Advanced Simulation Methods for the Reliability Analysis ...li/materials/lecture_NCEPU.pdf ·...

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

Contents

� Objective

� Reliability assessment of T-H passive systems

� Advanced Monte Carlo Simulation (MCS) methods

2

� Fast-running models: � Simplified T-H models� Bootstrapped Artificial Neural Networks (ANNs)

� Safety margins

� Conclusions

Objective 3

TO ADDRESS THE COMPUTATIONAL CHALLENGES RELATED TO THE MODELING AND RELIABILITY RELATED TO THE MODELING AND RELIABILITY

ASSESSMENT OF THERMAL-HYDRAULIC (T-H) PASSIVE SAFETY SYSTEMS

Contents

� Objective



4


� Safety margins

� Conclusions

5Reliability assessment of Thermal-Hydraulic (T-H) passive safety systems

Advantages:

� Simplicity

“Passive”= no need of external input (energy source) to operate

“Thermal-Hydraulic” = use of moving working fluids (e.g., naturalcirculation-based decay heat removal)

� 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

6Uncertainties in T-H passive system behavior and modeling

Mechanical componentsWELL-KNOWN

W

T3 L3

Г

“Passive” components:the natural elements

(e.g., natural circulation)

Lackof data oroperating experience

Natural forces(gravity) comparable to counter-forces(friction)

BEHAVIOR MODELING

UNCERTAIN

SENSITIVE TO SURROUNDINGS(i.e., to small random variations)

UNCERTAIN PARAMETERS(e.g., heat transfer coefficients)

7Reliability assessment �� (functional) failure probability evaluation by Monte Carlo (MC)

Monte Carlo (MC)sampling of uncertain parameters

(probability distributions)x = {x1, x2, …, xj, …, xn}

NT independent runs

Fue

l cla

ddin

gTe

mpe

ratu

re, Y(x

) FFailure threshold, αY

System model code(e.g. RELAP)

System performance indicator, Y(x)

Sample estimate of the (functional) failure probability P(F), ( ) ( ){ }ˆ Y

T

number Y xP F

N

α>=

Fue

l cla

ddin

gTe

mpe

ratu

re,

Time

S

8The Monte Carlo (MC)-based approach for (functional ) failure probability evaluation: drawbacks

( ) ( ){ }T

Y

N

xYnumberFP

α>=ˆ

SMALL NUMBER FORLONG CALCULATIONS!

SMALL NUMBER FORHIGHLY RELIABLE SYSTEMS!

S+

UNCERTAINTY/CONFIDENCE

9The Monte Carlo (MC)-based approach for (functional ) failure probability evaluation: contributions

ESTIMATION OF THE RELIABILITY (FUNCTIONAL FAILURE P ROBABILITY) OF T-H PASSIVE SYSTEMS

Long calculationsRare failure events Uncertainty/Confidence

1. Subset Simulation (SS)2. Line Sampling (LS)

Advanced MC Simulation methods

Fast-running models

2. Artificial Neural Networks(with bootstrap)

1. Simplified T-H models Order statistics(with bootstrap)

Safety margins

Contents

� Objective



10


� Safety margins

� Conclusions



Rare failure events

Advanced MC Simulation methods

1. Subset Simulation (SS)2. Line Sampling (LS)

121. Subset Simulation (SS)

x2Failure RegionF = F3

Target failure event, F

121 ... FFFFF mm ⊂⊂⊂⊂= −

Markov Chain Monte Carlo

Markov Chain Monte Carlo

(MCMC)

�� P(F3|F2)

m = 3

x1

F1

F2Standard

Monte Carlo Simulation(MCS)

Monte Carlo(MCMC)

�� P(F1)

�� P(F2|F1)

∏−

=+==

1

111 )|()()()(

m

iiim FFPFPFPFP

132. Line Sampling (LS)

x2 Failure Region, F

Key Idea: failure probability (P(F)) estimated using linespointing towards thefailure region F

j~P(F)(j)∑≈

NjFP

NFP )()(

1)(

x1

( )jx~ ~P(F)(j+2)

j+ 2~P(F)(j+1)

j+ 1 IF α ┴ FTHEN

VARIANCE ≈ 0

∑=

≈j

FPN

FP1

)()(

α

x⊥(α)

α = ‘important direction’

14Application: passive decay heat removal system of a Gas-cooled Fast Reactor (GFR)

Nine uncertain parameters, x (Gaussian):• Power• Pressure• Cooler wall temperature• Nusselt numbers (forced, mixed, free)• Friction factors (forced, mixed, free)• Friction factors (forced, mixed, free)

SYSTEM FUNCTIONAL FAILURE

Tout,core

( ){ }CxTx hotcoreout °>1200: ,

( ){ }CxTx avgcoreout °> 850: ,

F=

∩

15Application: results – (functional) failure probability estimation

( )FP̂( )FP̂

compt⋅=

2

1FOM

σ

Nloops= 3 (P(F) = 1.315·10-3)

FOM

Nloops= 4 (P(F) = 1.521·10-5)

FOM

Latin Hypercube Sampling (LHS) = benchmark simulation method in PRAStandard MCS

Comparison with:

( )FP̂( )FP̂ FOM

Standard MCS 1.600·10-3 216.00

SS 1.317·10-3 432.49

FOM

Standard MCS 0 2.31·104

SS 1.580·10-5 2.07·106

LHS 1.400·10-3 1.41·103

LS 1.314·10-3 1.00·106LHS 2.000·10-5 5.63·104

LS 1.510·10-5 9.87·108

E. Zio, N. Pedroni, “Estimation of the Functional Failure Probability of a Thermal-Hydraulic Passive System bySubset Simulation”,Nuclear Engineering and Design,2009, vol. 249(3), pp. 580-599.

E. Zio, N. Pedroni, “Reliability Analysis of Thermal-Hydraulic Passive Systems by Means of Line Sampling”,Reliability Engineering and System Safety, Vol. 9(11), 2009, pp. 1764-1781.

E. Zio, N. Pedroni, “How to effectively compute the reliability of a thermal-hydraulic passive system”,NuclearEngineering and Design, Volume 241, Issue 1, Jan. 2011, pp. 310-327.

NT = 1400 NT = 2300

16

1155 21450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 10-3

PDF

Cond. level 0

1155 21450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 10-3

Parameter x2 - Pressure [kPa]

Cond. level 1

1155 21450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 10-3 Cond. level 2

1650 1650 1650

F1 F2 F3 = F

Unconditional PDFs

Conditional PDFs

1.4

1.6

1.8

2

Cond. level 0

1.4

1.6

1.8

2

Cond. level 1

1.4

1.6

1.8

2

Cond. level 2

Pressure, x2

, j = 1, …, 9, i = 1, 2, 3( )ijj Fxq |

( )jj xq , j = 1, …, 9

Application: results – (global) Sensitivity analysis by SS (1)

0.6 1.40

0.5

1

1.5

2

2.5

3

3.5

PD

F

Cond. level 0

0.6 1.40

0.5

1

1.5

2

2.5

3

3.5

Parameter x8 - Friction factor error (mixed convection)

Cond. level 1

0.6 1.40

0.5

1

1.5

2

2.5

3

3.5

Cond. level 2

1 1 1

Unconditional PDFs

Conditional PDFs≠

x2 and x8 are moreimportant than x3

in affecting system failure

72.00 108.000

0.2

0.4

0.6

0.8

1

1.2

1.4

PD

F

72.00 108.000

0.2

0.4

0.6

0.8

1

1.2

1.4

Parameter x3 - Cooler wall temperature [°C]72.00 108.000

0.2

0.4

0.6

0.8

1

1.2

1.4

90.00 90.00 90.00

Friction factormixed, x8

Cooler walltemperature, x3

17

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

( ) ( )( ) ( )FPxP

FxPxFP

j

jj

|| = , j = 1, …, 9

Global information:� whole range of variability of xj is considered

� all other parameters (xk, k ≠ j) vary as well

Bayes’ theorem:

P(F

| x 2

)

Pressure, x2

Application: results – (global) Sensitivity analysis by SS (2)

1250 1300 1350 1400 1450 1500 1550 1600 16500

0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

70 75 80 85 90 95 100 1050

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

P(F

| x 3

)

P(F

| x 8

)

Friction factormixed, x8

Cooler walltemperature, x3

18

α tells which variables are more important in causing system failure

Application: results – (local) Sensitivity analysis by LS

LS important direction, α

Nloops α1 (x1) α2 (x2) α3 (x3) α4 (x4) α5 (x5) α6 (x6) α7 (x7) α8 (x8) α9 (x9)

4 + 0.0774 - 0.9753 + 0.0203 + 0.0032 - 0.1330 + 0.0008 + 0.0026 + 0.1534 - 0.0342

α tells which variables are more important in causing system failure

Agreement with SS and with reference case study of literaturePressure Nusselt mixed Friction mixed

E. Zio, N. Pedroni, “Monte Carlo Simulation-based Sensitivity Analysis of the model of a Thermal-Hydraulic PassiveSystem”, accepted for publication onReliability Engineering and System Safety, 2011.

19Line Sampling: technical issues

1. 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

2. Efficiency of LS with small sample sizes (e.g., < 100)�� needed with T-H codes requiring hours for a single simulation (Fong et al., 2009)

the computational cost associated to the identification of α� proposal of a new method to determine α, based on the minimizationof

the varianceof the LS failure probability estimator

Original contribution:� challenging the performance of LS in the estimation of small failure

probabilities (~10-4) with a small number of samples drawn (i.e., << 100)

20LS – Technical issue 1: accurate determination of th e ‘important direction’ α – Original method proposed

Constrained minimization of the variance of the LS failure probability estimator

21LS – Technical issue 1: accurate determination of the ‘important direction’ α - Results

Practical case: low number of T-H code simulations

3.45

3.5

3.55

3.6

3.65x 10-4

Fa

ilure

pro

babi

lity,

P(F

)

Ncode= 100

Accuracy (proposed method) ~ (3 – 7)·Accuracy (literature methods)

3.15

3.2

3.25

3.3

3.35

3.4

Proposed method

Fa

ilure

pro

babi

lity,

MCMC Design point Gradient

Precision (proposed method) ~ (5 – 7)·Precision (literature methods)

22Line Sampling: technical issues

1. 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

2. Efficiency of LS with small sample sizes (e.g., < 100)�� needed with T-H codes requiring hours for a single simulation (Fong et al., 2009)

the computational cost associated to the identification of α� proposal of a new method to determine α, based on the minimizationof

the varianceof the LS failure probability estimator

Original contribution:� challenging the performance of LS in the estimation of small failure

probabilities (~10-4) with a small number of samples drawn (i.e., << 100)

23LS – Technical issue 2: efficiency with small sample sizes - Results

Very smallsample size (ranging from 5 to 50)

1.2

1 .4

1 .6

1 .8

2x 10

-4

Fa

ilure

pro

babi

lity,

P(F

)

MAE = 5%

0 5 10 15 20 25 30 35 40 45 50 550.6

0 .8

1

Sample size, NT

Fa

ilure

pro

babi

lity,

MAE = 16%

MAE = 5%

MAE = 194%

95% CI = [0, 0.0582]Standard MCS with NT = 50

E. Zio, N. Pedroni, “An optimized Line Sampling method for the estimation of the failure probability of nuclear passivesystems”,Reliability Engineering and System Safety, Volume 95, Issue 12, Dec. 2010, pp. 1300-1313.

24Conclusions – Advanced MC Simulation methods: SS and LS

• SS and LS estimating the (functional) failure probability of T-H passive systems:

� Estimation of small failure probabilities (≤ 10-5)• Comparison with benchmark simulation methodsin PRA (standard MCS and LHS)

– SS and LS much more efficient than benchmark simulation methods in PRA

– LS performance almost independentof the failure probability � wide range of applications to real systems

• Optimization of the LS method → “important direction” based on minimization of the varianceof the LS failure probability estimator

– Combination of soft-computing methods (GA + ANN)

– More accurateand preciseestimates than other literature methods

• Successful LSwith 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 ”

Contents

� Objective



25


� Safety margins

� Conclusions



Long calculations

Fast-running models

2. Artificial Neural Networks(with bootstrap)

1. Simplified T-H models

1. Simplified T-H models

Passive Residual Heat Removal System in theHigh Temperature Reactor Pebble Modular (HTR-PM)[in collaboration with Institute of Nuclear and New Energy Technology (INET)- Tsinghua University, Beijing, China]

Safety Parameter

27

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 ofPSA2011, pp. 1718-1728,International Topical meeting on Probabilistic Safety Assessment and Analysis, March13-17, 2011, Wilmington, USA.

Application: passive Residual Heat Removal System i n the High Temperature Reactor Pebble Modular (HTR-PM)

N Parameter Distribution Note1 W Normal Residual heat power2 Ta,in Bi-Normal Temperature of inlet air in the air-cooled tower3 xi1 Uniform Resistance coefficient of elbow4 xi2 Uniform Resistance coefficient of header channel5 xiw Uniform Resistance coefficient of the water tank walls6

xia,in UniformSum of the resistance coefficients of inlet shutter and air coolingtower and silk net

7xia,out Uniform

Sum of the resistance coefficients of outlet shutter and air coolingtower and silk net

8 xia,narrow Uniform Resistance coefficient of the narrowest part of the tower9 Pa,in Uniform Pressure of the inlet air in the cooler tower10 dx Uniform Roughness of pipes11 Ha Normal Height of chimney12 La Normal Length of pipes in the exchanger13 Na Normal Total number of pipes in the air cooler14 Af Normal Air flow crossingarein thenarrowestpartof thetower

37 input parameters

14 Af Normal Air flow crossingarein thenarrowestpartof thetower15 Af,in Normal Inlet air flow crossing area in the tower16 Af,out Normal Outlet air flow crossing area from the tower17 Af,narrow Normal Crossing area in the narrowest part of the tower18 S1 Normal Distance between centers of adjacent pipes in horizontal direction19 S2 Normal Distance between centers of adjacent pipes in vertical direction20 S Normal Distance between fins in the ribbed pipe21 Da Normal Pipes inner diameter in the air cooling exchanger22 Do Normal Pipes outer diameter23 Douter Normal Rib outer diameter24 Pw Normal Water pressure in the pipes25 Hw Normal Elevatory height of water26 Nw Discrete Normal Number of water cooling pipes for each loop27 Lw Normal Length of the water cooling pipes28 Dw Normal Inner diameter of the water cooling pipes29 D1 Normal Inner diameter of the in-core and air cooler connecting pipes30 D2 Normal Inner diameter of the in-core header31 LC Normal Length of the in-core and air cooler connecting pipes (“cold leg”)32 LH Normal Length of the in-core and air cooler connecting pipes (“hot leg”)33 Ri Log-normal Thermal resistance of pipes inside of the heat exchanger34 Ro Log-normal Thermal resistance due to the dirt of the pipes fins35 Rg Log-normal Thermal resistance of the gap between fins36 Rf Log-normal Thermal resistance of fins37 lamd Normal Heat transfer coefficient of the pipes

1 output parameter: outlet water temperature

- 37 input parameters to be sampled- 3 Accidental scenarios:

A = 2 loops are failed

B = 1 loop is failed

X XX

29Application: passive Residual Heat Removal System i n the High Temperature Reactor Pebble Modular (HTR-PM)

C = 0 loops are failed

X

Acquire insights on the behavior of the system with respect to how much its output depends on the inputs

Model simplification by sensitivity analysis

Objective:

COMPUTATIONALLY BURDENSOMEDisadvantage:(several model computations)

comparison

30

VARIANCE DECOMPOSITION

SEVERAL MODEL

EVALUATIONS

Fast TH model of

RHR

ANALYTIC HIERARCHY PROCESS

SEVERAL MODEL

EVALUATIONSQualitative

resultsXY. 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 ofSAMO2010, Milano, July 2010, Procedia - Social andBehavioral Sciences, Volume 2, Issue 6, 2010, Pages 7772-7773.

Y. Yu, T. Liu, J. Tong, J. Zhao, F. Di Maio, E. Zio, A. Zhang, “Multi-Experts Analytic Hierarchy Process for the SensitivityAnalysis of Passive Safety Systems”,Proceedings of the 10th International Probabilistic Safety Assessment &Management Conference, PSAM10, Seattle, June 2010.

The simplified model of the Residual Heat Removal System in the HTR-PM

The standard Monte Carlo-driven fault engine inject s component failures at random times

1st loop of the RHR 2nd loop of the RHR 3rd loop of the RHR

31

Safety Parameter Safety Parameter

Availability of on-line instantaneous information

The Variance Decomposition Method

1 2( , )y m x x=Model

)]([)]([][ 11 2121XYVarEXYEVarYVar XXXX +=

][

)]([ 121

21

YVar

XYEVar XX=ηIndex of Importance of X1

][

)],([ 21,22,1

321

YVar

XXYEVar XXX=ηIndex of Importance of X1, X2(for a 3 inputs model X1,X2,X3)

Single parameter analysis: Index of importance 2Xη

3 accidental scenarios:

Index of Importance for single parameter contribution to the output variance

Group of parameters: index of Importance

3 accidental scenarios:

2η

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

The Analytic Hierarchy Process

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.,

35

- comparing pairwise the model inputs with respect to their importance on the top goal, e.g.

- computing the priority vectors , i.e., importance of the inputs

Input 1 Input 2

Input 1 1 3

Input 2 1/3 1

1 = A and B equally important

3 = A slightly more important than B

5 = A strongly more important than B

7 = A very strongly more important than B

9 = A absolutely more important than B

Comparison between Variance Decomposition method and Analytic Hierarchy Process (AHP)

� Power W is importantthe same with AHP� Inlet air temperature Ta,in is

importantthe same with AHP� Water pressure Pw not identified as

important by AHP

N Parameter Variance Decomposition Analytic Hierarchy Process [*]1 W X X2 Ta,in X X3 xi1

4 xi2

5 xiw

6 xia,in

7 xia,out

8 xia,narrow

9 Pa,in

10 dx11 Ha

12 La

13 Na

14 Af

15 Af,in

16 Af,out

17 A

� All other parameters have minor impacts with respect to W and Ta,in

the same with AHP

AHP and Variance Decompositionprovide coherent results

17 Af,narrow

18 S1

19 S2

20 S21 Da

22 Do

23 Douter

24 Pw X25 Hw

26 Nw

27 Lw

28 Dw

29 D1

30 D2

31 LC

32 LH

33 Ri

34 Ro

35 Rg

36 Rf

37 lamd

372. Fast-running empirical regression models: Bootstrapped Artificial Neural Networks (ANNs)

�ANNs = empirical regression models

� generate a reducednumber (50-100) of I/O data examplesby running the T-H code

� train the ANN model to fit the data

� usethe ANN (instead of the T-H code) to calculate the output

�ANNs for practical use in passive system reliability assessment:

ANN REGRESSION MODEL → ADDITIONAL UNCERTAINTYANN REGRESSION MODEL → ADDITIONAL UNCERTAINTY

CONFIDENCE INTERVAL FOR THE QUANTITY OF INTEREST

BOOTSTRAP (ENSEMBLE) OF ANN REGRESSION MODELS

� each modelof the ensemble provides an estimateof the output

� the empirical distribution of the bootstrapped estimates is built

in our work:

38Application: passive decay heat removal system of a Gas-cooled Fast Reactor (GFR)

Nine uncertain parameters, x (Gaussian):• Power• Pressure• Cooler wall temperature• Nusselt numbers (forced, mixed, free)• Friction factors (forced, mixed, free)• Friction factors (forced, mixed, free)

SYSTEM FUNCTIONAL FAILURE

Tout,core

( ){ }CxTx hotcoreout °>1200: ,

( ){ }CxTx avgcoreout °> 850: ,

F=

∩

39Application – Bootstrapped ANNs: functional failure probability estimation

4

5

6

7

8

9

10x 10

-4 ANN-based BBC 95% CIs for P(F)

4

5

6

7

8

9

10x 10

-4 Quadratic RS-based BBC 95% CIs for P(F)

Comparison with quadratic Response Surfaces (RSs)(Arul et al., 2009; Fong et al., 2009; Mathews et al., 2009)

ANNs Quadratic RSs

Fai

lure

pro

bab

ility

, P(F

)

Fai

lure

pro

bab

ility

, P(F

)Reference result obtained by Monte Carlo Simulation with NT = 250000 samples!

(~ 417 h)

Total CPU time (ANN/RS) << 1/100·Total CPU time (T-H model)

10 20 30 40 50 60 70 80 90 100 110

0

1

2

3

4

10 20 30 40 50 60 70 80 90 100 110

0

1

2

3

4

Fai

lure

pro

bab

ility

,

Fai

lure

pro

bab

ility

,

Training sample size Training sample size

N. Pedroni, E. Zio, G. E. Apostolakis, “Comparison of bootstrapped Artificial Neural Networks and quadratic ResponseSurfaces for the estimation of the functional failure probability of a thermal-hydraulic passive system”,ReliabilityEngineering and System Safety, 95(4), 2010, pp. 386-395.

40Bootstrapped ANNs: first-order global Sobol sensitivity indices – Results

0.7

0.75

0.8

0.85

0.9

0.95

1

2

0.7

0.75

0.8

0.85

0.9

0.95

1

2

Pressure, x2Pressure, x2

So

bo

l in

dex

So

bo

l in

dex

ANNs Quadratic RSs

10 20 30 40 50 60 70 80 90 100 1100.6

0.65

p10 20 30 40 50 60 70 80 90 100 110

0.6

0.65

Reference result obtained by Monte Carlo Simulation with NT = 110000 samples!(~ 92 h)

�ANNs produces more accurate and precise estimates than quadratic RSs� CPU time (ANN) ~ 1.5·CPU time (RS)

Training sample size Training sample size

E. Zio, G. E. Apostolakis, N. Pedroni, “Quantitative functional failure analysis of a thermal-hydraulic passive system bymeans of bootstrapped Artificial Neural Networks”,Annals of Nuclear Energy, Volume 37, Issue 5, 2010, pp. 639-649.

41Conclusions - Fast-running models

• Simplified MATLAB T-H model:� dependenceof the system response on the time and magnitude of

components and equipments failures

� influenceof the uncertaintiesand of components and equipments failures on the system function

� accuracyand speed of calculation �required coverage of scenarios for safety

• ANNs for substituting the T-H code in the estimation of the functional • 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 (~ 10-4)

• Small number (20-100) 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• Comparisonwith quadratic Response Surfaces

– better accuracies and precisions of ANNs

– slightly higher CPU cost associated to ANNs

Contents

� Objective



42


� Safety margins

� Conclusions



Uncertainty/Confidence

Order statistics(with bootstrap)

Safety margins

Safety margin quantification

( )2200MAXcladT F≤ °

( )cladT

Confidence Interval

Confidence Interval

Confidence Interval

44

Uncertainty in safety margins calculation by numerical code• Input values

• Modeling hypotheses

Interval

E. Zio, and F. Di Maio, “Bootstrap and Order Statistics for Quantifying Thermal-Hydraulic Code Uncertainties in theEstimation of Safety Margins”,Science and Technology of Nuclear Installations, Volume 2008, Article ID 340164, 9pages, doi:10.1155/2008/340164

- 37 input parameters to be sampled- 3 Accidental scenarios:

A = 2 loops are failed

B = 1 loop is failed

X XX

AIM: estimates of the 95th percentiles of the safety parameters distributions


� Bootstrapped Order Statistics (BOS)Minimum Minimum numbernumber ofof simulationssimulations N N forfor estimatingestimating the the distributiondistribution ofof the the safetysafety parameterparameter withwith a a givengiven confidenceconfidence, , accounting accounting alsoalso forfor the the uncertaintyuncertainty ofof the the empiricalempirical modelmodel usedused toto simulate the simulate the accidentalaccidental scenariosscenarios

COMPUTATIONAL BURDEN

C = 0 loops are failed

X

Results: Two RHR loops are enoughThe 3rd loop can be considered as a redundancy

Number of simulations N for each accidental scenario: 50Number of Bootstrap replications : 100


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, 2010, pp. 828–836

results in:Similarity of the 95th percentile point- estimates (reliability)

Narrowing the confidence intervals (robustness)

Approaching down the estimates to the true value (conservativeness)

… increasing the number of simulations


Contents

� Objective



48


� Safety margins

� Conclusions

Conclusions

• Objective:

• Contributions:

� Advanced Monte Carlo Simulation methods:

� SSand LS for the reliability analysis of a T-H passive system

� Optimization of the LS method (variance-minimizing search of “important direction” α)

49

To address the computational challenges related to the reliability analysis of Thermal-Hydraulic (T-H) passive safety systems

� Successful LS performancewith 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

� Percentileestimation by order statistics

� Percentile uncertainty quantification by bootstrap

Advanced Simulation Methods for the Reliability Analysis ...li/materials/lecture_NCEPU.pdf ·...

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