Statistical Prediction of Reactor Temperatures

c.scarrott@lancaster.ac.uk

http://www.maths.lancs.ac.uk/~scarrott

Maths and Stats Dept, Lancaster University

Carl Scarrott

Contents

• Project Objectives• Data• Statistical Model

• Identification• Estimation• Diagnostics

• Results & Problems• Conclusions• Further Research• References

Project Objectives

Main Objectives:• Risk Assessment• PANTHER Improvements• Detecting & Replacing Rogue Measurements

Data from 2 nuclear power stations:• Dungeness - subset of measurements• Wylfa - all measured

Wylfa used to validate methodology for Dungeness

Risk Assessment

• Assess risk of temperature

exceedance in Magnox reactors

• Establish safe operating limits

• Issues:

– subset of measurements

– used for control

– truncated upper tail

• Solution:

– predict unobserved temperatures

– physical model?

– statistical model?

• How to identify and model fixed

and random effects?

• Parameter estimation and diagnostics

Dungeness Temperatures

Risk Assessment - Dungeness

• Assess risk of temperature

exceedance in Magnox reactors

• Establish safe operating limits

• Issues:

– subset of measurements

– used for control

– truncated upper tail

• Solution:

– predict unobserved temperatures

– physical model?

– statistical model?

• How to identify and model fixed

and random effects?

• Parameter estimation and diagnostics Control Limit

Risk Assessment - Wylfa

Wylfa Temperatures

Control Thermocouples:

Population:

ControlLimit

HeavierTail

PANTHER Improvements

• PANTHER - deterministic reactor physics model

– Nuclear and thermal properties of reactor

– Complex parametric model

– Predicts reactor conditions, e.g. temperatures

• Future operation planning:

– fault studies

– refueling cycles

• Does not require temperatures

– unbiased by control action

• Transferable to other reactors

• Limited by our physical knowledge

• Reactors are ‘floppy’

– Sensitive to inputs

• Expensive computationally

• Cannot account for stochastic

variation

Benefits Drawbacks

PANTHER Improvement - WylfaTemperature Measurements PANTHER Simulation

• Cross-core tilt• Sensitive to neutron flux

PANTHER Improvement - WylfaTemperature Measurements

• Prediction RMS = 4• High frequency residual structure

PANTHER Prediction

Rogue Measurement Detection & Replacement

• Thermocouples used to measure temperature

• Occasional spuriously low/high or missing due to fault

• Important for risk assessment

• How to detect?– Subset of control thermocouples

• How to replace?– Temperature prediction:

• On-line tool => Statistical Prediction Spurious

– past values?– local interpolation?– Physical or Statistical?

Statistical Model Requirements

Risk Assessment:• Accurately predict temperatures• Use subset (3x3 sub-grid) of measurements• Unbiased by control action• Assess limits on remaining variation for risk

PANTHER Improvement:• Identify omitted & poorly modelled effects• Interpretable regressors

Detect & Replace Missing:• On-line => quick to compute

Wylfa Reactor Temperatures

• Magnox reactor

• Anglesey, Wales

• Reactor core

• Moderator:– Column of graphite bricks

• 6156 fuel channels

• Channel gas outlet temperatures (CGOT’s)

• Degrees C

• All Measured

Dungeness Reactors

• Magnox reactor

• Kent

• 3932 fuel channels

• Fixed Subset Measured:– 450 on 3x3 sub-grid

– 112 off-grid

• Used for reactor control

• Truncated upper tail

• What about unmeasured?

Wylfa Temperature Data

• Radial banding– flattened (inner)

– unflattened (outer)

• Smooth surface– control effect

– neutron diffusion

• Standpipes (4x4)– Measurement Error

• Chequerboard

• Triangles

• East to west ridge

• Missing

Spatial Structure:

Fuel Irradiation

• Fuel Age or Irradiation

• Main Explanatory Variable

• Old Fuel = Red

New Fuel = Blue

• Mega-Watt Days per tonne

• Standpipe Refuelling

• Chequerboard

• Triangles

• Regular & Periodic

Temperature and Irradiation Data

Statistical Model

• Predict Temperatures

• Explanatory Variables (Fixed Effects):

• Fuel Irradiation

• Reactor Geometry

• Operating Conditions - e.g. control rod insertion

• Stochastic/Non-deterministic Components (Random Effects):

• Control Effect - smooth assumption

• Measurement Errors - standpipe structure

• Random Errors

• See Scarrott and Tunnicliffe-Wilson (2001a)

Statistical Model

Tij=F(Xrs)+Gij+Nij+Sij+Eij

– Temperature at Channel (i,j)

– Fuel Irradiation for Channel (r,s)

– Direct and Neutron Diffusion Effect

– Linear Geometry

– Slowly Varying Spatial Component (Control)

– Standpipe Measurement Error

– Noise

Tij

Xrs

F(.)Gij

Nij

Sij

Eij

Fixed

Random

Response

Exploratory Analysis

• 2 dimensional spectral analysis

• Fuel irradiation & geometry effects are:

– regular

– periodic

• Easy to identify in spectrum

• Cross-spectrum used to examine the neutron diffusion effect

• Random effects also have spectral representation

• Multi-taper method developed for reactor region to minimise bias caused by spectral leakage (Scarrott and Tunnicliffe-Wilson, 2001b)

Main Tool:

Also:• PANTHER predictions & industry knowledge

• Graphical & non-parametric methods

Application - Temperature and Irradiation Data

Temperature Spectrum Irradiation Spectrum

• Geometry Effects• Standpipe Component

• Low Frequency• Similarities

Reactor Geometry - Example• Standpipe geometry

• Interstitial Channels

• Coolant Leakage

• Cools Adjacent Fuel Channels

• E-W Ridge of 2 channels

• Highlighted in this project• Significant improvements to PANTHER

Estimated Geometry Effects

• All Geometry Effects• Estimated in Statistical Model

• Spectrum of Geometry Effects

How to Model F(.)?Effect of Fuel Irradiation on Temperatures

• How to identify effect?– Empirical evidence– PANTHER predictions– Industry knowledge

• Suggest 2 components:– Direct Non-Linear Effect– Neutron Diffusion

• Direct non-linear:– increase for new fuel– flat for most of life– tail off at end

• Not observable as control smooths effect

Fuel Irradiation Against Temperature

• Hot flattened region

• Cold unflattened region

• Non-linear:– sharp increase for new

– constant most of life

– increase when old

• Weak relationship

• Scatter - Omitted Effects

• Statistical covariates:– Linear splines– Linear & exponential

• Exponential decay minimise cross-validation RMS

Neutron Diffusion Effect

Spatial Impulse Response Function • Spectral methods used to identify:

– Inverse transfer function

• Significance test by phase randomisation

(Scarrott and Tunnicliffe-Wilson, 2001b)

• Effect of irradiation on neighbours

• Features:

– Direct effect in centre

– Negative in adjacent channels

– Positive at further lags

• Modelled by:• Spatially lagged irradiation

• 2 kernel smoothers of irradiation: (bandwidths of 1 and 5 channels)

• Iterated until convergence

Smooth Random Effect

• Stochastic/Non-deterministic

• Control effect - main assumption

• Random smooth surface

• Hard to model deterministically

• Spectrum confirmed low frequency component:– Power law decay -

– Frequency cut-off -

– Variance -

• Spatial sinusoidal regressors:– harmonic and half harmonic cos/sin terms below cut-off

– 196 regressors

• Constrained coefficients by prior variance

• Dampens high frequency

• Prevents over-fitting

k

γ

Random Standpipe Effect

• Temperatures calibrated against standpipe mean

• Standpipe mean has measurement error

• Consistent random error within standpipe

• Clear evidence from prediction residuals

• Spectrum also confirmed residual standpipe structure

• Indicator for standpipes:– 392 extra regressors

• Constrain variance -

• Prevent over-fitting

Statistical Mixed Model

• Linear Mixed Model (Searle, 1982)

• Fixed & Random Effects

• Prior variance on

• Estimation by mixed model equations

• Use cross-validation predictions to prevent over-fitting

• Choose random effects parameters to minimize

cross-validation RMS

EZZXY2211 ββα

),,,( k

)β,β(21

Estimation Problems

• Inconsistency between random effects parameter from fit

with full data-set and 3x3 sub-grid

• Suggests model is mis-specified

• Similar to time series problem:

• model mis-specified

• multi-step ahead prediction criterion better than one-step ahead

• Conservative approach to use 3x3 sub-grid parameter

• Spectral domain approach!

Spectral Estimation

• Random Effects have spectral representation:

• Estimate parameters in frequency domain

• Iterative re-weighted least squares

• Similar parameters to 3x3 sub-grid

23223222 )...1)(...1(16

1),( gigififi eeeegfS

5.0,5.0),(I)]()2[(

42222

gfgf

gfn

otherwise0

1),(I

22 kgfgf

Prediction from Full Grid

Temperature Measurements Cross-validation Prediction

Residuals from Full Grid

Cross-Validation Residuals Residual Spectrum

• RMS = 2.2• Degrees of freedom = 440

• Remaining peaks are very small• Modulated effects• Omni-directional• Errors in variables?

Prediction from 3x3 Sub-Grid

Temperature Measurements Cross-validation Prediction

Residuals from 3x3 Sub-Grid

Prediction Residuals Residual Spectrum

• Off-grid Prediction RMS = 2.8• Degrees of freedom = 285

• Geometry effects • Low Frequency

Estimated Standpipe Effect• Correlation between Full and 3x3 = 0.72• Strong persistence• Temporal correlation: 0.96 (1 month) 0.56 (3 years)

Full Fit 3x3 Fit

Conclusions• Magnox Electric Plc. kindly provided 11 reactor snapshots

• Statistical model predicts very well:– RMS of 2.2 from full grid

– RMS of 2.8 from 3x3 sub-grid

– Physical Model RMS of 4 on full grid

• Strong temporal consistency:– fixed effects parameters

– residuals (0.87 to 0.46)

– standpipe errors (0.96 to 0.56)

• Enhancements to Physical Model - RMS < 2– residuals strongly consistent with statistical model

• Quick to compute - 10 minutes

• Could be used for near on-line measurement validation

Further Research

• Model remaining variation for risk assessment

• Predict number of exceedances of thresholds

• Fitting extreme value distribution:– Full & 3x3?

– On and off-grid?

– Regional?

– Temporal?

– Reactor?

• Spatio-temporal clustering of extreme residuals?

• Polar basis functions for smooth component

ReferencesScarrott, C.J. & Tunnicliffe-Wilson, G. (2001a). Building a statistical model to predict reactor temperatures. J. Appl. Stat 28(3), 497-511.

Scarrott, C.J. & Tunnicliffe-Wilson, G. (2001b). Spatial spectral estimation for reactor modeling and control. In Proc. of Amer. Stat. Assn. Q&P Sect.

Searle, S.R. (1982). Linear Models. Wiley, New York.

Further information:

c.scarrott@lancaster.ac.ukhttp://www.maths.lancs.ac.uk/~scarrott

g.tunnicliffe-wilson@lancaster.ac.uk