptance Citeion σˆ (x) (1 k m (x)) ( kˆ + λ σˆ ) B B B Uppe 95%/95% toleance limit of the calculated neuton multiplication facto k eff of the fuel system of inteest (application case) chaacteized by a set of paametes x = x,..., ( ) T 1 x n including all vaiations in k eff aising fom vaiations δx due to (e.g.) - bias and uncetainty in isotopic concentation and - manufactuing toleances dequate safety magin Uppe 95%/95% toleance limit of the bias of the employed citicality calculation pocedue (deived fom evaluation of epesentative benchmaks) 3 P L u ( α, γ) 0 df ( k ) ( 1 γ) = ( 1 α) eff (1-γ)=0.95 L u (α,γ) k eff

4 Paamete Vecto x J.C. Neube,. Hoefe, NEE-G Isotopic composition initial enichment (e) integal bunable absobes (Gd, E, IFB ods) bunup (B) cooling time (t) Bunup Cedit (BUC) Depletion conditions N / N(without CR) Relative change of isotopic numbe densities due to CR insetion (Zeo cooling time) 5 MWd/kg/U 10 MWd/kg U 15 MWd/kg U 20 MWd/kg U 25 MWd/kg U 30 MWd/kg U 35 MWd/kg U 40 MWd/kg U 45 MWd/kg U mo-95 tc-99 u-101 h-103 ag-109 cs-133 nd-143 nd-145 sm-147 sm-149 sm-150 sm-151 sm-152 eu-153 gd-155 u-234 u-235 u-236 u-238 np-237 pu-238 pu-239 pu-240 pu-241 pu-242 am-241 am-243

5 Paamete Vecto x J.C. Neube,. Hoefe, NEE-G Fuel design data cladding guide thimbles wate ods/channels fuel channels Geomety data Configuation 5

6 System design data Fuel aangement and geomety data Paamete Vecto x J.C. Neube,. Hoefe, NEE-G Neuton absobing mateials Stuctual mateials Pesence of modeato Wet Stoage Example: Convoy seies wet stoage ack (detail) Tanspot and Re-pocessing 6

7 Paamete Vecto x J.C. Neube,. Hoefe, NEE-G system conditions Nomal opeation bnomal / accidental conditions scenaios (final disposal) Isotopic composition Fuel design data System design data Range of paametes 7

8 kˆ (x) + λ σˆ (x) Paamete Vecto x (1 k m (x)) J.C. Neube,. Hoefe, NEE-G ( kˆ + λ σˆ ) B B B Most paametes chaacteized by some vaiability due to (fo instance) uncetainty in the isotopic composition of spent fuel due to uncetainties in depletion calculation and depletion conditions vaiability in spatial distibution of isotopic concentations (e.g., due to vaiability in axial and hoizontal bunup distibutions) specified manufactuing toleances (mateial compositions, dimensions fuel design, system design) paametes epesent andomized vaiables (as in isk-infomed appoaches) (e.g. analysis of final disposal: paametes descibing futue climates deteministic models pobability weighted by evaluating peiodicity of past climate changes) 8 Nuclea data uncetainties coss sections fission spectum neutons pe fission paametes decay constants, banching atios, yields

9 J.C. Neube,. Hoefe, NEE-G kˆ (x) + λ σˆ (x) (1 k ( ) Paamete Vecto x m (x)) kˆ B + λ B σˆ B Vaiability in calculated k eff value Vaiability in estimated bias x 3 x 2 Region of possible paamete values ( System egion ): Definition egion Confidence egion Toleance egion x 1 Taditional analysis (mostly) 9

10 J.C. Neube,. Hoefe, NEE-G Objective x 3 Ideal method: Monte Calo (MC) sampling on the paamete egion x 2 Sets of MC sampled paamete values (x s ) i = (x s1, x s2, x s3, ) i, i =1,,κ Pefoming κ citicality calculations x 1 kˆ (x) + λ σˆ (x) 10 k eff values (k eff ) i, i =1,,κ, distibution of k eff

11 J.C. Neube,. Hoefe, NEE-G Requiements MC sampling on a paamete egion equies a joint pobability density function (pdf) of the paametes Poblem: pdf usually unknown Necessay: pdf model deived fom empiical data Geneate MC samples x s unde the condition of empiical data X: Posteio pedictive p ( ) ( ) ( ) x X = p x Θ p Θ X dθ s s n x m data matix of n independent identically distibuted (iid) m-vaiate data x i = (x i1,x i2,,x im ) Θ pobability distibution model (e.g. nomal distibution: Θ = (µ,σ) paamete Θ unknown MC sampling on Θ unde the condition of the data X p ( Θ X) p( X Θ) p( Θ) 11 posteio knowledge about Θ distibution of X unde Θ pio knowledge about Θ

12 J.C. Neube,. Hoefe, NEE-G Obsevations Many paametes ae mutually independent due to thei meaning and due to manufactuing pocedues p G ( x Θ) = p ( ) g xg Θg g= 1 Isotopic composition Fuel design data System design data Fuel design data: Mutually independent due to manufactuing pocess (e.g.) pellet density pellet diamete cladding inne diamete cladding oute diamete System design data: Mutually independent due to manufactuing pocess (e.g.) bsobe content of neuton absobe panels Thickness of absobe panels Inside width of absobe channels fo fuel assemblies guide thimble inne diamete guide thimble oute diamete wate od inne diamete wate od oute diamete 12

13 J.C. Neube,. Hoefe, NEE-G Obsevations: pdf model fom empiical data Example: Cladding oute diamete of fuel ods ction limits fo manufactuing Consevative choice fo flooded fuel assembly: nomal density at lowe toleance limit Specified toleance inteval (MV± 0.05 mm) 13

14 J.C. Neube,. Hoefe, NEE-G Obsevations: pdf model fom empiical data Example: Cladding oute diamete of fuel ods impact change of tools ction limits fo manufactuing ction limits fo manufactuing 14

15 J.C. Neube,. Hoefe, NEE-G Obsevations: pdf model fom empiical data Example: Initial enichment distibution of pellets in a fuel assembly f e Specified toleance inteval (MV± 0.05 wt.-%) Don t teat the initial enichment as a andom vaiable, take uppe toleance limit Likewise (usually): pellet density (uppe toleance limit) absobe contents (lowe toleance limits) all paametes, fo which a pdf cannot be deived fom empiical data - paametes fo which the impact on eactivity is obvious by physics: choose bounding value (e.g. bounding axial pofile as a function of aveage bunup) - othewise: choose bounding theoetical pdf (e.g., unifom o tiangula on toleance inteval) 15

16 J.C. Neube,. Hoefe, NEE-G Example: Outcome fo wet stoage of fesh MOX fuel assemblies 16

17 J.C. Neube,. Hoefe, NEE-G Obsevations fo BUC: Impact of isotopic bias and uncetainty fom PIE (chemical assay) Calculated Isotopic coection facto ICF i = E i / C i, i := Isotope I, i= 1,,p ICF i andom numbe C i not independent (calculated in one calculation un) E i fo (at least) some of the isotopes neithe independent no uncoelated (due to measuement methods) measuement No. andom numbe dditionally: Missing data poblem Isotopes 17

18 J.C. Neube,. Hoefe, NEE-G Requiements Geneate MC samples x s unde the condition of incomplete empiical data X=(X obs, X mis ) fom the obseved data X obs Posteio pedictive p ( ) ( ) ( ) x X = p x Θ p Θ X dθ s obs s obs M missing-data indicato M ij = 1, X 0, X ij ij missing obseved measuement No. M is consideed as a set of andom vaiables X ij Isotopes 18

19 J.C. Neube,. Hoefe, NEE-G Obsevations: Natue of missingness Full poblem, i.e. full pobability density distibution of X and M unde Θ becomes: p ( ) ( ) ( ) X,M Θ, Φ = p M X, Φ p X Θ Φ natue of missingness p(m X,Φ): =distibution of the missing data mechanism complete-data pdf Missing data mechanism is ignoable if Usual case Missing data ae missing at andom (MR): missingness may depend on obseved data X obs, but does not depend on X mis p ( M X, Φ ) = p( M X, Φ) obs Θ and Φ ae a pioi independent ( Bayesian infeence): p(θ,φ) = p(θ) p(φ) 19 p ( Θ X ) p( X Θ) p( Θ) obs obs

20 J.C. Neube,. Hoefe, NEE-G Method (bief desciption): Data ugmentation Method Geneate MC samples x s unde the condition of incomplete empiical data X=(X obs, X mis ) fom the obseved data X obs Posteio pedictive p ( ) ( ) ( ) x X = p x Θ p Θ X dθ s obs s obs Intactable due to missingness solution of this poblem: Data ugmentation Method Complete-data pdf can be witten: p ( X Θ) = p( X X, Θ) p( X Θ) miss obs obs pedictive distibution of missing data 20

21 J.C. Neube,. Hoefe, NEE-G Method (bief desciption): Data ugmentation Given a guessθ p ( ) ( i+ 1) X X, Θ X () i mis obs () i mis Imputation step i = i+1 complete-data posteio pdf ( ( )) i+ 1 Θ Xobs, X mis Θ( i 1) p + Posteio step { () i Θ } ( ) () i, X mis,i = 1,2,3,... p Θ, X mis X in distibution obs { Θ, i = 1,2,3,... } in p( Θ ) () i X obs { () i X,i = 1,2,3,... } in p( X X ) mis distibution distibution mis obs = = p p ( X ) ( ) mis X obs, Θ p Θ X obs ( Θ X) p( X X ) mis = obs 21 Sufficient numbe of iteations samples x s can be dawn fom the model p(x Θ) afte each posteio step: ( ) () i p x Θ i x s ()

22 J.C. Neube,. Hoefe, NEE-G Example: Nomal distibution model p(x Θ) = N(x µ,σ) Multivaiate nomal distibution with unknown expectation µ and unknown covaiance matix Σ No pio infomation about µ and Σ available using a non-infomative pio (e.g. p(θ) = (detσ) -(m+1)/2 ) Posteio Step becomes W 1 ( ) 1 () () ( ) ( i n 1, Ψ with Ψ = n 1 S X,X ) () i i [ ] 1 obs mis Σ() i Inveted Wishat Distibution Sample covaiance matix and ( 1 xˆ ) () i, n Σ() i with xˆ () i ( () i X ) obs, Xmis () i N = xˆ µ Sample mean samples x s dawn afte each posteio step: N ( ) () i x µ, Σ x () i () i s Impotant: Sampling on covaiance 22

23 J.C. Neube,. Hoefe, NEE-G pplication to ICFs Since ICF s ae atios of positive numbes choose a multivaiate log nomal model Conside 12 actinides and 13 fission poducts ctinides Fission Poducts U-234 Pu-238 m-241 Mo-95 Cs-133 Nd-143 Sm-147 U-235 Pu-239 m-243 Tc-99 Nd-145 Sm-149 U-236 Pu-240 Ru-101 Sm-150 U-238 Pu-241 Rh-103 Sm-151 Pu-242 Sm-152 Np-237 Eu

24 J.C. Neube,. Hoefe, NEE-G pplication to ICFs 13 Empiical pdf of ICF fo Pu-239 Impotance of sampling on covaiance empiical pdf uncoelated sampling coelated sampling ICF Empiical pdf of ICF fo Pu-240 empiical pdf Empiical pdf of ICF fo Pu uncoelated sampling coelated sampling ICF uncoelated sampling 8 7 empiical pdf coelated sampling ICF 24

25 J.C. Neube,. Hoefe, NEE-G pplication to ICFs OECD Phase II-C CSK with xial Pofile B (27 complete empiical ICF data sets plus data sets containing ICF data fo U and Pu isotopes complete + 4 (U + Pu) 27 complete (U + Pu) uppe 95%/95% toleance limits pedictive posteio pdf fit (fee of binning): Johnson empiical distibution histogam of evaluated k eff values k eff

26 J.C. Neube,. Hoefe, NEE-G pplication to ICFs How many expeiments ae minimum equied? N empiical complete ICF data sets fo 25 isotopes: Monitoing functions fo estimating the minimum equied numbe N (MC sampling of 1000 ICF sets fom posteio pedictive pdf) 26 faction of sample values outside acceptance egion 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 3 2 1/2 1/3 n o outside n i inside f numbe of data sets N = acceptance egion [1/2, 2] acceptance egion [1/3, 3] standad deviation n i n + o n o 1.E+19 1.E+18 1.E+17 1.E+16 1.E+15 1.E+14 1.E+13 1.E+12 1.E+11 1.E+10 1.E+09 1.E+08 1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 standad deviation

27 J.C. Neube,. Hoefe, NEE-G pplication to ICFs How many expeiments ae minimum equied? N empiical ICF data sets fo 25 isotopes: 27 sets complete, N-27 sets with data fo the U and Pu isotopes only Monitoing functions fo estimating the minimum equied numbe N (MC sampling of 1000 ICF sets fom posteio pdf) 27 faction of sample values outside acceptance egion 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 3 2 1/2 1/3 n o outside n i inside acceptance egion [1/2, 2] acceptance egion [1/3, 3] standad deviation numbe of data sets N f = n i n + o n o 1.E+19 1.E+18 1.E+17 1.E+16 1.E+15 1.E+14 1.E+13 1.E+12 1.E+11 1.E+10 1.E+09 1.E+08 1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 standad deviation

28 J.C. Neube,. Hoefe, NEE-G pplication to ICFs k eff of OECD Phase II-C CSK with axial pofile B complete ICF data sets pedictive posteio pdf complete ICF data sets 58 complete ICF data sets fit (fee of binning): Johnson empiical distibution histogam of evaluated k eff values uppe 95%/95% toleance limits uppe 95%/95% toleance limit obtained with "consevative" ICFs k eff

29 J.C. Neube,. Hoefe, NEE-G pplication to ICFs OECD Phase II-C CSK with xial Pofile B Compaison of pedictive posteio k eff distibutions when using diffeent bunup cedit levels (31 complete empiical ICF data sets) 45 pedictive posteio pdf actinides + fission poducts actinides only U + Pu fissiles + U-238 uppe 95%/95% toleance limit fit (fee of binning): Johnson empiical distibution histogam of evaluated k eff values k eff

30 J.C. Neube,. Hoefe, NEE-G kˆ (x) + λ σˆ Citicality cceptance Citeion (x) (1 k m (x)) ( kˆ + λ σˆ ) B B B k k M M k B1 B2 Bn z z M M z n1 z z M M z n2 L L O L z z M M z 1m 2m nm Uppe 95%/95% toleance limit of the bias of the employed citicality calculation pocedue (deived fom evaluation of epesentative benchmaks) Sensitivity analysis m explanatoy vaiables n mmatix Z 30

31 J.C. Neube,. Hoefe, NEE-G Estimation of calculational bias Linea model: E [ k Z, β] = Zβ B Linea-least-squaes estimato: βˆ = T 1 T ( Z W Z) Z W x k T ( Z W ) 1 k V β = kz 1 ( ) T s k Zβˆ W ( k Zβˆ ) 2 = k n m 1 Σ k = W k W k σ = diag 2 ( W ) n T k = ( w,..., w ) 1 n σ 2 k a Inv χ 2 ( 2 n m;s ) daw σ 2 Insetion in egession fomula sample on k B 31 β σ 2 ( ) 2 β V, k an ˆ, σ β daw β Evaluation of the sample uppe 95%/95% toleance limit of k B

32 J.C. Neube,. Hoefe, NEE-G kˆ (x) + λ σˆ Uncetainty in nuclea data (x) (1 k m (x)) ( kˆ + λ σˆ ) B B B ( ) ( ) ( ) Obvious: p x, ξ Θ, Ψ = p x Θ p ξ Ψ pdf nuclea data Ideal method: MC sampling on nuclea data fo application case Uncetainty analysis esults fo application case fom fist ode evaluations: σ 2 d ( k) = cov( ξ ) i, ξ j i, j= 1 k ξ i k ξ j 32