# Difference between revisions of "Basic Statistical Analysis of LWR Pin Power Data"

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== Pin Power Dataset Representation == | == 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 | + | First, let's view the pin power dataset of each LWR as a 4D matrix ''P(i, j, k, l)'' where |

{| class="wikitable" | {| class="wikitable" | ||

|- | |- | ||

Line 16: | Line 16: | ||

| pin column | | pin column | ||

|- | |- | ||

− | ! scope="row" | + | ! scope="row"| k |

+ | | axial level | ||

+ | |- | ||

+ | ! scope="row"| l | ||

+ | | assembly number | ||

|} | |} | ||

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

+ | {| class="wikitable" | ||

+ | |- | ||

+ | ! scope="col"| Variable | ||

+ | ! scope="col"| Represents | ||

+ | |- | ||

+ | ! scope="row"| A | ||

+ | | reference power data set | ||

+ | |- | ||

+ | ! scope="row"| B | ||

+ | | alternate power data set | ||

+ | |- | ||

+ | ! scope="row"| 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 == | == 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<span style="color: red">[2]</span>{\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 == | == 1D Axial Power Results == | ||

+ | The axial power difference is defined by the formula below. | ||

+ | |||

+ | <math>C(k) = \sqrt<span style="color: red">[2]</span>{\frac{\sum_{ijl}(C{\cdot}W)}{{\sum_{ijl}}W}}</math> | ||

+ | |||

+ | The average axial power difference is defined as | ||

+ | |||

+ | <math>\frac{\sum_k<span style="color: red">[C(k){\cdot}dz]</span>}{{\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<span style="color: red">[2]</span>{\frac{\sum_k<span style="color: red">[C(k)^{2}{\cdot}dz]</span>}{{\sum_k}dz}}</math> | ||

+ | |||

== 2D Radial Power Results == | == 2D Radial Power Results == | ||

+ | The ''radial power difference'' is defined as | ||

+ | |||

+ | <math>C(i,j,l) = \sqrt<span style="color: red">[2]</span>{\frac{\sum_{k}(C{\cdot}W)}{{\sum_{k}}W}}</math> | ||

+ | |||

+ | The average radial power difference is defined as | ||

+ | |||

+ | <math>\frac{\sum_{ijl}<span style="color: red">[C(i,j,l){\cdot}dr]</span>}{{\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<span style="color: red">[2]</span>{\frac{\sum_k<span style="color: red">[C(i,j,l)^{2}{\cdot}dr]</span>}{{\sum_{ijl}}dr}}</math> | ||

+ | |||

== 3D Assembly Power Results == | == 3D Assembly Power Results == | ||

+ | The ''assembly power difference'' is defined as | ||

+ | |||

+ | <math>C(k,l) = \sqrt<span style="color: red">[2]</span>{\frac{\sum_{ij}(C{\cdot}W)}{{\sum_{ij}}W}}</math> | ||

+ | |||

+ | The average assembly power difference is defined as | ||

+ | |||

+ | <math>\frac{\sum_{kl}<span style="color: red">[C(k,l){\cdot}da]</span>}{{\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<span style="color: red">[2]</span>{\frac{\sum_{kl}<span style="color: red">[C(k,l)^{2}{\cdot}da]</span>}{{\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> | ||

+ | |||

+ | == Related == | ||

+ | |||

+ | [https://wiki.eclipse.org/ICE_Developer_Documentation#Documentation Developer Documentation] |

## Latest revision as of 13:57, 28 February 2015

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

Due to the migration of our articles from MediaWiki to Markdown, the formulas are not showing up properly. Unfortunately, Markdown does not have a lot of support for mathematical characters. We will soon be migrating back to MediaWiki pages, at which point this article will be back to normal.

## 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>