Spatial Spectral Estimation for Reactor Modeling and Control Carl Scarrott Granville Tunnicliffe-Wilson Lancaster University, UK

Contents Objectives Data Statistical Model Exploratory Analysis –2 Dimensional Spectral Analysis –Circular Multi-Taper Method Application Conclusions References

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?

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

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

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

Temperature and Irradiation Data

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

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

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

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

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

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

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

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

Tapering - 1 Raw Spectrum Leakage

Tapering - 2 Less Leakage Wider Bandwidth

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 Same Bandwidth Slightly More Leakage

Multi-Tapering on 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) How Calculate Continuous Tapers over a Disc?

Multi-Tapering on Reactor 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 How to Calculate Tapers for Square Grid over Reactor Region?

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

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

Compare Spectrum of Tapers N = 0 n = 0Average SpectrumNo Tapering Same Colour Scale

Application - Temperature Data 1 Raw SpectrumTapered Spectrum (1 taper)

Application - Temperature Data 2 Raw SpectrumMulti-Taper Spectrum (5 tapers)

Summary 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 For short series...

Application - Temperature and Irradiation Data Tapered Temperature SpectrumTapered Irradiation Spectrum

Application - Pre-whitened Temperature and Irradiation Data Tapered Pre-whitened Temperature SpectrumTapered Pre-whitened Irradiation Spectrum

Application - Temperature Corrected for Geometry & Fitted Irradiation Temperature Less Geometry EffectsDirect Irradiation Effect

Coherency - 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 - Temperature & Irradiation - 27 Tapers F value = 0.11 Clearly Insignificant at Geometry Effect Frequencies

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

Coherency Significance Test Phase Randomisation 100 Simulations 95% Tolerance Interval F Value Inappropriate Line Components

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

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

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

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 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, Thomson, D.J., (1990). Quadratic-inverse spectrum estimates: application to palaeoclimatology. Phil. Trans Roy. Soc. Lond. A, 332, 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, FOR MORE INFO... Carl Scarrott - Granville Tunnicliffe-Wilson -