Application of the MOCADATA Monte Carlo package to ...package to Uncertainty Analysis for...

Application of the MOCADATA Monte Carlo package to Uncertainty Analysis for Criticality Safety Assessment

Axel Hoefer, Oliver Buss, Jens Christian NeuberAREVA GmbH, PEPA-G (Offenbach, Germany)

Meeting on uncertainty propagations in the nuclear fuel cycle, Uppsala University, April 24-25, 2013

All rights are reserved, see liability notice.

Manufacturing Tolerances (materials, dimensions)

Nuclear data uncertainties

Uncertainties in Criticality Calculations

Isotopic Uncertaintyof spent fuel

Algorithmic uncertainty of criticality and depletion codesUncertainty

of calculated keff value

Validation of criticality code:criticality safety benchmarks

Validation of depletion code:post irradiation experiment

Application of the MOCADATA Monte Carlo package - Meeting on uncertainty propagations in the nuclear fuel cycle, Uppsala University, Sweden, 25/04/2013 - Axel Hoefer - AREVA GmbH Proprietary © AREVA - p.3

All rights are reserved, see liability notice.Application of the MOCADATA Monte Carlo package - Meeting on uncertainty propagations in the nuclear fuel cycle, Uppsala University, Sweden, 25/04/2013 - Axel Hoefer - AREVA GmbH Proprietary © AREVA - p.4

MC Sampling Procedure

kpxkk

tolisoTT

tolTiso xpxpxpx,xx

Tolerances and isotopic uncertainties → distribution of random vector

Neutron multiplication factor becomes random number

Distribution only accessible via Monte Carlo

Monte Carlo Procedure Repeatedly draw random samples from For each calculate with criticality code Order Statistic of Monte Carlo data → upper 95%/95% tolerance limit

MCx

MCxk

MCx

95/95k

95.0:fkP 9595/95

95-th percentile of p(k)Upper 95%/95% tolerance limit

itlim

?

95/95 kk

Maximum allowable keff

xp


x1

x2

x3

Method for Monte Carlo (MC) sampling on the parameter region

Sets of MC sampled parameter values (xs)i = (xs1, xs2, xs3, …)i, i =1,…,κ

keff values (keff)i, i =1,…,κ, distribution of keff

Performing κ criticality calculations

MC Sampling Procedure


1. Monte Carlo Sampling of parameters x1 ... xn from basic distribution models- Uniform distribution- piecewise uniform distribution- normal distribution- asymmetric normal distribution- triangular distribution- left/right saw tooth distribution- Bernoulli distribution- Gamma distribution- Beta distribution

2. Functions of parameters x1 ... xn : z = f1(f2(f3 ...(x1,...,xn)...))

k

1i

i_ei )x(ifunc_cterm

fi = (sum of all numerator terms)/(sum of all denominator terms)

func_i=”abs”, ”exp”, ”log”, ”sin”, ”cos”, ”id” (=identity)

MC sampling of manufacturing tolerances


BWR FA: Wall thickness of FA channels in different zones: corners, top, down, center (piecewise) uniform distributions; 1000 random draws




BWR FA: Center-to-Center Distance of Storage Positions: Saw-Tooth Distribution Saw-Tooth Distribution: 1000 random draws



inicalc

iniexp

xxxx

CEf

Measured isotopic concentration (PIE)

Calculated isotopic concentration

Initial isotopic concentration

Isotopic correction factor

calcinicorr xfxf1x

Corrected isotopic concentration

Benchmarks

Application Case

Isotopic Uncertainties


-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

U-235 Pu-240 Sm-149

ICF-

1

missing data

Isotopes Measurem

ent No.

Isotopic Uncertainties

Missing Data Problem


Draw with the aid of the Data Augmentation algorithm

From Depletion Code Validation: Matrix of Isotopic Correction Factors (ICFs)isotopes

benchmarks

misobs FFF ,:

Gaps: Missing Data Problem

obsF|fpf MC

Corrected isotopic concentrations for application case:

i,calcMCii,ini

MCi

MCi,corr xfxf1x MC

corrMCtol x,xk

for each considered isotope i

Isotopic Uncertainties: Data Augmentation


Each line vector of matrix is assumed to be a random observation from a log-normal distribution:

Unknown Model ParametersInformation on defined by observed data posterior distribution

if

F

Σ,FlogNflog i

Unknown vector of “true” ICFs

Unknown Covariance Matrix of(logarithmized) observed ICFs

ΣΘ ,Flog:

Θ obsFΘ |p




obsFΘ |p

p( | Fobs ) =∫p( | Fobs, Fmis ) p( Fmis | Fobs, ) p( | Fobs ) d dFmis

Complete Data Posterior

Prediction Observed Data

Posterior

Observed Data

Posterior

~ ~~Iterative Solution of Fixed Point Equation: Data Augmentation

(Tanner and Wong, The Calculation of Posterior Distributions by Data AugmentationJournal of the American Statistical Association, Vol. 82, No. 398. (Jun., 1987), pp. 528-540.)

► Due to missingness no analytic solution for




I-Step : p( Fmis | Fobs, (t-1) ) Fmis(t)

P-Step: p( | Fobs, Fmis(t) ) (t)

Convergence in distribution after sufficient number of Burn-in interations

Fmis,MC ~ p( Fmis | Fobs ) , MC ~ p( | Fobs )

Iterative Monte Carlo Sampling of missing data and model paramaters

TMC2

MC1MC ,...f,fF

Application to calculated number densities of application case




0102030405060708090

0.90 0.91 0.92 0.93 0.94 0.95E/C

Freq

uenc

y

Pu-240

0

10

20

30

40

50

60

70

80

0.90 0.92 0.94 0.96E/C

Freq

uenc

y

Pu-239

0

2

4

6

8

10

12

0.70 0.90 1.10E/C

Freq

uenc

y

U-238depletion

0

20

40

60

80

100

120

0.99 1.00 1.01 1.02 1.03E/C

Freq

uenc

y

U-235 depletion

0

10

20

30

40

50

60

70

0.98 1.00 1.02 1.04E/C

Freq

uenc

y

Pu-241




Bayesian combination of uncertaintiescovariance matrix

Σααα ,ˆNpMC

i

i,MCMCn

ˆ kk 1 Ti,MCi

i,MCMC

kˆˆ

nˆ kkkkΣ

1

1

Prior distribution of keff uncertainty: kpriorˆ,ˆN)(p Σkαkk

mean vector

1. MC sampling of nuclear data (NUDUNA):

2. keff calculations:

TMCMCMCAMCn,BMC,BMCMC ))(,(k,)(k,),(k:)(:B

αxααααkk 1

Crit. Benchmarks Appl. Case

MC draws of system parameters

3. Calculation of mean and covariance estimates:



Bayesian combination of uncertainties4. Evaluation of likelihood function of criticality benchmark measurements:

MMM ),(N)(|p Σαkαkkk

5. Bayesian updating of keff uncertainty

*M

*Mposterior ,N)(p)(|p Σkαkαkkk

Updated model parameters

keff of application case

prior posterior Impact of benchmark informationon application case keff predictiondetermined by similarity between benchmark and application case



Conclusions


Areva has the methods and tools to treat all uncertainties that appear in a criticality analysis System parameter uncertainties (materials + geometry) Isotopic uncertainties (depletion calculations) Nuclear data uncertainties (criticality + depletion) Algorithmic uncertainties (criticality + depletion)

The same mathematical framework and Monte Carlo methods can be applied to related applications, e.g. power distribution uncertainty analysis for reactor core designs

Application of the MOCADATA Monte Carlo package to ...package to Uncertainty Analysis for...

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