Spatial Spectral Estimation for Reactor Modeling and Control

1. Spatial Spectral Estimation for Reactor Modeling and Control Carl Scarrott Granville Tunnicliffe-Wilson Lancaster University, UK c.scarrott@lancaster.ac.uk g.tunnicliffe-wilson@lancaster.ac.uk

2. Contents • Objectives • Data • Statistical Model • Exploratory Analysis • 2 Dimensional Spectral Analysis • Circular Multi-Taper Method • Application • Conclusions • 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 model physical effects?

4. Magnox Reactor • Wylfa Reactors • Anglesey, Wales • Magnox Type • 6156 Fuel Channels • Fuel Channel Gas Outlet Temperatures (CGOT’s) • All Measured

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

6. Irradiation Data • Fuel Age or Irradiation • Old Fuel = Red New Fuel = Blue • Refuelled by Standpipe • Chequer-board Within Standpipe • Triangles • Regular & Periodic

7. Temperature and Irradiation Data

8. Irradiation Against Temperature • Hot Inner Region • Cold Outer Region • Scatter/Omitted Effects • Geometry • Control Action • Neutron Dispersion • Random Variation • Similar Behaviour • Sharp Increase • Constant • Weak Relationship

9. Pre-whitened Irradiation Against Temperature • Pre-whitening • Kernel Smoothing • Tunnicliffe-Wilson (2000) • Reveals Local Relationship • Near Linear • Less Scatter • Indirectly Corrects for Low Frequency Omitted Effects • Control Action • Neutron Dispersion

10. Statistical Model • Predict Temperatures • Explanatory Variables: • Fuel Irradiation (fixed) • Physical/Geometry Effects - (fixed) • Control - Smooth (stochastic) • Random Errors

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

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

13. Exploratory Analysis • 2 Dimensional Spectral Analysis • Fuel Irradiation & Geometry Effects are: • Regular • Periodic • Easy to Distinguish in Spectrum • Remove Geometry Effects • Rigorous Framework to Examine Both Aspect of Fuel Irradiation Effect

14. Problems • Raw Spectrum Estimates Biased by Spectral Leakage • Caused by Finite and Discrete Data or Edge Effects • Inconsistent Estimate of Spectrum • doesn’t improve with sample size

15. Solutions • Tapering of Data • Smoothing of Spectrum • Filtering • Parametric Methods • Multi-Taper Method (Thomson,1982)

16. Leakage Tapering - 1 Raw Spectrum

17. Tapering - 2 Wider Bandwidth Less Leakage

18. Multi-Taper Method • Thomson (1990) • Multiple Orthogonal Tapers • Maximise Spectral Energy in Bandwidth • Calculation - Eigen-problem • Average Tapered Spectra • Smoothed Estimate • K = No. of Tapers - Increases With N Slightly More Leakage Same Bandwidth

19. Multi-Tapering on a Disc How Calculate Continuous Tapers over a Disc? • Slepian (1968) • Continuous Function Over Unit Disc • Maximise Spectral Energy in Disc • Specify Bandwidth c in Frequency Domain • Seperable to 1-Dimensional Eigen-problem: • Solve for particular N and order eigenvalues by n • Want eigenvalues close to 1 • Discretized to Matrix Eigen-problem in Zhang (1994)

20. Multi-Tapering on Reactor How to Calculate Tapers for Square Grid over Reactor Region? • Define linear mapping A which: • calculates spectrum over reactor region • truncates spectrum outside of bandwidth W • transforms spectrum back onto reactor region • Want to find eigenvalues/vectors of A • Use continuous tapers as initial estimate • Apply Power Method: • repeated application of A on tapers • Resolves eigenvalues close to 1

21. sin cos Circular Tapers N = 0 n = 0 N = 1 n = 0 N = 2 n = 0 Only one taper for N = 0 as sin(0) = 0

22. Spectrum of Circular Tapers N = 2 n = 0 N = 0 n = 0 N = 1 n = 0 Same Color Axis

23. Compare Spectrum of Tapers No Tapering N = 0 n = 0 Average Spectrum • Same Colour Scale

24. Application - Temperature Data 1 Raw Spectrum Tapered Spectrum (1 taper)

25. Application - Temperature Data 2 Raw Spectrum Multi-Taper Spectrum (5 tapers)

26. Summary For short series... • One taper sufficient to remove leakage and clarify peaks • use this to identify geometry effects • Multiple tapers improve spectrum degrees of freedom and smooth continuous part of spectrum • required for cross-spectral analysis between irradiation and temperatures

27. Application - Temperature and Irradiation Data Tapered Temperature Spectrum Tapered Irradiation Spectrum

28. Application - Pre-whitened Temperature and Irradiation Data Tapered Pre-whitened Temperature Spectrum Tapered Pre-whitened Irradiation Spectrum

29. Application - Temperature Corrected for Geometry & Fitted Irradiation Temperature Less Geometry Effects Direct Irradiation Effect

30. - 27 Tapers • More Smoothing • Linear Association at Each Frequency • Squared Correlation • 1 - High Coherency 0 - Low Coherency • F Value = 0.11 • Spectra are Highly Related Coherency

31. Coherency Significance Test • Phase Randomisation • 100 Simulations • 95% Tolerance Interval • Robust Check on F Value • Line Components • Same Colour Axis • Confirms Significant Coherency

32. Spatial Impulse Response (SIR) • Inverse Fourier Transform of Transfer Function • Effect of Unit Increase in Fitted Fuel Irradiation on Temperatures • Direct Effect in Centre • Dispersion Effect • Negative Effect in Adjacent Channels

33. SIR - Tolerance Intervals • Phase Randomisation • 100 Simulations • 5 & 95% Tolerance Intervals • Smooth Function • Implies Only Direct and Adjacent Channel Effects are Significant

34. Conclusion • Developed MTM on a Disc • Adapted to Roughly Circular Region • Extended to Cross-Spectral Analysis • Tolerance Intervals by: • Phase Randomization • Jackknifing (Thomson et al,1990) • Identified Significant Geometry Effects • Evaluated Effect of Fuel Irradiation on Temperatures • Prediction RMS = 2.5 • Compared to Physical Model RMS = 4

35. References 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). Building a statistical model to predict reactor temperatures. Industrial Statistics in Action 2000 - Conference Presentation and Paper. Slepian, D.S., (1964). Prolate spheroidal wave functions, Fourier analysis and uncertainty - IV: Extension to many dimensions; generalized prolate spheroidal functions. Bell System Tech. J., 43, 3009-3057. Thomson, D.J., (1990). Quadratic-inverse spectrum estimates: application to palaeoclimatology. Phil. Trans Roy. Soc. Lond. A, 332, 539-597. Thomson, D.J. & Chave, A.D., (1990). Jacknifed Error Estimates for Spectra, Coherences and Transfer Functions in Advances in Spectrum Analysis (ed. Haykin, S.), Prentice-Hall. Zhang, X., (1994). Wavenumber specrum of very short wind waves: an application of two-dimensional Slepian windows to spectral estimation. J. of Atmos. and Oceanogr. Tec., 11, 489-505. FOR MORE INFO... Carl Scarrott - c.scarrott@lancaster.ac.uk Granville Tunnicliffe-Wilson - g.tunnicliffe-wilson@lancaster.ac.uk