Determination of Upperbound Failure Rate by Graphic Confidence Interval Estimate

1. LAUR-01-1671 Determination ofUpperbound Failure RatebyGraphic Confidence Interval Estimate K. S. Kim (Kyo) Los Alamos National Laboratory Los Alamos, NM 87545 E-mail: kyokim@lanl.gov Kim-1

2. 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 Kim-2

3. LAUR-01-1671 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) Kim-3

4. LAUR-01-1671 LANL Binning Example Kim-4

5. LAUR-01-1671 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 Kim-5

6. LAUR-01-1671 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 processcan 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 Kim-6

7. LAUR-01-1671 Chi-square Distribution Kim-7

8. LAUR-01-1671 Graphic Method • Zero-occurrence Events • Nonzero-occurrence Events Kim-8

9. LAUR-01-1671 Zero-occurrence Events Kim-9

10. LAUR-01-1671 Nonzero-occurrence Events Kim-10

11. LAUR-01-1671 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. Kim-11

12. Zero-occurrence Events LAUR-01-1671 (No occurrence for 3 components in 10 years, 80% Confidence Interval) C=3, T=10 yr U (80%)= Z/C*T =1.6/30 =0.053 /yr Spill frequency is less than 0.053/yr with 80% confidence Z=1.6 Kim-12

13. Nonzero-occurrence Events(4 occurrences for 5 components in 12 years, 95% Confidence interval) LAUR-01-1671 N=4, C=5, T=12 yrs U(95%) = R*(N/CT) = 2.3*0.067 = 0.15/yr Fire frequency is less than 0.15/yr with 95% confidence R=2.3 N=4 Kim-13

14. Concluding Remarks LAUR-01-1671 • 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%) Kim-14