#
Methodology for Sensitivity Analysis of Homogenized Cross-Sections to Instantaneous and Historical Lattice Conditions with Application to AP1000^{®} PWR Lattice

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## Abstract

**:**

^{®}reactor is presented where coolant density, fuel temperature, soluble boron concentration, and control rod insertion are the state variables of interest. The effects of considering the instantaneous values of the state variables, historical values of the state variables, and burnup-averaged values of the state variables are analyzed. Using these methods, it was found that a linear model that only considers the instantaneous and burnup-averaged values of state variables can fail to capture some variations in the homogenized cross-sections.

## 1. Introduction

^{®}reactor is presented where coolant density, fuel temperature, boron concentration, and control rod insertion are the state variables of interest. This approach will allow for importance ranking of state variables that can aid analysts in the formulation of models for HXS generation in core simulation. This analysis can aid engineers in developing models for HXS approximation from lattice simulations that may more effectively account for history effects in reactor lattices. With the many advanced reactor designs competing for near-term deployment that may require different parameterization of cross-section models, this methodology can help to identify how to best construct those models. In application to any reactor design, these results can be used to identify the sources of some errors that may arise when using the burnup-averaging methods for HXS approximation.

## 2. Methods and Models

#### 2.1. Lattice Model

^{®}reactor. Referred to in [23] as “Region B”, a pin map of the lattice with 1/8th symmetry is given in Figure 1. The lattice contains 264 UO

_{2}fuel rods with a uniform enrichment of 1.58% U-235 and ZIRLO

^{®}cladding. This lattice was selected because it is thought to represent the most simple lattice design that may be used in a PWR. In more complex lattice designs, there may be higher or lower sensitivities to the various state variables during burnup. There are 24 guide tubes that are either modeled with B

_{4}C control rod inserts or filled with water depending on the lattice state being modeled. There is also a single instrumentation tube in the center of the lattice that remains filled with water in all lattice configurations explored in this study. In terms of geometry, the pin pitch of the lattice is 1.26 cm. The fuel pellet inner radius is modeled to be 0.418 cm and in contact with the inner surface of the cladding. The cladding outer radius is 0.475 cm, and the guide tube and instrumentation tube have inner and outer radii of 0.561 cm and 0.612 cm, respectively.

#### 2.2. Sampling Approach

- coolant density;
- fuel temperature;
- boron concentration in coolant;
- control rod insertion.

#### 2.3. Sensitivity Analysis

- $Q=n-2$: This is the largest allowable value for Q. In this case, the RHM and CHM become identical in that the fitted $\mathbf{\gamma}$ parameter operates on the single $\overrightarrow{{x}_{1}}$, functioning identically as ${\mathbf{\beta}}_{n-1}$ in the CHM.
- $Q=0$: This is the smallest possible value for Q. In this case, only the current value of the state vector ${\overrightarrow{x}}_{n}$ is assigned an individual fitted vector. The remaining state vectors are averaged and assigned a single fitted vector:

- Full History Model (FHM): Uses every reactor state variable in the history leading up to the burnup at which the HXS is calculated to calculate sensitivities.
- Reduced History Model (RHM): Uses a reduction parameter to use burnup-averaged state variable values instead of considering the reactor state at every burnup step.

#### 2.4. Workflow Summary

## 3. Model Evaluation

^{2}values do not change significantly for lower burnups. Five different datasets are used, each composed of burnup sequences with sampled state parameters given in Table 1. For each dataset, there is a different set of state parameters that are perturbed, each according to the distributions given in Table 1. Three datasets vary for a single parameter: coolant density, fuel temperature, or boron concentration are each varied for 150 samples. One dataset varies coolant density, fuel temperature, and boron concentration simultaneously for 550 samples. The final dataset has all state parameters varied simultaneously for 400 samples. State parameters that are not varied are held at their nominal values. R

^{2}values suggest that the variability in all four of the HXS shown can be largely explained by the model for all datasets. The coefficient of determination can increase as more varied parameters are added (as is the case for ${\mathrm{\Sigma}}_{f,2}$) because the linearity in the relationships of these additional parameters can overshadow nonlinear relationships with those original state variables. To explain, consider a slightly nonlinear relationship between some state variable and an HXS. If a coefficient of determination were to be computed to describe the relationship between this state variable and the HXS, it would be low, indicating a weak linear relationship between the two variables. If a largely influential parameter with a linear relationship with the HXS is then added, the relationship described by R

^{2}will be be completely dominated by this new variable with strong linearity. Hence, the linearity arising from this new parameter will increase R

^{2}, overshadowing the nonlinear relationship with the original variable.

^{2}values for both energy groups of the homogenized fission cross-section for the lattice. The removal cross-section is not included because all R

^{2}values are above 0.98 with trends that will be discussed in later sections. A variety of RHMs are used with different Q and n values, the dataset with all state parameters varied, and 400 samples are being used. In this figure, BU${}_{n}$ is the x-axis which corresponds to the burnup at n. Each line in this figure corresponds to models with the same Q applied to different values of n. Models are shown with all possible Q values (up to $Q=18$), but, for clarity, only lines up to $Q=4$ are labeled. As n increases with burnup, the $Q<n-1$ relationship allows for models to be created with higher Q values.

^{2}of the model will decrease. These results demonstrate that some HXS prediction methods that rely strictly on the instantaneous values of the state variables and burnup-averaged quantities may fail to account for the effects of state variable changes over burnup. The explanation for this is the complex relationship between the production of fission products/consumption of fissile isotopes and neutron flux distribution in space and energy that results from the coupled Boltzman and Bateman equations. Methods that average burnup-dependent quantities may fail to account for burnup-dependent relationships between state variables and the production/consumption of isotopes. Overall, the Q value selected for parameterizing HXS generation models may be different for assorted lattice designs. Using an analysis like the one given above can provide an estimate for the importance of states at earlier burnup steps when predicting HXS at later burnup steps.

## 4. Sensitivity Results

#### 4.1. Parametric Sensitivity Analysis Using RHM

#### 4.2. Parametric Sensitivity Analysis Using CHM

#### 4.3. Burnup-Averaged State Variable Importance

#### 4.4. Stationarity Analysis

## 5. Conclusions

- Full History Model (FHM): Uses every reactor state variable in the history leading up to the burnup at which the HXS is calculated to calculate sensitivities.
- Reduced History Model (RHM): Uses a reduction parameter to use burnup-averaged state variable values instead of considering the reactor state at every burnup step.

^{®}reactor is presented where coolant density, fuel temperature, boron concentration in coolant, and control rod insertion are the state variables of interest. It was found that the linearity assumption was accurate over the domain of state variables chosen. Furthermore, it was found that fitting an RHM that only considers the burnup-averaged values of the state variables and value of the state variable at the time that the HXS is calculated can lead to notable losses in model accuracy as shown in Figure 4. The HXSs used in this application are the 2-group fission cross-section and the 2-group removal cross-section, but the methodology is certainly applicable to any other energy structure or reaction type. It was found that the historical values of state variables had a much larger effect on the fission cross-sections than the removal cross-sections. In addition, across all cross-sections considered, the fuel temperature seemed to have the least influence. A demonstration of the dependence of these sensitivities on burnup was also given, with the result that the instantaneous values of the state variables have a more constant sensitivity over burnup as compared to burnup-averaged quantities. Future work will include an application of this methodology to a broader set of LWR lattice designs and potentially advanced reactor designs. It is thought that the presence of burnable absorbers may significantly change these sensitivities due to the complex lattice behavior they introduce. Therefore, performing this sort of analysis on a broader set of LWR lattice designs may offer insight on the differences between lattices with—and without—burnable absorbers.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Structure of burnup sequences. (

**a**) depiction of coolant temperature and fuel temperature change over burnup for a single burnup sequence sample; (

**b**) structure of state vectors and evolution through burnup. Each component of $\overrightarrow{x}$ is independently sampled from their respective distributions.

**Figure 3.**Potential workflow to execute the proposed methodology to obtain sensitivity estimates for some HXS at some burnup step n.

**Figure 4.**Coefficient of determination for varying Q values when fitting the RHM to the relationship between all state variables and fission HXS over all burnups.

**Figure 5.**Sensitivity coefficients for previous and current values of state vector $\overrightarrow{x}$ for a burnups equal to 18 GWD/MTU and 30 GWD/MTU using the CHM. In the legend, g# indicates the energy group of the HXS for which the sensitivity measure corresponds.

**Figure 6.**Sensitivities of ${\mathrm{\Sigma}}_{f}$ to burnup-averaged values of state variables. Subfigure headers correspond to the burnup at which the HXS is calculated. In the legend, g# indicates the energy group of the HXS for which the sensitivity measure corresponds.

**Figure 7.**Mean behavior of HXS during burnup using $\alpha $ parameter from Equation (4). $\alpha $ demonstrates an estimation for the behavior of the HXS if all state variables were set to their mean value over the entire history.

**Figure 8.**Sensitivity measures for both groups of ${\mathrm{\Sigma}}_{f}$ from the SRC method for both burnup-averaged state variables ($\gamma $) and state variables at the burnup where the HXS is calculated (${\beta}_{0}$) as a function of burnup. Sensitivity measures calculated at each burnup step using the RHM where $Q=0$. In the legend, g# indicates the energy group of the HXS for which the sensitivity measure corresponds.

State Variable | Notation | Distribution | Distribution Specs. | Nominal Value | Unit |
---|---|---|---|---|---|

Coolant Density | $\rho $ | Uniform | [min = 0.652, max = 1.01] | 0.712 | $\mathrm{g}/{\mathrm{cm}}^{3}$ |

Fuel Temperature | ${T}_{f}$ | Uniform | [min = 293, max = 1793] | 900 | K |

Boron Concentration | b | Uniform | [min = 0, max = 3000] | 800 | ppm |

Control Rod Insertion | r | Bernoulli | [50% in/out] | out |

**Table 2.**Coefficient of determination using the CHM to fit to burnup sequence datasets with different varying state variables for the final burnup considered at 60 GWD/MTU. Checkmarks indicate varied state variables for each data set. I indicates the number of samples in the dataset.

I | $\mathit{\rho}$ | ${\mathit{T}}_{\mathit{f}}$ | b | r | ${\mathbf{\Sigma}}_{\mathit{f},1}$ | ${\mathbf{\Sigma}}_{\mathit{f},2}$ | ${\mathbf{\Sigma}}_{\mathbf{rm},1}$ | ${\mathbf{\Sigma}}_{\mathbf{rm},2}$ |
---|---|---|---|---|---|---|---|---|

150 | ✓ | 0.989 | 0.799 | 1.000 | 0.956 | |||

150 | ✓ | 0.995 | 0.997 | 0.984 | 0.997 | |||

150 | ✓ | 0.999 | 0.998 | 0.999 | 0.999 | |||

550 | ✓ | ✓ | ✓ | 0.989 | 0.970 | 1.000 | 0.973 | |

400 | ✓ | ✓ | ✓ | ✓ | 0.982 | 0.974 | 0.999 | 0.992 |

**Table 3.**SRC measures for the sensitivity of homogenized fission cross-section (${\mathrm{\Sigma}}_{f}$) and homogenized removal cross-section (${\mathrm{\Sigma}}_{rm}$) to instantaneous (${\beta}_{0}$) and burnup-averaged ($\gamma $) values for the coolant density ($\rho $), fuel temperature (${T}_{f}$), and boron concentration in coolant (b). Results are calculated using the RHM with $Q=0$. Numerical digit subscripts on cross-sections indicate neutron energy group.

${\mathbf{\Sigma}}_{\mathit{f},1}$ | ${\mathbf{\Sigma}}_{\mathit{f},2}$ | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\rho}$ | ${\mathbf{T}}_{\mathbf{f}}$ | $\mathit{b}$ | $\mathbf{\rho}$ | ${\mathbf{T}}_{\mathbf{f}}$ | $\mathit{b}$ | ||

BU${}_{n=3}$ = 9 GWD/MTU | ${\beta}_{0}$ | 0.769 | 0.110 | 0.305 | 0.796 | 0.125 | −0.391 |

$\gamma $ | −0.227 | 0.156 | 0.457 | −0.172 | 0.147 | 0.422 | |

BU${}_{n=10}$ = 30 GWD/MTU | ${\beta}_{0}$ | 0.714 | 0.148 | 0.152 | 0.325 | 0.329 | −0.327 |

$\gamma $ | −0.235 | 0.186 | 0.500 | -0.241 | 0.222 | 0.628 | |

${\mathrm{\Sigma}}_{\mathbf{rm},\mathbf{1}}$ | ${\mathrm{\Sigma}}_{\mathbf{rm},\mathbf{2}}$ | ||||||

BU${}_{n=3}$ = 9 GWD/MTU | ${\beta}_{0}$ | 1.000 | 0.016 | −0.002 | 0.504 | 0.032 | 0.812 |

$\gamma $ | −0.001 | −0.000 | −0.001 | 0.027 | 0.017 | 0.072 | |

BU${}_{n=10}$ = 30 GWD/MTU | ${\beta}_{0}$ | 1.001 | 0.016 | −0.001 | 0.480 | 0.069 | 0.833 |

$\gamma $ | −0.001 | 0.001 | 0.002 | −0.038 | 0.034 | 0.089 |

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**MDPI and ACS Style**

Price, D.; Folk, T.; Duschenes, M.; Garikipati, K.; Kochunas, B.
Methodology for Sensitivity Analysis of Homogenized Cross-Sections to Instantaneous and Historical Lattice Conditions with Application to AP1000^{®} PWR Lattice. *Energies* **2021**, *14*, 3378.
https://doi.org/10.3390/en14123378

**AMA Style**

Price D, Folk T, Duschenes M, Garikipati K, Kochunas B.
Methodology for Sensitivity Analysis of Homogenized Cross-Sections to Instantaneous and Historical Lattice Conditions with Application to AP1000^{®} PWR Lattice. *Energies*. 2021; 14(12):3378.
https://doi.org/10.3390/en14123378

**Chicago/Turabian Style**

Price, Dean, Thomas Folk, Matthew Duschenes, Krishna Garikipati, and Brendan Kochunas.
2021. "Methodology for Sensitivity Analysis of Homogenized Cross-Sections to Instantaneous and Historical Lattice Conditions with Application to AP1000^{®} PWR Lattice" *Energies* 14, no. 12: 3378.
https://doi.org/10.3390/en14123378