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Loca-Hybrid-Final
	ML12145A466
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Site:	South Texas
Issue date:	05/18/2012
From:	Morton D, Popova E; University of Texas at Austin
To:	Balwant Singal; Plant Licensing Branch IV
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Uncertainty modeling of LOCA frequencies and break size distributions for the STP GSI-191 resolution Elmira Popova, David Morton The University of Texas at Austin May 2012 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). The methodol-ogy of EPRI (1999) was used as the primary basis to develop the size and location-specific rupture frequencies 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 pa-per we propose a new approach to assign location-specific LOCA frequencies derived from the overall frequencies, as defined in Tregoning et al. (2008),

which we refer to as NUREG-1829.

The NUREG-1829 annual frequencies are neither plant specific nor 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 work with the six categories defined in Table 1, NUREG-1829 Volume 1, page xxi, as effective break size for the PWR 1

plants. Table 1 shows the mapping between the NUREG-1829 notation and ours. In addition, we use the term distribution to mean a distribution functioneither cumulative distribution function (CDF), probability density function (PDF), or probability mass function (PMF)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 South Texas Project PRA analysis uses only three LOCA categories, small, medium, and large. Our proposed methodol-ogy can be applied to any finite number of break-size categories.

In this document we will use the term location to represent a specific weld case, using the terminology of Fleming et al. (2011). Overall there are two distinct approaches to derive location- or weld-case-specific LOCA frequen-cies: bottom-up and top-down. The first approach requires location specific failure data to estimate the corresponding probability of a weld failure. Sup-pose a break occurs and assume there are Mj different welds in the plant where breaks of size catj can occur, weld1 , . . . , weldMj , then using the law of 2

total probability we can write:

Mj X

P [catj ] = P [catj lweldi ]P [weldi ], j = 1, 2, . . . , 6, i=1 where P [catj ] is the probability of a catj LOCA given that a break occurs, P [catj lweldi ] is the conditional probability of a catj LOCA given that the break occurs at weld i, and P [weldi ] is the probability that the break occurs at weld i, where again, weld i represents weld case i, using the terminology of Fleming et al. (2011).

In the bottom-up approach we first must determine P [catj lweldi ] (using estimation or expert elicitation). Then, if we assume that each location is equally likely to have the break, we can multiply by 1/Mj and sum the resulting probabilities to obtain the total probability the break is a catj LOCA. If the bottom-up approach is followed the resulting total catj LOCA probability will not equal the number provided in NUREG-1829 (or at least it is very unlikely to yield that number). This approach, taken by Fleming et al.

(2011), is an inherently bottom-up approach. In an attempt 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 using this as a stand alone methodology, which has led us to take a different path.

The approach that we propose to take is rooted in the combining the top-down and bottom-up approaches: We start with the NUREG-1829 fre-quencies and develop a way to distribute them across different locations pro-portionally to the frequencies estimated using the bottom-up approach. In this way, we maintain the NUREG-1829 frequencies overall but also allow for location-dependent differences. We should point out that we use the lo-3

cation specific tables as defined in Fleming et al. (2011). To our knowledge no other sources of location specific frequencies exist. If such information becomes available our proposed methodology can immediately incorporate that information.

For a top-down approach, we will use again the catj LOCA as an illus-trative example. The LOCA frequencies (NUREG-1829 Volume 1, page xxi, Table 1) are cumulative and so we compute the probability of a LOCA being in catj using the formula F requency[LOCA catj ] F requency[LOCA catj+1 ]

P [catj ] = ,

F requency[LOCA cat1 ]

for j = 1, . . . , 6 and where F requency[LOCA cat7 ] 0. Again we assume there are Mj different locations in the plant where breaks of size catj can occur, weld1 , . . . , weldMj . Assume, for the moment, given that we have a catj break, these Mj locations are equally likely to have the break, i.e.,

1 P [weldi lcatj ] = , i = 1, . . . , Mj .

Mj Then we have P [catj at weldi ] = P [catj ]P [weldi lcatj ] and so P [catj at weldi ] =

P [catj ]/Mj . Finally, applying the law of total probability, Mj X

P [catj ] = P [catj at weldi ],

i=1 we see that the resulting probability of a catj LOCA matches exactly the NUREG-1829 probability. The approach we propose in this document, fol-lows the steps we have just outlined, except we propose replacing the simple assumption of a catj break being equally likely to occur across all locations with an approach that uses location-specific conditional probabilities that we infer from Fleming et al. (2011).

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The above methodology distributes equally the LOCA frequencies as de-fined in NUREG-1829 Table 1 across all locations that can experience breaks from one or more of the six size categories. The six break size categories (from the NUREG-1829 Table 1) represent six bounded intervals. For a particular weld we need to be able to sample from the continuous interval 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 represent the distribution of the frequencies by reporting the mean, median, 5th and 95th percentiles. We will use this information to fit six continuous distributions for each break size category.

2 Proposed Methodology 2.1 Fitting distribution to the LOCA frequencies We first describe how we fit a distribution to the frequencies for each break size category. In theory, there are an infinite number of distributions that one can fit to the LOCA frequencies represented in NUREG-1829. For exam-ple, two split lognormal distributions are used in NUREG-1829 and gamma distributions are used in NUREG/CR 6928.

We choose to fit the bounded Johnson distribution, (Johnson, 1949) for the following reasons:

The Johnson has four parameters, which will allow us to match closely the distributional characteristics provided by NUREG-1829. In order to obtain the parameters of the Johnson distribution we solve an opti-mization problem with constraints defined by the distributional char-acteristics.

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The Johnson distribution allows for a variety of shapes. In particular, skewed, symmetric, bimodal, or unimodal shapes can be obtained.

The cumulative distribution function (CDF) of the bounded Johnson is:

F [x] = { + f [(x )/]} ,

where [x] is the 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)]. We restricted our attention to bell-shaped PDF curves only. This was achieved by imposing a constraint 1 in our fitting algorithm. 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. We note that the expert elicitation was for the 5%, 50% (median), and 95% quantiles, and did not involve eliciting the mean parameters. So we focus on matching the parameters elicited from the experts as indicated by the results in the final four columns of Table 3.

The six panels in Figures 1 and 2 show the fitted PDFs of the Johnson distribution for each category. Once the best fit is found, we sample the LOCA frequencies for each category to obtain F requency[LOCA catj ]a realization of the cumulative LOCA frequency to be in category j or larger.

2.2 Distribution of LOCA frequencies to different weld locations We first convert the sampled LOCA frequencies to probabilities using F requency[LOCA catj ] F requency[LOCA catj+1 ]

P [catj ] = , (1)

F requency[LOCA cat1 ]

where 6

Table 2: Fitted Johnson parameters Johnson Parameters Cat1 2.144623E+00 7.507774E-01 2.597254E-04 6.698783E-02 Cat2 1.365229E+00 4.195681E-01 4.822843E-06 3.646760E-03 Cat3 1.392766E+00 4.196874E-01 1.415507E-07 9.377713E-05 Cat4 1.701576E+00 4.554581E-01 5.609081E-09 1.302968E-05 Cat5 1.906196E+00 3.825140E-01 2.573251E-10 1.730668E-06 Cat6 2.525065E+00 3.816999E-01 1.853162E-11 8.868925E-07 Table 3: NUREG-1829 and fitted Johnson median, low and high quantiles values NUREG 1829 Values Fitted Johnson Relative Error 5th Median Mean 95th 5th Median Mean 95th 5th Median Mean 95th Cat1 6.90E-04 3.90E-03 7.30E-03 2.30E-02 6.89E-04 3.90E-03 6.68E-03 2.30E-02 0.20% 0.00% 8.56% 0.00%

Cat2 7.60E-06 1.40E-04 6.40E-04 2.40E-03 7.61E-06 1.39E-04 4.97E-04 2.41E-03 0.14% 0.38% 22.37% 0.62%

Cat3 2.10E-07 3.40E-06 1.60E-05 6.10E-05 2.09E-07 3.42E-06 1.24E-05 6.07E-05 0.52% 0.53% 22.69% 0.49%

Cat4 1.40E-08 3.10E-07 1.60E-06 6.10E-06 1.40E-08 3.09E-07 1.20E-06 6.12E-06 0.01% 0.28% 24.97% 0.25%

Cat5 4.10E-10 1.20E-08 2.00E-07 5.80E-07 4.18E-10 1.20E-08 9.91E-08 5.81E-07 1.99% 0.28% 50.47% 0.17%

Cat6 3.50E-11 1.20E-09 2.90E-08 8.10E-08 3.45E-11 1.21E-09 1.65E-08 8.04E-08 1.42% 0.43% 43.27% 0.75%

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200 4000 150000 150 3000 100000 PDF(x) PDF(x) PDF(x) 100 2000 50000 1000 50 0 0 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.000 0.001 0.002 0.003 0e+00 2e05 4e05 6e05 8e05 x x x (a) Cat1 (b) Cat2 (c) Cat3 1.5e+07 1200000 8e+06 1000000 6e+06 1.0e+07 800000 PDF(x) PDF(x) PDF(x) 600000 4e+06 5.0e+06 400000 2e+06 200000 0e+00 0.0e+00 0

0.0e+00 2.0e06 4.0e06 6.0e06 8.0e06 1.0e05 1.2e05 0.0e+00 5.0e07 1.0e06 1.5e06 0e+00 2e07 4e07 6e07 8e07 x x x (d) Cat4 (e) Cat5 (f) Cat6 Figure 1: Johnson PDF for each category 8

250 20000 8e+05 200 15000 6e+05 150 PDF(x) PDF(x) PDF(x) 10000 4e+05 100 50 5000 2e+05 0e+00 0

0 0.002 0.004 0.006 0.008 6.0e06 8.0e06 1.0e05 1.2e05 1.5e07 2.0e07 2.5e07 3.0e07 x x x (a) Cat1 (b) Cat2 (c) Cat3 4e+08 3.5e+09 6e+06 3.0e+09 5e+06 3e+08 2.5e+09 4e+06 2.0e+09 PDF(x) PDF(x) 2e+08 PDF(x) 3e+06 1.5e+09 2e+06 1.0e+09 1e+08 1e+06 5.0e+08 0e+00 0e+00 0.0e+00 5.0e09 1.0e08 1.5e08 2.0e08 2.5e08 3.0e08 1.0e10 1.5e10 2.0e10 2.5e10 3.0e10 3.5e10 4.0e10 0.0e+00 5.0e12 1.0e11 1.5e11 x x x (d) Cat4 (e) Cat5 (f) Cat6 Figure 2: Johnson PDF for each category, zoomed to a narrower range of frequencies near the modes of the distributions 9

J = {cat1 , cat2 , cat3 , ..., catB }: set of possible break types (categories),

P [catj ]: probability of observing a break that falls into category j given that a break was observed

F requency[LOCA catj ]: frequency of break of type j or larger, j J

F requency[LOCA catB+1 ] 0.

As we describe above, there are a total of B = 6 categories in NUREG-1829. Given P [catj ], the next step is to distribute that probability across all welds that can experience a break from that particular category. Not all types of welds can experience all types of breaks. We use Ij to denote the subset of locations that can have a break from category j.

We compute the probability that weld i will experience a break of type j using P [catj at weldi ] = wji P [catj ], where wji = P (weldi lcatj ) is the con-ditional probability of the break occurring at weld i given that we have a category j break. Restated, wji is the fraction that weld i contributes to cat-egory js total break frequency from the bottom-up approach for i Ij .

Computation of wji is straightforward. The bottom-up approach gener-ates the frequency of category j breaks at location i, which we denote F reqbu [LOCA catj at weldi ]. Given these frequencies, the wji values can be computed using:

F reqbu [LOCA catj at weldi ] F reqbu [LOCA catj+1 at weldi ]

wji = . (2)

F reqbu [LOCA cat1 at weldi ]

Given P [catj ] from equation (1) and wji from equation (2), we form P [catj at weldi ] = wji P [catj ]. (3)

Since the sum of all wji across i Ij is equal to one, with this approach we are guaranteed to match the NUREG-1829 specified values for P [catj ].

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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, we write breakSizeij U [minBreakji , maxBreakji ], j J, i Ij ,

where

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

weldsize i - actual weld size (it cannot experience break size larger than its diameter).

2.4 Methodology summary Our approach requires two sampling loops in our simulator CASA GRANDE, Letellier (2011). We need one sampling loop for the break size within each category and an outer loop that samples LOCA frequencies. Below is a step-by-step description of the procedure:

1. Input: N , the number of LOCA frequency samples, and S, the number of break size samples to generate.

2. Sample LOCA frequencies F requency[LOCA catj ], j = 1, 2, . . . , 6, from the fitted Johnson distributions for each break category; see Sec-tion 2.1.

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3. Distribute uncertainty across plant-specific welds as described in Sec-tion 2.2.

4. Sample break size for each possible weld / break-category combination as described in Section 2.3.

5. Estimate, and store, performance measures using CASA GRANDE.

6. Go to step 4 and repeat until we obtain S break-size samples.

7. Compute, and store, performance measures.

8. Go to step 2 and repeat until we have obtained N LOCA frequency samples.

9. Form the summary of aggregated performance measures.

3 Illustrative Example We illustrate the approach we describe in the first four steps from Section 2.4 using the following example. 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 sizes of 1.7, 2, and 2.5 inches and hence can experience only small breaks (category 1 and category 2). Two of those six (welds 4 and 5) are of medium size and have a diameter of 3.8 inches and thus can have small and medium breaks (category 1, category 2, and category 3 only; they cannot experience category 4, or larger, breaks). The last weld (weld

6) is large with a size of 35 inches and can have all types of breakssmall, medium, and large (category 1, . . . , category 6). A graphical representation of the system is shown in Figure 3.

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Figure 3: Example system depiction with six welds of various sizes that can each experience some subset of six types of breaks from category 1,. . . ,category 6.

Adapting the notation developed in Section 2 to this example 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 },

and 13