A Hybrid Approach to Modeling LOCA Frequencies and Break Sizes for the GSI-191 Resolution Effort at Vogtle

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1 A Hybrid Approach to Modeling LOCA Frequencies and Break Sizes for the GSI-191 Resolution Effort at Vogtle By David Morton 1, Ivilina Popova 1, and Jeremy Tejada 2 1 Proximira Inc. 2 SIMCON Solutions LLC October 4, 2013 Abstract This report answers three questions for the risk-informed approach to resolving GSI-191 at the Vogtle Nuclear Power Plant: 1. How should we model a continuous distribution of frequencies when NUREG-1829 (Tregoning et al., 2008) elicited three discrete percentiles? 2. How should we preserve the NUREG-1829 LOCA frequencies when distributing them across different welds in the plant? 3. How should we model, and sample from, a continuous distribution of break sizes when NUREG-1829 elicited percentiles at six discrete break sizes? Our answer to Question 1 is not plant-specific in that it relies only on the results in NUREG-1829 and the bounded Johnson distribution. We answer Question 2 by using the plant-specific frequencies of Fleming and Lydell (2013) and yet preserving the fleet-wide frequencies of NUREG We address Question 3 by using the conservative approach of linear interpolation, which amounts to assuming a uniform distribution between each of the NUREG-1829 break sizes. 1 Introduction As part of a risk-informed approach to resolving Generic Safety Issue 191 (GSI-191) at Vogtle Nuclear Power Plant, Fleming and Lydell (2013) built upon the established EPRI risk-informed in-service inspection program (EPRI, 1999). The methodology of EPRI (1999) was used as the primary basis to develop a joint probability distribution characterizing: (i) break-size frequencies and (ii) break locations at specific welds in the plant (Fleming and Lydell, 2013). As part of Vogtle s work on resolving GSI-191, we have implemented a modest modification of the approach of Fleming and Lydell (2013), which we term a hybrid approach, and we summarize that work in this report. Our hybrid approach ensures the joint distribution we form has marginal distributions for break-size frequencies that match the loss of coolant accident (LOCA) frequencies in Tregoning et al. (2008), which we refer to as NUREG The NUREG-1829 annual frequencies are from an expert elicitation based on knowledge of plant design, operation, and material performance (Tregoning et al., 2008) by members of 1

2 an expert panel. The elicited frequencies differ for PWR and BWR plant designs. That said, the frequencies are neither plant specific nor location specific within a plant. In carrying out GSI-191 analyses, arguments can be put forward that: (i) plant-specific information should be used for the overall LOCA initiating frequencies or (ii) it is desirable to preserve the LOCA frequencies elicited in NUREG In this document we provide a method for the latter approach. These LOCA frequencies, i.e., initiating event frequencies, are crucial input variables when using a risk-informed approach to resolving GSI-191. If these frequencies are not properly estimated, the effects will propagate through the remainder of the models and analysis. Important results such as estimates of: (i) the core damage frequency, (ii) the large early release frequency, and (iii) changes in these frequencies, relative to a base case model are sensitive to the LOCA frequencies. Thus it is essential to appropriately quantify the associated uncertainty and, given that our method preserves NUREG-1829 frequencies, it is important to distribute NUREG-1829 frequencies across specific welds in the plant, accounting for plant-specific design, pipe weld failure data, and degradation mechanisms. In this report, we use the six size categories defined in Table 7.19 (page 7-55) of Tregoning et al. (2008) as the effective break size for both the current-day estimate (per calendar year) and the end-of-plant-license estimate (per calendar year) for PWR plants. Table 1 shows the mapping between the NUREG-1829 notation and ours. In addition, we use the term distribution to mean a distribution function whether a cumulative distribution function (CDF), probability density function (PDF), or probability mass function (PMF) of a random variable used to model a specified uncertainty. 2

3 Table 1: LOCA categories notation map Effective break size (inch) Notation cat 1 cat 2 3 cat 3 7 cat 4 14 cat 5 31 cat 6 The Vogtle PRA analysis uses three LOCA categories: small, medium, and large, and these differ from the NUREG-1829 categories in Table 1. Nonetheless, our methodology is suitable because it can be applied to any finite number of break-size categories, and we discuss how we obtain distributions for the break sizes used in the Vogtle PRA. The bottom-up and top-down approaches are two distinct approaches to derive locationor weld-specific LOCA frequencies. The bottom-up approach defines weld categories, uses location-specific data on pipe weld failures, degradation mechanisms, and a Bayesian analysis of failure-rate uncertainty in order to estimate the corresponding frequencies of weld failures. This approach is described in Fleming and Lydell (2013) for Vogtle in detail. However, when estimating these frequencies for the welds in a specific plant, it is highly unlikely that when we aggregate the frequencies for a specific break size across all relevant welds in the plant that we will obtain the NUREG-1829 frequencies. Yet, as we indicate above, our goal is to provide a methodology that does preserve NUREG-1829 frequencies. In the top-down approach we begin with the NUREG-1829 break frequencies, and we are then faced with the question of how to distribute these frequencies across all relevant welds in the plant. In the absence of plant-specific information, one possibility is to assume that if a cat j break occurs (see Table 1) then it is equally likely to occur at each weld in the plant that can experience a break corresponding to the cat j size. This top-down approach preserves the NUREG-1829 frequencies, but it has the disadvantage of ignoring plant-specific information, ignoring service data on pipe failures at specific weld locations, and ignoring service data after the expert elicitation was performed in

4 The hybrid approach we describe in this report effectively combines the bottom-up and top-down approaches, both using the Vogtle plant-specific information from Fleming and Lydell (2013) and preserving the NUREG-1829 frequencies. Our approach starts with the NUREG-1829 frequencies, and we develop a way to distribute them across locations in proportion to the frequencies estimated using the bottom-up approach. This maintains the NUREG-1829 frequencies overall but allows for location-dependent differences. We use the location-specific tables given in Fleming and Lydell (2013) to implement our approach for Vogtle. We now begin to formalize these ideas using the following notation. Suppose a break occurs and assume that breaks of size cat j can occur in M j welds in the plant, weld 1,..., weld Mj. We use P [cat j ] to denote the probability of a cat j LOCA given that a break occurs, P [weld i ] to denote the probability that the break occurs at weld i, P [cat j weld i ] to denote the conditional probability of a cat j LOCA given that the break occurs at weld i, and P [weld i cat j ] to denote the conditional probability of the break occurring at weld i given that a cat j LOCA occurs. We will use the terms location and weld i to represent a specific weld case, using the terminology of Fleming and Lydell (2013). With this notation we can illustrate the top-down approach. The LOCA frequencies (Tregoning et al., 2008, Table 7.19, page 7-55) are cumulative, and so we compute the probability of a LOCA being in cat j using the formula P [cat j ] = F requency[loca cat j] F requency[loca cat j+1 ], F requency[loca cat 1 ] for j = 1,..., 6, where F requency[loca cat 7 ] 0. Again, we assume there are M j locations in the plant (weld 1,..., weld Mj ) where breaks of size cat j can occur. In the topdown approach we assume, given that we have a cat j break, that these M j locations are equally likely to have the break, i.e., P [weld i cat j ] = 1 M j, i = 1,..., M j. Then we have P [cat j at weld i ] = P [cat j ]P [weld i cat j ], and so P [cat j at weld i ] = P [cat j ]/M j. Finally, applying the law of total probability, M j P [cat j ] = P [cat j at weld i ] i=1 4

5 indicates that the resulting probability of a cat j LOCA precisely matches the NUREG-1829 probability. Our hybrid approach follows the steps just outlined, except that the simple assumption that a cat j break is equally likely to occur across all locations is replaced with the use of location-specific conditional probabilities that we infer from Fleming and Lydell (2013). The six break-size categories of Table 1 yield six bounded intervals for break sizes: [0.5, 1.625), [1.625, 3), [3, 7), [7, 14), [14, 31), and [31, DEGB max ), where all values are in inches and DEGB max (double-ended guillotine break) denotes the largest effective break size in the system under consideration. For a particular weld case, it is required that we be able to sample from the continuous interval of break-size values. In addition, we would like to be able to sample from a continuous distribution of the break frequencies. We accomplish the latter using the rows in Table 7.19 from NUREG-1829, which provides the median, 5th percentile, and 95th percentile frequencies, to fit a continuous distribution the bounded Johnson distribution for each break-size category for both the current-day and end-of-plant-license estimates. And we accomplish the former using a linear interpolation between the NUREG-1829 break sizes. The remainder of this report is organized as follows. Section 2 describes our methodology. Section 3 applies that methodology to Unit 1 at Vogtle and analyzes the results. Section 4 provides results for fitting the bounded Johnson distribution to the specific break-size categories used in the Vogtle PRA. Section 5 concludes, and the Appendix provides results for Unit 2 at Vogtle. 2 Methodology This section describes our methodology in three steps. First, we use the percentiles in the NUREG-1829 elicitation to fit a continuous distribution for the frequency for each break size. Second, we give the hybrid methodology sketched briefly above. Third, we describe a sampling-based implementation for continuous break sizes using linear interpolation. 5

6 2.1 Fitting the Johnson distribution to the LOCA frequencies We first describe how we fit a distribution to the frequencies for each break-size category. In theory, an infinite number of distributions can be fit to the LOCA frequencies represented in NUREG For example, a split lognormal distribution is used in NUREG-1829, and a gamma distribution is used in NUREG/CR We choose to fit the bounded Johnson distribution (Johnson, 1949), for the following reasons: The Johnson distribution has four parameters, which allows close matching of the distributional characteristics provided by NUREG The Johnson distribution allows for skewed, symmetric, bimodal, or unimodal shapes. Although a detailed analysis of various candidate distributions is beyond the scope of this report, unsuitable distributions include the gamma family of distributions (even though the gamma distribution is the conjugate prior to the Poisson distribution) and the so-called semibounded Johnson distribution (which is equivalent to a shifted log-normal distribution). The lack of suitability here arises because these families of distributions simply cannot reproduce, in a precise and reasonable manner, the quantiles specified in NUREG For example, even thought the semi-bounded Johnson distribution has three parameters, and three quantiles are specified, when we attempted a fit, the lower bound of the semi-bounded Johnson distribution corresponded to negative frequencies. The cumulative distribution function (CDF) of the bounded Johnson distribution is: F [x] = Φ {γ + δf[(x ξ)/λ]}, (1) where Φ[x] is the CDF of a standard normal random variable, γ and δ are shape parameters (with γ driving the distribution s skewness), ξ is a location parameter, λ is a scale parameter, and f(z) = log[z/(1 z)] for ξ x ξ + λ. We denote the bounded Johnson distribution with these four parameters as SB(γ, δ, ξ, λ). In general, δ > 0 and λ > 0. In our setting, frequencies are necessarily positive, and therefore ξ 0. We have proved the following fact for the bounded Johnson distribution, which is indispensable to the fitting procedure we describe below. 6

7 Fact 1. If X SB(γ, δ, ξ, λ) and a > 0, then ax SB(γ, δ, aξ, aλ). In order to obtain the parameters of the bounded Johnson distribution, we solve a nonlinear optimization problem, optimizing over the four parameters. For each break-size category, we minimize the sum of the squared deviations of the fitted values of the Johnson CDF at the NUREG-1829 LOCA frequencies from the NUREG-1829 percentiles (5th, 50th, and 95th). We enforce two constraints involving ξ and ξ + λ, which denote the lower and upper bounds for the bounded Johnson distribution, respectively. In addition to requiring that ξ and the width of the bounding interval (λ) be nonnegative, we require that ξ be smaller than the 5th percentile and that λ + ξ exceed the 95th percentile. Thus our optimization model is given by: min γ,δ,ξ,λ (F [x 0.05 ] 0.05) 2 + (F [x 0.5 ] 0.5) 2 + (F [x 0.95 ] 0.95) 2 (2a) s.t. ξ x 0.05 (2b) λ + ξ x 0.95 (2c) δ, ξ, λ 0. (2d) Of course, the CDF of the Johnson distribution, F [ ], appearing in the objective function (2a) depends on γ, δ, ξ, and λ as specified by equation (1). The values x 0.05, x 0.5, and x 0.95 are obtained from NUREG-1829 (Table 7.19) and are repeated in Tables 3 and 5 for current-day (25 years fleet average operation) and end-of-plant license (40 years fleet average operation). Fitting γ, δ, ξ, and λ to each of the six categories for both current-day and end-of-license values yields 12 instances of model (2). However, we should not attempt to solve model (2) directly, using the 5th, 50th, and 95th percentiles from Tables 3 and 5 because modern optimization software is ill-equipped to deal with numerical values smaller than (see Tables 3 and 5). Therefore, we use Fact 1 to rescale our optimization model. Specifically, F [x] = P(X x), where X SB(γ, δ, ξ, λ), and thus by Fact 1, F [ax] = P(aX ax), where ax SB(γ, δ, aξ, aλ) and where F [ ] is defined in (1). For each of the 12 instances 7

8 of the optimization model, we set a = 1/x 0.05 to obtain a well-scaled optimization model: min γ,δ,ξ,λ (F [ax 0.05 ] 0.05) 2 + (F [ax 0.5 ] 0.5) 2 + (F [ax 0.95 ] 0.95) 2 (3a) s.t. ξ ax 0.05 (3b) λ + ξ ax 0.95 (3c) δ, ξ, λ 0. (3d) Solving this model yields the correct values for γ and δ. To obtain the optimized values of ξ and λ for the original percentiles in Tables 3 and 5, we must divide by a. The fitted parameters of the Johnson distribution for each of the six categories for the current-day and end-of-plant-license estimates are given in Tables 2 and 4, respectively. The comparison between the NUREG-1829 distributional characteristics of the LOCA frequencies and the fitted ones for the current-day and end-of-plant-license estimates are presented in Tables 3 and 5, respectively. Because the NUREG-1829 expert elicitation was for the 5th, 50th (median), and 95th percentiles and did not involve eliciting the mean, we focus on matching the three distributional characteristics elicited from the experts as indicated by the results in the final set of columns of Tables 3 and 5. Figures 1-3 show the fitted PDFs of the Johnson distribution for both the current-day and end-of-plant-license estimates, denoted by Current and End for each category in the figures. Once the best fit is found, we can sample the LOCA frequencies for each category to obtain F requency[loca cat j ], corresponding to a realization of the cumulative LOCA frequency in category j or larger. Table 2: Fitted Johnson parameters for current day estimates Category Fitted Johnson Parameters γ δ ξ λ Cat E E E E-02 Cat E E E E-03 Cat E E E E-04 Cat E E E E-05 Cat E E E E-06 Cat E E E E-07 8

9 Table 3: NUREG-1829 (Table 7.19) and fitted Johnson median, 5th percentile, and 95th percentile values for current-day estimates of frequencies (events per reactor-calendar year) Category NUREG-1829 Values Fitted Johnson Distribution Relative Error 5th Median 95th 5th Median 95th 5th Median 95th Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Table 4: Fitted Johnson parameters for end-of-plant license estimates Category Fitted Johnson Parameters γ δ ξ λ Cat E E E E-02 Cat E E E E-03 Cat E E E E-04 Cat E E E E-05 Cat E E E E-06 Cat E E E E-07 Table 5: NUREG-1829 (Table 7.19) and fitted Johnson median, 5th percentile, and 95th percentile values for end-of-plant license estimates of frequencies (events per reactor-calendar year) Category NUREG-1829 Values Fitted Johnson Distribution Relative Error 5th Median 95th 5th Median 95th 5th Median 95th Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% Cat E E E E E E % 0.00% 0.00% 9

10 (a) Cat1 (b) Cat1 (zoomed) (c) Cat2 (d) Cat2 (zoomed) Figure 1: Johnson PDFs for category 1 and category 2 breaks, including detailed views of the frequencies near the mode of the distribution (x-axis has units of events per reactor-calendar year). Current denotes current-day (25 years fleet average operation) and End denotes end-of-plant license (40 years fleet average operation). 10

11 (a) Cat3 (b) Cat3 (zoomed) (c) Cat4 (d) Cat4 (zoomed) Figure 2: Johnson PDFs for category 3 and category 4 breaks, including detailed views of the frequencies near the mode of the distribution (x-axis has units of events per reactor-calendar year) 11

12 (a) Cat5 (b) Cat5 (zoomed) (c) Cat6 (d) Cat6 (zoomed) Figure 3: Johnson PDF for category 5 and category 6 breaks, including detailed views of the frequencies near the mode of the distribution (x-axis has units of events per reactor-calendar year) 12

13 2.2 Distribution of LOCA frequencies to weld locations After sampling a LOCA frequency from the Johnson distribution, we normalize to convert these to probabilities within each category using where P [cat j ] = F requency[loca cat j] F requency[loca cat j+1 ], (4) F requency[loca cat 1 ] J = {cat 1, cat 2, cat 3,..., cat B }: set of possible break types (categories) P [cat j ]: probability of observing a break that falls into category j given that a break was observed F requency[loca cat j ]: frequency of break of type j or larger, j J F requency[loca cat B+1 ] 0 (boundary condition). Example sampled LOCA frequencies which have been normalized and converted to probabilities for the current-day and end-of-plant-license estimates are given in Tables 6 and 7, respectively. This example uses the median values from the Johnson distributions. In actual implementation we use the sampled LOCA frequencies from the fitted bounded Johnson distributions. The right-most column of Tables 6 and 7 computes the probability mass for each category according to equation (4). Table 6: Example sampled LOCA frequencies and corresponding probabilities for currentday estimates. Frequency has units of events per reactor-calendar year. Category Break-Size Bins (in.) Frequency Probability 1 [0.5,1.625) 6.30E E-01 2 [1.625,3) 8.90E E-01 3 [3,7) 3.40E E-03 4 [7,14) 3.10E E-04 5 [14,31) 1.20E E-05 6 [31,41) 1.20E E-06 13

14 Table 7: Example sampled LOCA frequencies and corresponding probabilities for end-ofplant license estimates. Frequency has units of events per reactor-calendar year. Category Break-Size Bins (in.) Frequency Probability 1 [0.5,1.625) 7.20E E-01 2 [1.625,3) 1.20E E-01 3 [3,7) 7.60E E-03 4 [7,14) 6.60E E-04 5 [14,31) 2.80E E-05 6 [31,41) 2.90E E-06 As we describe in Section 1, there are a total of B = 6 categories in NUREG Given P [cat j ], the next step is to distribute that probability across all weld cases that can experience a break from that particular category. Not all types of welds can experience all types of breaks. We use I j 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 where w i j P [cat j at weld i ] = w i jp [cat j ], = P (weld i cat j ) is the conditional probability of the break occurring at weld i given a category j break. Restated, w i j is the fraction that weld i contributes to category j s total break frequency from the bottom-up approach for i I j. Computation of w i j is straightforward. The bottom-up approach generates the frequency of category j breaks at location i, which we denote F req bu [LOCA cat j at weld i ]. Also, there are different numbers of welds for each location i. We denote the number of welds for location i by n i. Given these frequencies and the number of welds for each location, the w i j values can be computed as w i j = (F req bu [LOCA cat j at weld i ] F req bu [LOCA cat j+1 at weld i ]) n i i I j (F req bu [LOCA cat j at weld i ] F req bu [LOCA cat j+1 at weld i ]) n i. (5) Given P [cat j ] from equation (4) and wj i from equation (5), we form P [cat j at weld i ] = wjp i [cat j ]. (6) Because the sum of all wj i across i I j is equal to one, this approach is guaranteed to match the NUREG-1829 specified values for P [cat j ]. 14

15 2.3 Sampling of the break size The final step is to sample the actual break size conditioned on the break category. The break size is assumed to have a uniform distribution within a given category, which is equivalent to performing a linear interpolation between the NUREG-1829 break frequencies. Formally, we write where minbreak i j = cat minbreak j breaksize i j U[minBreak i j, maxbreak i j], j J, i I j, maxbreakj i = min{cat maxbreak j, weld size i } cat minbreak j cat maxbreak j minimum break size that would put it into category j maximum break size that would put it into category j weld size i actual weld size (it cannot experience break size larger than its DEGB). 2.4 Methodology summary Our approach requires two sampling loops to simulate break sizes and frequencies. One sampling loop for the break size within each category, and an outer loop that samples LOCA frequencies. Below is the detailed 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 cat j ], j = 1, 2,..., 6, from the fitted Johnson distributions for each break category; see Section Distribute uncertainty across plant-specific welds as described in Section Sample break size for each possible weld and break category, as described in Section Estimate, and store, performance measures using a simulation model, such as the core damage frequency, large early release frequency, and changes in these frequencies. 15

16 6. Go to step 4 and repeat until S break-size samples are obtained. 7. Compute, and store, performance measures. 8. Go to step 2 and repeat until N LOCA frequency samples are obtained. 9. Form the summary of aggregated performance measures. 3 Results of Implementing the Hybrid Approach for Vogtle Unit 1 In this section, we present the results of implementing the hybrid approach for Vogtle for all 51 weld cases using the location-specific frequency tables from Fleming and Lydell (2013). (We note that the 51 weld cases do not include the small bore case, and eight of the 51 weld cases have zero counts. We follow Fleming and Lydell (2013) in including those eights cases in what we report below.) We assume that these weld cases cover all locations of interest where a break can occur. We label the weld cases using the same notation as Fleming and Lydell (2013). Vogtle has two units. These units differ only in the number of welds which fall into specific weld case categories, while the weld categories are identical for both units. We present the results for Unit 1 in this section, and analogous results for Unit 2 are presented in the Appendix. Table 8 summarizes some of the weld case characteristics defined in the tables from Fleming and Lydell (2013). In the table, DEGB is a double-ended guillotine break size (in inches), which indicates the largest break size a weld can have. Using the frequency tables for all 51 weld cases from Fleming and Lydell (2013), the frequencies and weights for the 51 welds are reported in Tables 9 and 10. The Frequency columns in the table correspond to F req bu [LOCA cat j at weld i ] and the Weight columns correspond to wj, i using our notation from Section 2.2. In these tables, an X indicates that the cell is inapplicable because the DEGB of the weld case is smaller than the category s size. In contrast a 0.00% indicates that the weld category could contribute but did not according to Fleming and Lydell (2013). (Weld cases with zero counts are reported in the latter manner, if the category applies based on the weld s size.) 16

17 Table 8: Characteristics of the 51 weld cases for both Units 1 and 2 RC Hot Leg RC Cold Leg Weld Damage Unit 1 Unit 2 Pipe DEGB Case Mechanisms # of Welds # of Welds Size (in.) Size (in.) 1A1 PWSCC, D&C A2 D&C B D&C C D&C A1 PWSCC, D&C A2 D&C B1 PWSCC, D&C B2 D&C C D&C D D&C E D&C F D&C G TASCS, D&C H D&C RC Hot Leg 3A1 PWSCC, D&C at S/G Inlet 3A2 D&C A1 PWSCC, TF, D&C PZR Surge 4A2 TF,D&C Line 4B TF, D&C C TF, D&C A1 PWSCC, D&C PZR PORV 5A2 D&C & SRV Lines 5C D&C G D&C PZR Spray 5B TT, D&C D1 PWSCC, D&C D2 D&C E TASCS, D&C F D&C I D&C SIR-RHR Lines 7A D&C B IGSCC, D&C C D&C D D&C E IGSCC, D&C SIR Safety 7F D&C Injection 7G TT, D&C Lines 7H D&C I D&C J TT, D&C K TASCS, D&C L D&C A VF,TT, D&C B VF,D&C C VF,TT, D&C Chemical & 8D VF, D&C Volume 8E VF, D&C Control 8F VF,TT, D&C G VF,TT, D&C H VF, D&C I VF, D&C Totals:

18 Table 9: Weld failure frequencies and weights for categories 1, 2, and 3 using the bottom-up approach. Frequency has units of events per weld-calendar year. Weld Case Category 1 Category 2 Category 3 Frequency Weight Frequency Weight Frequency Weight 1A1 1.78E % 2.22E % 1.75E % 1A2 0.00E % 0.00E % 0.00E % 1B 6.23E % 7.80E % 6.15E % 1C 3.11E % 3.90E % 4.64E % 2A1 5.01E % 6.99E % 4.10E % 2A2 0.00E % 0.00E % 0.00E % 2B1 5.00E % 6.97E % 4.09E % 2B2 0.00E % 0.00E % 0.00E % 2C 7.09E % 9.89E % 5.80E % 2D 4.25E % 5.93E % 3.48E % 2E 2.84E % 3.96E % 3.35E % 2F 1.42E % 1.98E % 1.67E % 2G 6.90E % 1.78E % X X 2H 3.36E % 8.66E % X X 3A1 6.21E % 7.78E % 6.13E % 3A2 0.00E % 0.00E % 0.00E % 4A1 0.00E % 0.00E % 0.00E % 4A2 4.96E % 1.10E % 8.56E % 4B 1.21E % 2.69E % 2.09E % 4C 3.90E % 8.66E % 6.75E % 5A1 0.00E % 0.00E % 0.00E % 5A2 2.43E % 4.33E % 1.72E % 5C 5.89E % 1.05E % 4.17E % 5G 2.77E % 4.95E % 2.31E % 5B 3.57E % 6.38E % 2.53E % 5D1 0.00E % 0.00E % 0.00E % 5D2 9.99E % 1.78E % 8.32E % 5E 2.38E % 4.26E % 1.99E % 5F 1.18E % 2.19E % 9.70E % 5I 2.54E % 6.66E % X X 7A 1.06E % 1.89E % 7.52E % 7B 1.31E % 2.34E % 9.30E % 7C 6.99E % 1.25E % 4.97E % 7D 1.84E % 3.29E % 1.31E % 7E 3.27E % 5.84E % 2.32E % 7F 1.12E % 1.99E % 7.93E % 7G 2.20E % 3.93E % 1.84E % 7H 3.69E % 6.58E % 3.07E % 7I 4.20E % 1.10E % X X 7J 2.54E % 6.66E % X X 7K 1.02E % 2.66E % X X 7L 1.18E % 3.10E % X X 8A 3.50E % 6.26E % 2.92E % 8B 1.32E % 2.36E % 1.10E % 8C 0.00E % 0.00E % X X 8D 4.45E % 1.17E % X X 8E 4.13E % 1.08E % X X 8F 1.54E % 2.75E % 1.28E % 8G 6.36E % 1.66E % X X 8H 7.10E % 1.27E % 5.92E % 8I 6.36E % 1.66E % X X Totals: 2.69E % E % 3.65E % 18

19 Table 10: Weld failure frequencies and weights for categories 4, 5, and 6 using the bottom-up approach. Frequency has units of events per weld-calendar year. Weld Case Category 4 Category 5 Category 6 Frequency Weight Frequency Weight Frequency Weight 1A1 4.88E % 3.08E % 9.80E % 1A2 0.00E % 0.00E % 0.00E % 1B 1.71E % 1.08E % 3.44E % 1C X X X X X X 2A1 9.77E % 5.95E % 2.40E % 2A2 0.00E % 0.00E % 0.00E % 2B1 9.76E % 5.95E % 2.40E % 2B2 0.00E % 0.00E % 0.00E % 2C 1.38E % 8.44E % 3.41E % 2D 8.30E % 5.06E % 2.05E % 2E X X X X X X 2F X X X X X X 2G X X X X X X 2H X X X X X X 3A1 1.71E % 1.08E % 3.43E % 3A2 0.00E % 0.00E % 0.00E % 4A1 0.00E % 0.00E % X X 4A2 2.33E % 8.60E % X X 4B 5.69E % 2.10E % X X 4C 1.83E % 6.77E % X X 5A1 0.00E % X X X X 5A2 3.03E % X X X X 5C 7.35E % X X X X 5G X X X X X X 5B 4.46E % X X X X 5D1 X X X X X X 5D2 X X X X X X 5E X X X X X X 5F X X X X X X 5I X X X X X X 7A 9.55E % 3.40E % X X 7B 1.18E % 4.20E % X X 7C 6.31E % 2.25E % X X 7D 2.26E % X X X X 7E 4.00E % X X X X 7F 1.37E % X X X X 7G X X X X X X 7H X X X X X X 7I X X X X X X 7J X X X X X X 7K X X X X X X 7L X X X X X X 8A X X X X X X 8B X X X X X X 8C X X X X X X 8D X X X X X X 8E X X X X X X 8F X X X X X X 8G X X X X X X 8H X X X X X X 8I X X X X X X Totals: 2.89E % 1.62E % 4.90E % 19

20 Before presenting the results of applying the method we describe in Section 2, we examine the relative contributions to the weights (wj i values) of specific welds overall, of specific welds within each break-size category, and of specific system-degb combinations. We do this to give insight into the weld cases most likely to experience a LOCA from each category. Tables summarize the top-five contributors to total failure frequency for each category and calculate the associated sum of the weights. Each category s top-five cases contribute more than 83% of its total failure frequency. These contributors consist of the following 13 weld cases: 1A1, 1B, 2A1, 2B1, 3A1, 7B, 7G, 7I, 7K, 7L, 8A, 8E, and 8F. If we further restrict attention to those top five weld cases with weight contributions exceeding 5% in their category, we end up with a total of 7 weld cases (1A1, 3A1, 7G, 7I, 7L, 8A, and 8E) that account for more than 76% of the weight in each of the six categories. Table 17 shows the 15 most important weld cases, taking into account all six categories simultaneously. With these 15 welds, we achieve a total weight of 90% in each column. Weights are again expressed as percentages, but we round to one decimal point. Here, Weight again represents the conditional probabilities, wj. i Perhaps the foremost observation from Table 8 and Tables concerns weld cases 1A1 and 3A1 from the Hot Leg and the Hot Leg at S/G Inlet, respectively. (In what follows we abbreviate Hot Leg at S/G Inlet by Hot Leg SG. ) There are 4 instances of each of these pipes in Unit 1 (and in Unit 2). These eight 29-inch pipes account for roughly 22%, 66%, 85%, and 90% of the weights in the respective categories 3-6; e.g., given that a category 5 break occurs, there is an 85% chance it occurred in either a 1A1 or 3A1 weld. The systems accounting for the majority of the weight (greater than 5%) for the weld cases in a given category in Tables are SIR Safety and CVCS welds for category 1 and category 2; Hot Leg SG and SIR Safety for category 3; and Hot Leg and Hot Leg SG welds for categories 4-6. The Hot Leg and Hot Leg SG contributions are primarily from weld cases 1A1 and 3A1. Fleming and Lydell (2013) discuss a number of degradation mechanisms (DMs), and for the 51 weld cases of Table 8 seven DMs apply: design and construction defects (D&C), inter-granular stress corrosion cracking (IGSCC), primary water stress corrosion cracking (PWSCC), thermal fatigue (TF), thermal stratification, cycling, and stripping (TASCS), 20

21 thermal transients (TT), and vibration fatigue (VF). Of these seven DMs, IGSCC and TASCS contribute less than 5% for all break categories. Fleming and Lydell (2013) indicate that for fatigue mechanisms or stress corrosion cracking... to develop into structural degradation or failure a crack initiation must occur and then grow into a surface connected crack and beyond. Design and construction (D&C) defects oftentimes provide for crack/flaw initiation, but not for crack/flaw growth. In addition, the D&C mechanism applies to all of the welds in Table 8. Hence, for the moment, we restrict attention to the remaining four mechanisms that appear in Table 17: PWSCC, TT, and VF. PWSCC, TT, and VF constitute the bulk of the weight for categories 1-3, and PWSCC dominates the contribution to categories 4-6. Table 11: Top five welds in failure-frequency weight for category 1 Weld Case DEGB (in.) Damage Mechanisms # of Welds Weight 7L SIR Safety 2.1 D&C % 7I SIR Safety 2.8 D&C % 8E CVCS 2.1 VF, D&C % 7G SIR Safety 4.2 TT, D&C % 7K SIR Safety 2.1 TASCS, D&C % Total % Table 12: Top five welds in failure-frequency weight for category 2 Weld Case DEGB (in.) Damage Mechanisms # of Welds Weight 7L SIR Safety 2.1 D&C % 7I SIR Safety 2.8 D&C % 8E CVCS 2.1 VF, D&C % 7G SIR Safety 4.2 TT, D&C % 7K SIR Safety 2.1 TASCS, D&C % Total % 21

22 Table 13: Top five welds in failure-frequency weight for category 3 Weld Case DEGB (in.) Damage Mechanisms # of Welds Weight 7G SIR Safety 4.2 TT, D&C % 3A1 Hot Leg SG 41.0 PWSCC, D&C % 8A SIR Safety 4.2 VF,TT, D&C % 1A1 Hot Leg 41.0 PWSCC, D&C % 8F CVCS 4.2 VF,TT, D&C % Total % Table 14: Top five welds in failure-frequency weight for category 4 Weld Case DEGB (in.) Damage Mechanisms # of Welds Weight 3A1 Hot Leg SG 41.0 PWSCC, D&C % 1A1 Hot Leg 41.0 PWSCC, D&C % 7B SIR Safety 14.1 IGSCC, D&C % 2A1 Cold Leg 43.8 PWSCC, D&C % 2B1 Cold Leg 38.9 PWSCC, D&C % Total % Table 15: Top five welds in failure-frequency weight for category 5 Weld Case DEGB (in.) Damage Mechanisms # of Welds Weight 3A1 Hot Leg SG 41.0 PWSCC, D&C % 1A1 Hot Leg 41.0 PWSCC, D&C % 2A1 Cold Leg 43.8 PWSCC, D&C % 2B1 Cold Leg 38.9 PWSCC, D&C % 7B SIR Safety 14.1 IGSCC, D&C % Total % Table 16: Top five welds in failure-frequency weight for category 6 Weld Case DEGB (in.) Damage Mechanisms # of Welds Weight 3A1 Hot Leg SG 41.0 PWSCC, D&C % 1A1 Hot Leg 41.0 PWSCC, D&C % 2A1 Cold Leg 43.8 PWSCC, D&C % 2B1 Cold Leg 38.9 PWSCC, D&C % 1B Hot Leg 41.0 D&C % Total % 22

23 Table 17: Failure frequency weights for the most important weld cases simultaneously accounting for all six categories Case DEGB (in.) Damage Mech. # of Welds Weight Cat 1 Cat 2 Cat 3 Cat 4 Cat 5 Cat 6 7L SIR Safety 2.1 D&C % 46.6% X X X X 7I SIR Safety 2.8 D&C % 16.5% X X X X 8E CVCS 2.1 VF, D&C % 16.3% X X X X 7K SIR Safety 2.1 TASCS, D&C % 4.0% X X X X 7G SIR Safety 4.2 TT, D&C 8 8.2% 5.9% 50.3% X X X 8A SIR Safety 4.2 VF,TT, D&C % 0.9% 8.0% X X X 8F CVCS 4.2 VF,TT, D&C 4 0.6% 0.4% 3.5% X X X 5F PZR Spray 5.7 D&C % 0.3% 2.7% X X X 5B PZR Spray 8.5 TT, D&C 9 0.1% 0.1% 0.7% 1.5% X X 7B SIR Safety 14.1 IGSCC, D&C 8 0.5% 0.4% 2.5% 4.1% 2.6% X 7A SIR-RHR 17.0 D&C % 0.3% 2.1% 3.3% 2.1% X 3A1 Hot Leg SG 41.0 PWSCC, D&C 4 2.3% 1.2% 16.8% 59.1% 66.4% 70.0% 1A1 Hot Leg 41.0 PWSCC, D&C 4 0.7% 0.3% 4.8% 16.9% 19.0% 20.0% 2A1 Cold Leg 43.8 PWSCC, D&C 4 0.2% 0.1% 1.1% 3.4% 3.7% 4.9% 2B1 Cold Leg 38.9 PWSCC, D&C 4 0.2% 0.1% 1.1% 3.4% 3.7% 4.9% Totals 93.5% 93.4% 93.5% 91.7% 97.5% 99.8% The 51 weld cases from Table 8 constitute a total of 25 distinct combinations of system type and DEGB size (because the table contains repeat DEGB sizes in the right-most column for a total of 14 distinct DEGB sizes that expand to 25 distinct combinations when coupled with system type). For example, in Table 8, weld cases 1A1 and 1B are of the same combination, as are weld cases 2B1 and 2D. All 25 combinations are shown in Table 18 and in Figures 4-6. The dominant role of welds 3A1 and 1A1 for categories 4-6 appears in the Hot Leg SG and Hot Leg (41-inch) rows of Table 18. The aggregated results displayed in the figures are consistent with the results for the 51 welds. By tracing changes as we scan Figures 4-6 we understand changes in the contribution to total weight for particular DEGB-system pairs. 23

24 Table 18: Aggregated weights by a combination of system and DEGB size DEGB (in.) # of Welds Weights Cat 1 Cat 2 Cat 3 Cat 4 Cat 5 Cat 6 SIR-RHR % 0.28% 2.06% 3.30% 2.10% X SIR Safety % 0.37% 2.68% 4.30% 2.73% X SIR Safety % 0.00% 0.04% 0.08% X X SIR Safety % 0.12% 0.85% 1.86% X X SIR Safety % 5.93% 50.34% X X X SIR Safety % 16.52% X X X X SIR Safety % 51.58% X X X X PZR Surge % 0.04% 0.57% 1.97% 1.30% X PZR Surge % 0.03% 0.42% 1.44% 0.95% X PZR Spray % 0.10% 0.69% 1.54% X X PZR Spray % 0.40% 3.22% X X X PZR Spray % 1.00% X X X X PZR PORV % 0.16% 1.19% 2.65% X X PZR PORV % 0.07% 0.63% X X X Hot Leg SG % 1.17% 16.78% 59.07% 66.43% 69.99% Hot Leg % 0.34% 4.81% 16.95% 19.06% 20.08% Hot Leg % 0.00% 0.01% X X X CVCS % 1.58% 13.42% X X X CVCS % 2.25% X X X X CVCS % 16.27% X X X X Cold Leg % 0.11% 1.14% 3.43% 3.73% 4.98% Cold Leg % 0.11% 1.13% 3.41% 3.70% 4.95% Cold Leg % 0.00% 0.01% X X X Cold Leg % 0.00% 0.00% X X X Cold Leg % 1.57% X X X X Totals: % 100.0% 100.0% 100.0% 100.0% 100.0% 24

25 Weight Weight 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% SIR-RHR SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety PZR Surge Category 1 Breaks PZR Surge PZR Spray PZR Spray PZR Spray PZR PORV PZR PORV Hot Leg SG Hot Leg and DEGB Break Size (in.) (a) Category 1 Hot Leg CVCS CVCS CVCS Cold Leg Cold Leg Cold Leg Cold Leg Cold Leg % 50% 40% 30% 20% 10% 0% SIR-RHR SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety PZR Surge Category 2 Breaks PZR Surge PZR Spray PZR Spray PZR Spray PZR PORV PZR PORV Hot Leg SG Hot Leg and DEGB Break Size (in.) (b) Category 2 Hot Leg CVCS CVCS CVCS Cold Leg Cold Leg Cold Leg Cold Leg Cold Leg Figure 4: Aggregated weight by system and DEGB size for categories 1 and 2 25

26 Weight Weight 60% Category 3 Breaks 50% 40% 30% 20% 10% 0% SIR-RHR SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety PZR Surge PZR Surge PZR Spray PZR Spray PZR Spray PZR PORV PZR PORV Hot Leg SG Hot Leg and DEGB Break Size (in.) Hot Leg CVCS CVCS CVCS Cold Leg Cold Leg Cold Leg Cold Leg Cold Leg (a) Category 3 60% Category 4 Breaks 50% 40% 30% 20% 10% 0% SIR-RHR SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety PZR Surge PZR Surge PZR Spray PZR Spray PZR Spray PZR PORV PZR PORV Hot Leg SG Hot Leg Hot Leg CVCS CVCS CVCS Cold Leg Cold Leg Cold Leg Cold Leg Cold Leg and DEGB Break Size (in.) (b) Category 4 Figure 5: Aggregated weight by system and DEGB size for categories 3 and 4 26

27 Weight Weight 70% 60% 50% 40% 30% 20% 10% 0% SIR-RHR SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety PZR Surge Category 5 Breaks PZR Surge PZR Spray PZR Spray PZR Spray PZR PORV PZR PORV Hot Leg SG Hot Leg and DEGB Break Size (in.) (a) Category 5 Hot Leg CVCS CVCS CVCS Cold Leg Cold Leg Cold Leg Cold Leg Cold Leg % 60% 50% 40% 30% 20% 10% 0% SIR-RHR SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety SIR Safety PZR Surge Category 6 Breaks PZR Surge PZR Spray PZR Spray PZR Spray PZR PORV PZR PORV Hot Leg SG Hot Leg and DEGB Break Size (in.) (b) Category 6 Hot Leg CVCS CVCS CVCS Cold Leg Cold Leg Cold Leg Cold Leg Cold Leg Figure 6: Aggregated weight by system and DEGB size for categories 5 and 6 27

28 Returning to the procedure of Section 2, we now distribute uncertainty across the 51 weld cases using the weights from Tables 9 and 10. With the sampled LOCA frequencies (for illustration, we take these as the median values from NUREG-1829), we compute P [cat j at weld i ] for each category-weld combination using equation (6). The results for these joint probability distributions are given in Tables 19 and 20. It is clear that the estimated P [cat j ] values in Tables 19 and 20 are the same as the NUREG-1829 P [cat j ] values in Tables 6 and 7, respectively, as shown in the last rows of Tables 19 and

29 Table 19: Joint weld case-category LOCA probabilities for the 51 weld cases for current-day estimates Weld Case Cat 1 Cat 2 Cat 3 Cat 4 Cat 5 Cat 6 1A1 5.66E E E E E E-07 1A2 0.00E E E E E E+00 1B 1.99E E E E E E-09 1C 9.93E E E-07 X X X 2A1 1.60E E E E E E-08 2A2 0.00E E E E E E+00 2B1 1.59E E E E E E-08 2B2 0.00E E E E E E+00 2C 2.26E E E E E E-09 2D 1.36E E E E E E-10 2E 9.04E E E-07 X X X 2F 4.52E E E-07 X X X 2G 2.20E E-04 X X X X 2H 1.07E E-03 X X X X 3A1 1.98E E E E E E-06 3A2 0.00E E E E E E+00 4A1 0.00E E E E E+00 X 4A2 1.58E E E E E-08 X 4B 3.86E E E E E-07 X 4C 1.24E E E E E-08 X 5A1 0.00E E E E+00 X X 5A2 7.74E E E E-07 X X 5C 1.88E E E E-05 X X 5G 8.84E E E-05 X X X 5B 1.14E E E E-06 X X 5D1 0.00E E E+00 X X X 5D2 3.18E E E-06 X X X 5E 7.60E E E-05 X X X 5F 3.77E E E-04 X X X 5I 8.11E E-03 X X X X 7A 3.37E E E E E-07 X 7B 4.17E E E E E-07 X 7C 2.23E E E E E-08 X 7D 5.88E E E E-07 X X 7E 1.04E E E E-06 X X 7F 3.56E E E E-06 X X 7G 7.03E E E-03 X X X 7H 1.18E E E-06 X X X 7I 1.34E E-02 X X X X 7J 8.11E E-03 X X X X 7K 3.24E E-03 X X X X 7L 3.77E E-02 X X X X 8A 1.12E E E-04 X X X 8B 4.21E E E-05 X X X 8C 0.00E E+00 X X X X 8D 1.42E E-03 X X X X 8E 1.32E E-02 X X X X 8F 4.90E E E-04 X X X 8G 2.03E E-04 X X X X 8H 2.26E E E-05 X X X 8I 2.03E E-04 X X X X Totals: 8.59E E E E E E-06 29

30 Table 20: Joint weld case-category LOCA probabilities for the 51 weld cases for end-of-plant license estimates Weld Case Cat 1 Cat 2 Cat 3 Cat 4 Cat 5 Cat 6 1A1 5.49E E E E E E-07 1A2 0.00E E E E E E+00 1B 1.93E E E E E E-09 1C 9.64E E E-06 X X X 2A1 1.55E E E E E E-07 2A2 0.00E E E E E E+00 2B1 1.55E E E E E E-07 2B2 0.00E E E E E E+00 2C 2.19E E E E E E-09 2D 1.32E E E E E E-09 2E 8.77E E E-07 X X X 2F 4.39E E E-07 X X X 2G 2.13E E-04 X X X X 2H 1.04E E-03 X X X X 3A1 1.92E E E E E E-06 3A2 0.00E E E E E E+00 4A1 0.00E E E E E+00 X 4A2 1.53E E E E E-07 X 4B 3.75E E E E E-07 X 4C 1.21E E E E E-07 X 5A1 0.00E E E E+00 X X 5A2 7.51E E E E-07 X X 5C 1.82E E E E-05 X X 5G 8.58E E E-05 X X X 5B 1.11E E E E-05 X X 5D1 0.00E E E+00 X X X 5D2 3.09E E E-06 X X X 5E 7.37E E E-05 X X X 5F 3.66E E E-04 X X X 5I 7.87E E-03 X X X X 7A 3.27E E E E E-07 X 7B 4.05E E E E E-07 X 7C 2.16E E E E E-08 X 7D 5.70E E E E-07 X X 7E 1.01E E E E-05 X X 7F 3.45E E E E-06 X X 7G 6.82E E E-03 X X X 7H 1.14E E E-06 X X X 7I 1.30E E-02 X X X X 7J 7.87E E-03 X X X X 7K 3.15E E-03 X X X X 7L 3.66E E-02 X X X X 8A 1.08E E E-04 X X X 8B 4.09E E E-05 X X X 8C 0.00E E+00 X X X X 8D 1.38E E-03 X X X X 8E 1.28E E-02 X X X X 8F 4.76E E E-04 X X X 8G 1.97E E-04 X X X X 8H 2.20E E E-04 X X X 8I 1.97E E-04 X X X X Totals: 8.33E E E E E E-06 30

31 We now aggregate the joint probability distributions of Tables 19 and 20 by system. Aggregating the 51 weld cases in this way yields results for each of the nine systems for each break-size category, and the results are shown in Tables and Figures 7-8. We can see that the aggregated results for the current-day estimates and the end-of-plant-license estimates are of the same shape but different scale for each category. We see that SIR Safety and CVCS welds have a significantly higher probability than other sets of welds for categories 1 and 2. For category 3, SIR Safety, CVCS, and Hot Leg SG contribute the majority of the probability. Categories 4-6 are similar, with the Hot Leg SG system dominating and the Hot Leg and Cold Leg systems providing smaller contributions. If we aggregate the joint probability distribution over the 51 weld cases to 25 unique sets of welds based on system type and DEGB size, we would obtain a set of plots for aggregated probabilities of the same shape but different scale as those in Figures 4-6 with the scale being P [cat j ] for j = 1, 2,..., 6; see equation (6). 31