Monte Carlo Simulation –Experimental and Theoretical View

Monte Carlo Simulation experimental and theoretical view

Introduction

Buffon

Gosset (Student)

Fermi, von Neumann, Ulam

Neutron transport

System Transport (RAMS)

Uncertainty and Sensitivity Analysis

Kelvin

The History of Monte Carlo Simulation

1707-

88

1908

1824-

1907

1950s

1930-

40s

1990s

44RAMS: evaluating the probability of system failure

xdxqxIFxPFP F )()()()(

: vector of the uncertain system parameters

: multidimensional probability density function (PDF) of

: failure region (in this case, F = {Y(x) > Y})

: indicator function -

nj xxxxx ...,,...,,, 21

n

j

jj xqxq1

)()( xnF

1,0:)( nF xI

Fx

FxxIF

if,0

if,1)(

(Y(x) Y)

(Y(x) > Y)

Y

Y

S

5Passive 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)

SYSTEM FUNCTIONAL FAILURE

Tout,core CxTx hotcoreout 1200: ,

CxTx avgcoreout 850: ,

F

=

The Isolation Condenser (IC) system is designed to remove excess sensible and core

decay heat from the Simplified Boiling Water Reactor (SBWR) by natural circulation,

when the normal heat removal system is unavailable

Isolation Condenser in SBWR

11 design and critical parameters 40 sub-critical parameters to be investigated

The Passive Residual Heat Removal (RHR) System of the HTR-PM

RHR System in HTR-PM

Legend:

1.Reactor

2.Vessel

3.Water cooling wall

4.Hot water header

5.Cold water header

6.Shade

7.Hot leg

8.Cold leg

9.Water tank

10.Air cooler

11.Air cooling tower

12.Inlet shutter

13.Outlet shutter

14.Inlet net

15.Outlet net

16.Wind shield

The Simulation Code Models

37 Inputs

Residual Heat in

Reactor Core Water-Cooled Wall

Air-Cooler in the

Tower

Atmosphere

radiation

Natural circulation

natural convection

RHR System in HTR-PM

Problem statement

Description of phenomena Adoption of mathematical models(which are for example turned into

operative computer codes for simulations)

Models Deterministic or Stochastic

In practice the system under analysis can not be characterized exactly

and the knowledge of the undergoing phenomena is incomplete

Uncertainty on both the values of the model parameters and

on the hypothesis underlying the model structure

Uncertainty propagates within the model and causes uncertainty

of the model outputs

How uncertainty propagates

y = m(x)

y = m(x)

x = vector of n uncertain input

parameters

m(x) = model structure

Deterministic Stochastic

y = output vector correspondent

to x and m(x)

(e.g., advection-dispersion equation, Newton law)

(e.g., Poisson model, exponential model)

m(x)

11Reliability assessment failure probability evaluation by Monte Carlo (MC)

Monte Carlo (MC)

sampling of uncertain parameters

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

System model code

(e.g. RELAP)

System

performance indicator, Y(x)

NT independent runs

Sample estimate of the failure probability P(F), YT

number Y xP F

N

Fuel

cla

ddin

g

Tem

per

ature

, Y

(x)

Time

F

Failure threshold, Y

S

12The Monte Carlo (MC)-based approach for failure probability

evaluation: drawbacks

T

Y

N

xYnumberFP

SMALL NUMBER FOR

HIGHLY RELIABLE SYSTEMS!

LONG CALCULATIONS!

S

+

UNCERTAINTY/CONFIDENCE

13The Monte Carlo (MC)-based approach for failure probability

evaluation: technical developments

1. Subset Simulation (SS)

2. Line Sampling (LS)

ESTIMATION OF THE RELIABILITY OF T-H SYSTEMS

Long calculations

Rare failure events

-Failure Domain

(FD) Identification

- Confidence on the

Safety Margin

estimation

Advanced MC Simulation

methods

Meta-models (e.g. Artificial Neural Networks)

1. Identification of important directions for pointing towards the failure region FD

2. Pre-selection of failure transients

Order statistics

TRANSIENTS

GENERATION

Issues: Solutions:

The experimental view

Simulation of system transport

Simulation for reliability/availability analysis of a component

Examples

Contents

Sampling Random Numbers

SAMPLING RANDOM

NUMBERS

Example: Exponential Distribution

Probability density function:

Expected value and variance:

00

0)(

t

tetf tT

2

1][

1][

TVar

TE

t

fT(t)

( ) tTf t e

Sampling Random Numbers from FX(x)

xX exF1

Sample R from UR(r) and find X:

RFX X1

xX exFR 1

RRFX X 1ln

11

Example: Exponential distribution

sample an

Graphically:

0 1, ,..., ,...kx x x

0

k

k k i

i

F P X x P X x

~ [0,1)R U

Sampling from discrete distributions

1- Calculate the analytic solution for the failure probability of the network,

i.e., the probability of no connection between nodes S and T

2- Repeat the calculation with Monte Carlo simulation

Arc number i Failure

probability Pi

1 0.050

2 0.025

3 0.050

4 0.020

5 0.075

Failure probability estimation: example

Analytic solution

21

43

1

[ 1] [ ] 3.07 10T jj

P X P M

3

1 1 4

3

2 2 5

4

3 1 3 5

5

4 2 3 4

[ ] 0.05 0.02 1 10

[ ] 0.025 0.075 1.875 10

[ ] 0.05 0.05 0.075 1.875 10

[ ] 0.025 0.05 0.02 7.5 10

P M p p

P M p p

P M p p p

P M p p p

1) Minimal cut sets

M2={2,5}

M3={1,3,5}

M4={2,3,4}

2) Network failure probability (rare event approximation):

The arrival state of arc i after it has undergone a transition can be

sampled by applying the inverse transform method to the set of discrete

probabilities of the mutually exclusive andexhaustive arrival

states

We calculate the arrival state for all the arcs of the newtork

As a result of these transitions, if the system falls in any configuration

corresponding to a minimal cut set, we add 1 to Nf

The trial simulation then proceeds until we collect N trials. We obtain

the estimation of the failure probability dividing Nf by N

0 1

R

S

B

B

1

31

B

B

1

21

S

,1i ip p

ip 1 ip

Monte Carlo simulation

N number of trials

NF number of trials corresponding to realization of a minimal cut set

The failure probability is equal to NF / N

clear all;

%arc failure probabilities

p=[0.05,0.025,0.05,0.02,0.075]; nf=0;

n=1e6; %number of MC simulations

for i=1:n

%sampling of arcs fault events

r=rand(1,5); s=zeros(1,5); rpm=r-p;

for j=1:5

if (rpm(1,j)= 1

nf=nf+1;

end

end

pf=nf/n; %system failure probability

For n=106, we obtain3[ 1] 3.04 10TP X

Monte Carlo simulation

SIMULATION OF SYSTEM

TRANSPORT

Monte Carlo simulation for system reliability

PLANT = system of Nc suitably connected components.

COMPONENT = a subsystem of the plant (pump, valve,...) which may stayin different exclusive (multi)states (nominal, failed, stand-by,... ). Stochastic

transitions from state-to-state occur at stochastic times.

STATE of the PLANT at t = the set of the states in which the Nccomponents stay at t. The states of the plant are labeled by a scalar which

enumerates all the possible combinations of all the component states.

PLANT TRANSITION = when any one of the plant components performs astate transition we say that the plant has performed a transition. The time at

which the plant performs the n-th transition is called tn and the plant state

thereby entered is called kn.

PLANT LIFE = stochastic process.

Random walk = realization of the system life generated by the underlying

state-transition stochastic process.

Plant life: random walk

Phase Space

( ) ( ) 0,R R MC t C t t T

( ) ( ) 1 ,R R MC t C t t T

( ) ( ) 1 ,R R MC t C t t T

( ) ( ) 0,R R MC t C t t T

( ) ( )R

T

C tF t

M

Example: System Reliability Estimation

Events at components

level, which do not entail

system failure

T(tt; k)dt = conditional probability of a transition at tdt, given that thepreceding transition occurred at t and that the state thereby entered wask.

C(k k; t) = conditional probability that the plant enters state k, given thata transition occurred at time t when the system was in state k. Both theseprobabilities form the trasport kernel:

K(t; k t; k)dt = T(t t; k)dt C(k k; t)

(t; k) = ingoing transition density or probability density function (pdf) of asystem transition at t, resulting in the entrance in state k

Stochastic Transitions: Governing

Probabilities

SIMULATION FOR COMPONENT

AVAILABILITY / RELIABILITY

ESTIMATION

1 2

m

State 2

State 1

State X=1 ONState X=2 OFF

One component with exponential

distribution of the failure time

1 2

m

State X=1 ONState X=2 OFF

One component with exponential

distribution of the failure time

values

3 10-3 h-1

m 25 10-3 h-1

Limit unavailability:

1/0.1071

1/ 1/U

m

m

Monte carlo simulation for estimating the

system availability at time t

- Nt time intervals t

- If the component fails in t+t, the counter increases cA(tj) =cA(tj) +1; otherwise, c

A(tj) = cA(tj)

Trial 1



- another trial

Trial 1

Trial i-th



- another trial

Trial 1

Trial i-th

Trial M-th



Trial 1

Trial i-th

Trial M-th

= unavailability at time tj

= mean value of cA(tj) at time tj

( ) { ( ) 1}A j

G t P X t

( )( )

A

j

A j

c tG t

M

The counter cA(tj) adds 1 until M trials have been sampled

Pseudo Code

%Initialize parameters

Tm=mission time; Nt=number of trials; Dt=bin length;

Time_axis=0:Dt:Tm;

FOR i=1:Nt

%parameter initialization for each trial

t=0; failure_time=0; repair_time=0; state=1;

while t=failure_time));

counter(i,lower_b)=1;

Else

t=t+exprnd(1/mu); state=1; repair_time=t;

upper_b=find(time_axis

SIMULATION FOR SYSTEM

AVAILABILITY / RELIABILITY

ESTIMATION

Phase Space

Arrival

Init

ial

1 2 3

1 -

2 -

3 -

)( 21BA

)(

31BA

( )2 1A B

)(

32BA

)( 13BA

)(

23BA

A

B

C

Components times of transition between states are exponentially distributed

( = rate of transition of component i going from its state ji to the state mi)i

mji 1

Indirect Monte Carlo: Example (1)

The components are initially (t=0) in their nominal states (1,1,1)One minimal cut set of order 1 (C in state 4:(*,*,4)) and one minimalcut set of order 2 (A and B in 3: (3,3,*)).

Arrival

1 2 3 4

Init

ial

1 -

2 -

3 -

4 -

C 21 C

31 C

41

C 12 C

32 C

42

C 13 C

23 C

43

C 14 C

24 C

34

Indirect Monte Carlo: Example (2)

SAMPLING THE TIME OF TRANSITION

The rate of transition of component A(B) out of its nominal state 1 is:

The rate of transition of component C out of its nominal state 1 is:

The rate of transition of the system out of its current configuration (1,1, 1) is:

We are now in the position of sampling the first system transition timet1, by applying the inverse transform method:

where Rt ~ U[0,1)

)(

31)(

21)(

1BABABA

CCCC

4131211

CBA111

1,1,1

)1ln(1

1,1,101 Rtt t

Analog Monte Carlo Trial

Assuming that t1 < TM (otherwise we would proceed to the successive trial), we now need to determine which transition has occurred, i.e. which

component has undergone the transition and to which arrival state.

The probabilities of components A, B, C undergoing a transition out of their initial nominal states 1, given that a transition occurs at time t1, are:

Thus, we can apply the inverse transform method to the discrete distribution

1,1,11

1,1,1

1

1,1,1

1 , ,

CBA

0

R

C 1,1,11

A

1,1,11

B

1,1,11

C

C

10

Sampling the kind of Transition (1)

Given that at t1 component B undergoes a transition, its arrival state can be sampled by applying the inverse transform method to the set of

discrete probabilities

of the mutually exclusive and exhaustive arrival states

B

B

B

B

1

31

1

21 ,

0 1

R

S

B

B

1

31

B

B

1

21

S

As a result of this first transition, at t1 the system is operating in configuration (1,3,1).

The simulation now proceeds to sampling the next transition time t2 with the updated transition rate

CBA131

1,3,1

Sampling the Kind of Transition (2)

RSU[0,1)

The next transition, then, occurs at

where Rt ~ U[0,1).

Assuming again that t2 < TM, the component undergoing the transition and its final state are sampled as before by application of the inverse trasform

method to the appropriate discrete probabilities.

The trial simulation then proceeds through the various transitions from one system configuration to another up to the mission time TM.

)1ln(1

1,3,112 Rtt t

Sampling the Next Transition

When the system enters a failed configuration (*,*,4) or(3,3,*), where the * denotes any state of the component,

tallies are appropriately collected for the unreliability and

instantaneous unavailability estimates (at discrete times tj [0, TM]);

After performing a large number of trials M, we canobtain estimates of the system unreliability and

instantaneous unavailability by simply dividing by M, the

accumulated contents of CR(tj) and CA(tj), tj[0,TM]

Unreliability and Unavailability Estimation

For any arbitrary trial, starting at t=0 with the system in nominal

configuration (1,1,1) we would sample all the transition times:

where

)1ln(1

1,

1

01 Rtti

mtim

im i

i

i

Cm

BAim

CBAi

i

i

ifor 4,3,2

,for 3,2

,,

)1,0[~1, URi

mt i

A

B

C

Direct Monte Carlo: Example (1)

These transition times would then be ordered in ascending order from

tmin to tmaxTM .

Let us assume that tmin corresponds to the transition of component A

to state 3 of failure. The current time is moved to t1= tmin in

correspondence of which the system configuration changes, due to

the occurring transition, to (3,1,1) still operational.

A

B

C

Direct Monte Carlo: Example (2)

The new transition times of component A are then sampled

and placed at the proper position in the timeline of the succession of

occurring transitions

The simulation then proceeds to the successive times in the list, incorrespondence of which a system transition occurs.

After each transition, the timeline is updated with the times of thetransitions that the component which has undergone the last transition

can do from its new state.

During the trial, each time the system enters a failed configuration, tallies are collected and in the end, after M trials, the unreliability and

unavailability estimates are computed.

)1ln(1

3,

3

13 RttA

mtAm

Am A

A

A

)1,0[~

2,1

3, UR

k

Amt A

Example (2)

The theoretical view

Contents

Sampling

Evaluation of definite integrals


Simulation for reliability/availability analysis

Buffons needle

Buffon considered a set of parallel straight lines a distance D apart onto a plane and computed the probability P that a needle of length L < D randomly

positioned on the plane would intersect one of these lines.

sinP P Y L

1( ) [0, ]

1( ) [0, ]

/

/ 2

Y

A

f y y DD

f

dy d L DP

D

rrRPrUR :cdf

1 :pdf dr

rdUru RR

0

Sampling (pseudo) Random Numbers

Uniform Distribution

~ [0,1)R U

1 mod i ix ax c m

, [0, 1]

1

a c m

m

0 0

1 1

15 15

16

22

16

11(5 2 1) mod 16 11

16

...

1313

16

2

x r

x r

x r

x

ii

xr

m


Uniform Distribution

Example: a = 5, c = 1, m = 16

Where

Sample R from UR(r) and find X:

RFX X1

Question: which distribution does X obey?

Application of the operator Fx to the argument of P above yields

Summary: From an R UR(r) we obtain an X FX(x)

xRFPxXP X 1

xFxFRPxXP XX

1

1 r 0

rUrR RPr xFX

R X x


Generic Distribution

Markovian system with two states (good, failed) hazard rate, = constant

cdf

pdf

Sampling a failure time T

tT etTPtF 1

dtedttTtPdttf tT

tTR etFrFR 1

RRFT T 1ln

11

Example: Exponential Distribution

hazard rate not constant

CDF

pdf

Sampling a failure time T

1 tTf t dt P t T t dt t e dt

1 tR TR F r F t e

1

1 1 ln 1TT F R R

Example: Weibull Distribution

1 tTF t P T t e

sample an

Graphically:

0 1, ,..., ,...kx x x

0

k

k k i

i

F P X x P X x

~ [0,1)R U

1 1

1 1

( ) ( )

~ [0,1) and ( )

k k R k R k

R

k k k k k k

P F R F F F F F

R U F r r

P F R F F F f P X x

Sampling by the Inverse Transform Method:

Discrete Distributions

Contents

Sampling




MC analog dart game: sample x from f(x)

the probability that a shot hits x dx is f(x)dx the award is g(x)

Consider N trials with result {x1, x2, ,xn}: the average award is

N

i

iN xgN

G1

1

1 ; 0 pdf

dxxfxfxf

dxxfxgGb

a

MC Evaluation of Definite Integrals (1D)

Analog Case

1

0

1

2 2

0

22

2

2cos 0.6366198

2

By setting: 1, g cos2

( )

1 ( ) cos

2 2

1 1 ( ) ( )

1 1 2 1 9

2

N

N

G x dx

f x x x

G E g x

E g x x dx

Var G Var g x E g x E g xN N

Var GN N

2

4

2 6

.47 10

for 10 histories, ~ [0,1) cos2

0.6342, 9.6 10N

i i i

N G

N x U g x x

G s


Example

MC biased dart game: sample x from f1(x)

the probability that a shot hits x dx is f1(x)dx the award is

N

i

iN xgN

Gxgxf

xfxg

1

11

1

1

1

The expression for G may be written

DDdxxfxgdxxfxg

xf

xfG 111

1


Biased Case

The pdf f1*(x) is:

From the normalization condition:

For the minimum value

* 2

1 ( )f x a bx

1 1

* 2

1

0 0

* 2

1

( ) 1 1 3( 1)3

( ) 3( 1)

bf x dx a bx dx a b a

f x a a x

31.5

2a

1

1

1 1

2

2

0 0 ( )( )

cos 12cos (1.5 1.5 )

2 1.5 1.5f x

g x

x

G xdx x dxx

1

0

2cos 0.6366198

2G x dx


Biased Case: Example

2

4

1 1

1 2 10.406275 9.9026 10NVar G

N N

Finally we obtain:

1

4 *

1 1 2

2 8

1

cos2

for 10 histories, ~1.5 1.5

0.6366, 9.95 10N

i

i i

i

N G

x

N x f g xx

G s


Biased Case: Example

Contents

Sampling




Monte Carlo simulation for system reliability

PLANT = system of Nc suitably connected components.

COMPONENT = a subsystem of the plant (pump, valve,...) which may stay in different exclusive (multi)states (nominal, failed, stand-by,... ).

Stochastic transitions from state-to-state occur at stochastic times.

STATE of the PLANT at t = the set of the states in which the Nccomponents stay at t. The states of the plant are labeled by a scalar

which enumerates all the possible combinations of all the component

states.

PLANT TRANSITION = when any one of the plant components performs a state transition we say that the plant has performed a

transition. The time at which the plant performs the n-th transition is

called tn and the plant state thereby entered is called kn.

PLANT LIFE = stochastic process.

Random walk = realization of the system life generated by the underlying

state-transition stochastic process.

Plant life: random walk

T(tt; k)dt = conditional probability of a transition at t dt, given that the preceding transition occurred at t and that the state thereby entered was k.C(k k; t) = conditional probability that the plant enters state k, given that a transition occurred at time t when the system was in state k.Both these probabilities form the trasport kernel:

K(t; k t; k)dt = T(t t; k)dt C(k k; t)

(t; k) = ingoing transition density or probability density function (pdf) of a system transition at t, resulting in the entrance in state k

Stochastic Transitions: Governing

Probabilities

The von Neumanns Approach and the Transport Equation

74

The transition density (t; k) is expanded in series of the

partial transition densities:

n(t; k) = pdf that the system performs the nth transition at t,

entering the state k.

Then,

Transport equation for the plant states

'

0

0

0

)','|,()','('),(

),(),(

k

t

t

n

n

ktktKktdtkt

ktkt

)**,,(),,(),(),( 110011000

*011

1

0

1

ktktKktktKktdtktk

t

t

),,()**,,(

),,(),(),(

112211*

1

112211

1

*122

2

1

2

1

2

ktktKktktKdt

ktktKktdtkt

k

t

t

k

t

t

),,(),,(*)*,,(...

...),(

11112211*

1

*2

,...,,*

1

2

1

121

nn

t

t

t

tn

kkk

t

tn

n

ktktKktktKktktKdt

dtdtktn

n

n

Initial Conditions: (t*, k*)

Formally rewrite the partial transition densities:

Monte Carlo Solution to the Transport

Equation (1)

MC analog dart game: sample x = (t1, k1; t2, k2; ...) from f(x)=

the probability that a shot hits x dx is f(x)dx the award is g(x)=1

Consider N trials with result {x1, x2, ,xn}: the average award is

N

i

iN xgN

G1

1

1 ; 0 pdf

dxxfxfxf

dxxfxgGb

a

),,(),,(*)*,,( 11112211 nn ktktKktktKktktK

MC Evaluation of Definite Integrals

Contents

Sampling




Monte Carlo Simulation in RAMS

k

t

k dtRktG0

),(),()(

= subset of all system failure states Rk(,t) = 1 G(t) = unreliability Rk(,t) = prob. system not exiting before t from the state k

entered at

References

Monte Carlo Simulation –Experimental and Theoretical View

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Transcript of Monte Carlo Simulation –Experimental and Theoretical View