Uncertainty and variability in weight loss forecasting J E Coote

Uncertainty and variability in weight loss forecastingJ E Coote 14th International Nuclear Graphite Specialists Meeting SeattleSeptember 2013

Contents • Background, terminology • WL modelling objectives • Architecture of FEAT-based ‘front-end’ methodology • Illustrative results • Review of model inputs • Discussion • Conclusions

Background • EDF’s UK AGR fleet requires regular safety case updates in support of continuing operation • Graphite weight loss (WL) forecasts are of central importance • Measured WL shows significant variability • Forecast WL (and analysis inputs) subject to uncertainty • Suitable interpretation of forecasts requires appropriate accounting of variability and uncertainty • Redundant modelling strategy: statistical and physically-based • This talk will describe how present physically-based methods address these objectives.

Terminology • Variability (describes a system): • a spread in system outputs in response to a fixed set of inputs • cannot be reduced with more measurements • e.g. weight loss measurements Example measurements, showing variability

Terminology • Uncertainty (describes an observer): • incomplete knowledge of a system • can be reduced if more data is available • e.g. model uncertainty (inability to determine beyond all doubt the model form that underlies a set of observations) Example of model uncertainty; different possible fits to same data:

Basis for physical modelling • ‘FEAT’ solver for FE structural/fluids/heat transfer modelling • Entirely deterministic (cannot itself account for variability or uncertainty) Fixed inputs Specific to reactor and brick layer Describe a generic channel FEAT analysis (single brick per run) Results through time Mesh values for density, weight loss, and deformed geometry. Scalar tuning parameters ARM (modifies oxidation rate) and Ascale (modifies shrinkage rate) Extract quantities of interest E.g. WL at a node average density over specified volume % of brick volume over a WL threshold. Channel-specific modifier Scales dosimetry to channel of interest

Desired forecast types Specific predictions: “WL=10.1% at fuel channel, brick 6, time t” Volume-average forecasts Probabilistic forecasts cdf

Analysis input data Calculated by MCBEND code Fixed Dosimetry Reactor power history Temperatures Gas coolant concentrations Pressure boundary conditions Brick geom & mesh Station records Calculated by PANTHER code (based ultimately on a limited set of reactor thermocouple measurements) Bulk concentrations from reactor data; local values not measurable but back-fitted via reasoning/judgement Not directly measurable; inferred from bulk flows, geometry, and model back-fitting. FEAT model Probabilised Virgin density and bore diameter ARM and Ascale parameters (modelling rate variability in density and shrinkage) Channel power Known geometry at start of life (within tolerances). Mesh developed for efficiency while matching/surpassing fidelity of other inputs. Routinely tuned Bulk measurements of virgin bricks Dosimetry calculations based on fuelling history WL measurements

Modelling variability (1) • Define variable parameters as distributions e.g. ARM~N(μ,σ) and run FEAT within Monte-Carlo analysis • FEAT is not quick; need alternative means to carry out 100000s of runs…

FEAT emulation Before: After: Quantity of interest e.g. volume-average density at slice 2 of upper trepanning location =1.61g.cm-3 Data extraction process FEAT FE analysis (slow) Nodal density & deformation results ARM=1.15 Ascale=1.02 F=0.97 (Tuning parameters) Emulate FEAT via response function lookups (fast) Quantity of interest e.g. volume-average density at slice 2 of upper trepanning location =1.61+ε g.cm-3 ( ε=small residual) ARM=1.15 Ascale=1.02 F=0.97

Variability in WL and shrinkage rates Assume distribution form for • ARM (WL rate modifier) • Ascale (shrinkage rate modifier) and optimise. Alternative distribution forms plausible: this indicates model uncertainty.

Bootstrapping, to estimate uncertainty Original measurements Tuning Best estimate [ξ]0 nx1 Tuning [ξ]1 nx1 Tuning [ξ]2 nx1 Tuning Resample n times e.g. 1000 [ξ]3 nx1 [ξ]BS nx1000 x1000 [ξ]4 nx1 Tuning Etc. Tuning [ξ]1000 nx1

Virgin bore/density distributions • Virgin bore and density was variable: model this variability explicitly to reduce ARM / Ascale spread • Model output is adjusted via randomly-sampled correction term • Bore distribution based on manufacturing tolerances • Density distribution based on bulk virgin measurements

Analysis input data Calculated by MCBEND code Fixed Dosimetry Reactor power history Temperatures Gas coolant concentrations Pressure boundary conditions Brick geom & mesh Station records Calculated by PANTHER code (based ultimately on a limited set of reactor thermocouple measurements) Bulk concentrations from reactor data; local values unmeasurable but back-fitted via reasoning/judgement Not directly measurable; inferred from bulk flows, geometry, and model back-fitting. FEAT model Probabilised Virgin density and bore diameter ARM and Ascale parameters (modelling rate variability in density and shrinkage) Channel power Known geometry at start of life (within tolerances). Mesh developed for efficiency while matching or surpassing fidelity of other inputs. Routinely tuned Bulk measurements of virgin bricks Dosimetry calculations based on fuelling history WL measurements

Model uncertainties / approximations FEAT model: • Assume neighbouring channels all at same power* • Assume Darcy Law model for permeable flow* • Assume particular model for coolant chemistry (under review) • Uncertain accuracy where no measurements exist Wider front-end methodology: • Assume generic channel axial dose profile • Assume independent and time-invariant distributions of given forms for shrinkage and WL rate corrections, and for virgin bore and density (subsequently tested) • Assume WL/density/bore variability is absolute rather than fractional (subsequently tested) *Methodology being implemented to remove assumption

Reassurances • FEAT represents best-available physical model following decades of development; front-end methodology offers unprecedented understanding of results. • Physical modelling should be the best option when predicting temporally and spatially beyond basis of current measurements. • Ongoing innovations in FEAT and the front-end methodology will address some current assumptions explicitly. • New inspection data and measurement techniques will reduce uncertainty & improve ability to characterise spatial variability. • Comparability against 3rd-party predictions via independent methods. • Core Component Condition Assessment Validation Protocol ensures that ongoing model performance is understood.

Conclusions • The most quantifiable elements of variability and uncertainty are now included explicitly in front-end methodology. • Uncertainties and assumptions expected to be reduced as more data and measurement techniques become available. • Some further sources of uncertainty remain outside; continue to handle via reasoning, judgement & sensitivity studies. • Important to retain an appreciation of how ALL uncertainties may propagate through cascaded analyses and affect any end conclusions.

www.fnc.co.uk

Uncertainty and variability in weight loss forecasting J E Coote