Statistical Prediction of Reactor Temperatures lancaster.ac.uk 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: Control Limit Heavier Tail

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 BenefitsDrawbacks

PANTHER Improvement - Wylfa Temperature MeasurementsPANTHER Simulation Cross-core tilt Sensitive to neutron flux

PANTHER Improvement - Wylfa Temperature 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 T ij =F(X rs )+G ij +N ij +S ij +E ij – 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 T ij X rs F(.) G ij N ij S ij E ij 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 SpectrumIrradiation 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

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

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

Prediction from Full Grid Temperature MeasurementsCross-validation Prediction

Residuals from Full Grid Cross-Validation ResidualsResidual 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 MeasurementsCross-validation Prediction

Residuals from 3x3 Sub-Grid Prediction ResidualsResidual 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 Fit3x3 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

References Scarrott, C.J. & Tunnicliffe-Wilson, G. (2001a). Building a statistical model to predict reactor temperatures. J. Appl. Stat 28(3), 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: lancaster.ac.uk lancaster.ac.uk