Advanced Simulation Methods for the Reliability Analysis of Nuclear Passive Systems

Transcription

1 Advanced Simulation Methods for the Reliability Analysis of Nuclear Passive Systems Francesco Di Maio, Nicola Pedroni, Enrico Zio* Politecnico di Milano, Department of Energy, Nuclear Division *Chaire SSDE-Foundation Europeenne pour l Energie Nouvelle, EDF Ecole Centrale Paris and Supelec

2 Contents 2 Objective Reliability assessment of T-H passive systems Advanced Monte Carlo Simulation (MCS) methods Fast-running models: Simplified T-H models Bootstrapped Artificial Neural Networks (ANNs) Safety margins Conclusions

3 Objective 3 TO ADDRESS THE COMPUTATIONAL CHALLENGES RELATED TO THE MODELING AND RELIABILITY ASSESSMENT OF THERMAL-HYDRAULIC (T-H) PASSIVE SAFETY SYSTEMS

4 Contents 4 Objective Reliability assessment of T-H passive systems Advanced Monte Carlo Simulation (MCS) methods Fast-running models: Simplified T-H models Bootstrapped Artificial Neural Networks (ANNs) Safety margins Conclusions

5 Reliability assessment of Thermal-Hydraulic (T-H) passive safety systems 5 Passive = no need of external input (energy source) to operate Thermal-Hydraulic = use of moving working fluids (e.g., natural circulation-based decay heat removal) Advantages: Simplicity Reduction of human interaction Reduction or avoidance of external electrical input of power/signals Drawbacks: Lower economic competitiveness (with respect to active systems) UNCERTAINTY IN BEHAVIOR AND MODELING RELIABILITY (FAILURE PROBABILITY) ASSESSMENT

6 Uncertainties in T-H passive system behavior and modeling 6 T3 W L3 Mechanical components WELL-KNOWN Г BEHAVIOR Natural forces (gravity) comparable to counter-forces (friction) Passive components: the natural elements (e.g., natural circulation) UNCERTAIN MODELING Lack of data or operating experience SENSITIVE TO SURROUNDINGS (i.e., to small random variations) UNCERTAIN PARAMETERS (e.g., heat transfer coefficients)

7 Reliability assessment (functional) failure probability evaluation by Monte Carlo (MC) 7 N T independent runs Monte Carlo (MC) sampling of uncertain parameters (probability distributions) x = {x, x 2,, x j,, x n } System model code (e.g. RELAP) Fuel cla adding Temperat ture, Y(x) F Failure threshold, α Y S System performance indicator, Y(x) Time Sample estimate of the (functional) failure probability P(F), Pˆ ( F ) = number Y x { ( ) > αy } N T

8 The Monte Carlo (MC)-based approach for (functional) failure probability evaluation: drawbacks 8 LONG CALCULATIONS! + Pˆ ( F ) number = N { Y ( x) > α } SMALL NUMBER FOR HIGHLY RELIABLE SYSTEMS! S T Y UNCERTAINTY/CONFIDENCE

9 The Monte Carlo (MC)-based approach for (functional) failure probability evaluation: contributions 9 ESTIMATION OF THE RELIABILITY (FUNCTIONAL FAILURE PROBABILITY) OF T-H PASSIVE SYSTEMS Rare failure events Long calculations Uncertainty/Confidence Advanced MC Simulation methods Fast-running models Safety margins. Subset Simulation (SS) 2. Line Sampling (LS). Simplified T-H models Order statistics 2. Artificial Neural Networks (with bootstrap) (with bootstrap)

10 Contents 0 Objective Reliability assessment of T-H passive systems Advanced Monte Carlo Simulation (MCS) methods Fast-running models: Simplified T-H models Bootstrapped Artificial Neural Networks (ANNs) Safety margins Conclusions

11 The Monte Carlo (MC)-based approach for (functional) failure probability evaluation: contributions ESTIMATION OF THE RELIABILITY (FUNCTIONAL FAILURE PROBABILITY) OF T-H PASSIVE SYSTEMS Rare failure events Advanced MC Simulation methods. Subset Simulation (SS) 2. Line Sampling (LS)

12 . Subset Simulation (SS) 2 Markov Chain Monte Carlo (MCMC) F Markov Chain Monte Carlo (MCMC) Target failure event, F Fm Fm... F2 F = x 2 m = 3 Failure Region F = F 3 P(F 3 F 2 ) P(F 2 F ) Standard Monte Carlo Simulation (MCS) P( F) = P( F m ) = P( F F 2 F m ) P( Fi + Fi ) i= x P(F )

13 2. Line Sampling (LS) 3 Key Idea: failure probability (P(F)) estimated using lines pointing towards the failure region F x 2 Failure Region, F ~P(F) (j) j P ( F ) N N j= P ( F ) ( j) ~P(F) (j+) j+2 ( j ) x~ j+ IF α F ~P(F) (j+2) THEN α VARIANCE 0 x (α) x α = important direction

14 Application: passive decay heat removal system of a Gas-cooled Fast Reactor (GFR) 4 Nine uncertain parameters, x (Gaussian): Power Pressure Cooler wall temperature Nusselt numbers (forced, mixed, free) Friction factors (forced, mixed, free) SYSTEM FUNCTIONAL FAILURE F = T out,core hot x T ( x) { >200 C} : out, core ( x) { avg x T > 850 C} : out, core

15 Application: results (functional) failure probability estimation 5 Comparison with: Standard MCS Latin Hypercube Sampling (LHS) = benchmark simulation method in PRA FOM = σ 2 N loops = 3 (P(F) = ) Pˆ ( F ) FOM Standard MCS SS t comp LHS LHS LS N loops = 4 (P(F) = ) Pˆ ( F ) FOM Standard MCS SS LS N T = 400 N T = 2300 E. Zio, N. Pedroni, How to effectively compute the reliability of a thermal-hydraulic passive system, Nuclear Engineering and Design, Volume 24, Issue, Jan. 20, pp E. Zio, N. Pedroni, Reliability Analysis of Thermal-Hydraulic Passive Systems by Means of Line Sampling, Reliability Engineering and System Safety, Vol. 9(), 2009, pp E. Zio, N. Pedroni, Estimation of the Functional Failure Probability of a Thermal-Hydraulic Passive System by Subset Simulation, Nuclear Engineering and Design, 2009, vol. 249(3), pp

16 Application: results (global) Sensitivity analysis by SS () 6 F F 2 F 3 = F x 0-3 Cond. level 0 x 0-3 Cond. level x 0-3 Cond. level PDF PDF Parameter x 2 - Pressure [kpa] Cond. level Cond. level Cond. level Parameter x 3 - Cooler wall temperature [ C] Pressure, x 2 Cooler wall temperature, x 3 Unconditional PDFs ( ) j x j q, j =,, 9 Conditional PDFs q ( x F ), j =,, 9, i =, 2, 3 j j i Conditional PDFs Unconditional PDFs Cond. level 0 Cond. level Cond. level 2 PDF Friction factor mixed, x 8 x 2 and x 8 are more important than x 3 in affecting system failure Parameter x 8 - Friction factor error (mixed convection)

17 Application: results (global) Sensitivity analysis by SS (2) 7 P(F x 2 ) Pressure, x Bayes theorem: P P = P( F ), j =,, 9 ( F x ) j ( x j F ) P( x ) Global information: j whole range of variability of x j is considered all other parameters (x k, k j) vary as well P(F x 3 ) Cooler wall temperature, x 3 P(F x 8 ) Friction factor mixed, x

18 Application: results (local) Sensitivity analysis by LS 8 α tells which variables are more important in causing system failure LS important direction, α N loops α (x ) α 2 (x 2 ) α 3 (x 3 ) α 4 (x 4 ) α 5 (x 5 ) α 6 (x 6 ) α 7 (x 7 ) α 8 (x 8 ) α 9 (x 9 ) Pressure Nusselt mixed Friction mixed Agreement with SS and with reference case study of literature E. Zio, N. Pedroni, Monte Carlo Simulation-based Sensitivity Analysis of the model of a Thermal-Hydraulic Passive System, accepted for publication on Reliability Engineering and System Safety, 20.

19 Line Sampling: technical issues 9. Determination of the important direction α additional runs of the T-H model code ( overall CPU cost) Original contributions: comparison of three literature methods for identifying α use of Artificial Neural Networks (instead of the T-H code) to reduce the computational cost associated to the identification of α proposal of a new method to determine α, based on the minimization of the variance of the LS failure probability estimator 2. Efficiency of LS with small sample sizes (e.g., < 00) needed with T-H codes requiring hours for a single simulation (Fong et al., 2009) Original contribution: challenging the performance of LS in the estimation of small failure probabilities (~0-4 ) with a small number of samples drawn (i.e., << 00)

20 LS Technical issue : accurate determination of the important direction α Original method proposed 20 Constrained minimization of the variance of the LS failure probability estimator

21 LS Technical issue : accurate determination of the important direction α - Results 2 Practical case: low number of T-H code simulations 3.65 x 0-4 Failure proba ability, P(F) N code = Proposed method MCMC Design point Gradient Accuracy (proposed method) ~ (3 7) Accuracy (literature methods) Precision (proposed method) ~ (5 7) Precision (literature methods)

22 Line Sampling: technical issues 22. Determination of the important direction α additional runs of the T-H model code ( overall CPU cost) Original contributions: comparison of three literature methods for identifying α use of Artificial Neural Networks (instead of the T-H code) to reduce the computational cost associated to the identification of α proposal of a new method to determine α, based on the minimization of the variance of the LS failure probability estimator 2. Efficiency of LS with small sample sizes (e.g., < 00) needed with T-H codes requiring hours for a single simulation (Fong et al., 2009) Original contribution: challenging the performance of LS in the estimation of small failure probabilities (~0-4 ) with a small number of samples drawn (i.e., << 00)

23 LS Technical issue 2: efficiency with small sample sizes - Results 23 2 x 0-4 Very small sample size (ranging from 5 to 50) Failure probability, P(F) MAE = 5% 0.8 MAE = 6% Standard MCS with N T = 50 Sample size, N T MAE = 94% 95% CI = [0, ] E. Zio, N. Pedroni, An optimized Line Sampling method for the estimation of the failure probability of nuclear passive systems, Reliability Engineering and System Safety, Volume 95, Issue 2, Dec. 200, pp

24 Conclusions Advanced MC Simulation methods: SS and LS 24 SS and LS estimating the (functional) failure probability of T-H passive systems: Estimation of small failure probabilities ( 0-5 ) Comparison with benchmark simulation methods in PRA (standard MCS and LHS) SS and LS much more efficient than benchmark simulation methods in PRA LS performance almost independent of the failure probability wide range of applications to real systems Optimization of the LS method important direction based on minimization of the variance of the LS failure probability estimator Combination of soft-computing methods (GA + ANN) More accurate and precise estimates than other literature methods Successful LS with very small sample sizes (5-50) Sensitivity analysis SS: global information based on a large amount of conditional samples LS: local information based on the important direction

25 Contents 25 Objective Reliability assessment of T-H passive systems Advanced Monte Carlo Simulation (MCS) methods Fast-running models: Simplified T-H models Bootstrapped Artificial Neural Networks (ANNs) Safety margins Conclusions

26 The Monte Carlo (MC)-based approach for (functional) failure probability evaluation: contributions 26 ESTIMATION OF THE RELIABILITY (FUNCTIONAL FAILURE PROBABILITY) OF T-H PASSIVE SYSTEMS Long calculations Fast-running models. Simplified T-H models 2. Artificial Neural Networks (with bootstrap)

27 . Simplified T-H models 27 Passive Residual Heat Removal System in the High Temperature Reactor Pebble Modular (HTR-PM) [in collaboration with Institute of Nuclear and New Energy Technology (INET)- Tsinghua University, Beijing, China] Safety Parameter Transparent and fast T-H MATLAB model (embedded within a Monte Carlo-driven fault injection engine to sample component failures) F. Di Maio, E. Zio, L. Tao, J. Tong, Passive System Accident Scenario Analysis by Simulation, proceedings of PSA20, pp , International Topical meeting on Probabilistic Safety Assessment and Analysis, March 3-7, 20, Wilmington, USA.

28 Application: passive Residual Heat Removal System in the High Temperature Reactor Pebble Modular (HTR-PM) 37 input parameters N Parameter Distribution Note W Normal Residual heat power 2 T a,in Bi-Normal Temperature of inlet air in the air-cooled tower 3 x i Uniform Resistance coefficient of elbow 4 x i2 Uniform Resistance coefficient of header channel 5 x iw Uniform Resistance coefficient of the water tank walls 6 Sum of the resistance coefficients of inlet shutter and air cooling x ia,in Uniform tower and silk net 7 Sum of the resistance coefficients of outlet shutter and air cooling x ia,out Uniform tower and silk net 8 x ia,narrow Uniform Resistance coefficient of the narrowest part of the tower 9 P a,in Uniform Pressure of the inlet air in the cooler tower 0 dx Uniform Roughness of pipes H a Normal Height of chimney 2 L a Normal Length of pipes in the exchanger 3 N a Normal Total number of pipes in the air cooler 4 A f Normal Air flow crossing are in the narrowest part of the tower 5 A f,in Normal Inlet air flow crossing area in the tower 6 A f,out Normal Outlet air flow crossing area from the tower 7 A f,narrow Normal Crossing area in the narrowest part of the tower 8 S Normal Distance between centers of adjacent pipes in horizontal direction 9 S 2 Normal Distance between centers of adjacent pipes in vertical direction 20 S Normal Distance between fins in the ribbed pipe 2 D a Normal Pipes inner diameter in the air cooling exchanger 22 D o Normal Pipes outer diameter 23 D outer Normal Rib outer diameter 24 P w Normal Water pressure in the pipes 25 H w Normal Elevatory height of water 26 N w Discrete Normal Number of water cooling pipes for each loop 27 L w Normal Length of the water cooling pipes 28 D w Normal Inner diameter of the water cooling pipes 29 D Normal Inner diameter of the in-core and air cooler connecting pipes 30 D 2 Normal Inner diameter of the in-core header 3 L C Normal Length of the in-core and air cooler connecting pipes ( cold leg ) 32 L H Normal Length of the in-core and air cooler connecting pipes ( hot leg ) 33 R i Log-normal Thermal resistance of pipes inside of the heat exchanger 34 R o Log-normal Thermal resistance due to the dirt of the pipes fins 35 R g Log-normal Thermal resistance of the gap between fins 36 R f Log-normal Thermal resistance of fins 37 lamd Normal Heat transfer coefficient of the pipes output parameter: outlet water temperature

29 Application: passive Residual Heat Removal System in the 29High Temperature Reactor Pebble Modular (HTR-PM) - 37 input parameters to be sampled - 3 Accidental scenarios: A = 2 loops are failed B = loop is failed X X X C = 0 loops are failed

30 Model simplification by sensitivity analysis 30 Objective: Acquire insights on the behavior of the system with respect to how much its output depends on the inputs Disadvantage:COMPUTATIONALLY BURDENSOME (several model computations) comparison VARIANCE DECOMPOSITION SEVERAL MODEL EVALUATIONS Fast TH model of RHR ANALYTIC HIERARCHY PROCESS X SEVERAL MODEL EVALUATIONS Qualitative results Y. Yu, T. Liu, J. Tong, J. Zhao, F. Di Maio, E. Zio, A. Zhang, Variance Decomposition Sensitivity Analysis of a Passive Residual Heat Removal System Model, Proceedings of SAMO200, Milano, July 200, Procedia - Social and Behavioral Sciences, Volume 2, Issue 6, 200, Pages Y. Yu, T. Liu, J. Tong, J. Zhao, F. Di Maio, E. Zio, A. Zhang, Multi-Experts Analytic Hierarchy Process for the Sensitivity Analysis of Passive Safety Systems, Proceedings of the 0th International Probabilistic Safety Assessment & Management Conference, PSAM0, Seattle, June 200.

31 The simplified model of the Residual Heat Removal System in the HTR-PM 3 The standard Monte Carlo-driven fault engine injects component failures at random times st loop of the RHR 2 nd loop of the RHR 3 rd loop of the RHR Safety Parameter Safety Parameter Availability of on-line instantaneous information

32 The Variance Decomposition Method Model y = m( x, x ) 2 Var[ Y ] = VarX [ EX ( Y X )] + EX [ VarX ( Y )] X 2 2 Index of Importance of X Index of Importance of X, X 2 (for a 3 inputs model X,X 2,X 3 ) η η 2 2,2 = = Var X Var [ EX ( Y X)] 2 Var[ Y ] X, X 2 [ EX ( Y X, X 2)] 3 Var[ Y ]

33 Single parameter analysis: Index of importance 2 η X 3 accidental scenarios: Index of Importance for single parameter contribution to the output variance

34 Group of parameters: index of Importance 2 η 3 accidental scenarios: Index of Importance for group parameter contribution to the output variance The group sensitivity analysis allows underlining important physical and modelling aspects related to the system behaviour

35 The Analytic Hierarchy Process 35 Experts are asked to independently build the judgment matrix by: - defining the top goal of the hierarchy, e.g., capability of the RHR system in removing the core decay power - defining the hierarchy structure, e.g., - comparing pairwise the model inputs with respect to their importance on the top goal, e.g. Input Input 2 Input 3 Input 2 /3 = Aand Bequally important 3 = Aslightly more important than B 5 = Astrongly more important than B 7 = Avery strongly more important than B 9 = Aabsolutely more important than B - computing the priority vectors, i.e., importance of the inputs

36 Comparison between Variance Decomposition method and Analytic Hierarchy Process (AHP) N Parameter Variance Decomposition Analytic Hierarchy Process [*] W X X 2 T a,in X X 3 x i 4 x i2 5 x iw 6 x ia,in 7 x ia,out 8 x ia,narrow 9 P a,in 0 dx H a 2 L a 3 N a 4 A f 5 A f,in 6 A f,out 7 A f,narrow 8 S 9 S 2 20 S 2 D a 22 D o 23 D outer 24 P w X 25 H w 26 N w 27 L w 28 D w 29 D 30 D 2 3 L C 32 L H 33 R i 34 R o 35 R g 36 R f 37 lamd Power W is important the same with AHP Inlet air temperature T a,in is important the same with AHP Water pressure P w not identified as important by AHP All other parameters have minor impacts with respect to W and T a,in the same with AHP AHP and Variance Decomposition provide coherent results

37 2. Fast-running empirical regression models: Bootstrapped Artificial Neural Networks (ANNs) 37 ANNs = empirical regression models ANNs for practical use in passive system reliability assessment: generate a reduced number (50-00) of I/O data examples by running the T-H code train the ANN model to fit the data use the ANN (instead of the T-H code) to calculate the output ANN REGRESSION MODEL ADDITIONAL UNCERTAINTY in our work: BOOTSTRAP (ENSEMBLE) OF ANN REGRESSION MODELS each model of the ensemble provides an estimate of the output the empirical distribution of the bootstrapped estimates is built CONFIDENCE INTERVAL FOR THE QUANTITY OF INTEREST

38 Application: passive decay heat removal system of a Gas-cooled Fast Reactor (GFR) 38 Nine uncertain parameters, x (Gaussian): Power Pressure Cooler wall temperature Nusselt numbers (forced, mixed, free) Friction factors (forced, mixed, free) SYSTEM FUNCTIONAL FAILURE F = T out,core hot x T ( x) { >200 C} : out, core ( x) { avg x T > 850 C} : out, core

39 Application Bootstrapped ANNs: functional failure probability estimation Comparison with quadratic Response Surfaces (RSs) (Arul et al., 2009; Fong et al., 2009; Mathews et al., 2009) ANNs 0 x 0-4 ANN-based BBC 95% CIs for P(F) Quadratic RSs 0 x 0-4 Quadratic RS-based BBC 95% CIs for P(F) 39 Failure probability, P(F) Failure probability, P(F) Training sample size Reference result obtained by Monte Carlo Simulation with N T = samples! (~ 47 h) Total CPU time (ANN/RS) << /00 Total CPU time (T-H model) 0 Training sample size N. Pedroni, E. Zio, G. E. Apostolakis, Comparison of bootstrapped Artificial Neural Networks and quadratic Response Surfaces for the estimation of the functional failure probability of a thermal-hydraulic passive system, Reliability Engineering and System Safety, 95(4), 200, pp

40 Bootstrapped ANNs: first-order global Sobol sensitivity indices Results ANNs 2 Pressure, x 2 Pressure, x Quadratic RSs Sobol index Training sample size Sobol index Training sample size Reference result obtained by Monte Carlo Simulation with N T = 0000 samples! (~ 92 h) ANNs produces more accurate and precise estimates than quadratic RSs CPU time (ANN) ~.5 CPU time (RS) E. Zio, G. E. Apostolakis, N. Pedroni, Quantitative functional failure analysis of a thermal-hydraulic passive system by means of bootstrapped Artificial Neural Networks, Annals of Nuclear Energy, Volume 37, Issue 5, 200, pp

41 Conclusions - Fast-running models 4 Simplified MATLAB T-H model: dependence of the system response on the time and magnitude of components and equipments failures influence of the uncertainties and of components and equipments failures on the system function accuracy and speed of calculation required coverage of scenarios for safety ANNs for substituting the T-H code in the estimation of the functional failure probability of T-H passive systems: Estimation of small failure probabilities (~ 0-4 ) Small number (20-00) of T-H code runs to build the models CPU cost by two orders of magnitude Bootstrap of ANN models uncertainties (confidence intervals) of the estimates Estimation of first-order global Sobol sensitivity indices Comparison with quadratic Response Surfaces better accuracies and precisions of ANNs slightly higher CPU cost associated to ANNs

42 Contents 42 Objective Reliability assessment of T-H passive systems Advanced Monte Carlo Simulation (MCS) methods Fast-running models: Simplified T-H models Bootstrapped Artificial Neural Networks (ANNs) Safety margins Conclusions

43 The Monte Carlo (MC)-based approach for (functional) failure probability evaluation: contributions 43 ESTIMATION OF THE RELIABILITY (FUNCTIONAL FAILURE PROBABILITY) OF T-H PASSIVE SYSTEMS Uncertainty/Confidence Safety margins Order statistics (with bootstrap)

44 Safety margin quantification 44 ( ) T clad MAX ( Tclad 2200 F ) Confidence Confidence Interval Interval Confidence Interval Uncertainty in safety margins calculation by numerical code Input values Modeling hypotheses E. Zio, and F. Di Maio, Bootstrap and Order Statistics for Quantifying Thermal-Hydraulic Code Uncertainties in the Estimation of Safety Margins, Science and Technology of Nuclear Installations, Volume 2008, Article ID 34064, 9 pages, doi:0.55/2008/34064

45 Application: passive Residual Heat Removal System in the 45High Temperature Reactor Pebble Modular (HTR-PM) AIM: estimates of the 95 th percentiles of the safety parameters distributions - 37 input parameters to be sampled - 3 Accidental scenarios: A = 2 loops are failed B = loop is failed X X X C = 0 loops are failed COMPUTATIONAL BURDEN Bootstrapped Order Statistics (BOS) Minimum number of simulations N for estimating the distribution of the safety parameter with a given confidence, accounting also for the uncertainty of the empirical model used to simulate the accidental scenarios

46 Application: passive Residual Heat Removal System in the 46High Temperature Reactor Pebble Modular (HTR-PM) Number of simulations N for each accidental scenario: 50 Number of Bootstrap replications: 00 Results: Two RHR loops are enough The 3rd loop can be considered as a redundancy E. Zio, F. Di Maio, J. Tong, Safety Margins Confidence Estimation for a Passive Residual Heat Removal System, Reliability Engineering and System Safety, Vol. 95, 200, pp

47 47 Application: passive Residual Heat Removal System in the High Temperature Reactor Pebble Modular (HTR-PM) increasing the number of simulations results in: Similarity of the 95 th percentile point- estimates (reliability) Approaching down the estimates to the true value (conservativeness) Narrowing the confidence intervals (robustness)

48 Contents 48 Objective Reliability assessment of T-H passive systems Advanced Monte Carlo Simulation (MCS) methods Fast-running models: Simplified T-H models Bootstrapped Artificial Neural Networks (ANNs) Safety margins Conclusions

49 Conclusions 49 Objective: To address the computational challenges related to the reliability analysis of Thermal- Hydraulic (T-H) passive safety systems Contributions: Advanced Monte Carlo Simulation methods: SS and LS for the reliability analysis of a T-H passive system Optimization of the LS method (variance-minimizing search of important direction α) Successful LS performance with very small sample sizes Fast-running models: MATLAB implementation of a simplified T-H model ANNs for the reliability analysis of a T-H passive system ANN regression model uncertainty quantification by bootstrap Safety margins Percentile estimation by order statistics Percentile uncertainty quantification by bootstrap