NUREG/IA-0428 

International 

 Agreement Report

Performing Uncertainty Analysis of IIST

Facility SBLOCA by TRACE and DAKOTA 

Prepared by:

Jong-Rong Wang, Chiung-Wen Tsai*, Hao-Tzu Lin, Chunkuan Shih*

Institute of Nuclear Energy Research, Atomic Energy Council, R.O.C.

1000, Wenhua Rd., Chiaan Village, Lungtan, Taoyuan, 325, TAIWAN

 

*Institute of Nuclear Engineering and Science, National Tsing Hua University,

101 Section 2, Kuang Fu Rd., Hsinchu, TAIWAN

K. Tien, NRC Project Manager

 Division of Systems Analysis

Office of Nuclear Regulatory Research

U.S. Nuclear Regulatory Commission

Washington, DC 20555-0001

Manuscript Completed: March 2013

Date Published: September 2013

Prepared as part of 

The Agreement on Research Participation and Technical Exchange

Under the Thermal-Hydraulic Code Applications and Maintenance Program (CAMP)

Published by

U.S. Nuclear Regulatory Commission
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NUREG/IA-0428 

International 

 Agreement Report

Performing Uncertainty Analysis of IIST

Facility SBLOCA by TRACE and DAKOTA 

Prepared by:

Jong-Rong Wang, Chiung-Wen Tsai*, Hao-Tzu Lin, Chunkuan Shih*

Institute of Nuclear Energy Research, Atomic Energy Council, R.O.C.

1000, Wenhua Rd., Chiaan Village, Lungtan, Taoyuan, 325, TAIWAN

 

*Institute of Nuclear Engineering and Science, National Tsing Hua University,

101 Section 2, Kuang Fu Rd., Hsinchu, TAIWAN

K. Tien, NRC Project Manager

 Division of Systems Analysis

Office of Nuclear Regulatory Research

U.S. Nuclear Regulatory Commission

Washington, DC 20555-0001

Manuscript Completed: March 2013

Date Published: September 2013

Prepared as part of 

The Agreement on Research Participation and Technical Exchange

Under the Thermal-Hydraulic Code Applications and Maintenance Program (CAMP)

Published by

U.S. Nuclear Regulatory Commission

iii 

ABSTRACT

Nowadays, the increasing computing speed realizes the quantification of propagation of input 

uncertainties to output data with Monte Carlo simulation or modified simulation methods. The 

best estimate plus uncertainty (BEPU) methods have been proposed to be used instead of 

typical conservative methodologies. Based on the CAMP activity, this project demonstrates the 

capability of SNAP-TRACE-DAKOTA for 2 % small break LOCA (SBLOCA) of IIST experiment. 

The number of samples was determined by Wilks’ formula to generate the upper bound of peak 

cladding temperature (PCT) with 95/95 confidence level and probability. The PCTs by IIST 

experiment and best-estimate calculation are 804 K and 861 K respectively. The mean value and 

standard deviation of the 59 trial by SNAP-TRACE-DAKOTA are 938.7 K and 63.6 K respectively, 

and the maximum value of PCT is 1054 K.
 iv 

 
v 

FOREWORD

The US NRC (United States Nuclear Regulatory Commission) is developing an advanced 

thermal hydraulic code named TRACE for nuclear power plant safety analysis. The development 

of TRACE is based on TRAC, integrating RELAP5 and other programs. NRC has determined 

that in the future, TRACE will be the main code used in thermal hydraulic safety analysis, and no 

further development of other thermal hydraulic codes such as RELAP5 and TRAC will be 

continued. A graphic user interface program, SNAP (Symbolic Nuclear Analysis Program) which 

processes inputs and outputs for TRACE is also under development. One of the features of 

TRACE is its capacity to model the reactor vessel with 3-D geometry. It can support a more 

accurate and detailed safety analysis of nuclear power plants. TRACE has a greater simulation 

capability than the other old codes, especially for events like LOCA. 

Taiwan and the United States have signed an agreement on CAMP (Code Applications and 

Maintenance Program) which includes the development and maintenance of TRACE. INER 

(Institute of Nuclear Energy Research, Atomic Energy Council, R.O.C.) is the organization in 

Taiwan responsible for the application of TRACE in thermal hydraulic safety analysis, for 

recording user’s experiences of it, and providing suggestions for its development. In this report, 

the GRS method is applied to perform the uncertainty analysis for IIST 2 % SBLOCA transient. 

All steps of analysis procedure including random sampling, data communication, TRACE 

execution, and DAKOTA post-analysis are integrated via SNAP. 
 vi 

 
vii 

CONTENTS

Page

ABSTRACT ............................................................................................................................... iii 

FOREWORD ............................................................................................................................... v 

CONTENTS .............................................................................................................................. vii 

FIGURES ................................................................................................................................. viii 

TABLES .................................................................................................................................... ix 

EXECUTIVE SUMMARY ........................................................................................................... xi 

ABBREVIATIONS ................................................................................................................... xiii 

1. INTRODUCTION ................................................................................................................ 1-1 

2. IIST FACILITY AND SBLOCA EXPERIMENT ................................................................... 2-1 

2.1 DESCRIPTION OF IIST FACILITY ............................................................................ 2-1 

2.2 DESCRIPTION OF IIST SBLOCA EXPERIMENT ...................................................... 2-1 

3. IIST TRACE MODEL AND UNCERTAINTY EVALUATION ............................................... 3-1 

3.1 IIST TRACE MODEL ................................................................................................. 3-1 

3.2 UNCERTAINTY EVALUATION .................................................................................. 3-1 

4. RESULTS AND DISCUSSIONS ........................................................................................ 4-1 

5. CONCLUSIONS ................................................................................................................. 5-1 

6. REFERENCES ................................................................................................................... 6-1
viii 

FIGURES

Page

Fig.1.1 PROPAGATION OF INPUT UNCERTAINTIES ................................................. 1-2 

Fig.1.2 PROPAGATION OF OUTPUT UNCERTAINTIES ............................................. 1-3 

Fig.2.1 THE SCHEMA OF IIST FACILITY ..................................................................... 2-4 

Fig.3.1 THE SIMULATION OF RPV IN MODEL A ......................................................... 3-4 

Fig.3.2 THE SIMULATION OF RPV IN MODEL B ......................................................... 3-5 

Fig.3.3 THE NODOLIZATION OF MODEL B ................................................................. 3-6 

Fig.3.4 THE LOOSELY-COUPLED INTERFACE .......................................................... 3-7

Fig.3.5 THE INTEGRATION OF SNAP-TRACE-DAKOTA ............................................. 3-8 

Fig.4.1 UNCERTAINTY CONFIGURATION INTERFACE ............................................. 4-5 

Fig.4.2 THE SNAP JOB STREAM FOR UNCERTAINTY ANALYSIS ............................ 4-6 

Fig.4.3 THE HISTOGRAMS OF THE INPUT PARAMETERS AND RESULTANT 

PCTS.... ............................................................................................................ 4-7 

Fig.4.4 THE PCTS DURING SBLOCA .......................................................................... 4-8 

Fig.4.5 THE PARTIAL RANK CORRELATION COEFFICIENTS BETWEEN INPUT 

PARAMETERS AND PCT... .............................................................................. 4-9

Fig.4.6 PCT DISTRIBUTION AND CONFIDENCE INTERVAL .................................... 4-10 

Fig.4.7 POWER COMPONENT AND INITIAL POWER ............................................... 4-11 
ix 

TABLES

Page

Table 2.1 THE COMPARISONS OF KEY PARAMETERS BETWEEN MAANSHAN NPP 

AND IIST FACILITY ......................................................................................... 2-2 

Table 2.2 THE INITIAL CONDITIONS OF IIST SBLOCA .................................................. 2-3 

Table 3.1 MINIMUM NUMBER OF CODE RUNS FOR ONE-SIDE AND TWO-SIDE 

TOLERANCE LIMITS ........................................................................................ 3-3 

Table 4.1 THE KEY PARAMETERS FOR THE UNCERTAINTY ANALYSIS ..................... 4-3 

Table 4.2 THE UPPER BOUNDS OF PCT BY DIFFERENT METHODS .......................... 4-4 
 x 

 
xi 

EXECUTIVE SUMMARY

A RHRP IIST facility has been established for safety studies of the Westinghouse three loops 

PWR. The scaling factors of the IIST facility for height and volume in the RCS are approximately 

1/4 and 1/400, respectively. The maximum operating pressure of the IIST facility is 2.1 MPa. The 

IIST facility has three loops as well as all the systems which are about studying Westinghouse 

PWR plant system transients. An experiment of the IIST facility was finished which simulated a 

2% cold-leg-break LOCA with total HPI failure. This break was located in loop 2 of IIST facility, 

which is one of the two loops that do not have a pressurizer. 

The TRACE model of IIST facility has been developed which described in the NUREG report 

(IA-0252). Besides, comparing the results of TRACE and IIST data, it indicates that they are in 

reasonable consistency. In this report, the GRS method is applied to perform the uncertainty 

analysis for IIST 2 % SBLOCA transient. 

The GRS method was used to investigate the propagation of input uncertainties to output data. 

The input parameters with uncertainties of TRACE IIST model were generated randomly based 

on specified PDFs. The number of samples was determined by Wilks’ formula to generate the 

upper bound of PCT with 95/95 confidence level and probability. All TRACE runs were defined 

and executed through SNAP job streams, and TRACE calculation results were read by AptPlot 

script. The data interactions and communications between TRACE and DAKOTA were 

controlled by SNAP. 

The analysis results indicate that the upper bound of PCT is 1054 K by GRS method. The 

ranking coefficients indicate that the break area is the most sensitive among 5 selected input 

parameters (thermal power, U-tube heat transfer area, heater heat transfer area, feedwater 

temperature, break area). However, users are not able to define all considered input parameters 

as SNAP UDN variables under SNAP 2.0.6 environment due to the limitation of SNAP numerics 

module; several important parameters such as initial water level and pressure, and cell volume 

are not able to be involved in uncertainty analysis via SNAP. 

xiii 

ABBREVIATIONS

 

ASTRUM Automated Statistical Treatment of Uncertainty Method

BEPU Best Estimate Plus Uncertainty

CAMP Code Applications and Maintenance Program

DAKOTA Design Analysis Kit for Optimization and Terascale Applications 

IAEA International Atomic Energy Agency 

IIST Institute of Nuclear Energy Research Integral System Test

LOCA Loss Of Coolant Accident

NPP Nuclear Power Plant 

PCT Peak Cladding Temperature 

PDF Probability Distribution Function

PWR Pressurized Water Reactor

RCS Reactor Coolant System

RHRP Reduced-High and Reduced-Pressure 

SBLOCA Small Break Loss Of Coolant Accident

SNAP Symbolic Nuclear Analysis Program

U+S analysis Uncertainty and Sensitivity analysis

 

 

 

 

 

 

 

 

 

 

1-1 

1. INTRODUCTION

Recently, the trend of nuclear reactor safety analysis reveals an increasing interest to substitute 

best estimate plus uncertainty (BEPU) for conservative methodologies which may apply 

conservative codes or the combination of best-estimate codes and conservative initial and 

boundary conditions to achieve the safety margins and regulate the licensing and operations of 

nuclear reactors. 

Compared with conservative methodologies, the methodologies of BEPU adopt best estimate 

codes and realistic input data with uncertainties to quantify the limiting values i.e., peak cladding 

temperature (PCT) for loss of coolant accidents (LOCAs). According to the key report of IAEA 

[1], the methodologies of BEPU are divided into two approaches which evaluate the problems 

based on either (a) propagation of input uncertainties or (b) extrapolation of output uncertainties. 

For the propagation of input uncertainties (Fig. 1.1), i.e., GRS method [2], the uncertainty effects 

are involved by identifying the uncertain input parameters with specified probability distribution 

functions (PDFs) followed by sample runs. For the extrapolation of output uncertainties (Fig. 1.2), 

i.e., CIAU [3], uncertainty is determined by the comparison between numerical results and 

experimental data. The review of accident analysis and BEPU approaches are referred to 

Pourgol-Mohammad [4], Glaeser [5], and D’Auria, et al [6]. So far, BEPU approaches have been 

noticeably adopted by venders. Westinghouse proposed a methodology named Automated 

Statistical Treatment of Uncertainty Method (ASTRUM) [7,8] for realistic large break LOCA 

(LBLOCA) analysis. AREVA implemented the GRS method to evaluate the convolution of 

LBLOCA uncertainty contributors to PCT [9]. 

Not only the vender’s codes but several best estimate codes have been involved in BEPU 

methodologies. One of the best estimate thermal-hydraulic codes, TRACE, has been applied for 

BEPU evaluation. Jaeger, et al. [10] established the combined usage of TRACE and the 

uncertainty and sensitivity (U+S) analysis tool SUSA to investigate the applicability of TRACE to 

supercritical water related thermal-hydraulic properties. The tool SUSA is a stand-alone code, 

providing the capabilities of random sampling of input parameters, determination of output 

bounds with 95/95 confidence level and probability, and measurement of sensitivity of code 

results to input uncertainties. SUSA is also an interface to exchange data with TRACE. On the 

CAMP 2011 spring meeting, it was announced that modified SNAP is integrated with the toolkit 

DAKOTA to perform input parameter sampling, statistical analysis and reporting [11]. Jaeger 

[12], et al., assessed the performance of SNAP-TRACE-DAKOTA against the results of 

TRACE-SUSA. The comparison shows the agreement between SNAP-TRACE-DAKOTA and 

TRACE-SUSA results. Now, the uncertainty analysis user’s manual is available [13]. 

Based on the previous CAMP activity, the current framework of this project is to demonstrate the 

capability of SNAP-TRACE-DAKOTA for 2 % small break LOCA (SBLOCA) of IIST experiment. 

The GRS method was used to investigate the propagation of input uncertainties to output data. 

The input parameters with uncertainties of TRACE IIST model were generated randomly based 

on specified PDFs. The number of samples was determined by Wilks’ formula [14] to generate 

the upper bound of PCT with 95/95 confidence level and probability. All TRACE runs were 

defined and executed through SNAP job streams, and TRACE calculation results were read by 

AptPlot script. The data interactions and communications between TRACE and DAKOTA were 

controlled by SNAP. 
1-2 

Thermo-hydraulic 

codes

Multiple input 

parameters

Selected input 

parameters with 

uncertainties Prediction of 

performance

Multiple output 

parameters

Uncertainty distribution 

for each sampling

Fig. 1.1 Propagation of input uncertainties 
1-3 

Fig. 1.2 Propagation of output uncertainties 

2-1 

2. IIST FACILITY AND SBLOCA EXPERIMENT 

The IIST facility is a reduced-high and reduced-pressure (RHRP) test facility to simulate the 

thermal hydraulics of a Westinghouse 3-loop pressurized water reactor (PWR) at Maanshan 

nuclear power plant (NPP) since 1992 [15]. The comparisons of key parameters between 

Maanshan NPP and IIST facility are listed in Table 2.1. The research purposes of the IIST 

facility are: (a) to enhance the understanding of thermal hydraulics during transients [16,17] as 

well as SBLOCAs [18], (b) to contribute to the evaluations and developments of safety computer 

codes [19,20], (c) to validate the emergency operation procedures during the transients [21]. 

2.1 DESCRIPTION OF IIST FACILITY 

The scaling factors of the IIST facility for height and volume in the reactor coolant system (RCS) 

are approximately 1/4 and 1/400, respectively, and the maximum operating pressure is 2.1 MPa. 

The scaling of hot leg is based on the Froude number criterion to simulate the transition of flow 

regimes in the horizontal pipes during transients and accidents. The key parameters of IIST 

facility are listed in Table 1. As shown in Fig. 2.1 [19], the IIST facility consists of a pressure 

vessel and 3 loops. The pressure vessel has 3 inlet and 3 outlet nozzles. Coolant enters the 

vessel through the inlet nozzles and flows down through the downcomer, and flows up through 

the heater rods to the outlet nozzles. The bypass flow from the upper plenum to the downcomer 

is simulated by three external tubes connected with the valves. Each loop has a steam generator 

and a coolant pump, and the 3 loops are identical, except that there is a pressurizer in the loop 1. 

The pressurizer connected with loop 1 equips an electrical heater, spray nozzle and pressure 

relief valves. The capacity of electrical heater is 10 kW, and the penetrations of spray nozzle and 

pressure relief valves are located on the top of pressurizer. There are 30 U-tubes in each steam 

generator. However, the steam dome of a steam generator doesn’t contain separators and 

dryers, because the steam velocity in the steam dome is not strong enough to entrain liquid into 

seam line at the low core power during simulation of the decay heat level. The secondary 

feedwater flow rate is controlled by flow control valve actuated by the water level controller of 

each steam generator. The IIST facility incorporates a data acquisition system which measures 

temperature, pressure, flow rate, liquid level, and differential pressure. 

2.2 DESCRIPTION OF IIST SBLOCA EXPERIMENT 

This experiment was performed to investigate 2 % cold leg break with total failure of high 

pressure injection [18]. The horizontal break nozzle was installed in the cold leg of loop 2 which 

is not connected with pressurizer. The initial conditions of this experiment are listed in Table 2.2. 

The break was occurred at time zero, and the primary pressure dropped until it become only a 

little higher than the secondary side pressure. This experiment was terminated at 1734 s 

because the uncovering of the core was caused by continuous boil-off of vessel coolant 

inventory without the actuation of coolant makeup system. 
2-2 

Table 2.1 The comparisons of key parameters between Maanshan NPP and IIST facility[19]

Parameters IIST Maanshan IIST/Maanshan

Design pressure (MPa) 2.1 15.6 0.135 

Maximum core power (MW) 0.45 2775 1.62×10-4

Core 

Height (m) 1.0 3.6 0.277 

Hydraulic diameter (m) 0.108 1.22×10-28.85 

Bypass area (m2) 7.2×10-5

 1.54×10-2 4.67×10-3

Hot leg 

Inner diameter, D (m) 5.25×10-2

 7.35×10-1 7.13×10-2

Length, L (m) 2.0 7.28 2.75×10-1

L/D0.5 (m0.5) 8.72 8.48 1.03 

Cold leg 

Inner diameter, D (m) 5.25×10-2

 7.87×10-1 6.67×10-2

Length, L (m) 5.0 15.7 3.18×10-1

L/D0.5 (m0.5) 21.8 17.69 1.22 

U-tube in one SG 

Number 30 5626 5.33×10-3

Average length (m) 4.08 16.85 2.24×10-1

Inner diameter (mm) 15.4 15.4 1.0 

Pressurizer 

Volume (m3) 9.32×10-239.64 2.35×10-3

Surge-line flow area (m2) 3.44×10-4

 6.38×10-2 5.39×10-3
2-3 

Table 2.2 The initial conditions of IIST SBLOCA[19]

Parameter Value 

Core power (kW) 126 

PZR pressure (MPa) 0.958 

PZR water level (mm) 1459 

Loop flow rate (kg/s) 

Loop1 0.210 

Loop2 0.217 

Loop3 0.217 

Hot leg temp. (K) 

Loop1 450 

Loop2 449 

Loop3 451 

Cold leg temp. (K) 

Loop1 409 

Loop2 408 

Loop3 409 
2-4 

Fig. 2.1 The schema of IIST facility 

 
3-1

3. IIST TRACE MODEL AND UNCERTAINTY EVALUATION 

3.1 IIST TRACE MODEL

The IIST TRACE model was developed based on the relevant documents [15,17, 21], and two 

generations of IIST facility TRACE model has been assessed against experimental data. The 

major difference of these two models is the simulation of reactor pressure vessel (RPV). The 

model A simulates the RPV by pipe components (Fig. 3.1), while the model B simulates the RPV 

by a 3-D vessel component (Fig. 3.2). The assessment results indicate that the predictions by 

model B are better than those by model A in the primary system pressure and break flow [22]. 

Therefore, the vessel modeling of model B was adopted for the uncertainty analysis. 

Fig. 3.3 shows the nodolization of model B, which consists of 101 hydraulic components, 212 

control blocks, 39 heat structures and a power component. The primary loops include hot legs, 

steam generator U-tubes, crossover leg, coolant pump and cold legs. These loops are identical 

except that the pressurizer is located in loop1. The break area is controlled by a valve 

component and located in loop 2. A break component is used to simulate ambient condition. 

Each of the 3 identical steam generators consists of downcomer, boiling section and steam 

dome. The feedwater flow rates are simulated by time-dependent junctions, and the downstream 

condition of each steam line is simulated by a break component with constant boundary 

condition. 

3.2 UNCERTAINTY EVALUATION 

3.2.1 FUNDAMENTAL METHODOLOGY 

The GRS method was applied to investigate the uncertainty effect propagating from input 

parameters through TRACE to PCT, as shown in Fig. 1.1. Because the required minimum 

number of TRACE runs is dependent of the values of confidence level and probability, Wilks’ 

formula [14] was employed to determinate the minimum number of runs. The correlations 

between number of code runs, confidence level, and probability of Wilks’ formula are defined in 

Eq. (3-1) and Eq. (3-2) for one-side tolerance limit and two-sided tolerance limit respectively. 

The minimum number of code runs is tabulated in Table 3.1. 

1-α

n≧β Eq. 3-1

1-α

n

-n(1-α)α

n-1≧β Eq. 3-2

Where α is probability, β is the confidence level, and n denotes the number of code runs. 

Since the value of PCT is the safety criterion to ensure the integrity of fuel assemblies for 

LOCAs, the minimum number of 59 was used to generate the maximum bound of PCT which 

achieve 95/95 criterion. Finally, correlations between input parameters and PCTs are calculated 

for sensitivity study and ranking to investigate what input parameters dominate the contribution 

of uncertain distribution of PCT. 

3.2.2 DAKOTA TOOLKIT 

The DAKOTA [23] toolkit was applied for the sampling of input parameters and the calculation of 

correlations and ranking of input parameters. The uncertainty quantification package [24] of 

DAKOTA provides Monte Carlo sampling and Latin Hypercube sampling methods combined with 

various PDFs including normal, lognormal, uniform, logunifrom, hypergeometric, and 
3-2

user-supplied histograms. As for correlations mentioned in the previous section, four types of 

correlations including simple and partial raw correlations and simple and partial rank correlations 

are returned in DAKOTA output files [24]. The coefficients of first two correlations are obtained 

by Pearson’s correlation shown in Eq. 3-3, and the other two are calculated by Spearman’s rank 

correlation. 

Eq. 3-3

where r is the Pearson’s correlation coefficient, n is the number of samples, and x and y denote 

two quantities. 

The formula of Spearman’s rank correlation is the same as Pearson’s (Eq. 3-3); however, the 

difference is that Spearman’s rank correlation employs the rank data which substitute the ranked 

values for raw data. 

3.2.3 UNCERTAINTY ANALYSIS PROCEDURE 

DAKOTA provides users an interface to couple other codes for uncertainty analysis. Fig. 3.4 

illustrates the concept of a loosely-coupled interface [24] between DAKOTA and simulation 

codes (i.e., TRACE) by which data can be exchanged between DAKOTA and other simulation 

codes. Thanks for the modified SNPA, it is able to integrate TRACE and DAKOTA via SNAP job 

stream. The integration of SNAP-TRACE-DAKOTA is shown in Fig. 3.5 where the Extract Data 

is a plug-in tool to read TRACE output data. 
3-3

Table 3.1 Minimum number of code runs for one-side and two-side tolerance limits 

 One-side tolerance limits Two-side tolerance limits 

 α

β

0.90 0.95 0.99 0.90 0.95 0.99 

0.90 22 45 230 38 77 388 

0.95 29 59 299 46 93 473 

0.99 44 90 459 64 130 662 
3-4

Fig. 3.1 The simulation of RPV in model A 
3-5

Fig. 3.2 The simulation of RPV in model B 
3-6

Fig. 3.3 The nodolization of model B 
3-7

Fig. 3.4 The loosely-coupled interface
3-8

SNAP DAKOTA job stream

DAKOTA DAKOTA 

input file

Random 

variates

TRACE input 

files

Stream 

manager

TRACE solver

TRACE solver

TRACE solver

Extract Data

TRACE 

output resluts

DAKOTA post process

DAKOTA

DAKOTA output 

file and report

Fig. 3.5 The integration of SNAP-TRACE-DAKOTA 
4-1

4. RESULTS AND DISCUSSIONS 

The initial conditions of IIST 2% SBLOCA are listed in Table 2.2. As mentioned in section 3.2.3, 

the setting of input uncertainties and the execution of uncertainty analysis was performed via 

SNAP. The built-in graphical user interface (GUI) of uncertainty configuration shown in Fig. 4.1 

provides several tabs to define the number of samples, variables, and PDFs. Table 4.1 lists the 

5 key parameters taken into account in the uncertainty analysis, which are defined as the SNAP 

user-defined numerics (UDN) variables and linked with uncertainty configuration to generate 59 

TRACE input files. Fig. 4.2 shows the overall SNAP job stream for uncertainty analysis. 

Fig. 4.3 shows the histograms of the 5 input parameters and 59 resultant PCTs. Fig. 4.4 displays 

the 59 PCTs as a function of time. According to Wilks’ formula, the maximum value (1054 K at 

1734 s) from the 59 trials represents the upper-side tolerance limit with a confidence level of 95 

% and probability of 95 %. The PCTs by IIST experiment and best-estimate calculation are 804 

K and 861 K respectively. The mean value and standard deviation of the 59 trial are 938.7 K and 

63.6 K respectively. The partial rank correlation coefficients between input parameters and PCT 

shown in Fig. 4.5 indicate that break area is the most sensitive parameter. 

Assuming the PDF of PCT is a normal distribution, two approaches were applied to confirm the 

upper bound of PCT derived by the GRS method. The first approach used the mean value and 

standard deviation of the 59 trial to calculate the PCT which cover 95 % area of the PCT 

distribution (Fig. 4.6), which is calculated by Eq. 4-1. 

PCT95 = PCTmean + 1.645σ Eq. 4-1

where PCTmean is the mean value of PCT, σ is the standard deviation of PCT 

The second approach applies the t distribution and chi-squared distribution to estimate the 

population mean and population standard deviation of PCT from 59 sample PCT data. 

Consequently, the upper bound of PCT covers 95 % probability is estimated by the above 

population mean and population standard deviation. 

The ratio t defined in Eq. 4-2 follows the t distribution, which estimates the population mean with 

a specific confidence level by the number of samples, sample mean, and sample standard 

deviation. 

 Eq. 4-2

where is sample mean, μ is population mean, is standard deviation of sample mean by 

sample mean defined in Eq. 4-3. 

 Eq. 4-3

where S is sample standard deviation, N is the number of samples. 

Similarly, the ratio X defined in Eq. 4-4 follows the Chi-squared distribution, which estimates the 

population standard deviation in terms of sample standard deviation and number of samples. 
4-2

Eq. 4-4

where N is the number of samples, S is sample standard deviation, σ is population standard 

deviation. 

Table 4.2 lists the different upper bounds with 95/95 criterion estimated by GRS method and the 

other two approaches mentioned above. The comparison shows that the GRS method provides 

a reasonable estimation to quantify the propagation of input uncertainties on output results. 

Although the uncertainty analysis procedure is integrated via SNAP job stream, there is a major 

limitation. All input parameters associated with uncertainties are defined as UDN variables to 

generate the values with specified uncertainties. Fig. 4.7 illustrate that the initial thermal power 

is defined as a UDN variable. However, not all input parameters are able to be UDN variables; 

only five input parameters listed in Table 4.1 were used in IIST uncertainty analysis because of 

this limitation. 
4-3

Table 4.1 The key parameters for the uncertainty analysis

Input parameters Nominal 

values 

Uncertainty 

 range 

PDFs 

Thermal power 126 (kW) [-8, +8] (%) 

Uniform distribution

U-tube heat transfer area 100 (%) [-15, +15] (%) 

Heater heat transfer area 100 (%) [-15, +15] (%) 

Feedwater temperature 399.4 (K) [-10, +10] (%) 

Break area 2 (%) [2, 2.1] (%) 
4-4

Table 4.2 The upper bounds of PCT by different methods

Methods Upper bound (K) Confidence level/probability 

GRS 1054 95/95 

Eq. 4-1 1159 

Eq. 4-1 + t distribution + 

Chi-squared distribution 

1136 
4-5

Fig. 4.1 Uncertainty configuration interface
4-6

Fig. 4.2 The SNAP job stream for uncertainty analysis 
4-7

Fig. 4.3 The histograms of the input parameters and resultant PCTs 
4-8

Fig. 4.4 The PCTs during SBLOCA 
4-9

Fig. 4.5 The partial rank correlation coefficients between input parameters and PCT 
4-10

1.645

95 % 5 %

Fig. 4.6 PCT distribution and confidence interval 
4-11

Fig. 4.7 Power component and initial power 

5-1

5. CONCLUSIONS

The GRS method is applied to perform the uncertainty analysis for IIST 2 % SBLOCA. All steps 

of analysis procedure including random sampling, data communication, TRACE execution, and 

DAKOTA post-analysis are integrated via SNAP. The upper bound of PCT is 1054 K by GRS 

method. The ranking coefficients indicate that the break area is the most sensitive among 5 

selected input parameters. However, users are not able to define all considered input 

parameters as SNAP UDN variables under SNAP 2.0.6 environment due to the limitation of 

SNAP numerics module; several important parameters such as initial water level and pressure, 

and cell volume are not able to be involved in uncertainty analysis via SNAP. 

6-1 

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UNITED STATES 

NUCLEAR REGULATORY COMMISSION

WASHINGTON, DC 20555-0001

--------------------

OFFICIAL BUSINESS
NUREG/IA

-0428 Performing Uncertainty Analysis of IIST Facility SBLOCA 

by TRACE and DAKOTA