1. Building a Statistical Model to Predict Reactor Temperatures Carl Scarrott Granville Tunnicliffe-Wilson Lancaster University c.scarrott@lancaster.ac.uk g.tunnicliffe-wilson@lancaster.ac.uk

2. Outline Objectives Data Statistical Model Exploratory Analysis Results Conclusion References

3. Project 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?

4. Wylfa Reactors Magnox Type Anglesey, Wales Reactor Core 6156 Fuel Channels Fuel Channel Gas Outlet Temperatures (CGOTs) Degrees C All Measured

5. Dungeness Reactors Magnox Type Kent, England 3932 Fuel Channels Fixed Subset Measured: –450 on 3 by 3 sub-grid –112 off-grid What About Unmeasured?

6. Temperature Data Radial Banding Smooth Surface Standpipes (4x4) Chequer-board Triangles East to West Ridges Missing Spatial Structure:

7. Irradiation Data Fuel Age or Irradiation Main Explanatory Variable Old Fuel = Red New Fuel = Blue Standpipe Refuelling Chequer-board Triangles Regular & Periodic

8. Temperature and Irradiation Data

9. Statistical 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

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

11. Exploratory 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)

12. Application - Temperature and Irradiation Data Temperature SpectrumIrradiation Spectrum

13. Reactor 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

14. Geometry Regressors - 1 Brick Size Chequer-board Heat Differential Coolant Leakage into Interstitial Holes Cools Adjacent Fuel Channels E-W Ridge of 2 channels

15. 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 - 2

16. Geometry Spectra Brick Chequer-board E-W Ridge E-W Ridge and Chequer-boardControl Rod Hole Indicator

17. Estimated Geometry Effect All Geometry Effects Estimated in Model Fit

18. How to Model F(.)? Effect of Fuel Irradiation on Temperatures Direct Non-Linear Effect Neutron Diffusion We know there is:

19. Irradiation 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

20. Pre-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

21. Direct Irradiation Effect Linear Splines (0:1000:7000) Linear & Exponential Choose exponential decay to minimise cross-validation RMS Use fitted effect to examine cross- spectrum with temperatures

22. Spatial 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

23. Irradiation 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)

24. Smooth 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

25. Mixed Model Linear Model Fixed & Random Effects Mixed Model Formulation: –Snedecor and Cochrane (1989) – – has constrained variance Use cross-validation predictions to prevent over-fitting

26. Prediction from Full Grid Cross-validation Prediction RMS of 2.34

27. Residuals from Full Grid Few Large Residuals Noisy Spectrum No Low Frequency Some Residual Structure

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

29. Residuals from 3x3 Grid Larger Residuals Some Low Frequency

30. Conclusion 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

31. Physical or Statistical Model? Nuclear properties of reactor Transferable to other reactors Reactor operation planning: –refueling patterns –fault studies Limited by our physical knowledge Cant 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

32. Further 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 doesnt improve on 3 by 3 sub-grid –expect only 2 measurements per standpipe –competes with smooth random component

33. 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 - c.scarrott@lancaster.ac.uk Granville Tunnicliffe-Wilson - g.tunnicliffe-wilson@lancaster.ac.uk