Bayesian Calibration of Thermal-Hydraulics Model with Time-Dependent Output

Bayesian Calibration of Thermal-Hydraulics Model with Time-Dependent Output

Application of statistical calibration combining data and statistical modelling to learn model parameters

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Damar Wicaksono

October 11, 2016
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    WIR SCHAFFEN WISSEN – HEUTE FÜR MORGEN Bayesian Calibration of

    Thermal-Hydraulics Model with Time-Dependent Output Damar Wicaksono*, Omar Zerkak, and Andreas Pautz NUTHOS-11 Conference, Gyeongju, Korea, 11.10.2016 *damar.wicaksono@epfl.ch
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 2 / 15 ) Uncertainty Quantification, Forward

    and Backward Given data from a Separate Effect Test facility… Inverted Annular Film Boiling (IAFB) Dispersed Flow (DFFB) Transitionary (Inverted Slug) TRACE Post-CHF Closure Laws Tclad evolution at different axial levels 4.114 [m] FEBA (KIT, Germany) 1 28 ⋮ TRACE Model …derive the uncertainties of TRACE reflood-related model parameters Forward Model : , → Backward Model Uncertain inputs (controllable & model parameters) Uncertainty Propagation Uncertain output Experimental data ? Statistical Analysis & Decision Making Inverse Quantification
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 3 / 15 ) Origin of Model

    Parameters Uncertainty FLECHT-SEASET Westinghouse, USA ACHILLES Winfrith, UK FEBA KIT, Germany Excerpt from the TRACE Code Theory Manual: 1. “…the approximate value of the coefficient in Eq. (4-119) was determined from data comparisons with FLECHT- SEASET high flooding rate reflood data…” (pp. 164) 2. “In TRACE, the above interfacial drag coefficient has been reduced by a factor of ¾ to better match FLECHT-SEASET high flooding rate reflood data, so…” (pp. 166) 3. “…in simulations of FLECHT-SEASET high flooding rate tests, this value appeared to be too large and thus was reduced to…” (pp. 223) RBHT PSU, USA What if the calibration was done on other, similar, separate effect test facilities?
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 4 / 15 ) True, Measured, and

    Predicted Values: a Formulation for Model Calibration exp ; , = true ; , + ϵ true ; , = model , �; , + ; , Model Discrepancy, unknown Experimental Data is a distorted information of the true value Model prediction is a distorted information of the true value Controlled/Design Variables, known (e.g., pressure, heat flux, etc.) spatio-temporal process, high-dimensional, correlated Measurement error, random Model Parameters; True or Best-Fitted, but unknown Computer Model, expensive to run Given ; , and other unknowns, what is � and its uncertainties?
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 5 / 15 ) Unknown ⇒ Uncertain:

    Bayesian Framework exp ; , = model , �; , + ; , + observed (data) Code runs; high-dimensional & expensive random ⇒ uncertain statistical meta-modeling ⇒ uncertain unknown ⇒ uncertain unknown ⇒ uncertain (unknown code input-output relationship for limited budget of runs) would like to know more about Each unknown is uncertain, treated probabilistically by assigning prior distribution to be updated with data
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 6 / 15 ) Region of high

    nonlinearities/ interaction Many initial parameters, but only handfuls are important (in some sense) 26 Parameters to specify the TRACE model: • 4 boundary conditions (Pressure, inlet, etc.) • 9 material properties (, , etc.) • 2 spacer grid model (HTCenh, Δ) • 10 Post-CHF closure relations (HTC, , etc.) • 1 quench temperature DFFB Interf. Drag Grid HT Enhc. IAFB Interf. Drag DFFB Wall HTC DFFB Vapor-Interf. HTC Power Inlet Velocity Quench Temp. Outlet Pressure IAFB Wall HTC Sensitivity analysis on average temperature by screening method (Morris) With flat independent prior uncertainties, either in linear or log scales Total effect sensitivity Indication of nonlinearities/ interaction Focus on the most important ones Tclad at mid-assembly
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 7 / 15 ) Gaussian Process is

    prior distribution for functions , ∗ ≡ Cov , ∗ = � exp − | − ∗ | () =1 f ~𝐺𝐺 (), 2 , ∗ Probability Distribution over function (of ) values Mean function, the usual: zero, constant, linear, etc. Common variance correlation function power-exponential correlation correlation length power parameter multi-parameters construction Gaussian Process Gaussian process is Gaussian with infinite variates: ∈ ℝ; ∈ ℝK Prior Data Posterior Learn about () and Used in Bayesian Non-linear Regression
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 8 / 15 ) Output Dimension Reduction:

    Principal Component Analysis (PCA) Full temperature transient output of FEBA TRACE Simulation is an “image”: × 𝟐𝟐 = 𝟑𝟑𝟑𝟑′𝟔𝟔𝟔𝟔 dimensions Dimension reduction by Principal Component Analysis (PCA) ′ = ⋅ ⋅ = ⋅ PC1 = 75% PC 2 = 9% PC 3 = 5% Mean SVD Concatenate and centered outputs Eigenvectors (principal components) PC scores (depends on parameter values) K
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 9 / 15 ) Principal Component –

    based Gaussian Process Meta-modelling model ; , = �model , + � , ⋅ + =1 PCA is Proper Orthogonal Decomposition (or KL-transform, Empirical OF, etc.) Only the scores are function of truncation error GP Meta-model construction on : 1. Run TRACE train times following a design 2. Carry Out PCA 3. Regress PC Scores on 4. Validate on valid data (500 points) Fitting 1is easy But higher is harder PC 1 PC 8 Benefit of using more PC in reconstruction increasingly marginal time to evaluate TRACE ; , output: 6-10 [] → ~ 𝟎𝟎−𝟑𝟑[s] PC Reconstruction Error DOE Coefficient of determination, 2 (1 is good!)
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 10 / 15 ) Modelling Discrepancy Model

    Discrepancy is the difference between model with known, perfect parameters (but with missing physics, error, etc.) and perfect data. Error tends to be correlated. ; , = true ; , − model , �; , unknown Why? Avoid overfitting, otherwise parameters will be used to compensate model bias in a given dataset would like to know Difficulty: Too many unknowns ⋅ and �, identifiability issue Solution: Modular Bayesian Approach* - estimate bias using GP with data and prediction with best- estimate parameters values, and allow for uncertainties Bottom Assembly Top Assembly *Bias defined up until quenching Model with the true/best/perfect parameters value ; , ~𝐺𝐺 , Σ
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 11 / 15 ) Normal Likelihood and

    Flat Prior: Posterior Formulation Bayes’ Theorem � exp = exp} � × � ∫ exp} � × � � � Posterior Normalization constant, multi-D integral, not necessary with MCMC Likelihood Prior The Metropolis-Hastings (MH) Algorithm to sample from an un-normalized density � exp , in a nutshell* (use emcee package): 1.Start at any random value 2.Generate sample close to the current value 3.Accept or reject based on (new)/ old exp} � ~ � + , Σ + > > > + Σ + (exp 2 +other 2 ) GP meta-model Principal Comp. truncation model discrepancy experiment and other Multiple sources of variance, avoid overfitting Likelihood exp , = model �; , + , + Gaussian Gaussian Gaussian given unknown Consider only data from 1 FEBA Test. No in formulation
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 12 / 15 ) Most parameters uncertainties

    are constrained by data – model formulation Spacer Grid HT Enhancement DFFB Wall HTC DFFB Interfacial Drag IAFB Interfacial Drag Quenching Temperature -- are prior ranges 50’000 MCMC Samples Some ranges simply implausible Indication of bias at particular instance Most parameters are constrained and not strongly correlated given data and statistical model
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 13 / 15 ) Posterior Predictive Checks:

    uncertainty propagation Uncertainty Propagation using 200 Monte Carlo Samples drawn from posterior distributions Calibration : Test 216, sys = 4.1 bar & inlet = 𝟑𝟑. cm ⋅ s−1 Validation : Test 214, sys = 4.1 bar & inlet = . cm ⋅ s−1  Nominal Run Posterior Runs Bottom Assembly (0.8 [m]) Mid Assembly (2.4 [m]) Top Assembly (3.5 [m]) Prior Runs Experiment
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 14 / 15 ) Open Issues and

    Future Work ? ? ? ?  1. Quantification was done on one FEBA Test (No. 216). The effect of experimental control variables (pressure and inlet velocity) was not taken into account (under investigation) 2. Some parameters were found to be collinear in initial 7-parameter model calibration: the effect of one compensates the effect of the other (non-identifiable). Solution: Subset the parameters or, perhaps, introduce additional code response – data pair (under investigation) 70’000 MCMC Samples
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    TRACE reflood model parameters prior uncertainties can be constrained with

    data from the FEBA facility and assumed statistical calibration model …various sources of uncertainties, some are correlated, were taken into account to hedge against overfitting …Propagation of posterior uncertainty on test case seems to be consistent (narrower but still enveloping) uncertainty bands But, …the effect of experimental condition was not yet taken into account …parameter sensitivity is not enough to guarantee identifiability, additional output-data pairs needed to break collinearities Wir schaffen Wissen – heute für morgen Thanks a lot for your attention! Acknowledgments: • Swiss Federal Nuclear Safety Inspectorate (ENSI) • Swiss Federal Office of Energy (BFM)
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 16 / 15 ) ETHZ-EPFL MSc in

    Nuclear Engineering • Swiss joint MSc program between ETHZ and EPFL • Established in 2008, more than 100 students • Two-year program, 120 ECTS credits • Students attend two top-ranked universities: • ETH Zurich #9 • EPF Lausanne #14 (based on QS) • Scientific support and research at the Paul Scherrer Institute • Grants and scholarships available
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 17 / 15 ) ETHZ-EPFL MSc in

    Nuclear Engineering Curriculum focuses on: • Neutronics • Thermohydraulics • Nuclear Material Science • Nuclear Safety • Waste Management • Radiation Protection …and more Unique world-class facilities: • CROCUS research reactor • Swiss Light Source synchrotron • Hot Lab facility • Proton therapy center • Numerous TH experimental facilities Curriculum includes: • Three-month industrial internship • Research project • Master thesis
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 18 / 15 ) ETHZ-EPFL MSc in

    Nuclear Engineering For more information visit master-nuclear.ch
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 19 / 15 ) Initial Parameters Values

    and Ranges of Uncertainties No. Parameters Name Physical Model Described PDF PDF Parameter Nominal Value Unit 1 BREAK40_ptb Outlet Pressure Uniform, Multiplicative [0.9, 1.1] 1.0 [-] 2 FILL40_tltb Inlet Coolant Temperature Uniform, Additive [-5.0, +5.0] 0.0 [K] 3 FILL40_vmtbm Inlet Coolant Velocity Uniform, Multiplicative [0.9, 1.1] 1.0 [-] 4 POWER99_rpwtbr Heater Rods Power Uniform, Multiplicative [0.9, 1.05] 1.0 [-] 5 MATPROP51_cond Thermal Cond. (Nichrome) Uniform, Multiplicative [0.9, 1.05] 1.0 [-] 6 MATPROP51_cp Specific Heat (Nichrome) Uniform, Multiplicative [0.9, 1.05] 1.0 [-] 7 MATPROP51_emis Emissivity (Nichrome) Uniform, Multiplicative [0.9, 1.0] 0.95 [-] 8 MATPROP50_cond Thermal Cond. (MgO) Uniform, Multiplicative [0.9, 1.2] 1.0 [-] 9 MATPROP50_cp Specific Heat (MgO) Uniform, Multiplicative [0.9, 1.2] 1.0 [-] 10 VESSEL1_epsw Wall Roughness Uniform [6.1E-7, 2.44E-6] 1.525E-6 [m] 11 MATPROP52_cond Thermal Cond. (SS) Uniform, Multiplicative [0.95, 1.05] 1.0 [-] 12 MATPROP52_cp Specific Heat (SS) Uniform, Multiplicative [0.95, 1.05] 1.0 [-] 13 MATPROP52_emis Emissivity (SS) Uniform, Multiplicative [0.5625, 0.9375] 0.84 [-] • 26 uncertain input parameters are considered • Below are the uncertain input parameters related to boundary conditions, initial condition, and material properties
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 20 / 15 ) Initial Parameters Values

    and Ranges of Uncertainties No. Parameters Name Physical Model Described PDF PDF Parameter Nominal Value Units 14 gridHTEnhSV Spacer grid convective HTC Enhancement Log-Uniform, Multiplicative [0.50, 2.00] 1.0 [-] 15 kGridSV Spacer grid pressure losses Uniform, Multiplicative [0.25, 1.75] 1.0 [-] 16 iAFBWallHTCSV Wall-fluid HTC (IAFB) Log-Uniform, Multiplicative [0.5, 2.00] 1.0 [-] 17 dFFBWallHTCSV Wall-fluid HTC (DFFB) Log-Uniform, Multiplicative [0.5, 2.00] 1.0 [-] 18 iAFBLIHTCSV Liquid-Interface HTC (IAFB) Log-Uniform, Multiplicative [0.25, 4.00] 1.0 [-] 19 iAFBVIHTCSV Vapor-Interface HTC (DFFB) Log-Uniform, Multiplicative [0.25, 4.00] 1.0 [-] 20 dFFBLIHTCSV Liquid-Interface HTC (IAFB) Log-Uniform, Multiplicative [0.25, 4.00] 1.0 [-] 21 dFFBVIHTCSV Vapor-Interface HTC (DFFB) Log-Uniform, Multiplicative [0.25, 4.00] 1.0 [-] 22 iAFBIntDragSV Interfacial Drag (IAFB) Log-Uniform, Multiplicative [0.125, 8.00] 1.0 [-] 23 dFFBIntDragSV Interfacial Drag (DFFB) Log-Uniform, Multiplicative [0.25, 4.00] 1.0 [-] 24 iAFBWallDragSV Wall Drag (IAFB) Log-Uniform, Multiplicative [0.50, 2.00] 1.0 [-] 25 dFFBWallDragSV Wall Drag (DFFB) Log-Uniform, Multiplicative [0.50, 2.00] 1.0 [-] 26 tQuenchSV Quench Temperature Uniform, Additive [-50.0, +50.0] TRACE [K] • 26 uncertain input parameters are considered • Below are the uncertain input parameters related to model parameters
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 21 / 15 ) Morris Screening Method

    𝐸𝐸 ≡ Δ Δ = 1 , 2 , … , i + Δ , … , − 1 , 2 , … , , … , Δ For many trajectories (𝒓𝒓 ), the basic statistics of the elementary effects can be calculated and interpreted as global sensitivity measures: «Standarized Elementary Effect» or EE Interpretation on Input Parameter Importance or ∗ Low High Low Non- influential Influential Linear, Additive High Non- influential Influential Non-linearities, Interactions
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    http://www.psi.ch/stars 2016.10.11/STARS/WD41 ( 22 / 15 ) Validating Gaussian Process

    Meta-Model 2 , � = 1 − ∑ − � 𝑣𝑣 =1 2 ∑ − 2 𝑣𝑣 =1 Coefficient of determination Training Design (35 Optimized Latin Hypercube Design) Validation Design (100 Select points for Hammersley Sequence) Residual Sum of Squares Total Sum of Squares