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# Basic Statistical Analysis of LWR Pin Power Data

This article details the basic mathematical formulas used to statistically compare pin power data sets from a reference and an alternate LWR.

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## Contents

## Pin Power Dataset Representation

First, let's view the pin power dataset of each LWR as a 4D matrix *P(i, j, k, l)* where

Variable | Represents |
---|---|

i | pin row |

j | pin column |

k | axial level |

l | assembly number |

In order to compare two of these pin power datasets, we shall define the following 4D pin power datasets:

Variable | Represents |
---|---|

A | reference power data set |

B | alternate power data set |

C | power difference data set |

where

_C = B - A _ (a **basic power difference**)

or

_C = (B - A) / A _ (a **relative power difference**)

In addition, each element in *C* can be weighted by a 4D matrix *W*.

## Derived Quantities

There are four general derived quantities used in pin power analysis. The first is the *simple difference* which is simply the dataset C. The second is the absolute difference which is the absolute value of *C* or |*C*|. The third is the average difference is a weighted average calculated with the following formula.

<math>\frac{\sum(C{\cdot}W)}{{\sum}W}</math>

Finally, the *root-mean-square difference* is a weighted RMS difference determined by the formula below.

<math>\sqrt[2]{\frac{\sum(C^{2}{\cdot}W)}{{\sum}W}}</math>

The following sections detail how to calculate the above quantities over a variety of dimensions to develop statistical results from pin power differences.

## 1D Axial Power Results

The axial power difference is defined by the formula below.

<math>C(k) = \sqrt[2]{\frac{\sum_{ijl}(C{\cdot}W)}{{\sum_{ijl}}W}}</math>

The average axial power difference is defined as

<math>\frac{\sum_k[C(k){\cdot}dz]}{{\sum_k}dz}</math>

where the dz, the axial weight, is <math>dz = mesh(k+1) - mesh(k)</math>. Here mesh is a 1D array containing the actual physical distance between each axial level k.

The RMS axial power difference is defined by the formula below.

<math>\sqrt[2]{\frac{\sum_k[C(k)^{2}{\cdot}dz]}{{\sum_k}dz}}</math>

## 2D Radial Power Results

The *radial power difference* is defined as

<math>C(i,j,l) = \sqrt[2]{\frac{\sum_{k}(C{\cdot}W)}{{\sum_{k}}W}}</math>

The average radial power difference is defined as

<math>\frac{\sum_{ijl}[C(i,j,l){\cdot}dr]}{{\sum_{ijl}}dr}</math>

where the *dr*, the *radial weight*, is <math>dr(i,j,l) = \sum_kW</math>

The RMS radial power difference is defined by the formula below.

<math>\sqrt[2]{\frac{\sum_k[C(i,j,l)^{2}{\cdot}dr]}{{\sum_{ijl}}dr}}</math>

## 3D Assembly Power Results

The *assembly power difference* is defined as

<math>C(k,l) = \sqrt[2]{\frac{\sum_{ij}(C{\cdot}W)}{{\sum_{ij}}W}}</math>

The average assembly power difference is defined as

<math>\frac{\sum_{kl}[C(k,l){\cdot}da]}{{\sum_{kl}}da}</math>

where the *da*, the assembly *weight*, is <math>da(k,l) = \sum_{ij}W</math>

The RMS assembly power difference is defined by the formula below.

<math>\sqrt[2]{\frac{\sum_{kl}[C(k,l)^{2}{\cdot}da]}{{\sum_{kl}}da}}</math>

In addition to the above quantities, the *form factor* found at each pin for the reference dataset is defined as

<math>ff_{A}(i,j,k,l) = A(i,j,k,l) / A(k,l)</math>

and for the alternate dataset

<math>ff_{B}(i,j,k,l) = B(i,j,k,l) / B(k,l)</math>

where the form *factor difference* is

<math>ff_{C} = ff_{B} - ff_{A}</math>