Building a Statistical Model to Predict Reactor Temperatures Carl Scarrott Granville Tunnicliffe-Wilson Lancaster University
[email protected] [email protected]
Outline l Objectives l Data l Statistical
Model l Exploratory Analysis l Results l Conclusion l References
Project Objectives l l l
Assess risk of temperature exceedance in Magnox nuclear reactors Establish safe operating limits Issues: – Subset of measurements – Control effect – Upper tail censored
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Solution: – Predict unobserved temperatures – Physical model – Statistical model
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How to identify and model physical effects? How to model remaining stochastic variation?
Wylfa Reactors l l l l l
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Magnox Type Anglesey, Wales Reactor Core 6156 Fuel Channels Fuel Channel Gas Outlet Temperatures (CGOT’s) Degrees C All Measured
Dungeness Reactors l l l l
Magnox Type Kent, England 3932 Fuel Channels Fixed Subset Measured: – 450 on 3 by 3 sub-grid – 112 off-grid
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What About Unmeasured?
Temperature Data Spatial Structure: l l l l l l l
Radial Banding Smooth Surface Standpipes (4x4) Chequer-board Triangles East to West Ridges Missing
Irradiation Data l l l l l l l
Fuel Age or Irradiation Main Explanatory Variable Old Fuel = Red New Fuel = Blue Standpipe Refuelling Chequer-board Triangles Regular & Periodic
Temperature and Irradiation Data
Statistical Model l l
Predict Temperatures Explanatory Variables (Fixed Effects): – Fuel Irradiation – Reactor Geometry – Operating Conditions
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Stochastic (Non-deterministic) Components: – Smooth Variation Resulting from Control Action
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Random Errors
Statistical Model T = F( X ) + G + N + E ij
T
ij
X
rs
F(.) G N E
ij ij ij
rs
ij
ij
ij
– Temperature at Channel (i,j) – Fuel Irradiation for Channel (r,s) – Direct and Neutron Diffusion Effect – Linear Geometry – Slowly Varying Spatial Component – Random Error
Exploratory Analysis l l
2 Dimensional Spectral Analysis Fuel Irradiation & Geometry Effects are: – Regular – Periodic
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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 Data Temperature Spectrum
Irradiation Spectrum
Reactor Geometry l l
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Standpipe Geometry Fuel Channels in Holes Through Graphite Bricks Interstitial Holes Along Central 2 Rows: – Control Rod – Fixed Absorber
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2 Brick Sizes: – Octagonal – Square
Geometry Regressors - 1 l l
Brick Size Chequer-board Heat Differential
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Coolant Leakage into Interstitial Holes Cools Adjacent Fuel Channels E-W Ridge of 2 channels
Geometry Regressors - 2 l l
Brick Size Chequer-board in Central 2 Rows Larger Bricks Cooled More as Greater Surface Area
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Control Rod Hole Larger Adjacent Channels Cooled More
Geometry Spectra Brick Chequer-board
E-W Ridge
E-W Ridge and Chequer-board
Control Rod Hole Indicator
Estimated Geometry Effect l
All Geometry Effects
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Estimated in Model Fit
How to Model F(.)? l
Effect of Fuel Irradiation on Temperatures
We know there is: l Direct
Non-Linear Effect l Neutron Diffusion
Irradiation Against Temperature l l
Hot Inner Region Cold Outer Region Similar Behaviour – Sharp Increase – Constant
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Weak Relationship Scatter/Omitted Effects – – – –
Geometry Control Action Neutron Diffusion Random Variation
Pre-whitened Irradiation Against Temperature l
Indirectly Correct for Low Frequency Omitted Effects – Control Action – Neutron Diffusion
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Reveals Local Relationship Kernel Smoothing Tunnicliffe-Wilson (2000) Near Linear Correlation = 0.6 Less Scatter
Direct Irradiation Effect l Linear
Splines (0:1000:7000) l Linear & Exponential l Choose exponential decay to minimise cross-validation RMS l Use fitted effect to examine crossspectrum with temperatures
Spatial Impulse Response l
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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 Effects l
Neutron Diffusion: – negative effect within 2 channels – small positive effect beyond 2 channels
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Modelled by: – 2 spatial kernel smoothers of irradiation (bandwidths of 2 and 6 channels) – lagged irradiation regressors (symmetric, up to 6 channels)
Smooth Component l l l l l l l l
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 Model l Linear
Model l Fixed & Random Effects l Mixed Model Formulation: – Snedecor and Cochrane (1989) – Y = Xa + Zß + e – ß has constrained variance l Use
cross-validation predictions to prevent over-fitting
Prediction from Full Grid l l
Cross-validation Prediction RMS of 2.34
Residuals from Full Grid l l
Few Large Residuals Noisy Spectrum
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No Low Frequency Some Residual Structure
Prediction from 3 by 3 Sub-grid l l
Fixed Effects from Full Model RMS of 2.64
Residuals from 3x3 Grid l
Larger Residuals
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Some Low Frequency
Conclusion l
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
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Identified significant geometry effects Enhancements to Physical Model Can be used for on-line measurement validation
Physical or Statistical Model? Physical Model l l l
Nuclear properties of reactor Transferable to other reactors Reactor operation planning: – refueling patterns – fault studies
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Limited by our physical knowledge Can’t account for stochastic variations Expensive computationally
Statistical Model l l l l l
Empirical Requires data Non-transferable Account for all regular variation Improve accuracy of Physical Model: – identify omitted effects
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Rapid on-line prediction Rigorous framework for Risk Assessment
Further Investigation l l l l
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
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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
References Box, 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] Granville Tunnicliffe-Wilson -
[email protected]
