Building a Statistical Model to Building a Statistical Model to Predict Reactor TemperaturesPredict Reactor Temperatures

Carl Scarrott

Granville Tunnicliffe-WilsonLancaster University

c.scarrott@lancaster.ac.ukg.tunnicliffe-wilson@lancaster.ac.uk

OutlineOutline

ObjectivesDataStatistical ModelExploratory AnalysisResultsConclusionReferences

Project ObjectivesProject Objectives

Assess risk of temperature exceedance in Magnox nuclear reactors

Establish safe operating limits Issues:

– Subset of measurements

– Control effect

– Upper tail censored Solution:

– Predict unobserved temperatures

– Physical model

– Statistical model How to identify and model physical effects? How to model remaining stochastic variation?

Wylfa ReactorsWylfa Reactors

Magnox Type Anglesey, Wales Reactor Core 6156 Fuel Channels Fuel Channel Gas

Outlet Temperatures (CGOT’s)

Degrees C All Measured

Dungeness ReactorsDungeness Reactors

Magnox Type Kent, England 3932 Fuel Channels Fixed Subset

Measured:– 450 on 3 by 3 sub-grid– 112 off-grid

What About Unmeasured?

Temperature DataTemperature Data

Radial Banding Smooth Surface Standpipes (4x4) Chequer-board Triangles East to West

Ridges Missing

Spatial Structure:

Irradiation DataIrradiation Data

Fuel Age or Irradiation

Main Explanatory Variable

Old Fuel = Red New Fuel = Blue

Standpipe Refuelling

Chequer-board Triangles Regular & Periodic

Temperature and Irradiation DataTemperature and Irradiation Data

Statistical ModelStatistical Model

Predict Temperatures Explanatory Variables (Fixed Effects):– Fuel Irradiation– Reactor Geometry– Operating Conditions

Stochastic (Non-deterministic) Components:– Smooth Variation Resulting from Control Action

Random Errors

Statistical ModelStatistical Model

ijijijrsijENGXT )F(

– Temperature at Channel (i,j)– Fuel Irradiation for Channel (r,s)– Direct and Neutron Diffusion Effect– Linear Geometry– Slowly Varying Spatial Component– Random Errorij

ij

ij

rs

ij

E

N

G

X

T

F(.)

Exploratory AnalysisExploratory Analysis

2 Dimensional Spectral Analysis Fuel Irradiation & Geometry Effects are:

– Regular– Periodic

Easy to Identify in Spectrum Cross-Spectrum used to Examine the Fuel

Irradiation Diffusion Effect Multi-tapers Used to Minimise Bias Caused by

Spectral Leakage Scarrott and Tunnicliffe-Wilson(2000)

Application - Temperature and Irradiation DataApplication - Temperature and Irradiation Data

Temperature Spectrum Irradiation Spectrum

Reactor GeometryReactor Geometry

Standpipe Geometry Fuel Channels in

Holes Through Graphite Bricks

Interstitial Holes Along Central 2 Rows:– Control Rod– Fixed Absorber

2 Brick Sizes:– Octagonal– Square

Geometry Regressors - 1Geometry Regressors - 1

Brick Size Chequer-board Heat Differential

Coolant Leakage into Interstitial Holes

Cools Adjacent Fuel Channels E-W Ridge of 2 channels

Brick Size Chequer-board in Central 2 Rows

Larger Bricks Cooled More as Greater Surface Area

Control Rod Hole Larger Adjacent Channels Cooled More

Geometry Regressors - 2Geometry Regressors - 2

Geometry SpectraGeometry Spectra

Brick Chequer-board E-W Ridge

E-W Ridge and Chequer-board Control Rod Hole Indicator

Estimated Geometry EffectEstimated Geometry Effect

All Geometry Effects Estimated in Model Fit

How to Model F(.)?How to Model F(.)?

Effect of Fuel Irradiation on Temperatures

Direct Non-Linear EffectNeutron Diffusion

We know there is:

Irradiation Against TemperatureIrradiation Against Temperature

Hot Inner Region Cold Outer Region

Similar Behaviour– Sharp Increase

– Constant Weak Relationship Scatter/Omitted

Effects– Geometry

– Control Action

– Neutron Diffusion

– Random Variation

Pre-whitened Irradiation Against TemperaturePre-whitened Irradiation Against Temperature

Indirectly Correct for Low Frequency Omitted Effects– Control Action

– Neutron Diffusion Reveals Local

Relationship Kernel Smoothing Tunnicliffe-Wilson

(2000) Near Linear Correlation = 0.6 Less Scatter

Direct Irradiation EffectDirect Irradiation Effect

Linear Splines (0:1000:7000)Linear & ExponentialChoose exponential decay to

minimise cross-validation RMSUse fitted effect to examine cross-

spectrum with temperatures

Spatial Impulse ResponseSpatial Impulse Response

Inverse Transfer Function between Fitted Irradiation and Temperature Spectrum Corrected for Geometry

Effect of Unit Increase in Fuel Irradiation on Temperatures

Direct Effect in Centre Diffusion Effect Negative Effect in Adjacent

Channels Due to Neutron Absorption in Older Fuel

Irradiation Diffusion EffectsIrradiation Diffusion Effects

Neutron Diffusion:– negative effect within 2 channels– small positive effect beyond 2 channels

Modelled by:– 2 spatial kernel smoothers of irradiation

(bandwidths of 2 and 6 channels)– lagged irradiation regressors

(symmetric, up to 6 channels)

Smooth ComponentSmooth Component

Stochastic/Non-deterministic Square Region Spatial Sinusoidal Regressors Periods Wider than 12 Channels Constrained Coefficients Dampen Shorter Periods Prevents Over-fitting Fits a Random Smooth Surface

Mixed ModelMixed Model

Linear ModelFixed & Random EffectsMixed Model Formulation:– Snedecor and Cochrane (1989)– – has constrained variance

Use cross-validation predictions to prevent over-fitting

εβα ZXYβ

Prediction from Full GridPrediction from Full Grid Cross-validation Prediction RMS of 2.34

Residuals from Full GridResiduals from Full Grid Few Large Residuals Noisy Spectrum

No Low Frequency Some Residual Structure

Prediction from 3 by 3 Sub-gridPrediction from 3 by 3 Sub-grid Fixed Effects from Full Model RMS of 2.64

Residuals from 3x3 GridResiduals from 3x3 Grid

Larger Residuals Some Low Frequency

ConclusionConclusion

Statistical model predicts very well:– RMS of 2.34 from full grid– RMS of 2.64 from 3 by 3 sub-grid

(assuming fixed effects known)– Physical Model RMS of 4 on full grid

Identified significant geometry effects Enhancements to Physical Model Can be used for on-line measurement

validation

Physical or Statistical Model?Physical or Statistical Model?

Nuclear properties of reactor Transferable to other

reactors Reactor operation planning:

– refueling patterns

– fault studies Limited by our physical

knowledge Can’t account for stochastic

variations Expensive computationally

Empirical Requires data Non-transferable Account for all regular

variation Improve accuracy of

Physical Model:– identify omitted effects

Rapid on-line prediction Rigorous framework for Risk

Assessment

Physical Model Statistical Model

Further InvestigationFurther Investigation Prediction on full circular reactor region Accurate estimation of geometry effects from sub-grid Cross-validation - justification as estimation criterion instead

of ML/REML Smooth random component specification:

– parameters optimized to predictive application

– differ slightly between full and 3 by 3 sub-grid

– signals some mis-specification of spatial error correlation Stochastic standpipe effect caused by measurement errors

within a standpipe:– reduces RMS on full grid to 2.02

– RMS doesn’t improve on 3 by 3 sub-grid

– expect only 2 measurements per standpipe

– competes with smooth random component

ReferencesReferencesBox, G.E.P. & Jenkins, G.M. (1976). Time Series Analysis, Forecasting and Control. Holden-Day.

Logsdon, J. & Tunnicliffe-Wilson, G. (2000). Prediction of extreme temperatures in a reactor using measurements affected by control action. Technometrics (under revision).

Scarrott, C.J. & Tunnicliffe-Wilson, G. (2000). Spatial Spectral Estimation for Reactor Modeling and Control. Presentation at Joint Research Conference 2000 - Statistical Methods for Quality, Industry and Technology. Available from http://www.maths.lancs.ac.uk/~scarrott/Presentations.html.

Snedecor, G.W. and Cochrane, W.G. (1989). Statistical Methods(eighth edition). Iowa State University Press, Ames.

Thomson, D.J., (1990). Quadratic-inverse spectrum estimates: application to palaeoclimatology. Phil. Trans Roy. Soc. Lond. A, 332, 539-597.

FOR MORE INFO...

Carl Scarrott - c.scarrott@lancaster.ac.ukGranville Tunnicliffe-Wilson - g.tunnicliffe-wilson@lancaster.ac.uk