spatial spectral estimation for reactor modeling and control

Spatial Spectral Estimation for Spatial Spectral Estimation for Reactor Modeling and ControlReactor Modeling and Control

Carl Scarrott

Granville Tunnicliffe-WilsonLancaster University, UK

c.scarrott@lancaster.ac.ukg.tunnicliffe-wilson@lancaster.ac.uk

ContentsContents

Objectives Data Statistical Model Exploratory Analysis

– 2 Dimensional Spectral Analysis– Circular Multi-Taper Method

Application Conclusions References

Project ObjectivesProject 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 ReactorMagnox Reactor

Wylfa Reactors Anglesey, Wales Magnox Type 6156 Fuel Channels Fuel Channel

Gas Outlet Temperatures (CGOT’s)

All Measured

Temperature DataTemperature Data

Radial Banding Standpipes (4x4) Chequer-board Triangles East to West

Ridges Missing

Spatial Structure:

Irradiation DataIrradiation Data

Fuel Age or Irradiation

Old Fuel = Red New Fuel = Blue

Refuelled by Standpipe

Chequer-board Within Standpipe

Triangles Regular & Periodic

Temperature and Irradiation DataTemperature and Irradiation Data

Irradiation Against TemperatureIrradiation 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 TemperaturePre-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 ModelStatistical Model

Predict TemperaturesExplanatory Variables:– Fuel Irradiation (fixed)– Physical/Geometry Effects - (fixed)– Control - Smooth (stochastic)

Random Errors

Statistical ModelStatistical Model

ijijijrsij ENGXT )(F– Temperature at Channel (i,j)– Fuel Irradiation for Channel (r,s)– Direct and Neutron Dispersion Effect– Linear Geometry– Slowly Varying Spatial Component– Random Errorij

ij

ij

rs

ij

E

N

G

X

T

(.)F

How to Model F(.)?How to Model F(.)?

Effect of Fuel Irradiation on Temperatures

Direct Non-Linear EffectNeutron Dispersion

We know there is:

Exploratory AnalysisExploratory 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

ProblemsProblems

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

2

2χ )(S ~f

SolutionsSolutions

Tapering of DataSmoothing of SpectrumFilteringParametric Methods

Multi-Taper Method (Thomson,1982)

Tapering - 1Tapering - 1

2

xcos

y

Raw Spectrum

Leakage

Scalelog 10

Tapering - 2Tapering - 2

Less Leakage

Wider Bandwidth

Multi-Taper MethodMulti-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

22Kχ )(S ~f

Same Bandwidth

Slightly MoreLeakage

Multi-Tapering on a DiscMulti-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?

)sin(

)cos()()(

N

NrRr,ψ N,nN,n

,2,1,0,,2,1,0 nN

1

0 ,,, d)()( rrrRrrcJR nNNnNnN

2,, nNnN c

nN ,

Multi-Tapering on ReactorMulti-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 overReactor Region?

Circular TapersCircular Tapers

Only one taperfor N = 0 as sin(0) = 0

N = 0 n = 099952.0λ

π 2c

0.02272/88 W

N = 1 n = 099087.0λ

N = 2 n = 092732.0λ

99087.0λ 92372.0λ

sin

cos

Spectrum of Circular TapersSpectrum of Circular Tapers

N = 0 n = 0 N = 1 n = 0 N = 2 n = 0

π 2c 0.0227W Scalelog10

Same Color Axis

Compare Spectrum of TapersCompare Spectrum of Tapers

N = 0 n = 0 Average SpectrumNo Tapering

Same Colour Scale

π 2c

Application - Temperature Data 1Application - Temperature Data 1

Raw Spectrum Tapered Spectrum (1 taper)

Application - Temperature Data 2Application - Temperature Data 2

Raw Spectrum Multi-Taper Spectrum (5 tapers)

SummarySummary

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 DataApplication - Temperature and Irradiation Data

Tapered Temperature Spectrum Tapered Irradiation Spectrum

Application - Pre-whitened Temperature and Application - Pre-whitened Temperature and Irradiation Data Irradiation Data

Tapered Pre-whitened Temperature Spectrum Tapered Pre-whitened Irradiation Spectrum

Application - Temperature Corrected for Geometry Application - Temperature Corrected for Geometry & Fitted Irradiation & Fitted Irradiation

Temperature Less Geometry Effects Direct Irradiation Effect

CoherencyCoherency

- 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

π 4c

Coherency Significance TestCoherency Significance Test

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

Coherency

Spatial Impulse Response (SIR)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 IntervalsSIR - Tolerance Intervals

Phase Randomisation 100 Simulations 5 & 95% Tolerance

Intervals Smooth Function Implies Only Direct and

Adjacent Channel Effects are Significant

ConclusionConclusion

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

ReferencesReferences

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.ukGranville Tunnicliffe-Wilson - g.tunnicliffe-wilson@lancaster.ac.uk