ML120540570

Uncertainty Modeling of LOCA Frequencies and Break Size Distributions for the STP GSI-191 Resolution

ML120540570
Person / Time
Site:	South Texas
Issue date:	02/22/2012
From:	Galenko A, Popova E University of Texas at Austin
To:	Plant Licensing Branch IV
Singal, B K, NRR/DORL, 301-415-301
Shared Package
ML120540492	List: ML120540509 ML120540557 ML120540570 ML120540604 ML120540610 ML120540667 ML120540694 ML120540727 ML120540758 ML120540784 ML120540816
References
TAC ME7735, GSI-191, TAC ME7736
Download: ML120540570 (28)

Text

Uncertainty modeling of LOCA frequencies and break size distributions for the STP GSI-191 resolution Elmira Popova, Alexander Galenko The University of Texas at Austin

Background

• The Pilot Project for the STP GSI-191 resolution started in January 2011. One of the distinguishing characteristics of the proposed solution approach is the inclusion of uncertainties. Therefore, their proper modeling is required required.

• Major input to the Pilot Project are the LOCA frequencies.

• The initial quantification used models and results from the EPRI risk-informed in service inspection program.

Background

• The Pilot Project is an unique combination of existing PRA infrastructure and Casa Grande, a new simulator that models and executes events, physics, statistical sampling, i.e. performs everything that the existing PRA cant.

• Both of them have the same input - LOCA frequencies.

• The LOCA frequencies used in PRA are not location-specific, whereas Casa Grande needs them to be location-specific.

• In this presentation, location will be equivalent to weld locations inside STP.

NUREG-1829

• The expert elicitation study in NUREG-1829 gives the LOCA frequencies for 6 break size categories. Bellow is Table 1 from that study.

Current-day Estimate (per cal. yr)

LOCA Size Eff. Break Plant Type (25 yr fleet average operation)

(gpm) Size (inch) 5th Per. Median Mean 95th Per.

>100 1/2 6.90E-04 3.90E-03 7.30E-03 2.30E-02

>1,500 1 5/8 7.60E-06 1.40E-04 6.40E-04 2.40E-03

>5,000 3 2.10E-07 3.40E-06 1.60E-05 6.10E-05 PWR

>25K 7 1.40E-08 3.10E-07 1.60E-06 6.10E-06

>100K 14 4.10E-10 1.20E-08 2.00E-07 5.80E-07

>500K 31 3.50E-11 1.20E-09 2.90E-08 8.10E-08

NUREG-1829 results and our major challenges

• The LOCA frequencies are not plant specific or plant-location specific. In our analysis we need to initiate a break at a random location (weld) inside STP. (Call this Problem 1)

• The six break size categories (columns in the table) are ranges bounded by six discrete points.

points For a particular weld we need to be able to sample from the continuous range of break size values. (Call this Problem 2)

• The LOCA table provides four distributional characteristics (rows in the table) for each break size category: mean, median, 5th and 95th percentiles. We would like to sample from distributions that match these values. (Call this Problem 3)

Main objective

• In this communication we will discuss possible solutions to each of the three problems. We consider Problem 1 to be the critical one and we hope to find a mutual agreeable path path.

• Main assumption - we conserve the values from NUREG-1829 and use them as input to both Case Grande and PRA analysis.

Proposed solution to Problem 1

• The LOCA frequencies are not plant specific or plant-location specific. In our analysis we need to initiate a break at a random location (weld) inside STP.

• The solution to Problem 1 in the initial quantification was a bottom-up approach: model the uncertainties at each weld location location, and accumulate them to get the total LOCA frequency. By following this path one cannot hope to get the NUREG-1829 numbers as the resulting total cumulative frequencies.

• Our current methodology is top-down: we start with the NUREG-1829 numbers and distribute them to all weld locations. This way the conservation of the LOCA frequencies is guaranteed.

Proposed solution to Problem 1

• The distribution of the LOCA numbers will be equally likely.

• Small breaks are equally likely to occur in small, medium, or large welds; medium breaks are equally likely to occur in medium and large welds; large breaks can occur only in large welds.

• Each location (weld) is equally likely to be chosen.

• These assumptions and the law of total probability leads to the preservation of the total LOCA frequencies.

Proposed solution to Problem 2

• The six break size categories (columns in the NUREG-1829 table) are ranges bounded by six discrete points. For a particular weld we need to be able to sample from the continuous range of break size values.

values

• We will connect the bounding points with straight lines.

• This implies that the values within each range are equally likely. Hence we are modeling each break size as Uniform distribution over the range.

Resulting Break Size Cumulative Distribution Function for weld 6 from the Illustrative Example Weld6 1.01 1

0.99 CDF(Break Size) 0.98 0.97 0.96 0.95 0 5 10 15 20 25 30 35 40 Break Size

Proposed solution to Problem 3

• The LOCA table provides four distributional characteristics (rows in the table) for each break size category: mean, median, 5th and 95th percentiles. We would like to sample from distributions that match these values.

• NUREG-1829 used two split p Lognormal g distributions.

• NUREG/CR 6828 used Gamma distribution.

• There are infinite number of distributions one can potentially fit.

• If we want to match the four characteristics as close as possible we need a parametric distribution with four parameters.

• The bounded Johnson distribution fits this requirement. Many different shapes: skewed, symmetric, bimodal, unimodal are all possible.

Johnson distribution

• CDF(x) of Bounded Johnson Distribution F[x]={ + f [(x ) / ]},

[x]cdf of a standard Normal(0,1) random variable

, - shape parameters

- location parameter (left bound)

- scale parameter

+ right bound

Description of the estimation procedure

• For each break size category we solve a nonlinear optimization problem.

• The objective function is the weighted squared error.

error

• Have four constraints that correspond to matching each of the NUREG-1829 characteristics: mean, median, 5th, and 95th percentiles.

Estimated Parameters of Johnson Distribution Johnson Parameters gamma delta xi lambda Category1 0.7288246 0.3893326 0.000634494 0.02449228 Category2 6.95E-01 2.40E-01 7.41E-06 2.44E-03 Category3 7.24E-01 2.44E-01 2.06E-07 6.24E-05 Category4 7.14E-01 2.39E-01 1.36E-08 6.19E-06 Category5 4.73E-01 2.69E-01 1.87E-10 5.93E-07 Category6 4.75E-01 2.73E-01 1.77E-14 8.52E-08

Comparison with NUREG-1829 NUREG-1829 Fitted Johnson Error 5% Mean 95% 5% Mean 95% 5% Mean 95%

Category1 6.90E-04 7.30E-03 2.30E-02 6.89E-04 7.30E-03 2.30E-02 0.08% 0.01% 0.00%

Category2 7.60E-06 6.40E-04 2.40E-03 7.56E-06 6.42E-04 2.40E-03 0.59% 0.38% 0.04%

Category3 2.10E-07 1.60E-05 6.10E-05 2.09E-07 1.60E-05 6.12E-05 0.40% 0.23% 0.38%

Category4 1.40E-08 1.60E-06 6.10E-06 1.40E-08 1.59E-06 6.08E-06 0.26% 0.65% 0.39%

Category5 4.10E-10 2.00E-07 5.80E-07 4.14E-10 1.98E-07 5.86E-07 0.94% 0.94% 0.98%

Category6 3.50E-11 2.90E-08 8.10E-08 3.59E-11 2.84E-08 8.41E-08 2.60% 2.03% 3.78%

Methodology summary

1. Set N - number of LOCA frequency samples and S - number of break size samples to generate.

2. Sample LOCA frequencies from the fitted Johnson distributions (or different distributions).

3. Distribute frequency across plant specific welds (equally likely or based on predefined weights).

4

4. Sample actual break size for each possible weld / break category combination using Uniform distribution (or a different distribution).

5. Estimate performance measures, store them.

6. If we ran S break sizes samples go to the next step, otherwise go to step 4.

7. Compute the performance measures summary, store them.

8. If we ran N LOCA frequencies samples go to the next step, otherwise, go to step 2.

9. Make aggregated performance measures summary.

Illustrative example - first four steps, N=1, S=1

• Welds:

- Total number of welds: 6

- Small welds (2.5 inches): 3

- Medium welds (4 inches): 2

- Large welds (35 inches): 1

• Failure categories:

- Total number of failure categories: 6

- Small welds: category1 and category2

- Medium welds: category1, category2 and category3

- Large welds: category1, category2, category3, category4, category5 and category6

Example: System Description Example: Failure Categories BreakSize(inches)

Category1 0.5"-1.625" Category2 1.625"-3" Category3 3"-7" Category4 7"-14" Category5 14"-31" Category6 31"-41"

Example: Global Failure frequencies NUREG-1829 5% quantile Mean 95% quantile Category1 6.90E-04 7.30E-03 2.30E-02 Category2 7.60E-06 6.40E-04 2.40E-03 Category3 2.10E-07 1.60E-05 6.10E-05 Category4 1.40E-08 1.60E-06 6.10E-06 Category5 4.10E-10 2.00E-07 5.80E-07 Category6 3.50E-11 2.90E-08 8.10E-08

Example: Global Failure Probabilities Sampled Frequencies Probabilities 3.90E-03 9.64E-01 1.40E-04 3.46E-02 3 40E 06 3.40E-06 8 41E 04 8.41E-04 3.10E-07 7.67E-05 1.20E-08 2.97E-06 1.20E-09 2.97E-07 P[cat j ]= Frequency[cat j ] / Frequency[catl ]

lJ

Example: Global Failure Probabilities

• For example, the probability that a failure will fall into category 3 is given by 3.40E-6 P[catcategory3 ]= =

3.90E-3+1.40E-4 + 3.40E-6 + 3.10E-7 +1.20E-8 +1.20E-9

= 8.41E-4

Example: Local Failure Probabilities Weld weld1 weld2 weld3 weld4 weld5 weld6 Actual Target Category1 1.61E-01 1.61E-01 1.61E-01 1.61E-01 1.61E-01 1.61E-01 9.64E-01 9.64E-01 Category2 5.77E-03 5.77E-03 5.77E-03 5.77E-03 5.77E-03 5.77E-03 3.46E-02 3.46E-02 Category3 2.80E-04 2.80E-04 2.80E-04 8.41E-04 8.41E-04 Category4 7.67E-05 7.67E-05 7.67E-05 Category5 2.97E-06 2.97E-06 2.97E-06 Category6 2.97E-07 2.97E-07 2.97E-07 P[cat j at locationi ]= P[cat j ] / M j

Example: Local Failure Probabilities

• For example, the probability that weld4 will experience a break of category 3 is given by M category3 = 3 P[catcategory3 ] =8.41E-4 8.41E-4 P[catcategory3 at locationweld 4 ]= P[catcategory3 ] / M category3 = = 2.80E-04 3

Example: Break Size Distribution Weld Parameters Category1 Category2 Category3 Category4 Category5 Category6 a 0.5 1.625 weld1 b 1.625 2.5 a 0.5 1.625 weld2 b 1.625 2.5 a 0.5 1.625 weld3 b 1.625 2.5 a 0.5 1.625 3 weld4 b 1.625 3 4 a 0.5 1.625 3 weld5 b 1.625 3 4 a 0.5 1.625 3 7 14 31 weld6 b 1.625 3 7 14 31 35 breakSizeij ~Uniform[min Break ij , max Break ij ]

min Break ij = cat min j

Break max Break ij = min[cat max j Break

, weldi size

]

Example: Break Size Distribution

• For example, the distribution that covers break size for weld4 in category 3 is given by well 4 min i Break B k min Breakcategory3 = catcategory3 =3 weld 4 max Break size max Breakcategory3 = min[catcategory3 , weldweld 3 ] = min[7, 4] = 4 weld 4 weld 4 weld 4 breakSizecategory3 ~Uniform[min Breakcategory3 max Breakcategory3 ] ~ Uniform[3, 4]

Example: Sampled Break Sizes Weld weld1 weld2 weld3 weld4 weld5 weld6 Category1 1.1 0.6 0.87 1.34 0.79 1.23 g y Category2 2.4 1.9 2.1 2.9 1.75 2.36 Category3 3.56 3.14 5.97 Category4 11.67 Category5 25.68 Category6 32.67

Presenters

• Elmira Popova, PhD, Professor, Robert and Jane Mitchell Endowed Faculty Fellowship in Engineering, Department of Mechanical Engineering, g g Operations p Research and Industrial Engineering, The University of Texas at Austin

• Alexander Galenko, PhD, Research Associate, Department of Mechanical Engineering, Operations Research and Industrial Engineering, The University of Texas at Austin