A Computer Program for Evaluating the Alpha Factor Model Parameters Using the Bayesian Operation

Transcription

1 Computer Program for Evaluating the lpha actor odel Parameters Using the Bayesian Operation Baehyeuk Kwon a, oosung Jae a*, and Dong Wook Jerng b a Department of Nuclear Engineering, Hanyang University 17 Haengdang-Dong, Sungdong-u, Seoul, Korea b Department of Energy Systems Engineering, Chung-ng University 84 Heukseok-Ro, Dongjak-u, Seoul, Korea *Correspondence: jae@hanyang.ac.kr bstract: The assessment of common cause failure (CC) is necessary for reducing the uncertainty during the process of probabilistic safety assessment. basic unavailability assessment method is an approach for the quantitative analysis of CC modeling using Bayesian probability, in which the estimation of parameters is more accurate by combining the failure information from system, component and cause level. This study describes the CC evaluation program which has been developed for assessing the α-factor common cause failure parameters. Examples are presented to demonstrate the calculation process and necessary databases are presented. s a result, the posterior s for α-factors model parameter are obtained using the conjugate family s as well as general s for conducting a numerical estimation. Due to the fact that CC is one of the significant factors to affect both core damage frequency and large early release frequency, the appropriate evaluation for the relevant parameters is essential, though there are rare the CC data. In the previous study, the ultiple reek etter model () had been used for modeling the common cause failures in the OPR 1000 reactors. In the future modeling for the reactors, the α-factors approach might be employed for simulating the common cause failures as well as it will be quantified using the computer program developed by the C# language. The main operation to quantify the α-factors parameters is Bayesian which combines the prior and the likelihood function to produce the posterior. It is expected that this program might contribute to enhancing the quality of probabilistic safety assessment and to reducing common cause failure uncertainty. Keywords: common cause failure, ultiple reek etter model, α-factors model, Bayesian, parameter uncertainty, data analysis 1. INTRODUCTION Since the quantitative analysis about the Common Cause ailure (CC) was first attempted in WSH-1004 [1], the various quantitative CC have been suggested. These for the CC analysis were synthetically analyzed in NURE/CR-4780 [2-3] and have been mostly used in the probability safety assessment (PS). The Common Cause ailure events occur when one or more components fail simultaneously or around same time due to a common cause. Due to the fact that CC is one of the significant factors to affect Core Damage requency and arge Early Release requency, systematic and accurate evaluation of the CC parameter is essential. However, it is practically impossible to perform CC analysis accurately with the lack of CC data just for one Nuclear Power Plant because CC is rare event. To address this problem, NURE/CR-5485 [4] reported the various CC parameter estimation methods based on the Bayesian method, which can reduce the CC parameter uncertainty using CC data not only of the reference Nuclear Power Plants but also of other NPPs. Up to now, PS in Korea has performed the CC analysis using beta factor model or ultiple reek etter () model and mainly used the CC parameter data provided by USNRC. The CC parameter data provided by USNRC, however, is currently the CC parameter data of lpha-actor odel (). Thus, the CC parameters used for the CC analysis in Korea have used parameters converted form parameters to parameter. In this case, the PS researchers have not performed the Bayesian updating for reducing uncertainty of the parameters due to the Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

2 fact that conversions of these CC parameters cannot know the parameters. or this reason, therefore the parameter has been used popularly to perform PS for Korean plants. In this study, therefore, the Bayesian-based CC parameters estimation program using α-factors parameter data has been developed for evaluating the CC parameters and applied for the system of the reference plant to obtain the uncertainty data of the parameters in model. The calculation algorithm of this program using the Bayesian method has been developed and programmed by the C# language in this study. 2. ETHODOOY 2.1. CC parametric The Bayesian-based CC parameters estimation program developed in this study consists of various CC. ultiple reek etter () model and lpha-actor odel () as shown in table 1 are important in this study. The estimation approaches of each parameter of the CC model are different. However, these CC have same value when parameters of the model and model can be converted to parameter of Basic Parameter odel (BP) following the conversion formulas described in table 2. Depending on the reason of these, the developed program in this study uses the relation of and. CC Table 1. Parameters of the CC model CC parameter Non-Staggered testing (NS) Staggered testing (S) ( 1,,,,..., 0) m 1 Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

3 CC Table 2. CC conversion parameters to Basic Parameter odel CC parameter Non-Staggered testing (NS) Staggered testing (S) ( 1,,,,..., 0) m CC ultiplier The conversion formulas in Table 2, however, cannot be practically used due to fact that the total failure probability of each component (Q t ) is unavailable generally. Due to the fact that the CC events rarely occur, the values of Q t and Q 1 are almost the same as shown in Eq. (1). Therefore, for calculation of the failure probability involving k specific components (Q k ), the CC multiplier ( k ) is first calculated. Q k is converted to the approximated equations to be practically used as shown in Eq. (2). In these equations, k and Q 1 are CC multiplier and single failure probability, respectively. The program developed in this study was designed to easily calculate CC multipliers for both and as shown in Table 3. (2) (1) Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

4 CC Table 3. CC ultipliers ormula CC parameter Non-Staggered testing (NS) Staggered testing (S) ( 1,,,,..., 0) m Bayes Theory Up to now there has been no detail guidance for estimating the CC parameters using the Bayesian method in an uncertainty analysis. In this study the Bayesian method has been introduced and applied for reducing the uncertainty of the common cause failures. The basic equation of the Bayesian theory [5-6] is described as follows. (3) Where, ( / E) is posterior of given evidence E, ( ) 0 is of prior to the evidence, and E ( / ) is likelihood function or the probability of the evidence E for the given. Table 4. Conjugate family s of Bayes operation ikelihood Conjugate prior Posterior Binomial Distribution Beta Beta ulti-nominal Dirichlet Dirichlet The likelihood function for common cause failure parameters depends on the common cause failure and has the corresponding conjugate as shown in Table 4. The posterior is proportional to the product of prior and the likelihood function as shown in Eq. 4. By applying the normalization factors into denominator, the Bayesian data analysis for the common cause failure parameters can be performed to result in determining the posterior as shown in Eq. (2). Bayesian analysis can be used as an efficient data analysis method among moments method, the curve fitting method, maximum likelihood method and Bayes method. (4) Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

5 2.4. Parameter s The C# program named COCP (Common Cause ailure Parameter Estimation for PS), has been developed and applied to obtain the common cause failure parameters. The parameters of the common cause failure can be obtained using the Beta conjugate family as sown in Table 4. The Bayesian approach results are shown in Table 5. It involves the systematic approach about how to combine the prior and the likelihood function to produce the posterior. Thus, the developed program, COCP, can not only estimate the CC parameter but also perform the Bayesian updating of the CC parameters of the important systems in the reference nuclear power plants. CC Table 5. Distribution of the CC parameter CC parameter Beta (a, b) Non-Staggered (NS) testing Staggered (S) Testing Beta (a, b) 3. RESUTS 3.1. Computer Program, COCP To evaluate the parameter s and the common cause failure uncertainty, a computer program named COCP (Common Cause ailure Parameter Estimation for PS), has been developed using the C# language. igure 1 shows the main screen of the program developed in this study. This program might contribute to evaluating easily the common cause failure analysis related to the ultiple reek etter model and lpha-actor odel. The developed program involves the Common Cause Component roup (CCC = m) which may be applied from a value of 2 to 8. The output data of the user-selected CC parameters offered in this program provides prior, likelihood and posterior for the parameters of the ultiple reek etter model and lpha-actor odel as shown in igure 2. Each provides the confidence interval such as 5 %, 50 %, and 95 %, mean values for the Beta (a, b). The COCP computer program consists of four functions which are Input module, Conversion module, Run module and Help module as shown in table 6. The Input module can deal with USNRC lpha-actor odel CC parameters [7] in which users can select the appropriate CC parameters considering the relevant system, failure mode, type of component, and system size. The Conversion Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

6 function module can convert from parameters to parameters easily. The common cause failure multiplier function supplies the function to calculate the multiplying values for the CC parameters selected by the analyzing user. The Bayesian analysis function module supplies the function to perform the Bayesian-updating for the CC parameter when the user allocates the actual number of the CC events in likelihood menu of the computer program. Table 6 : ain functions used in the computer program, COCP ain function Input data Conversion Bayesian analysis Description Connect database Selection of NRC CC data (2007) to conversion (CC parameter) to conversion (CC parameter) Bayesian analysis of / parameter / multiplier calculation igure 1. ain screen of the program (a) parameter output frame (b) parameter output frame igure 2. Parameter Output obtained from the program 3.2 Execution of the computer program Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

7 The common cause failure parameters have been evaluated using the program, COCP developed in this study under the particular assumed conditions. or demonstrating the program execution process, the prior data of the auxiliary feedwater system, motor driven pump, and fail to start for CCC=3 are utilized. The USNRC CC parameter estimations used in this study are shown in Table 7 while the likelihood data are shown in the Table 8. To execute the developed program, the parameters data of the CC have been calculated individually using the model and model and they are compared each other. Table 9 shows the Bayesian updating results showing that the calculated CC multiplier values are exactly identical regardless of the model types depending on the conditions. It is shown that the program, COCP might evaluate the parameter s of the model and model properly without any problems. Table 7. Prior data CCC ail mode α-factors lpha 1 lpha 2 lpha 3 Parameter of a 1.89E E E+00 beta b 8.16E E E+02 m=3 ail to 5% 9.33E E E-03 Start Percentiles 50% 9.60E E E-02 95% 9.79E E E-02 ean 9.59E E E-02 Table 8. ikelihood data CCC ail mode α-factors lpha 1 lpha 2 lpha 3 Parameter of a 5.00E E E+00 beta b 7.00E E E+01 m=3 ail to 5% 7.99.E E E-03 Start Percentiles 50% 8.82.E E E-02 95% 9.40.E E E-02 ean 8.77.E E E-02 Table 9. Posterior data and CC multiplier CC CC Non- Staggered CCC m=3 CCC m=3 Staggered m=3 ail mode ail to Start ail mode ail to Start ail to Start CC parameters lpha 1 lpha 2 lpha 3 Parameter of beta a 2.39.E E E+00 b 1.52.E E E+02 5% 9.14.E E E-02 Percentiles 50% 9.41.E E E-02 95% 9.63.E E E-02 ean 9.40.E E E-02 CC multiplier NS 8.86E E E-02 S 9.40E E E-02 CC parameters β γ Parameter of beta a 1.00.E E-01 b 1.31.E E-01 5% 1.67.E E-01 Percentiles 50% 1.32.E E-01 95% 3.63.E E+01 ean 2.39.E E+01 CC multiplier NS 8.86E E E-02 Parameter of beta a 3.74.E E-01 b 5.85.E E-01 5% 8.59.E E-01 Percentiles 50% 5.97.E E-01 95% 1.52.E E+00 ean 2.39.E E+00 CC multiplier S 9.40E E E-02 Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii

8 4. CONCUSION The common cause failures in nuclear power plants are one of the significant factors to affect both the values of Core Damage requency and arge Early Release requency. Several investigators have made many efforts to attempt to reduce the common cause failure uncertainty. In this study a methodology using the Bayesian operation has been developed and applied for reducing the uncertainty of the common cause failures. The Bayesian-based CC parameters estimation program called COCP has also been developed for evaluating the common cause failures parameters. It is shown that the COCP program can obtain the parameter s for various common cause failure for both the non-staggered and the staggered tests. It is also shown that this program might contribute to assessing the safety measures such as core damage frequency and large early release frequency. It is expected that the developed program might contribute to supplying the efficient risk assessment procedures by evaluating the uncertainty of the relevant common cause failures parameters of the analyzing systems. cknowledgement This work was supported by the KHNP Central Research Institute s Project and partly by Korea Radiation Safety oundation (rant Code: SB110). References [1] U.S. NRC (Nuclear Regulatory Commission), Reactor Safety Study: n ssessment of ccident Risks in U.S. commercial Nuclear Power Plant, WSH-1400, NURE , [2] osleh and K. N. leming, Pickard, owe, and arrick,., U.S. NRC, Procedure for Treating Common Cause ailures in Safety and Reliability Studies, Procedural ramework and Examples, NURE/CR-4780 Vol 1, [3] osleh and K. N. leming, Pickard, owe and arrick,., U.S. NRC, Procedure for Treating Common Cause ailures in Safety and Reliability Studies, nalytical Background and Techniques, NURE/CR-4780 Vol 2, [4] osleh,., INEE, uidelines on odelling Common-Cause ailures in Probabilistic Risk ssessment, NURE/CR-5485, [5] lfredo H-S. ng, Wilson H. Tang, Probability Concepts in Engineering Planning and Design, Volume I-Basic Principles, pp , [6] Peter. ee, Bayesian Statistics n Introduction, Third Edition, pp , [7] U.S.NRC (Nuclear Regulatory Commission), 2008, CC Parameter Estimations, 2007 Update. Probabilistic Safety ssessment and anagement PS 12, June 2014, Honolulu, Hawaii