Determination ofUpperbound Failure Rate

by

Graphic Confidence Interval Estimate

K. S. Kim (Kyo)Los Alamos National Laboratory

Los Alamos, NM 87545

E-mail: kyokim@lanl.gov

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If you believe that selecting Power Ball numbersis a random process, that is, a Poisson process,then your chance of winning is 1 in 1000000. But considering your horoscope today and invokingthe Bayesian theorem, your chance can be 1 in 5.Of course, there are sampling errors of plus-minus….

Gee, I wonder whatis the odd of gettingmy money back

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DOE Hazard Analysis Requirement

DOE Order 5480.23 requires Hazard Analysis for all Nuclear Facilities

Hazard Analysis entails estimation of Consequence and Likelihood (or Frequency) of potential accidents

Potential Accidents are “Binned” according to Consequence & Frequency for determination of further analysis and necessary Controls

DOE-STD-3009 provides Example for Binning

LANL Binning Matrix (risk matrix)

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LANL Binning Example

F R E Q U E N C Y

Decreasing Likelihood ---->

I II III IV V

A

1 1 2 2 3

B

1 2 2 3 3

C

1 2 3 3 4

D

3 3 3 4 4

C O N S E Q U E N C E

Increasi

ng S

ev

erit

y

--->

E

4 4 4 4 4

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Method for Frequency Determination

Historical Record of Event Occurrence (number of

events per component-time or N/C*T) • A simple division of N/C*T ignores uncertainty (1 event in 10

component-yrs and 100 events per 1,000 component-yrs would be represented by the same frequency value of 0.1/yr)

• Not useful for a type of accident that has not occurred yet (Zero-occurrence events)

Fault Tree/Event Tree Method (for PRA) can be used for Overall Accident Likelihood: Historical record is used for estimation of initiating event frequency or component failure rate/frequency

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Statistical Inference Primer

Typical occurrences of failure (spill, leaks, fire, etc.) are considered as random discrete events in space and time (Poisson process), thus Poisson distribution can be assumed for the Failure Rate (or Frequency)

Classical Confidence Intervals have the property that Probability of parameters of interest being contained within the Confidence Interval is at least at the specified confidence level in repeated samplings

Upperbound Confidence Interval for Poisson process can be approximated by Chi-square distribution function

U (1-P) is upper 100(1-P)% confidence limit (or interval) of ,P is exceedance probability,2(2N+2; 1-P) is chi-square distribution with 2N+2 degrees of

freedom

T*C2

)P1 ; 2N2( )P1(

2

U

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Chi-square Distribution

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Graphic Method

Zero-occurrence Events

Nonzero-occurrence Events

P)/2-1 2;(χ multipliera is Z where

TC

Z

TC2

)P1 ; 2(χ )P1(

2

2

U

2)/N(2N2 χ multipliera is Rwhere

TC

N R

TC2

)P1 ; 2N2(χ )P1(

2

2

U

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Zero-occurrence Events

Z values for Zero-occurrence Events

0.5

1

1.5

2

2.5

3

3.5

50 55 60 65 70 75 80 85 90 95 100

Confidence Interval (%)

Z V

alue

s

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Nonzero-occurrence Events

R values for Nonzero-occurrence Events

1

1.5

2

2.5

3

3.5

4

4.5

5

1 10 100 1000

Number of Failures (N)

R V

alue

s

95% Confidence

90% Confidence

80% Confidence

70% Confidence

50% Confidence

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Examples

Upperbound frequency estimate of a liquid radwaste spill of more than 5 gallons for a Preliminary Hazard Analysis (desired confidence level is set as 80% or exceedance probability of 0.2). No such spill has been recorded for 3 similar facilities in 10 years.

Upperbound frequency estimate of a fire lasting longer than 2 hours for Design Basis Accident Analysis (desired confidence level is set as 95% or exceedance probability of 0.05). Four (4) such fires have been recorded in 5 similar facilities during a sampling period 12 years.

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Z values for Zero-occurrence Events

0.5

1

1.5

2

2.5

3

3.5

50 55 60 65 70 75 80 85 90 95 100

Confidence Interval (%)

Z V

alu

es Z=1.6

C=3, T=10 yr

U (80%)= Z/C*T =1.6/30=0.053 /yr

Spill frequency is less than0.053/yr with 80% confidence

Zero-occurrence Events(No occurrence for 3 components in 10 years, 80% Confidence Interval)

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R values for Nonzero-occurrence Events

1

1.5

2

2.5

3

3.5

4

4.5

5

1 10 100 1000

Number of Failure (N)

R V

alu

es

95% Confidence

90% Confidence

80% Confidence

70% Confidence

50% Confidence

N=4

R=2.3

N=4, C=5, T=12 yrsU(95%) = R*(N/CT)

= 2.3*0.067= 0.15/yr

Fire frequency is less than 0.15/yrwith 95% confidence

Nonzero-occurrence Events(4 occurrences for 5 components in 12 years, 95% Confidence interval)

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• Setting Confidence Level depends on analysts

• Higher Level for events with sparse historical data (infrequent or rare events)

• Higher Level for Conservative Design Analysis (95% for DBA)

• Lower Level for expected or best estimate analysis (50%)

Concluding Remarks

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