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

e 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). e paper describes the main features of the UPM model (hybrid method) compared with other methodologies. e requested output provided by UPM is presented, and it is discussed regarding the results of other methodologies.


Introduction to the UAM/Exercise I-1b
"Pin-Cell Burn-Up Benchmark" e 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.e SCALE-6.0/1covariance 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 �les 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., crosssection) uncertainties.e 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.Speci�cations of the "Pin-Cell Burn-Up Benchmark".e speci�cation of this pin-cell benchmark is given in Tables 1 and 2 (geometry and material speci�cations), showing a typical con�guration of a TMI-1 PWR unit cell.
e linear fuel density (gU/cm) calculated according to values taken from Tables 1 and 2 is 6.2784 gU/cm.e average power density (W/gU) can be assumed to be equal to 33.58 W/gU.e fuel sample is burned for a unique complete cycle, the length of the burn time, and subsequent cooling time is given in Table 3. e speci�c power and the �nal cumulative burnup are also given, 61.28 GWd/MTU.
Concerning boundary conditions, the following type of boundary conditions can be used: (a) for a "cylindrical pincell" model, re�ective boundary conditions are utili�ed 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, re�ective boundary conditions on all surfaces are applied.For depletion, it can be considered an in�nite burn-up spectrum mode.
Fuel temperature (K) 900.0 Cladding temperature (K) 600.0 Moderator (coolant) temperature (K) 562.0 Moderator (coolant) density (g/cm 3  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 crosssections: (a) Reaction rates (capture and �ssion) and uncertainties for major isotopes:  142,143,145,146,148,150 Nd; 147,148,149,150,151,152,154 Sm; 151,153,154,155 Eu; 154,155,156,158,160 Gd.

Summary of Propagation Uncertainty
Methodologies in Burn-Up Calculations e �rst 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.e results were presented at the Sixth Workshop (UAM-6) of OECD Benchmark for Uncertainty Analysis in Best-Estimate Modelling (UAM).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 crosssection 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 �gure, the original 235 U COVERX/SCALE6.1 �le 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.
(1) 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 �les or a single �le with full covariance information.��S will generate a set of multigroup libraries in SCALE6 format; this sampling is done with the SCALE6.1/44groupscovariance library using XSUSA code.
(2) e sensitivity/uncertainty procedure is based on a �rst order Taylor series approach.So, the number density can be written as where  eff  = ∑       .�e can de�ne the sensitivity coe�cients as   = [  /  ]   eff , and   = ( eff  −  eff  ) is the error in the 1-group effective cross-sections.is 1-group error depends explicitly on the uncertainty of cross-sections, and implicitly on the neutron-�ux uncertainty, Here,    is the error due to nuclear data and   is the error due to neutron-�ux.e variance in the number density can be obtained using the sandwich formula: e �rst term propagates the multigroup cross-section uncertainty with no uncertainty in the neutron �ux.And, the second term propagates the effect of this uncertainty with the uncertainty in the neutron �ux.
If the uncertainty in the neutron �ux 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 burnup 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 (  , twogroup cross-sections (  abs1    abs1   and reaction rates (RR U235cap  RR U235cap  .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:  eff it is explicitly dependent on the nuclear data (e.g., crosssections, nu-bar, ) and implicitly dependent on the number density which characterizes the system: is the sensitivity coefficient explicitly of cross-sections (ΔΔ and   is the sensitivity coefficient of number density, (ΔΔ; 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 239 Pu at high burnups, mainly for nu-bar nuclear reaction.For 238 U, (  and (  ′  reactions are the most important for all burnup.For 235 U, sensitivity decreases with burn-up, being nu-bar with the highest value.Evolution of 135 Xe(  is also shown.Some "�ssion-gamma" cross-correlations for 239 Pu and 235 U are also illustrated.Figure 5 shows the integrated sensitivities,   , for the most important isotopes related with criticality: 239240241 Pu, 235238 U. Also, some important �ssion products are shown: 135 Xe and 103 Rh.   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, �ssion yield and/or decay data.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.

Results with the Hybrid Method
In Table 6,  eff and their associated uncertainty for PWR unitcell are summari�ed at four different burnups.e �ve most important nuclide reactions that contribute to uncertainty are identi�ed: (i) for fresh fuel, U 238 ( , U 235 ( and ( , U 238 (  ′ , U 235 ( �ss-, and (ii) for high burnup: Pu 239 ( U 238 (  and (  ′ , Pu 239 ( �ss, and ( �ss-.In addition, the contribution of number density uncertainty, var(  , is evaluated, being the cross-sections and �ssion yields the most important contributions, and it can be concluded that the contribution of decay data uncertainty is negligible. Table 7 shows the uncertainty of two-group crosssections: ∑ abs-1  ∑ abs-2  ∑ �s-1  ∑ �s-2   ∑ �s-1   ∑ �s-2  diff 1 , and diff 2 (subscript 1 refers to fast group and subscript 2 to the thermal group).e low contribution of the uncertainty due to number density uncertainty except for thermal groups can be seen.e total uncertainty is about 1%, and the contribution due to the uncertainty in �ssion yields is negligible.
As an example of integrated sensitivities of macroscopic two-group cross-sections, Figures 7 and 8 show these values for ∑ abs-1 . 238U is the most important contributor with the (  ′  and (  reactions.
Table 8 shows the uncertainty for the following capture and �ssion reaction rates: 235238 U and 239240241 Pu. e total uncertainty is in the range of 1%-3%.In general, the uncertainty contribution due to the uncertainty in the number density (var(RR   is below the contribution due to cross-section (var(RR  , except for 240 Pu and 241 Pu reaction rates where this contribution is larger. In Table 9, it can be seen that the number density uncertainty for some major and minor actinides due to crosssection data remains below 3%.Larger uncertainties are predicted for minor actinides (e.g., 246 Cm) and the uncertainty throughout irradiation period rises.And, it can be concluded that the uncertainty due to decay data uncertainty is negligible.
In Table 10, the uncertainty in the number of �ssion products due to cross-sections, decay, and �ssion yields data has been predicted.Some isotopes, 155 Gd, 154155 Eu, and 149 Sm 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 �ssion yields remain below 3%, except for 95 Mo with 4.5% (with high sensitivity to 95 Zr �ssion yield) and 149 Sm with 4.7% (with high sensitivity to 149 Pm �ssion yield) �11].For decay data uncertainties, the isotope 151 Eu reaches a maximum uncertainty of 3.2% as a consequence of the 6.7% relative error in the half-life of 151 Sm.

Conclusions and Comparison with Other Methodologies
ere has been a very small contribution of participants in the pin-cell burn-up benchmark, Exercise I-1b, with only (2) e importance of different source of cross-section uncertainty has been evaluated by SNU&KAERI.us, for fresh fuel the  eff relative uncertainty is 0.79% or 0.30%, with uncertainty cross-section data ENDF/B.VII.1 or JENDL/-3.3,respectively.
(3) Comparing results between UPM and GRS (using both institutions similar uncertainty data and codes), it can be concluded that the linear approximation used by UPM neglects the possible correlation between the prediction of number densities and neutron transport calculation.At high burn-up the lower uncertainty in  eff predicted by GRS (0.75%) with respect to UPM (0.89%) shows a possible negative correlation between these terms.var(Σ  ) 0.9 0.9 0.9 0.9 var(Σ  )_ ΔXS 0.0 0.1 0.1 0.1 var(Σ  )_ ΔFY 0.0 0.0 0.0 0.0 Total Uncertainty 0.9 0.9 0.9 0.9 Relative uncertainty (%) var(Σ  ) 0.9 0.9 0.9 0.9 var(Σ  )_ ΔXS 0.0 0.0 0.0 0.0 var(Σ  )_ ΔFY 0.0 0.0 0.0 0.0 Total Uncertainty 0.9 0.9 0.9 0.9 (5) For two group cross-sections and reaction rates, the relative uncertainty is in the range of 1-2%.And comparing GRS between UPM, a positive correlation in two-group cross-sections between the prediction of number densities and neutron transport calculation due to the highest uncertainty values predicted by GRS is shown.For reaction rates, a negative correlation is found.NRG predicts larger uncertainties above 2% because of the TENDL2011 library.
(6) e uncertainty in the number density of major isotopes ( 35 U, 39 Pu,…) is in the range of 1-3% increasing with burnup; higher uncertainty is predicted by GRS/XSUSA (3.5% for 4 Pu).For minor actinides, the highest uncertainty value is for 46 Cm T 8: Uncertainty in reaction rates.Cross-section uncertainties are taken from SCALE6.�e contribution of �ssion yield data uncertainty is also studied by UPM; the predicted number density uncertainty is in the range of 1-4%, with a maximum value for  Mo with 4.8%.And, for uncertainty crosssection, the largest uncertainties found by UPM are  Gd (15.4%),  Eu (18.6%), and  Sm (15.5%).For these isotopes, GRS predicts lower uncertainty:  Gd (5.3%),  Eu (5.5%), and  Sm (2.5%).
(8) In the case of  Gd (generated by -decay of  Eu), it shows higher sensitivities to 3, Eu(,  reaction It is expected that new contributions for this benchmark will supply additional information to de�ne the output range of uncertainty of this Exercise I-1b.And, as complete covariance data in ENDF/B-VII.1, JEFF-3.2, and JENDL-4.xbecome available, exercise I-1b can be performed as originally designed and results compared with the SCALE6/44-GROUPS library supplying additional valuable information. Finally, a general recommendation of this work should be the de�nition of input uncertainties for the following UAM Exercises.In particular one of the next steps in the roadmap of OECD LWR UAM benchmark is Phase II ("Core Phase") and in particular is the "Exercise II-2: Time-Dependent Neutronics": where neutron kinetics and fuel depletion stand-alone performance will be assessed.From the point of view of burn-up calculations, it can be considered a long-term time phenomena described by fuel assembly depletion performance (used for core design and fuel management).e objective of this Exercise II-2 will be to determine the uncertainty in predicting the relative power over time of a core aer a short-term reactivity change as well as during longer-term depletion cases.
T 7: Uncertainty in two-group cross-section data.Cross-section uncertainties are taken from SCALE6.1/44-GROUP.
T 10: Uncertainty in number density of some important �ssion products.Cross-section uncertainties are ta�en from SCALE6.1/44-GROUP (ΔXS).Fission yields (ΔFYs) and decay data (ΔDD) source of uncertainty from JEFF-3.1.1.For number density of �ssion products, NRG predicts larger uncertainty values than UPM and GRS, with a maximum uncertainty in  Sm of 31.7%.For this isotope, GRS and UPM predict an uncertainty approximately 2%.So, the in�uence of TENDL2011 in the prediction of �ssion products is �uite large.Decay data uncertainty is analyzed by UPM showing only an important uncertainty of 3.3% in  Eu.