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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>