Spatial Spectral Estimation for Reactor Modeling and Control
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Transcript of Spatial Spectral Estimation for Reactor Modeling and Control
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Spatial Spectral Estimation for Spatial Spectral Estimation for Reactor Modeling and ControlReactor Modeling and Control
Carl Scarrott
Granville Tunnicliffe-WilsonLancaster University, UK
[email protected]@lancaster.ac.uk
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ContentsContents
Objectives Data Statistical Model Exploratory Analysis
– 2 Dimensional Spectral Analysis– Circular Multi-Taper Method
Application Conclusions References
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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?
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Magnox ReactorMagnox Reactor
Wylfa Reactors Anglesey, Wales Magnox Type 6156 Fuel Channels Fuel Channel
Gas Outlet Temperatures (CGOT’s)
All Measured
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Temperature DataTemperature Data
Radial Banding Standpipes (4x4) Chequer-board Triangles East to West
Ridges Missing
Spatial Structure:
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Irradiation DataIrradiation Data
Fuel Age or Irradiation
Old Fuel = Red New Fuel = Blue
Refuelled by Standpipe
Chequer-board Within Standpipe
Triangles Regular & Periodic
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Temperature and Irradiation DataTemperature and Irradiation Data
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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
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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)
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Statistical ModelStatistical Model
Predict TemperaturesExplanatory Variables:– Fuel Irradiation (fixed)– Physical/Geometry Effects - (fixed)– Control - Smooth (stochastic)
Random Errors
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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
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How to Model F(.)?How to Model F(.)?
Effect of Fuel Irradiation on Temperatures
Direct Non-Linear EffectNeutron Dispersion
We know there is:
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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
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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
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SolutionsSolutions
Tapering of DataSmoothing of SpectrumFilteringParametric Methods
Multi-Taper Method (Thomson,1982)
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Tapering - 1Tapering - 1
2
xcos
y
Raw Spectrum
Leakage
Scalelog 10
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Tapering - 2Tapering - 2
Less Leakage
Wider Bandwidth
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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
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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 ,
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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?
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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
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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
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Compare Spectrum of TapersCompare Spectrum of Tapers
N = 0 n = 0 Average SpectrumNo Tapering
Same Colour Scale
π 2c
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Application - Temperature Data 1Application - Temperature Data 1
Raw Spectrum Tapered Spectrum (1 taper)
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Application - Temperature Data 2Application - Temperature Data 2
Raw Spectrum Multi-Taper Spectrum (5 tapers)
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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...
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Application - Temperature and Irradiation DataApplication - Temperature and Irradiation Data
Tapered Temperature Spectrum Tapered Irradiation Spectrum
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Application - Pre-whitened Temperature and Application - Pre-whitened Temperature and Irradiation Data Irradiation Data
Tapered Pre-whitened Temperature Spectrum Tapered Pre-whitened Irradiation Spectrum
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Application - Temperature Corrected for Geometry Application - Temperature Corrected for Geometry & Fitted Irradiation & Fitted Irradiation
Temperature Less Geometry Effects Direct Irradiation Effect
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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
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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
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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
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SIR - Tolerance IntervalsSIR - Tolerance Intervals
Phase Randomisation 100 Simulations 5 & 95% Tolerance
Intervals Smooth Function Implies Only Direct and
Adjacent Channel Effects are Significant
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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
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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 - [email protected] Tunnicliffe-Wilson - [email protected]