Specific objectives: Contribute with updated results to Ex I-3, computing uncertainties in k-eff, radial power distributions and peak factors; Compare nodal and pin-by-pin results; Evaluate the probability distribution of core parameters

Castro González, Emilio

Cuervo Gómez, Diana

García Herranz, Nuria

Sánchez-Cervera Huerta, Santiago

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017
Results for Exercise I-3 with COBAYA:
Analysis of the peaking power factors
E. Castro, S. Sánchez-Cervera, N. García-Herranz, D. Cuervo
emilio.castro@upm.es
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017
• Background
• Objectives
• Methodology
• Results of keff
• Results of power distributions
• Results of peaking power factors
• Conclusions
2
Outline
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 3
Background
• UPM methodology for Uncertainty Quantification is based in two tools:
• SCALE 6.2.1: NEWT and Sampler.
• COBAYA: core simulator with nodal and pin-by-pin capabilities
• Random few-group libraries for COBAYA can be generated in an 
automatic way using SCALE by stochastic sampling of nuclear data with 
Sampler(NEWT) → presented in UAM-9
• UQ capabilities in COBAYA → presented in UAM-9
• 1st. Order PT (full covariance matrix required)
• Sampled-based (set of random few-group libraries required)
• Preliminary results of Exercise I-3 → presented in UAM-10
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 4
Objectives
• Exercise I-3: UQ in TMI-1 full core calculations using 
SCALE(NEWT)/COBAYA at both nodal and pin-by-pin levels
• Specific objectives:
• Contribute with updated results to Ex I-3, computing uncertainties in k-
eff, radial power distributions and peak factors.
• Compare nodal and pin-by-pin results.
• Evaluate the probability distribution of core parameters.
• TMI-1 ARI core definition: FA Enrichment(w/o) BP(%) Gd (pins)
1 4.00 - -
2 4.95 3.5 4
3 5.00 - 4
4 4.40 - -
5 5.00 3.5 4
6 4.85 - 4
7 4.95 - 4
8 5.00 - 8
9 4.95 - 8
10 4.95 3.5 -
11 5.00 - -
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 5
Methodology
SAMPLER / NEWT FA 1
SAMPLER / NEWT FA 2
...
SAMPLER / NEWT FA 27
Perturbed 
NEMTAB cross 
section libraries
Perturbed 
COBAYA 
simulations
Perturbed core 
results
NEWT2NEMTAB 
tool
A sampling-based approach using SCALE and COBAYA was used to 
propagate uncertainties in nuclear data to core parameters.
Advantages:
• Very easy to implement.
• Any kind of observable can be analyzed.
• Does not involve linear approximations.
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 6
Methodology
SAMPLER / NEWT FA 1
SAMPLER / NEWT FA 2
...
SAMPLER / NEWT FA 27
Perturbed 
NEMTAB cross 
section libraries
Perturbed 
COBAYA 
simulations
Perturbed core 
results
NEWT2NEMTAB 
tool
NEWT (SCALE 6.2.1) was used as lattice code.
• SAMPLER was used to produce perturbed NEWT inputs.
• 56g-v7.1 cross sections and covariance library.
• 11 unrodded FA, 7 rodded FA, 1 reflector.
• 900 random samples for each fuel assembly type:
• 900 * 19 lattice calculations.
• Nodal homogenization.
• Pin-wise homogenization.
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 7
Methodology
SAMPLER / NEWT FA 1
SAMPLER / NEWT FA 2
...
SAMPLER / NEWT FA 27
Perturbed 
NEMTAB cross 
section libraries
Perturbed 
COBAYA 
simulations
Perturbed core 
results
NEWT2NEMTAB 
tool
NEWT outputs are grouped together in NEMTAB-formatted libraries:
• NEWT2NEMTAB tool.
• Parametrized libraries as a function of coolant density, coolant 
temperature, fuel temperature and boron concentration.
• Nodal library contains 19 materials.
• Pin-by-pin library contains 1153 materials.
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 8
Methodology
SAMPLER / NEWT FA 1
SAMPLER / NEWT FA 2
...
SAMPLER / NEWT FA 27
Perturbed 
NEMTAB cross 
section libraries
Perturbed 
COBAYA 
simulations
Perturbed core 
results
NEWT2NEMTAB 
tool
COBAYA core simulator:
• Solves the multigroup neutron diffusion equation corrected by 
interface discontinuity factors.
• Nodal solver: Analytic Coarse-Mesh Finit-Difference (ACMFD).
• Pin-by-pin solver: Fine-Mesh Finite-Difference (FMFD).
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 9
Methodology
SAMPLER / NEWT FA 1
SAMPLER / NEWT FA 2
...
SAMPLER / NEWT FA 27
Perturbed 
NEMTAB cross 
section libraries
Perturbed 
COBAYA 
simulations
Perturbed core 
results
NEWT2NEMTAB 
tool
A statistical analysis of the 900 core results is performed:
• Mean values.
• Standard deviations.
• Moments of higher order.
• Histograms.
• Analysis of convergence.
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 10
Results of keff
keff ∆k/k(%)
Nodal 1.00373 0.515
Pin-by-pin 1.00350 0.515
Good agreement between the two solvers.
Results normally distributed.
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 11
Results of keff
With 200 samples: 62 pcm
difference between the mean 
and the unperturbed values
Unperturbed
(nominal) 
value in blue Final relative SD 
value in blue
Final skewness
value in blue
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 12
Results of power distributions
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 13
Results of peaking power factors
Nodal Pin-by-pin
F∆H node/pin 1.71 (2.1%) 1.91 (2.8%)
F∆H assembly 1.64 (0.4%) 1.60 (1.1%)
Fxy node/pin 2.11 (0.5%) 2.46 (0.7%)
Fxy assembly 2.05 (0.5%) 2.12 (0.5%)
Fq node/pin 2.56 (2.3%) 2.74 (2.7%)
Fq assembly 2.45 (0.5%) 2.30 (1.1%)
• Assembly averaged results can be used to compare both solvers.
• Mean values are consistent between solvers.
• Standard deviations show large differences.
• Are they normally distributed?
𝐹𝐹∆𝐻𝐻 = 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚�𝑃𝑃
𝐹𝐹𝑚𝑚𝑥𝑥 = 𝑚𝑚𝑚𝑚𝑚𝑚 𝑃𝑃𝐷𝐷(𝑧𝑧)𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝐷𝐷(𝑧𝑧)
𝐹𝐹𝑞𝑞 = 𝑃𝑃𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚𝑃𝑃𝐷𝐷
Relative standard deviation in brackets.
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 14
Results of peaking power factors
Pin-by-pin solver yields non-normal peaking factors, in contrast to the nodal solution.
Peaking factors uncertainty seems very sensitive to spatial meshing.
Assembly-averaged peaking factors
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017 15
Results of peaking power factors
Pin peaking factors
Again, pin peaking power factors show non-normal distributions.
This implies that providing a mean value and a standard deviation is not enough to 
properly describe the uncertainty.
What is the reason for these behaviors?
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017
Results of peaking power factors
Hottest fuel assembly position in PBP simulations:
• Position A: 77 % of the random samples.
• Position B: 23 % of the random samples.
Focusing on the assembly-averaged F∆H obtained using the pin-by-pin solver.
F∆H = Normalized power of the hottest fuel assembly.
Power of assemblies in positions A and B follow a normal distribution, but they 
overlap, causing the right tail in the peaking factor distribution.
16
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017
• SCALE/COBAYA is a suitable tool for UQ at both nodal and pin levels.
It has been applied to Exercise I-3 / TMI-1
• Uncertainties in keff
• Consistent values between nodal and pin-by-pin simulations (∆k ≈ 500 pcm)
• Normally distributed results -> confirms that the mean value and the standard
deviation permit establishing confidence intervals
• A large number of simulations mandatory to obtain a mean value in perfect
agreement with the nominal value.
• Uncertainties in peaking power factors
• The spatial mesh impacts the assembly-averaged peaking factors.
• Values computed with the pin solver (both pin- and assembly-averaged) do not
follow a normal distribution -> mean value and the standard deviation are not
sufficient and their PDF are mandatory to construct the confidence intervals required
for safety analysis.
• This behavior highlights the need of performing full core calculations
at the pin-level to get reliable estimates of the uncertainties in
peaking factors.
• It would be of interest the comparison to the uncertainty predictions provided by
nodal solvers with pin-power reconstruction methods.
Conclusions
17
UAM-11, AREVA (Erlangen, Germany) May 10-12, 2017
Thank you for your attention!
This work was supported by the 
Spanish Nuclear Safety Council 
(CSN)


Results for Exercise I-3 with COBAYA: Analysis of the peaking power factors

http://oa.upm.es/49833/1/INVE_MEM_2017_256569.pdf

Results for Exercise I-3 with COBAYA: Analysis of the peaking power factors

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