Building a Statistical Model to Predict Reactor Temperatures
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Building a Statistical Model to Building a Statistical Model to Predict Reactor TemperaturesPredict Reactor Temperatures
Carl Scarrott
Granville Tunnicliffe-WilsonLancaster University
[email protected]@lancaster.ac.uk
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OutlineOutline
ObjectivesDataStatistical ModelExploratory AnalysisResultsConclusionReferences
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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?
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Wylfa ReactorsWylfa Reactors
Magnox Type Anglesey, Wales Reactor Core 6156 Fuel Channels Fuel Channel Gas
Outlet Temperatures (CGOT’s)
Degrees C All Measured
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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?
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Temperature DataTemperature Data
Radial Banding Smooth Surface Standpipes (4x4) Chequer-board Triangles East to West
Ridges Missing
Spatial Structure:
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Irradiation DataIrradiation Data
Fuel Age or Irradiation
Main Explanatory Variable
Old Fuel = Red New Fuel = Blue
Standpipe Refuelling
Chequer-board Triangles Regular & Periodic
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Temperature and Irradiation DataTemperature and Irradiation Data
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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
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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(.)
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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)
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Application - Temperature and Irradiation DataApplication - Temperature and Irradiation Data
Temperature Spectrum Irradiation Spectrum
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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
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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
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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
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Geometry SpectraGeometry Spectra
Brick Chequer-board E-W Ridge
E-W Ridge and Chequer-board Control Rod Hole Indicator
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Estimated Geometry EffectEstimated Geometry Effect
All Geometry Effects Estimated in Model Fit
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How to Model F(.)?How to Model F(.)?
Effect of Fuel Irradiation on Temperatures
Direct Non-Linear EffectNeutron Diffusion
We know there is:
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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
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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
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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
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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
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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)
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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
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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
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Prediction from Full GridPrediction from Full Grid Cross-validation Prediction RMS of 2.34
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Residuals from Full GridResiduals from Full Grid Few Large Residuals Noisy Spectrum
No Low Frequency Some Residual Structure
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Prediction from 3 by 3 Sub-gridPrediction from 3 by 3 Sub-grid Fixed Effects from Full Model RMS of 2.64
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Residuals from 3x3 GridResiduals from 3x3 Grid
Larger Residuals Some Low Frequency
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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
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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
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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
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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 - [email protected] Tunnicliffe-Wilson - [email protected]