South Texas Project, Units 1 & 2, Response to NRC Request for … · 2014. 1. 24. · ,6 and where Freqvency[LOCA > cat7] =0 . Again we assume there are Mj different locations in

Nuclear Operating Company

South Texas Pro/ect Electric Generating Station P.. Box 289 Wadsworth. Texas 77483 ,

December 23, 2013NOC-AE-1 300307010 CFR 50.1210 CFR 50.90

U. S. Nuclear Regulatory CommissionAttention: Document Control DeskWashington, DC 20555-0001

South Texas ProjectUnits 1 & 2

Docket Nos. STN 50-498, STN 50-499Response to NRC Request for Reference Document

For STP Risk-Informed GSI-191 Application(TAC NOs MF2400 and MF2401)

Reference: Letter, G. T. Powell, STPNOC, to NRC Document Control Desk, "Supplement1 to Revised STP Pilot Submittal and Requests for Exemptions and LicenseAmendment for Risk-Informed Approach to Resolving Generic Safety Issue(GSI)-191, " November 13, 2013, NOC-AE-1 3003043 (ML1 3323A1 83)

In a phone conversation on December 19, 2013, the NRC staff requested clarification forreferences used in Enclosure 4-3 of the referenced correspondence supporting the assignmentof pipe break frequencies to welds in the containment. In this discussion the NRC staffdetermined that Reference 8 to Enclosure 4-3 needed to be submitted on the STP docket forthe staff's review. The requested document is attached.

There are no regulatory commitments in this letter.

If there are any questions, please contact Ken Taplett at 361-972-8416.

I declare under penalty of perjury that the foregoing is true and correct.

Executed on: /).CA•1,,?,.

Vice President, Corporate Servicesawh A•-,.

Attachment: University of Texas at Austin, Modeling and Sampling LOCA Frequency andBreak Size for STP GSI-191 Resolution, September 2012

ST133802775

NOC-AE-1 3003070Page 2 of 2

cc:

(paper copy) (electronic copy)

Regional Administrator, Region IVU. S. Nuclear Regulatory Commission1600 East Lamar BoulevardArlington, TX 76011-4511

Balwant K. SingalSenior Project ManagerUS. Nuclear Regulatory CommissionOne White Flint North (MS 8 B13)11555 Rockville PikeRockville, MD 20852

NRC Resident InspectorU, S. Nuclear Regulatory CommissionP. O. Box 289, Mail Code: MN116Wadsworth, TX 77483

Jim CollinsCity of AustinElectric Utility Department721 Barton Springs RoadAustin, TX 78704

Steven P. Frantz, EsquireA. H. Gutterman, EsquireMorgan, Lewis & Bockius LLP

Balwant K. SingalMichael MarkleyJohn StangU. S. Nuclear Regulatory Commission

John RaganChris O'HaraJim von SuskilNRG South Texas LP

Kevin PolioRichard PehaCity Public Service

Peter NemethCrain Caton & James, P.C.

C. MeleCity of Austin

Richard A. RatliffRobert FreeTexas Department of State Health Services

NOC-AE-1 3003070Attachment

University of Texas at Austin

Modeling and Sampling LOCA Frequency and Break Size for STP GSI-191 Resolution

September 2012

(47 pages)

Modeling and sampling LOCA frequency and breaksize for STP GSI-191 resolution

Elmira Popova, David Morton, Ying-An PanThe University of Texas at Austin

September 2012

1 Introduction

In the initial quantification (Crenshaw, 2012), Fleming et al. (2011) performed a, substan-

tial study designed to build 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 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 report 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 NUR.EG-1829 annual frequencies are neither plant specific nor location specific

within a plant. 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 report we work with the six categories defined in Table 7.19 (page 7-55) of Trego-

ning 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-either cumulative distribution function

(CDF), probability density function (PDF), or probability mass function (PMF)-of a ran-

dom variable used to model a specified uncertainty.

We should point out that South Texas Project PRA analysis uses only three LOCA

categories, small, medium, and large. Our proposed methodology can be applied to any

1

Table 1: LOCA categories notation map

Effective break size(inch) Notation

2 cat11-.5 cat 283 cat 37 cat 4

14 cat531 cat6

finite number of break-size categories.

In this report we will use the term location to represent a specific weld. Overall there

are two distinct approaches to derive location- or weld-specific LOCA frequencies: bottom-

up and top-down. The first approach requires location-specific failure data, to estimate the

corresponding probability of a weld failure. Suppose a, break occurs and assume there are

My different welds in the plant where breaks of size catj can occur, weld1 , , weldA.1, then

using the law of total probability we can write:

Al1

P[catj] = • P[catjIweld]P[weld], j = 1,2,., 6,t=1

where P[catj] is the probability of a catj LOCA given that a break occurs, P[catjIweld.j]

is the conditional probability of a catj LOCA given that the break occurs at weld i, and

P[weldj] is the probability that the break occurs at weld i.

In the bottom-up approach we first must determine P[catjjwiedj] (using estimation or

expert elicitation). Then, if we assume that each location is equally likely to have the

break, we can multiply by 1/MAIj 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 value 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 combining the top-down and bottom-

up approaches: We start with the NUREG-1829 frequencies and develop a way to distribute

them across different locations proportionally to the frequencies estimated using the bottom-

up approach. In this wvay, we maintain the NUREG-1829 frequencies overall but also allow

for location-dependent differences. We should point out that we use the location-specific

tables given 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

incorporate that information.

For a top-down approach, we will use again the catj LOCA as an illustrative example.

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 catj using the formulaP[cat] =Frequency[LOCA > catj] - F'requency[LOCA > catji+1]

Frequency[LOCA > cat1]

for j = 1 ... ,6 and where Freqvency[LOCA > cat7 ] =0 . Again we assume there are

Mj different locations in the plant where breaks of size catj can occur, 'weld1 ,. . . , weldi. 3 .

Assume, for the moment, given that we have a caty break, and these Mj locations are equally

likely to have the break, i.e.,

:1P[wehlilcatj] • i = 1,...,IMyl.

Then we have P[catj at weld1 ] = P[catj]P[weldjjcatj] and so P[catj at weld.] = P[catj]/IAlj.

Finally, applying the law of total probability,

Al1j

P[catj] = P[catj at 'weld1 ],

we see that the resulting probability of a catj LOCA matches exactly the NUREG-1829

probability. The approach we propose in this report, follows the steps we have just outlined,

except we propose replacing the simple assumption of a catj break being equally likely to oc-

cur across all locations with an approach that uses location-specific conditional probabilities

that we infer from Fleming et al. (2011).

3

The above methodology distributes equally the LOCA frequencies across all locations

that can experience breaks from one or more of the six size categories. The six break size

categories of Table 1 yield six bounded intervals for break sizes; i.e., [0.5, 1.625), [1.625,3),

[3, 7), [7,14), [14,31), and [31, DEGBI,,x), where all values are in inches and DEGBmxa

(double-ended guillotine break) denotes the largest effective break size in the system under

consideration. 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 7.19 from NUREG-1829 represent information on the

distribution of the frequencies by reporting the median, 5th, and 95th percentiles. We will

use this information to fit a. continuous distribution for each b.'reak size ca:tegory for both the

current-day and end-of-plant-license estimates.

2 Proposed Methodology

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, there are an infinite number of distributions that one can fit to the LOCA frequencies

represented in NUREG-1829. For example, 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 allows us to match closely the distributional

characteristics provided by NUREG-1829.

" The Johnson distribution allows for a variety of shapes. In particular, skewed, sym-

metric, bimodal, or unimodal shapes can be obtained.

While a detailed analysis of other possibilities is beyond the scope of this report, we do

note that while the gamma distribution is the conjugate prior to the Poisson distribution,

the gamma family of distributions is note consistent with the NUREG-1829 percentiles.

4

A similar conclusion holds for the so-called semi-bounded Johnson distribution (which is

equivalent to a shifted log-normal distribution).

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

F[x] = ýD {m + 6f[(z - ý)/A]}, (1)

where 4)[x] is the CDF of a. standard normal random variable, 7Y and 3 are shape parameters

(with -y driving the distribution's skewness), ,c is a location parameter, A is a, scale parameter,

and f(z) = log[z/(1 - z)] for ý < x < ý + A. We denote the bounded Johnson distribution

with these four parameters as SB(7, 6,,c, A). In general, 3 > 0 and A > 0. In our setting,

we know frequencies are necessarily positive and so we also have ý > 0. WVe have proved

the following fact for the bounded Johnson distribution. This fact helps with the fitting

procedure we describe below.

Fact 1. If X - SB(7y 6;,,A) and a > 0 then aX - SB(-.6, ac,,aA\).

In order to obtain the parameters of the bounded Johnson distribution, we solve a. nonlin-

ear optimization problem., optimizing over the four parameters. For each break-size category,

we minimize the suim of the squared deviations of the fitted values of the Johnson CDF at the

NUREG-1829 LOCA frequencies from the NUREG-1829 percentiles (5%, 50%, and 95%).

We enforce two constraints involving ,c and c + A, which denote the lower and upper bounds

for the bounded Johnson distribution. In addition to requiring that the lower bound of the

Johnson distribution (c) and the width of the bounding interval (A) be nonnegative, we

require (c be smaller than the 5% percentile and that A + c exceed the 95% percentile. As

we indicate above the shape parameter 3 is necessarily nonnegative. Thus the optimization

model we formulate is given by:

minm (F[Xoo5] - 0.05))2 + (F[a70.5] - 0.5)2 + (F[XO.95] - 0.95)2 (2a)

s.t. -T -. 05 (2b)

A + _> : 0 .9• (2c)

3,A > 0. (2d)

The values X0 j.5, o..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

5

(40 years fleet average operation). The CDF of the Johnson distribution, F[x], depends on

the parameters .y, 3, ý, A, as specified in equation (1).

Fitting -y, , J, and ,\ to each of the six categories for both current-day and end-of-license

values yields 12 total instances of model (2). However, attempting to solve model (2) directly,

using the 5%, 50%, and 95% percentiles from Tables 3 and 5 is not a good idea. Modern

optimization software is ill-equipped to deal with numerical values smaller than 1 x 10. (see

Tables 3 and 5). So, we make use of Fact 1 to rescale our optimization model. Specifically,

F[x] = P(X < x), where X SB(y 3, ,A), and thus by Fact 1 F[ax] = P(aX < ax),

aX -SB(-y, 5, ri(, aA). For each of the 12 instances of the optimization model we face, we

set a = 1/X0.0 5 to obtain a well-scaled optimization model:

min (F[axo.05] - 0.05)2 + (F[axo.5] - 0.5)2 + (F[axO.9 5] - 0.95)2 (3a)

SA.t. < axo.o.5 (3b))

A + _> axo.95 (3c)

J, $, A > 0. (3d)

Solving this model yields the correct values for -' and 6 and to obtain the optimized values

of c and A 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. We note that the NUREG-1829 expert elicitation was for the

5%, 50% (median), and 95% quantiles, and did not involve eliciting the mean. So we focus on

matching the three distributional characteristics elicited from the experts as indicated by the

results in the final four 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. Once the best fit is found, we sample the

LOCA frequencies for each category to obtain Frequericy[LOCA > catj--a realization of

the cumulative LOCA frequency to be in category j or larger.

6

Table 2: Fitted Johlson parameters for current-day estimates

Johnson Parameters- A

Cat1 1.650950E+00 5.256964E-01 4.117000E-05 1.420000E-02Cat2 1.646304E+00 4.593913E-01 2.530000E-06 3.200000E-03Cat3 1.646605E+00 4.589467E-01 1.200000E-07 1.220000E-04Cat4 1.645739E+00 4.487957E-01 6.023625E-09 1.220000E-05Cat5 1.645211E+00 3.587840E-01 2.892430E-10 1.160000E-06Cat6 1.645072E+00 3.343493E-01 2.636770E-11 1.600000E-07

Table 3: NUREG-1829 and fitted Johnson mean, median, 5% and 95% quantiles values forcurrent-day estimates

NUREG 1829 Values Fitted Johnson Relative Error5th Median Mean 95th 5th Median Mean 95th 5th Median Mean 95th

Catl 6.80E-05 6.30E-04 1.90E-03 l.1OE-03 6.80E-05 6.30E-04 1.62E-03 7.10E-03 0.00% 0.00% 14.48% 0.00%Cat2 5OOF-06 8.90E-05 4.20E-04 1.60E-03 5.OOE-06 8.90E-05 3.21E-04 1.60E-03 0.00% 0.00% 23.49% 0.00%Cat3 2.10E-07 3.40E-06 1.60E-05 6.10E-05 2.10E-07 3.40E-06 1.23E-05 6.lOE-05 0.00% 0.00% 23.38% 0.00%Cat4 1.40E-08 3.10E-07 1.60E-066. 10E-06 1.40E-O8 3.10E-07 1.20E-06 6.10E-06 0.00% 0.00% 25.10% 0.00%Cat5 4.10E-10 1. 0E -0 2.OOE-07 I 5.80E-O7 4.1E-0 1.20E-08 9.54E-08 5.80E-07 1.00% 0.00% 52.31% 0.00%Cat6 3.50E-11 1.20E-09 2.90E-05 8 1OE-08 3.50E-11 1.20E-09 1.25E-08 8.1OE-08 0.00% 0.00% 56.86,0.00%

Table 4: Fitted Johnson parameters for end-of-plant-license estimates

Johnson Parameters

__y_ 6 A

Cat1 1.649918E+00 5.325856E-01 3.756000E-05 1.580000E-02Cat2 1.646018E+00 4.574212E-01 2.800000E-06 4.400000E-03Cat3 1.646650E+00 4.55211SE-01 2.8000OOE-07 2.800000E-04Cat4 1.645624E+00 4.396424E-01 1.000000E-08 2.800000E-05Cat5 1.645225E+00 3.559738E-01 7.288800E-10 2.800000E-06Cat6 1.645069E+00 3.2951667E-01 6.762640E-11 4.200000E-07

7

u~ C0 Ca U'

- Current- End

o

0.000 0.005 0.010 0.015

(a) CatI

0- Current-- End

0.000 0.001 0,002 0.003 0.004

le-04 2e-04 3e-04 4e-04 5e-04

(b) Catl (zoomned)

0_

t2

gg

gg

gg0.

O

le-05 2e-05 3e-05 4e-05 5e-05

(c) Cat2 (d) Cat2 (zoomed)

Figure 1: Johnson PDFs for category 1 and category 2 break, and each isnarrower range of frequencies near the mode of the distribution

zoomed to a

8

00.

o ~oD C

Current- End

a

1 1

0.00000 0.00005 0.00010 0.00015 0.00020 0.00025

(a) Cat3

2e-07 4e-07 6e-07 8e-07 le-0G

(b) Cat3 (zoomed)

CurrentEnd

0

0.0e4-00 S.Oe-06 1.0e-05 1.5e-05I I

2.0e-05 2.5e-05 2e-08 4e-08 6e-O8 Sn-OS le-07

(c) CaI4 (d) Cat4 (zoomed)

Figure 2: Johnson PDFs for category 3 and category 4 break, and each is zoomed to anarrower range of frequencies near the mode of the distribution

9

CL C*

- Current- End

CC S~ CC

O.OeudO 5.Oe-07 1Oe0 1.5e-06 2O0e-06 2.5e-06 5.0e-10 1.0e-09 1.5e-09 2.0e-09

(a) Cat5 (b) Cat5 (zoomed)

0

- Current- End

0.

Oe*OO le-07 2e-07 3e-07 4e-07 4e-1l 6e-11 8e-11 le-10

(c) Cat6 (d) Cat6 (zoomed)

Figure 3: Johnson PDF fornarrower range of frequencies

category 5 and category 6 break, and each isnear the mode of the distribution

zoomed to a

10

Table 5: NUREG-1829 anid fitted Johnson mean, median, 5% and 95% quantiles values forend-of-plant-license estimates

NUREG 1829 Values Fitted Johnson Relative Error5th Median Mean 95th 5th Median Mean 95th 5th Median Mean 95th

Catl 7.OOE-05 7.20E-04 2.10E-03 7.90E-03 7.OOE-05 7.20E-04 1.82E-03 7.90E-03 0.00% 0.00% 13.19% 0.00%Cat2 6.10E-06 1.20E-04 5.80E-04 2.20E-03 6.10E-06 1.20E-04 4.40E-04 2.20E-03 0.00% 0.00% 24.19% 0.00%Cat3 4.80E-07 7.60E-06 3.60E-05 1.40E-04 4.80E-07 7.60E-06 2.79E-05 1.40B-04 0.00% 0.00% 22.37% 0.00%Cat4 2.80E-08 6.60E-0 7 3.60E-06 1.40E-05 2.80E-08 6.60E-07 2.70E-06 1.40E-05 0.00% 0.00% 25.00% 0.00%Cat5 1.OOE-09 2.80E-08 4.80E-07 1.40E-G6 1.OOBE-09 2.80E-08 2.29E-07 1.40E-06 0.00%I 0.00% 52.30% 0.00%Cat6 8.70E-11 2.90E-09 7.50E-08 2.O1E-07 8.70E-11 2.90E-09 3.25E-08 2.E]O-07 0.00% 0.00% 56.64% 0.00%

2.2 Distribution of LOCA frequencies to different weld locations

We first convert the sampled LOCA frequencies to probabilities using

P[lat] -Frequerncy[LOCA > catj] - Frequency[LOCA > catj+](4[cay= Frequency[LOCA > cat,]

where

" J {cat,, cat2, cat3 , ... , cat.B}: set of possible break types (categories)

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

was observed

" Frequency[LOCA > catj]: frequency of break of type j or larger, j E J

" Frequency[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] = u§P[catj],

where w. = P(weld.icatj) is the conditional probability of the break occurring at weld i

given that we have a category j break. Restated, 'wy7 is the fraction that weld i contributes to

11

category j's total break frequency from the bottom-up approach for i C Ij. Computation of

wj is straightforward. The bottom-up approach generates the frequency of category j breaks

at location i, which we denote Freq,,[LOCA > catj at weldi]. Also, there are different

numbers of welds for each category i. We denote the number of welds for category i by ni.

Given these frequencies and the number of welds for each category, the t' values can be

computed using:

- (Freqbu[LOCA > catj at weld1 ] - Freqb6 ,.[LOCA > catj+1 at weldj]) x n . (5)' ic j (Freqb,,[LOCA > catj at weldi] - Freqb,,[LOCA > catj+l at weldi]) x ni

Given P[catj] from equation (4) and ' from equation (5), we form

P[catj at weld1 ] = wjP[catj]. (6)

Since the sum of all wt across i E Ij is equal to one, with this approach we are guaranteed

to match the NUREG-1829 specified values for P[caty].

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

breakSize'- UJ[mniniB7-eal§ý,i, ?n.aBrealcj'], j G Jji E Ij,

where

SininBreak, = cat , i"B"ca:

* maxBreak' = mmi cat" • wedl• }

" catntinBrcak - minimum break size that would put it into category j

" catw d e - maximum break size that would put it into category

" 'weld"i2e - actual weld size (it cannot experience break size larger than its DEGB).

12

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 Frequency[LOCA > cats], j = 1, 2,..., 6, from the fitted

Johnson distributions for each break category; see Section 2.1.

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

4. Sample break size for each possible weld and 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 5 break-size samples.

7. Compute, and store, performance measures.

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

9. Form the summary of aggregated performance measures.

In Section 4 we apply our approach to the 45 weld cases of Fleming et al. (2011). However,

before doing so, in the next section we apply the ideas developed above to a, small example

with just six types of welds so that it is easier to follow along without, the details of many

types of welds.

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 types of welds (Fleming et al. (2011)

13

use the term weld cases) 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 2, 2.8, and 2.83 inches and hence

can experience small breaks (category 1 and category 2). Two of those six (welds 4 and 5)

are medium with sizes of 4.24 and 5.66 inches and thus can have small and medium breaks

(category 1, category 2, and category 3). The last weld (weld 6) is large with a size of 41.01

inches and can have all types of breaks--small, medium, and large (category 1, ... , category

6). A graphical representation of the system is shown in Figure 4.

283 •

Onif

IFigure 4: Example system depiction with six welds of various sizes that can each experiencesome subset of six types of breaks from category 1,... ,category 6.

Adapting the notation developed in Section 2 to this example we have:

J = {cot,. ca 2., cat 3. cat 4 , cat 5 , cat 6 l},

Icatl = {weldm, weld2 , weld 3, weld 4, weld%. weld },

Icat2 = {weld 1 , weld 2, weld:, weld4 , weld5 , weld 6},

Icat3 ={ weld4, weld 5 , weld 6 },

,cat4 = {weld6}, 1cat5 {w=ld6 }, Icat6 = {weld6},

and

14

B7 reak,5 I

Break Si ze"M

l'eai• Z ~ cat.1

BreakSi ze"Ieldu

cat2

I. wed4

BreakSi zze 15

Break Si euceld5cat2

Break Size""di

BreakSi., e"2

Break Si wel~d.,

cat2

Br~eak Si ze""14

Break Si zeat-?.eaED~l.• cat2 •

.at2

Break,5i -.ecou,~I r ,,.:S. t2e':.•

BreakoSizelaet

Cat3

Br eak Si ze""W5J•

.bX l••cat3G

U[0.5, 1.625]

- U[0.5,1.625]

- U[0.5,1.625]

- U[0.5,1.625]

U[0.5, 1.625]

U[0.5,1.625]

- U[1.625, 2]

- U[1.625, 2.8]

U[1.625, 2.83]

U[1.625, 3]

- U[1.625,3]

- U[1.625,3]

- U[3, 4.24]

- U [3, 5.66]

- U[3, 7]

- U[7,14]

U[14,31]

- U[31,41].

(7a.)

(7b)

(7c)

(7d)

(7e)

(7f)

(7g)

(7h)

(7i)

(Tj)

(7k)

(71)

(7m)

(7n)

(70)

(7p)

(7q)

(7r)

Below we enumerate the first four steps of the procedure from Section 2.4 for this example

system.

1. Assume S = 1, N = 1.

2. Sampled LOCA frequencies using the fitted Johnson distributions for the current-day

and end-of-plant-license estimates as given in Tables 6 and 7, respectively. For this

illustration we simply use the median values from the Johnson distributions. In actual

implementation we use the sampled LOCA frequencies from the fitted bounded Johnson

15

distributions. The right-most column of Tables 6 and 7 computes the probability mass

for each category according to equation (4).

3. Break frequency tables for the six welds obtained from the bottom-up approach can

be found in Tables 8-10. Tables 6 and 7 contain bins defining the break categories, as

derived from Table 1. The associated categories for each break size are indicated in

Tables 8-10.

Using Tables 8-10 we compute weights for each weld and report results in Tables 11-13.

To describe the derivation of these weights we begin with Table 11. The weld 1 fre-

quency value in that table is the difference between the cumulative frequencies from the

0.5-inch row and the 2-inch row from Table 8 times the number of welds for weld case 1.

The weld 2 frequency value is the difference between the frequencies from the 0.5-inch

row and the 1.99-inch row from Table 8 times the number of welds for location 2. The

weld 3, wel(-d 4, weld 5, and weld 6 frequenciees are sinilarly the difference betwee.en the

frequencies from the 0.5-inch rows and the 2-inch rows from Tables 8-10 times the re-

spective number of welds for location 3, 4, 5, and 6. Finally, we normalize the resulting

values using eqluation (5) to compute the weights w.eld1 . . .j/dd6 Tables 12 and 13"(cut u . .t c p cat l •a 2

contain the results of the analogous calculations for category 2 and category 3. There

is no need to form the corresponding frequency values for category 4, . ., category 6

because these categories only occur for weld 6, and hence these weights are simply

100%.

Using equation (6) we now compute P[catj at weldj] for each category-weld combina-

tion for both the current-day and end-of-plant-license estimates. The results are given

in Tables 14 and 15. It is obvious that the estimated P[catj] in Tables 14 and 15 is the

same as the initial P[cat9] in Tables 6 and 7 respectively. Note that there are 1n1 = 0

instances of weld 1 in the system under consideration (see Table 8) and this is reflected

in Tables 11-15.

4. We simulate break sizes for each weld within each category using the uniform distri-

butions with the parameters specified in equation (7). The sample is shown in Table

16

16.

Table 6: Sampled LOCA frequencies and corresponding probabilities for current-day esti-mates

Failure Type Category Break Size Bins (in.) Frequency Probabilitysmall 1 [0.5,1.625) 6.30E-04 8.59E-01small 2 [1.625,3) 8.90E-05 1.36E-01

medium 3 [3,7) 3.40E-06 4.90E-03medium 4 [7,14) 3.10E-07 4.73E-04

large 5 [14,31) 1.20E-08 1.71E-05large 6 [31,41) 1.20E-09 1.90E-06

Table 7: Sampled LOCA frequencies and corresponding probabilities for end-of-plant-licenseestimates

Failure Type Category Break Size Bins (in.) Frequency Probabilitysmall 1 [0.5,1.625) 7.20E-04 8.33E-01small 2 [1.625,3) 1.20E-04 1.56E-01

medium 3 [3,7) 7.60E-06 9.64E-03medium 4 [7,14) 6.60E-07 8.78E-04

large 5 [14.31) 2.80E-08 3.49E-05large 6 [31,41) 2.90E-09 4.03E-06

Finally, we note that our assumptions lead to a piecewise linear CDF governing the break

size for a, given weld. For example, consider weld 6. The CDF of the break size for that weld

has six pieces with the slopes determined by the P[catj at weld6] values and break points at

the cat? naxBreca bin boundaries of 1.625, 3, 7, 14, 31, and 41 inches, see Figure 5.

17

Table 8: Frequency tables for small welds from bottom-up approach

SMALLweld 1 weld 2s weld3

System SIR. Small Bore PressurizerSize Case (in.) 1.5 2 2DEGB (in.) 2.121320:344 2.828427125 2.828427125Weld Type B-.] B-J B-JDM D&C VR, SC, D&C TF, D&CNo. Welds 0 16 2

X, Break Size (in.) F(LOCA >_ X) X, Break Size (in.) F(LOCA > X.) X, Break Size (in.) F(LOCA > X)0.5 (catI) 1.14E-08 0.5 (cat1) 1.22E-06 0.5 (cat1) 4.59E-08

0.75 (catl) 6.84E-09 0.75 (cat1) 7.18E-07 0.75 (catl) 2.76E-081 (carl 4.85E-09 1 (catl) 5.00E-07 1 (catl) 1.96E-08

1.5 (catl) 3.07E-09 1.4 (cati) 3.30E-07 1.5 (catl) 1.24E-082 (cat2) 1.65E-09 1.5 (catl) 3.OSE-07 2 (cat2) 6.64E-09

1.99 (cat2) 1.75E-07 2.83 (cat2) 3.13E-092.0 (cat2) 1.73E-072.8 (cat2) 8.66E-08

Table 9: Frequency tables for medium welds from bottom-up app)roach

MEDIUMweld 4 weld5,

System Pressurizer CVCSSize Case (in.) 3 4DEGB (in.) 4.242640687 5.656854249Weld Type B-3 BCDM TF, D&C TF, D&¶CNo. Welds 14 4

X, Break Size (in.) F(LOCA >_ X) X, Break Size (in.) F(LOCA >_ X)0.5 (cat1) 4.59E-08 0.5 (catl) 7.98E-08

0.75 (catl) 2.76E-08 0.75 (catl) 4.79E-081 (catl) 1.96E-08 1 (catl) 3.40E-08

1.5 (catl) 1.24E-08 1.5 (catl) 2.15E-082 (cat2) 6.64E-09 2 (cat2) 1.12E-083 (cat3) 2.75E-09 3 (cat3) 4.51E-09

4.24 (cat3) 1.30E-09 4 (cat3) 2.34E-09•_5.66 (cat3) 1.08E-09

18

Table 10: Frequency tables for large welds from bottom-up approach

LARGE1-

wLd'Co,System Sc filetSize Case (in.) 29DEGB (in.) 41.01219331Weld Type B-FDM SC, D&CNo. Welds 4

X, Break Size (in.) F(LOCA4 > X)0.5 (catl) 1.98E-061.5 (carl) 4.5932E-07

2 (cat2) 3.4469E-073 (cat3) 2.3061E-074 (cat3) 1.5971E-076 (cat3) 9.5224E-08

6.75 (cat3) 8.1186E-0814 (cat5) 3.3453E-0820 (cat5) 1.8122E-0829 (cat5) 9.5661E-09

31.5 (cat6) 8.3016E-0941.01 (cat6) 5.2422E-09

Table 11: Category 1 weld weights in total failure frequency using bottom-up approach

Catl weldl weld2 weld3 weld4 weld5 weld6 TotalFrequency 0.00E+00 1.68E-05 7.85E-08 5.49E-07 2.74E-07 6.53E-06 2.42E-05

Weight 0.00% 69.31% 0.32% 2.27% 1.13% 26.96% 100.00%


Cat2 weldl weld2 weld3 weld4 weld5 weld6 TotalFrequency 0.00E+00 2.80E-06 1.33E-08 7.47E-08 2.69E-08 4.56E-07 3.37E-06

Weight 0.00% 83.04% 0.39% 2.22% 0.80% 13.54% 100.00%


Cat3 weld4 weld5 weld6 TotalFrequency 3.86E-08 1.80E-08 7.89E-07 8.45E-07

Weight 4.56% 2.13% 93.31% 100.00%

19

Table 14: Distributed LOCA probabilities among all welds for current-day estimates

weldl weld2 weld3 weld4 weld5 weld6 Estimated P[catj]Cati 0.OOE+00 5.95E-01 2.78E-03 1.95E-02 9.72E-03 2.32E-01 8.59E-01Cat2 0.OOE+00 1.13E-01 5.36E-04 3.01E-03 1.09E-03 1.84E-02 1.36E-01Cat3 X X X 2.24E-04 1.05E-04 4.58E-03 4.90E-03Cat4 X X X X X 4.73E-04 4.73E-04Cat5 X X X X X 1.71E-05 1.71E-05Cat6 X X X X X 1.90E-06 1.90E-06

Table 15: Distributed LOCA probabilities among all welds for end-of-plant-license estimates

weldl weld2 weld3 weld4 weld5 weld6 Estimated P[catj]Cat1 0.00E+00 5.78E-01 2.70E-03 1.89E-02 9.43E-03 2.25E-01 8.33E-01Cat2 0.00E+00 1.30E-01 6.15E-04 3.46E-03 1.25E-03 2.11E-02 1.56E-01Cat3 X X X 4.40E-04 2.06E-04 8.99E-03 9.64E-03Cat4 X X X X X 8.78E-04 8.78E-04Cat5 X X X X X 3.49E-05 3.49E-05Cat6 X X X X X 4.03E-06 4.03E-06

Table 16: Sampled break sizes (inches) for all welds within each break category

Weld 1 2 3 4 5 6Catl 1.40 1.53 0.94 0.84 1.04 1.49Cat2 1.93 1.73 2.34 2.97 1.64 2.89Cat3 X X X 3.17 4.96 5.13Cat4 X X X X X 8.20Cat5 X X X X X 19.32Ca.t6 X X X X X 31.27

20

0 10 20 30 40 0 10 20 30 40

(a) current-day estimates (b) end-of-plant-license estimates

Figure 5: CDF of break size for weld 6

4 The 45 Weld Cases of Fleming et al. (2011)

We perform a similar analysis to the previous section but we now consider all 45 weld cases

of the location-specific frequency tables from Fleming et al. (12011). We assume that these

45 weld cases cover all locations of interest where a break can occur. We label the 45 welds

as woeldl, weld.,, ... , wveld 45 and summarize some of their characteris tics defined in the tables

from Fleming et al. (2011) in Table 17. In the table, DEGB is again a double-ended guillotine

break size (in inches) which indicates the largest break size a weld can have. However, we

notice that there is a. discrep~ancy with the DEGB sizes (in inches) and the largest break sizes

(in inches) in the locationl-specific frequency tables from Flemning et al. (2011). For example,

for Calculation Case SA, the DEGB size is 2.83 inches but the largest break size is 3 inches.

In this report, we will use the largest break sizes (in inches) in the tables from Fleming et al.

(2011) as otw DEGB sizes. That is, we treat the last column of Table 17 as our DEGB sizes.

Assume S = 1 and N = 1 from the procedure in Section 2.4. Using the frequency tables

for all 45 weldl cases from Fleming et al. (2011), the freqiuencies and weights for the 45 welds

21

Table 17: Summary of the 45 welds

Weld Calculation Case System DEGB (in.) Number of Welds Largest Break Size (in.)1 IA Hot Leg 41.01219331 4 41.012 1B Hot Leg 41.01219331 11 41.013 1C Hot Leg 41.01219331 1 41.014 2 SG Inlet 41.01219331 4 41.015 3A Cold Leg 38.89087297 4 38.896 3B Cold Leg 43.84062043 4 43.80

7 3C Cold Leg 38.89087297 12 38.898 3D Cold Leg 43.84062043 24 43.809 4A Surge Line 22.627417 1 22.63

10 4B Surge Line 22.627417 7 22.6311 4C Surge Line 22.627417 2 22.6312 4D Surge Line 3.535533906 6 3.5413 5A Pressurizer 8.485281374 29 8.49

14 5B Pressurizer 4.242640687 14 4.24

15 5C Pressurizer 5.656854249 53 5.6616 5D Pressurizer 4.242640687 4 4.2417 5E Pressurizer 8.485281374 29 8.4918 5F Pressurizer 8.485281374 0 8.4919 5C Pressurizer 8.485281374 4 8.4920 5H Pressurizer 5.656854249 2 5.6621 5I Pressurizer 2.828427125 2 2.8322 5J1 Pressurizer 8.485281374 0 8.4923 6A Small Bore 2.828427125 16 2.8324 6B Small Bore 1.414213562 193 1.4125 7A SIR 16.97056275 21 16.9726 7B SIR 11.3137085 9 11.3127 7C SIR 11.3137085 3 11.31

28 7D SIR, 16.97056275 3 16.9729 7E SIR. 16.97056275 57 16.9730 7F SIR 14.14213562 30 14.1431 7G SIR 11.3137085 42 11.3132 7H SIR 8.485281374 23 8.4933 71 SIR 5.656854249 5 5.6634 7J SIR 4.242640687 9 4.2435 7K SIR 2.8284271.25 10 2.83

36 7L SIR. 2.121320344 0 2.0037 7M ACC 16.97056275 0 16.9738 7N ACC 16.97056275 35 16.97

39 70 ACC 16.97056275 15 16.9740 8A CVCS 2.828427125 10 3.0041 8B CVCS 5.656854249 19 5.6642 8C CVCS 2.828427125 47 3.0043 8D CVCS 5.656854249 6 5.66

44 8E CVCS 5.656854249 4 5.6645 8F CVCS 5.656854249 1 5.66

22

are reported in Tables 18 and 19. In these tables an "X" indicates a weld case is not capable

of contributing a positive frequency to this category because its DEGB 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 et al. (2011).

Before we continue with carrying out the steps of our methodology described in Sec-

tion 2.4, we examine the relative contributions to the weights of specific welds overall, of

specific welds within each break-size category, and of specific system-DEGB combinations.

We do this to try to give insight into the weld cases are most likely to experience a LOCA

from each category. We illustrate the weld weights in total failure frequency for the six

categories in Figure 6. We see that weld - 4, weld,,.5, weld 26, weld, , and weld 4 have consid-

erable weight as we look across the six categories. The results of these five weld cases are

summarized in Table 20. Note that the total weight associated with these five weld cases

exceeds 80% in four of the six categories and exceeds 67% in all categories, as shown in the

last row of Table 20.

We also summarize the top five contributors to total failure frequency for each category

in Tables 21-26 and calculate the associated sum of the weights. VVAe see that the five top

weld cases contribute more than 84% for all six categories, and there are 14 welds in total

(weldl, wveld 4, Weld 5 , uweld 6, weld8 , weldg, weldl0 , weld13 , weld2,3 , tweld94, weld25 , 'weld2 6,

weld2 7, and weld38 ) that account for this large total weight. If we restrict attention to those

top five weld cases with weight contributions exceeding 5% for each category, we end up

with a total of 10 welds (weldl, weld 4, weld5, weld6 , weld9 , weld.2 3, weld.2 4, weld,5. weld2 ,6

and weld.2 7) that account for more than 80% of the weight for all the six categories.

When we focus on the type of system associated with the weld cases in Tables 21-26,

Small Bore and SIR welds taken together account for a large portion of weight for both

category 1 and category 2; Surge Line, SIR, and SG Inlet welds account for much of the

weight for both category 3 and category 5; SIR welds are dominant for category 4; and, SG

Inlet and Hot Leg welds account for much of category 6. Note that there are other welds not

indicated in these tables but from the same systems: SIR, Surge Line, ACC, Pressurizer, Hot

Leg, and Cold Leg. Thus, the total weight of the weld cases from the system types indicated

in Tables 21-26 is at least the number shown in these tables. In addition, SIR welds have

23

a. considerable total weight for categories 1 ... , 5. In particular, 'weld25, weld26 , and weld27

account for a total weight of more than 21%, 64%, 40%, and 93% for category 1, category 2,

category 3, and category 4, respectively, and also weld 25 accounts for more than 31% weight

for category 5.

Continuing to focus on the contributions due to welds by their type of system, we aggre-

gate the 45 weld cases to 23 unique sets of weld cases with different combinations of system

types and DEGB sizes. For example, looking at Table 17 and restricting attention to the

cold leg system, we aggregate weld cases 3A and 3C and we aggregate cases 3B and 3D

because they have the same respective DEGB sizes. The results are reported in Table 27

and are also illustrated in Figures 7-9. These aggregated results are consistent with the

results for the 45 welds but with the same or larger weight for each combination because we

now aggregate rather than restrict attention to a subset of welds. By tracing changes as we

scan Figures 7-9 we understand changes in the contribution to total weight for particular

DEGB-system pairs.

We now return to the procedure of Section 2.4 to distribute uncertainty across the 45 weld

cases using the weights from Tables 18 and 19. With the sampled LOCA frequencies from

Tables 6 and 7 in Section 3 (we again take these as the median from NUREG-1829 for illustra-

tive purposes here, while these are sampled in implementation), we compute P[catj at weldc]

for each category-weld combination using equation (6). The results for these joint probabil-

ity distributions are given in Tables 28 and 29. As in Section 3, it is clear that the estimated

P[catj] values in Tables 28 and 29 are the same as the initial P[catj] in Tables 6 and 7

respectively.

We now take the joint probability distributions of Tables 28 and 29 and perform aggre-

gatiuns in order to understand the contributions of different svstemns and of different weld

DEGB sizes. First we aggregate the 45 welds to have 15 sets of the weld cases with different

DEGB sizes. The results are shown in Figures 10-11. WVe 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 of the six categories. This is because the weights used for the

current-day and end-of-plant-license estimates are the same-they are simply rescaled by

different probabilities for the categories from NUREG-1829 for current-day versus end-of-

24

Table 18: WVeld weights in total failure frequency for category 1 to category 3 using bottom-tip approach

Cat I Cat2 Cat3Frequency Weight Frequency Weight Frequency Weight

weldl 1.33E-06 0.37% 9.24E-08 0.58% 3.65E-08 0.71%weld2 1.78E-08 0.00% 1.23E-09 0.01% 2.09E-09 0.04%weld3 1.03E-08 0.00% 7.1SE-10 0.00% 1.21E-09 0.02%weld4 6.53E-06 1.82% 4.56E-07 2.85% 7.89E-07 15.35%weld5 5.09E-07 0.14% 3.83E-08 0.24% 4.87E-08 0.95%weld6 5.09E-07 0.14% 3.83E-08 0.24% 4.67E-08 0.95%weld7 2.82E-08 0.01% 2.12E-09 0.01%Y, 2.70E-09 0.05%5/Kweld8 5.64E-08 0.02% 4.24E-09 0.03% 5.40E-09 0.11%weld9 7.33E-06 2.04% 8.48E-07 5.29% 1.46E-06 28.44%

weldlO 3.91.E-07 0.11% 4.53E-08 0.28% 7.SOE-08 1.52%weldll 1.82E-07 0.05% 2.11E-08 0.13% 3.644E-08 0.71%weld12 3.35E-07 0.09% 3.85E-08 0.24% 7.23E-08 1.41%weldl3 1.14E-06 0.32% 1.13E-07 0.70% 7.22E-08 1.41%weldl4 5.49E-07 0.15% 5.44E-08 0.34% 3.86E-08 0.75%weldl5 7.79E-07 0.22% 7.71E-08 0.48% 5.46E-08 1.06%weldl6 5.88E-08 0.02% 5.82E-09 0.04% 4.12E-09 0.08%weldl7 4.20E-07 0.12% 4.22E-08 0.26% 2.70E-08 0.53%weldl8 0.00E+00 0.00% 0.60E+00 0.00% 0.OOE+00 0.00%weldl9 5.95E-08 0.02% 5.90E-09 0.04% 3.78E-09 0.07%weld20 2.94E-08 0.01% 2.91E-09 0.02% 2.06E-09 0.04%weld2l 7.85E-08 0.02% 1.33E-08 0.08% X x

weld22 0.OOE+00 0.00% 0.OOE+00 0.00% 0.OOE+00 0.00%weld23 1.68E-05 4.68% 2.ROE-06 17.45% X Xweld24 2.36E-04 65.81% X X X Xweld25 4.99E-05 13.88% 6.64E-06 41.43% 1.32E-06 25.67%weld26 2.14E-05 5.95% 2.85E-06 17.76% 5.65E-07 11.00%weld27 7.95E-06 2.21% 1.06E-06 6.60% 2.10E-07 4.09%weld28 9.09E-07 0.25% 1.21E-07 0.75% 2.40E-08 0.47%weld29 5.56E-07 0.15% 7.41E-08 0.46% 1.47E-08 0.29%weld30 2.93E-07 0.08% 3.90E-08 0.24% 7.74E-09 0.15(7,,weld3l 4.10E-07 0.11% 5.46E-08 0.34% 1.08E-08 0.21%weld32 2.24E-07 0.06% 2.99E-08 0.19% 5.93E-09 0.12%weld33 4.88E-S 0.01% 6.50E-09 0.04% 1.75E-09 0.03%weld34 8.78E-08 0.02% 1.17E-08 0.07% 3.14E-09 0.06%weld35 9.75E-08 0.03% 1.65E-08 0.10% X Xweld36 0.OOE+00 0.00% 0.OOE+00 0.00% X Xweld37 0.OOE+00 0.00% 0.OOE+00 0.00% 0.OOE+00 0.00%weld38 1.55E-06 0.43% 2.04E-07 1.27% 4.41E-08 0.86%weld39 8.03E-08 0.02% 1.05E-08 0.07% 2.28E-09 0.04%weld40 3.68E-07 0.10% 3.61E-08 0.23% 2.42E-08 0.47%weld4l 6.99E-07 0.19% 6.86E-08 0.43% 4.59E-08 0.89%weld42 7.56E-07 0.21.% 7.42E-08 0.46% 4.97E-08 0.97%weld43 9.65E-08 0.03% 9.47E-09 0.06% 6.34E-09 0.12%weld44 2.74E-07 0.08% 2.69E-08 0.17% I.SO-Os 0.35%weld45 1.61E-08 0.00% 1.58E-09 0.01% 1.06E-09 0.02%Total 3.59E-04 100.00% 1.60E-05 100.00% 5.14E-06 100.00V,.

25

Table 19: Weld weights in total failure frequency for category 4 to category 6 using bottom-up approach

Cat4 Cat5 Cat6Frequency Weight Frequency Weight Frequency Weight

weldl 0.OOE+00 0.00% 2.15E-08 5.43% 6.57E-09 14.67%weld2 0.0OE+00 0.00% 2.87E-10 0.07% 8.77E-11 0.20%weld3 0.OOE+f00 0.0)0% 1.67E-10 0.04% 5.11E-11 0.11%weld4 0.OOE+00 0.00% 1.01E-07 25.42% 3.32E-08 74.14%

weld5 0.OOE+00 0.00% 5.77E-09 1.46% 2.25E-09 5.02%weld6 0.OOE+00 0.00% 5.77E-09 1.46% 2.25E-09 5.02%weld7 0.OOE+00 0.00% 3.20E-10 0.08% 1.25E-10 0.28%weld8 0.OOE+00 0.00% 6.39E-10 0.16% 2.49E-10 0.56%weld9 0.0OE+00 0.00% 1.18E-07 29.93% X XweldlO 0.OOE+00 0.00% 6.32E-09 1.60% X Xweldll 0.OOE+00 0.00% 2.95E-09 0.74% X Xweldl2 X X N X X Xweld13 7.66E-09 1.17% X X X Xweld14 X X " X X Xweld15 X X X X X Xweldl6 X X X X X Xweldl7 2.87E-09 0.44% X X X Xweldl8 0.00E+00 0.00% X X X Xweldl9 4.01E-10 0.06% X X X Xweld20 X X X X X Xweld2l X X X X X Xweld22 0.00E+00 0.00% X X X Xweld23 X X X X X Xweld24 X X X X X Xweld25 3.42E-07 52.09% 1.25E-07 31.47% X Xweld26 2.OOE-07 30.47% N N N Xweld27 7.43E-08 11.33% X X X Xweld28 6.22E-09 0.95% 2.27E-09 0.57% X Xweld29 3.81E-09 0.58% 1.39E-09 0.35% X Xweld30 2.00E-09 0.31% 7.31E-10 0.18% X Xweld3l 3.83E-09 0.58% X X X Xweld32 2.10E-09 0.32% x X X Xweld33 X X X X x Xweld34 X X X X X Xweld35 x X X X X Xweld36 X X X X X Xweld37 0.OOE+1j0 0.00% 0.OOE+00 0.00% X Xweld38 1.06E-08 1.62% 3.88E-09 0.98% X Xweld39 5.50E-10 0.08% 2.OOE-10 0.05% X Xweld40 X X X X X Xweld4l X X x X x xweld42 X X X X X xweld43 X X X X X Xweld44 X X X X X Xweld45 X X X X X XTotal 6.56E1-07 100.00% 3.96E-07 100.00% 4.48E-08 100.00%

26

Figure 6: Weld weights in total failure frequency for the six categories

27

Table 20: Weights for weld 24 , weldc2.5, weld2 6 , weld9 , and weld4 in total failure frequency for

category 1 to category 6

Weight

Weld CaIc. Case System DEGB (in.) No. Welds Cati Cat2 Cat3 Cat4 Cat5 Cat624 6B Small Bore 1.41 193 65.81% X X X X X25 7A SIR 16.97 21 13.88% 41.43% 25 67% 52.09% 31.471 X26 7B SIR 11.31 9 5. 95% 17.76% 11.00% 30.47% X X

9 4A Surge Line 22.63 1 2.04% 5.29% 28.44% 0.00% 29.93% X4 21 SG Inlet 41.01 4 1.82%X 2.85-/o 1.5. 3'X 0.10('VU( )25 49% 74 147c

Total 89.5% 67.337 80.457 82.56% j682% 74.14/

Table 21: Top five weld weights in total failure frequency for category 1

Weld Calc. Case System DEGB (in.) No. Welds Weight24 6B Small Bore 1.41 193 65.81%25 7A SIR. 16.97 21 13.88%26 7B SIR 11.31 9 5.95%23 6A Small Bore 2.83 16 4.68%27 7C SIR. 11.31 3 2.21%

Total 92.53%


Weld Calc. Case System DEGB (in.) No. Welds Weight25 7A SIR. 16.97 21 41.43%26 7B SIR 11.31 9 T7.7623 6A Small Bore 2.83 16 17.45%27 7C SIR. 11.31 3 6.60%9 4A Surge Line 22.63 1 5.29%

Total 88.54%


Weld Calc. Case System DEGB (in.) No. Welds Weight9 4A Surge Line 22.63 1 28.44%25 7A SIR. 16.97 21 25.67%4 2 SG Inlet 41.01 4 15.35%

26 7B SIR 11.31 9 11.00%27 7C SIR 11.31 3 4.09%

Total 84.54%

28


Weld Calc. Case System DEGB (in.) No. Welds Weight25 7A SIR 16.97 21 52.09%26 7B SIR. 11.31 9 30.47%27 7C SIR 11.31 3 11.33%38 7N ACC 16.97 35 1.62%13 5A Pressurizer 8.49 29 1.17%

Total 96.68%


Weld Cale. Case System DEGB (in.) No. Welds Weight25 7A SIR. 16.97 21 31.47%9 4A Surge Line 22.63 1 29.93%4 2 SG Inlet 41.01 4 25.42%

1 1A Hot Leg 41.01 4 5.43%10 4B Surge Line 22.63 7 1.60%

Total 93.84%


Weld Calc. Case System DEGB (in.) No. Welds Weight4 2 SG Inlet 41.01 4 74.14%1 IA Hot Leg 41.01 4 14.67%5 3A Cold Leg 38.89 4 5.02%6 3B Cold Leg 43.84 4 5.02%8 3D Cold Leg 43.84 24 0.56%

Total 99.41%

29

Table 27: Aggregated weight by system and DEGB

WeightSystem DEGB (in.) No. Welds Catl Cat2 Cat3 Cat4 Cat5 Cat6

ACC 16.97 50 0.45% 1.34% C).90% 1.70% 1.03% XCVCS 3 57 0.31% 0.69% 1.44% X X XCVCS 5.66 30 0.30% 0.67% 1.38% X X X

Cold Leg 38.89 16 0.15% 0.25% 1.00% 0.00% 1.54% 5.30%Cold Leg 43.8 28 0.16% 0.27% 1.06% 0.00% 1.62% 5.58%Hot Leg 41.01 16 0.37% 0.59% 0.77% 0.00% 5.54, 14.98%

Pressurizer 2.83 2 0.02% 0.08% X X X XPressurizer 4.24 18 0.17% 0.38% 0.837 X X XPressurizer 5.66 55 0.23% 0.50% 1.10% X X XPressurizer 8.49 62 0.46% 1.00% 2.01% 1.67% X X

SG Inlet 41.01 4 1.82% 2.85% 15.35% 0.00% 25.42% 74.14%SIR 2 0 0.00% 0.00% X X X XSIR 2.83 10 0.03% 0.10% X X X XSIR 4.24 9 0.02% 0.07%1 0.06% X X XSIR 5.66 5 0.01% 0.04% 0.03% X X XSIR 8.49 23 0.06% 0.19% 0.127 0.32% X XSIR 11.31 54 8.27% 24.70% 15.30% 42.38% X XSIR 14.14 30 0.08% 0.24% 0.15% 0.31T 0.18% X

SIR 16.97 81 14.28% 26 26.43% 53.62% 32.39% XSmall Bore 1.41 193 65.81% X X X X XSmall Bore 2.83 16 4.68% 17.45% X X X XSurge Line 3.54 6 0.09% 0.24% 1.41% X X XSurge Line 22.63 10 2.20% 5.70% 30.67% 0.00% 32.27% X

30

.. 8 I99)169 .1 461 148 1• ý,0 849 .24 5.1 • 1 1 .1• 41.4 51 849 1ý• 14) - , 1 41 21 . 51 1 2

9200% ~ ~ ~ / 09 ~ 90~ (a) Catl 90 .0 0

99, -ý - ý 41 1 -ý 1 U1 . 1 14 . 1 -

45-8(2.

0982803

U - - - - - - - mE2k- 2~' 36 46 46 80 ~' 00~ *30 6 6

0 C2

c~ 350' 50 ~0' & 996 99 00'3. 3.0 09 090 09 80 40

40 5' 9' 9'

-1) -1 1 - 1~09

(b) Cat2

Figure 7: Aggregated weight by system and DEGB size for category 1 and category 2

31

DE4S (.,.)1697 1 5 66 1ý I9 4•1 411 2 t 4 1 1 1•B4 1U 2 83 4 24 5 66 4 1ý A A 1 i 1697 ýA 4 2 23 1 14 22-1

745454.4

I I I.-. E m . .mII li l ml

(a) Cat 3

8! 610 41 1 2 .1 ý 1 1- 11 1 14 11 1-7 L.41 2.- 1 04 228

1ý -

WOICI,4 40 00¾

I

I.."-.1

'P 44 44 444 44) ~40

(b) Cat4

Figure 8: Aggregated weight by system and DEGB size for category 3 and category 4

32

16,97 4.)4.47 24.44 48.84 446 42.04 284 424 564 846 4141 2 284 424 2.64 844 4244I 14.24 2442 4.424•86444 4244

I

(a) Cat5

16.97 1 14 1. .1 .1 2 424 544 844 4-. 1 1422 2.8 4.24 44 88 4. 14144 1697 A1.4 2.84 54.4 2263

W-ght 4G O

NW43

4 422

(b) Cat6

Figure 9: Aggregated weight by systemn and DEGB size for category 5 and category 6

33

Table 28: Distributed LOCA probabilities among 45 weld cases for current-day estimates

Catl Cat2 Cat3 Cat4 Cat5 Cat6weldl 3.18E-03 7.83E-04 3.48E-05 0.OOE+00 9.30E-07 2.79E-07

weld2 4.25E-05 1.04E-05 1.99E-06 0.OOE+00 1.24E-08 3.73E-09weld3 2.47E-05 6.08E-0(; 1LE-06 0.00E+00 7.23E-09 2.17E-09weld4 1.56E-02 3.87E-03 7.53E-04 0.O0E+00 4.36E-06 1.41E-06weld5 1.22E-03 3.25E-04 4.65E-05 0.00E+00 2.50E-07 9.57E-08weld6 1.22E-03 3.25E-04 4.65E-05 0.00E+(0 2.50E-07 9.57E-OSweld7 6.75E-05 1S.E-05 2.58E-06 0.OOE+00 1.38E-08 5.30E-09weld8 1.35E-04 3.60E-05 5.15E-06 0.00E+00 2.77E-08 1.06E-08

weld9 1.75E-02 7.19E-03 1.39E-o3 O.OOE+00 5.13E-06 XweldlO 9.35E-04 3.84E-04 7.45E-05 O.OOE+00 2.74E-07 N

weldll 4.36E-04 1.79E-04 3.47E-05 O.OOE+00 1.28E-07 Xweldl2 8.02E-04 3.29E-04 6.90E-05 X X Xweldl3 2.72E-03 9.55E-04 6.89E-05 5.53E-06 X Xweld14 1.31E-03 4.61E-04 3.68E-05 X X Xweldl5 1.66E-03 6.54E-04 5.22E-f)5 X X Xweld16 1.40E-04 4.93E-05 3.94E-06 X X Xweldl7 1.02E-f:3 3.58E-04 2.58E-05 2.07E-06 X Xweldl8 0.0U0E+00 0.OOE+±O O.OOE+00 O.UOE+00 X Xweldl9 1.42E-04 5.0E-05 - 3.61E-06 2.89E-07 X Xweld20 7.02E-05 2.47E-05 1.97E-06 X X Xweld2l 1.88E-04 1.113E-04 X X X Xweld22 O.OOE+00 O.OOE+00 O.0OE+00 O.OOE+00 X Xweld23 4.02E-02 2.37E-02 X X X Xweld24 5.65E-01 X X X X Xweld25 1.19E-01 5.63E-02 1.26E-03 2.46E-04 5.39E-06 Xweld26 5.11E-02 2.41E-02 5.40E-04 1.44E-04 x Xweld27 1.9OE-02 8.97E-03 2.01E-04 5.36E-05 X Xweld28 2.17E-03 1.03E-03 2.29E-05 4.49E-06 9.83E-08 Xweld29 1.33E-03 6.28E-04 1.40E-05 2.75E-06 6.01E-08 Xweld30 7.O0E-04 3.30E-04 7.39E-06 1.45E-06 3.17E-08 Xweld3l 9.79E-04 4.63E-04 1.03E-05 2.76E-06 X Xweld32 5.36E-04 2.53E-04 5.66E-06 1.51E-06 X Xweld33 1.17E-04 5.51E-05 1.67E-06 X X Xweld34 2.10E-04 9.91E-05 3.OOE-06 X X Xweld35 2.33E-04 1.40E-04 X X X Xweld36 O.0OE+00 0.0OE+00 X X X Xweld37 0.OOE+00 O.OOE+00 O.OOE+00 O.00E+00 O.OOE+00 Xweld38 3.71E-03 1.73E-03 4.21E-05 7.67E-06 1.68E-07 Xweld39 1.92E-04 8.93E-05 2.18E-06 3.97E-07 8.68E-09 Xweld40 8.80E-04 3.06E-04 2.31E-05 X X xweld4l 1.67E-03 5.81E-04 4.39E-05 X X X

weld42 i.S1E-03 6.29E-04 4.74E-05 X X Xweld43 2.31E-04 8.03E-05 6.06E-06 X X Xweld44 6.56E-04 2.28E-04 1.72E-05 X X X

weld45 3.85E-05 1.34E-05 l.OlE-06 X X XEstimated P[catj] 8.59E-01 1.36E-01 4.90E-03 4.73E-04 1.71E-05 1.90E-06

34

Table 29: Distributed LOCA probabilities among 45 weld cases for end-of-plant-license esti-mates

Cat1 Cat2 Cat3 Cat4 Cat5 Cat6

weldl 3.09E-03 8.99E-04 6.84E-05 O.OOE+00 1.89E-06 5.91E-07weld2 4.12E-05 1.20E-05 3.92E-06 O.OOE+00 2.53E-08 7.89E-09weld3 2.40E-05 6.99E-06 2.28E-06 O.OOE+00 1.47E-08 4.59E-09weld4 1.52E-02 4.44E-03 1.48E-03 0.00E+-00 8.S6E-06 2.99E-06weld5 1.18E-03 3.73E-04 9.14E-05 0.00E+00 5.08E-07 2.02E-07weld6 1.18E-03 3.73E-04 9.14E-05 O.OOE+00 5.08E-07 2.02E-07weld7 6.55E-05 2.07E-05 5.06E-06 O.OOE+00 2.82E-08 1.12E-08weld8 1.31E-04 4.13E-05 1.01E-05 O.OOE+00 5.63E-08 2.24E-08weld9 1.70E-02 8.26E-03 2.74E-03 0.OOE+00 1.04E-05 X

weldlO 9.08E-04 4.41E-04 1.46E-04 O.OOE+00 5.57E-07 Xweldll 4.23E-04 2.06E-04 6.82E-05 O.OOE+00 2.60E-07 XweIdl2 7.78E-04 3.78E-04 1.36E-04 X X Xweldl3 2.64E-03 1.IE-03 1.35E-04 1.03E-05 X Xweldl4 1.27E-03 5.30E-04 7.23E-05 x X Xweldl5 1.81E-03 7.51E-04 1.03E-04 X X Xweldl6 1.36E-04 5.67E-05 7.74E-06 X X Xweldl7 9.SSE-04 4.11E-04 5.07E-05 3.84E-06 X XweIdl8 0.O0E+O O.OOE+00 O.O0E+00 O.OOE+00 X Xweldl9 1.38E-04 5.74E-05 7.09E-06 5.36E-07 X Xweld20 6.82E-05 2.83E-05 3.87E-06 X X Xweld2l 1.82E-04 1.29E-04 X X X Xweld22 O.OOE+00 O.OOE+00 O.OOE+00 O.OOE+00 X Xweld23 3.90E-02 2.72E-02 X X X Xweld24 5.48E-01 X X X X Xweld25 1.16E-01 6.47E-02 2.47E-03 4.57E-04 1.1OE-05 Xweld26 4.96E-02 2.77E-02 1.06E-03 2.67E-04 X Xweld27 1.84E-02 1.03E-02 3.94E-04 9.95E-05 X Xweld28 2.11E-03 1.18E-03 4.51E-05 8.33E-06 2.OOE-07 Xweld29 1.29E-03 7.21E-04 2.76E-05 5.10E-06 1.22E-07 Xweld30 6.79E-04 3.80E-04 1.45E-05 2.68E-06 6.44E-08 Xweld3l 9.50E-04 5.31E-04 2.03E-05 5.13E-06 X Xweld32 5.20E-04 2.91E-04 1.11E-05 2.81E-06 X Xweld33 1.13E-04 6.33E-05 3.27E-06 X X Xweld34 2.04E-04 1.14E-04 5.89E-06 X X Xweld35 2.26E-04 1.61E-04 X X X Xweld36 O.OOE+00 O.OOE+00 X X X Xweld37 O.OOE+00 O.O0E+±O O.OOE+00 O.OOE+00 0.0()E+00 Xweld38 3.60E-03 1.9:E-03 8.27E-05 1.42E-05 3.41E-07 Xweld39 1.86E-04 1.03E-04 4.28E-06 7.36E-07 1.77E-08 Xweld40 8.54E-04 3.52E-04 4.54E-05 X X Xweld4l 1.62E-03 6.68E-04 8.62E-05 X X Xweld42 1.75E-03 7.22E-04 9.32E-05 X X Xweld43 2.24E-04 9.22E-05 1.19E-05 X X Xweld44 6.36E-04 2.62E-04 3.38E-05 X X Xweld45 3.73E-05 1.54E-05 1.98E-06 X X X

Estimated P[catjI 8.33E-01 1.56E-01 9.64E-03 8.78E-04 3.49E-05 4.03E-06

35

plant-license. The welds with DEGB size 1.41 inches have a significantly higher aggregated

LOCA probability (more than 0.5 which is more than half of the estimated P[catl]) than

those with larger DEGB sizes for category 1. From the figures, the shape of the aggregated

LOCA probabilities for category 2 is roughly a rescaling of that for category 1 with DEGB

size of at least 1.625 inches. Also, the shape of the aggregated LOCA probabilities for cat-

egory 3 is (roughly) a rescaling of that for category 2 with DEGB size of at least 3 inches

except that the welds with DEGB size 22.63 inches now have a relatively larger portion of

the aggregated LOCA probability. Category 4 is a bit unusual in that there are relatively

fewer weld cases in Fleming et al. (2011) with frequencies that fall in the category 4 bin.

This is reflected in the figures in that welds with DEGB size of at least 22.62 inche-s have

zero contribution in category 4. Comparing the aggregated LOCA probabilities for category

5 to that for category 3, they roughly remain the same shape with DEGB sizes below 14

inches are now zero, and the welds with DEGB sizes 16.97, 22.62, and 41.01 inches have

closer and a relative high proportion of the aggregated LOCA probabilities. Similarly, the

shape of the aggregated LOCA probabilities for category 6 is a rescaling of that for category

5 with DEGB sizes of at least 31 inches.

We now consider an aggregation of the 45 weld cases by system type rather than by

DEGB size. This yields nine sets of welds with different system types. The results are

shown in Figures 12-13. For the same reason that we discuss above, 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 can see that Small Bore welds have a

significantly higher aggregated LOCA probability (more than 0.5 which is more than half of

the estimated P[cati]) than other sets of the welds for category 1. Moving from category 1 to

category 2, the Small Bore welds having a DEGB size of at least 1.625 inches have relatively

small aggregated LOCA probability. For category 3, the SIR and Surge Line systems are the

largest contributors followed by SG Inlet and the Pressurizers. The sparsity of observations

from Fleming et al. (2011) for category 4 is again reflected in the figures, with the SIR.

system dominating. Comparing the aggregated LOCA probabilities for category 5 to those

for category 3, we again see that SIR and Surge Line systems are the largest contributors

but the SG Inlet now has a larger relative contribution, followed by the Hot Leg and Cold

36

to•

0 2

01

15

ý23

05

°

03

I -1 . L I ý ý141 2 2.83 3 3.54 4.24 5.66 8.49 1131 14.1416.9722.63388941.01 43.8

DEGB (iW)

(a) Call (b) Cat,2

.24 5.66 849 1131 14141697226338.894101 43.8 141 2 2,83 3 3.54 424 566 849 11.31 14,14 16,97 22.63 38.894101 41.80EGBf in) DEGB (

(c) Cat3 (d) Cat4

1.0

1A

1,2

!10•8

0.6

04

02

I(2. l141 2 2.83 3 3.54 4.24 566 8.4 1131 14.4 16.97 22,633889 41.01 438

De3GB (in.)

(e) Cat5

1.41 2 2.83 3 3.54 4.24 5.66 8.49 11.31 14.14 16.97 22.63 38.89 41.01 43.8DEGB (in)

(f) Cat6

Figure 10: Aggregated distributed LOCA probability by DEGB size for each category forcurrent-day estimates 37

04-

03

14 2 z, 3 35 4z45,•, B9111,1 14ý14 1§9•371•91] -3 I4 n II• -t j 1 - • - .4" 1D E G B (in ') 4E G6 On.;

(a) CatI (b) Cat2

1.

0'

01

DEGB O14 04061(i41

(c) Ca~t3 (d) Cat.4

35

3

2.5

2"

1.5

0.5

I1. ..41 2 2 .5 .2 5.4 6.451.31.1 4.72.43691.1 4.m141 2 283 3 354 4.24 566 8.49 11.31 141416,9722,63 38,8940,31 438

DEGB (in.

(e) Cat5

1 Al 2 2.83 3 3 54 4.24 5.66 8,49 11.31 14,1416.97 22,6338 8941,01 418DEGB (in)

(f) Cat6

Figure 11: Aggregated distributed LOCA probabilityend-of-plant-license estimates 38

by DEGB size for each category for

Leg. The aggregated probabilities for category 6 are dominated by the SG Inlet, followed

by the Hot Leg and Cold Leg. While SIR. welds have the largest contributions for categories

2-5, SIR does not contribute to category 6. Furthermore, SG Inlet welds have relatively high

aggregated LOCA probabilities for category 5 and category 6. Also, Surge Line welds have

relatively high aggregated LOCA probabilities for categories 3 and 5.

Note that if we aggregate the joint probability distributions over the 45 welds to have

23 unique sets of the welds with different combinations of system types and DEGB sizes,

we obtain a set of plots for aggregated LOCA probabilities of the same shape but different

scale as that in Figures 7-9 with the scale being the estimated P[catj] for j = 1, 2, .. , 6 (see

equation (6)). Finally, we simulate break sizes for each weld within each category as the

procedure stated in Section 2.3. A sample is shown in Table 30.

39

0

01

007

0 06

005I

003

001F

ACC CVCS Cold Leg Hot Leg P=eSunGee SO Slt SIR Small Bote-Slge UneSyst( e

(b) Cat2(a) CatI

(c) Cat3 (d) Cat4

System System

(e) Cat5 (f) Cat6

Figure 12: Aggregated distributed LOCA probability by system for each category for current-day estimates

40

(a) Catl (b) Cat2

(c) Cat3 (d) Cat4

Sy.-m ySye

(e) Cat5 (f) Cat6

Figure 13: Aggregated distributed LOCA probability by system for each category for end-of-plant-license estimates

41

Table 30: Sampled break sizes (inches) for the 45 welds within each break category

Weld Catl Cat2 Cat3 Cat4 Cat5 Cat6

1 0.89 1.91 4.10 9.93 28.35 38.97

2 1.46 2.98 3.65 12.55 22.89 40.033 1.25 2.61 5.69 9.45 24.08 31.36

4 0.94 2.84 5.88 8.95 28.38 40.45

5 1.37 2.81 4.65 9.84 22.49 37.96

6 1.38 2.20 4.29 8.63 20.58 31.49

7 0.67 2.99 5.26 9.59 14.19 32.838 1.50 2.22 4.19 9.36 15.49 36.77

9 1.22 2.15 4.16 11.50 19.97 X

10 1.30 2.42 6.33 11.35 14.48 X

11 1.55 2.79 6.52 11.22 19.51 X

12 0.51 2.63 3.22 X X X

13 0.52 2.60 4.35 8.26 X X

14 1.17 2.17 3.90 X X X

15 1.11 2.50 3.12 X X X16 1.45 2.32 4.19 X x x

17 1.44 1.95 6.52 7.81 X X18 1.62 1.94 3.17 7.51 X X

19 0.87 2.26 4.22 8.26 X X20 1.49 2.67 4.59 X X x21 1.16 2.67 x X X X22 1.12 2.98 4.61 8.26 X X

23 1.16 2.74 X X X X24 0.57 X X X X N25 0.72 1.66 5.71 8.94 16.93 X

26 0.95 2.82 6.36 9.93 X X

27 1.37 1.68 3.55 11.19 X X

28 1.09 2.53 5.99 12.57 14.21 X29 0.92 2.51 5.34 11.48 16.77 x30 0.56 1.68 6.38 11.89 14.10 X

31 1.12 1.74 3.86 8.45 X X32 1.15 2.17 4.42 8.22 X X33 0.64 2.71 4.65 X X X

34 0.90 2.91 3.38 X X X

35 0.60 2.58 X X X X:36 1.18 2.10 X X X X37 0.96 2.74 4.47 13.96 15.29 X38 0.86 2.53 4.36 11.27 14.03 X

39 1.13 2.02 3.78 12.65 15.63 X

40 0.79 2.36 X X X x41 0.93 2.94 5.36 X X X

42 1.30 2.55 X X X X

43 1.43 1.81 5.07 X X X

44 1.26 2.14 5.23 X X X

45 0.57 1.71 .5.04 X X X

42

5 Fitting the Johnson Distribution for PRA Break Sizes

The PRA for South Texas Project uses three break-size intervals corresponding to intervals

of 0.5-2 inches, 2-6 inches, and 6 inches up to the largest DEGB in the system. (Break sizes

smaller than 0.5 inches are excluded in the PRA.) Thus the break sizes used in the PRA do

not align with the break sizes associated with NUREG-1829. In order to sample initiating

frequencies for the PRA, we require fits of the Johnson family to both the 2-inch LOCA

frequencies and to the 6-inch break LOCA frequencies. To do so, we interpolate the 5%,

50%, and 95% percentiles associated with the LOCA frequencies between category 2 (1-5

inches) and category 3 (3 inches) from NUREG-1829 (Table 7.19) to obtain the interpolated

2-inch break LOCA frequencies. Similarly, we interpolate the LOCA frequencies between

category 3 (3 inches) and category 4 (7 inches) to obtain the interpolated 6-inch break LOCA

frequencies. The results are reported in Table 31. As we can see from Tables 3 and 5, the

frequencies drop off more quickly than linearly as the break size grows. Hence, the linear

interpolation is conservative in that it overestimates the frequencies associated with 2-inch

and 6-inch break sizes that would have presumably been elicited, had the expert elicitation

of NUREG-1829 been done at the 2-inch and 6-inch sizes.

We fit the bounded Johnson distribution as stated in Section 2.1. The fitted parameters

of the bounded Johnson distribution for the 2-inches and 6-inches break for the current-

day and end-of-plant-license estimates are reported in Tables 32 and 34, respectively. The

comparison between the interpolated distributional characteristics of the LOCA frequencies

and the fitted ones for the current-day and end-of-plant-license estimates are presented in

Tables 33 and 35, respectively. Again, the NUREG-1829 expert elicitation was for the 5%,

50% (median), and 95% quantiles, and did not involve eliciting the mean. So we focus on

matching the three distributional characteristics elicited from the experts as indicated by the

results in the corresponding right-most columns of Tables 33 and 35. We also note that the

fitted values of the Johnson parameters in Tables 32 and 34 are consistent with the trends

in Tables 2 and 4 for the NUREG-1829 categories. Figure 14 shows the fitted PDFs of the

bounded Johnson for both the current-day and end-of-plant-license estimates denoted by

"Current" and "End" for the 2-inch and 6-inch break sizes.

43

Table 31: Interpolated mean, median, low and high quantiles values

Current-Day Estimate End-of-Plant-License Estimate(per cal. year) (per cal. year)

5th Median Mean 95th 5th Median Mean 95th2-inches 3.69E-06 6.57E-05 3.10E-04 1.18E-03 4.57E-06 8.93E-05 4.32E-04 1.64E-036-inches 6.30E-08 1.08E-06 5.20E-06 1.98E-05 1.41E-07 2.40E-06 1.17E-05 4.55E-05

Table 32: Fitted Johnson parameters for current-day estimates

Johnson Parameters-Yj A

2-inches 1.646308E+00 4.593851E-01 1.870000E-06 2.360550E-036-inches 1.646403E+00 4.566256E-01 3.OOOOOOE-08 3.965000E-05

Table 33: Interpolated values and fitted Johnson mean, median, 5% and 95% quantiles valuesfor current-day estimates

I Interpolated Values Fitted Johnson Relative Error5th IMediani Mean 95th 5th Medianl Mean 95th 5th IMedian Mean 95th

2-inches 3.69E-06 6.57E-05 3.1OE-04 1.18E-03 3.69E-06 6.57E-05 2.37E-04 1.18E-:3 0.00% 0.00% 23.49% 0.00%6-inches 6.30E-08 1.08E-06 5.20E-06 1.98E-05 6.30E-08 1.0SE-06 3.96E-06 1.98E-05 0.00% 0.00% 23.86% 0.00%

Table 34: Fitted Johnson parameters for end-of-plant-license estimates

Johnson Parameters

2-inches 1.646032E+00 4.573699E-01 2.110000E-06 3.276360E-036-inches 1.64641-7E+00 4.516761E-01 8.OOOOOOE-08 9.100000E-05

Table 35: Interpolated values and fitted Johnson mean, median, 5% and 95W, quantiles valuesfor end-of-plant-license estimates

Interpolated Values Fitted Johnson Relative Error

5th Median Mean 95th 5th Median Mean 95th 5th MedianI Mean 95th2-inches 4.57E-06 8.93E-05 4.32E-04 1.64E-03 4.57E-06 8.93E-05 3.27E-04 1.64E-03 0.00% 0.00% 24.15-A 0.00%6-inches 1.41E-07 2.40E-06 1.17E-05 4.55E-05 1.41E-07 2.40E-06 9.01E-06 4.55E-05 0.00% 0.00% 22.95% 0.00%

44

0

CurrentI- End

o 0

0,0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 Oe+00 le-05 2e-05 3e-05 4e-05 5e-05

(a,) 2-inches (b) 2-inchies (zoomed)

- CurrentEnd

00_

I

Oe'-00 2e-05 4e-05 6e-05 8e-05 Oe+00 2e-07 4e-07 6e-07 8e-07 le-0S

(c) 6-incihes (d) 6-inches (zoomed)

Figure 14: Johnson PDF for 2-inch and 6-inch break sizes, and each is zoomed to a narrowerrange of frequencies near the mode of the distribution

45

Conclusion

In this report we present solutions to three problems:

1. How should we preserve the NUREG-1829 LOCA frequencies when distributing them

across different locations (welds) in a. nuclear power plant. The approach that we

propose to take is rooted in combining the top-down and bottom-up approaches: \'Ve

start with the NUREG-1829 frequencies and develop a way to distribute them to

different locations proportionally 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.

2. The six break size categories (from the NUREG-1829 Table 7.19) represent six intervals.

For a particular weld we need to be able to sample from the continuous interval of break

size values. We propose to use linear interpolation which is equivalent to assigning

equally likely probabilities (more specifically, a uniform distribution) within each break

size category.

3. How to model the distribution of the LOCA frequencies. We propose to fit the bounded

Johnson distribution to the NUREG-1829 quantiles of 5%, 50%, and 95%, minimizing

the sum of squared deviations from these elicited percentiles.

Acknowledgements

We thank Alexander Galenko for work on an early version of this report.

46

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. 0. Lydell., and D. Chrun (2011, July). Development of LOCA Initiating

Event Frequencies for South Texas Project GSI-191. Technical Report, KnF Consulting

Services, LLC, Spokane, 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 Estimation Method-

ology. Letter Report Revision 0, University of Maryland, College Park, MA.

Tregoning, R.., P. Scott, and A. Csontos (2008, April). Estimating Loss-of-Coolant Acci-

(lent (LOCA) Frequencies Through the Elicitation Process: Main Report (NUREG-1829).

NUREG 1829, NRC, Washington, DC.

47

Documents

South Texas Project, Units 1 & 2, Response to NRC Request for … · 2014. 1. 24. · ,6 and where Freqvency[LOCA > cat7] =0 . Again we assume there are Mj different locations in