Science and Technology of Nuclear Installations

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Uncertainty Analysis in Reactor Physics Modeling

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Research Article | Open Access

Volume 2013 |Article ID 790206 |

O. Cabellos, "Presentation and Discussion of the UAM/Exercise I-1b: “Pin-Cell Burn-Up Benchmark” with the Hybrid Method", Science and Technology of Nuclear Installations, vol. 2013, Article ID 790206, 12 pages, 2013.

Presentation and Discussion of the UAM/Exercise I-1b: “Pin-Cell Burn-Up Benchmark” with the Hybrid Method

Academic Editor: Kostadin Ivanov
Received06 Sep 2012
Accepted10 Dec 2012
Published28 Jan 2013


The aim of this work is to present the Exercise I-1b “pin-cell burn-up benchmark” proposed in the framework of OECD LWR UAM. Its objective is to address the uncertainty due to the basic nuclear data as well as the impact of processing the nuclear and covariance data in a pin-cell depletion calculation. Four different sensitivity/uncertainty propagation methodologies participate in this benchmark (GRS, NRG, UPM, and SNU&KAERI). The paper describes the main features of the UPM model (hybrid method) compared with other methodologies. The requested output provided by UPM is presented, and it is discussed regarding the results of other methodologies.

1. Introduction to the UAM/Exercise I-1b “Pin-Cell Burn-Up Benchmark”

The general frame of the OECD LWR UAM benchmark consists of three phases with different exercises for each phase [1]. In the Phase I (“Neutronics Phase”), the Exercise 1 (I-1) “Cell Physics” is focused on the derivation of the multigroup microscopic cross-section libraries. Since the OECD LWR UAM benchmark establishes a framework for propagating cross-section uncertainties in LWR design and safety calculations, the objective of the extension of this Exercise I-1 to I-1b (cell burn-up physics) is to address the uncertainties in the depletion calculation due to the basic nuclear data as well as the impact of processing of nuclear and covariance data. The SCALE-6.0/1 covariance library [2] is the recommended source of cross-section data uncertainty. However, covariance data coming from other source of uncertainty together with evaluated nuclear data files can be used without any inconvenience.

To address this problem different sensitivity/uncertainty (S/U) tools can be used to propagate nuclear data (e.g., cross-section) uncertainties. The requested output of Exercise I-1b is criticality value, reactions rates, collapsed cross-sections and nuclide concentrations as well as their uncertainties for depletion in a PWR pin-cell model.

1.1. Specifications of the “Pin-Cell Burn-Up Benchmark”

The specification of this pin-cell benchmark is given in Tables 1 and 2 (geometry and material specifications), showing a typical configuration of a TMI-1 PWR unit cell.

Fuel temperature (K) 900.0
Cladding temperature (K) 600.0
Moderator (coolant) temperature (K) 562.0
Moderator (coolant) density (g/cm3) 0.7484
Reactor power (MWt) 2772.0
Total number of fuel assemblies in the reactor core 177
Number of fuel rods per fuel assembly 208
Active core length (mm) 3571.20

Unit cell pitch (mm) 14.427
Fuel pellet diameter (mm) 9.391
Fuel pellet material UO2
Fuel density (g/cm3) 10.283
Fuel enrichment (w/o) 4.85
Cladding outside diameter (mm)10.928
Cladding thickness (mm) 0.673
Cladding materialZircaloy-4
Cladding density (g/cm3) 6.55
Gap materialHe
Moderator material H2O

The linear fuel density (gU/cm) calculated according to values taken from Tables 1 and 2 is 6.2784 gU/cm. The average power density (W/gU) can be assumed to be equal to 33.58 W/gU. The fuel sample is burned for a unique complete cycle, the length of the burn time, and subsequent cooling time is given in Table 3. The specific power and the final cumulative burnup are also given, 61.28 GWd/MTU.

Operating cycle 1
Burn time (days) 1825.0
Final Burnup (GWd/MTU) 61.28
Downtime (days) 1870.0
Specific power (kW/kgU) 33.58

Concerning boundary conditions, the following type of boundary conditions can be used: (a) for a “cylindrical pin-cell” model, reflective boundary conditions are utilized at the center-line boundary while white boundary conditions are applicable at the peripheries of the cell model; (b) for a “square pin-cell” model, reflective boundary conditions on all surfaces are applied. For depletion, it can be considered an infinite burn-up spectrum mode.

1.2. Requested Output of the “Pin-Cell Burn-Up Benchmark”

Results and associated uncertainties are provided at eight burn-up steps: 0, 10, 20, 30, 40, 50, 60, and shutdown (61.28) GWd/MTU. And, six additional decay steps are required at 1, 3, 5, 10, 50, and 100 years of cooling time. The requested output can be summarized in the following three sets of information:(i)criticality values: Kinf and nuclide reactions that contribute the most to the uncertainty in kinf;(ii)reaction rates and collapsed macroscopic cross-sections:(a)Reaction rates (capture and fission) and uncertainties for major isotopes: 235,238U and 239,240,241Pu;(b)Two-group macroscopic cross-sections, fast and thermal, and associated uncertainties for the homogenized pin cell: absorption, fission, nu-fission, and diffusion coefficient. The thermal energy cutoff is 0.625 eV. (iii)Number densities:(a)actinides (15): 233,234,235,236,238U; 237Np; 238,239,240,241,242Pu; 241,243Am; 244,246Cm;(b)fission products (36): 95Mo; 99Tc; 101,106Ru; 103Rh; 109Ag; 133,134,135,137 Cs; 139La; 140,142,144Ce; 142,143,145,146,148,150Nd; 147,148,149,150,151,152,154Sm; 151,153,154,155Eu; 154,155,156,158,160Gd.

2. Summary of Propagation Uncertainty Methodologies in Burn-Up Calculations

The first phase of participation in this exercise was completed in April 2012 with a total of 4 participants: GRS, NRG, UPM, and SNU&KAERI. Table 4 summarizes the main calculation methodologies and nuclear data libraries and their uncertainties. The results were presented at the Sixth Workshop (UAM-6) of OECD Benchmark for Uncertainty Analysis in Best-Estimate Modelling (UAM).

Methodology/codes (Nuclear data library)Covariance data libraryCriticality calculation methodologyUncertainty methodology

(44 groups)
2-D-SNSampling multigroups libraries based on uncertainties in nuclear data libraries
(The Netherlands)
XS and FYs
Monte CarloTotal Monte Carlo
(South Korea)
Monte CarloMonte Carlo uncertainty propagation
Hybrid method (TSUNAMI)
FY & Decay data
2-D-SNHybrid method:
TSUNAMI and sampling in inventory calculations

On one hand, depletion calculations are performed by GRS and UPM with SCALE6 code system [3], while NRG uses SERPENT code [4] and SNU&KAERI participates in the benchmark with its own McCARD code [5], both Monte Carlo codes. On the other side, for uncertainty calculations, GRS and NRG use Monte Carlo techniques, GRS with a sampling methodology (XSUSA [6]) of multigroup cross-section libraries provided in SCALE6 format and NRG using the technique of Total Monte Carlo [7] with TENDL2011. UPM applies a hybrid method [8] based on determining the sensitivity coefficients with TSUNAMI code [9] and performing a Monte Carlo sampling to determine the uncertainty of the number densities; these uncertainties are computed with ACAB code [10]. McCARD code makes use of the technique of Adjoint Weighted Perturbation (AWP) method to predict the sensitivity coefficients.

Regarding cross-section covariance data, GRS, SNU&KAERI, and UPM use SCALE6/COVA-44 groups. In addition, SNU&KAERI provides results with uncertainties coming from JENDL3.3 and ENDF/B-VII.0. Figure 1 shows an example of cross-section covariance data taken from SCALE6.1/COVA-44G. In this figure, the original 235U COVERX/SCALE6.1 file is processed with ANGELO, LAMBDA, and NJOY codes to visualize the correlation matrix. NRG uses TENDL2011 and their uncertainty for cross-section data libraries. In addition, NRG and UPM have carried out some calculations with the uncertainty provided in Fission Yields (TENDL2011, JEFF-3.1.1) and Decay Data (JEFF-3.1.1) libraries.

Next, the main characteristics of the uncertainty propagation methodologies used in this Benchmark are summarized, and the uncertainty propagation in number density is used as an example in the following Figures 2, 3, and 6. Figure 2 shows the calculation scheme of the Monte Carlo methodologies. NRG uses for each sampling a different nuclear data library TENDL2011; the generation of this library is done using the TASMAN code [7]. TASMAN is a computer code for the production of covariance data using results of the nuclear model code TALYS, and for automatic optimization of the TALYS results with respect to experimental data. It is assumed that each nuclear model (i.e., TALYS input) parameter has its own uncertainty; running TALYS many times, it provides a sampling of ENDF files or a single file with full covariance information. GRS will generate a set of multigroup libraries in SCALE6 format; this sampling is done with the SCALE6.1/44-groups covariance library using XSUSA code. The sensitivity/uncertainty procedure is based on a first order Taylor series approach. So, the number density can be written as where .

We can define the sensitivity coefficients as , and is the error in the 1-group effective cross-sections. This 1-group error depends explicitly on the uncertainty of cross-sections, and implicitly on the neutron-flux uncertainty, Here, is the error due to nuclear data and is the error due to neutron-flux. The variance in the number density can be obtained using the sandwich formula: The first term propagates the multigroup cross-section uncertainty with no uncertainty in the neutron flux. And, the second term propagates the effect of this uncertainty with the uncertainty in the neutron flux.

If the uncertainty in the neutron flux can be considered negligible, a simple scheme of S/U can be illustrated in Figure 3. In this case, TRITON code [3] is run to determine the number densities at different burn-up steps, as a reference or nominal calculation without uncertainties. And, the number densities calculated in the nominal case are used to generate TSUNAMI [9] inputs at each burn-up step. With TSUNAMI code, S/U analysis can be provided for criticality , two-group cross-sections and reaction rates . However, number density sensitivities are not calculated with TSUNAMI code.

Once, the sensitivity coefficients are calculated by TSUNAMI code, the criticality uncertainty analysis based on “nuclear data uncertainties” can be formulated as follows: it is explicitly dependent on the nuclear data (e.g., cross-sections, nu-bar, …) and implicitly dependent on the number density which characterizes the system: is the sensitivity coefficient explicitly of cross-sections and is the sensitivity coefficient of number density, ; both are calculated by TSUNAMI code. Figures 4 and 5 show the -eff integrated sensitivity coefficients for cross-section and number density at each burn-up step. In Figure 4, the evolution of shows the importance of 239Pu at high burnups, mainly for nu-bar nuclear reaction. For 238U, and reactions are the most important for all burnup. For 235U, sensitivity decreases with burn-up, being nu-bar with the highest value. Evolution of 135Xe is also shown. Some “fission-gamma” cross-correlations for 239Pu and 235U are also illustrated. Figure 5 shows the integrated sensitivities, , for the most important isotopes related with criticality: 239,240, 241Pu, 235,238U. Also, some important fission products are shown: 135Xe and 103Rh.

is the covariance cross-section data taken from SCALE6.1/COVA, and is the covariance number densities predicted by ACAB code. It can be calculated with the uncertainty due to cross-section, fission yield and/or decay data.  Our ACAB code is used to propagate nuclear data uncertainty (cross-section, fission yield, and decay data) in the prediction of number density uncertainty: ACAB accounts for the impact of nuclear data uncertainty as follows (see Figure 6). (i) In a first step, a coupled neutron-depletion calculation (without uncertainties) is carried out only once, taken the best-estimated values for all the parameters involved in the problem. (ii) In a second step, ACAB performs a simultaneous random sampling of the probability density functions (PDF) of all these variables: cross-section, fission yield, and decay data. Then, ACAB computes the isotopic concentrations at the end of each burn step, taking the fluxes halfway through each burn step determined in the best-estimated calculation. Then, only the depletion calculations are repeated or run many times. A statistical analysis of the results allows assessing the uncertainty in the calculated number density and determining . Table 5 shows an example of this type of information.



3. Results with the Hybrid Method

In Table 6, and their associated uncertainty for PWR unit-cell are summarized at four different burnups. The five most important nuclide reactions that contribute to uncertainty are identified: (i) for fresh fuel, , and , , , and (ii) for high burnup: and , , and . In addition, the contribution of number density uncertainty, var, is evaluated, being the cross-sections and fission yields the most important contributions, and it can be concluded that the contribution of decay data uncertainty is negligible.

Mean eff
Rel. std. dev. (%)Mean eff 1.25 Rel. std. dev. (%)Mean eff 1.08rel. std. dev. (%)Mean eff 0.90 Rel. std. dev. (%)

U238 ( , )0.28U238 ( , )0.26Pu239 (nubar)0.39Pu239 (nubar)0.54
U235 (nubar)0.26Pu239 (nubar)0.20U238 ( , )0.26U238 ( , )0.26
U235 ( , )0.21U235 (nubar)0.20U238 ( , )0.19U238 ( , )0.25
U238 ( , )0.12U235 ( , )0.15Pu239 (fis)0.13Pu239 (fis)0.21
U235 ( , -fiss)0.10U238 ( , )0.15U235 (nubar)0.13Pu239 ( , -fiss)0.18

Total var( )0.490.510.630.79

var( )_XS/SCALE0.
var( )_FYs0.
var( )_Decay0.

Total var( )0.490.580.710.89

Table 7 shows the uncertainty of two-group cross-sections: , and (subscript 1 refers to fast group and subscript 2 to the thermal group). The low contribution of the uncertainty due to number density uncertainty except for thermal groups can be seen. The total uncertainty is about 1%, and the contribution due to the uncertainty in fission yields is negligible.


abs-1Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

abs-2Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

fis-1Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

fis-2Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

fis-1Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

fis-2Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

diff-1Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

diff-2Mean value

Relative uncertainty (%)var( )
var( )_ XS0.
var( )_ FY0.

Total Uncertainty0.

As an example of integrated sensitivities of macroscopic two-group cross-sections, Figures 7 and 8 show these values for . 238U is the most important contributor with the and reactions.

Table 8 shows the uncertainty for the following capture and fission reaction rates: 235,238U and 239,240,241Pu. The total uncertainty is in the range of 1%–3%. In general, the uncertainty contribution due to the uncertainty in the number density is below the contribution due to cross-section , except for 240Pu and 241Pu reaction rates where this contribution is larger.


235U-capMean value

Relative uncertainty (%)var( )
var( )_ XS0.

235U-fisMean value

Relative uncertainty (%)var( )
var( )_ XS0.

238U-capMean value

Relative uncertainty (%)var( )
var( )_ XS0.

238U-fisMean value

Relative uncertainty (%)var( )
var( )_ XS0.

239Pu-capMean value

Relative uncertainty (%)var( )
var( )_ XS0.90.70.7

239Pu-fissMean value

Relative uncertainty (%)var( )
var( )_ XS0.90.70.7

240Pu-capMean value

Relative uncertainty (%)var( )
var( )_ XS2.41.31.3

240Pu-fissMean value

Relative uncertainty (%)var( )
var( )_ XS3.12.22.1

241Pu-capMean value

Relative uncertainty (%)var( )
var( )_ XS2.01.71.5

241Pu-fissMean value

Relative uncertainty (%)var( )
var( )_ XS2.91.71.5

In Table 9, it can be seen that the number density uncertainty for some major and minor actinides due to cross-section data remains below 3%. Larger uncertainties are predicted for minor actinides (e.g., 246Cm) and the uncertainty throughout irradiation period rises. And, it can be concluded that the uncertainty due to decay data uncertainty is negligible.

MeanMean Rel. std. dev. (%)Mean Rel. std. dev. (%)Mean Rel. std. dev. (%)

233U 3.10.0 2.10.0 2.10.0
234U 1.00.0 1.90.0 3.10.0
235U 0.20.0 0.30.0 0.60.0
236U 0.80.0 0.50.0 0.40.0
238U 0.00.0 0.10.0 0.10.0
237Np 1.20.0 0.80.0 0.80.0
238Pu 2.30.0 1.40.0 0.90.0
239Pu 1.20.0 1.10.0 1.30.0
240Pu 3.10.0 2.10.0 1.90.0
241Pu 2.90.0 1.70.0 1.50.0
242Pu 3.70.0 1.90.0 1.40.0
241Am 3.30.1 1.90.1 1.80.1
243Am 5.90.0 3.00.0 1.90.0
244Cm 6.90.0 3.40.0 2.10.0

In Table 10, the uncertainty in the number of fission products due to cross-sections, decay, and fission yields data has been predicted. Some isotopes, 155Gd, 154,155Eu, and 149Sm show a relative error above 10%, being the high uncertainty in cross-section data, the reason of this large uncertainty. In general, the uncertainty due to fission yields remain below 3%, except for 95Mo with 4.5% (with high sensitivity to 95Zr fission yield) and 149Sm with 4.7% (with high sensitivity to 149Pm fission yield) [11]. For decay data uncertainties, the isotope 151Eu reaches a maximum uncertainty of 3.2% as a consequence of the 6.7% relative error in the half-life of 151Sm.

MeanMean Rel. std. dev. (%)Mean Rel. std. dev. (%)Mean Rel. std. dev. (%)