application of the mocadata monte carlo package to
TRANSCRIPT
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
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Performing κ criticality calculations
keff values (keff)i, i =1,…,κ, distribution of keff
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.5
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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
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.7
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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
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.8
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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
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.9
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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.
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missing data
ICF-
1
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.10
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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:
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.11
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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
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.12
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Isotopic Uncertainties: Data Augmentation
► 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
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.13
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Isotopic Uncertainties: Data Augmentation
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
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.14
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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
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.15
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Isotopic Uncertainties: Data Augmentation
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.16
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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
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.17
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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
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.18
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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
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.19