Building a Statistical Model to Predict Reactor Temperatures

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 (CGOT’s) • 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 Spatial Structure: • Radial Banding • Smooth Surface • Standpipes (4x4) • Chequer-board • Triangles • East to West Ridges • Missing

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 Spectrum Irradiation 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 • Coolant Leakage into Interstitial Holes • Cools Adjacent Fuel Channels • E-W Ridge of 2 channels • Brick Size Chequer-board • Heat Differential

15. Geometry Regressors - 2 • Brick Size Chequer-board in Central 2 Rows • Larger Bricks Cooled More as Greater Surface Area • Control Rod Hole Larger • Adjacent Channels Cooled More

16. Geometry Spectra Brick Chequer-board E-W Ridge E-W Ridge and Chequer-board Control 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 We know there is: • Direct Non-Linear Effect • Neutron Diffusion

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. 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 or Statistical Model? 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 doesn’t 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