The study explored the calculation of uncertainty based on available crosssection covariance data and computational tool on fuel lattice levels, which included pin cell and the fuel assembly models. Uncertainty variations due to temperatures changes and different fuel compositions are the main focus of this analysis. Selected assemblies and unit pin cells were analyzed according to the OECD LWR UAM benchmark specifications. Criticality and uncertainty analysis were performed using TSUNAMI2D sequence in SCALE 6.1. It was found that uncertainties increase with increasing temperature, while
The demand for the best estimate calculations in nuclear reactor core modeling and design has increased in recent years. Uncertainty analysis has been highlighted as an important part of the design and safety analysis of modern nuclear reactors. The modeling aspects of uncertainty analysis and sensitivity analysis are to be further developed and validated on scientific grounds in support of their performance. The Organization for Economic Cooperation and Development (OECD)/Nuclear Energy Agency (NEA) initiated the Benchmark for Uncertainty Analysis in Modeling, Design, Operation, and Safety Analysis of Light Water Reactor (OECD LWR UAM benchmark). The general objective of the benchmark is to propagate the uncertainty through complex coupled multiphysics and multiscale simulations. The benchmark is divided into three phases with Phase I highlighting the uncertainty propagation in neutronics calculations, while Phases II and III are focused on uncertainty analysis of reactor core and reactor system, respectively.
In Phase I of the OECD LWR UAM benchmark, the exercises are divided into three parts: cell physics (Exercise I), lattice physics (Exercise II), and core physics (Exercise III) [
In general, uncertainty is calculated based on covariance matrix and weighting factor coefficients [
In order to obtain the uncertainty of the response of interest, which may be the critical eigenvalue, the reactivity difference between two reactor states, or the ratio of reactions rates, sensitivity coefficients (
The evaluation of nuclear data induced uncertainty is possible by the use of nuclear crosssection variance and covariance data. By including the uncertainty or covariance information, the analyst can propagate crosssection data uncertainties through sensitivity studies to the final calculated quantities of interest. The covariance data files provide the estimated variance for the individual data as well as any correlation that may exist. In principle, the covariance matrices can be now selfshielded in the same way as the crosssections, although in practice this is rarely done. The impact of this treatment on the obtained covariance matrices and their dependence on energy group structure needs to be studied. The SCALE 6.1/TSUNAMI2D [
The SCALE 6.1 covariance library data corresponds also to 44group relative uncertainties assembled from a variety of sources, including evaluations from ENDF/BVII, ENDF/BVI, JENDL3.3, and more than 300 approximated uncertainties from a collaborative project performed by Brookhaven National Laboratory (BNL), Los Alamos National Laboratory (LANL), and Oak Ridge National Laboratory (ORNL).
It is assumed that the same relative (rather than absolute) uncertainties can be applied to all crosssection libraries, even if these are not strictly consistent with the nuclear data evaluations. In addition, the assumption that there are no covariance correlations between energy groups is applied [
For light water reactors, two components of sensitivity coefficient are needed. Explicit sensitivity represents the sensitivity of the calculated
The SCALE 6.1/TSUNAMI2D sequence is used to perform the study. First, the ENDF/BVII.0 based 238group microscopic crosssection data library is processed using BONAMIST and CENTRM/PMC. Next, forward and adjoint calculations are performed using NEWT, a 2D transport solver. Finally, sensitivity coefficients are calculated, and uncertainty data is generated by SAMS using the default covariance data library in 44 groups (44groupcov).
The SCALE 6.1 sensitivity and uncertainty methodology is based on the firstorder perturbation theory to calculate response sensitivity coefficients, which are then folded with nuclear data covariances to obtain the response uncertainty. TSUNAMI2D applies the generalized perturbation theory (GPT) to generate uncertainties associated with the fewgroup assembly homogenized neutron crosssection data [
The study begins with specification of fuel pin cells of three light water reactor types and two critical experiments provided by the OECD LWR UAM benchmark within Exercise I1 [
Uncertainty in
Fuel  Operating conditions 

Uncertainty in 
Largest uncertaintycontributing reaction 

BWR  HZP  1.3382  0.52 
^{
238}U( 
HFP (40% void)  1.2208  0.62 
^{
238}U( 

PWR  HZP  1.4206  0.48 
^{
238}U( 
HFP  1.4017  0.49 
^{
238}U( 

VVER  HZP  1.3448  0.51 
^{
238}U( 
HFP  1.3270  0.52 
^{
238}U( 

KRITZ 2.1  HZP  1.2323  0.59 
^{
238}U( 
HFP  1.1837  0.63 
^{
238}U( 

KRITZ 2.13  HZP  1.2642  0.55 
^{
238}U( 
HFP  1.2329  0.58 
^{
238}U( 
Observations of the results showed that
The neutron flux of PWR unit cell at two different operating conditions.
The resonance absorption due to Doppler broadening is reflected in the calculation of the sensitivity, mainly the implicit sensitivity. This implicit sensitivity accounts for the selfshielding effect. Figure
Relative change in the sensitivity of ^{238}U(
In addition, three fuel assemblies of three light water reactor types provided by the OECD LWR UAM benchmark were analyzed. The details of the specifications are readily available [
Assembly
Fuel  Operating conditions 

Uncertainty in 
Largest uncertaintycontributing reaction 

BWR  HZP  1.1116  0.50 
^{
238}U( 
HFP (40% void)  1.0779  0.56 
^{
238}U( 

PWR  HZP  1.4130  0.46 
^{
23}U( 
HFP  1.3968  0.47 
^{
238}U( 

VVER  HZP  1.3164  0.47 
^{
238}U( 
HFP  1.3115  0.47 
^{
238}U( 
Similar to the fuel pin models, in fuel assembly models, the uncertainty in
The fuel assembly analysis (as part of Exercise I2) includes the propagation of multigroup crosssection uncertainties (multigroup covariance matrix) to twogroup homogenized crosssection uncertainties (twogroup covariance matrix). The twogroup crosssection uncertainties are obtained using the SCALE6.0 44group covariance matrix as input to the TSUNAMI2D sequence with GPT in SCALE 6.1. The obtained results are shown in Table
Twogroup crosssection uncertainty in LWR fuel assembly.
Response crosssection  Uncertainty (% 
Uncertainty (% 
Uncertainty (% 


BWR  PWR  VVER  
HZP  HFP  HZP  HFP  HZP  HFP  

0.84  0.91  0.87  0.88  0.81  0.82 

0.13  0.15  0.14  0.14  0.12  0.12 

0.84  0.91  0.87  0.88  0.81  0.82 

0.13  0.15  0.14  0.14  0.12  0.12 

0.78  0.83  0.86  0.87  0.81  0.82 

0.20  0.22  0.22  0.22  0.21  0.21 

0.68  0.72  0.36  0.36  0.47  0.47 

0.32  0.32  0.32  0.32  0.32  0.32 

0.84  0.91  0.87  0.87  0.81  0.81 

1.10  1.22  1.20  1.21  1.03  1.03 

0.27  0.34  0.30  0.33  0.26  0.29 

0.13  0.16  0.15  0.15  0.13  0.13 
One can define ninedimensional response vector
BWR HZP covariance matrix.
BWR HFP covariance matrix.
TMI HZP covariance matrix.
TMI HFP covariance matrix.
The obtained results for different LWR types and cases indicate the following tendencies.
Group 1 (fast) crosssection uncertainty is ~23 times larger than Group 2 (thermal) crosssections uncertainty.
Uncertainty contributions:
a major contributor to Group 1 (fast) crosssection uncertainties is U238 inelastic scattering;
U238 inelastic scattering uncertainty is quite large;
40% void (and higher) exhibit larger uncertainty in
Uncertainty (correlation) contribution:
U238 inelastic scattering uncertainty is quite large and dominates correlation coefficient.
Selected fuel pin cells and four types of fuel assemblies from a representative Generation III LWR (GENIII) specification were analyzed for the purpose of comparing effect of the compositions on the uncertainty calculations. The specifications of the GENIII unit cells and fuel assemblies are readily available [
Three types of unit cells were analyzed at Hot Full Power; these include MOX, UOX, and UOX with Gd_{2}O_{3}. The multiplication factors and their uncertainties are presented in Table
Uncertainty in
Fuel  Compositions 

Uncertainty in 
Largest uncertaintycontributing reaction 

MOX  9.8% ^{239}Pu  1.0921  0.94 
^{
238}U( 
6.5% ^{239}Pu  1.0540  0.97 
^{
238}U( 

3.7% ^{239}Pu  1.0115  0.99 
^{
238}U( 

 
UOX  4.2% ^{235}U  1.2431  0.51 
^{
238}U( 
3.2% ^{235}U  1.1741  0.54 
^{
238}U( 

2.1% ^{235}U  1.0490  0.59 
^{
238}U( 

 
UOXGd_{2}O_{3}  2.2% ^{235}U  0.2166  1.79 
^{
238}U( 
1.9% ^{235}U  0.1997  1.94 
^{
238}U( 
For each group of the fuel cells, several factors influence the changes in the uncertainty in
For each unit cell, the calculated
For the MOX fuel cells, the amount of ^{238}U is reduced as the amount of ^{239}Pu, the fissile material, is increased. The reduction in the amount of ^{238}U means that there is less neutrons absorption by ^{238}U nuclides. This is later found to be the most important nuclide contributor to uncertainty in
However, the uncertainties of the MOX fuel cells were nearly twice than that of UOX fuel. The presence of ^{239}Pu plays an important role in the increase in uncertainty. Figure
Plot of ^{239}Pu absorption crosssections compared to ^{235}U absorption crosssections.
As a consequence, from the fact that more neutrons are absorbed by ^{239}Pu than ^{235}U, more neutrons are produced by fission due to ^{239}Pu. Table
Macroscopic fission crosssections in MOX fuel cells.
Fuel  Composition 

^{
235}U 
^{
239}Pu 

MOX  3.7% Pu329  1.0115  6.70  13.74 
MOX  6.5% Pu329  1.0540  4.65  8.40 
MOX  9.8% Pu329  1.0921  3.69  5.88 
LWR neutronics parameters [
Parameter  ^{ 235}U  ^{ 239}Pu 

Average 
2.4  2.9 
Average 
2.0  1.9 
Average 
280 barns  790 barns 
Neutrons produced by ^{239}Pu, in general, will have higher energy than that of ^{235}U. In this case, the neutron spectrum is harder because more neutrons with higher energies are produced by ^{239}Pu, the dominant fission nuclide.
On the other hand, for UOX fuel cell with Gd_{2}O_{3} added, similar finding (hardening of the neutron spectrum) occurred but due to a very different mechanism. The uncertainties in
The uncertainties in fuel cells with harder neutron spectrum seemed to be higher than fuel cells with softer neutron spectrum. This is due to the absorption of the ^{238}U. The resonance absorption of ^{238}U that occurs at higher neutron energy is very large.
The absorption of thermal neutrons by ^{239}Pu dominates the fission process. Since fission in ^{239}Pu produces more fast neutrons, the neutron spectrum becomes harder. The harder the spectrum, the higher the ^{238}U(
The plot of inelastic crosssection of ^{238}U as a function of energy.
The neutron flux spectra of three unit cell compositions.
The sensitivity profiles were compared in Figure
The sensitivity of ^{238}U(
The covariance matrix of ^{238}U(
The 2D plot of ^{238}U(
The 2D plot of ^{238}U(
Four types of fuel assembly from a representative Generation III LWR specification (as part of Exercise I2) were analyzed at Hot Full Power condition:
type 1 (UOX
type 2 (UOX
type 3 (UOX
type 4 (MOX).
Uncertainty in
Fuel  Compositions 

Uncertainty in 
Largest uncertaintycontributing reaction 

GEN III  Type 1  1.2501  0.49 
^{
238}U( 
Type 2  1.1228  0.49 
^{
238}U( 

Type 3  0.9564  0.53 
^{
238}U( 

Type 4  1.0700  0.97 
^{
238}U( 
Results show that
Additionally, the twogroup crosssection uncertainties are presented in Table
Twogroup crosssection uncertainty (%
Response crosssection  GEN III type 1  GEN III type 2  GEN III type 3  GEN III type 4 


0.90  0.90  0.90  0.97 

0.14  0.14  0.14  0.14 

0.90  0.90  0.90  0.97 

0.14  0.14  0.14  0.14 

0.89  0.89  0.89  1.00 

0.21  0.19  0.19  0.24 

0.37  0.37  0.50  0.44 

0.32  0.32  0.33  0.62 

0.90  0.90  0.90  0.97 

1.25  1.25  1.24  1.47 

0.33  0.32  0.31  0.37 

0.15  0.15  0.15  0.16 
These results show, as expected, a larger uncertainty for the MOX fuel assembly (type 4) than for the UOX assemblies (types 1 through 3).
Figure
The neutron flux spectra of GENIII LWR fuel assembly.
Sensitivity profiles of ^{238}U(
The sensitivity of ^{238}U(
Sensitivity studies have been performed using SCALE [
increasing temperature leads to increasing uncertainty in
decreasing ^{238}U in fuel composition leads to decreasing uncertainty in
major contributor to uncertainty is affected by the neutron spectrum.
The future studies will be focused on Exercise I3—propagation of fewgroup crosssection uncertainties to the core steady state standalone neutronics calculations using statistical methodology similar to the one reported in [
The authors would like to acknowledge the two Ph.D. fellowship grants, one at the Technical University of Catalonia (sponsored by the Spanish Nuclear Security Council) and the other at the Pennsylvania State University (sponsored by the US Nuclear Regulatory Commission), which supported the graduate students performing the studies reported in this paper.