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

2. If you believe that selecting Power Ball numbers is a random process, that is, a Poisson process, then your chance of winning is 1 in 1000000. But considering your horoscope today and invoking the Bayesian theorem, your chance can be 1 in 5. Of course, there are sampling errors of plus-minus…. Gee, I wonder what is the odd of getting my money back Kim-2

3. 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 LAUR-01-1671

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

5. 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 LAUR-01-1671

6. 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 Kim-6 LAUR-01-1671

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

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

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

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

11. 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 LAUR-01-1671

12. Z=1.6 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 Zero-occurrence Events (No occurrence for 3 components in 10 years, 80% Confidence Interval) Kim-12 LAUR-01-1671

13. N=4 R=2.3 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 Nonzero-occurrence Events (4 occurrences for 5 components in 12 years, 95% Confidence interval) Kim-13 LAUR-01-1671

14. 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 Kim-14 LAUR-01-1671