determination of upperbound failure rate by graphic confidence interval estimate
DESCRIPTION
LAUR-01-1671. Determination of Upperbound Failure Rate by Graphic Confidence Interval Estimate. K. S. Kim (Kyo) Los Alamos National Laboratory Los Alamos, NM 87545 E-mail: [email protected]. Kim-1. - PowerPoint PPT PresentationTRANSCRIPT
Determination ofUpperbound Failure Rate
by
Graphic Confidence Interval Estimate
K. S. Kim (Kyo)Los Alamos National Laboratory
Los Alamos, NM 87545
E-mail: [email protected]
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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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