Application of the MOCADATA Monte Carlo package to

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

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

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

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Manufacturing Tolerances (materials, dimensions)

Nuclear data uncertainties

Uncertainties in Criticality Calculations

Isotopic Uncertainty of 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

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MC Sampling Procedure Tolerances and isotopic uncertainties → distribution of random vector T xiso , xtol p x iso p xtol x T T p x

Neutron multiplication factor becomes random number

Distribution only accessible via Monte Carlo k kx pk

Monte Carlo Procedure

Repeatedly draw random samples

from

px xMC For each

calculate

with criticality code xMC k xMC Order Statistic of Monte Carlo data → upper 95%/95% tolerance limit k95 / 95

Upper 95%/95% 95-th percentile of p(k) Maximum allowable keff tolerance limit ?

Pk95 / 95 f 95 : 0.95 k95 / 95 klim it 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

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MC Sampling Procedure x3

Method for Monte Carlo (MC) sampling on the parameter region x2

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

x1


Performing κ criticality calculations

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


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MC sampling of manufacturing tolerances 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)...)) fi = (sum of all numerator terms)/(sum of all denominator terms)

k e _ i func_i=”abs”, ”exp”, ”log”, ”sin”, ”cos”, term c func_i(xi ) ”id” (=identity)

i1 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.6

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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


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MC sampling of manufacturing tolerances

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


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Isotopic UncertaintiesMeasured isotopic

Initial isotopicconcentration (PIE)

inicalc

iniexp

xx xx

C E

f

concentration

Calculated isotopic Isotopic correction factor concentration

Corrected isotopic concentration

Benchmarks

calcinicorr xfx1x f Application Case


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Isotopic Uncertainties

Missing Data Problem

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

Isotopes Measurem

ent No.


missing data

ICF-

1


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Isotopic Uncertainties: Data Augmentation From Depletion Code Validation: Matrix of Isotopic Correction Factors (ICFs)

Gaps: Missing Data Problem

Fobs,Fmis f MC pf | Fobs

Data Augmentation algorithm

Corrected isotopic concentrations for application case: MC MC MC MC MCxcorr,i 1 fi xini,i fi xcalc,i kxtol , xcorr

for each considered isotope i

Draw with the aid of the

isotopes

benchmarks

F:


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Isotopic Uncertainties: Data Augmentation

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

log fi NlogF ,Σ

Unknown vector Unknown Covariance Matrix of of “true” ICFs (logarithmized) observed ICFs

Unknown Model Parameters Θ : logF ,Σ

Information on Θ defined by observed data posterior distribution pΘ | Fobs


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► Due to missingness no analytic solution for pΘ | Fobs

Iterative Solution of Fixed Point Equation: Data Augmentation (Tanner and Wong, The Calculation of Posterior Distributions by Data Augmentation

Journal of the American Statistical Association, Vol. 82, No. 398. (Jun., 1987), pp. 528-540.)

~ ~ p( | Fobs ) =∫ p( | Fobs, Fmis ) p( Fmis

~

| Fobs, ) p( | Fobs ) d dFmis

Observed Complete Data Prediction Observed Data Posterior Data

Posterior Posterior


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Iterative Monte Carlo Sampling of missing data and model paramaters

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

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

Convergence in distribution after sufficient number of Burn-in interations

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

MC MC Application to calculated number FMC f1 , f2 ,...T

densities of application case


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120

100

Freq

uenc

y

Isotopic Uncertainties: Data Augmentation70

U-235 depletion

80

Freq

uenc

y

Freq

uenc

y

Pu-241

20 20

Pu-239

1010

000 0.99 1.00 1.01 1.02 1.03 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04

E/C E/C E/C 12 90

6070

5060 80 50 40 60 40 30

3040 20

0

2

4

6

8

10 U-238 depletion

0.70 0.90 1.10 E/C

0

30 40 50 60 70 80

Pu-240

0.90 0.91 0.92 0.93 0.94 0.95 E/C

Freq

uenc

y

Freq

uenc

y

20 10


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Bayesian combination of uncertainties mean vector covariance matrix

1. MC sampling of nuclear data (NUDUNA): αMC pα Nα̂,Σ

2. keff calculations: MC draws of system parameters

k : k(α ) : k (α ),,k (α ) , k (α ,x (α ))T MC MC B,1 MC B,nB MC A MC MC MC

Crit. Benchmarks Appl. Case

3. Calculation of mean and covariance estimates: 1 1 T

k̂ kMC,i Σ̂k 1 kMC,i k̂ kMC,i k̂

nMC i nMC i

Prior distribution of keff uncertainty: k pk(α) Nk̂,Σ̂k prior


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prior posterior

Bayesian combination of uncertainties 4. Evaluation of likelihood function of criticality benchmark measurements:

kM pkM | k(α) Nk(α),ΣM

5. Bayesian updating of keff uncertainty Updated model parameters

* *k pk | k(α) pk(α) Nk ,Σ posterior M M

keff of application case Impact of benchmark information on application case keff prediction determined by similarity between benchmark and application case


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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


Documents

Application of the MOCADATA Monte Carlo package to