A comparison of bounding approach with isotopic correction factors and Monte Carlo sampling in burnup credit method

One of the methodologies used in criticality safety analysis is burnup credit method, which allows considering fuel burnup in models with spent fuel. This removes excessive conservatism from the analysis, but it also brings new uncertainties originating from computational prediction of spent fuel composition. The burnup credit method offer several possibilities of how to deal with this problem, e.g. using bounding approach with correction factors on nuclide concentrations, which is simple, but still very conservative approach. Another option is Monte Carlo sampling, which aims at receiving the most realistic result as possible, but is very computationally demanding. In this work, we have analyzed correction factors for selected nuclides and compared the results of both methods on model of spent fuel storage pool. The results show how much conservative the bounding approach is – in this case, the multiplication factor was higher by almost 0.03 than in Monte Carlo sampling, exceeding the standard deviation by more than 5.4 times.


I. INTRODUCTION
HEN evaluating the nuclear safety of a system, a criticality analysis is performed in order to determine whether the system meets the requirements of the local regulations. The regulation requirements could be summarized in simplified form as: where is calculated value of effective multiplication factor of the evaluated model, which should be less than or equal to limit value (usually defined by legislation), if all biases and uncertainties marked as ∆ are included.
Burnup credit is an approach that credits the reduction in reactivity due to fuel burnup. The calculation using burnup credit method consists of two main steps: a burnup calculation, which estimates nuclide concentrations in the spent fuel, and criticality calculation, which uses the nuclide concentrations determined in the first step. However, computational prediction of fuel composition introduces an additional source of bias or uncertainty. There are several methods that deal with these uncertainties and in general, they can be divided into two main categories: bounding methods and best-estimate methods.
The bounding approach aims to get the most conservative result by adjusting the concentrations of nuclides in a system. The concentration of selected nuclides in the model is multiplied by correction factors (referred to also as the isotopic correction factors) so that concentrations of fissile nuclides are increased and concentrations of absorbing nuclides are decreased. This leads to a more reactive system with the conservative estimate of multiplication factor keff.
The best-estimate methods aim to determine the uncertainty in spent fuel in a more accurate and realistic way. This can be done by using Monte Carlo sampling of the different parameters in a model based on their uncertainty and selected distribution. Other options, such as direct-difference method or sensitivity calculations, are also possible.
In this paper, we have tried to investigate how different the results of both methods are and what impact this could have on the possible compliance with regulatory limits.

II. CORRECTION FACTORS
Correction factors (CF) are part of validation of each calculation code. The factors are calculated from difference between the experiment and the calculation (E/C) based on PIE experiments results. The mean E/C value of a given nuclide is increased or decreased (depending whether it is fissile or absorbing nuclide) by the value of its standard deviation multiplied by the tolerance-limit factor. The value of tolerance-limit factor is determined by the number of measured samples in PIE experiments. If more than 10 measurement data are available for a nuclide, a two-sided tolerance-limit factor is used at statistics with 95 % certainty and 68.3 % population. For fewer than 10 measurement data, one-sided tolerance-limit factor is used with 95 % certainty and 95 % population. In our case, the number of samples varied from 8 to 46, which is equivalent to tolerance factor of 3.14 (for 8 samples) to 1.22 (for 46 samples).
In this work, the correction factors were determined for a set of 40 nuclides based on PIE experiments [5] , [6] and [7] . The correction factors were analyzed for calculation codes from SCALE code system, the results are shown in Table I.
During this analysis, some modifications have been made. As the nuclide 103 Rh was not measured in PIE experiments, its correction factor was estimated to be the same as in the case of Ag, which is the closest fission product in terms of weight while also having low value of its correction factor. Another problem was caused by isotopes 151 Eu and 153 Eu, which were measured with high deviation and thus leading into negative value of their correcting factor. Their correction factor was therefore set to zero. Additional analysis was also made for isotopes of gadolinium contained in pins with Gd2O3 burnable absorber, as these data are not part of PIE experiments. The Gd crosssections were conservatively modified by their 3σ deviation from uncertainty library, the impact of modified crosssections was then studied during burnup calculation and correction factors values were determined as a function of burnup, as can be seen in Fig. 2.

III. CALCULATION
The comparison of both methods was made on TVSA-T Gd-2M+ 4.76 fuel assemblies. In burnup calculation, a 2D model of the fuel assembly was used. The fuel pins were divided into 15 groups with respect to 30°symmetry. The isotopic composition of the fuel was then calculated by TRITON deterministic code from SCALE code system and ENDF/B-VII.1 library in 252-group broadened energy structure.
In criticality calculations, only 47 nuclides were considered in spent fuel compositionthis includes 40 nuclides listed in Table I, 6 more isotopes of gadolinium and nuclide 16 O, which is part of UO2 and Gd2O3 compounds. The calculations were made on simplified 3D model of spent fuel pool. It consists of lattice filled with fuel assemblies with same symmetry as in burnup calculation. Each assembly is inserted into boron steel hexagonal tube, as is pictured in Fig. 2. The lattice is placed on stainless steel support plate, under which is a layer of moderator and concrete with steel lining. The pool is flooded with clean water with density conservatively assumed as 1.0 g/cm 3 .

A. Bounding Approach
In bounding approach, the concentrations of nuclides were adjusted using correction factors CF mentioned above. The criticality calculation was then made for 21 steps of fuel burnup, calculated by KENO-VI code from SCALE system. The results are shown in Fig. 3. The maximum of effective multiplication factor is in this case at 8000 MWd/MTU, with a value of: = 0.94309 ± 0.00016.
As this is the highest achievable value, it is also a crucial point from the nuclear safety perspective and was therefore investigated further in case of Monte Carlo sampling.

B. Monte Carlo Sampling
In Monte Carlo sampling, the correction factors were not used. Instead, the concentrations of nuclides were multiplied by a factor from a set of pseudorandom numbers, which were generated with normal distribution based on E/C value and its standard deviation as shown in Table I. Each nuclide in each pin has therefore its own unique concentration, while also satisfying its probability distribution (see Fig. 5).
As the sampling method is more computationally demanding than the standard bounding approach described above, we have analyzed only the case with maximum multiplication factor, which is according to the previous calculation at burnup of 8000 MWd/MTU. A total of 5000 calculation cases were created, from which the first case was sampled in unperturbed state, where nuclide concentrations were multiplied only by mean E/C value and the deviations were not considered. All other cases are perturbed, i.e. they were sampled according to the procedure described above.
The samples were calculated in criticality calculation by transportation code KENO-VI from SCALE code system. The results are individual values of multiplication factor of sampled cases . The mean value ̅ , as well as corrected sample standard deviation , of all perturbed cases was determined: This leads to a final resultthe effective multiplication factor of all perturbed cases: = 0.91374 ± 0.00542, which is close to multiplication factor of unperturbed case 0 : 0 = 0.91872 ± 0.00013.