ML120540604
| ML120540604 | |
| Person / Time | |
|---|---|
| Site: | South Texas  |
| Issue date: | 02/22/2012 |
| From: | Galenko A, Popova E University of Texas at Austin |
| To: | Plant Licensing Branch IV |
| Singal, B K, NRR/DORL, 301-415-301 | |
| Shared Package | |
| ML120540492 | List: |
| References | |
| TAC ME7735, GSI-191, TAC ME7736 | |
| Download: ML120540604 (19) | |
Text
Uncertainty modeling of LOCA frequencies and break size distributions for the STP GSI-191 resolution Alexander Galenko, Elmira Popova agalenko@gmail.com, elmira@mail.utexas.edu The University of Texas at Austin 1 Introduction In the initial quantification (Crenshaw, 2012), Fleming et al. (2011) per-formed a substantial study designed to build upon the established EPRI risk-informed in service inspection program (EPRI, 1999). This method-ology was used as primary basis to develop the size and location-specific rupture frequencies (events per year) for the initial quantification. Although the overall methodology appears to be sound based on peer review (Mosleh, 2011) and reasonableness of the values obtained, NRC feedback in the Pilot Project reviews has resulted in further review of the approach. In this paper we propose a new approach to assign location specific LOCA frequencies that are derived from the overall frequencies, as defined in Tregoning et al. (2008).
We refer to this source reference as NUREG-1829.
The NUREG-1829 annual frequencies are not plant specific or plant-location specific. Yet they are used throughout the nuclear industry as an important input to PRA analyses, and therefore, they need to be preserved.
Conservation of the NUREG-1829 break frequencies is our guiding principle.
In this document we will work with the six break size categories, defined in Table 1, NUREG-1829 Volume 1, page xxi as effective break size for the 1
PWR plants. Table 1 shows the mapping between NUREG-1829 notation and ours. In addition, the use of the term distribution will be equivalent to a distribution function (either cumulative distribution or probability density function) of a random variable used to model a specified uncertainty.
Table 1: LOCA categories notation map Effective break size (inch) for PWR Notation 1
2 cat1 1 58 cat2 3 cat3 7 cat4 14 cat5 31 cat6 We should point out that the South Texas Project PRA analysis uses only 3 LOCA categories, small, medium, and large. Our proposed methodology can be applied to any finite number of categories.
In this document we will use the term location to represent weld loca-tions. Overall there are two ways to come up with location-specific LOCA frequencies: bottom-up and top-down. The bottom-up approach requires location specific failure data to estimate the corresponding probability of a weld failure. If we assume there are Mj different locations in the plant where breaks of size catj can occur, location1 , . . . , locationMj , then using the law of total probability we can write:
Mj X
P [catj ] = P [catj llocationi ]P [locationi ], j = 1, 2, . . . , 6 i=1 where P [catj ] is the total probability of a catj LOCA, P [catj llocationi ] is the 2
conditional probability of a catj LOCA given location i has been chosen, and P [locationi ] is the probability that location i is chosen. In our application we will assume that all locations are equally likely to be selected, i.e.
1 P [locationi ] = , i = 1, . . . , Mj Mj In the bottom-up approach we first must determine P [catj llocationi ] (us-ing estimation or expert elicitation). If that happens, then we can multiply these probabilities by 1/Mj , sum them up, and we will obtain the total prob-ability of a catj LOCA. If the bottom-up approach is followed we claim that the resulting total catj LOCA probability will NOT equal the number pro-vided in NUREG-1829 (or at least it is very unlikely to get that number).
The approach taken by Fleming et al. (2011) is an inherently bottom-up approach. To preserve the NUREG-1829 frequencies Fleming et al. (2011) developed an approximation scheme. In their review, the NRC technical team raised several questions about the bottom-up approach, which has lead us to propose a different methodology.
The method that we propose is rooted in the top-down approach: start with the NUREG-1829 frequencies and develop an intuitive way to distribute the cumulative frequencies among different weld locations.
We will use again the catj LOCA as an illustrative example. Assume that we have computed the total probability of a catj LOCA from the NUREG-1829 frequency table using the formula F requency[catj ]
P [catj ] = P6 i=1 F requency[cati ]
Again we assume there are Mj different locations in the plant where breaks of size catj can occur, (location1 , . . . , locationMj ), and they are equally likely, 3
i.e.
1 P [locationi ] = , i = 1, . . . , Mj .
Mj Note that the denominator is the total frequency of all LOCA sizes. We will assume that all P [catj llocationi ] = P [catj ], i = 1, . . . , Mj . Then applying the law of total probability we see that the resulting probability of a catj LOCA equals exactly the NUREG-1829 probability and P [catj at locationi ] =
P [catj ]/Mj .
The above methodology distributes equally the LOCA frequencies as de-fined in NUREG-1829 Table 1 between all locations that can experience breaks from one or more of the six size categories. The six break size cate-gories (columns in NUREG-1829 Table 1) are ranges bounded by six discrete points. For a particular weld we need to be able to sample from the con-tinuous range of break size values. In addition, we would like to be able to sample from the distribution of the frequencies. The rows in Table 1 from NUREG-1829 represents the distribution of the frequencies by reporting the mean, median, 5the and 95the percenttiles. We will use this information to fit six continuous distributions for each break size category.
2 Proposed methodology 2.1 Fitting distributions to the LOCA frequencies We will first describe the distribution fit to the frequencies for each break size category. In theory, there are infinite number of distributions that one can fit to the LOCA frequencies represented in NUREG-1829: two split Lognormal distributions are used in NUREG-1829 and Gamma distributions are used in NUREG/CR 6928.
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We chose to fit the bounded Johnson distribution, Johnson (1949) for the following reasons:
- It has four parameters that will allow us to match closely the four distri-butional characteristics provided by NUREG-1829. In order to get the parameters of the Johnson distribution we solve an optimization prob-lem with constraints defined by the four distributional characteristics:
the 5th percentile, the mean, the median, and the 95th percentile.
- It can have variety of shapes. In particular, skewed, symmetric, bi-modal, or unimodal shapes can be obtained.
The cumulative distribution function (CDF) of the bounded Johnson is:
F [x] = { + f [(x )/]} ,
where [x] - CDF of a standard Normal (0,1) random variable, and are shape parameters, is a location parameter, is a scale parameter, and f (z) = log[z/(1 z)]. The fitted parameters of the Johnson distribution for each of the six categories are given in Table 2. The comparison between the NUREG-1829 distributional characteristics of the LOCA frequencies and the fitted ones are presented in Table 3. The largest error in our estimation is 3.78% which we consider to be small enough.
Figures 1 and 2 show the fitted CDFs and probability density functions (PDFs) of the Johnson distribution for each category.
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Table 2: Fitted Johnson Parameters Johnson Parameters Cat1 0.7288246 0.3893326 0.00063449 0.02449228 Cat2 6.95E-01 2.40E-01 7.41E-06 2.44E-03 Cat3 7.24E-01 2.44E-01 2.06E-07 6.24E-05 Cat4 7.14E-01 2.39E-01 1.36E-08 6.19E-06 Cat5 4.73E-01 2.69E-01 1.87E-10 5.93E-07 Cat6 4.75E-01 2.73E-01 1.77E-14 8.52E-08 Table 3: NUREG-1829 and fitted Johnson mean, 5th and 95th percentile values NUREG-1829 Fitted Johnson Error 5% Mean 95% 5% Mean 95% 5% Mean 95%
Cat1 6.90E-04 7.30E-03 2.30E-02 6.89E-04 7.30E-03 2.30E-02 0.08% 0.01% 0.00%
Cat2 7.60E-06 6.40E-04 2.40E-03 7.56E-06 6.42E-04 2.40E-03 0.59% 0.38% 0.04%
Cat3 2.10E-07 1.60E-05 6.10E-05 2.09E-07 1.60E-05 6.12E-05 0.40% 0.23% 0.38%
Cat4 1.40E-08 1.60E-06 6.10E-06 1.40E-08 1.59E-06 6.08E-06 0.26% 0.65% 0.39%
Cat5 4.10E-10 2.00E-07 5.80E-07 4.14E-10 1.98E-07 5.86E-07 0.94% 0.94% 0.98%
Cat6 3.50E-11 2.90E-08 8.10E-08 3.59E-11 2.84E-08 8.41E-08 2.60% 2.03% 3.78%
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1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 CDF(x) CDF(x) CDF(x) 0.4 0.4 0.4 0.2 0.2 0.2 0.000 0.005 0.010 0.015 0.020 0.025 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0e+00 1e05 2e05 3e05 4e05 5e05 6e05 x x x (a) Cat1 (b) Cat2 (c) Cat3 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 CDF(x) CDF(x) CDF(x) 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0e+00 1e06 2e06 3e06 4e06 5e06 6e06 0e+00 1e07 2e07 3e07 4e07 5e07 6e07 0e+00 2e08 4e08 6e08 8e08 x x x (d) Cat4 (e) Cat5 (f) Cat6 Figure 1: Johnson CDFs for each category 7
400 140000 3500 120000 3000 300 100000 2500 2000 80000 PDF(x) 200 PDF(x) PDF(x) 1500 60000 1000 40000 100 500 20000 0 0 0.000 0.005 0.010 0.015 0.020 0.025 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0e+00 2e05 4e05 6e05 8e05 x x x (a) Cat1 (b) Cat2 (c) Cat3 1.2e+07 1200000 8e+07 1.0e+07 1000000 6e+07 800000 8.0e+06 PDF(x) PDF(x) PDF(x) 600000 6.0e+06 4e+07 400000 4.0e+06 2e+07 200000 2.0e+06 0
0e+00 2e06 4e06 6e06 8e06 0e+00 1e07 2e07 3e07 4e07 5e07 6e07 0e+00 2e08 4e08 6e08 8e08 1e07 x x x (d) Cat4 (e) Cat5 (f) Cat6 Figure 2: Johnson PDFs for each category 8
Once the best fit is found, we sample the LOCA frequencies for each category to get F requency[catj ] - realization of LOCA frequency for category j.
2.2 Distribution of LOCA frequencies to different weld locations We first convert LOCA frequencies to probabilities using X
P [catj ] = F requency[catj ]/ F requency[catl ],
lJ where
- P [catj ] - probability of observing a break that falls into category j given a break was observed
- F requency[catj ] - frequency of failure of type j where j J
- J = {cat1 , cat2 , cat3 , ..., catB } - set of possible break types (categories)
Given P [catj ] the next step is to distribute that probability among all welds that can experience a break from the same category. We compute P [catj at locationi ] - the probability that weld i will experience break of type j using P [catj at locationi ] = P [catj ]/Mj , where Mj = lIj l - number of welds that can experience category j breaks, i Ij , Ij - set of welds that can experience break category j. Here we assume that every weld that can experience a break of category j has equal probability of actually experiencing it.
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2.3 Sampling of the break size The final step is to sample the actual break size conditioned on the break category. Here we assume that the break size has a uniform distribution within a given category, formally breakSizeij U [minBreakji , maxBreakji ], j J, i Ij ,
where
- U [minBreakji , maxBreakji ] is the Uniform distribution bounded by minBreakji and maxBreakji
- minBreakji = catminBreak j
- maxBreakji = min[catmaxBreak j , weldsize i ]
- catminBreak j - minimum break size that would put it into category j
- catmaxBreak j - maximum break size that would put it into category j
2.4 Methodology summary This methodology will require two sampling loops in our simulator CASA Grande, Letellier (2011). We need one sampling loop for the break size within each category and a second loop that samples LOCA frequencies from their fitted distributions. Below is a step-by-step description of that procedure:
- 1. Set N - number of LOCA frequency samples and S - number of break size samples to generate 10
- 2. Sample LOCA frequencies F requency[catj ] from the fitted Johnson dis-tributions, see Section 2.1
- 3. Distribute frequency across plant specific welds as described in Section 2.2
- 4. Sample actual break size for each possible weld / break category com-bination as described in Section 2.3
- 5. Estimate performance measures, store them
- 6. If we ran S break sizes samples go to the next step, otherwise go to step 4
- 7. Compute the performance measures summary, store them
- 8. If we ran N LOCA frequencies samples go to the next step, otherwise, go to step 2
- 9. Make aggregated performance measures summary 3 Illustrative example We illustrate our approach described in the first four steps from Section 2.4 using the following example, see Figure 3. Assume we have a total of six welds and these are the only locations where a break can occur. Three of them (welds 1, 2 and 3) are small and have a size of 2.5 inches and hence can experience only small breaks (category1 and category2). Two of those six (welds 4 and 5) are of medium size and have a size of 4 inches and thus can have small and medium breaks (category1, category2 and category3 only, 11
they cant experience category 4 break). The last weld (weld 6) is large and has a size of 35 inches and can have all three types of breaks - small, medium and large (category1, category2, category3, category4, category5 and category6).
Figure 3: Illustrative Example
- 1. Assume S = 1, N = 1.
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- 2. The sampled LOCA frequencies (using the fitted Johnson distributions) are given in Table 4 Table 4: Sampled LOCA frequencies and corresponding probabilities Failure Type Category Break Size Frequency Probability small 1 0.5-1.625 3.9E-03 9.64E-01 small 2 1.625-3 1.4E-04 3.46E-02 medium 3 3-7 3.4E-06 8.41E-04 medium 4 7-14 3.1E-07 7.67E-05 large 5 14-31 1.2E-08 2.97E-06 large 6 31-41 1.2E-09 2.97E-07
- 3. We have J = {cat1 , cat2 , cat3 , cat4 , cat5 , cat6 },
Icat1 = {weld1 , weld2 , weld3 , weld4 , weld5 , weld6 },
Icat2 = {weld1 , weld2 , weld3 , weld4 , weld5 , weld6 },
Icat3 = {weld4 , weld5 , weld6 },
Icat4 = {weld6 }, Icat5 = {weld6 }, Icat6 = {weld6 }.
Mcat1 =6, Mcat2 =6, Mcat3 =3, Mcat4 =1, Mcat5 =1, Mcat6 =1.
BreakSizeweld cat1 U [0.5, 1.625]
1 BreakSizeweld cat1 U [0.5, 1.625]
2 BreakSizeweld cat1 U [0.5, 1.625]
3 BreakSizeweld cat1 U [0.5, 1.625]
4 13
BreakSizeweld cat1 U [0.5, 1.625]
5 BreakSizeweld cat1 U [0.5, 1.625]
6 BreakSizeweld cat2 U [1.625, 2.5]
1 BreakSizeweld cat2 U [1.625, 2.5]
2 BreakSizeweld cat2 U [1.625, 2.5]
3 BreakSizeweld cat2 U [1.625, 3]
4 BreakSizeweld cat2 U [1.625, 3]
5 BreakSizeweld cat2 U [1.625, 3]
6 BreakSizeweld cat3 U [3, 4]
4 BreakSizeweld cat3 U [3, 4]
5 BreakSizeweld cat3 U [3, 7]
6 BreakSizeweld cat4 U [7, 14]
6 BreakSizeweld cat5 U [14, 31]
6 BreakSizeweld cat6 U [31, 35]
6 Using the formula for P [catj at locationi ] we compute probabilities for each weld. The results are given in Table 5. We see that the sum of the distributed probabilities and the targeted probabilities are the same.
- 4. We simulate break sizes for each weld within each category using uni-14
form distribution with the specified above parameters. The sample is shown in Table 6.
We find it is worth mentioning that our assumptions lead to a piece-wise linear CDF function of the break size distribution for a given weld. For example, consider weld 6. The CDF of the Break size for that weld will have 6 break points with the slopes determined by the P [catj at locationweld6 ] values and break points at catmaxBreak j values (1.625, 3, 7, 14, 31), see Figure 4.
Table 5: Distributed LOCA probabilities among all welds Weld 1 2 3 4 5 6 Actual Target Cat1 1.61E-01 1.61E-01 1.61E-01 1.61E-01 1.61E-01 1.61E-01 9.64E-01 9.64E-01 Cat2 5.77E-03 5.77E-03 5.77E-03 5.77E-03 5.77E-03 5.77E-03 3.46E-02 3.46E-02 Cat3 2.80E-04 2.80E-04 2.80E-04 8.41E-04 8.41E-04 Cat4 7.67E-05 7.67E-05 7.67E-05 Cat5 2.97E-06 2.97E-06 2.97E-06 Cat6 2.97E-07 2.97E-07 2.97E-07 Table 6: Sampled break sizes for all welds within each break category Weld 1 2 3 4 5 6 Cat1 1.1 0.6 0.87 1.34 0.79 1.23 Cat2 2.4 1.9 2.1 2.9 1.75 2.36 Cat3 4.56 6.54 5.97 Cat4 9.67 Cat5 25.68 Cat6 32.67 15
Figure 4: CDF function of break size distribution for weld 6 Conclusion In this document we are presenting solutions to three problems:
- 1. How to distribute the NUREG 1829 LOCA frequencies to different locations (welds) in a nuclear plant. We make the simple assumption that small breaks are equally likely to occur in small, medium, or large welds; medium breaks are equally likely to occur in medium and large welds; large breaks can occur only in large welds. This allows us to preserve the NUREG 1829 LOCA frequencies.
- 2. The six break size categories (columns in NUREG-1829 Table 1) are ranges bounded by six discrete points. For a particular weld we need to be able to sample from the continuous range of break size values. We 16
propose to use the linear interpolation which is equivalent to assigning equally likely probabilities within each break size category.
- 3. How to model the distribution of the LOCA frequencies - we propose and fit the Johnson distributions.
We believe that the first problem is the most important that we need to agree on its solution. The other two can be modeled with different distribu-tions. We have already implemented the Gamma distributions fit from the NUREG/CR 6928 and working on fitting a set of Beta distributions. This will give a portfolio of options to apply.
In this document we do not discuss the different sampling techniques needed. Popova and Galenko (2011) describe all the sampling methodologies that we implement.
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References Crenshaw, J. W. (2012, January). South Texas Project Units 1 and 2 Docket Nos. STN 50-499, Summary of the South Texas Project Risk-Informed Approach to Resolve Generic Safety Issue (GSI-191). Letter from John W.
Crenshaw to USNRC.
EPRI (1999). Revised Risk-Informed In-Service Inspection Procedure. TR 112657 Revision B-A, Electric Power Research Institute, Palo Alto, CA.
Fleming, K. N., B. O. Lydell, and D. Chrun (2011, July). Development of LOCA Initiating Event Frequencies for South Texas Project GSI-191.
Technical report, KnF Consulting Services, LLC, Spokance, WA.
Johnson, N. (1949). Systems of frequency curves generated by methods of translations. Biometrika 36, 149-176.
Letellier, B. (2011). Risk-Informed Resolution of GSI-191 at South Texas Project. Technical Report Revision 0, South Texas Project, Wadsworth, TX.
Mosleh, A. (2011, October). Technical Review of STP LOCA Frequency Es-timation Methodology. Letter Report Revision 0, University of Maryland, College Park, MA.
Popova, E. and A. Galenko (2011, Deecember). Uncertainty Quantification (UQ) Methods, Strategies, and Illustrative Examples Used for Resolving the GSI-191 Problem at South Texas Project. Technical Report Revision 0, The University of Texas at Austin, Austin, TX.
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Tregoning, R., P. Scott, and A. Csontos (2008, April). Estimating Loss-of-Coolant Accident (LOCA) Frequencies Through the Elicitation Process:
Main Report (NUREG-1829). NUREG 1829, NRC, Washington, DC.
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