Technical Basis for 

Regulatory Guidance on 

the Alternate Pressurized 

Thermal Shock Rule

Draft Report for Comment

Office of Nuclear Regulatory Research

NUREG-2163
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Technical Basis for 

Regulatory Guidance on 

the Alternate Pressurized 

Thermal Shock Rule

Draft Report for Comment

Manuscript Completed: November 2014

Date Published: February 2015

Prepared by: 

Gary L. Stevens, Mark T. Kirk, and Mohammad Modarres* 

*University of Maryland

Gary L. Stevens, NRC Project Manager

Office of Nuclear Regulatory Research

NUREG-2163

 
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iii

1 ABSTRACT

2 During plant operation, the walls of reactor pressure vessels (RPVs) are exposed to neutron 

3 radiation, resulting in embrittlement of the vessel steel and weld materials in the area of the 

4 RPV adjacent to the core. If an embrittled RPV had a flaw of critical size and certain severe 

5 system transients were to occur, the flaw could rapidly propagate through the vessel, resulting 

6 in a through-wall crack and, thereby, challenging the integrity of the RPV. The severe transients 

7 of concern, known as pressurized thermal shock (PTS), are characterized by a rapid cooling of 

8 the internal RPV surface in combination with repressurization of the RPV. Advancements in 

9 understanding and knowledge of materials behavior, the ability to realistically model plant 

10 systems and operational characteristics, and the ability to better evaluate PTS transients to 

11 estimate loads on vessel walls led the U.S. Nuclear Regulatory Commission (NRC) to develop a 

12 risk-informed revision of the existing PTS Rule that was published in Section 50.61a, “Alternate 

13 Fracture Toughness Requirements for Protection against Pressurized Thermal Shock Events,” 

14 of Title 10, “Energy,” of the Code of Federal Regulations (10 CFR 50.61a).

15

16 This report explains the basis for the requirements that establish the entry conditions to permit 

17 use of 10 CFR 50.61a and describes methods by which the following four requirements can be 

18 met: (1) criteria relating to the date of construction and design requirements, (2) criteria relating 

19 to evaluation of plant-specific surveillance data, (3) criteria relating to inservice inspection (ISI) 

20 data and non-destructive examination (NDE) requirements, and (4) criteria relating to alternate 

21 limits on embrittlement.

22

v

1 FOREWORD

2 The reactor pressure vessel (RPV) is exposed to neutron radiation during normal operation. 

3 Over time, the RPV steel becomes progressively more brittle in the region adjacent to the core. 

4 If a vessel had a preexisting flaw of critical size and certain severe system transients occurred, 

5 this flaw could propagate rapidly through the vessel, resulting in a through-wall crack. The 

6 severe transients of concern, known as pressurized thermal shock (PTS), are characterized by 

7 rapid cooling (i.e., thermal shock) of the internal RPV surface that may be combined with 

8 repressurization. The simultaneous occurrence of critical-size flaws, embrittled steel, and a 

9 severe PTS transient is a low-probability event. U.S. pressurized-water reactors (PWRs) are 

10 not projected to approach the levels of embrittlement to make them susceptible to PTS failure, 

11 even during extended operation beyond the original 40-year design life.

12

13 Advancements in understanding and knowledge of materials behavior, the ability to realistically 

14 model plant systems and operational characteristics, and the ability to better evaluate PTS 

15 transients to estimate loads on vessel walls led to the development of a risk-informed revision of 

16 the existing PTS Rule that was published in Section 50.61a, “Alternate Fracture Toughness 

17 Requirements for Protection against Pressurized Thermal Shock Events,” of Title 10, “Energy,”

18 of the Code of Federal Regulations (10 CFR 50.61a).

19

20 The “Alternate PTS Rule” contained in 10 CFR 50.61a provides revised PTS screening criteria in 

21 the form of an embrittlement reference temperature, RTMAX-X, which characterizes the RPV

22 material’s resistance to fracture from initiated flaws. The Alternate PTS Rule is based on more 

23 comprehensive analysis methods than the existing PTS Rule contained in 10 CFR 50.61, “Fracture 

24 Toughness Requirements for Protection against Pressurized Thermal Shock Events.” This 

25 alternate rule became desirable because the existing requirements contained in 10 CFR 50.61 are 

26 based on unnecessarily conservative assumptions. The Alternate PTS Rule reduces regulatory 

27 burden for those PWR licensees who expect to exceed the 10 CFR 50.61 embrittlement 

28 requirements before the expiration of their operating licenses, while still maintaining adequate safety

29 margins. PWR licensees may choose to comply with the Alternate PTS Rule as a voluntary 

30 alternative to complying with the requirements contained in 10 CFR 50.61.

31

32 This document explains the basis for the requirements that establish the entry conditions to 

33 permit use of the Alternate PTS Rule. It also describes methods by which the following four

34 requirements can be met:

35

36 1. Criteria relating to the date of construction and design requirements;

37 2. Criteria relating to evaluation of plant-specific surveillance data;

38 3. Criteria relating to inservice inspection (ISI) data and non-destructive examination (NDE) 

39 requirements; and

40 4. Criteria relating to alternate limits on embrittlement.

41

42

43

44 Brian W. Sheron, Director

45 Office of Nuclear Regulatory Research

46 U.S. Nuclear Regulatory Commission

47

vii

1 TABLE OF CONTENTS

2 ABSTRACT................................................................................................................................iii

3 FOREWORD ..............................................................................................................................v

4 TABLE OF CONTENTS............................................................................................................ vii

5 LIST OF FIGURES ....................................................................................................................ix

6 LIST OF TABLES.......................................................................................................................xi

7 EXECUTIVE SUMMARY ......................................................................................................... xiii

8 ABBREVIATIONS AND ACRONYMS .......................................................................................xv

9 SYMBOLS AND EXPRESSIONS............................................................................................ xvii

10 1. INTRODUCTION............................................................................................................... 1

11 1.1 Background......................................................................................................... 1

12 1.2 Scope of this Report ........................................................................................... 3

13 2. OVERVIEW OF THE ALTERNATE PTS RULE................................................................. 7

14 2.1 Background......................................................................................................... 7

15 2.2 Applicability of the Alternate PTS Rule................................................................ 7

16 2.3 Updated Embrittlement Correlation ..................................................................... 8

17 2.4 ISI Volumetric Examination and Flaw Assessments............................................ 8

18 2.5 NDE-Related Uncertainties ................................................................................10

19 2.6 Surveillance Data...............................................................................................10

20 3. RESPONSES TO STAKEHOLDER FEEDBACK..............................................................11

21 4. GUIDANCE FOR CRITERIA RELATING TO THE DATE OF CONSTRUCTION AND 

22 DESIGN REQUIREMENTS..............................................................................................19

23 4.1 Requirements in the Alternate PTS Rule............................................................19

24 4.2 Regulatory Guidance .........................................................................................19

25 5. GUIDANCE FOR CRITERIA RELATING TO THE EVALUATION OF PLANT-SPECIFIC 

26 SURVEILLANCE DATA....................................................................................................21

27 5.1 Embrittlement Trend Curve ................................................................................22

28 5.2 Data Used in Statistical Tests ............................................................................24

29 5.3 Statistical Evaluation of Surveillance Data .........................................................26

30 5.4 How to Perform Statistical Tests of Surveillance Data........................................29

31 5.4.1 Type A Deviations .........................................................................................29

32 5.4.2 Type B Deviations .........................................................................................30

33 5.4.3 Type D Deviations .........................................................................................31

34 5.4.4 Comments on the Statistical Tests.................................................................32

35 5.5 Evaluation of Plant Data Relative to the Statistical Tests ...................................33

36 5.6 Considerations When Statistical Tests Are Failed..............................................36

37 5.6.1 Factors to Consider When Statistical Tests Are Failed ..................................36

38 5.6.2 Specific Procedures.......................................................................................37

39 5.7 Summary and Conclusions ................................................................................38

40 6. GUIDANCE RELATING TO ISI DATA AND NDE REQUIREMENTS................................41

41 6.1 Background........................................................................................................41

42 6.2 Guidance on Criteria Relating to Alternate Limits on Embrittlement ...................46

43 6.2.1 Guidance on a Procedure to Preclude Brittle Fracture...................................46

44 6.2.2 Guidance on Considerations to Include in a Plant-Specific PFM Analysis......49

45 6.3 Guidance for Initial Evaluation of NDE Data.......................................................52

46 6.4 Guidance for Further Evaluation of NDE Data....................................................61

47 6.4.1 Guidance on the Elements and NDE Techniques Associated with ASME Code 

48 Examinations.................................................................................................61
viii

1 6.4.2 Guidance on a Procedure to Adjust NDE Data and Comparison to Flaws 

2 Assumed in PFM Calculations .......................................................................62

3 6.4.2.1 Guidance on Application of NDE Uncertainties....................................64

4 7. GUIDANCE ON CRITERIA RELATING TO ALTERNATE LIMITS ON 

5 EMBRITTLEMENT...........................................................................................................67

6 8. SUMMARY.......................................................................................................................69

7 9. REFERENCES.................................................................................................................71

8 APPENDIX A: DERIVATION OF THE STATISTICAL TEST FOR TYPE D DEVIATIONS......A-1

9 APPENDIX B: REGULATORY ASSESSMENT OF STATISTICAL SURVEILLANCE

10 TESTS............................................................................................................B-1

11 APPENDIX C: FLAW DEPTH AND DENSITY DISTRIBUTIONS CONSIDERING

12 PROBABILITY OF DETECTION AND BIAS IN NDE DATA WITH 

13 APPLICATIONS .............................................................................................C-1

14 APPENDIX D: SENSITIVITY-STUDY RESULTS ON FLAW DISTRIBUTIONS

15 CONSIDERING VFLAW DATA, POD, AND MEASUREMENT ERRORS

16 IN NDE DATA.................................................................................................D-1

17
ix

1 LIST OF FIGURES

2 Figure 1. Four Types of Deviation between Heat-Specific Surveillance Data

3 and a DT30 ETC.........................................................................................................28

4 Figure 2. DT30 Prediction Residuals Based on Eqn. (1). Low flux points are from

5 power reactors while high flux points are from test reactors. Similar trends

6 exist for both plates and welds [15]. ..........................................................................29

7 Figure 3. Procedure to Assess Type B Deviations. ..................................................................30

8 Figure 4. Specific Procedures to Account for Failure of the Mean Test (left) or Low

9 Fluence Outlier Statistical Test (right)........................................................................38

10 Figure 5. Flow Diagram with Recommendations for Meeting the Requirements of the

11 Alternate PTS Rule. ..................................................................................................45

12 Figure 6. ASME Code Section XI Examination and Flaw Evaluation Process and

13 Identification of Flaws for Comparison to Alternate PTS Rule....................................55

14 Figure 7. Sample Appendix VIII Examination Results for Plant J..............................................56

15

xi

1 LIST OF TABLES

2 Table 1. RTMAX-X PTS Embrittlement Limits from the Alternate PTS Rule................................ 2

3 Table 2. Flaw Limit Tables from the Alternate PTS Rule......................................................... 5

4 Table 3. EPRI Recommendations for Addressing Alternate PTS Rule Requirements [10].....12

5 Table 4. Standard Deviation of Residuals about Eqn. (1) ......................................................23

6 Table 5. Independent Variables in the Eqn. (1) ETC and the Ranges and Mean Values

7 of the Calibration Dataset ........................................................................................23

8 Table 6. α = 1% Student’s t-Values .......................................................................................31

9 Table 7. α = 1% Threshold Value for the Outlier Test............................................................32

10 Table 8. Heats of Material that Exhibit Statistically Significant Deviations (α = 1%) of

11 Type A, B, or D........................................................................................................35

12 Table 9. Proportion of Material Heats in the Current Surveillance Database that Exhibit 

13 Statistically Significant Deviations (α = 1%) of Type A, B, or D ................................35

14 Table 10. Sample Brittle Fracture Assessment.........................................................................48

15 Table 11. Summary of the Revised Flaw Depth and Flaw Density VFLAW Parameters

16 to be Used in a Revised PFM Analysis for Beaver Valley 2 (from Appendix C)........51

17 Table 12. Determination of Total Weld Length Examined for Plant J........................................57

18 Table 13. Determination of Total Plate Surface Area Examined for Plant J..............................58

19 Table 14. Determination of the Flaw Positions for Plant J ........................................................59

20 Table 15. Comparison of Flaws to Alternate PTS Rule Flaw Tables for Plant J........................60

21 Table 16. Example for Plant J After Applying an Initial Alternate PTS Rule Flaw Table

22 Check Using the Procedure in Section 6.3...............................................................64

23

xiii

1 EXECUTIVE SUMMARY

2 In early 2010, the U.S. Nuclear Regulatory Commission (NRC) promulgated the Alternate 

3 Pressurized Thermal Shock (PTS) Rule as 10 CFR 50.61a, “Alternate Fracture Toughness 

4 Requirements for Protection against Pressurized Thermal Shock Events,” which amended existing 

5 regulations to provide alternate embrittlement requirements for protection against PTS events for 

6 pressurized-water reactor (PWR) pressure vessels. These requirements are based on more 

7 comprehensive, accurate, and realistic analysis methods than those used to establish the limits in 

8 10 CFR 50.61, “Fracture Toughness Requirements for Protection against Pressurized Thermal 

9 Shock Events.” This alternate rule became desirable because the existing requirements, as 

10 contained in 10 CFR 50.61, are based on unnecessarily conservative assumptions. The Alternate 

11 PTS Rule reduces regulatory burden for those PWR licensees who expect to exceed the 

12 10 CFR 50.61 embrittlement requirements before the expiration of their operating licenses while still 

13 maintaining adequate safety margins. PWR licensees may choose to comply with the Alternate 

14 PTS Rule as a voluntary alternative to complying with the requirements contained in 10 CFR 50.61.

15

16 The Alternate PTS Rule provides revised PTS screening criteria in the form of embrittlement 

17 reference temperatures, RTMAX-X, that characterize the RPV material’s resistance to fracture initiating 

18 from flaws. The RTMAX-X embrittlement limits may be used by licensees provided that the following 

19 criteria are met:

20

21 1. Criteria relating to the date of construction and design requirements: The Alternate 

22 PTS Rule is applicable to licensees whose construction permits were issued before 

23 February 3, 2010, and whose RPVs were designed and fabricated to the 1998 Edition (or an 

24 earlier edition) of the ASME (formerly the American Society of Mechanical Engineers) Boiler 

25 and Pressure Vessel Code (“Code”). The reason for this applicability restriction is because 

26 the structural and thermal hydraulic analyses that established the basis for the Alternate 

27 PTS Rule embrittlement limits only represented plants constructed before this date. It is the 

28 responsibility of a licensee to demonstrate that the risk-significant factors controlling PTS for 

29 any plant constructed after February 3, 2010, are adequately addressed by the 

30 technical-basis calculations developed in support of the Alternate PTS Rule. Chapter 4 of 

31 this document describes methods by which licensees can satisfy these criteria and identifies 

32 factors to be considered in such an evaluation.

33 2. Criteria relating to evaluation of plant-specific surveillance data: The Alternate PTS 

34 Rule includes statistical tests that must be performed on RPV surveillance data to determine 

35 whether the surveillance data are sufficiently close to the predictions of an embrittlement 

36 trend curve (ETC) that the predictions of the ETC are valid for use. From a regulatory 

37 perspective, it is of particular interest to determine whether plant-specific surveillance data 

38 deviate significantly from the predictions of the ETC in a manner that suggests that the ETC 

39 is likely to underpredict plant-specific data trends. Chapter 5 of this document describes 

40 guidance by which licensees can assess the closeness of plant-specific data to the ETC 

41 using statistical tests, including:

42  A detailed description of the mathematical procedures to use to assess 

43 compliance with the three statistical tests in the Alternative PTS Rule.

44  A list of factors to consider in diagnosing the reason that particular surveillance 

45 data sets might fail these statistical tests.

46  A description of certain situations in which adjustments of the ETC predictions 

47 can be made.
xiv

1 3. Criteria relating to inservice inspection (ISI) data and nondestructive examination 

2 (NDE) requirements: The Alternate PTS Rule describes a number of tests of and 

3 conditions on the collection and analysis of ISI data that are intended to provide reasonable 

4 assurance that the distribution of flaws that was assumed to exist in the probabilistic fracture 

5 mechanics (PFM) calculations that provided the basis for the RTMAX-X limits provide an 

6 appropriate, or bounding, model of the population of flaws in the RPV of interest. Chapter 6

7 of this document provides guidance by which licensees can satisfy these criteria. The 

8 guidance discussed in this NUREG includes the following components:

9  Guidance for initial evaluation of NDE data obtained from qualified ISI 

10 examinations.

11  Guidance for further evaluation of NDE data obtained from qualified ISI 

12 examinations, as follows:

13 i. Elements and NDE techniques associated with the qualified ASME 

14 Code, Section XI, Mandatory Appendix VIII ISI examinations performed 

15 to assess compliance with the requirements of the Alternate PTS Rule.

16 ii. A mathematical procedure that can be used to adjust NDE data to 

17 account for flaw detection and sizing errors and comparison of the 

18 adjusted data to the population of flaws assumed in PFM technical basis 

19 for the Alternate PTS Rule.

20  Guidance for plants with RPV flaws that fall outside the applicability of the flaw 

21 tables in the Alternate PTS Rule, including:

22 i. A mathematical procedure that can be used to preclude brittle fracture 

23 based on RTNDT information.

24 ii. A mathematical procedure that can be used to combine the NDE data 

25 with the population of flaws assumed in the PFM calculations to estimate 

26 the total flaw distribution that is predicted to exist in the RPV, and 

27 guidance on the use of this total flaw distribution as part of a PFM 

28 calculation using the Fracture Analysis of Vessels—Oak Ridge (FAVOR) 

29 computer code.

30 4. Criteria relating to alternate limits on embrittlement: Guidance is provided by which 

31 licensees can estimate a plant-specific value of through-wall cracking frequency (TWCF) for 

32 cases in which the RTMAX-X limits of the Alternate PTS Rule are not satisfied. Chapter 7 of 

33 this document describes these two sets of guidance so that licensees can satisfy 

34 embrittlement acceptability criteria.

35

36 This document provides guidance and the associated technical basis for methods by which these 

37 requirements can be met.
xv

1 ABBREVIATIONS AND ACRONYMS

Abbreviation Definition

AP Advanced Passive

APWR Advanced Pressurized-Water Reactor

ASME (Not an abbreviation now; formerly stood for the American Society of Mechanical 

Engineers)

BWR boiling-water reactor

CFR Code of Federal Regulations

E energy

EPR Evolutionary Power Reactor

EPRI Electric Power Research Institute

ETC embrittlement trend curve

FAVOR Fracture Analysis of Vessels—Oak Ridge

FRN Federal Register Notice

IGSCC intergranular stress-corrosion cracking

ISI inservice inspection

MeV million electron-Volts

MRP Materials Reliability Program

NDE non-destructive examination

NRC United States Nuclear Regulatory Commission

NRR Office of Nuclear Reactor Regulation

NSSS Nuclear Steam Supply System

ORNL Oak Ridge National Laboratory

PDF probability density function

PDI Performance Demonstration Initiative

PFM Probabilistic Fracture Mechanics

PNNL Pacific Northwest National Laboratory

POD probability of detection

PRA probabilistic risk assessment

PTS pressurized thermal shock

PVRUF Pressure Vessel Research User Facility

PWR pressurized-water reactor

RES Office of Nuclear Regulatory Research

RPV reactor pressure vessel

TLR Technical Letter Report

TWCF through-wall cracking frequency

TWE through-wall extent

UT ultrasonic testing

2

3

xvii

1 SYMBOLS AND EXPRESSIONS

Symbol Definition

A Multiplier for determining DT30 based on product form

ADJ An adjustment in the embrittlement prediction to account for a failure of the mean 

test

B Multiplier for determining DT30 based on product form

C1 and C2 Critical values of the outlier test

CRP Intermediate term for determining DT30 in degrees Fahrenheit or Celsius

Cu Copper content in weight-percent

Cue Effective copper content in weight-percent

f(Cue,P) Intermediate term for determining DT30 based on effective copper content and 

phosphorus content

g(Cue,Ni, φte) Intermediate term for determining DT30 based on effective copper content, nickel 

content, and effective fluence

m Slope of ΔT30 prediction residuals plotted vs. the base-10 logarithm of fluence

Max(Cue) Intermediate term for determining DT30 based on effective copper content

MD Intermediate term for determining DT30 based on degrees Fahrenheit or Celsius

Mn Manganese content in weight percent

n Number of ΔT30 observations in a plant-specific surveillance dataset

Ni Nickel content in weight percent

P Phosphorus content in weight percent

r* Normalized residual

r*1 and r*2 Calculated values of residuals for the outlier test

rLIMIT(1) and rLIMIT(2) Critical values of the outlier test (same as C1 and C2)

rmax Maximum permissible ΔT30 prediction residual

rmean Mean ΔT30 prediction residual for a plant-specific surveillance dataset

R(max1) The largest ΔT30 prediction residual for a plant-specific surveillance dataset

R(max2) The second-largest ΔT30 prediction residual for a plant-specific surveillance dataset

RTMAX-X

Any or all of the material properties RTMAX–AW, RTMAX–PL, RTMAX–FO, RTMAX–CW, or 

sum of RTMAX–AW and RTMAX–PL for a particular reactor vessel, expressed as a 

temperature in degrees Fahrenheit (°F) or Celsius (°C)

RTMAX-AW

Material property, expressed as a temperature in degrees Fahrenheit (°F) or 

Celsius (°C), which characterizes the reactor vessel’s resistance to fracture initiating 

from flaws found along axial weld fusion lines

RTMAX-PL

Material property, expressed as a temperature in degrees Fahrenheit (°F) or 

Celsius (°C), which characterizes the reactor vessel’s resistance to fracture initiating 

from flaws found in plates remote from welds

RTMAX-FO

Material property, expressed as a temperature in degrees Fahrenheit (°F) or 

Celsius (°C), which characterizes the reactor vessel’s resistance to fracture initiating 

from flaws in forgings remote from welds

RTMAX-CW

Material property, expressed as a temperature in degrees Fahrenheit (°F) or 

Celsius (°C), which characterizes the reactor vessel’s resistance to fracture initiating 

from flaws found along circumferential weld fusion lines

RTNDT Nil ductility transition temperature in degrees Fahrenheit (°F) or Celsius (°C)

RTNDT(u)

Unirradiated nil ductility transition temperature in degrees Fahrenheit (°F) or 

Celsius (°C)

RTPTS

The RTNDT value at end of license, as defined in 10 CFR 50.61, in degrees 

Fahrenheit (°F) or Celsius (°C)

S Distance of flaw below surface in inches (in) or millimeters (mm)

se(m) Standard error of m

te Effective time in seconds

TC Irradiated (coolant) temperature in degrees Fahrenheit (°F) or Celsius (°C)
xviii

1

2 SYMBOLS AND EXPRESSIONS (concluded)

3

Symbol Definition

TCRIT(α) Critical value of Student’s t-distribution

Tm T-statistic for m

t, TWALL Wall thickness in inches (in) or millimeters (mm)

t(...) Student’s t-distribution

TWE Through-wall extent in inches (in) or millimeters (mm)

TWEMAX Maximum through-wall extent in inches (in) or millimeters (mm)

TWEMIN Minimum through-wall extent in inches (in) or millimeters (mm)

α Statistical significance level

ΔT30

Increase in the Charpy-V notch energy transition temperature, in degrees 

Fahrenheit (°F) or Celsius (°C), at 30 foot-pounds (ft-lb) or 41 Joules (J) caused by 

neutron-irradiation embrittlement

ΔT30(ADJ)

An adjusted value of ΔT30 in degrees Fahrenheit (°F) or Celsius (°C) that accounts 

for a failure in the mean slope test

ΔT30i(measured) A measured value of ΔT30 in degrees Fahrenheit (°F) or Celsius (°C)

ΔT30i(ETC-mean) A value of ΔT30 predicted by Eqn. (1) in degrees Fahrenheit (°F) or Celsius (°C)

ΔYS Change in yield strength in thousands of pounds per square inch (ksi) or 

Megapascals (MPa)

φ Flux in neutrons per square centimeter per second (n/cm2

-sec)

φt Fluence in neutrons per square centimeter (n/cm2

)

φte Effective fluence in neutrons per square centimeter (n/cm2

)

σ Standard deviation in degrees Fahrenheit (°F) or Celsius (°C)

4

5
1

1 1. INTRODUCTION

2 1.1 Background

3 The United States Nuclear Regulatory Commission (NRC) promulgated Section 50.61a, “Alternate 

4 Fracture Toughness Requirements for Protection against Pressurized Thermal Shock Events,” of 

5 Title 10, “Energy,” of the Code of Federal Regulations (10 CFR 50.61a) [1] on January 4, 2010, as 

6 reported in the Federal Register Notice (FRN) 75 FR 13 [2]. 10 CFR 50.61a amended existing 

7 regulations to provide alternate fracture toughness requirements for protection against pressurized 

8 thermal shock (PTS) events for pressurized-water reactor (PWR) pressure vessels.

9

10 The “Alternate PTS Rule” contained in 10 CFR 50.61a provides alternate embrittlement 

11 requirements based on more comprehensive analysis methods. This action became desirable 

12 because the existing requirements, as contained in 10 CFR 50.61, “Fracture Toughness 

13 Requirements for Protection against Pressurized Thermal Shock Events” [3], are based on 

14 unnecessarily conservative assumptions. The Alternate PTS Rule reduces regulatory burden for 

15 those PWR licensees who expect to exceed the 10 CFR 50.61 embrittlement requirements before 

16 the expiration of their operating licenses, while maintaining adequate safety. PWR licensees may 

17 choose to comply with the Alternate PTS Rule as a voluntary alternative to complying with the 

18 requirements contained in 10 CFR 50.61.

19

20 The Alternate PTS Rule provides revised PTS screening criteria in the form of embrittlement

21 reference temperatures, RTMAX-X, that characterize the reactor pressure vessel (RPV) material’s 

22 resistance to fracture initiation from flaws. The PTS screening criteria are provided in Table 1 of the 

23 Alternate PTS Rule, and are shown in Table 1 of this document. The values shown in Table 1

24 below are based on up-to-date understandings and models of the many factors affecting the 

25 operating safety of PWRs. Further discussion on the provisions and use of the Alternate PTS Rule 

26 appear in Chapter 2 of this report.
2

1

2

Table 1. RTMAX-X PTS Embrittlement Limits from the Alternate PTS Rule 3

4

5

6

7
3

1 1.2 Scope of this Report

2 The RTMAX-X embrittlement limits shown in Table 1 may be used by licensees provided that 

3 certain criteria are met, as outlined in the Alternate PTS Rule. This document explains the basis 

4 for these requirements (which establish the entry conditions to permit use of the Alternate PTS 

5 Rule) and describes methods by which the following four requirements can be met:

6

7 1. Criteria relating to the date of construction and design requirements: 

8 Paragraph (b) of 10 CFR 50.61a restricts the applicability of the Alternate PTS Rule to 

9 reactors for which a construction permit was issued before February 3, 2010, and whose 

10 RPV was designed and fabricated to the 1998 Edition, or earlier, of the ASME (formerly 

11 the American Society of Mechanical Engineers) Boiler and Pressure Vessel Code 

12 (“Code”). Chapter 4 of this document describes the criteria related to the date of plant 

13 construction by which licensees can make use of the Alternate PTS Rule.

14

15 2. Criteria relating to evaluation of plant-specific surveillance data: Paragraph (f) of 

16 the Alternate PTS Rule requires that the licensee verify that plant-specific surveillance 

17 data for the RPV in question satisfy three statistical tests described in the Alternate PTS 

18 Rule. These tests assess whether or not the plant-specific surveillance data are 

19 adequately predicted by the embrittlement trend curve (ETC) used in the Alternate PTS 

20 Rule. If this verification cannot be made, the licensee is required to submit an evaluation 

21 of plant-specific surveillance data to the Director of the NRC Office of Nuclear Reactor 

22 Regulation (NRR) that proposes a method to account for plant-specific surveillance data 

23 when assessing the subject RPV relative to the Alternate PTS Rule limits on RTMAX-X. 

24 Chapter 5 of this document describes methods by which licensees can satisfy these 

25 criteria.

26

27 3. Criteria relating to inservice inspection (ISI) data and non-destructive examination 

28 (NDE) requirements: Paragraph (e) of the Alternate PTS Rule requires that the 

29 licensee verify that the flaw density and size distributions detected within the beltline 

30 region of the RPV during a qualified examination under Section XI of the ASME Code [4]

31 are bounded by flaw tables contained within the Alternate PTS Rule. The Alternate PTS 

32 Rule requires that any flaws detected within the inner 1 inch or 10 percent of the wall 

33 thickness of the vessel base material, whichever is greater, do not exceed the limits 

34 shown in Table 2. If this verification cannot be made, the licensee must demonstrate 

that the RPV will have a through-wall cracking frequency (TWCF) of less than 1 × 10-6 35

36 per reactor year. Chapter 6 of this document describes methods by which licensees can 

37 satisfy these criteria.

38

39 4. Criteria relating to alternate limits on embrittlement: Chapter 7 of this document 

40 describes criteria by which licensees can assess plant-specific TWCF for cases in which

41 the RTMAX-X limits of the Alternate PTS Rule are not satisfied.

42

43 References to the “beltline” region of the RPV throughout this report refer to all regions of the 

RPV adjacent to the reactor core that are exposed to a fluence of 1 × 1017 n/cm2 44 or higher 

45 during the operating lifetime of the reactor [25]. Fluence values should be determined in 

46 accordance with methodology consistent with that specified in Regulatory Guide 1.190 [27] or 

47 using methods otherwise acceptable to the staff.

48

49 The NRC solicited input from interested stakeholders regarding an Alternate PTS regulation

50 during three public meetings in 2011 [17]. The Materials Reliability Program (MRP) developed 
4

1 recommended technical methods or approaches in seven areas. Those recommended 

2 technical methods are described in MRP-334 [10] and were provided to reduce the resources 

3 needed by utilities and the NRC to implement the Alternate PTS Rule, as well as to provide a 

4 consistent and acceptable level of safety subsequent to Rule implementation, especially in 

5 those instances in which alternate evaluations are required to demonstrate compliance with the 

6 Alternate PTS Rule. NRC responses to the recommendations made in MRP-334 are discussed 

7 in Chapter 3.

8

9
1

5

Table 2. Flaw Limit Tables from the Alternate PTS Rule 2

3

4

5

6

7

7

1 2. OVERVIEW OF THE ALTERNATE PTS RULE

2 2.1 Background

3 PTS events are system transients in a PWR during which there is a rapid cooldown that results in 

4 cold vessel temperatures with or without repressurization of the RPV. The rapid cooling of the 

5 inside surface of the RPV causes thermal stresses which can combine with stresses caused by high 

6 pressure. The aggregate effect of these stresses is an increase in the potential for fracture if a 

7 pre-existing flaw is present in a region of the RPV having significant embrittlement.

8

9 The PTS Rule, described in 10 CFR 50.61, establishes screening criteria below which the potential 

10 for a RPV to fail because of a PTS event is deemed to be acceptably low. These screening criteria 

11 effectively define a limiting level of embrittlement beyond which operation cannot continue without 

12 further plant-specific compensatory action or analysis unless the licensee receives an exemption 

13 from the requirements of 10 CFR 50.61. Some compensatory actions are neutron-flux reduction, 

14 plant modifications to reduce the PTS event probability or severity, and RPV annealing, which are 

15 addressed in 10 CFR 50.61(b)(3), (b)(4), and (b)(7) and in 10 CFR 50.66, “Requirements for 

16 Thermal Annealing of the Reactor Pressure Vessel” [5].

17

18 Currently, no operating PWR RPV is projected to exceed the 10 CFR 50.61 screening criteria before 

19 the expiration of its original 40-year operating license. However, several PWR RPVs are 

20 approaching the screening criteria, while others are likely to exceed the screening criteria during the 

21 period of license renewal.

22

23 The NRC developed technical bases that support updating the PTS regulations (see References [6], 

24 [7], [8], [9], and [14]). These technical bases concluded that the risk of through-wall cracking 

25 because of a PTS event is much lower than previously estimated. This finding indicated that the 

26 screening criteria in 10 CFR 50.61 are unnecessarily conservative. Therefore, the NRC developed 

27 10 CFR 50.61a, the Alternate PTS Rule, which provides alternate screening criteria based on the

28 updated technical bases. These technical bases covered the following topics:

29

30 a. Applicability of the Alternate PTS Rule;

31 b. Updated embrittlement correlation;

32 c. ISI volumetric examination and flaw assessments;

33 d. NDE-related uncertainties; and

34 e. Surveillance data.

35

36 A brief overview of these topics is provided in the following sections.

37

38 In addition, seven subsequent requirements that licensees must satisfy as a part of implementation 

39 of the Alternate PTS Rule are defined in paragraph (d) of 10 CFR 50.61a.

40

41 2.2 Applicability of the Alternate PTS Rule

42 The Alternate PTS Rule is based, in part, on analysis of information from three currently operating 

43 PWRs. Because the severity of the risk-significant transient classes (e.g., primary-side pipe breaks

44 and stuck-open valves on the primary side that may later reclose) is controlled by factors that are 

45 common to PWRs in general, the NRC concluded that the results and screening criteria developed

46 from these analyses could be applied with confidence to the entire fleet of operating PWRs. This
8

1 conclusion was based on an understanding of the characteristics of the dominant transients that 

2 drive their risk significance and on an evaluation of a larger population of high-embrittlement PWRs. 

3 This evaluation revealed no design, operational, training, or procedural factors that could credibly

4 increase either the severity of these transients or the frequency of their occurrence in the general 

5 PWR population above the severity and frequency characteristic of the three plants that were 

6 modeled in detail. The NRC also concluded that PTS events that were insignificant risk contributors

7 in these analyses are not expected to become dominant contributors in other plants.

8

9 The Alternate PTS Rule is applicable to licensees whose construction permits were issued before 

10 February 3, 2010, and whose RPVs were designed and fabricated to the 1998 Edition or an earlier

11 edition of the ASME Code.

12

13 2.3 Updated Embrittlement Correlation

14 The technical basis for the Alternate PTS Rule used many different models and parameters to 

15 estimate the yearly probability that a PWR will develop a through-wall crack as a consequence of 

16 PTS loading. One of these models was a revised ETC that uses information on the chemical

17 composition and neutron exposure of low-alloy steels in the RPV beltline region to estimate the 

18 fracture-mode transition temperature of these materials. Although the general trends predicted by 

19 the embrittlement models in 10 CFR 50.61 and the Alternate PTS Rule are similar, the mathematical 

20 form of the revised ETC in the Alternate PTS Rule differs substantially from the ETC in 

21 10 CFR 50.61. The ETC in the Alternate PTS Rule was updated to more accurately represent the

22 substantial amount of RPV surveillance data that has accumulated between the last revision of the 

23 10 CFR 50.61 ETC in the mid-1980s and the database supporting the 10 CFR 50.61a ETC, which

24 was finalized in 2002. The specifics of the updated ETC used in the Alternate PTS Rule are 

25 discussed in Section 5.1.

26

27 2.4 ISI Volumetric Examination and Flaw Assessments

28 The Alternate PTS Rule differs from 10 CFR 50.61 in that it requires licensees who choose to follow 

29 its requirements to analyze the results from ISI volumetric examinations done in accordance with 

30 Section XI, “Inservice Inspection of Nuclear Power Plant Components,” of the ASME Code. The 

31 analyses of ISI volumetric examinations may be used to determine whether the flaw density and 

32 size distribution in the licensee’s RPV beltline region are bounded by the flaw density and size 

33 distribution used in the development of the Alternate PTS Rule.

34

35 The Alternate PTS Rule was developed using a flaw density, spatial distribution, and size 

36 distribution determined from experimental data, as well as from physical models and expert

37 judgment. The experimental data were obtained from samples removed from RPV materials from 

38 cancelled plants (i.e., from the Shoreham and the Pressure Vessel Research Users Facility 

39 (PVRUF) vessels). The NRC considers that comparison of the results from qualified ASME Code

40 Section XI ISI volumetric examination is needed to confirm that the flaw density and size 

41 distributions in the RPV to which the Alternate PTS Rule may be applied are consistent with the flaw 

42 density and size distribution used in the development of the Alternate PTS Rule.

43

44 Paragraph (g)(6)(ii)(C) in 10 CFR 50.55a, “Codes and Standards” [11], requires licensees to 

45 implement ISI examinations in accordance with Supplements 4 and 6

46 [12] to Mandatory Appendix VIII to Section XI of the ASME Code. Supplement 4 contains 

47 qualification requirements for the RPV ISI volume from the clad-to-base-metal interface to the 

48 inner 15 percent of the RPV base material wall thickness. Supplement 6 contains qualification 
9

1 requirements for RPV weld volumes that lie within the outer 85% of the RPV base material wall 

2 thickness.

3

4 A simplified representation of the flaw density and size distribution used in the development of the 

5 Alternate PTS Rule is summarized by numerical values in Tables 2 and 3 of the Alternate PTS Rule 

6 for weld and plate/forging materials, respectively, as duplicated in Table 2. Hereafter, Tables 2 

7 and 3 of the Alternative PTS rule are referred to, collectively, as “the flaw tables.” These limits 

8 represent the number of flaws in each size range that were evaluated in the underlying technical 

9 bases. If a distribution of flaws having size and density greater than that of the flaw tables is found 

10 in a RPV, those flaws must be evaluated to ensure that they are not causing the TWCF to exceed a 

value of 1 × 10-6 11 .

12

13 The technical basis for the Alternate PTS Rule also indicated that flaws buried more deeply than 

14 1 inch from the clad-to-base interface are not as susceptible to brittle fracture as flaws of similar size 

15 located closer to the inner surface. Therefore, the Alternate PTS Rule does not require an 

16 assessment of the density of these flaws, but still requires large flaws, if discovered, to be evaluated 

17 for contributions to TWCF if they are within the inner three-eighths of the vessel base material wall 

18 thickness. Section II, “Discussion,” of the January 4, 2010, Federal Register Notice [2] for the 

19 Alternate PTS Rule indicates that the limitation for flaw acceptance, specified in Table IWB-3510-1

20 in Section XI of the ASME Code, approximately corresponds to the threshold for the sizes of flaws

21 that can make a significant contribution to TWCF if they are present in RPV material at this depth. 

22 Therefore, the Alternate PTS Rule requires that flaws exceeding the size limits in Table IWB-3510-1 

23 be evaluated for contribution to TWCF in addition to the other evaluations for such flaws that are

24 prescribed in the ASME Code.

25

26 The Alternate PTS Rule also clarifies that, to be consistent with Mandatory Appendix VIII to 

27 Section XI of the ASME Code, the smallest flaws that must be sized are 0.075 inch in through-wall 

28 extent (TWE). For each flaw detected that has a TWE equal to or greater than 0.075 inch, the

29 licensee is required to document the dimensions of the flaw, its orientation, its location within the 

30 RPV, and its depth from the clad-to-base-metal interface. Those planar flaws for which the major 

31 axis of the flaw is identified by a circumferentially-oriented ultrasonic transducer must be categorized 

32 as ‘‘axial.’’ All other planar flaws may be categorized as ‘‘circumferential.’’ If there is uncertainty 

33 about which flaw dimension constitutes the major axis for a given flaw identified with an ultrasonic 

34 transducer oriented in the circumferential direction, it should be considered as an axial flaw. The 

35 NRC may also use this information to evaluate whether plant-specific information gathered suggests 

36 that the NRC staff should generically re-examine the technical basis for the Alternate PTS Rule.

37

38 Surface cracks that penetrate through the RPV stainless steel clad and more than 0.070 inch into 

39 the welds or the adjacent RPV base metal were not included in the technical basis for the Alternate 

40 PTS Rule because these types of flaws have not been observed in the beltline of any operating 

41 PWR vessel. However, flaws of this type were observed in a boiling-water reactor (BWR) RPV

42 head in 1990 that were attributed to intergranular stress-corrosion cracking (IGSCC) of the stainless 

43 steel cladding. BWRs are not susceptible to PTS events and hence are not subject to the 

44 requirements of 10 CFR 50.61. However, if similar cracks were found in the beltline region of a

45 PWR, they would be a significant contributor to TWCF because of their size and location. As a 

46 result, the Alternate PTS Rule requires licensees to determine whether cracks of this type exist in 

47 the beltline weld region as a part of each required ASME Code Section XI ultrasonic examination.

48
10

1 2.5 NDE-Related Uncertainties

2 The flaw sizes shown in Table 2 represent actual flaw dimensions, while the results from ASME 

3 Code Section XI examinations are estimated dimensions. The available information indicates that,

4 for most flaw sizes in Table 2, qualified inspectors will oversize flaws. Comparing oversized flaws to 

5 the size and density distributions in Table 2 is conservative, but not necessary.

6

7 As a result of stakeholder feedback received during the NRC’s solicitation for public comments on a 

8 preliminary draft of the Alternate PTS Rule, the final published Alternate PTS Rule permits licensees 

9 to adjust the flaw sizes estimated by inspectors qualified under Supplements 4 and 6 to Mandatory 

10 Appendix VIII to Section XI of the ASME Code. The NRC also determined that licensees should be 

11 allowed to consider other NDE uncertainties, such as probability of detection (POD) and flaw density 

12 and location, because these uncertainties may affect the ability of a licensee to demonstrate 

13 compliance with the Alternate PTS Rule. As a result, the language in 10 CFR 50.61a(e) allows

14 licensees to account for the effects of NDE-related uncertainties in meeting the flaw size and density 

15 requirements of Table 2. The Alternate PTS Rule does not provide specific guidance on a 

16 methodology to account for the effects of NDE-related uncertainties, but notes that accounting for 

17 such uncertainties may be based on data collected from ASME Code inspector-qualification tests or 

18 any other tests that measure the difference between the actual flaw size and the size determined

19 from the ultrasonic examination. Because collecting, evaluating, and using data from ASME Code 

20 inspector-qualification tests requires extensive engineering judgment, the Alternate PTS Rule 

21 requires that the methodology used to adjust flaw sizes to account for the effects of NDE-related 

22 uncertainties be reviewed and approved by the Director of NRR.

23

24 2.6 Surveillance Data

25 Paragraph (f) of the Alternate PTS Rule defines the process for calculating the values for the 

26 reference temperature, RTMAX–X, for a particular RPV material. These values may be based on the 

27 RPV material’s copper (Cu), manganese (Mn), phosphorus (P), and nickel (Ni) weight percentages, 

28 reactor cold-leg coolant temperature, and fast neutron flux and fluence values, as well as the

29 unirradiated nil-ductility transition reference temperature, RTNDT.

30

31 The Alternate PTS Rule includes a procedure by which the RTMAX–X values, which are predicted 

32 for plant-specific RPV materials using an ETC, are compared to heat-specific 

33 surveillance data that are collected as part of surveillance programs under 

34 Appendix H, “Reactor Vessel Material Surveillance Requirements”

35 [13], to 10 CFR Part 50. The purpose of this comparison is to assess how well the surveillance data 

36 are represented by the ETC. If the surveillance data are close (closeness is assessed statistically) 

37 to the ETC, the predictions of this ETC are used. This is expected to be the case most often. 

38 However, if the heat-specific surveillance data deviate significantly and non-conservatively from the 

39 predictions of the ETC, this indicates that alternative methods (i.e., other than the ETC) may be 

40 needed to reliably predict the temperature-shift trend (and to estimate RTMAX–X) for the conditions 

41 being assessed. Therefore, the Alternate PTS Rule includes three statistical tests to determine the 

42 significance of the differences between heat-specific surveillance data and the ETC.

43
11

1 3. RESPONSES TO STAKEHOLDER FEEDBACK

2 The NRC solicited input from interested stakeholders regarding an Alternate PTS 

3 Implementation Regulatory Guide during three of public meetings in 2011 [17]. Based on those 

4 meetings, the Electric Power Research Institute’s (EPRI’s) Materials Reliability Program (MRP) 

5 developed recommended technical methods or approaches that would be useful for Alternate 

6 PTS Rule implementation in seven areas. Those recommended technical methods are 

7 described in MRP-334 [10]. That report provided fifteen specific recommendations that might 

8 reduce the resources needed by utilities and the NRC to implement the Alternate PTS Rule. In 

9 addition, EPRI stated that the recommendations would provide a consistent and acceptable 

10 level of safety subsequent to Alternate PTS Rule implementation, especially in those instances 

11 where evaluations are required to demonstrate compliance with the Alternate PTS Rule.

12

13 EPRI’s recommendations from MRP-334 are included in Table 3. The NRC responses to the 

14 recommendations made in MRP-334 are shown in the last column of the table. As indicated by 

15 some of the responses to EPRI’s recommendations, some of the guidance established in this 

16 document was adjusted for further clarification.

17

18
12

Table 3. EPRI Recommendations for Addressing Alternate PTS Rule Requirements [10] 1

No.

10 CFR 50.61a 

Paragraph

MRP-334

Recommendation

MRP-334

Justification

NRC

Respon

1 (f)(6), (a)(10) Adjustments should not

be made to sister-plant 

∆T30 values for the

purposes of performing

the surveillance data

tests in 10 CFR 50.61a.

The 10 CFR 50.61a ETC directly takes

into consideration the material properties

and the irradiation temperature in the 

calculation of ∆T30 and the residual for

each point of plant or sister-plant data.

Additional detail is provided in Section 3.1.

The NRC agrees with th

Sister-plant data are use

surveillance data set (se

surveillance procedure), 

to the data are made.

2 (f)(6), (a)(10) Limitations should be

placed on the variation in 

chemistry within a weld

heat that must be 

considered when

considering sister-plant

data.

An example of a weld heat with significant

variation is shown in Appendix A of this

report. Additional discussion is provided 

in Section 3.1.

The NRC agrees that va

composition should be co

a need to explain/unders

identified by the statistica

required for surveillance 

pointed out in Step 2(d)(i

entitled “Factors to Cons

Statistical Tests of Step 2 

the heading “Compositio

Variables.” Specific limit

these should be justified 

basis.

3 (f)(6)(vi), (a)(11) If the surveillance data 

tests cannot be met 

using exact time 

weighted average 

TC values, a tolerance 

that is equivalent to plus 

or minus one half of the 

range of the operating 

temperatures over the 

irradiation time should be 

considered. If the 

surveillance data tests 

can be satisfied by using 

a temperature within this 

tolerance range on TC,

adjustment to 

∆T30 values should not 

be required.

Large differences in irradiation 

temperature can also have significant 

effects on the calculated ∆T30 values. 

10 CFR 50.61a defines TC as a 

time-weighted average of the cold-leg 

temperature under normal full-power 

operating conditions. However, the values 

used in the development of the ETC may 

not have reflected an accurate 

time-weighted average for each specific 

plant. This tolerance accounts for the 

uncertainty in the actual operating 

temperature when the largest changes in 

the embrittlement index occurred. 

Additional discussion is provided in 

Section 3.1.

The NRC agrees that va

temperature should be c

a need to explain/unders

identified by the statistica

required for surveillance 

pointed out in Step 2(d)(

entitled “Factors to Cons

Statistical Tests of Step 

the heading “Compositio

Variables.” Specific limit

these should be justified 

basis.
13

No.

10 CFR 50.61a 

Paragraph

MRP-334

Recommendation

MRP-334

Justification

NRC

Respon

4 (f)(6)(vi), 

(f)(6)(v)(A), 

(f)(6)(v)(C)

A procedure is 

recommended in

Section 3.2 of the

report for calculating the

∆T30 adjustment to 

meet the requirements

of Paragraph (f)(6)(vi) 

when either the mean

or outlier statistical tests 

are failed when

applying the

requirements of

Paragraph (f)(6)(v) of 

10 CFR 50.61a.

The surveillance capsule data (SCD)

statistical tests in (f)(6) can be overly

conservative, so any needed 

adjustments to ∆T30 and RTMAX for

failing the SCD statistical tests should

be minimal. The proposed procedure

is simple to implement and addresses

the concerns with deviation from the

ETC without excessive conservatism.

Additional discussion is provided in 

Section 3.2.

The NRC adopted this re

to address failures of the 

Refer to Step 2(b)(i).

The NRC adopted a diffe

address outlier failures c

fluence datum. Refer to 

The NRC does not disag

proposed procedure for 

but considers that use o

should be justified on a c

basis.

5 (f)(6)(vi), 

(f)(6)(v)(B)

An industry-consensus 

ETC that addresses 

irradiation-induced 

shifts caused by fluence 

values up to 

1 × 1020 n/cm2

(E > 1 MeV) should be 

used to calculate the 

predicted values of ∆T30

if the slope test is not 

satisfied.

Data from several plants, with fluence 

exposures of up to 8.5 × 1019 n/cm2, 

have been evaluated and no slope-test

deviations were encountered. It is 

possible that high-fluence surveillance 

capsule data could fail the statistical 

slope test. The industry has developed 

a Coordinated Surveillance program in 

order to obtain high fluence data from 

power reactors. The intent is to use 

this data to develop an 

industry-consensus ETC that 

addresses irradiation-induced shifts 

caused by fluence values up to 

1 × 1020 n/cm2 (E > 1 MeV). Additional 

discussion is provided in Section 3.3.

As of the current date, a

by broad consensus bet

and industry stakeholde

Also, the data from the c

surveillance program wil

in large quantities until 2

The NRC agrees that, in 

slope-test failure, the be

understanding of irradiat

mechanics should be us

explain the failure and s

actions. However, beca

consensus on this matte

reached, the NRC is not 

guidance on this matter.

6 (f)(6)(vi), 

(f)(6)(v)(A),

(f)(6)(v)(B), 

(f)(6)(v)(C)

Recommendations for 

possible criteria for not

adjusting ∆T30 values in 

accordance with 

(f)(6)(vi) when the tests 

of (f)(6)(v)(A), (B), and 

(C) cannot be satisfied 

are provided in 

Section 3.4.

There are several scenarios where an 

adjustment to ∆T30 values would not be 

appropriate. These scenarios and 

criteria are discussed in Section 3.4.

As discussed in Step 2(d

Regulatory Guide, the N

there are many situation

adjustment of the ΔT30 v

needed nor justifiable. S

situations are discussed 

However, specific excep

provided; these should b

case-specific basis.
14

No.

10 CFR 50.61a 

Paragraph

MRP-334

Recommendation

MRP-334

Justification

NRC

Respon

7 (d)(4), (c)(3) A provision should be 

included in the 

proposed Regulatory 

Guide for direct 

calculation of 

TWCF95-TOTAL using the 

plant-specific RTMAX-XX

values for cases in 

which the RTMAX-XX

limits in Table 1 of 

10 CFR 50.61a cannot 

be met.

There are some conservatisms in the 

RTMAX-XX screening limits. Also, plate 

and forging limited plants are likely to 

reach the more restrictive limit for 

circumferential welds before reaching 

the plate or forging limits. Additional 

discussion is provided in Section 3.5

The NRC agrees with th

recommendation. Guida

to the Regulatory Guide 

that refers to the approp

and methods in Section 

NUREG-1874.

8 (e)(1) Any small flaws 

detected by ISI 

within +0.5 inch of the 

outer boundaries of the 

ASME Section XI 

inspection volume can 

be treated as potential 

plate flaws. All other 

flaws within the 

inspection volume 

should be treated as 

weld flaws.

The plate flaw limits in Table 3 of 

10 CFR 50.61a should only be 

evaluated for RPV base metal flaws 

not affected by welding that are 

detected by ISI. Additional discussion 

is provided in Section 3.6.

The NRC agrees with th

Guidance for establishin

flaws are in the plate or 

included in this NUREG 

Section 6.1).

9 (e)(1) Plate flaws that have a 

depth (TWE) of 0.4 inch

or more should be 

evaluated as weld 

flaws.

This is needed to ensure that the small 

flaws created during fabrication of the 

plates and forgings, which is what was 

simulated in the FAVOR probabilistic 

fracture mechanics analysis code, 

have not been expanded to 

larger-sized flaws by the effects of 

welding, which were simulated by 

FAVOR as weld flaws. Additional 

discussion is provided in Section 3.6.

The NRC disagrees with 

recommendation becaus

A case-specific justificat

to be developed to demo

plate flaw should be reca

weld flaw.
15

No.

10 CFR 50.61a 

Paragraph

MRP-334

Recommendation

MRP-334

Justification

NRC

Respon

10 (e)(4)(i) The proposed 

Regulatory Guide 

should provide criteria 

or methods for 

determining whether 

the deviation in flaw 

limits will have a 

detrimental effect on 

the applicability of the 

PTS Screening Criteria 

in Table 1 of 

10 CFR 50.61a.

Such criteria would reduce burden on 

both the NRC and industry while 

providing an acceptable level of safety 

and quality. Additional discussion is 

provided in Section 3.7 and 

Appendix E.

The NRC agrees with th

Both simplified and more 

procedures are given in 

Regulatory Guide that a

evaluations that may be 

the flaw limits are excee

industry may choose to j

alternatives to these pro

case-specific basis.

11 (e)(4)(i) Deviations to the flaw 

limits for plants that 

have low embrittlement 

should be generically 

allowed. Low 

embrittlement can be 

defined as having 

RTMAX-X values that are 

equivalent to or less 

than a TWCF of 1% of 

the PTS risk objective, 

or approximately 

1 × 10-8 events per 

year.

The vessel material at these plants 

would still have high enough fracture 

toughness that even large increases in 

the number of flaws above the 

10 CFR 50.61a limits would not result 

in reaching the PTS risk objective of 

1 × 10-6 events per year. Additional 

discussion is provided in Section 3.7.

The NRC disagrees with 

recommendation and the 

justification. If the flaw l

would be inappropriate t

relationships between TW

RTMAX-X that depend on 

being correct to address 

Simplified criteria for ass

fracture are provided in t

Guide that may be used 

or, alternatively, another 

case-specific solution m
16

No.

10 CFR 50.61a 

Paragraph

MRP-334

Recommendation

MRP-334

Justification

NRC

Respon

12 (e)(1) The flaw limits in 

Tables 2 and 3 of 

10 CFR 50.61a should 

be applied to axial flaws 

only.

This proposed approach addresses 

what the real technical concern should 

be; that is, the maximum number of 

axial flaws that could lead to potential 

vessel failure. Additional discussion is 

provided in the flaw-orientation 

discussion in Section 3.7.1.

The NRC disagrees with 

All flaws, whether axial o

circumferential, are cons

comparison to Tables 2 

10 CFR 50.61a because 

subpart (e), “Examinatio

Assessment Requireme

paragraph (e)(1)(iii) state

“(iii) For each flaw detec

inspection volume descr

paragraph (e)(1) with a t

extent equal to or greate

0.075 inches, the license

document the dimension

including through-wall ex

whether the flaw is axial 

circumferential in orienta

location within the reacto

including its azimuthal a

and its depth embedded 

clad-to-base-metal interf

Paragraph (e)(4) states, 

“(4) The licensee shall p

to demonstrate that the 

will have a TWCF of less 

per reactor year if the AS

Section XI volumetric ex

required by paragraph (c

this section indicates an

following:

“(i) The flaw density and 

inspection volume descr

paragraph (e)(1) exceed 

Tables 2 or 3 of this sec
17

No.

10 CFR 50.61a 

Paragraph

MRP-334

Recommendation

MRP-334

Justification

NRC

Respon

13 (e)(1), (e)(4) A qualitative 

assessment process 

should be provided for 

addressing a flaw that 

exceeds the maximum 

size allowed in the flaw 

limit tables but is less 

than 0.875" in 

through-wall extent.

A qualitative process that considers 

flaw orientation, flaw location, flaw 

size, and material properties is 

proposed in Section 3.7.1.

The NRC agrees with th

recommendation is addr

of the Regulatory Guide

14 (e)(1), (e)(4), 

(e)(5)

A quantitative 

assessment process 

should be provided 

(a) for instances in 

which multiple flaws 

exceed the flaw limit 

tables or (b) if the 

requirements of the 

qualitative process 

cannot be satisfied.

A quantitative process that considers 

the contribution of individual flaws to 

the total vessel TWCF is provided in 

Section 3.7.2. If needed, an additional 

process that reduces some of the 

conservatisms in the Section 3.7.2 

process is provided in Appendix F.

The NRC agrees with th

recommendation is addr

of the Regulatory Guide

15 (e)(1) When assessing the 

flaw-limit tables, 

multiple flaws that are 

combined into one flaw 

in accordance with the 

ASME Section XI 

proximity rules should 

not be counted as 

multiple ISI flaws.

The ASME Section XI proximity rules 

were considered in the development of 

the flaw distributions used in the 

technical basis for 10 CFR 50.61a. 

Additional discussion is provided in 

Section 3.7.2.

The NRC agrees with th

Flaw proximity is consid

Regulatory Guide, as ind

Figure 3.

19

1 4. GUIDANCE FOR CRITERIA RELATING TO THE DATE OF 

2 CONSTRUCTION AND DESIGN REQUIREMENTS

3 The Alternate PTS Rule is applicable to plants with construction permits issued before 

4 February 3, 2010, and with RPVs designed and fabricated in accordance with the 1998 Edition or an 

5 earlier edition of the ASME Code. This chapter provides guidelines and supporting bases for this 

6 requirement.

7

8 4.1 Requirements in the Alternate PTS Rule

9 Paragraph (b), “Applicability,” of the Alternate PTS Rule identifies that the Alternate PTS Rule is 

10 applicable for plants for which a construction permit was issued after February 3, 2010. Section II, 

11 “Discussion,” of the January 4, 2010, Federal Register Notice [2] includes the following:

12

13 ...The final rule is applicable to licensees whose construction permits were issued before 

14 February 3, 2010 and whose reactor vessels were designed and fabricated to the 

15 American Society of Mechanical Engineers Boiler and Pressure Vessel Code (ASME 

16 Code), 1998 Edition or earlier. This would include applicants for plants such as Watts 

17 Bar Unit 2 who have not yet received an operating license. However, it cannot be 

18 demonstrated, a priori, that reactor vessels that were not designed and fabricated to the 

19 specified ASME Code editions will have material properties, operating characteristics,

20 PTS event sequences and thermal hydraulic responses consistent with those evaluated 

21 as part of the technical basis for this rule. Therefore, the NRC determined that it would 

22 not be prudent at this time to extend the use of the rule to future PWR plants and plant 

23 designs such as the Advanced Passive (AP) 1000, Evolutionary Power Reactor (EPR) 

24 and U.S. Advanced Pressurized Water Reactor (US–APWR). These designs have 

25 different reactor vessels than those in the currently operating plants, and the fabrication 

26 of the vessels based on these designs may differ from the vessels evaluated in the 

27 analyses that form the bases for the final rule. Licensees of reactors who commence 

28 commercial power operation after the effective date of this rule or licensees with reactor 

29 vessels that were not designed and fabricated to the 1998 Edition or earlier of the ASME 

30 Code may, under the provisions of § 50.12, seek an exemption from § 50.61a(b) to apply 

31 this rule if a plant-specific basis analyzing their plant operating characteristics, materials 

32 of fabrication, and welding methods is provided....

33

34 Regulatory guidance for the above requirements is discussed in the next section.

35

36 4.2 Regulatory Guidance 

37 The purpose of the restriction given in Section 4.1 on the applicability of the Alternate PTS Rule 

38 embrittlement limits is that the structural and thermal hydraulic analyses that established the 

39 basis for the Alternate PTS Rule only represented plants constructed before February 3, 2010. 

40 A licensee that applies the Alternate PTS Rule to a plant with a construction permit issued after 

41 February 3, 2010, must demonstrate that the risk-significant factors controlling PTS for the plant 

42 in question are adequately addressed by the technical-bases calculations, which are detailed in 

43 NUREG-1806 [6] and its supporting technical references. Factors to be considered in this 

44 evaluation should include, but might not be limited to, the following:

45
20

1 • the event sequences that may lead to overcooling of the RPV;

2

3 • the thermal-hydraulic response of the nuclear steam supply system (NSSS) in response 

4 to such sequences;

5

6 • Characteristics of the RPV design (e.g., vessel diameter, vessel wall thickness, and 

7 operating pressure) that influence the stresses that develop in the beltline region of the 

8 vessel (as defined in Section 1.2) in response to the event sequences; and

9

10 • Characteristics of the RPV materials and their embrittlement behavior.

11
21

1 5. GUIDANCE FOR CRITERIA RELATING TO THE EVALUATION OF

2 PLANT-SPECIFIC SURVEILLANCE DATA

3 The information in this chapter was developed and published in support of the development of 

4 the Alternate PTS Rule. Relevant information from this work is repeated here for completeness 

5 and clarity [14]. In some cases, the information in Reference [14] is further clarified here 

6 compared to the technical bases supporting the Alternate PTS Rule. Additionally, the 

7 information presented in this chapter adds to that from Reference [14] in two respects: in 

8 Section 5.2 procedures for data grouping (e.g., treatment of “sister plant” data) before 

9 performing the statistical tests are described, and Section 5.6.2 describes procedures that can 

10 be used if the statistical tests are failed.

11

12 In this chapter, procedures are described by which the plant-specific surveillance data that are 

13 collected as part of surveillance programs (in accordance with Appendix H to 10 CFR 50) can 

14 be analyzed to assess how well they are represented by the ETC for DT30 adopted in the 

15 Alternate PTS Rule. If the surveillance data are “close” (closeness is assessed statistically) to 

16 the prediction of the ETC, the predictions of the ETC are used. Statistically significant 

17 differences between plant-specific data sets and the ETC prediction identify situations in which

18 more focused attention is warranted, and are taken as an indication that methods other than, or 

19 in addition to, the ETC may be needed to predict DT30 trends. While standard statistical 

20 procedures exist to assess the significance of differences between individual data sets and ETC 

21 predictions, similarly standard procedures are not as commonly available to assess the practical 

22 importance of such differences, or to adjust the data to account for these differences. This

23 chapter concludes with discussions of (1) factors that may be considered if the statistical tests 

24 are failed and (2) procedures that could be used to adjust the predictions of the ETC in certain 

25 circumstances.

26

27 The remainder of this chapter is divided into six sections, as follows:

28

29 • Section 5.1 describes the ΔT30 ETC. Details on the development of this ETC appear in 

30 ORNL/TM-2006/530 [9].

31

32 • Section 5.2 describes entry conditions that need to be met to use the proposed 

33 surveillance assessment procedure.

34

35 • Section 5.3 describes different types of deviations between plant-specific surveillance 

36 data and the trends represented by the DT30 ETC from Reference [9].

37

38 • Section 5.4 describes procedures that statistically assess the deviations described in 

39 Section 5.3.

40

41 • Section 5.5 applies the procedures of Section 5.4 to the surveillance database that 

42 supported the development of the ETC [9].

43

44 • Section 5.6 discusses factors that should be considered if the statistical tests of 

45 Section 5.4 are failed; it also describes procedures that could be used to adjust the 

46 predictions of the ETC in certain circumstances.

47
22

1 5.1 Embrittlement Trend Curve

2 As detailed in ORNL/TM-2006/530 [9], the numerical coefficients in the following equations were 

3 determined by fitting them to the DT30 data collected in 10 CFR 50 Appendix H surveillance 

4 programs that have been reported to the NRC through approximately 2002. This DT30 ETC, 

5 which is used in the Alternate PTS Rule, has the following form:

6

DT30 = MD + CRP (1)

7

0.5

e

2.471

C MD = A × (1- 0.001718 × T ) × (1+ 6.13 × P × Mn ) × φt (2)

8















=

−

−

−

1.417x10 for welds

1.561x10 for plates

1.140x10 for forgings

A

7

7

7

(3)

9

( ) ( ) ( e) e e

1.191 CRP = B × 1+ 3.77 ×Ni × f Cu ,P × g Cu ,Ni,φt (4)

10















=

155.0 for welds

135.2 for plates in vessels manufactured by Combustion Engineering

102.5 for plates in vessels not manufactured by Combustion Engineering

102.3 for forgings

B (5)

11







>

≤ = MIN[Cu, MAX(Cu )] for Cu 0.072 wt%

0 for Cu 0.072 wt% Cu

e

e (6)

12





 = 0.301 for all other materials

0.243 for Linde 80 welds MAX(Cu ) e (7)

13

( ) [ ]

[ ] 









− + × > >

− > ≤

≤

=

Cu 0.072 1.359 (P - 0.008) for Cu 0.072 and P 0.008

Cu 0.072 for Cu 0.072 and P 0.008

0 for Cu 0.072

f Cu ,P

0.668

e

0.668

e e (8)

14

( ) ( )











 φ + × − × − φ = +

0.629

[log t (1.1390 Cu ) (0.448 Ni) 18.120] g Cu ,Ni, t 0.5 (0.5 x tanh 10 e e e e (9)

15















φ < × 















φ

×

φ ×

φ φ ≥ ×

φ = for 4.39 10 n / cm / sec

4.39 10 t

t for 4.39 10 n / cm / sec

t

10 2

0.2595 10

10 2

e (10)

16

17 The dependent variable, DT30, is estimated by Eqn. (1) in degrees Fahrenheit. The standard 

18 deviation of residuals about Eqn. (1) for different product forms and copper contents appears in 
23

1 Table 4. The units and descriptions of independent variables in these equations are given in 

2 Table 5. As with any equation calibrated to empirical data, inaccuracies have a greater 

3 tendency to occur at the extremes of, or beyond this limits of, the calibration dataset. Users of 

4 Eqn. (1) should therefore exercise caution when applying it to conditions near to or beyond the 

5 extremes of its calibration dataset, which appear in Table 5. For example, for materials having 

6 copper contents below 0.072 weight percent, Eqn. (1) predicts negative DT30 values when the 

7 irradiation temperature exceeds 582.1°F; clearly, negative shift values are incorrect.

8

9 Table 4. Standard Deviation of Residuals about Eqn. (1) 10

Product Form Standard Deviation, σ (°F)

Cu ≤ 0.072 wt % Cu > 0.072 wt %

Weld

18.6

26.4

Plate 21.2 (a)

Forging 19.6

11 a. Includes the standard reference materials.

12

13 Table 5. Independent Variables in the Eqn. (1) ETC and the Ranges and Mean Values of the 

14 Calibration Dataset

Variable Symbol Units

Values of Surveillance Database

Average Standard 

Deviation Minimum Maximum

Neutron Fluence

(E > 1 MeV) φt n/cm2 1.24E+19 1.19E+19 9.26E+15 1.07E+20

Neutron Flux (E > 1 MeV) φ n/cm2/sec 8.69E+10 9.96E+10 2.62E+08 1.63E+12

Irradiation Temperature TC °F 545 11 522 570

Copper Content Cu weight % 0.140 0.084 0.010 0.410

Nickel Content Ni weight % 0.56 0.23 0.04 1.26

Manganese Content Mn weight % 1.31 0.26 0.58 1.96

Phosphorus Content P weight % 0.012 0.004 0.003 0.031

15

16
24

1 5.2 Data Used in Statistical Tests

2 Surveillance data used to perform the statistical tests outlined in this chapter should be as

3 follows:

4

5 1. Materials Evaluated (i.e., the Meaning of “Plant-Specific”)

6 a) When performing the statistical tests required by the Alternate PTS Rule, each shell 

7 and weld material in the RPV beltline region for which 10 CFR 50 Appendix H 

8 surveillance data exist should be evaluated. The statistical tests should be 

9 performed separately for each heat.

10 b) For each heat for which data are available, the ΔT30 values used in the statistical 

11 tests should include data from both of the following sources:

12 • Data obtained for the heat of material in question as part of a 10 CFR 50 

13 Appendix H surveillance program conducted for the plant in question; and

14 • Data obtained for the heat of material in question as part of a 10 CFR 50 

15 Appendix H surveillance program conducted for any other plant that is operating, 

16 or has operated, under a license issued by the NRC. Data from this source is 

17 often referred to as having come from a “Sister Plant.” Such data may have 

18 different best-estimate chemistry values and different irradiation temperatures 

19 (e.g., the Cu and T values for Heat “123” at Plant “XYZ” may be 0.23 and 553°F, 

20 respectively, whereas the Cu and T values for Heat “123” at Plant “ABC” may be 

21 0.21 and 544°F, respectively). The statistical tests described in Sections 5.3

22 and 5.4 operate on the residuals (i.e., the difference between the surveillance 

23 measurement of ΔT30 and the prediction of the ETC). In so doing, the ETC takes 

24 account of these plant-specific variations in chemistry and/or temperature, so

25 there is no need to make further adjustments to account for so-called “Sister 

26 Plant” data (see Reference [10]).

27 2. Data-Quantity Requirements: To perform these statistical tests, at least three 

28 plant-specific DT30 values measured at three different fluence levels should be available. 

29 If this condition is not met, the ETC described in the Alternate PTS Rule should be used 

30 to estimate DT30.

31 3. Data-Binning Requirements

32 a) As discussed in Item 1, data obtained for the heat of material in question as part of a 

33 10 CFR 50 Appendix H surveillance program conducted for any plant that is 

34 operating, or has operated, under a license issued by the NRC should be binned 

35 together and considered in these statistical evaluations.

36 b) For plates and forgings, Charpy data is often obtained for different orientations of the 

37 notch relative to the primary working direction of the plate or forging. For the 

38 purpose of these statistical tests, ΔT30 values for a particular plate or forging should 

39 be computed from unirradiated specimens having the same notch orientation. Once 

40 ΔT30 values are calculated, different notch orientation data from the same heat of 

41 material should be binned together for the purpose of the statistical tests described in 

42 Sections 5.3 and 5.4. This is appropriate because the differences in unirradiated 

43 transition temperature caused by notch orientation will be subtracted out when

44 ΔT30 values are calculated.

45 4. Data-Characterization Requirements: For all materials meeting the requirements of the 

46 three preceding items, the following information is needed:

47 • heat identification

48 • plant identification

49 • capsule identification
25

1 • product form

2 • notch orientation

3 • the unirradiated reference temperature, RTNDT(U)

4 • DT30

5 • Charpy V-notch energy data used to estimate ΔT30

6 • fluence

7 • operating time

8 • cold-leg temperature under normal full-power operating conditions (Tc)

9 o Note: Tc (°F) is determined as the time-weighted average coolant 

10 temperature of the cold leg of the reactor coolant system covering the 

11 time period from the start of full-power operation through the end of 

12 licensed operation.

13 • copper (Cu) content

14 • nickel (Ni) content

15 • phosphorus (P) content

16 • manganese (Mn) content

17 • citation

18 The values of Cu, Ni, P, and Mn are the best-estimate values for the material. For a 

19 plate or forging, the best-estimate value is normally the mean of the measured values for 

20 that plate or forging. For a weld, the best-estimate value is normally the mean of the 

21 measured values for a weld deposit made using the same weld wire heat number as the 

22 critical vessel weld. If these values are not available, either the upper limiting values 

23 given in the material specifications to which the vessel material was fabricated, or 

24 conservative estimates (i.e., mean plus one standard deviation) based on generic data

25 should be used. Table 4 of 10 CFR 50.61a provides upper-bound estimates for P 

26 and Mn. Similarly, upper bound estimates for Cu and Ni are provided in 10 CFR 50.61.
26

1 5.3 Statistical Evaluation of Surveillance Data

2 In developing this surveillance assessment procedure, consideration was given to the 

3 development of statistical tests capable of detecting the four types of deviations between 

4 heat-specific surveillance data and the DT30 ETC expressed by Eqns. (1) through (10). These 

5 deviations are illustrated in Figure 1 and are identified as Types A, B, C, and D. The potential 

6 origins and implications of these types of statistical deviations are described briefly in the 

7 following sections. Only situations in which the data suggest that the DT30 ETC may provide a 

8 non-conservative prediction (i.e., situations in which the data exceed the ETC prediction) are 

9 illustrated in Figure 1. The opposite situation is also possible; i.e., situations in which the data 

10 are underestimated by the ETC. However, from a regulatory standpoint, only non-conservative 

11 predictions are important.

12

13 • Type A Deviations: For Type A deviations, measurements differ from the mean ETC 

14 prediction more or less uniformly at all fluence levels. Additionally, the magnitude of this 

15 deviation is larger than would be expected based on the population of data used to 

16 calibrate the ETC. Potential origins of Type A deviations may include, but are not limited 

17 to, errors in the chemical composition values or errors in the unirradiated DT30 value 

18 associated with the surveillance sample. The implication of a statistically significant 

19 Type A deviation (procedures for evaluating statistical significance are described in 

20 Section 5.4.1) is that the ETC may systematically underestimate the value of DT30 for the 

21 heat of steel being evaluated.

22

23 • Type B Deviations: For Type B deviations, measurements differ from the mean ETC 

24 prediction by an amount that increases as fluence increases. Additionally, the 

25 magnitude of this deviation is larger than would be expected based on the population of 

26 data used to calibrate the ETC. Potential origins of Type B deviations may include, but 

27 are not limited to, (1) errors in the temperature value associated with the surveillance 

28 sample or (2) the existence in the surveillance sample of an embrittlement mechanism 

29 that is not represented in the ETC calibration dataset. A recent review of high-fluence 

30 data has revealed the possible existence of Type B deviations in a large empirical 

31 database (see Figure 2 and Reference [15]). The implications of a statistically significant 

32 Type B deviation (procedures for evaluating statistical significance are described in 

33 Section 5.4.2) are that the ETC may systematically underestimate the value of DT30 for 

34 the heat of steel being evaluated and that the magnitude of this underestimation will 

35 increase as the plant operation continues.

36

37 • Type C Deviations: For Type C deviations, measurements differ from the mean ETC by 

38 exhibiting more scatter than would be expected based on the population of data used to 

39 calibrate the ETC. Potential origins of Type C deviations may include, but are not limited 

40 to, errors made in testing, labeling, or controlling the notch orientation of surveillance 

41 specimens. The implication of statistically significant Type C deviations is not that the 

42 ETC may systematically underpredict the true embrittlement, but rather that the ability of 

43 the surveillance data to provide insight into embrittlement trends is called into question 

44 for a specific heat of material.

45

46

47 • Type D Deviations: For Type D deviations, one or more of the DT30 measurements differ 

48 significantly from the mean ETC prediction even though all other measurements for the 

49 heat of steel being evaluated agree well with that prediction. The magnitude of the 
27

1 deviation of these outliers is larger than would be expected based on the population of 

2 data used to calibrate the ETC. Potential origins of Type D deviations may include, but 

3 are not limited to, (1) a large measurement error in the single datum or (2) the rapid 

4 emergence in the surveillance sample of an embrittlement mechanism that is not 

5 represented in the calibration dataset (e.g., rapid emergence of a Type B deviation). 

6 The implication of a statistically significant Type D deviation (procedures for evaluating 

7 statistical significance are described in Section 5.4.3) is that the ETC may systematically 

8 underestimate the value of DT30 for the heat of steel being evaluated.

9

10 If they are statistically significant, Type A, B, and D deviations all give rise to concerns that the 

11 embrittlement trends predicted by the ETC may produce non-conservative estimates of the 

12 embrittlement experienced by materials used to construct the RPV that is being evaluated. For 

13 this reason, methods to assess the statistical significance of these deviations are described in 

14 Sections 5.4.1 through 5.4.3. Type C deviations, if they are statistically significant, suggest that 

15 the surveillance program for the material in question may not provide a reliable indication of 

16 embrittlement trends for that material. Because Appendix H to 10 CFR 50 requires the 

17 performance of surveillance on the “limiting” (meaning “most irradiation-sensitive”) materials 

18 used to construct the RPV beltline, the existence of a Type C deviation is important from a 

19 regulatory viewpoint, but not in the context of indicating a potential non-conservatism in the 

20 predictions of the DT30 ETC adopted in the Alternate PTS Rule. For this reason, statistical 

21 procedures to detect Type C deviations were not included in the Alternate PTS Rule (see 

22 Reference [1]) and, therefore, are not described in Section 5.4.
28

1

2

3

4

Figure 1. Four Types of Deviation between Heat-Specific Surveillance Data and a DT30 ETC. 5

– 30(MEAS)

T

D

30(ETC,mean)

T

D

φt

30

T

D

φt

Mean ETC Prediction 

Surveillance 

Measurements 

ETC Confidence Bounds

– 30(MEAS)

T

D

30(ETC,mean)

T

D

φt

30

T

D

φt

– 30(MEAS)

T

D

T

D

T

D

– 30(MEAS)

T

D

30(ETC,mean)

T

D

φt

30

T

D

φt

Type A

(measurements 

uniformly offset 

from ETC)

Type B

(measurements diverge 

from ETC; different 

fluence trend)

Type C

(measurements have 

more uncertainty than 

ETC calibration data)
29

1

2

3 Figure 2. DT30 Prediction Residuals Based on Eqn. (1). Low flux points are from power reactors 

4 while high flux points are from test reactors. Similar trends exist for both plates 

5 and welds [15].

6

7

8 5.4 How to Perform Statistical Tests of Surveillance Data

9 This section describes how to perform the Type A (mean test), Type B (slope test), and Type D 

10 (outlier test) assessments of surveillance data that are required by the Alternate PTS Rule.

11

12 5.4.1 Type A Deviations

13 As illustrated by Figure 1, Type A deviations are characterized by measurements that differ from 

14 the mean ETC prediction more or less uniformly at all fluence levels. The following procedure 

15 can be used to detect Type A deviations when the DT30 measurements for a specific heat of 

16 material are uniformly underpredicted by Eqn. (1). A statistical significance level of α = 1%

17 (i.e., 2.33 standard deviations) is recommended for this one-sided test; Section 5.4.4 discusses 

18 the rationale for this selection.

19

20 A-1. Ensure that the entry conditions of Section 5.2 have been met.

21 A-2. Estimate the residual for each datum using the following formula:

T30(measured) T30(predicted) r = ∆ − ∆ (11)

where DT30(measured) represents each individual measurement, and DT30(predicted) 22 is 

23 the value of DT30 predicted using Eqn. (1) and best-estimate composition and 

24 exposure values for the plant from which the companion DT30(measured) value was 

25 obtained.

26

27

Product Form = Forgings, Prediction = RM-9

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

0.0E+00 4.0E+19 8.0E+19 1.2E+20 1.6E+20 2.0E+20

Φ [n/cm2

]

Predicted - Measured 

D

T30 [oF]

High Flux

Low Flux

Product form = Forgings

Prediction = 10 CFR 50.61a
30

1

2

3 A-3. Estimate the mean residual (rmean) for the DT30 dataset using the following 

4 formula:

∑{ } =

=

n

i 1

mean ir

n

1

r (12)

5 where n is the number of DT30 measurements for the specific heat of material 

6 being assessed.

7 A-4. Estimate the maximum allowable residual (rmax) using the following formula:

n

2.33

rmax

σ = (13)

8 where n is the number of DT30 measurements for the specific heat of material 

9 being assessed and σ is the population standard deviation, which is taken from

10 Table 4.

11 A-5. If rmean exceeds rmax, the subject dataset is judged to show a Type A deviation.

12

13 5.4.2 Type B Deviations

14 As illustrated by Figure 1, Type B deviations are characterized by measurements that differ from 

15 the ETC prediction by an amount that increases as fluence increases. The following procedure 

16 is used to detect Type B deviations. Similarly to the test for Type A deviations, a statistical 

17 significance level of α = 1% is recommended for this one-sided test; Section 5.4.4 discusses the 

18 rationale for this selection.

19

20 B-1. Ensure that the entry conditions of Section 5.2 have been met.

21 B-2. For each measured DT30 value, calculate the difference between the measured 

22 and predicted value of DT30 using Eqn. (11). As illustrated in Figure 3, plot r vs.

23 the log10 value of fluence. The abscissa is expressed in this way because 

24 embrittlement, as quantified by DT30, increases approximately linearly with the 

25 logarithm of fluence.

26

27 Figure 3. Procedure to Assess Type B Deviations.

28

29 B-3. Using the method of least squares, estimate the slope (m) of the data plotted as 

30 shown in Figure 3. Also estimate the standard error of the estimated value of the 

31 slope, se(m).

 

Log10(φt)

m

r
31

1 B-4. Estimate the T-statistic for the slope as follows:

se(m)

m Tm = (14)

2 B-5. Establish the critical T-value as follows:

T t(α,n 2) CRIT(α) ≡ − (15)

3 where t(...) represents Student’s t-distribution, α is the selected significance 

4 level, and n is the number of DT30(Measured) values. Adoption of α = 1% is 

5 recommended for regulatory implementation of this procedure (Section 5.4.4

discusses the rationale for this selection). Table 6 provides values of TCRIT(1%) 6 .

7

8 Table 6. α = 1% Student’s t-Values

Number of 

DT30 Values, n

n-2 One-Tailed TCRIT

(1%, n-2)

3 1 31.82

4 2 6.96

5 3 4.54

6 4 3.75

7 5 3.36

8 6 3.14

9 7 3.00

10 8 2.90

11 9 2.82

12 10 2.76

13 11 2.72

14 12 2.68

15 13 2.65

9

B-6. If Tm exceeds TCRIT(α) 10 ,the subject dataset is judged to show a Type B deviation.

11

12 5.4.3 Type D Deviations

13 As illustrated by Figure 1, Type D deviations are characterized by one or more of the 

14 DT30 measurements differing significantly from the mean ETC prediction even though all other 

15 measurements for the heat of steel being evaluated agree well with the ETC. Similarly to the 

16 tests for both Type A and Type B deviations, a statistical significance level of α = 1% is 

17 recommended for this one-sided test; Section 5.4.4 discusses the rationale for this selection.

18

19 D-1. Ensure that the entry conditions of Section 5.2 have been met.

20 D-2. Estimate the normalized residual, r*, for each or the n observations in the 

21 DT30 dataset using the following formula:

σ = r

r* (16)

22 where r is defined from Eqn. (11) and σ is the population standard deviation 

23 taken from Table 4.

24 D-3. Find the largest and second largest r* values from Step D-2; designate these r*1

25 and r*2 respectively.
32

1 D-4. Find the limit values rLIMIT(1) and rLIMIT(2) corresponding to n in Table 7. These 

2 threshold values correspond to a significance level of α = 1%. Appendix A of this 

3 document describes how the values of C1 and C2 that appear in Table 7 were 

4 derived.

D-5. If r*1 ≤ rLIMIT(1) and r*2 ≤ rLIMIT(2) 5 , the dataset is judged to not show a Type D 

6 deviation. Otherwise the surveillance dataset is judged to show a Type D 

7 deviation.

8

9 Table 7. α = 1% Threshold Value for the Outlier Test

n rLIMIT(2) rLIMIT(1)

3 1.55 2.71

4 1.73 2.81

5 1.84 2.88

6 1.93 2.93

7 2.00 2.98

8 2.05 3.02

9 2.11 3.06

10 2.16 3.09

11 2.19 3.12

12 2.23 3.14

13 2.26 3.17

14 2.29 3.19

15 2.32 3.21

17 2.37 3.24

26 2.53 3.36

64 2.83 3.62

Note: In Appendix A, rLIMIT(1) is referred to as 

C2 and rLIMIT(2) is referred to as C1. The 

notation is changed here to improve clarity.

10

11 5.4.4 Comments on the Statistical Tests

12 The significance level recommended for regulatory implementation of the Type A, B, and D tests 

13 is α = 1%. At this significance level, there is less than a 1% chance that the underlying cause of 

14 the detected difference (Type A, B, or D) between the plant-specific surveillance data and the 

15 value of DT30 predicted by Eqn. (1) has occurred as a result of chance alone. The following 

16 considerations informed this recommendation:

17

18 • A 1% significance level makes it more difficult for plant-specific surveillance data to be 

19 declared “different” from the predictions of the ETC than has traditionally been the case. 

20 For example, both 10 CFR 50.61 [3] and Revision 2 of Regulatory Guide 1.99 [16] use a 

21 “2σ” criterion which, for a one-sided test, implies an α = 2.5% significance level. The 

22 staff views this change in significance level from 2.5% to 1% as appropriate in view of 

23 the greater physical and empirical support of Eqn. (1) than was available at the time both 

24 10 CFR 50.61 and Revision 2 of Regulatory Guide 1.99 were adopted.

25 • The selection of the α = 1% significance level represents a conscious tradeoff between 

26 the competing goals of providing high confidence in the determination of a statistically 

27 significant difference between plant-specific surveillance data and the predictions of 

28 Eqn. (1) versus limiting the risk of judging that a particular set of plant-specific 
33

1 surveillance data are similar to the DT30 ETC when, in fact, they are not. The former 

2 goal is achieved by adopting a small value for α, whereas the latter goal is achieved by 

3 increasing the α value.

4 • If a plant-specific data set is determined by these statistical tests to be “different” from 

5 the predictions of Eqn. (1), the alternatives that are available to licensees to either 

6 troubleshoot the cause of the difference or develop new/replacement data are limited, 

7 expensive, or have long lead times. Consequently, while the NRC recognizes that it is 

8 important to compare plant-specific data to generic trends and to take action when such 

9 comparisons show differences, the NRC believes it is technically justified to reserve 

10 such actions for those cases in which the differences are the most clear and are the 

11 most practically significant.

12

13 Considering all of these factors, a significance level for Type A, B, and D deviations of α = 1% 

14 was adopted in the Alternative PTS Rule.

15

16 5.5 Evaluation of Plant Data Relative to the Statistical Tests

17 The information presented in this section appeared previously in Reference [14]. It is repeated 

18 here for clarity and completeness.

19

20 The plant surveillance data used in ORNL/TM-2006/530 [9] to calibrate Eqn. (1) were evaluated 

21 according to the statistical tests detailed in Section 5.4. After filtering to remove heats having 

22 less than three DT30 observations at three different fluence values, 159 data sets remained for 

23 evaluation. These sets included data from both PWRs and BWRs, as well as from plants that 

24 are no longer in operation. While the BWR and ex-plant data are not directly pertinent to the 

25 Alternate PTS Rule, they were retained in this analysis for information. The detailed results of 

26 these analyses appear in Appendix B.

27
34

1 Table 8 summarizes the 14 heats of material from 12 plants that show a statistically significant 

2 deviation of Type A, B, or D, while Table 9 summarizes the proportion of heats in the 

3 surveillance database that exhibit statistically significant deviations at the α = 1% level. The 

4 information in Table 9 demonstrates that both the mean and the outlier tests exhibit a rate of 

5 deviation above the expected value for a population of 159 data sets (i.e., 1% of 159, or 1.59),

6 while the deviation rate of the slope test is close to this expected value. These observations 

7 suggest that, for the mean and outlier tests, the assumption that the residuals are distributed 

8 normally about the ETC (Eqn. (1)) may be incorrect. One possible explanation for this situation 

9 could be the presence of systematic biases in some of the unirradiated T30 values (arising from, 

10 for example, measurement error or imprecision). Such biases, if present, would have no effect 

11 on the detection rate of the slope test because it operates on the differences between 

12 DT30 residuals, not on their absolute values. Thus, any systematic biases in the unirradiated 

13 DT30 values would be subtracted out of the DT30 difference values that are used to perform the 

14 slope test and, consequently, could not affect the outcome of the test. Conversely, the mean 

15 and outlier tests both operate on the absolute values of DT30 residuals. Consequently, 

16 systematic biases in the unirradiated values of DT30 could affect the normalcy of the 

17 DT30 residuals and, thereby, the detection rate of the mean and outlier tests.

18
35

1 Table 8. Heats of Material that Exhibit Statistically Significant Deviations (α = 1%) of Type A, B, 

2 or D

Plant Name Heat ID Product 

Form

Population 

σ (°F)

Number 

of DT30

Values

Test Results

A -

Mean 

Test

B -

Slope 

Test

D -

Outlier 

Test

 Operating PWRs

San Onofre Unit 3 PSO301 Plate 18.6 3 FAIL PASS FAIL

D.C. Cook 2 PCK201 Plate 21.2 8 FAIL PASS PASS

Beaver Valley 1 PBV101 Plate 21.2 8 FAIL PASS FAIL

Callaway WCL101 Weld 18.6 4 FAIL PASS FAIL

Surry 1 WSU101 Weld 26.4 3 FAIL PASS FAIL

Indian Point 2 PIP203 Plate 21.2 3 FAIL PASS PASS

Sequoyah 1 FSQ101 Forging 19.6 8 FAIL PASS PASS

Sequoyah 1 WSQ101 Weld 26.4 4 FAIL PASS PASS

Sequoyah 2 WSQ201 Weld 26.4 4 PASS PASS FAIL

 Operating BWRs

River Bend 1 and Oyster 

Creek WRB01 Weld 18.6 3 FAIL PASS FAIL

 Decommissioned PWRs

Maine Yankee PMY01 Plate 21.2 6 FAIL PASS PASS

Zion 1 WZN101 Weld 18.6 5 PASS FAIL PASS

 Decommissioned BWRs

Big Rock Point PBR01 Plate 21.2 5 FAIL FAIL FAIL

3

4

5

6 Table 9. Proportion of Material Heats in the Current Surveillance Database that Exhibit 

7 Statistically Significant Deviations (α = 1%) of Type A, B, or D

Test Type

Data Sets that Show Statistically 

Significant (α = 1%) Deviation

Count Percent

A Mean Test 11 6.9%

B Slope Test 2 1.3%

D Outlier Test 7 4.4%

8
36

1 5.6 Considerations When Statistical Tests Are Failed

2 Failure of plant-specific surveillance data to pass any of these statistical tests indicates a 

3 situation in which Eqn. (1) may underestimate the embrittlement magnitude; use of Eqn. (1) in 

4 such situations without additional justification is not advised. Such failures suggest that more 

5 focused assessment of the surveillance data is warranted, and are taken as an indication that 

6 methods other than, or in addition to, the ETC may be needed to reliably predict DT30 trends. 

7 However, the most appropriate approach may not be a heat-specific adjustment of the ETC 

8 predictions in all cases. For example, statistically significant differences may indicate situations 

9 in which the available data (i.e., the ΔT30 measurements and/or the composition and exposure 

10 values associated with the ΔT30 measurements) are incorrect, making adjustment of the ETC 

11 predictions to match these data unwise.

12

13 The discussion of this section is divided into two parts. First, a list of factors is provided that 

14 could be helpful in diagnosing the reason that the surveillance data failed the statistical tests. 

15 This is followed by a discussion of situations in which adjustments of the ETC predictions may

16 be made using simple procedures.

17

18 5.6.1 Factors to Consider When Statistical Tests Are Failed

19 Before adjustments of the ETC predictions to match surveillance data are considered, a detailed 

20 assessment of the accuracy and appropriateness of the DT30 data is suggested. Such an 

21 assessment should consider, but not be limited to, the following factors:

22

23 • Unirradiated RTNDT value: As noted in Section 5.5, the occurrence of mean and outlier 

24 failures in the available surveillance data exceeds the statistically expected value. Both 

25 of these tests are sensitive to the accuracy of the unirradiated RTNDT value. In these 

26 cases, a records investigation of the unirradiated RTNDT value, and/or the performance of 

27 additional testing of archival material may provide a more accurate estimate of RTNDT, 

28 which may explain the reason for failure of the mean and/or outlier tests.

29 • Irradiated DT30 values: While most CVN energy vs. temperature curves from which

30 DT30 values are estimated are based on 8 to 12 individual measurements, some data 

31 sets are more limited, which can increase uncertainty in the DT30 estimate. If any of the 

32 statistical tests are not satisfied, a review of the individual CVN energy vs. temperature 

33 curves may help to reveal the cause of the failure.

34 • Composition and exposure variables: The input variables to Eqn. (1) are subject to 

35 variability and are often based on limited data. However, the predictions of Eqn. (1)

36 depend on these input variables, particularly Cu content, fluence, temperature, and 

37 Ni content. If a sensitivity analysis reveals that small variations of the values input to 

38 Eqn. (1) rationalize the failure of the statistical tests, this might indicate that more refined 

39 information concerning input values (e.g., additional measurements) could explain the 

40 reason for the failure of the statistical tests.

41 • Notch orientation: The DT30 value for plate and forging materials is sensitive to the 

42 orientation of the notch in the CVN specimen relative to the primary working direction of 

43 the plate or forging. Differences in notch orientation between the unirradiated DT30 value 

44 and the DT30 value of all the irradiated specimens could help to explain why the mean 

45 test is not satisfied. Similarly, differences in notch orientation between the unirradiated 

46 DT30 value and the DT30 value of the irradiated specimens in a single capsule could help 

47 to explain why the outlier test is not satisfied. In these situations, the outcome of a 
37

1 records search or metallurgical investigation of the tested specimens might be useful to 

2 explain the reason for the failure of the statistical tests.

3 • Comparative trends analysis: In addition to CVN specimens, surveillance capsules also 

4 contain tensile specimens that are part of the capsule testing program. Like ΔT30, the 

5 increase in yield strength with irradiation (ΔYS) also follows certain trends. If the 

6 ΔYS data for a particular material that failed the statistical tests follows the trends 

7 exhibited by ΔYS data for a similar composition, this information might indicate that the 

8 CVN specimens were taken from the wrong material, or have been somehow 

9 mislabeled.

10

11 5.6.2 Specific Procedures

12 If a statistical test is failed, it is permissible to use any of the following procedures:

13

14 1. Mean Test (Type A) Failure: A procedure to adjust ETC predictions to account for a 

15 failure of the mean test is illustrated in Figure 4 (left side). This procedure is as follows:

16

17 a. Calculate the value ADJ as follows:

max ADJ r r = mean − (17)

18 b. Adjust the prediction of Eqn. (1) as follows:

T MD CRP ADJ D 30( ADJ ) = + + (18)

c. Use the value ΔT30(ADJ) 19 in place of ΔT30 in all calculations required by the Alternate 

20 PTS Rule for the material that failed the mean test.

21

22 2. Slope Test (Type B) Failure: One procedure for adjusting ETC predictions to account for 

23 a failure of the slope test is to adjust the ETC predictions, Eqn. (1), based on the greater 

24 increase of embrittlement with fluence suggested by the plant-specific data. The specific 

25 procedure used should be technically justified and documented.

26

27 3. Outlier Test (Type D) Failure (Not Satisfied at Low Fluence): Figure 4 (right side) 

28 illustrates a situation in which a ΔT30 value measured at low fluence is the cause of 

29 failing the outlier test. Such a failure is not considered structurally relevant to a PTS 

30 evaluation, and may therefore be ignored, provided both of the following conditions are 

31 satisfied:

32 a. The fluence of the datum that caused the outlier test failure, φtLOW, is less than 

33 10% of the fluence at which the PTS evaluation is being performed, φtEVAL, and

34 b. After elimination of the datum measured at φtLOW, the entry conditions for the 

35 surveillance tests are still met (i.e., at least three data points measured at three 

36 different fluence levels remain) and all three statistical tests are satisfied with the 

37 reduced data set.

38

39 Other approaches to assessment of surveillance data in which all surveillance measurements 

40 are bounded are subject to review and approval by the NRC.

41
38

1

2 Figure 4. Specific Procedures to Account for Failure of the Mean Test (left) or Low Fluence 

3 Outlier Statistical Test (right).

4

5

6 5.7 Summary and Conclusions

7 This chapter discusses the three statistical tests required by the Alternate PTS Rule and 

8 methods by which data can be evaluated relative to these tests. The aim of performing these 

9 tests is to determine whether the surveillance data are sufficiently “close” to the predictions of 

10 the ETC that the predictions of the ETC may be used. From a regulatory perspective, it is of 

11 particular interest to determine whether plant-specific surveillance data deviate significantly from 

12 the predictions of the ETC in a manner that suggests that the ETC is likely to underpredict 

13 plant-specific trends. To this end, a statistical significance level of α = 1% was adopted, which 

14 represents a conscious tradeoff between the competing goals of providing high confidence in 

15 the determination of a statistically significant difference between plant-specific surveillance data 

16 and the DT30 ETC versus limiting the risk of judging that a particular set of plant-specific 

17 surveillance data are similar to the ETC when, in fact, they are not. The α = 1% tests will show 

18 fewer statistically significant deviations than did either 10 CFR 50.61 or Revision 2 of Regulatory 

19 Guide 1.99, both of which use a “2σ” criterion that, for a one-sided test, implies an α = 2.5% 

20 significance level. The staff views this change in significance level (from 2.5% to 1%) as 

21 appropriate in view of the greater empirical support for the ETC used in the Alternate PTS Rule 

22 than was available at the time either 10 CFR 50.61 or Regulatory Guide 1.99 (Revision 2) were 

23 adopted.

24

D

T30(MEAS)

–

D

T30(ETC,mean)

φt

D

T30

φt

Mean ETC Prediction 

Surveillance 

Measurements 

ETCConfidence Bounds

D

T30(MEAS)

–

D

T30(ETC,mean)

φt

D

T30

φt

Mean Test Failure

rmean

rmax

ADJ = rmean + rmax

C2

r(max1)

φ φtEVAL tLOW

Low Fluence Outlier Test Failure
39

1 Three tests were adopted in the Alternate PTS Rule because they collectively indicate when an 

2 underprediction of embrittlement magnitude by the ETC may have occurred. In Appendix B,

3 these statistical tests are applied to the operating plant surveillance database assembled 

4 through (approximately) 2002. Of the heats of material considered, only 14 heats from 

5 12 plants fail one or more of the three statistical tests. Failure of plant-specific surveillance data 

6 to pass any of the statistical tests indicates a situation in which the ETC may underestimate the 

7 embrittlement magnitude; use of Eqn. (1) in such situations without additional justification is not 

8 advised. Such failures suggest that more focused assessment of the surveillance data is 

9 warranted, and are taken as an indication that methods other than, or in addition to, the ETC 

10 may be needed to reliably predict DT30 trends. However, the most appropriate approach may 

11 not be a heat-specific adjustment of the ETC predictions in all cases. For example, statistically 

12 significant differences may indicate situations in which the available data (i.e., the 

13 ΔT30 measurements and/or the composition and exposure values associated with the 

14 ΔT30 measurements) are incorrect, making adjustment of the ETC predictions to match these 

15 data unwise. This chapter includes both a list of factors that could be helpful in diagnosing the 

16 reason that the surveillance data fail the statistical tests and a discussion of certain situations for 

17 which adjustments of the ETC predictions can be made.

18

41

1 6. GUIDANCE RELATING TO ISI DATA AND NDE REQUIREMENTS

2 6.1 Background

3 The technical basis for the Alternate PTS Rule concludes that flaws as small as 0.1 inch in 

4 through-wall extent contribute to the TWCF, and nearly all of the contributions come from flaws 

5 buried less than 1 inch below the inner diameter surface of the reactor vessel wall. Therefore, the 

6 Alternate PTS Rule specifies flaw limits for such flaws as indicated in Table 2 in this document. For 

7 weld flaws that exceed the sizes prescribed in Table 2, the risk analyses indicated that a single flaw 

contributes a significant fraction of the 1 × 10-6 8 per reactor year limit on TWCF. Therefore, if a flaw 

9 that exceeds the sizes and quantities described in the flaw tables is found in a reactor vessel, it is 

10 important to assess it individually.

11

12 The technical basis for the Alternate PTS Rule also indicated that flaws buried deeper than 1 inch

13 from the clad-to-base interface did not contribute as significantly to total risk as flaws of similar size 

14 located closer to the inner surface. Therefore, the Alternate PTS Rule does not require the 

15 comparison of the density of these flaws, but still requires large flaws, if any are discovered, to be 

16 evaluated for contributions to TWCF if they are within the inner three-eighths of the vessel wall’s

17 thickness. Because flaws greater than three-eighths of the vessel wall’s thickness from the inside 

18 surface do not contribute to TWCF, flaws greater than three-eighths of the vessel wall’s thickness 

19 from the inside surface need not be analyzed.

20

21 The limitation for flaw acceptance, specified in Table IWB-3510-1 in Section XI of the ASME Code, 

22 approximately corresponds to the threshold for flaw sizes that can make a significant contribution to 

23 TWCF if flaws of such sizes are present in reactor vessel material at this depth. Therefore, the final 

24 rule requires that flaws exceeding the size limits in Table IWB-3510-1 be evaluated for their 

25 contribution to TWCF in addition to the other evaluations for such flaws that are prescribed in the 

26 ASME Code.

27

28 Paragraph (e) of the Alternate PTS Rule describes a number of tests and conditions on the 

29 collection and analysis of NDE data that are intended to provide reasonable assurance that the 

30 distribution of flaws assumed to exist in the probabilistic fracture mechanics (PFM) calculations that 

31 provided the basis for the RTMAX-X limits in Table 1 in this document provide an appropriate, or 

32 bounding, model of the population of flaws in the RPV of interest. These tests and conditions, the 

33 totality of which are illustrated by the diagram in Figure 5 in this document, go beyond a simple 

34 comparison of the NDE data to the flaw table limits in Table 2.

35

36 A summary of the process depicted in Figure 5 is as follows:

37

38 Step A: All plant-specific recordable flaw data (see Figure 6) should be collected for the inner 

39 three-eighths of the wall thickness (3/8t) for the base material and weld metal 

examination volumes† 40 within the RPV beltline region by performing ultrasonic test (UT) 

41 volumetric examinations and by using procedures, equipment, and personnel required 

42 by Supplements 4 and 6 to Mandatory Appendix VIII to Section XI of the ASME Code.

 † Any flaws that are detected within the ultrasonic transducer scan paths but are located outside the 

examination volume required by Section XI of the ASME Code should also be included in the flaw table 

evaluation.
42

1 Step B: The plant-specific flaw data from Step A should be evaluated for axial flaw surface 

2 connection. Any flaws with a through-wall extent greater than or equal to 0.075 inch, 

3 axially oriented and located at the clad-to-base-metal interface, should be verified to not 

4 be connected to the RPV inner surface using surface-examination techniques capable 

5 of detecting and characterizing service-induced cracking of the RPV cladding. Eddy 

6 current and visual examination methods are acceptable to the staff for detection of 

7 cladding cracks. An appropriate quality standard shall be implemented to ensure that 

8 these examinations are effective at identification of surface cracking as required by 

9 Criterion IX, “Control of Special Processes,” in Appendix B to 10 CFR 50. This

10 requires, in part, that measures are established to assure that special processes, 

11 including nondestructive testing, are controlled and accomplished by qualified 

12 personnel using qualified procedures in accordance with applicable codes, standards, 

13 specifications, criteria, and other special requirements. Appropriate quality standards 

14 for implementation of surface examinations are identified in Section V, “Nondestructive 

15 Examination,” and Section XI of the ASME Code.

16 Step C: If the results of Step B are acceptable, the plant-specific flaw data should be evaluated 

17 for acceptability in accordance with the flaw-acceptance standards in 

18 Table IWB-3510-1 in Section XI of the ASME Code.

19 Step D: If the results of Step C are satisfactory, or if the results of Step F (if applicable) are 

20 acceptable, the plant-specific flaw data should be compared to Tables 2 and 3 of the 

21 Alternate PTS Rule. A specific example of how this step may be performed is shown in 

22 Section 6.3.

23 Step E: If the results of Step B indicate that any axial flaw with a TWE greater than 0.075 inch

24 are connected to the RPV inner surface, or if the results of Step F (if applicable) are not 

25 acceptable, other plant-specific assessment is required and the provisions of the 

26 Alternate PTS Rule may not be used.

27 Step F: If the evaluation associated with Step C is not successful (i.e., if any flaws exceed the 

28 flaw-acceptance standards in Table IWB-3510-1), the flaws must be evaluated and 

29 found to be acceptable in accordance with the flaw-evaluation methods in Section XI of 

30 the ASME Code, and the flaws must be evaluated for acceptability according to 

31 10 CFR 50.61a (see Step I).

32 Step G: If the results of Step D are not acceptable, NDE uncertainties may be accounted for in 

33 the evaluation. Appendix C describes the development and application of one 

34 methodology acceptable to the staff that accounts for uncertainties in NDE data. This 

35 method may be used for the purpose of developing more realistic vessel-specific flaw 

36 depth and density distributions for comparison to Tables 2 and 3 of the Alternate PTS 

37 Rule, as well as for use in a plant-specific PFM analysis. The methodology considers 

38 flaw-sizing error, a flaw-detection threshold, POD, and a prior flaw-distribution 

39 assumption. It uses a Bayesian updating methodology to combine the observed NDE 

40 data with the available flaw data and models used as part of the PTS re-evaluation 

41 effort. The licensee must submit the adjustments made to the volumetric test data to 

42 account for NDE-related uncertainties as described in 10 CFR 50.61a(c)(2).

43 Step H: The revised flaw distribution results of Step G should be used to compare the revised 

44 plant-specific flaw data to Tables 2 and 3 of the Alternate PTS Rule.
43

1 Step I: If the results of Step H are not acceptable, all flaws should be evaluated for 

2 acceptability using one of the following approaches:

3 1. Preclusion of brittle fracture. Satisfactory demonstration of upper shelf behavior, 

4 which precludes brittle fracture, can be based on keeping temperature above 

5 RTNDT + 60°F using the following steps:

6 i. Compute the irradiated RTNDT for all flaws as follows:

• Determine the unirradiated value of RTNDT and RTNDT(u) 7 for the material at 

8 each flaw location.

9 • Determine the fluence at each flaw location.

10 • Compute DT30 for each flaw using Eqn. (1) and the fluence at each flaw 

11 location.

12 • Compute the flaw-specific value of RTNDT as RTNDT(u) + DT30 for each 

13 flaw.

14 ii. Assuming a lower-bound PTS transient temperature of 75°F, upper shelf 

15 behavior is assured if RTNDT + 60 ≤ 75°F. Therefore, the flaw-specific value 

16 of RTNDT should be less than or equal to 15°F.

17 iii. The evaluation associated with Step I is acceptable if the flaw-specific value 

18 of RTNDT is less than or equal to 15°F for all flaws.

19 2. Calculate the plant-specific TWCF using a plant-specific PFM analysis. 

20 A plant-specific PFM analysis to calculate TWCF is complex, and there are many 

21 variations of inputs possible for such an analysis. Therefore, specific guidance for 

22 plant-specific PFM analysis to calculate TWCF is not included in this document. 

23 General considerations to include in a plant-specific PFM analysis are provided in 

24 Section 6.2.2. A discussion of the methodology that was used in performing TWCF 

25 calculations for PTS may be found in NUREG-1806 [6], NUREG-1807 [21], and 

26 NUREG/CR-6854 [22]. The steps associated with conducting a plant-specific PFM 

27 calculation are as follows:

28 i. Perform a Bayesian update of the flaw distribution:

29 • Apply the procedures of Appendix C and obtain revised flaw-depth and 

30 flaw-density parameters (similar to those shown in Table 16).

31 ii. Calculate the TWCF using a PFM computer code (e.g., the code in 

32 ORNL/TM-2012/567, “Fracture Analysis of Vessels – Oak Ridge 

33 FAVOR, v12.1, Computer Code: Theory and Implementation of Algorithms, 

34 Methods, and Correlations” [20]):

35 • Run the generalized procedure for generating flaw-related inputs for the 

36 FAVOR Code described in NUREG/CR-6817 [19] using the revised 

37 flaw-depth and flaw-density parameters.

38 • Develop necessary plant-specific input using the guidance in 

39 NUREG-1806 [6], NUREG-1807 [21], and NUREG/CR-6854 [22].

40 • Run a plant-specific PFM analysis.

41 • Calculate the TWCF.

42 iii. Compare the plant-specific TWCF to the TWCF limit specified in the 

43 Alternate PTS Rule:

44 • The evaluation associated with Step I is acceptable if the calculated 

TWCF is less than or equal to the limit of 1 × 10-6 45 events per reactor year 

46 specified in the Alternate PTS Rule.

47 Step J: If the results of Step I are not acceptable, the licensee must perform a plant-specific 

48 assessment for PTS and submit the assessment to the Director of the Office of Nuclear 

49 Reactor Regulation for review and approval.
44

1 Step K: If the results of Step D or (if applicable) Step H or (if applicable) Step I are satisfactory, 

2 the screening criteria contained in Table 1 of the Alternate PTS Rule may be applied to 

3 the plant in question. The plant-specific assessment, including explicit details and 

4 results, must be submitted to the Director of the Office of Nuclear Reactor Regulation 

5 for review and approval in the form of a license amendment at least three years before 

6 RTMAX–X is projected to exceed the Alternate PTS Rule screening criteria.

7

8 Based on this process, the guidelines discussed in this chapter include the following components:

9

10 1. Guidance for plants for the case in which RPV flaws fall outside the applicability of the flaw 

11 tables in the Alternate PTS Rule, including:

12 i. Guidance on a procedure to preclude brittle fracture.

13 ii. Guidance on considerations to include in a plant-specific PFM analysis.

14 2. Guidance for initial evaluation of NDE data obtained from qualified ISI examinations.

15 3. Guidance on methods for further evaluation of NDE data obtained from qualified ISI 

16 examinations, including:

17 i. Guidance on the elements and NDE techniques associated with the qualified ISI 

18 examinations performed in accordance with Mandatory Appendix VIII to Section XI 

19 of the ASME Code to assess compliance with the requirements of the Alternate PTS 

20 Rule.

21 ii. Guidance on a mathematical procedure that can be used to adjust NDE data to 

22 account for flaw-detection and sizing errors, as well as guidance on comparison of 

23 the adjusted data to the population of flaws assumed in the PFM analysis used to 

24 develop the Alternate PTS Rule.

25

26 Guidance for these topics is described in detail in the following sections.

27
45

1

2

3

4

Figure 5. Flow Diagram with Recommendations for Meeting the Requirements of the Alternate PTS Rule. 5

6
46

1 6.2 Guidance on Criteria Relating to Alternate Limits on

2 Embrittlement

3 This section describes guidance for situations in which plant-specific NDE results do not meet the 

4 acceptance standards of Table IWB-3510-1 in Section XI of the ASME Code or are not within the 

5 limits prescribed by the flaw tables in the Alternate PTS Rule. In such situations, additional efforts 

6 beyond those described in Section 6.1 may still allow application of the PTS Screening Criteria 

7 shown in Table 1 in this document.

8

9 The guidelines discussed in this section include the following components:

10

11 • Guidance on a procedure that can be used to preclude brittle fracture based on RTNDT

12 information for RPV flaws that fall outside the applicability of the Alternate PTS Rule flaw 

13 tables.

14 • Guidance on a procedure that can be used to combine the NDE data with the population of 

15 flaws assumed in the TWCF calculations to estimate a total flaw distribution that is predicted 

16 to exist in the RPV, as well as guidance on the use of the total flaw distribution if a licensee 

17 chooses to conduct a PFM calculation.

18

19 These topics are described in detail in the following sections.

20

21 6.2.1 Guidance on a Procedure to Preclude Brittle Fracture

22 Guidance on a mathematical procedure to preclude brittle fracture for flaws that exceed the 

23 acceptance standards of Table IWB-3510-1, as identified by Step I in Figure 5 in this document, is 

24 provided in this section. Such guidance is based on Section II, “Discussion,” provided in the 

25 Supplemental Information section of the Federal Register Notice for the Alternate PTS Rule [2], 

26 which includes the following discussion with respect to ISI Volumetric Examination and Flaw 

27 Assessments:

28

29 The technical basis for the final rule also indicates that flaws buried deeper than 1 inch from 

30 the clad-to-base interface are not as susceptible to brittle fracture as similar size flaws 

31 located closer to the inner surface. Therefore, the final rule does not require the comparison 

32 of the density of these flaws, but still requires large flaws, if discovered, to be evaluated for 

33 contributions to TWCF if they are within the inner three-eighths of the vessel thickness. The 

34 limitation for flaw acceptance, specified in ASME Code, Section XI, Table IWB-3510-1, 

35 approximately corresponds to the threshold for flaw sizes that can make a significant 

36 contribution to TWCF if present in reactor vessel material at this depth. Therefore, the final 

37 rule requires that flaws exceeding the size limits in ASME Code, Section XI, 

38 Table IWB-3510-1 be evaluated for contribution to TWCF in addition to the other evaluations 

39 for such flaws that are prescribed in the ASME Code.

40

41 A simplified procedure is described here for the situation in which the as-found NDE data for a 

42 plant reveals that one or more flaws fall outside the maximum range of the Alternate PTS Rule 

43 flaw tables, and that the flaws do not satisfy ASME Code Section XI flaw-acceptance criteria. 

44 Therefore, the following situation results:

45

46 • The NDE data was obtained using a qualified examination in accordance with Appendix VIII 

47 of Section XI of the ASME Code, thereby satisfying Step A in Figure 5.
47

1 • All recordable flaws are located away from the clad-to-base-metal interface. Therefore, 

2 Step B is satisfied with a result of “NO” in Figure 5.

3 • Some of the recordable flaws do not meet ASME Code Section XI flaw-acceptance criteria. 

4 Therefore, Step C is not satisfied with a result of “NO” in Figure 5.

5

6 Thus, based on the above, Step F in Figure 5 requires that flaw evaluation be performed with 

7 acceptable results in accordance with ASME Code Section XI for those recordable flaws that do not 

8 meet ASME Code Section XI flaw-acceptance criteria. In addition, further evaluation for 

9 acceptability using Step I is required. For the purposes of this example, it is assumed that flaw 

10 evaluation in accordance with ASME Code Section XI is successful and is not described in the 

11 assessment that follows.

12

13 For the further evaluation required by Step I in Figure 5, one approach is to determine whether brittle 

14 fracture can be precluded. For this situation, a recommended evaluation could be as follows:

15

16 • Step I - Preclude Brittle Fracture Option:

17 o Satisfactory demonstration of upper shelf behavior, which precludes brittle 

18 fracture, can be based on maintaining temperature above RTNDT + 60°F [23].

19 o Compute the irradiated RTNDT for all flaws as follows:

o Determine the unirradiated value of RTNDT and RTNDT(u) 20 for the material at 

21 each flaw location.

22 o Determine the fluence at each flaw location.

23 o Compute DT30 for each flaw using Eqn. (1) and the fluence at each flaw 

24 location.

25 o Compute the flaw-specific value of RTNDT as RTNDT(u) + DT30 for each flaw.

26 o Assuming a lower-bound PTS transient temperature of 75°F, upper shelf behavior is 

27 assured if RTNDT + 60 ≤ 75°F. Therefore, the flaw-specific value of RTNDT should be 

28 less than or equal to 15°F.

29 o Flaws are acceptable if the flaw-specific value of RTNDT is less than or equal 

30 to 15°F. For this situation, the decision at Step F is “ACCEPTABLE” in Figure 5.

31 o Flaws are not acceptable if the flaw-specific value of RTNDT is greater than 15°F. For 

32 this situation, the decision at Step F is “NOT ACCEPTABLE” in Figure 5, other 

33 assessment is required (Step E in Figure 5), and the provisions of the Alternate PTS 

34 Rule may not be used.

35 • For the case in which Step F is “ACCEPTABLE”, all recordable flaws less than the 

36 maximum of 0.1t or 1.0" must be checked against the flaw tables (Step D in Figure 5).

37 • If the result of Step D is “PASS,” the RTMAX-X limits of the Alternate PTS Rule may be used 

38 (Step K in Figure 5).

39

40 A sample calculation demonstrating acceptability by the “Preclude Brittle Fracture” option in Step I is 

41 demonstrated in Table 10.

42

43
48

Table 10. Sample Brittle Fracture Assessment 1

2

3

4

STEP D: Preclude Brittle Fracture for J-2 Flaws

Height on 

RPV (inches)

RPV Circumferential 

Position (°)

Depth Below Clad, 

x (inches)

X 235.00 150.1 1.66 1.02E+19 0.6714 6.85E+18 54.0 4.02E+09

Y 235.14 157.7 2.70 1.02E+19 0.5231 5.34E+18 54.0 3.13E+09

RTNDT(u) (°F) Cu

(wt %)

Ni

(wt %)

Mn

(wt%)

P

(wt%)

X -40 0.09 0.48 1.300 0.004

Y -40 0.09 0.48 1.300 0.004

Flaw No. Temperature

(°F)

φ te

Effective Fluence 

(n/cm2

)

A B Max(Cue) Cue f(Cue,P) g(Cue,Ni,φ te) Mfterm

(°F)

CRPte

(°F

X 550.0 1.272E+19 1.417E-07 155.0 0.2430 0.0900 0.0683 9.412E-01 29.2 25.

Y 550.0 1.057E+19 1.417E-07 155.0 0.2430 0.0900 0.0683 9.253E-01 26.6 25.

Notes: a. Determined from a plant-specific fluence map.

b. From RPV surveillance program.

Flaw No.

Attenuated Fluence 

at Flaw Location 

(n/cm2

)

LINDE 80 Weld Chemistryb

End-of-Life 

Effective Full 

Power Years 

(EFPY)

Attenuated 

Flux at Flaw 

Location 

(n/cm2

-sec)

Flaw Position Flaw No. Fluence at Clad/Base 

Metal Interface at 

Flaw Location (n/cm2

)

a

Attenuation 

Factor,

e-0.24x
49

1 6.2.2 Guidance on Considerations to Include in a Plant-Specific PFM Analysis

2 Guidance on the key considerations that should be included in a plant-specific PFM analysis to 

3 calculate TWCF, as identified by Step I in Figure 5, is provided in this section. Such guidance 

4 includes a mathematical procedure to combine NDE data with the PFM flaw distribution used to 

5 develop the Alternate PTS Rule. A plant-specific TWCF analysis might be necessary based on 

6 Section d(4) of the Alternate PTS Rule [1], where the following is stated:

7

8 (4) If the analysis required by paragraph (d)(3) of this section indicates that no reasonably 

9 practicable flux reduction program will prevent the RTMAX–X value for one or more reactor 

10 vessel beltline materials from exceeding the PTS screening criteria, then the licensee shall 

11 perform a safety analysis to determine what, if any, modifications to equipment, systems, 

12 and operation are necessary to prevent the potential for an unacceptably high probability of 

13 failure of the reactor vessel as a result of postulated PTS events. In the analysis, the 

14 licensee may determine the properties of the reactor vessel materials based on available 

15 information, research results and plant surveillance data, and may use probabilistic fracture 

16 mechanics techniques...

17

18 The PFM computer code, FAVOR [20], was developed by Oak Ridge National Laboratory 

19 (ORNL) under NRC funding to predict failure probabilities for embrittled vessels subject to PTS 

20 transients. FAVOR was used in the underlying TWCF analyses performed as a part of the 

21 development of the Alternate PTS Rule. Critical inputs to FAVOR are the number and sizes of 

22 fabrication flaws in the RPVs of interest and the characteristics of cooldown scenarios. The 

23 TWCF analysis further requires the expected frequencies of the cooldown scenarios. Work on 

24 flaw distributions was coordinated with another NRC research program conducted by PNNL to 

25 perform examinations of RPV materials to detect and measure the numbers and sizes of 

26 fabrication flaws in welds and base metal. To supplement the limited data from flaw detection 

27 and measurements, PNNL applied an expert judgment elicitation process and the PRODIGAL 

28 flaw simulation model developed in the United Kingdom by Rolls-Royce and Associates. 

29 PNNL’s experimental work on flaw distributions provided fabrication flaw data from 

30 nondestructive and destructive examinations, which were used to develop statistical 

31 distributions to characterize the numbers and sizes of flaws in the various regions of RPVs. 

32 Based on these statistical distributions, PNNL developed a computer program, VFLAW, which 

33 generated flaw distributions that were used as inputs to the FAVOR computer code [19]. These 

34 input files for FAVOR describe flaw distributions based on PNNL’s research activities. The 

35 VFLAW program and the incorporation of plant-specific NDE data are therefore critical input 

36 elements for performing plant-specific TWCF analyses.

37

38 A recommended methodology for combining plant-specific NDE data with the PFM flaw 

39 distribution used to develop the Alternate PTS Rule is provided in Appendix C. For the sample 

40 application to Beaver Valley 2 discussed in Appendix C (Table C-4), the results of which are 

41 repeated in this section in Table 11, revised flaw parameters that represent a plant-specific flaw 

42 distribution are shown. The revised parameters on the flaw uncertainty distribution may be used 

43 to generate a revised flaw distribution for Beaver Valley 2 that is consistent with the flaw 

44 distribution used in the PTS FAVOR analyses. The revised parameters will lead to revised flaw 

45 distributions that may be input to a FAVOR PFM analysis. These parameters have been 

46 customized using the Appendix C procedure based on Beaver Valley 2’s specific geometry and 

47 as-found NDE data. The revised flaw distributions may be generated and used as input to 

48 FAVOR by re-running the VFLAW program with the revised parameters shown in Table 11. 

49 Re-running the VFLAW program allows adjustment of the generic VFLAW flaw distribution used 

50 to develop the Alternate PTS Rule into a plant-specific flaw distribution based on the updated 
50

1 knowledge of flaws obtained from plant-specific ISI. As shown in Table 11, this yields a 

2 Bayesian updated flaw distribution for use in the plant-specific PFM required at Step I in 

3 Figure 5.

4

5 As described in Appendix C, it is important to note that the VFLAW data, while representing 

6 generic values of flaw characteristics (based on expert judgment and the PVRUF and 

7 Shoreham RPVs), should be specialized to a specific RPV by using RPV-specific weld length, 

8 weld bead thickness, geometry, and other weld characteristics. Therefore, the specialized 

9 VFLAW data that results from this evaluation is the most representative information that can be 

10 used to describe a prior distribution of flaw depth and flaw density for a plant-specific PFM 

11 assessment. If other prior information is available, such information may also be used instead 

12 of, or in addition to, the specialized VFLAW data.

13

14 A PFM analysis that calculates TWCF also requires (1) the thermal-hydraulic characteristics of 

15 the cooldown scenarios as input to FAVOR and (2) the frequency of those scenarios as input to 

16 the TWCF calculation. The characteristics and frequency of excessive cooldown scenarios are 

17 not developed by existing probabilistic risk assessments (PRAs), which only model the end 

18 states of core damage and containment failure. The PTS rulemaking relied on extensive PRA 

19 analyses to explore, identify, and characterize excessive cooldown end states.

20

21 A plant-specific PFM analysis that calculates TWCF is complex, and there are many variations 

22 of inputs possible for such an analysis. Therefore, guidance for plant-specific PFM analysis to 

23 calculate TWCF is not included in this report. A discussion of the methodology that was used in 

24 performing a PFM analysis to calculate TWCF for PTS may be found in NUREG-1806 [6], 

25 NUREG-1807 [21], and NUREG/CR-6854 [22]. As indicated in the Alternate PTS Rule, 

26 plant-specific PFM analysis to calculate TWCF and the description of the modifications made to 

27 the plant-specific inputs must be submitted to the Director of NRR in the form of a license 

28 amendment at least three years before RTMAX–X is projected to exceed the PTS screening 

29 criteria.

30

31 The steps associated with conducting a plant-specific PFM evaluation are as follows:

32

33 • Bayesian Update of Flaw Distribution:

34 o Apply the procedures of Appendix C and obtain revised flaw depth and flaw density 

35 parameters (similar to those shown in Table 11).

36 • Calculate TWCF Using FAVOR:

37 o Run VFLAW using the revised flaw depth and flaw density parameters.

38 o Develop necessary plant-specific input using the guidance in NUREG-1806, 

39 NUREG-1807, and NUREG/CR-6854.

40 o Run plant-specific FAVOR analysis.

41 o Calculate the TWCF.

42 • Compare Plant-Specific TWCF from FAVOR Analysis to Limit:

o If the calculated TWCF value is less than or equal to the limit of 1 × 10-6 43 per reactor 

44 year specified in the Alternate PTS Rule, the NDE data are acceptable and the PTS 

45 Screening Criteria in Table 1 may be used (Step K in Figure 5).

o If the TWCF value is greater than the limit of 1 × 10-6 46 per reactor year specified in the 

47 Alternate PTS Rule, the NDE data are unacceptable and the PTS Screening Criteria 

48 in Table 1 may not be used. Alternate actions are required to resolve PTS 

49 embrittlement issues, which may include changes to the facility to reduce the 

50 likelihood and/or severity of the cooldown transients. Plant-specific PTS assessment 

51 is required for review and approval by the Director of NRR (Step J in Figure 5).
51

1

2

3

4 Table 11. Summary of the Revised Flaw Depth and Flaw Density VFLAW Parameters to be 

5 Used in a Revised PFM Analysis for Beaver Valley 2 (from Appendix C)

6

Case Flaw Size 

Category

Original VFLAW Parameters of 

Uncertainty Distribution Used in 

PTS Work

(Flaw Depth Parameters Specialized to 

Beaver Valley 2)

Revised Parameters of Uncertainty 

Distribution Based on Beaver 

Valley 2 NDE Data

1 Small

(a ≤ Δ)

α3 = 0.180

α4 = 1419

α'3 = 0.230

α'4 = 1909

2 Large

(a > Δ)

α3 = 0.180

α4 = 4

α'3 = 0.230

α'4 = 5

3 Small

(a ≤ Δ)

U1 = 34

U2 = 8

U3 = 1

U'1 = 513.75

U'2 = 17.71

U'3 = 1.54

4 Large

(a > Δ)

α1 = 4.615

α2 = 4

α'1 = 4.563

α'2 = 5

7

8

9
52

1 6.3 Guidance for Initial Evaluation of NDE Data

2 Guidance for initial evaluation of NDE data obtained from qualified ISI examinations, as identified in

3 Steps A through F in Figure 5, are provided in this section. These steps are expected to be the 

4 most common use of the Alternate PTS Rule, and they represent a comparison of the as-reported 

5 NDE results with the flaw tables. Successful comparison of the NDE data to the flaw tables 

6 provides reasonable assurance that the plant-specific flaw distribution is bounded by the flaw 

7 distribution used in the development of the Alternate PTS Rule, therefore justifying application of the 

8 alternate PTS screening limits given in Table 1 to the plant in question.

9

10 As indicated under Item (e), “Examination and Flaw Assessment Requirements,” of the Alternate 

11 PTS Rule, the volumetric examination results evaluated under paragraphs (e)(1), (e)(2), and (e)(3) 

12 must be acquired using procedures, equipment, and personnel that have been qualified under 

13 Supplements 4 and 6 to Mandatory Appendix VIII to Section XI of the ASME Code, as specified in 

14 10 CFR 50.55a(b)(2)(xv). Paragraph (g)(6)(ii)(C) of 10 CFR 50.55a requires licensees to implement 

15 Supplements 4 and 6 to Appendix VIII to Section XI of the ASME Code. Supplement 4 contains 

16 qualification requirements for the RPV ISI volume from the clad-to-base-metal interface to the inner 

17 15 percent of the RPV base-metal wall thickness. Supplement 6 contains qualification requirements

18 for RPV weld volumes that lie within the outer 85% of the RPV base-metal wall thickness.

19

20 Figure 6 shows the ASME Code Section XI examination and flaw-evaluation process and identifies 

21 the flaws from that process that should be used for comparison to the Alternate PTS Rule flaw 

22 tables. As noted in the figure, the process used to identify flaws for comparison to the flaw tables is 

23 a subset of the process outlined in the ASME Code. All recordable flaws, subsequent to application

24 of the flaw proximity rules of Subarticle IWA-3300 in Section XI of the ASME Code, are used for 

25 comparison to the Alternate PTS Rule flaw tables [17b].

26

27 A sampling of RPV weld examinations (done in accordance with Appendix VIII to Section XI of the 

28 ASME Code) from thirteen PWR RPVs is included in Reference [18]. This sampling is a compilation 

29 of several RPV beltline weld examinations performed since 2000 that were provided by 

30 Westinghouse in response to a request from the NRC. These results form the basis for the example 

31 comparison to the flaw tables that is shown in Figure 7 and in Table 12 through Table 15 in this 

32 document. A discussion of this figure and these tables is provided in the following paragraphs.

33

34 In the example shown in Figure 7 and in Table 12 through Table 15, the following evaluation 

35 process was used (Steps A through D and K in Figure 5):

36

37 Step A - Qualified Examination in Accordance with Appendix VIII to Section XI of the ASME 

38 Code

39 o The Appendix VIII examination results are shown for Plant J in Figure 7.

40 o Eleven welds (two girth welds and nine intersecting axial welds) were examined in 

41 the RPV beltline region. A total of eight combined recordable flaws (as defined by 

42 the process shown in Figure 6) were detected.

43 Step B - Do Axial Flaws > 0.075" in Depth at the Clad-to-Base-Metal Interface Open to the RPV 

44 Inside Surface?

45 o Based on the S dimension (distance of flaw below the clad-to-base-metal interface) 

46 shown in Figure 7 for all flaws, no flaws at the clad-to-base-metal interface require 

47 supplemental examination to confirm that they are not connected to the RPV inside 

48 surface.

49 o If supplemental examination is required, perform a demonstrated surface or visual 

50 examination of the clad surface within the required ASME Code Section XI 
53

1 examination volume (e.g., Figures IWB-2500-1 and IWB-2500-2) to identify potential 

2 axial indications in the cladding surface that may extend into the base metal. 

3 A demonstrated visual or surface examination is one that has been shown to be 

4 capable of detecting and characterizing inservice cracking in a clad surface 

5 representative of the reactor vessel cladding process. Eddy current test is one 

6 examination method acceptable to the staff for performing this verification.

7 o A method for determining the uncertainty in locating the position of axial flaws 

8 identified by surface or visual examination and the location of axial flaws identified by 

9 UT examination shall be documented. Any axial flaws located by UT within the 

10 uncertainty bounds of an axial crack shall be considered to be the same axial flaw.

11 o If it is confirmed that no axial flaws are connected to the RPV inside surface,

12 Step B = “NO” in Figure 5.

13 Step C - Are All Flaws Acceptable According to Table IWB-3510-1 in Section XI of the ASME 

14 Code?

15 o As indicated in Figure 7 (in the column titled “ASME Code Disposition” in the second 

16 table), all recordable flaws are acceptable for meeting the requirements of

17 Table IWB-3510-1 in Section XI of the ASME Code. The details of this 

18 determination are not shown.

19 o Therefore, Step C = “YES” in Figure 5.

20 Step D - For Flaws Located in the Inner 1.0" or 0.1t (Whichever Is Greater), Check Flaw Tables

21 o Compute the Total Weld Length Examined. The total weld examination length is 

22 determined in Table 12 based on the circumference of the RPV and the percentage 

23 of the weld examined (i.e., the coverage amount) for each of the eleven welds.

24 o Compute the Total Plate Surface Area Examined. The total plate surface area 

25 examined is determined in Table 13 based on the surface area of the ASME Code 

26 Section XI examination volume less the surface area of the weld for each of the 

27 eleven welds examined.

28 o Determine the Position of Flaws. The position of all flaws is determined in 

29 Table 14 as follows:

30  Determine whether the flaw resides in plate or weld material. This 

31 determination is based on the flaw position with respect to the weld 

centerline and the maximum crown width‡ 32 on either surface of the inspected 

33 weld; if the flaw position resides within the span of the weld centerline ± 

34 one-half of the weld crown width, the flaw was considered to be a WELD 

35 flaw. Otherwise, it was treated as a PLATE flaw.

36  Determine whether the flaw resides within the inner one inch or 10% of the 

37 base-metal wall thickness, whichever is greater, for comparison to the 

38 Alternate PTS Rule flaw tables.

39  Determine whether the flaw resides between the inner one inch or 10% of 

40 the base-metal thickness, whichever is greater, and the inner 37.5% of the 

41 base-metal wall thickness (3/8t), for potential evaluation of brittle fracture.

 ‡ The weld crown width was used in this example for the entire RPV base-metal wall thickness, which 

can potentially classify flaws located mid-wall as weld flaws rather than plate flaws in single-groove or 

double-V-groove weld configurations. This approach is conservative with respect to the flaw limits in 

Table 2. Consideration should also be given to the proximity of the flaw to the weld’s heat-affected 

zone (HAZ). An example might be that any flaw located in the plate material, but within some 

technically justified minimum distance from the edge of the weld, should be considered to be affected 

by the HAZ and therefore also considered to be a weld flaw. In addition, if there are any weld-repair 

areas located within the examination volume, any flaws detected within those areas should be classified 

as WELD flaws.
54

1  Based on the results shown in Table 14, two flaws should be compared to 

2 the Weld Flaw Table (Table 2) of the Alternate PTS Rule and one flaw 

3 should be compared to the Plate/Forging Flaw Table (Table 3) of the 

4 Alternate PTS Rule. Although two flaws reside between the inner one 

5 inch or 10% of the base-metal thickness, whichever is greater, and the 

6 inner 37.5% of the base-metal wall thickness (3/8t) from the 

7 clad/base-metal interface, they do not require further assessment for 

8 acceptability because they were found to be acceptable in accordance 

9 with Table IWB-3510-1.

10 o Flaw Assessment:

11  The flaws determined in the previous step to be located in the inner 1.0" or 

12 0.1t (whichever is greater) are compared to the flaw tables from the Alternate 

13 PTS Rule in Table 15 and found to be acceptable.

14  The remaining two flaws located between the inner one inch or 10% of 

15 the base-metal thickness, whichever is greater, and the inner 37.5% of 

16 the base-metal wall thickness (3/8t) from the clad/base-metal interface

17 are acceptable in accordance with Table IWB-3510-1.

18  Therefore, Step D = “PASS” in Figure 5.

19 Step K - PTS Screening Criteria (Table 1 of 10 CFR 50.61a) Is Applicable - Submittal Required

20 o Plant J is allowed to use the PTS Screening Criteria shown in Table 1. The licensee 

21 must submit their PTS evaluation to the NRC for review and approval.

22

23
55

1

2

3 Figure 6. ASME Code Section XI Examination and Flaw Evaluation Process and Identification 

4 of Flaws for Comparison to Alternate PTS Rule.

5

6
56

1

2

3 Figure 7. Sample Appendix VIII Examination Results for Plant J.

4

Step A: Plant J Examination Results: (see Westinghouse Letter LTR-AMLRS-11-71 [18])

 Circumferential Weld Axial Weld
57

Table 12. Determination of Total Weld Length Examined for Plant J 1

2

3

4

5

STEP D: Compute Total Weld Length Examined

Weld ID Direction

Radius, Ri

(inches)

Length

(inches) Comments

Examination 

Coverage

Ex

L

J-1 Circumferential 86.5 543.5 Length computed as 2πRi 100% 

J-2 Circumferential 86.5 543.5 Length computed as 2πRi 100% 

J-3 Axial N/A 70 Length assumed 100% 

J-4 Axial N/A 70 Length assumed 100% 

J-5 Axial N/A 70 Length assumed 100% 

J-6 Axial N/A 70 Length assumed 100% 

J-7 Axial N/A 70 Length assumed 100% 

J-8 Axial N/A 70 Length assumed 100% 

J-9 Axial N/A 70 Length assumed 77.8% 

J-10 Axial N/A 70 Length assumed 77.8% 

J-11 Axial N/A 70 Length assumed 77.8% 

Total, L = 
58

Table 13. Determination of Total Plate Surface Area Examined for Plant J 1

2

Notes: (1) The weld width, WW, is defined as the maximum weld crown width from both sides of the RPV wall. From Figure 7, 3

circumferential welds and 1.38" for axial welds. 4

(2) The examination width, WE, is defined as the ASME Code Section XI examination volume width, which is one-half o5

both sides of the weld. From Figure 7, the RPV base-metal wall thickness, which is equal to WE, is 8.60". 6

7

8

STEP D: Compute Total Plate Surface Area Examined

Weld ID

Examined 

Length, L

Weld Width, 

WW

(1) 

(inches)

Examination 

Width, WE

(2) 

(inches)

P

J-1 543.5 1.65 8.60 

J-2 543.5 1.65 8.60 

J-3 70.0 1.38 8.60 

J-4 70.0 1.38 8.60 

J-5 70.0 1.38 8.60 

J-6 70.0 1.38 8.60 

J-7 70.0 1.38 8.60 

J-8 70.0 1.38 8.60 

J-9 54.5 1.38 8.60 

J-10 54.5 1.38 8.60 

J-11 54.5 1.38 8.60 

Total, A = 

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Comments

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)

Plate surface area computed as L x (WE - WW)
59

Table 14. Determination of the Flaw Positions for Plant J 1

2

3

STEP D: Determine Position of Flaws

Height on RPV 

(inches)

RPV Circumferential 

Position (°)

Depth Below 

Clad (inches)

J-2 1 235 in 1.65 235.00 150.1 1.66 WELD NO YES 

2 235 in 1.65 235.14 157.7 2.70 WELD NO YES 

3 235 in 1.65 235.80 129.3 8.60 WELD NO NO 

4 235 in 1.65 235.50 332.1 4.49 WELD NO NO 

J-7 1 120 ° 1.38 123.60 123.6 0.63 PLATE YES N/A 

J-9 1 60 ° 1.38 256.50 60.3 0.11 WELD YES N/A 

2 60 ° 1.38 240.30 60.4 0.79 WELD YES N/A 

J-11 1 300 ° 1.38 312.61 300.3 3.60 WELD NO NO 

Flaw Position Thro

Exte

= 2a 

Weld or Plate Flaw?

Is Flaw Within First 

1.0" or 0.10t?

Is Flaw Within 

First 3/8t?

Weld Width

(inches) Weld Centerline Indication No. Weld No.
60

Table 15. Comparison of Flaws to Alternate PTS Rule Flaw Tables for Plant J 1

2

3

4

5

STEP D: Compare Flaws to 10 CFR 50.61a Flaw Tables

Table 2 Comparison = Weld Flaws

(Allowable) (Cumulative)

Bin Minimum TWE Maximum TWE Max. # of Flaws # of Flaws # of Flaws / L

No. (inches) (inches) per 1,000 inches Detected x 1,000

1 0 0.075 No limit 0 0.00 YES

2 0.075 0.475 166.70 2 1.20 YES 

3 0.125 0.475 90.80 2 1.20 YES 

4 0.175 0.475 22.82 2 1.20 YES 

5 0.225 0.475 8.66 2 1.20 YES 

6 0.275 0.475 4.01 2 1.20 YES 

7 0.325 0.475 3.01 1 0.60 YES 

8 0.375 0.475 1.49 0 0.00 YES

9 0.425 0.475 1.00 0 0.00 YES

10 0.475 Infinite 0.00 0 0.00 YES

Table 3 Comparison = Plate/Forging Flaws

(Allowable) (Cumulative)

Bin Minimum TWE Maximum TWE Max. # of Flaws # of Flaws # of Flaws / A

No. (inches) (inches) per 1,000 inches Detected x 1,000

1 0 0.075 No limit 0 0.00 YES

2 0.075 0.375 8.05 1 0.08 YES 

3 0.125 0.375 3.15 1 0.08 YES 

4 0.175 0.375 0.85 1 0.08 YES 

5 0.225 0.375 0.29 0 0.00 YES

6 0.275 0.375 0.08 0 0.00 YES

7 0.325 0.375 0.01 0 0.00 YES

8 0.375 Infinite 0.00 0 0.00 YES

Acceptab

Acceptab
61

1 6.4 Guidance for Further Evaluation of NDE Data

2 Guidance for further evaluation of NDE data (Steps G through I in Figure 5) is provided in this 

3 section. Such guidance is based on Section II, “Discussion,” provided in the Supplemental 

4 Information section of the Federal Register Notice for the Alternate PTS Rule [2], where the 

5 following discussion is included with respect to NDE-Related Uncertainties:

6

7 The flaw sizes in Tables 2 and 3 represent actual flaw dimensions while the results from the 

8 ASME Code examinations are estimated dimensions. The available information indicates 

9 that, for most flaw sizes in Tables 2 and 3, qualified inspectors will oversize flaws. 

10 Comparing oversized flaws to the size and density distributions in Tables 2 and 3 is 

11 conservative and acceptable, but not necessary.

12

13 As a result of stakeholder feedback received on the NRC solicitation for comments 

14 published in the August 2008 supplemental proposed rule, the final rule will permit licenses 

15 to adjust the flaw sizes estimated by inspectors qualified under the ASME Code, Section XI, 

16 Appendix VIII, Supplement 4 and Supplement 6.

17

18 The NRC determined that, in addition to the NDE sizing uncertainties, licensees should be 

19 allowed to consider other NDE uncertainties, such as probability of detection and flaw 

20 density and location, because these uncertainties may affect the ability of a licensee to 

21 demonstrate compliance with the rule. As a result, the language in § 50.61a(e) will allow 

22 licensees to account for the effects of NDE-related uncertainties in meeting the flaw size and 

23 density requirements of Tables 2 and 3. The methodology to account for the effects of 

24 NDE-related uncertainties must be based on statistical data collected from ASME Code 

25 inspector qualification tests or any other tests that measure the difference between the 

26 actual flaw size and the size determined from the ultrasonic examination...

27

28 Specific guidance on various elements of NDE data evaluation is provided in the following sections.

29

30 6.4.1 Guidance on the Elements and NDE Techniques Associated with ASME 

31 Code Examinations

32 The Alternate PTS Rule was developed using a flaw density, spatial distribution, and size 

33 distribution determined from experimental data, as well as from physical models and expert 

34 judgment. To implement the Alternate PTS Rule, actual flaw densities and distributions need to 

35 be estimated from the results of periodic ISI performed on RPV welds and adjacent base 

36 material. The method for these examinations is ultrasonic testing (UT). As discussed in 

37 Section 6.3, the data used for evaluation of the Alternate PTS Rule must be acquired using 

38 procedures, equipment and personnel that are qualified under Supplements 4 and 6 to 

39 Appendix VIII to Section XI of the ASME Code. Appendix VIII provides requirements for 

40 performance demonstration for UT examination procedures, equipment, and personnel used to 

41 detect and size flaws. Supplement 4 specifies qualification requirements for examination of the 

42 inner 15% of clad ferritic reactor vessels, and may also be applied to the inner 15% of unclad 

43 ferritic reactor vessels. Supplement 6 specifies qualification requirements for examination of 

44 unclad ferritic components and the outer 85% of clad ferritic components.

45

46 Supplement 4 provides performance-demonstration rules intended to produce effective 

47 procedures, personnel, and equipment for detection and sizing of flaws typical of those that 

48 might be expected to form as a result of service conditions. These rules result in targeting flaws 
62

1 that are primarily planar in orientation and emanate from the inside surface (clad-to-base-metal 

2 interface) of the RPV specimens that are used for the qualification process. The aspect ratio of 

3 flaws used for performance demonstration also closely follows the acceptance criteria of 

4 Article IWB-3000 in Section XI of the ASME Code. These flaws are the focus of the 

5 performance demonstration because they are of structural significance under standard 

6 operating conditions. Therefore, the objective of Supplement 4 qualification is to demonstrate 

7 UT capabilities on planar flaws. Examinations using Appendix VIII qualification may also detect 

8 larger fabrication flaws; this is often the case in practice because no known credible subcritical 

9 cracking mechanisms affect the RPV material for the current fleet of U.S. reactors.

10

11 The PFM analyses supporting the Alternate PTS Rule assumed all flaws were planar in nature. 

12 Because the empirical evidence on which the flaw distribution used in these analyses was 

13 based included both planar flaws and fabrication flaws [19], this assumption was viewed as 

14 being conservative because it produced a higher density of planar flaws than typically exist in 

15 nuclear RPVs. Fabrication flaws are considerably smaller than would be expected to exist 

16 within current Supplement 4 qualification specimens and are not typically connected to the 

17 clad-to-base-metal interface. Nevertheless, it is probable that parameters associated with 

18 qualified UT methods may be optimized to enhance the method’s capability to detect and size 

19 small fabrication flaws in the inner one inch of RPV base material. Therefore, licensees should 

20 consider enhancements to Appendix-VIII-qualified procedures to ensure accurate detection and 

21 sizing of the flaws inferred by the Alternate PTS Rule while maintaining the essential variables 

22 for which the procedure was qualified.

23

24 Within the context of a qualified Appendix VIII RPV examination, various elements and NDE 

25 techniques associated with the examination might be able to be varied to provide better NDE 

26 data for comparison with the flaw limits included in Tables 2 and 3 of the Alternate PTS Rule.

27 Such elements and techniques may include, but are not be limited to, the following:

28

29 • Reducing the scan index;

30 • Use of an ultrasonic straight-beam examination technique; or

31 • Use of an enhanced recording criterion (lower threshold).

32

33 NRC staff have been working with UT/NDE experts at Pacific Northwest National Laboratory 

34 (PNNL) to assess RPV examination and implementation of the Alternative PTS Rule. The 

35 assessment, including enhancements to procedures and examination techniques, is addressed 

36 in a Technical Letter Report (TLR) [24]. Due consideration of the above elements should be 

37 given when establishing the protocols and procedures for Appendix VIII examinations used by 

38 licensees who choose to apply the Alternate PTS Rule in order to maximize the usefulness of 

39 the resulting NDE data and provide reasonable assurance of the acceptability of the NDE data 

40 comparisons required by the Alternate PTS Rule.

41

42 6.4.2 Guidance on a Procedure to Adjust NDE Data and Comparison to Flaws

43 Assumed in PFM Calculations

44 Guidance on a mathematical procedure that can be used to adjust NDE data to account for 

45 flaw-detection and -sizing errors, as identified by Step G in Figure 5, is provided in this section. This 

46 evaluation adjusts the as-found NDE data by taking into account flaw-sizing errors, POD

47 uncertainties, and prior flaw distributions such as those used in the development of the Alternate 

48 PTS Rule (i.e., VFLAW [19]). Any or all of these uncertainties may be considered, depending on the 

49 level of detail needed for flaw assessment. After such adjustment, the adjusted NDE results may be 
63

1 compared with the flaw tables again (Step H in Figure 5) because NDE techniques tend to oversize 

2 smaller flaws, thereby distributing detected flaws into larger bins where the allowed number of flaws 

3 is smaller. In such cases, adjustment for NDE flaw-detection and -sizing uncertainties may result in

4 a less conservative distribution of flaw sizes, possibly allowing a comparison of the adjusted NDE 

5 data to the flaw tables to be successful. A successful comparison of the revised NDE data to the 

6 flaw tables would be viewed by the staff as providing reasonable assurance that the plant-specific 

7 flaw distribution is bounded by the flaw distribution used in the development of the Alternate PTS 

8 Rule, indicating that it is appropriate to apply the alternate PTS screening limits (Step K in Figure 5).

9

10 Appendix C describes the development and application of a methodology to account for 

11 uncertainties in NDE data and for analyzing such data for the purpose of developing more 

12 realistic vessel-specific flaw depth and density distributions for comparison to the flaw tables, as 

13 well as for plant-specific PFM analysis (discussed in Section 6.2.2). The methodology 

14 considers POD, flaw-measurement error, and flaw-detection threshold in its application and 

15 uses Bayesian updating to combine the observed NDE data with the available flaw data and 

16 models used as part of the PTS re-evaluation effort. The Bayesian framework used by this

17 methodology is described in Appendix C, followed by application details of the NDE data 

18 uncertainties (i.e., POD and measurement/sizing error) and the Bayesian updating procedure. 

19 Application of the methodology is demonstrated in Appendix C using the ultrasonic NDE data 

20 obtained from a previous ISI examination of the Beaver Valley 2 RPV and a MATLAB software 

21 routine that uses the Bayesian equations developed in Appendix C. The MATLAB software 

22 listing is also included in Appendix C.

23 As an example of this procedure, assume that Plant J performs an initial NDE assessment using 

24 the guidance of Section 6.3 with the results shown in Table 16. These results indicate that

25 Plant J fails the Alternate PTS Rule flaw-table check based on violating the flaw-acceptance

26 limit for Flaw Bin #2, i.e., 170.10 actual flaws per 1,000 inches of weld vs. 166.70 allowed flaws 

27 per 1,000 inches of weld. Therefore, Step D in Figure 5 yields a “FAIL” decision, thereby 

28 necessitating the evaluations associated with Step G in Figure 5. As a result, procedures 

29 similar to those described in Appendix C should be applied.

30

31 A key input for this procedure is a POD and associated sizing error data appropriate to the 

32 methods and techniques used to examine the RPV. Example PODs and flaw-sizing error data 

33 have been previously developed by the Electric Power Research Institute (EPRI) based on 

34 Performance Demonstration Initiative (PDI) qualification data [26]. Similar information should be 

35 obtained for the examination technique being applied and technically justified for use as a part 

36 of applying the procedures described in Appendix C.

37

38
64

1 Table 16. Example for Plant J After Applying an Initial Alternate PTS Rule Flaw Table Check 

2 Using the Procedure in Section 6.3.

3

Flaw

Bin #

Flaw Depth

(inch)

Observed 

(Detected) 

Number of Flaws 

from NDE ISI

10 CFR 50.61a 

Flaw Limits

(# of Flaws per 

1,000" of 

Weld)

Acceptable?

(Is No. of 

Flaws < 

Limit?)

1 0.000 < a ≤ 0.075 234.45 No limit Yes

2 0.075 < a ≤ 0.475 170.10 166.70 No

3 0.125 < a ≤ 0.475 11.23 90.80 Yes

4 0.175 < a ≤ 0.475 0.90 22.82 Yes

5 0.225 < a ≤ 0.475 0.05 8.66 Yes

6 0.275 < a ≤ 0.475 0.01 4.01 Yes

7 0.325 < a ≤ 0.475 0.00 3.01 Yes

8 0.375 < a ≤ 0.475 0.00 1.49 Yes

9 0.425 < a ≤ 0.475 0.00 1.00 Yes

4

5 6.4.2.1 Guidance on Application of NDE Uncertainties

6 Appendix D discusses the results of sensitivity analyses that apply the Appendix C Bayesian 

7 updating methodology to systematically assess the effect of NDE uncertainties such as POD 

8 and measurement (sizing) error on the estimated flaw populations. The sensitivity analyses 

9 were performed for a base case and twelve sensitivity cases that investigated variations in 

10 observed NDE data and application of prior probability density functions (PDFs), POD, and 

11 flaw-sizing error.

12 As a part of the sensitivity studies, lower NDE detection limits of 0.04" and 0.075" were also 

13 evaluated and found to have no significant impact on the observed data.

14 The results obtained from the twelve sensitivity cases consistently show that small (i.e., 10%) 

15 overpopulation of flaws in Bins 2 and 3 of the Alternate PTS Rule flaw tables, as might be 

16 expected from actual plant inspections because of oversizing of small flaws, would be shifted to 

17 Bins 1 and 2 after accounting for the measurement error in the Bayesian inference. When POD 

18 was also considered, the effects of small flaws that might be missed by NDE methods were

19 clearly seen in Bins 1 and 2, with an additional number of flaws in the posterior estimates as 

20 compared to the observed flaws. However, the results are sensitive to the POD used, 

21 especially the portion of the POD for smaller flaw sizes. It is therefore important that, if a POD 

22 is used, the POD must be sufficiently justified.

23 The effects of the consideration and choice of the prior distributions of flaw density and depth 

24 were significant. When no prior information was used to describe the flaw-density and 

25 flaw-depth distributions, POD and measurement error were very sensitive, and significantly 

26 amplified the number of flaws that resulted in Bins 1 and 2. However, when prior PDFs were 
65

1 used, the posteriors were significantly moderated by the existence of the prior PDFs, and the 

2 POD and measurement errors played less significant roles.

3 The results of the sensitivity analysis documented in Appendix D reveal the following:

4 • Neglecting consideration of prior PDFs in the evaluation provides a conservative 

5 assessment.

6 • Neglecting consideration of flaw-sizing error in the evaluation provides a conservative 

7 assessment.

8 • POD has a significant impact in the evaluation and should be included for the case in 

9 which PDFs are not considered. For the case in which PDFs are evaluated, the impact 

10 of POD is relatively small and may be neglected.

11 • Consideration of flaw-sizing error, POD, or prior PDFs (identified as “NDE uncertainties” 

12 in the Alternate PTS Rule) using methods similar to those shown in Appendix C is

13 successful in removing conservatisms that may unnecessarily prevent a licensee from 

14 passing the Alternate PTS Rule flaw tables.

15

67

1 7. GUIDANCE ON CRITERIA RELATING TO ALTERNATE LIMITS ON 

2 EMBRITTLEMENT

3 Paragraph (c)(3) of 10 CFR 50.61a states the following:

4

5 Each licensee shall compare the projected RTMAX–X values for plates, forgings, axial 

6 welds, and circumferential welds to the PTS screening criteria in Table 1 of this section, 

7 for the purpose of evaluating a reactor vessel’s susceptibility to fracture due to a PTS 

8 event. If any of the projected RTMAX–X values are greater than the PTS screening 

9 criteria in Table 1 of this section, then the licensee may propose the compensatory 

10 actions or plant-specific analyses as required in paragraphs (d)(3) through (d)(7) of 

11 this section, as applicable, to justify operation beyond the PTS screening criteria in 

12 Table 1 of this section.

13

14 This section describes one method by which licensees could perform the plant-specific analyses 

15 indicated by the highlighted text. This method is acceptable to the staff provided that the 

16 requirements of 10 CFR 50.61a concerning flaw evaluations and surveillance assessment are

17 satisfied.

18

19 The RTMAX-X limits in Table 1 of 10 CFR 50.61a were established in NUREG-1874 to ensure that the 

TWCF remains below 1 × 10-6 20 per reactor year. As described in NUREG-1874, the RTMAX-X limits 

21 are based on the results of PFM analyses; they account for the combined TWCF contributions from 

22 various flaw populations in the RPV. Some simplifications of the PFM data underlying these limits 

23 were necessary to permit their expression in a tabular form. As an example, the TWCF attributable 

to circumferentially-oriented flaws occurring in circumferential welds was held below 1 × 10-8 24 per 

reactor year rather than 1 × 10-6 25 per reactor year. This simplification was made for expedience and 

26 was not intended to address a safety concern. Therefore, the following procedure, which eliminates

27 similar simplifying assumptions, can be used to demonstrate compliance with the RTMAX-X limits in 

28 Table 1 of 10 CFR 50.61a. This procedure was originally described in Section 3.5.1 of 

29 NUREG-1874:

30

31 1. Determine RTMAX-X for all axial welds (RTMAX-AW), plates (RTMAX-PL), circumferential 

32 welds (RTMAX-CW), and forgings (RTMAX-FO) in the RPV beltline region according to the 

33 requirements of 10 CFR 50.61a. These RTMAX-X values must be expressed in units 

34 of Rankine (R) (degrees Fahrenheit (°F) plus 459.69).

2. Use the RTMAX-X values from Step 1 to estimate the 95th 35 percentile TWCF 

36 contribution from each component in the beltline using the following formulas:

37

TWCF95−AW = exp{5.5198⋅ ln(RTMAX −AW − 616)− 40.542}⋅ β (19)

TWCF95−PL = exp{23.737 ⋅ ln(RTMAX −PL − 300)−162.38}⋅ β (20)

TWCF95−CW = exp{9.1363⋅ ln(RTMAX −CW − 616)− 65.066}⋅ β (21)

TWCF95−FO = exp{23.737 ⋅ ln(RTMAX −FO − 300) −162.38}⋅ β

+η ⋅{ × ⋅ }⋅ β − − ⋅ 137 0.185 RTMAX FO 1.3 10 10 (22)

38

39 where:

40

41 η = 0 if the forging is compliant with Regulatory Guide 1.43; otherwise η = 1.

42
68

1 β = 1 if TWALL ≤ 9.5 inches

2 β = 1 + 8 × (TWALL – 9.5) if 9.5 < TWALL < 11.5 inches

3 β = 17 if TWALL ≥ 11.5 inches

4

3. Estimate the total 95th 5 percentile TWCF for the RPV using the following formulae 

6 (noting that, depending on the type of vessel in question, certain terms in the 

7 following formula will be zero).

8

























⋅

⋅ +

⋅ +

⋅ +

=

−

−

−

−

−

FO FO

CW CW

PL PL

AW AW

TOTAL

TWCF

TWCF

TWCF

TWCF

TWCF

95

95

95

95

95

α

α

α

α

(23)

9

10 where:

11

12 α = 2.5 if RTMAX-xx ≤ 625 R

( 625) 250

1.5 13 α = 2.5 − RTMAX −xx − if 625 R < RTMAX-xx < 875 R

14 α = 1 if RTMAX-xx ≥ 875 R

15

16

4. If TWCF95-TOTAL from Step 3 is less than 1 × 10-6 17 per reactor year, the requirements of 

18 Table 1 of 10 CFR 50.61a are met.

19
69

1 8. SUMMARY

2 In early 2010, the NRC promulgated the Alternate PTS Rule in 10 CFR 50.61a, which amended 

3 existing regulations to provide alternate embrittlement requirements for protection against PTS 

4 events for PWR RPVs. These requirements are based on more comprehensive, accurate, and 

5 realistic analysis methods than those used to establish the limits in 10 CFR 50.61. This action 

6 became desirable because the existing requirements, as contained in 10 CFR 50.61, are based on 

7 unnecessarily conservative assumptions. While still maintaining adequate safety margins, the 

8 Alternate PTS Rule reduces regulatory burden for those PWR licensees who expect to exceed the 

9 10 CFR 50.61 embrittlement requirements before the expiration of their operating licenses. PWR 

10 licensees may choose to comply with the Alternate PTS Rule as a voluntary alternative to complying 

11 with the requirements contained in 10 CFR 50.61.

12

13 The Alternate PTS Rule provides revised PTS screening criteria in the form of embrittlement 

14 reference temperatures, RTMAX-X, that characterize the RPV material’s resistance to fracture initiating 

15 from flaws. The RTMAX-X embrittlement limits may be used by licensees provided that the following

16 criteria are met:

17

18 1. Criteria relating to the date of construction and design requirements: The 

19 Alternate PTS Rule is applicable to licensees whose construction permits were issued 

20 before February 3, 2010, and whose RPVs were designed and fabricated to the 

21 1998 Edition or an earlier edition of the ASME Code. The reason for this applicability 

22 restriction is that the structural and thermal hydraulic analyses that established the basis 

23 for the Alternate PTS Rule’s embrittlement limits only represented plants constructed 

24 before this date. It is the responsibility of a licensee to demonstrate that the 

25 risk-significant factors controlling PTS for any plant constructed after February 3, 2010,

26 are adequately addressed by the technical basis calculations developed in support of 

27 the Alternate PTS Rule. Chapter 4 of this document describes methods by which 

28 licensees can satisfy these criteria and identifies factors to be considered in such an

29 evaluation.

30

31 2. Criteria relating to plant-specific surveillance data: The Alternate PTS Rule includes 

32 statistical tests that must be performed on RPV surveillance data to determine whether

33 the surveillance data are sufficiently “close” to the predictions of an ETC that the 

34 predictions of the ETC are valid for use. From a regulatory perspective, it is of particular 

35 interest to determine whether plant-specific surveillance data deviate significantly from 

36 the predictions of the ETC in a manner that suggests that the ETC is very likely to 

37 underpredict plant-specific data trends. Chapter 5 of this document describes guidance

38 by which licensees can assess the closeness of plant-specific data to the ETC using 

39 statistical tests. This guidance includes the following, including:

40  A detailed description of the mathematical procedures to use to assess 

41 compliance with the three statistical tests in the Alternative PTS Rule.

42  A list of factors to consider in diagnosing the reason that particular 

43 surveillance data sets may fail these statistical tests.

44  A description of certain situations in which routine adjustments of the ETC 

45 predictions can be made.

46

47 3. Criteria relating to ISI data and NDE requirements: The Alternate PTS Rule 

48 describes a number of tests of and conditions on the collection and analysis of ISI data 

49 that are intended to provide reasonable assurance that the distribution of flaws that was 
70

1 assumed to exist in the PFM calculations that provide the basis for the RTMAX-X limits 

2 provide an appropriate, or bounding, model of the population of flaws in the RPV of 

3 interest. Chapter 6 of this NUREG includes guidance by which licensees can satisfy 

4 these criteria. The guidance discussed in this chapter includes the following 

5 components:

6  Guidance for plants for the case in which RPV flaws fall outside the 

7 applicability of the flaw tables in the Alternate PTS Rule, including:

8 i. A mathematical procedure that can be used to preclude brittle 

9 fracture based on RTNDT information.

10 ii. A mathematical procedure that can be used to combine the NDE 

11 data with the population of flaws assumed in the PFM calculations to 

12 estimate the total flaw distribution that is predicted to exist in the 

13 RPV, as well as guidance on the use of this total flaw distribution as 

14 part of a PFM calculation using the FAVOR computer code.

15  Guidance for initial evaluation of NDE data obtained from qualified ISI 

16 examinations.

17  Guidance for further evaluation of NDE data obtained from qualified ISI 

18 examinations, as follows:

19 i. The elements and NDE techniques associated with the qualified ISI 

20 examinations performed in accordance with Mandatory Appendix VIII 

21 to Section XI of the ASME Code to assess compliance with the 

22 requirements of the Alternate PTS Rule.

23 ii. A mathematical procedure that can be used to adjust NDE data to 

24 account for flaw-detection and -sizing errors and comparison of the 

25 adjusted data to the population of flaws assumed in the PFM 

26 technical basis for the Alternate PTS Rule.

27

28 4. Criteria relating to alternate limits on embrittlement: Guidance is provided by which 

29 licensees can estimate a plant-specific value of TWCF for cases in which the RTMAX-X

30 limits of the Alternate PTS Rule are not satisfied. Chapter 7 of this document describes 

31 these two sets of guidance so that licensees can satisfy embrittlement acceptability 

32 criteria.

33

34 This document provides guidance and the associated technical basis for methods by which the

35 above criteria can be satisfied.

36
71

1 9. REFERENCES

2 [1] U.S. Code of Federal Regulations, Title 10, “Energy,” Chapter I, Section 50.61a,

3 “Alternate Fracture Toughness Requirements for Protection Against Pressurized 

4 Thermal Shock Events” (10 CFR 50.61a).

5 [2] U.S. Nuclear Regulatory Commission, “Alternate Fracture Toughness 

6 Requirements for Protection Against Pressurized Thermal Shock Events,” Federal 

7 Register, Vol. 75, No. 1, January 4, 2010, pp. 13–29 (75 FR 13).

8 [3] U.S. Code of Federal Regulations, Title 10, “Energy,” Chapter I, Section 50.61, 

9 “Fracture Toughness Requirements for Protection Against Pressurized Thermal 

10 Shock Events” (10 CFR 50.61).

11 [4] American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, 

12 2013 edition, Section XI, “Rules for Inservice Inspection of Nuclear Power Plant 

13 Components,” New York, NY.

14 [5] U.S. Code of Federal Regulations, Title 10, “Energy,” Chapter I, Section 50.66,

15 “Requirements for Thermal Annealing of the Reactor Pressure Vessel” (10 CFR 

16 50.66).

17 [6] U.S. Nuclear Regulatory Commission, “Technical Basis for Revision of the 

18 Pressurized Thermal Shock (PTS) Screening Limit in the PTS Rule 

19 (10 CFR 50.61): Summary Report,” NUREG-1806, August 2007.

20 [7] U.S. Nuclear Regulatory Commission, “Recommended Screening Limits for 

21 Pressurized Thermal Shock (PTS),” NUREG-1874, March 2010.

22 [8] Elliot, B.J., Senior Materials Engineer, U.S. Nuclear Regulatory Commission, 

23 memorandum to M.A. Mitchell, Branch Chief, U.S. Nuclear Regulatory 

24 Commission, ‘‘Development of Flaw Size Distribution Tables for Draft Proposed 

25 Title 10 of the Code of Federal Regulations (10 CFR) 50.61a,’’ April 3, 2007, 

26 Agencywide Documents Access and Management System (ADAMS) Accession 

27 No. ML070950392.

28 [9] Eason, E.D., et al., “A Physically Based Correlation of Irradiation-Induced 

29 Transition Temperature Shifts for RPV Steels,” ORNL/TM-2006/530, 

30 November 2007, Oak Ridge National Laboratory, Oak Ridge, TN, ADAMS 

31 Accession No. ML081000630.

32 [10] Electric Power Research Institute (EPRI), “Materials Reliability Program: Proposed 

33 Resolutions to the Analytical Challenges of Alternate PTS Rule (10 CFR 50.61a) 

34 Implementation (MRP-334),” Technical Update No. 1024811, January 2012, Palo 

35 Alto, CA.

36 [11] U.S. Code of Federal Regulations, Title 10, “Energy,” Chapter I, Section 50.55a, 

37 “Codes and Standards” (10 CFR 50.55a).

38

39 [12] ASME, Boiler and Pressure Vessel Code, Mandatory Appendix VIII, “Performance 

40 Demonstration for Ultrasonic Examination Systems,” to Section XI, “Rules for 

41 Inservice Inspection of Nuclear Power Plant Components,” New York, NY, 

42 2013 edition.

43
72

1 [13] U.S. Code of Federal Regulations, Title 10, “Energy,” Chapter I, Part 50, “Domestic 

2 Licensing of Production and Utilization Facilities,” Appendix H, “Reactor Vessel 

3 Material Surveillance Program Requirements.”

4 [14] U.S. Nuclear Regulatory Commission, ‘‘Statistical Procedures for Assessing 

5 Surveillance Data for 10 CFR Part 50.61a,’’ June 9, 2008, ADAMS Accession 

6 No. ML081290654.

7 [15] U.S. Nuclear Regulatory Commission, ‘‘Technical Basis for Revision of Regulatory 

8 Guide 1.99: NRC Guidance on Methods to Estimate the Effects of Radiation 

9 Embrittlement on the Charpy V-Notch Impact Toughness of Reactor Vessel 

10 Materials,’’ draft NUREG report, October 1, 2007, ADAMS Accession 

11 No. ML081120289.

12 [16] U.S. Nuclear Regulatory Commission, “Radiation Embrittlement of Reactor Vessel 

13 Materials,” Regulatory Guide 1.99, Rev. 2, May 1988, ADAMS Accession 

14 No. ML003740284.

15 [17] NRC Public Meeting Summary Reports:

16 a. U.S. Nuclear Regulatory Commission, “Category 2 Public 

17 Meeting - Discussion of Reactor Pressure Vessel Integrity Issues for 

18 Operating Nuclear Power Plants,” August 5, 2011, ADAMS Accession 

19 No. ML112170262.

20 b. U.S. Nuclear Regulatory Commission, “Category 2 Public 

21 Meeting - Discussion of Nondestructive Examination Aspects of Reactor 

22 Pressure Vessel Integrity Issues for Operating Nuclear Power Plants,” 

23 October 6, 2011, ADAMS Accession No. ML112790501.

24 c. U.S. Nuclear Regulatory Commission, “Category 2 Public 

25 Meeting - Discussion of Reactor Pressure Vessel Integrity Issues for 

26 Operating Nuclear Power Plants,” November 18, 2011, ADAMS Accession 

27 No. ML113220295.

28 [18] Palm, Nathan A., Fellow Engineer, and Michael G. Semmler, Acting Manager, 

29 Westinghouse Electric Company, letter to Dr. Aladar Csontos, Branch Chief, 

30 U.S. Nuclear Regulatory Commission, “Inspection Data for Use in Development of 

31 Alternate Pressurized Thermal Shock (PTS) Rule Implementation Regulatory 

32 Guide,” LTR-AMLRS-11-71, September 7, 2011, ADAMS Accession 

33 No. ML112560145.

34 [19] U.S. Nuclear Regulatory Commission, “A Generalized Procedure for Generating 

35 Flaw-Related Inputs for the FAVOR Code,” NUREG/CR-6817 (PNNL-14268), 

36 March 2004, ADAMS Accession No. ML040830499.

37 [20] Oak Ridge National Laboratory, “Fracture Analysis of Vessels - Oak Ridge 

38 FAVOR, v12.1, Computer Code: Theory and Implementation of Algorithms, 

39 Methods, and Correlations,” ORNL/TM-2012/567, November 2012, Oak Ridge, TN, 

40 ADAMS Accession No. ML13008A015.

41 [21] U.S. Nuclear Regulatory Commission, “Probabilistic Fracture Mechanics - Models, 

42 Parameters, and Uncertainty Treatment Used in FAVOR Version 04.1,” 

43 NUREG-1807, June 2007.

44 [22] U.S. Nuclear Regulatory Commission, “Fracture Analysis of Vessels - Oak Ridge 

45 FAVOR, v04.1, Computer Code: Theory and Implementation of Algorithms, 
73

1 Methods, and Correlations,” NUREG/CR-6854 (ORNL/TM-2004/244), 

2 August 2007.

3 [23] Pellini, W.S., and P.P. Puzak, “Fracture Analysis Diagram Procedures for the 

4 Fracture-Safe Engineering Design of Steel Structures,” Bulletin 88, May 1963, 

5 Welding Research Council, Shaker Heights, OH.

6

7 [24] Pacific Northwest National Laboratory, “Evaluation on the Feasibility of Using

8 Ultrasonic Testing of Reactor Pressure Vessel Welds for Assessing Flaw 

9 Density/Distribution per 10 CFR 50.61a, Alternate Fracture Toughness 

10 Requirements for Protection Against Pressurized Thermal Shock,” PNNL-19666, 

11 June 2014, ADAMS Accession No. ML14162A001.

12 [25] U.S. Nuclear Regulatory Commission, “Evaluation of the Beltline Region for 

13 Nuclear Reactor Pressure Vessels,” NRC Technical Letter 

14 Report TLR-RES/DE/CIB-2013-01, November 14, 2014, ADAMS Accession 

15 No. ML14318A177.

16

17 [26] EPRI, “Nondestructive Evaluation: Probabilistic Analysis of Performance 

18 Demonstration Ultrasonic Flaw Detection and Through-Wall Sizing Results for 

19 Reactor Pressure Vessel Inspections,” EPRI Report 1025787, September 2012, 

20 Palo Alto, CA.

21

22 [27] U.S. Nuclear Regulatory Commission, “Calculational and Dosimetry Methods for 

23 Determining Pressure Vessel Neutron Fluence,” Regulatory Guide 1.190, 

24 March 2001, ADAMS Accession No. ML010890301.

25

A-1

1 APPENDIX A: DERIVATION OF THE STATISTICAL TEST FOR TYPE D 

2 DEVIATIONS

3 C2 is defined so that the probability that all the normalized residuals are less than C2 is 

4 1 - α = 0.99, assuming the embrittlement shift model is correct. Under this assumption, the 

5 normalized residuals all have a standard normal distribution with a mean of 0 and standard 

6 deviation of 1. The cumulative distribution function of the standard normal distribution, denoted

7 by F, is then:

8

C2 = F-1 (0.99)1/n Eqn. A-1

9

10 For any n and C2 as determined above, define:

11

p = Prob {C1 < X ≤ C2} Eqn. A-2

12

13 for any C1 < C2 and where X has a standard normal distribution. Then:

14

1-p = Prob {X ≤ C1} + Prob {X > C2} Eqn. A-3

15

16 However, the second term in (1-p) is negligible compared to the first term. In fact, from Table 7

17 in the main body of the text, C2 ≥ 2.71 for n ≥ 3 and therefore Prob {X > C2} < 0.0034 for n ≥ 3. 

18 Thus:

19

1-p ≈ Prob {X ≤ C1} Eqn. A-4

20

21 The probability that the subject dataset does not show a Type D deviation can be expressed in 

22 terms of p. Because all of the n normalized residuals should be less than or equal to C1 to pass 

23 the Outlier Test, the following may be written:

24

Prob {x[1] ≤ C1} = (1-p)

n Eqn. A-5

25

26 Also, the Outlier test states that it is acceptable to have a single normalized residual between C1

27 and C2 while the other (n – 1) normalized residuals are all less than C1. Therefore:

28

Prob {x[2] < C1 ≤ x[1] ≤ C2} = np (1-p)

n-1 Eqn. A-6

29

30 Because Eqn. A-5 and Eqn. A-6 are mutually exclusive, the sum of their probabilities is the 

31 probability of a Type D deviation, or 1 - α = 0.99. This sum, denoted by G(p), is:

32

G(p) = (1-p)

n + np (1-p)n-1 = (1-p)n-1 [1 + (n-1)p] Eqn. A-7

33

34 By iteration, the value p0 may be found that yields G(p0) = 0.99. To calculate C1 in Table 7,

35 Prob {X ≤ C1} is set to 1-p0. Then:

36

C1 = F-1 (1-p0) Eqn. A-8

B-1

APPENDIX B: REGULATORY ASSESSMENT OF STATISTICAL SURVEILLANCE T1

Table B-1. Type A Deviations (Mean Test) 2

General Information Type A De

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] rmean [

o

F] 

Arkansas Nuclear 1 PWR PAN101 P 6 21 -13 

Arkansas Nuclear 1 PWR WAN101 W 3 26 6 

Arkansas Nuclear 2 PWR PAN201 P 3 21 -3 

Beaver Valley 1 PWR PBV101 P 8 21 23.0 

Beaver Valley 1 PWR WBV101 W 4 26 -3 

Beaver Valley 2 PWR PBV201 P 6 19 -7 

Beaver Valley 2 PWR WBV201 W 3 26 -37 

Braidwood 1 PWR FBD101 F 6 19 -7 

Braidwood 1 PWR WBD101 W 3 19 3 

Braidwood 2 PWR FBD201 F 6 19 -9 

Braidwood 2 PWR WBD201 W 3 19 -8 

Brunswick 1 BWR PBR_01 P 5 21 31.3 

Brunswick 1 BWR WBR_01 W 3 26 35 

Byron 1 PWR FBY101 F 6 19 15 

Byron 1 PWR WBY101 W 3 19 0 

Byron 2 PWR FBY201 F 6 19 -5 

Byron 2 PWR WBY201 W 3 19 -1 

Callaway PWR PCL101 P 8 19 -2 

Callaway PWR WCL101 W 4 19 27.0 

Calvert Cliffs 1 PWR PCC103 P 3 21 -6 

Calvert Cliffs 1 PWR WCC101 W 3 26 -17 

Calvert Cliffs 2 PWR PCC202 P 3 21 -7 

Catawba 1 PWR FCB101 F 6 20 -16 

Catawba 1 PWR WCB101 W 3 19 -8 

Catawba 2 PWR PCB201 P 6 21 -1 

Catawba 2 PWR WCB201 W 3 19 1 

Commanche Peak 1 PWR PCP101 P 4 19 -5 

Connecticut Yankee ex-PWR PCTY04 P 3 21 -4 
B-2

General Information Type A De

Plant Name Unit Reactor 

Type Heat Product Form

n

σ [

o

F] 

rmean [

o

F]

Connecticut Yankee ex-PWR PCTY02P3 21-13 

Crystal River3 PWR PCR301P4 21 17 

Crystal River3 PWR WCR301W4 26 12 

Crystal River3 PWR WHSS66W4 26-20 

Crystal River3 PWR WHSSCRW3 26-6 

Crystal River & Surry 3 & 2 PWR WHSS65W3 263 

D.C. Cook1 PWR PCK101P8 21-4 

D.C. Cook1 PWR WCK101W3 26-103 

D.C. Cook2 PWR PCK201P8 21 24.6 

D.C. Cook2 PWR WCK201W4 19 17 

Davis Besse PWR FDB102F4 19-7 

Davis Besse PWR WDB101W5 261 

Davis Besse PWR PRS101P3 21-6 

Davis Besse PWR WRS101W3 26-14 

Diablo Canyon1 PWR PDC103P3 21-24 

Diablo Canyon1 PWR WDC101W3 26-2 

Diablo Canyon2 PWR PDC201P8 21-5 

Diablo Canyon2 PWR WDC201W4 26-3 

Dresden3 BWR WDR302W3 26-17 

Dresden2 BWR PDR201P5 21-32 

Dresden3 BWR PDR301P6 21-10 

Dresden3 BWR WDR301W5 26-5 

Duane Arnold BWR PDAC01P3 21 10 

Duane Arnold BWR WDAC01W3 19-4 

Farley1 PWR PFA101P8 21 13 

Farley1 PWR WFA101W4 26-7 

Farley2 PWR PFA201P8 21 15 

Farley2 PWR WFA201W4 19-51 

FitzPatrick BWR PFTZ01P6 21-11 

Fort Calhoun PWR PFC101P5 21-10 

Fort Calhoun PWR WFC101W4 26-1 

Ginna PWR FGIN02F4 19-18 
B-3

General Information Type A De

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] rmean [

o

F] 

Ginna PWR FGIN01 F 4 19 17 

Ginna PWR WGIN01 W 4 26 13 

H.B. Robinson 2 PWR WHB201 W 3 26 6 

Indian Point 2 PWR PIP202 P 3 21 -1 

Indian Point 2 PWR PIP203 P 3 21 32.6 

Indian Point 3 PWR PIP304 P 5 21 13 

Indian Point 3 PWR WIP301 W 3 26 33 

Kewaunee PWR FKWE02 F 4 19 -7 

Kewaunee PWR FKWE01 F 4 19 -9 

Kewaunee PWR WKWE01 W 4 26 16 

Maine Yankee ex-PWR PMY_01 P 6 21 28.5 

Maine Yankee ex-PWR SHSS01 SRM 17 21 1 

Maine Yankee ex-PWR WMY_01 W 4 26 -8 

McGuire 1 PWR PMC101 P 8 21 13 

McGuire 1 PWR WMC101 W 4 26 -11 

McGuire 2 PWR FMC201 F 8 20 1 

McGuire 2 PWR WMC201 W 4 19 3 

Millstone 1 ex-BWR PML101 P 4 21 -6 

Millstone 2 PWR PML201 P 5 21 13 

Millstone 2 PWR WML201 W 3 26 -27 

Nine Mile Point 1 BWR PNM101 P 5 21 -10 

North Anna 1 PWR FNA101 F 6 20 -14 

North Anna 1 PWR WNA101 W 3 26 21 

North Anna 2 PWR FNA201 F 6 20 -18 

North Anna 2 PWR WNA201 W 3 26 -23 

Oconee 1 PWR POC102 P 6 21 5 

Oconee 1 PWR SHSS02 SRM 64 21 -14 

Oconee 1 PWR WOC101 W 3 26 5 

Oconee 2 PWR FOC201 F 5 19 -3 

Oconee 2 PWR WOC201 W 3 26 -23 

Oconee 3 PWR FOC301 F 4 19 2 

Oconee 3 PWR FOC302 F 3 19 21 
B-4

General Information Type A De

Plant Name Unit Reactor 

Type Heat Product Form

n

σ [

o

F] 

rmean [

o

F]

Oconee3 PWR WOC301W4 26-28 

Oyster Creek BWR PGG_01P4 192 

Oyster Creek BWR PCPR02P4 21-3 

Oyster Creek BWR PEP2JPP5 19-3 

Oyster Creek BWR WGG_01W4 19-8 

Oyster Creek BWR WML101W6 26-23 

Oyster Creek BWR WQC102_1W4 26 26 

Oyster Creek BWR WQC201_1W4 26 17 

Oyster Creek BWR WRB_01W3 19 34.6 

Palisades PWR PPAL01P7 21-33 

Palisades PWR WPAL01W4 26-5 

Point Beach2 PWR FPB202F4 20 19 

Point Beach1 PWR PPB102P4 21-28 

Point Beach1 PWR PPB101P4 219 

Point Beach1 PWR WPB101W4 26-42 

Point Beach2 PWR FPB201F4 19 16 

Point Beach2 PWR WPB201W4 266 

Prarie Island1 PWR FPI101F8 19-2 

Prarie Island1 PWR WPI101W4 26-19 

Prarie Island2 PWR FPI201F8 20-4 

Prarie Island2 PWR WPI201W4 26-11 

Quad Cities1 BWR PQC101P6 211 

Quad Cities1 BWR WQC101W4 26-16 

Quad Cities2 BWR PQC201P4 21-18 

Quad Cities2 BWR WQC201W4 26 18 

Saint Lucie1 PWR PSL101P5 21-19 

Saint Lucie1 PWR WSL101W3 26-18 

Saint Lucie2 PWR PSL201P3 21 14 

Salem1 PWR PSA103P3 21-30 

Salem1 PWR PSA101P3 216 

Salem2 PWR PSA201P8 21 10 

Salem2 PWR WSA201W4 26-18 
B-5

General Information Type A De

Plant Name Unit Reactor 

Type Heat Product Form

n

σ [

o

F] 

rmean [

o

F]

San Onofre2 PWR PSO201P3 21-7 

San Onofre3 PWR PSO301P3 19 40.1 

Seabrook1 PWR PSB101P4 19 19 

Sequoyah1 PWR FSQ101F8 20 18.3 

Sequoyah1 PWR WSQ101W4 26 32.4 

Sequoyah2 PWR FSQ201F8 207 

Sequoyah2 PWR WSQ201W4 26 23 

South Texas1 PWR PST101P4 193 

South Texas2 PWR PST201P4 194 

Surry1 PWR PSU101P3 215 

Surry1 PWR WSU101W3 26 43.5 

Surry2 PWR PSU201P6 210 

Surry2 PWR WSU201W3 26-14 

Three Mile Island1 PWR PTM101P3 21-17 

Trojan ex-PWR PTRO01P6 211 

Trojan ex-PWR WTRO01W3 198 

Turkey Point3 PWR FTP302F3 20 10 

Turkey Point3 PWR SASTM SRM 26 215 

Turkey Point3 PWR WTP301W4 26 13 

Virgil Summer1 PWR PVS101P8 21-20 

Virgil Summer1 PWR WVS101W4 190 

Votgle1 PWR PVO101P6 19-1 

Votgle1 PWR WVO101W3 19-4 

Votgle2 PWR PVO201P6 19-14 

Votgle2 PWR WVO201W3 19-13 

Waterford3 PWR PWF301P3 19-13 

Watts Bar1 PWR FWB101F4 20-6 

Wolf Creek1 PWR PWC101P8 198 

Wolf Creek1 PWR WWC101W4 19 17 

Zion1 ex-PWR PZN101P8 21-2 

Zion1 ex-PWR WZN101W5 26-6 

Zion2 ex-PWR PZN201P6 213 
B-6

General Information Type A De

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] rmean [

o

F] 

Zion 2 ex-PWR WZN201 W 3 26 3 

1

2

3
B-7

Table B-2. Type B Deviations (Slope Test) 1

2

General Information Type B Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] m se(m) Tm 

Arkansas Nuclear 1 PWR PAN101 P 6 21 -26.06 10.09 -2.5

Arkansas Nuclear 1 PWR WAN101 W 3 26 -37.69 19.02 -1.9

Arkansas Nuclear 2 PWR PAN201 P 3 21 -5.72 16.85 -0.3

Beaver Valley 1 PWR PBV101 P 8 21 -37.91 21.46 -1.7

Beaver Valley 1 PWR WBV101 W 4 26 -56.91 15.54 -3.6

Beaver Valley 2 PWR PBV201 P 6 19 4.78 6.63 0.7

Beaver Valley 2 PWR WBV201 W 3 26 -50.14 20.08 -2.5

Braidwood 1 PWR FBD101 F 6 19 26.30 21.44 1.2

Braidwood 1 PWR WBD101 W 3 19 8.25 8.49 0.9

Braidwood 2 PWR FBD201 F 6 19 14.24 22.91 0.6

Braidwood 2 PWR WBD201 W 3 19 6.66 19.20 0.3

Brunswick 1 BWR PBR_01 P 5 21 67.73 13.28 5.1

Brunswick 1 BWR WBR_01 W 3 26 41.69 45.09 0.9

Byron 1 PWR FBY101 F 6 19 7.34 15.79 0.4

Byron 1 PWR WBY101 W 3 19 26.54 8.12 3.2

Byron 2 PWR FBY201 F 6 19 0.03 19.60 0.0

Byron 2 PWR WBY201 W 3 19 27.60 8.32 3.3

Callaway PWR PCL101 P 8 19 -8.04 14.97 -0.5

Callaway PWR WCL101 W 4 19 -41.68 12.80 -3.2

Calvert Cliffs 1 PWR PCC103 P 3 21 13.74 2.75 4.9

Calvert Cliffs 1 PWR WCC101 W 3 26 37.90 2.00 18.9

Calvert Cliffs 2 PWR PCC202 P 3 21 11.27 12.00 0.9

Catawba 1 PWR FCB101 F 6 20 24.31 20.63 1.1

Catawba 1 PWR WCB101 W 3 19 7.91 2.47 3.2

Catawba 2 PWR PCB201 P 6 21 0.93 10.27 0.0

Catawba 2 PWR WCB201 W 3 19 91.48 9.59 9.5

Commanche Peak 1 PWR PCP101 P 4 19 -19.21 15.89 -1.2

Connecticut Yankee ex-PWR PCTY04 P 3 21 -63.22 2.55 -24.7

Connecticut Yankee ex-PWR PCTY02 P 3 21 -47.44 1.52 -31.2
B-8

General Information Type B Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] m se(m) Tm 

Crystal River 3 PWR PCR301 P 4 21 22.02 11.98 1.8

Crystal River 3 PWR WCR301 W 4 26 46.92 13.65 3.4

Crystal River 3 PWR WHSS66 W 4 26 -110.03 117.85 -0.9

Crystal River 3 PWR WHSSCR W 3 26 -116.65 31.01 -3.7

Crystal River & Surry 3 & 2 PWR WHSS65 W 3 26 2.10 14.59 0.1

D.C. Cook 1 PWR PCK101 P 8 21 -8.31 11.44 -0.7

D.C. Cook 1 PWR WCK101 W 3 26 -27.17 98.72 -0.2

D.C. Cook 2 PWR PCK201 P 8 21 1.58 17.98 0.0

D.C. Cook 2 PWR WCK201 W 4 19 -14.88 17.64 -0.8

Davis Besse PWR FDB102 F 4 19 33.72 23.70 1.4

Davis Besse PWR WDB101 W 5 26 10.23 58.55 0.1

Davis Besse PWR PRS101 P 3 21 19.87 18.23 1.0

Davis Besse PWR WRS101 W 3 26 24.78 24.12 1.0

Diablo Canyon 1 PWR PDC103 P 3 21 -2.29 33.31 -0.0

Diablo Canyon 1 PWR WDC101 W 3 26 23.50 74.96 0.3

Diablo Canyon 2 PWR PDC201 P 8 21 -1.49 7.64 -0.1

Diablo Canyon 2 PWR WDC201 W 4 26 -34.17 7.34 -4.6

Dresden 3 BWR WDR302 W 3 26 -25.60 231.23 -0.1

Dresden 2 BWR PDR201 P 5 21 2.15 5.51 0.3

Dresden 3 BWR PDR301 P 6 21 -2.52 4.90 -0.5

Dresden 3 BWR WDR301 W 5 26 7.60 7.66 0.9

Duane Arnold BWR PDAC01 P 3 21 -2.02 13.71 -0.1

Duane Arnold BWR WDAC01 W 3 19 20.21 6.54 3.0

Farley 1 PWR PFA101 P 8 21 21.46 8.51 2.5

Farley 1 PWR WFA101 W 4 26 -27.48 10.40 -2.6

Farley 2 PWR PFA201 P 8 21 20.33 15.54 1.3

Farley 2 PWR WFA201 W 4 19 -25.24 19.69 -1.2

FitzPatrick BWR PFTZ01 P 6 21 8.83 10.75 0.8

Fort Calhoun PWR PFC101 P 5 21 -22.33 33.89 -0.6

Fort Calhoun PWR WFC101 W 4 26 -73.25 49.46 -1.4

Ginna PWR FGIN02 F 4 19 39.26 74.20 0.5

Ginna PWR FGIN01 F 4 19 -33.49 29.11 -1.1
B-9

General Information Type B Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] m se(m) Tm 

Ginna PWR WGIN01 W 4 26 13.77 24.57 0.5

H.B. Robinson 2 PWR WHB201 W 3 26 7.86 33.97 0.2

Indian Point 2 PWR PIP202 P 3 21 -59.41 46.88 -1.2

Indian Point 2 PWR PIP203 P 3 21 -17.19 2.14 -8.0

Indian Point 3 PWR PIP304 P 5 21 2.30 29.72 0.0

Indian Point 3 PWR WIP301 W 3 26 -0.18 79.92 0.0

Kewaunee PWR FKWE02 F 4 19 5.09 6.32 0.8

Kewaunee PWR FKWE01 F 4 19 -3.54 28.08 -0.1

Kewaunee PWR WKWE01 W 4 26 -15.96 19.26 -0.8

Maine Yankee ex-PWR PMY_01 P 6 21 -6.49 15.82 -0.4

Maine Yankee ex-PWR SHSS01 SRM 17 21 -4.95 15.22 -0.3

Maine Yankee ex-PWR WMY_01 W 4 26 10.14 12.08 0.8

McGuire 1 PWR PMC101 P 8 21 24.92 19.19 1.3

McGuire 1 PWR WMC101 W 4 26 -43.26 13.24 -3.2

McGuire 2 PWR FMC201 F 8 20 4.68 13.50 0.3

McGuire 2 PWR WMC201 W 4 19 -34.72 13.01 -2.6

Millstone 1 ex-BWR PML101 P 4 21 -46.63 28.45 -1.6

Millstone 2 PWR PML201 P 5 21 23.17 31.56 0.7

Millstone 2 PWR WML201 W 3 26 -56.09 3.80 -14.7

Nine Mile Point 1 BWR PNM101 P 5 21 41.45 22.11 1.8

North Anna 1 PWR FNA101 F 6 20 8.37 20.75 0.4

North Anna 1 PWR WNA101 W 3 26 -40.22 70.03 -0.5

North Anna 2 PWR FNA201 F 6 20 18.17 12.92 1.4

North Anna 2 PWR WNA201 W 3 26 -5.18 23.10 -0.2

Oconee 1 PWR POC102 P 6 21 -19.62 25.32 -0.7

Oconee 1 PWR SHSS02 SRM 64 21 1.87 4.39 0.4

Oconee 1 PWR WOC101 W 3 26 41.08 14.09 2.9

Oconee 2 PWR FOC201 F 5 19 -15.62 13.86 -1.1

Oconee 2 PWR WOC201 W 3 26 30.52 12.69 2.4

Oconee 3 PWR FOC301 F 4 19 -2.79 5.78 -0.4

Oconee 3 PWR FOC302 F 3 19 -21.76 32.80 -0.6

Oconee 3 PWR WOC301 W 4 26 41.81 33.82 1.2
B-10

General Information Type B Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] m se(m) Tm 

Oyster Creek BWR PGG_01 P 4 19 21.14 22.39 0.9

Oyster Creek BWR PCPR02 P 4 21 -8.62 42.21 -0.2

Oyster Creek BWR PEP2JP P 5 19 -32.14 8.29 -3.8

Oyster Creek BWR WGG_01 W 4 19 39.71 61.15 0.6

Oyster Creek BWR WML101 W 6 26 10.03 20.07 0.5

Oyster Creek BWR WQC102_1 W 4 26 6.48 11.60 0.5

Oyster Creek BWR WQC201_1 W 4 26 57.46 59.89 0.9

Oyster Creek BWR WRB_01 W 3 19 20.94 63.55 0.3

Palisades PWR PPAL01 P 7 21 -77.49 15.02 -5.1

Palisades PWR WPAL01 W 4 26 -42.23 5.32 -7.9

Point Beach 2 PWR FPB202 F 4 20 41.30 12.36 3.3

Point Beach 1 PWR PPB102 P 4 21 -76.91 8.77 -8.7

Point Beach 1 PWR PPB101 P 4 21 -55.54 23.06 -2.4

Point Beach 1 PWR WPB101 W 4 26 -12.46 53.53 -0.2

Point Beach 2 PWR FPB201 F 4 19 -0.70 8.03 -0.0

Point Beach 2 PWR WPB201 W 4 26 38.35 27.60 1.3

Prarie Island 1 PWR FPI101 F 8 19 -0.54 17.08 -0.0

Prarie Island 1 PWR WPI101 W 4 26 57.84 38.91 1.4

Prarie Island 2 PWR FPI201 F 8 20 8.36 9.37 0.8

Prarie Island 2 PWR WPI201 W 4 26 -40.19 12.22 -3.2

Quad Cities 1 BWR PQC101 P 6 21 7.70 2.14 3.5

Quad Cities 1 BWR WQC101 W 4 26 10.12 7.81 1.3

Quad Cities 2 BWR PQC201 P 4 21 -10.04 3.17 -3.1

Quad Cities 2 BWR WQC201 W 4 26 -3.89 4.21 -0.9

Saint Lucie 1 PWR PSL101 P 5 21 -5.59 16.17 -0.3

Saint Lucie 1 PWR WSL101 W 3 26 -46.07 12.84 -3.5

Saint Lucie 2 PWR PSL201 P 3 21 49.57 1.95 25.4

Salem 1 PWR PSA103 P 3 21 -6.76 19.49 -0.3

Salem 1 PWR PSA101 P 3 21 5.06 24.98 0.2

Salem 2 PWR PSA201 P 8 21 -3.87 21.39 -0.1

Salem 2 PWR WSA201 W 4 26 -45.51 9.34 -4.8

San Onofre 2 PWR PSO201 P 3 21 11.38 26.39 0.4
B-11

General Information Type B Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] m se(m) Tm 

San Onofre 3 PWR PSO301 P 3 19 15.75 21.22 0.7

Seabrook 1 PWR PSB101 P 4 19 -6.15 14.03 -0.4

Sequoyah 1 PWR FSQ101 F 8 20 31.83 17.01 1.8

Sequoyah 1 PWR WSQ101 W 4 26 -8.36 4.54 -1.8

Sequoyah 2 PWR FSQ201 F 8 20 12.09 14.40 0.8

Sequoyah 2 PWR WSQ201 W 4 26 -71.71 89.12 -0.8

South Texas 1 PWR PST101 P 4 19 -9.15 21.18 -0.4

South Texas 2 PWR PST201 P 4 19 16.33 4.60 3.5

Surry 1 PWR PSU101 P 3 21 10.46 20.36 0.5

Surry 1 PWR WSU101 W 3 26 9.51 7.62 1.2

Surry 2 PWR PSU201 P 6 21 -2.98 13.63 -0.2

Surry 2 PWR WSU201 W 3 26 7.37 24.69 0.3

Three Mile Island 1 PWR PTM101 P 3 21 -41.07 2.89 -14.2

Trojan ex-PWR PTRO01 P 6 21 -2.31 12.00 -0.1

Trojan ex-PWR WTRO01 W 3 19 -14.97 10.82 -1.3

Turkey Point 3 PWR FTP302 F 3 20 52.36 70.45 0.7

Turkey Point 3 PWR SASTM SRM 26 21 10.59 8.87 1.1

Turkey Point 3 PWR WTP301 W 4 26 15.55 35.61 0.4

Virgil Summer 1 PWR PVS101 P 8 21 -12.14 16.33 -0.7

Virgil Summer 1 PWR WVS101 W 4 19 -20.29 23.75 -0.8

Votgle 1 PWR PVO101 P 6 19 20.85 19.05 1.0

Votgle 1 PWR WVO101 W 3 19 -41.81 19.81 -2.1

Votgle 2 PWR PVO201 P 6 19 15.57 10.12 1.5

Votgle 2 PWR WVO201 W 3 19 25.29 19.30 1.3

Waterford 3 PWR PWF301 P 3 19 -87.52 64.25 -1.3

Watts Bar 1 PWR FWB101 F 4 20 41.28 90.57 0.4

Wolf Creek 1 PWR PWC101 P 8 19 -8.38 9.54 -0.8

Wolf Creek 1 PWR WWC101 W 4 19 4.70 8.95 0.5

Zion 1 ex-PWR PZN101 P 8 21 14.89 14.97 0.9

Zion 1 ex-PWR WZN101 W 5 26 43.23 8.52 5.0

Zion 2 ex-PWR PZN201 P 6 21 19.41 15.57 1.2

Zion 2 ex-PWR WZN201 W 3 26 17.59 31.94 0.5

1
B-12

Table B-3. Type D Deviations (Outlier Test) 1

2

General Information Type D Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] ri(max1) ri(max2) C1 

Arkansas Nuclear 1 PWR PAN101 P 6 21 0.98 -0.13 1.9

Arkansas Nuclear 1 PWR WAN101 W 3 26 1.47 0.03 1.5

Arkansas Nuclear 2 PWR PAN201 P 3 21 0.36 -0.32 1.5

Beaver Valley 1 PWR PBV101 P 8 21 2.52 2.26 2.0

Beaver Valley 1 PWR WBV101 W 4 26 0.83 0.14 1.7

Beaver Valley 2 PWR PBV201 P 6 19 -0.04 -0.06 1.9

Beaver Valley 2 PWR WBV201 W 3 26 -0.61 -1.39 1.5

Braidwood 1 PWR FBD101 F 6 19 0.41 0.23 1.9

Braidwood 1 PWR WBD101 W 3 19 0.42 0.03 1.5

Braidwood 2 PWR FBD201 F 6 19 0.85 0.41 1.9

Braidwood 2 PWR WBD201 W 3 19 0.04 -0.59 1.5

Brunswick 1 BWR PBR_01 P 5 21 4.09 2.99 1.8

Brunswick 1 BWR WBR_01 W 3 26 2.56 1.53 1.5

Byron 1 PWR FBY101 F 6 19 1.67 1.58 1.9

Byron 1 PWR WBY101 W 3 19 0.38 0.37 1.5

Byron 2 PWR FBY201 F 6 19 0.46 0.36 1.9

Byron 2 PWR WBY201 W 3 19 0.59 -0.10 1.5

Callaway PWR PCL101 P 8 19 1.06 0.70 2.0

Callaway PWR WCL101 W 4 19 3.03 0.98 1.7

Calvert Cliffs 1 PWR PCC103 P 3 21 -0.10 -0.19 1.5

Calvert Cliffs 1 PWR WCC101 W 3 26 -0.14 -0.71 1.5

Calvert Cliffs 2 PWR PCC202 P 3 21 -0.14 -0.38 1.5

Catawba 1 PWR FCB101 F 6 20 0.14 0.05 1.9

Catawba 1 PWR WCB101 W 3 19 -0.27 -0.46 1.5

Catawba 2 PWR PCB201 P 6 21 0.30 0.26 1.9

Catawba 2 PWR WCB201 W 3 19 1.75 0.74 1.5

Commanche Peak 1 PWR PCP101 P 4 19 0.38 -0.15 1.7

Connecticut Yankee ex-PWR PCTY04 P 3 21 1.02 -0.60 1.5

Connecticut Yankee ex-PWR PCTY02 P 3 21 0.17 -0.45 1.5

Crystal River 3 PWR PCR301 P 4 21 1.46 1.03 1.7
B-13

General Information Type D Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] ri(max1) ri(max2) C1 

Crystal River 3 PWR WCR301 W 4 26 1.12 1.00 1.7

Crystal River 3 PWR WHSS66 W 4 26 0.71 -0.78 1.7

Crystal River 3 PWR WHSSCR W 3 26 0.77 0.06 1.5

Crystal River & Surry 3 & 2 PWR WHSS65 W 3 26 0.23 0.18 1.5

D.C. Cook 1 PWR PCK101 P 8 21 0.57 0.28 2.0

D.C. Cook 1 PWR WCK101 W 3 26 -2.30 -3.73 1.5

D.C. Cook 2 PWR PCK201 P 8 21 2.27 1.68 2.0

D.C. Cook 2 PWR WCK201 W 4 19 1.47 1.28 1.7

Davis Besse PWR FDB102 F 4 19 0.98 -0.60 1.7

Davis Besse PWR WDB101 W 5 26 2.01 0.32 1.8

Davis Besse PWR PRS101 P 3 21 -0.05 -0.12 1.5

Davis Besse PWR WRS101 W 3 26 -0.26 -0.36 1.5

Diablo Canyon 1 PWR PDC103 P 3 21 -0.53 -1.24 1.5

Diablo Canyon 1 PWR WDC101 W 3 26 1.14 -0.69 1.5

Diablo Canyon 2 PWR PDC201 P 8 21 0.21 -0.04 2.0

Diablo Canyon 2 PWR WDC201 W 4 26 0.54 -0.15 1.7

Dresden 3 BWR WDR302 W 3 26 0.74 -1.02 1.5

Dresden 2 BWR PDR201 P 5 21 -0.23 -1.56 1.8

Dresden 3 BWR PDR301 P 6 21 0.49 -0.20 1.9

Dresden 3 BWR WDR301 W 5 26 1.01 0.43 1.8

Duane Arnold BWR PDAC01 P 3 21 0.66 0.39 1.5

Duane Arnold BWR WDAC01 W 3 19 0.14 -0.32 1.5

Farley 1 PWR PFA101 P 8 21 1.39 1.20 2.0

Farley 1 PWR WFA101 W 4 26 0.38 -0.46 1.7

Farley 2 PWR PFA201 P 8 21 1.92 1.22 2.0

Farley 2 PWR WFA201 W 4 19 -2.12 -2.13 1.7

FitzPatrick BWR PFTZ01 P 6 21 -0.03 -0.16 1.9

Fort Calhoun PWR PFC101 P 5 21 0.03 -0.11 1.8

Fort Calhoun PWR WFC101 W 4 26 0.59 0.32 1.7

Ginna PWR FGIN02 F 4 19 1.35 -0.34 1.7

Ginna PWR FGIN01 F 4 19 2.19 1.10 1.7

Ginna PWR WGIN01 W 4 26 1.17 0.58 1.7
B-14

General Information Type D Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] ri(max1) ri(max2) C1 

H.B. Robinson 2 PWR WHB201 W 3 26 0.96 0.05 1.5

Indian Point 2 PWR PIP202 P 3 21 1.26 -0.57 1.5

Indian Point 2 PWR PIP203 P 3 21 1.81 1.53 1.5

Indian Point 3 PWR PIP304 P 5 21 1.41 1.11 1.8

Indian Point 3 PWR WIP301 W 3 26 1.90 1.54 1.5

Kewaunee PWR FKWE02 F 4 19 -0.08 -0.43 1.7

Kewaunee PWR FKWE01 F 4 19 -0.04 -0.05 1.7

Kewaunee PWR WKWE01 W 4 26 1.10 0.74 1.7

Maine Yankee ex-PWR PMY_01 P 6 21 2.36 1.58 1.9

Maine Yankee ex-PWR SHSS01 SRM 17 21 1.48 0.96 2.3

Maine Yankee ex-PWR WMY_01 W 4 26 0.05 -0.12 1.7

McGuire 1 PWR PMC101 P 8 21 1.98 1.16 2.0

McGuire 1 PWR WMC101 W 4 26 0.54 -0.68 1.7

McGuire 2 PWR FMC201 F 8 20 1.52 0.13 2.0

McGuire 2 PWR WMC201 W 4 19 1.32 0.14 1.7

Millstone 1 ex-BWR PML101 P 4 21 0.43 0.18 1.7

Millstone 2 PWR PML201 P 5 21 1.94 1.09 1.8

Millstone 2 PWR WML201 W 3 26 -0.27 -1.13 1.5

Nine Mile Point 1 BWR PNM101 P 5 21 0.05 -0.24 1.8

North Anna 1 PWR FNA101 F 6 20 0.88 -0.65 1.9

North Anna 1 PWR WNA101 W 3 26 2.07 0.94 1.5

North Anna 2 PWR FNA201 F 6 20 -0.12 -0.17 1.9

North Anna 2 PWR WNA201 W 3 26 -0.59 -0.63 1.5

Oconee 1 PWR POC102 P 6 21 1.53 1.41 1.9

Oconee 1 PWR SHSS02 SRM 64 21 1.47 1.40 2.8

Oconee 1 PWR WOC101 W 3 26 0.87 0.31 1.5

Oconee 2 PWR FOC201 F 5 19 0.94 0.00 1.8

Oconee 2 PWR WOC201 W 3 26 -0.09 -1.17 1.5

Oconee 3 PWR FOC301 F 4 19 0.37 0.16 1.7

Oconee 3 PWR FOC302 F 3 19 2.53 0.98 1.5

Oconee 3 PWR WOC301 W 4 26 0.89 -1.03 1.7

Oyster Creek BWR PGG_01 P 4 19 0.23 -0.05 1.7
B-15

General Information Type D Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] ri(max1) ri(max2) C1 

Oyster Creek BWR PCPR02 P 4 21 0.48 -0.17 1.7

Oyster Creek BWR PEP2JP P 5 19 0.86 0.12 1.8

Oyster Creek BWR WGG_01 W 4 19 0.62 -0.32 1.7

Oyster Creek BWR WML101 W 6 26 0.07 -0.70 1.9

Oyster Creek BWR WQC102_1 W 4 26 1.14 1.02 1.7

Oyster Creek BWR WQC201_1 W 4 26 1.16 1.03 1.7

Oyster Creek BWR WRB_01 W 3 19 2.27 1.87 1.5

Palisades PWR PPAL01 P 7 21 0.11 -0.48 

Palisades PWR WPAL01 W 4 26 0.21 0.07 1.7

Point Beach 2 PWR FPB202 F 4 20 1.64 1.35 1.7

Point Beach 1 PWR PPB102 P 4 21 -0.37 -0.52 1.7

Point Beach 1 PWR PPB101 P 4 21 1.08 1.03 1.7

Point Beach 1 PWR WPB101 W 4 26 -0.46 -1.58 1.7

Point Beach 2 PWR FPB201 F 4 19 1.06 0.94 1.7

Point Beach 2 PWR WPB201 W 4 26 0.00 0.00 1.7

Prarie Island 1 PWR FPI101 F 8 19 0.77 0.77 2.0

Prarie Island 1 PWR WPI101 W 4 26 0.89 -0.43 1.7

Prarie Island 2 PWR FPI201 F 8 20 0.56 0.11 2.0

Prarie Island 2 PWR WPI201 W 4 26 0.57 -0.47 1.7

Quad Cities 1 BWR PQC101 P 6 21 0.74 0.60 1.9

Quad Cities 1 BWR WQC101 W 4 26 0.63 -0.63 1.7

Quad Cities 2 BWR PQC201 P 4 21 -0.11 -0.38 1.7

Quad Cities 2 BWR WQC201 W 4 26 1.07 0.81 1.7

Saint Lucie 1 PWR PSL101 P 5 21 -0.55 -0.72 1.8

Saint Lucie 1 PWR WSL101 W 3 26 -0.28 -0.68 1.5

Saint Lucie 2 PWR PSL201 P 3 21 2.02 -0.01 1.5

Salem 1 PWR PSA103 P 3 21 -1.09 -1.33 1.5

Salem 1 PWR PSA101 P 3 21 0.91 0.01 1.5

Salem 2 PWR PSA201 P 8 21 1.58 1.20 2.0

Salem 2 PWR WSA201 W 4 26 -0.12 -0.24 1.7

San Onofre 2 PWR PSO201 P 3 21 0.02 -0.10 1.5

San Onofre 3 PWR PSO301 P 3 19 2.43 2.34 1.5
B-16

General Information Type D Deviation 

Plant Name Unit Reactor 

Type Heat Product Form n σ [

o

F] ri(max1) ri(max2) C1 

Seabrook 1 PWR PSB101 P 4 19 1.39 1.19 1.7

Sequoyah 1 PWR FSQ101 F 8 20 2.37 1.73 2.0

Sequoyah 1 PWR WSQ101 W 4 26 1.39 1.31 1.7

Sequoyah 2 PWR FSQ201 F 8 20 1.57 1.36 2.0

Sequoyah 2 PWR WSQ201 W 4 26 3.62 1.44 1.7

South Texas 1 PWR PST101 P 4 19 0.71 0.59 1.7

South Texas 2 PWR PST201 P 4 19 0.67 0.39 1.7

Surry 1 PWR PSU101 P 3 21 0.80 0.02 1.5

Surry 1 PWR WSU101 W 3 26 1.88 1.58 1.5

Surry 2 PWR PSU201 P 6 21 0.55 0.47 1.9

Surry 2 PWR WSU201 W 3 26 0.00 -0.58 1.5

Three Mile Island 1 PWR PTM101 P 3 21 0.38 -1.29 1.5

Trojan ex-PWR PTRO01 P 6 21 0.74 0.14 1.9

Trojan ex-PWR WTRO01 W 3 19 0.77 0.55 1.5

Turkey Point 3 PWR FTP302 F 3 20 1.05 0.67 1.5

Turkey Point 3 PWR SASTM SRM 26 21 1.98 1.94 2.5

Turkey Point 3 PWR WTP301 W 4 26 1.30 0.36 1.7

Virgil Summer 1 PWR PVS101 P 8 21 0.12 -0.15 2.0

Virgil Summer 1 PWR WVS101 W 4 19 0.95 0.16 1.7

Votgle 1 PWR PVO101 P 6 19 0.70 0.56 1.9

Votgle 1 PWR WVO101 W 3 19 0.67 -0.02 1.5

Votgle 2 PWR PVO201 P 6 19 -0.17 -0.23 1.9

Votgle 2 PWR WVO201 W 3 19 -0.17 -0.49 1.5

Waterford 3 PWR PWF301 P 3 19 0.53 -0.87 1.5

Watts Bar 1 PWR FWB101 F 4 20 0.94 0.85 1.7

Wolf Creek 1 PWR PWC101 P 8 19 1.10 0.88 2.0

Wolf Creek 1 PWR WWC101 W 4 19 1.20 1.20 1.7

Zion 1 ex-PWR PZN101 P 8 21 0.43 0.33 2.0

Zion 1 ex-PWR WZN101 W 5 26 0.31 0.02 1.8

Zion 2 ex-PWR PZN201 P 6 21 1.20 0.27 1.9

Zion 2 ex-PWR WZN201 W 3 26 0.66 0.04 1.5
C-1

1 APPENDIX C: FLAW DEPTH AND DENSITY DISTRIBUTIONS 

2 CONSIDERING PROBABILITY OF DETECTION AND BIAS IN NDE 

3 DATA WITH APPLICATIONS

4

5 Prepared for the U.S. Nuclear Regulatory Commission under 

6 Subcontract 4000099247with UT-Battelle, LLC (Oak Ridge National Laboratory)

7

8

9

10

11

12

13 Mohammad Modarres

14 University of Maryland

15

16 October 2011

17
C-2

1 Table of Contents

2 TABLE OF CONTENTS..........................................................................................................C-2

3 LIST OF TABLES....................................................................................................................C-3

4 LIST OF FIGURES .................................................................................................................C-3

5 ABBREVIATIONS...................................................................................................................C-4

6 SYMBOLS ...................................................................................................................................

7 .......................................................................................................................................C-5

8 C.1 Introduction and Background ...........................................................................C-7

9 C.2 Probabilistic Approach to Combine Uncertain NDE Flaw Data with

10 Prior Flaw Distributions ....................................................................................... 9

11 C.2.1 The Bayesian Framework ............................................................................C-9

12 C.2.2 Probability of Detection and Measurement Error........................................C-11

13 C.2.2.1...............................................................................................Probability of Detection

14 ..................................................................................................................C-11

15 C.2.2.2.......................................................................................Measurement (Sizing) Error

16 ..................................................................................................................C-12

17 C.2.3 The Methodology for Updating the NDE Data ............................................C-13

18 C.2.3.1................................................................Updating Flaw Depth Distribution Models

19 ..................................................................................................................C-13

20 C.2.3.1.1Likelihood Function of Exact Flaw Depth Measurements.................C-16

21 C.2.3.1.2Likelihood Function of Interval Flaw Depth Measurements..............C-16

22 C.2.3.1.3Bayesian Updating of Parameters of the Flaw Depth Distribution....C-17

23 C.2.3.2..............................................................................Updating the Flaw Density Model

24 ..................................................................................................................C-17

25 C.3 Application Example ......................................................................................C-19

26 C.3.1 Description of the NDE Data Used as Evidence to Build the Likelihood 

27 Function ....................................................................................................C-19

28 C.3.2 Description of Flaw Depth and Density Information Used in VFLAW

29 as Prior PDFs in this Example ...................................................................C-20

30 C.3.3 Detection and Sizing Error Models .............................................................C-23

31 C.3.4 Bayesian Updating of the Parameters of the Flaw Depth Distributions.......C-24

32 C.3.4.1............Bayesian Updating of the Parameters of Large Flaw Depth Distribution

33 Based on the 3/8t UT Inspection Data .......................................................C-25

34 C.3.4.2............Bayesian Updating of the Parameters of Small Flaw Depth Distribution

35 Based on the 3/8t UT Inspection Data .......................................................C-27

36 C.3.4.3..................Results of Updating Parameters of Flaw Density Distribution Based

37 on the 3/8t UT Inspection Data ..................................................................C-29

38 C.3.4.3.1Posterior Density of Large Flaws .....................................................C-29

39 C.3.4.3.2Posterior Density of Small Flaws .....................................................C-30

40 C.3.4.3.3Sensitivity of the Flaw Density Results to a Lower Bound of

41 0.075” for the Interval Data Observed in the 3/8t Region .................C-33

42 C.3.5 Updating the 10 CFR 50.61a Tables ..........................................................C-33

43 C.4 References ....................................................................................................C-37

44

45
C-3

1 List of Tables

2 Table C-1. Summary of Likelihood Functions........................................................................C-10

3 Table C-2. Flaw Depth (a) and Flaw Density (b) Distributions and Hyper-PDF

4 Parameters Used in VFLAW for PVRUF..............................................................C-22

5 Table C-3. Updated VFLAW Distribution for Small Flaws......................................................C-28

6 Table C-4. Summary of the Posterior Parameter PDFs of Flaw Depth and Flaw

7 Density to be used in VFLAW and FAVOR Runs for Beaver Valley-2..................C-32

8 Table C-5. Alternate PTS Rule Flaw Table Assessment using Mean Number-of-Flaws

9 by Size Interval for Beaver Valley-2 (Assuming a Lower Limit of 0.04” for

10 Observed Data

11 Interval) ...............................................................................................................C-35

12 Table C-6. Alternate PTS Rule Flaw Table Assessment using Mean Number-of-Flaws

13 by Size Interval for Beaver Valley-2 (Assuming a Lower Limit of 0.075” for

14 Observed Data Interval).......................................................................................C-35

15

16

17

18 List of Figures

19 Figure C-1. A simple description of the Bayesian updating process .......................................C-8

20 Figure C-2. Measurement error distribution..........................................................................C-13

21 Figure C-3. Beaver Valley-2 weld map [C-14] ......................................................................C-20

22 Figure C-4. Flaw depth and flaw density distributions used in VFLAW and their

23 corresponding parameter hyper-PDFs ..............................................................C-21

24 Figure C-5. POD vs. flaw depth............................................................................................C-23

25 Figure C-6. Measurement error vs. flaw depth .....................................................................C-24

26 Figure C-7. Bayesian updating elements of the Beaver Valley-2 example............................C-25

Figure C-8. Prior and posterior distributions of the flaw depth intensity of parameter λl 27 ........C-26

28 Figure C-9. Bayesian prior and posterior PDFs of λs (per unit inch)......................................C-28

29 Figure C-10. Marginal POD independent of small flaw depth ...............................................C-28

30 Figure C-11. Prior and posterior flaw density (a) and intensity (b) for large size flaw............C-30

31 Figure C-12. Prior and posterior flaw density (a) and intensity (b) for small size flaws..........C-31

32 Figure C-13. Posterior number of flaws in the 3/8t region in various flaw size intervals

33 including epistemic uncertainties .....................................................................C-33

34 Figure C-14. Posterior number of flaws in the Supplement 4 inspection volume in

35 various flaw size intervals including epistemic uncertainties ............................C-36

36

37

38

39

40
C-4

1 Abbreviations

2

Abbreviation Definition

PDF Probability Density Function

CDF Cumulative Density Function

ISI In-Service Inspection

NDE Nondestructive Evaluation

POD Probability of Detection

PTS Pressurized Thermal Shock

RPV Reactor Pressure Vessel

SAW Submerged Arc Weld

SMAW Shielded Metal Arc Weld

UT Ultrasonic Test

3

4

5
C-5

1 Symbols and Expressions

2

Symbol Definition

a True flaw depth (inches)

A Random variable representing flaw depth

aˆ Logarithm of signal-response amplitude in flaw-detection NDE

a* NDE measured (or observed) flaw depth (inches)

ath Threshold flaw depth for detection below which flaw detection is beyond the 

capability of the NDE technology used

D The event that a flaw is detected

D The event that a flaw is not detected

EM Model error of the NDE measurement error, represented by a normal 

distribution with mean of zero and known standard deviation

fa(a*-Mε|Φ) PDF of the true flaw depth, given the vector of PDF parameters represented in 

terms of observed flaw depth a* but corrected for the measurement error Mε

( ) l tr f a | Φ, a ≥ t Conditional PDF of large flaw depth

( ) s tr f a | Φ, a < t Conditional PDF of small flaw depth

f(.), g(.), h(.), 

l(.), m(.), x(.), 

g’(.). k(.), γ(.)

PDF functions of model parameters, flaw depth, or flaw density

g(Mε) PDF of the measurement error (in terms of a*)

L(a|Φ) Likelihood of true flaw depth given the vector of parameters Φ

L(Data|θ) Likelihood of observed NDE data conditioned on (given) the unknown 

parameter θ

L(n | n )j

* Likelihood of observing n* flaws given there are a total of nj true flaws

Mε NDE measurement error (combined random and systematic errors)

m*i Number of flaw depths observed (reported in NDE) in the interval i

n Number of flaw-depth intervals (reported in NDE)

n* Number of exact flaw depths observed (reported in NDE)

















*

j

n

n Combination of n* observed flaws out of total flaws nj (observed or detected, 

and unobserved or not detected flaws) 

N Mean of the PDF of the number of flaws in volume v

N(0;σ) Normal PDF with mean zero and standard deviation of σ

POD(a) Probability of detection of a flaw of depth a

POD(a*-Mε) POD of true flaw depth represented in terms of observed flaw depth a* by 

correcting for the measurement error Mε

Pr(.) Probability

Pr(D) POD independent of flaw depth

Pr(D) Probability of no detection regardless of size

Pr(n | n )

*

j Probability of total flaws nj conditioned on observing n* flaws in NDE

Pr(Φ, ttr) POD independent of flaw size (i.e., Pr(D))

ttr Transition flaw depth which separates large flaw depth from small flaw depth

v Volume of inspected weld

βi Parameter i of the POD model

Δ Weld bead thickness
C-6

Symbol Definition

ε Random error of the model

λ Flaw-depth intensity parameter (inch-1) of the exponential distribution a f(a) e−λ = λ representing flaw-depth distribution, also flaw intensity (flaws/ft3

) 

representing Poisson flaw density distribution:

N!

( v) Pr(N) e

N v λ = −λ

π0(θ) Prior PDF of an unknown parameter θ

π1(θ|Data) Posterior PDF of an unknown parameter θ given observed NDE data

Φ Vector of parameters of the flaw-depth PDF

θ An unknown parameter of a model

Θ Vector of parameters of the POD model

ρ Poisson distribution parameter representing the volumetric intensity of flaws 

(flaws per unit volume)

σPOD(a) Standard deviation of PDF of the POD model error vs. flaw depth

γ(.|α1, α2) Gamma PDF with parameters α1 and α2

1

2
C-7

1 C.1 Introduction and Background

2 In 2010, the NRC amended its regulations to provide alternative screening methods for 

3 protection against Pressurized Thermal Shock (PTS) events that could affect the Reactor 

4 Pressure Vessels (RPVs) of the operating Pressurized-Water Reactors (PWRs). The new rule, 

5 10 CFR 50.61a [C-1], provides a means for determining whether the flaws found through 

6 In-Service Inspection (ISI) of a particular RPV are consistent with the assumptions regarding the 

7 number and size of flaws used in the PTS analyses that provided the technical basis for the new 

8 rule. To address this requirement, 10 CFR 50.61a includes two tables (one table for weld 

9 material and one table for plate/forging material) that express the maximum flaw density and 

10 flaw depths that are allowed in the beltline of an RPV.

11 The new rule relies on flaw characteristics (flaw depth and density) that were simulated and 

12 used by the probabilistic fracture mechanics computer code FAVOR [C-2] to develop the 

13 10 CFR 50.61a flaw tables. These flaw characteristics were proposed by Simonen et al. in 

14 NUREG/CR-6817 [C-3] and are used by the VFLAW code to generate the input flaws for 

15 FAVOR. The data include distribution functions that represent the initial fabrication flaws in 

16 some RPVs. The FAVOR computer code fully incorporates these distributions by sampling from 

17 each using Monte Carlo techniques.

18 The flaw distributions reported in NUREG/CR-6817 are based on information obtained from 

19 destructive and very precise techniques used to experimentally detect and size the flaws. 

20 These tables have been viewed as the permissible limit for distribution of the “true” flaw depths 

21 and flaw densities. These depth and densities may be contrasted with those “observed” in an 

22 ISI using Non-Destructive Evaluation (NDE) techniques such as the Ultrasonic Test (UT). The 

23 NDE results used to assess compliance with the 10 CFR 50.61a tables would be more 

24 representative of the specific vessel inspected than the flaw distributions in NUREG/CR-6817 

25 and could be used to support development of vessel-specific flaw-size and -density 

26 distributions. However, the NDE results are influenced by detection limits, detection errors, and 

27 sizing and measurement errors. Development of the vessel-specific distribution of the true flaw 

28 density and depth should account for such uncertainties.

29 This report describes the development and application of a methodology to account for 

30 uncertainties in NDE data and to analyze such data for the purpose of developing more realistic 

31 vessel-specific flaw-depth and -density distributions for comparison to the 10 CFR 50.61a 

32 screening tables, as well as for use as an input to FAVOR analysis. The methodology 

33 considers detection probability, flaw-measurement error, and flaw-detection threshold in its 

34 application. Application of the methodology is demonstrated using the ultrasonic NDE data 

35 obtained from an ISI of the Beaver Valley 2 RPV.

36 Flaw distributions developed and reported in NUREG/CR-6817 are based on destructive 

37 evaluation of two cancelled RPVs supplemented by expert judgment data. In contrast, the 

38 vessel-specific inspection data come from NDE inspections using, for example, UT methods. 

39 The variability in both the characteristics of the data (size, range, and number of flaws 

40 measured) and the UT system performance (probability of detection and measurement/sizing 

41 errors) highlighted a need for analyzing the inspection results on a vessel-specific or 

42 data-set-specific basis. For this purpose, traditional methods were inadequate, and therefore a 

43 new methodology that could accept a very small number of flaws (typical of vessel-specific NDE 

44 results), including inspection-system flaw-detection reliability, flaw-sizing accuracy 

45 (measurement error), and flaw-detection threshold, was developed.
C-8

1 The methodology is demonstrated here through an application using the UT data reported for 

2 the Beaver Valley 2 RPV [C-13]. The objective was to provide a probabilistic description (in 

3 terms of the probability density function) of the actual flaw depth and flaw density of the welds 

4 based on the detected flaws from NDE examinations of this RPV using UT. Because the NDE 

5 data are uncertain and contain measurement errors, data developed as part of the PTS study 

6 (the so-called VFLAW data) were specialized to the Beaver Valley 2 RPV and used as the prior 

7 flaw information. Combination of the prior information and the NDE results of the Beaver 

8 Valley 2 RPV led to the development of the posterior flaw information, which represents the 

9 distribution of the true flaw depth and flaw density of the Beaver Valley 2 RPV welds. See 

10 Figure C-1 for an illustration of the Bayesian updating process. It is important to note that the 

11 VFLAW data, while representing generic values of flaw characteristics (based on expert 

12 judgment and on the PVRUF and Shoreham RPVs), may be specialized to a specific RPV, for 

13 example by using RPV-specific weld length, bead thickness, geometry and other weld 

14 characteristics of (in this case) Beaver Valley 2. Therefore, the specialized VFLAW data would 

15 be the most representative information that can be used to describe prior flaw-depth and 

16 flaw-density distributions and characteristics. If other prior information is available, such 

17 information may also be used instead of or in addition to the specialized VFLAW data.

18

19 Figure C-1. A simple description of the Bayesian updating process.

20

21 In the remainder of this report, the methodology is described first, followed by the application to 

22 the Beaver Valley 2 ISI data. The results are used to compare the Beaver Valley 2 RPV to the 

23 10 CFR 50.61a screening tables. As such, the VFLAW-based flaw models are updated to more 

24 realistically represent this RPV.

25

UT data 

considered as

evidence of flaw

distribution

True flaw 

distributions

(depth & 

density)

Prior data used as 

VFLAW flaw 

distribution 

(depth & density)
C-9

1 C.2 Probabilistic Approach to Combine Uncertain NDE Flaw Data 

2 with Prior Flaw Distributions

3 In this section, a Bayesian updating methodology is proposed to combine the observed NDE 

4 data with the available flaw data and models used as part of the PTS re-evaluation effort. First, 

5 the Bayesian framework in general will be discussed, followed by the NDE data uncertainties 

6 (i.e., probability of detection and measurement/sizing error), and finally the developed Bayesian 

7 updating procedure will be discussed.

8 C.2.1 The Bayesian Framework

9 Suppose that background (prior) information is available about a model (e.g., the probability 

10 density functions representing flaw depth and flaw density). If new evidence becomes available, 

11 for example vessel-specific NDE data, it is possible to update the prior information 

12 (e.g., distribution models) in light of the new NDE-based evidence. The process of updating 

13 prior distribution models may be formally done through Bayesian inference. Bayesian inference 

14 is conceptually simple and logically consistent. It provides a powerful approach to combine 

15 observed or measured data with background (prior) information.

16 In Bayesian inference, the state of knowledge (uncertainty) of a random variable of unknown 

17 value that is of interest is quantified by assigning a probability distribution or Probability Density 

18 Function (PDF) to its possible values. Bayesian inference provides the mathematical formalism 

19 by which this uncertainty can be updated in light of any new evidence.

20 In the framework of the Bayesian approach, the parameters of the model of interest (e.g., the 

21 intensity parameter, λ, of an exponential distribution representing flaw depth) are treated as 

22 random variables, the true values of which are unknown. Thus, a PDF can be assigned to 

23 represent the parameter values. In practice, however, some prior information about the 

24 parameters may be known, including any prior data and subjective judgments regarding the 

25 parameter values that may be available. For example, the intensity parameter, λ, of an 

exponential distribution representing the variable flaw depth (e.g., in inch-1 26 ) may be known from 

27 related observations and data of similar vessels. If new NDE observations provide limited but 

28 more specific data about a particular RPV, it is possible to update any prior PDF of the 

29 flaw-depth intensity parameter by combining the NDE data with the prior PDF. In this case, the 

30 prior knowledge is combined with the specific NDE data to build a posterior PDF that is more 

31 representative of the true intensity parameter and thus the true flaw-depth distribution that 

32 accounts for the uncertainties in the observed NDE data.

33 Let θ be a parameter of interest (for example the flaw intensity parameter). Assume that θ is a 

34 continuous random variable, such that the prior and posterior PDFs of θ are likewise 

35 continuous. Also, let L(Data|θ) express how likely is it to observe the data (e.g., the NDE data 

36 measured) in light of given values of parameter θ. Certain values of θ show that the data 

37 observed are more likely to support the PDF whose parameter is θ. Then, according to 

38 Bayesian inference [C-4], the posterior PDF that represents a properly weighted combination of 

39 the prior PDF and the likelihood of the parameter θ will be:

( ) ( ) ( )

( ) ( ) ∫

θ

π θ θ θ

π θ θ

π θ = L Data d

L Data

Data

0

0

1

Eqn. (C-1)
C-10

1 The posterior PDF, π1(θ|Data), represents the updated prior PDF of θ, π0(θ), in light of the 

2 observed data (shown by its likelihood function L(Data|θ)). The denominator of Eqn. (C-1) is 

3 called the marginal density of the data or the normalization constant. The most difficult part of a 

4 Bayesian analysis, besides describing the likelihood function, is the computational challenge of 

5 determining the normalizing constant that often requires multidimensional numerical integration. 

6 The prior PDF π0(θ) reveals knowledge of parameter θ before data are used. The posterior 

7 PDF π1(θ|Data) is called posterior because it reflects the PDF of θ after the data are used.

8 Certain prior PDFs are called “conjugate” [C-4]. This is the class of prior PDFs that yields the 

9 same functional form for the posterior distribution. For example, if flaw-depth distribution is 

10 represented by an exponential distribution, a gamma PDF representing the intensity 

11 parameter, λ, of the exponential distribution is a conjugate distribution. This means that when 

12 updated by new NDE flaw data, the posterior PDF of λ will also be a gamma distribution. 

13 Hamada et al. [C-5] present a comprehensive overview of the mathematical steps for updating 

14 the conjugate distributions used in the Bayesian analyses. In simple problems, the conjugate 

15 distribution makes posterior PDF calculations simple because it eliminates the complex, 

16 computationally challenging integrations in Eqn. (C-1).

17 For example, consider NDE flaw observations (data). In this case, the likelihood of all such 

18 data, given a parameter θ of the flaw-depth PDF L(Data| θ), would be expressed as the 

19 probability of the intersections of the individual flaw measurements; that is, the probability of an 

20 event consisting of flaw-measurement-1 flaw-measurement-2 flaw-measurement-3 ... .

21 The likelihood, therefore, would be the product of the probability of observing each flaw 

depth, ai 22 , as shown by Eqn. (C-2). The likelihood function is independent of the order of each 

23 data point and is given by:

L(Data | θ) = c∏L (a | θ) i

i

i Eqn. (C-2)

24 where c is a combinatorial constant that quantifies the number of combinations in which the 

observed NDE data points, ai 25 , could have occurred. The constant c cancels from the numerator 

26 and denominator of Eqn. (C-1), and therefore is usually not included in the expressions of the 

27 likelihood function. Table C-1 summarizes the likelihood functions for different types of flaw 

28 data observations, if the PDF of the variable of interest (flaw depth in this case) is described by 

f(a|θ) with the corresponding Cumulative Density Function (CDF) of ∫ θ = θ

a

0

29 F(a | ) f(x | )dx .

30 Table C-1. Summary of Likelihood Functions

31

Type of Observation Likelihood Function Example Description

Exact Flaw Depth f(ai|θ) Exact flaw depth ai is reported

Right Censored Flaw 1-F(aR|θ) Flaw depth exceeds aR

Left Censored Flaw F(aL|θ) Flaw depth is less than aL

Interval Censored F(aR|θ) - F(aL|θ) Flaw depth is between aL and aR

Left Truncated f(ai|θ) / [1-F(aC|θ)] Flaw depth is ai where flaw observations 

are not possible below aC

32
C-11

1 C.2.2 Probability of Detection and Measurement Error

2 Interpretation of NDE data requires an understanding of the associated uncertainties. For 

3 example, one inspector may miss a specific flaw found by another. For this reason, the 

4 reliability of an inspection is customarily expressed in terms of the Probability of Detection 

5 (POD) of a given flaw size and expressed by curves of POD vs. flaw size. Determining detailed 

6 POD curves can also be difficult. Flaw detection depends on several factors and it can be 

7 difficult to produce statistical data to estimate the POD considering variations in material types, 

8 inspection methods, skill levels, equipment imprecision, and other factors.

9 Additionally, sizing or measurement errors occur and measurement model uncertainties exist. 

10 The measurement errors are generally composed of some combination of stochastic (random) 

11 errors and systematic errors. Random errors are intrinsic to any measurement and are caused 

12 by instrumentation imprecision and variability, among other sources. Systematic errors are 

13 biases in NDE measurement resulting in the mean of many separate measurements differing

14 consistently and appreciably from the true value of the measured flaw. Biases or systematic 

15 errors are often significant contributors to flaw data uncertainties. For example, in UT 

16 application to RPVs, there is a general trend toward overestimating the size of small flaws while 

17 underestimating the size of larger flaws [C-6].

18 C.2.2.1Probability of Detection

19 Formally, the POD can be defined as the probability that the NDE system detects a flaw of 

20 specific depth a, if it exists, and is denoted by POD(a) [C-6]. POD is generally modeled as a 

21 function of through-wall extent of the flaw. The POD also depends on other factors such as 

22 material test equipment, measurement method, and geometry. For example, POD generated 

23 for a particular material thickness might not be true for the same material with a different 

24 thickness. The POD should also consider appropriate adjustments for shallow surface flaws 

25 adjacent to the clad.

26 Consider a binary random variable D indicating a flaw detected (or D indicating a flaw not 

detected). Probability of detection of a flaw of depth ai 27 is:

POD(a ) Pr(D | a a ) i = = i Eqn. (C-3)

28 The data from which POD(a) functions are generated can be categorized into two types: 

29 hit/miss and signal-response amplitude. The hit/miss data type shows whether a flaw is 

30 detected or not. This type of data is subjective in nature depending on the operator experience. 

31 In this method, the smallest and largest flaw sizes detected should be identified. Any size below 

32 the smallest size is never detected, whereas a size above the largest size is always detected. 

33 The POD is then calculated as the ratio of the number of successful detections over the total 

34 number of inspections performed for a particular flaw size and is called the averaged POD.

35 Different forms of POD curves have been used in the literature (see References [C-7], [C-8], 

36 and [C-9]). The logistic POD model for hit/miss data is a common model and is represented as:











 β β β β 0 o therwise

logistics(a | , ,a ) for a > a POD(a | , ,a ) = 1 2 th th

1 2 th Eqn. (C-4)
C-12

1 where a is the flaw depth and N(0;σ(a)) is the random error represented by a normal distribution 

2 with a constant standard deviation σ or a deviation which may be a function of flaw depth (that 

3 is, σ(a)). The random error accounts for the POD model error. Model parameters β1 and β2

4 may be uncertain (epistemic) and would be represented by the bivariate PDF k(Θ), where Θ is 

5 the vector of the parameters, Θ = {β1, β2}. The PDF function m(σ(a)) may also be used to 

6 express any epistemic uncertainties by treating σ(a) as a random variable. The parameter ath in 

7 Eqn. (C-4) represents the threshold (the flaw size below which detection is not possible).

8 The other type of POD data represents measurements of the amplitude of signal response 

9 recorded by the NDE system, such as in a UT system. For signal-response data, much more 

10 information is supplied in the signal for further analyses than is in the hit/miss data. In the 

11 signal-response approach, the most important parameters are the inspection threshold (noise 

12 level) and the decision threshold. All responses less than the inspection threshold are ignored.

13 In the signal-response approach, it is generally assumed that the logarithm of the 

14 signal-response amplitude, , is linearly correlated to the logarithm of the flaw depth, a, as 

15 shown in Eqn. (C-5):

log(aˆ) = β + β log(a) + ε 3 4 Eqn. (C-5)

16 where ε is the random error and β3 and β4 are the regression parameters. The random error 

17 can be assumed to have a normal distribution with mean zero and a constant or variable 

18 standard deviation. Based on this assumption, it can be shown that the POD curve can be 

19 modeled using a lognormal CDF. The corresponding POD is shown as:

POD(a) Pr(aˆ aˆ ) Pr(log(aˆ) log(aˆ )) = > th = > th Eqn. (C-6)

20 where ath is the decision (detection) threshold signal. The decision threshold is chosen to be a 

21 bit higher than the inspection threshold in order to minimize the probability of a false call (for 

22 cases in which the decision threshold is not intentionally chosen to allow a certain probability of 

23 a false call) in the POD estimates. The â vs. a data can also be converted into hit/miss data by 

24 using the decision threshold, and an averaged POD can be determined.

25 C.2.2.2 Measurement (Sizing) Error

26 Measurement results are always associated with errors of varying magnitudes. Measurement 

27 error is defined as the difference between the measured and the true flaw depth. Measurement 

error is defined by Eqn. (C-7), where Mε is the measurement error, a* 28 is the measured value of 

29 flaw depth, and a is the true value of flaw depth:

Mε = a* - a Eqn. (C-7)

30 The measurement error has two components: systematic error (bias) and random error. 

31 Random error may be represented by a normal distribution with zero mean and constant 

32 standard deviation. Systematic error in most cases is a major contribution to flaw 

33 measurements and cannot be ignored. In its simplest form, the measurement error can be 

34 represented by a linear function of true size as shown in Eqn. (C-8), where m is the slope and 

c is the intercept of the line representing measurement error, Mε 35 , versus true flaw depth, a. If 

36 available, PDFs of m and c can be expressed to represent the epistemic uncertainty in these 
C-13

parameters. Also, Mε, may be represented as a linear function of measured flaw depth a* 1

because all data available are in terms of a* 2 .

*

M M M

* ' '

M

* M *

M

*

M M M

M

and E' N (0, ' ) of M vs. a

where, a measured depth, m slope, c intercept,

m'a c' E' m 1

E

m 1

c

a

m 1

m M

M m(a M ) c E ; or

or based on Eqn. (7) :

N (0, ),

M ma c E

where, a true flaw depth and E

e

e

e e

e

= s

= = =

= + +

+

+

+

+

+ =

= − + +

= s

= + +

=

Eqn. (C-8)

3 The aleatory random measurement error E'M is shown by the normal distribution NM(0; σ'M) in 

4 Eqn. (C-8). As such, the total measurement error may be represented as the normal distribution 

g(Mε) = N(m'a* 5 +c'; σ'M). Figure C-2 shows a conceptual example of the measurement error as a 

6 function of true flaw size (one can also show Mε vs. a*). According to Eqn. (C-7), the 

relationship between the true flaw depth, a, and measured depth, a*

, is a = a − Mε

* 7 .

8

9 Figure C-2. Measurement-error distribution

If the measurement error’s given flaw depth, a, is Mε 10 , EM describes the PDF of random or 

11 stochastic error. Clearly, Figure C-2 depicts a case in which the standard deviation of the PDF 

12 of EM is not constant; rather, it is a monotonically increasing line.

13 C.2.3 The Methodology for Updating the NDE Data

14 This section discusses the details of the Bayesian updating approach, which combines UT data 

15 (including their uncertainties) with the prior PDF models of flaw depth and density (used in 

16 VFLAW to generated FAVOR input) to arrive at posterior distributions of flaw characteristics.

17 C.2.3.1 Updating Flaw-Depth Distribution Models

18 The ultimate objective of this analysis is to describe PDFs of the flaw depth and density 

19 considering UT measurement of flaws (which contain modeling uncertainty and stochastic 

20 variability in form of the POD and measurement error). We start with the Bayesian estimation of 

21 the (large and small) flaw depths, followed by the same estimation for flaw density.
C-14

1 In the Bayesian framework described by Eqn. (C-1), prior information on the flaw depth PDF (for 

2 example, data and expert elicitation described in NUREG/CR-6817) may be combined with the 

3 uncertain measurement data from NDE to assess a posterior distribution of the flaw depths. 

4 Consider the PDF of the flaw depth, a, as f(a | Φ), where Φ is a vector of all the PDF 

5 parameters.

6 The data reported in NUREG/CR-6817 show that the small and large flaw depths are 

7 represented using different distributions. For example, the models proposed and used in 

8 VFLAW rely on the exponential distribution for large flaw depths but use the multinomial 

9 distribution to model small flaw depths. The flaw depth that separates the two distributions is 

10 defined as the transition flaw depth. Therefore, a given PDF may only be true up to a limit from 

which it transitions to another distribution. For the transition flaw depth ttr 11 , the PDF that 

12 describes the flaw depth should be conditioned on exceeding or falling below the transition limit. 

13 Therefore:

( )

( ) Pr(a t )

f(a | ) f a | , a t

Pr(a t )

f(a | ) f a | , a t

tr

s tr

tr

l tr

<

Φ Φ < =

≥

Φ Φ ≥ =

Eqn. (C-9)

In VFLAW, for example, ttr 14 for the weld regions was selected as the flaw depth of about the bead 

15 thickness, Δ, of the corresponding weld. Further, in VFLAW the flaw depth is normalized to the 

16 dimensionless random variable a/Δ instead of to a. The approach discussed herein applies to 

17 PDFs of both a and a/Δ, but the respective equations of the approach are only shown in terms 

18 of the PDF similar to Eqn. (C-9).

19 Consider the PDF of NDE measurement error Mε (represented by Eqn. (C-8)) of a measured 

flaw depth a* 20 with stochastic random variable E. The flaw-depth distribution corrected for the 

measurement error, Mε 21 , by assuming that true flaw depth, a, may be represented by correcting 

the measured depth a* using the relationship a = a* - Mε 22 [C-10]. As stated earlier in Eqn. (C-8), 

the PDF of Mε will be g(Mε) = N(m'a* 23 + c'; σ'M). If there are epistemic uncertainties about 

24 parameters m' and c', they may be represented by the bivariate PDF x(Ω), where Ω is the vector 

25 of the parameters m' and c'. The expected distribution of measurement error would be:

= Ω Ω ∫

Ω

g'(Mε ) g(Mε )x( )d Eqn. (C-10)

26 While the measurement-error-corrected PDF adjusts the flaw depths to true values, the flaws 

27 missed, as characterized by the POD, should also be accounted for. That is, for every 

measured flaw a* 28 , a corresponding probability (including uncertainties) that additional flaw(s) 

29 may be missed should be accounted for.

30 If the POD is represented as a mathematical function of the flaw size, a, the vector of 

31 parameters Θ of the POD function may be associated with a known multivariate PDF k(Θ) in 

32 order to account for the uncertainties. It is also possible to add a stochastic model error term, 

33 represented by a normal distribution with the mean zero and the standard deviation σPOD(a). 

34 For simplicity, it is also possible to assume that the error term is absorbed in the PDF of the 

35 POD model parameters (i.e., k(Θ)). The marginal POD independent of the random variables Θ

36 and σPOD(a) would be:
C-15

∫∫

σ Θ

= Θ σ Θ σ Θ σ

, 

POD POD POD

POD

POD(a) POD(a | , (a))k( )m( (a))d d

Eqn. (C-11)

Marginal POD, independent of flaw depth conditional on Φ (see Eqn. (C-9)) and ttr 1 , is then 

2 expressed as:

∫

∫

∞

= Φ ≥ = Φ ≥

= Φ < = Φ <

tr

tr

t

l tr l tr

t

0

s tr s tr

Pr (D) POD( , a t ) POD(a)f (a | , a t )da

Pr (D) POD( , a t ) POD(a)f (a | , a t )da

Eqn. (C-12)

3 where Pr(D) expresses the probability that a small or large flaw is detected regardless of size,

which in general is a function of Φ and ttr; that is, POD(Φ, t 4 tr). The probability of not detecting a 

flaw would be Pr(DD�)=1-Pr(D) ���������������� 5 . Let the random variable A represent the true 

6 flaw depth. Then, according to the Bayesian formulation by Celeux et al. [C-11], the probability 

7 that a detected flaw has a depth in the vicinity of a, inside an interval Δa, may be expressed as:

Eqn. (C-13)

8 where D is the event that a flaw is detected. The limit of Eqn. (C-13) as the flaw depth 

9 interval Δa approaches zero may be found after dividing Eqn. (C-13) by Δa. The left term limit 

10 represents the likelihood that a detected flaw will have the depth a. Further, by rearranging the 

11 right side, the PDF of flaw depth independent of detection and POD, given a flaw of depth a, 

12 may be found:

Pr(D)

1 . lim Pr(D | a A a a). aPr(a A a a) lim

Pr(D) a

Pr(a A a a).Pr(D | a A a a) L(a | D) lim

a

Pr(a A a a | D) lim

POD(a)

a 0

f(a| ,t )

a 0

a 0 a 0

tr

))))))))( ))))))))))( < < + D

D

< < + D

= D

< < + D < < + D = = D< < + D

D →

Φ

D →

D → D →

Eqn. (C-14)

13 Using Eqn. (C-14), the corresponding likelihood that a detected flaw is of depth a may be 

14 expressed as:

POD( ,t )

f(a | ,t ).POD(a) L(a | )

tr

tr

Φ

Φ Φ = Eqn. (C-15)

15 Eqn. (C-15) forms the basis for development of the likelihood function, which would be 

16 necessary in the Bayesian framework (as discussed in Section C.2.1) to estimate the posterior 

17 PDF of flaw depth given the observed NDE flaw data and the prior information about the 

18 characteristics of flaw PDF.
C-16

1 C.2.3.1.1 Likelihood Function of Exact Flaw-Depth Measurements

2 Suppose that the NDE reports exact flaw depths. Based on Eqn. (C-2), the likelihood function 

of n* flaws of known depth (containing measurement errors) a , a , a , . . .

*

3

*

2

* 3 1 would be 

4 . It is important to correct for POD and the measurement error and 

5 parameter uncertainties according to Eqn. (C-10) and to represent flaw depths in terms of the 

measured value using Eqn. (C-7) so that a* = Mε + a 1 6 . Therefore, using Eqn. (C-15), the 

7 likelihood of exact flaw measurements reported may be expressed as [C-10]:

[ ] ∏ ∫ =

ε ε ε ε

ε

− − Φ

Φ

Φ =

*

*

n

i 1 M

tr

*

i

*

i n

tr

tr POD(a M )f((a M ) | ,t )g'(M )dM

POD( ,t )

1 L(Allexactmeasureddata | ,t ) Eqn. (C-16)

8 Because NDE data are measured and have associated measurement error and other 

9 uncertainties, but the POD and measurement error discussed earlier are modeled to be relevant 

to the true flaw depth, a, in Eqn. (C-16) the true flaw depth is replaced with its equivalent (a* 10 -Mε). 

11 The expectation of the numerator of Eqn. (C-15), independent of the measurement error, is 

found by multiplying the term f(a | ,t ).POD(a) 12 Φ tr by the expected PDF of the measurement error, 

13 g'(Mε), and integrating this over all values of the measurement error.

14 C.2.3.1.2 Likelihood Function of Interval Flaw-Depth Measurements

15 If in addition to or instead of the exact flaw-depth data, interval data are reported (such as the 

16 number of flaws observed less than a given depth or between an upper and lower limit), the 

likelihood of mi

* 17 flaws reported in the interval i, corrected for the measurement error but 

18 independent of the error, would be [C-12]:

*

* i

i

*

i 1

m a

a

*

M

* *

tr

tr

*

i POD(a M )f((a M ) | ) g'(M ) dM da

POD( ,t )

1 L(Intervaliconsistingof m me asuredflaw depths | ,t ) 















− − Φ

Φ

Φ = ∫ ∫

− e

e e e e

Eqn. (C-17)

19 If n such flaw-depth intervals were reported, the corresponding likelihood function using 

20 Eqn. (C-17) would be:

∏ ∫ ∫ =

ε ε ε ε 















− − Φ

Φ

Φ =

− ε

n

i 1

m a

a

*

M

* *

tr

tr

*

* i

i

*

i 1

POD(a M )f((a M ) | )g'(M )dM da

POD( ,t )

1 L(ninterval sof measureddepths | ,t )

Eqn. (C-18)

 1 Note that if X and Y are two continuous random variables with probability density functions f(x) and g(y), 

the random variable Z = X + Y will have a probability density function h(Z) such that:

∫

∞

−∞ h(z) = f(z − y)g(y)dy
C-17

1 If exact and interval NDE data are both reported, the likelihood function would be a multiple of 

2 Eqn. (C-16) and Eqn. (C-18). That is:

L(nInterval MeasuredFlaw Depths | , t ) x L(n * Exact MeasuredFlaw Depths | , t )

L(Mix MeasuredFlaw Depths | , t ) =

tr tr

tr

F F

F 3

4 C.2.3.1.3 Bayesian Updating of Parameters of the Flaw Depth Distribution

5 Regardless of the form of the data (exact flaw-depth and/or interval data), the Bayesian 

6 inference of the vector of parameters Φ of the flaw-depth PDF would be obtained (according to 

Eqn. (C-1)) from ( | Data,t ) L(Data | , t ) ( ) π1 Φ tr ∝ Φ tr π0 Φ , where ( | Data) 7 π1 Φ is the posterior 

multivariate PDF of vector Φ, and ( ) 8 π0 Φ is the prior multivariate PDF of Φ.

Determination of the posterior ( | Data) 9 π1 Φ requires complex integrations. Further, integration 

10 of the denominator of the Bayesian inference in Eqn. (C-1) also requires multi-dimensional 

11 integration that in most cases can only be performed numerically.

12 It is also critically important to note that Eqn. (C-9) through Eqn. (C-20) are all in terms of the 

13 actual flaw depth, a, measured in English or metric units. As stated before, it is also possible to 

14 express the equations in terms of the normalized flaw depth, a/Δ, instead. This is simply done 

15 by replacing flaw depth, a, with a/Δ in all PDFs, measurement error, and POD equations. 

16 Clearly the parameters of the equations should be adjusted accordingly.

17 C.2.3.2 Updating the Flaw Density Model

18 If flaws are detected with a probability given by Eqn. (C-11) and Eqn. (C-12) as 

Pr(D) = POD(Φ, t 19 tr) (or not detected with a probability of Pr(D) = 1− Pr(D)), the probability that a 

20 specific total number of flaws n* observed in a given volume v out of the true number of flaws nj

21 follows the binomial distribution (see References [C-11] and [C-12]):

[ ] [ ]

[ ] [ ] * j *

* j *

n n

tr

n n

tr *

j

j

*

n n n n

*

j

j

*

POD( ,t ) 1-POD( , t )

n

n

L(n | n ) =

or

Pr(D) 1-Pr(D)

n

n

L(n | n ) =

Φ Φ 































Eqn. (C-19)

where 

n !(n n )!

n !

n

n

*

j

*

j

*

j

− =













 22 . The best estimate of the true number of flaws would be to assume that 

n* 23 represents the mean of Eqn. (C-19). Accordingly, the mean estimate of the true number of 

24 flaws, N, would be found from:

POD( , t )

n N

tr

*

Φ = Eqn. (C-20)

25 A more formal way to estimate the number (and thus density) of flaws would be to use the 

Bayesian updating of nj

, which allows the estimation of the posterior distribution of nj 26 to describe

27 the true number of flaws:
C-18

∑

=

nj

j j

*

j j

*

*

j L(n | n )Pr(n )

L(n | n )Pr(n ) Pr(n | n ) Eqn. (C-21)

Using Eqn. (C-19) (representing the likelihood of observing n* flaws out of the unknown nj 1 flaws)

2 and a Poisson prior flaw density model, the posterior probability distribution of the number of 

3 flaws would be obtained from:

[ ] [ ]

∑ [ ] [ ] ∞

=

−ρ −

−ρ −

ρ Φ Φ 















ρ Φ Φ 















=

* j

j * j *

j * j *

n n j

v n

n n

tr

n n

tr *

j

j

v n

n n

tr

n n

tr *

j

*

j

]

n !

e ( v) POD( ,t ) 1- POD( , t ) [

n

n

]

n !

e ( v) POD( ,t ) 1- POD( , t ) [

n

n

Pr(n | n ) Eqn. (C-22)

4 where ρ is the volumetric flaw intensity and ρv is the expected number of true flaws in the 

5 inspected volume v. If the mean number of flaws from Eqn. (C-20) is large, the computer 

6 rounding issues associated with Eqn. (C-22) may be avoided by performing the analysis for a 

smaller volume (for example, 1/100th 7 of the inspected volume) and later prorating nj accordingly.

8 An alternative approach that is approximate, but simpler than Eqn. (C-19) through Eqn. (C-22), 

9 would be to update the parameter ρ itself based on the mean number of flaws. For example, if 

10 the volume of the UT inspection is v, the volumetric flaw intensity ρ of the flaws associated with 

the mean estimated N true flaws from Eqn. (C-20) in this volume would be .

v

N 11 ρ = Assuming 

12 that the mean occurrence intensity of flaws remains constant over the volume of inspection, the 

13 Poisson distribution may be used to represent the likelihood of N flaws in volume v, given the 

14 volumetric flaw intensity ρ as shown by Eqn. (C-23):

N!

( v) L(N| ) = e

N -( v) ρ ρ ρ

Eqn. (C-23)

15 where v is the inspection volume and ρ is the flaw intensity (flaws per unit volume) associated 

16 with the mean estimated number of flaws N.

17 If prior information about the flaw intensity, ρ, were available, it would be desirable to update the 

18 flaw intensity by using such prior information. For example, assume that the prior volumetric 

19 flaw intensity is described by the PDF, π0(ρ). Then, the posterior PDF, π1(ρ|N), can be 

20 estimated using the likelihood function in Eqn. (C-23) from:

whereL(N | ) L(N | ) 

( | N) L(N | ) ( )

j

j

1 0

r = ∏ r

π r ∝ r π r

Eqn. (C-24)

21 If the prior PDF of the flaw intensity is expressed by the gamma distribution with parameters

, as −α ρ

α α −

Γ α

α ρ γ ρ = ρ α α = 1

2 2

e

( ) ( ) gamma( | , )

2

1

1

0 1 2 22 , the posterior PDF is a conjugate 

distribution described by ( ) gamma( | v ,n ) 1 ρ = ρ + a1 j + a2 23 g . The alternative form of the gamma 
C-19

1 distribution used by the MATLAB tool (to be used later in the following example) uses 

parameter β instead of parameter α1 , so that , ) 1

gamma( | 2

1

a

a

2 ρ β = .

3 C.3 Application Example

4 As part of a cooperative effort between the Electric Power Research Institute (EPRI) and the 

5 NRC to provide guidance on evaluation of PTS, NDE analyses were performed for a PWR RPV 

6 (Beaver Valley 2). In this section, the Beaver Valley 2 NDE data will be described first. Next, 

7 the VFLAW data described in NUREG/CR-6817 will be used as prior information, followed by 

8 the application of the Bayesian updating procedure discussed in Section C.2 to determine 

9 flaw-depth and flaw-density distributions specific to Beaver Valley 2. The resulting posterior 

10 distribution of the parameters of the flaw depth and flaw density, as described in Section C.2, 

11 will be derived and discussed. The results of flaw-depth and flaw-density PDFs will be used to 

12 compare against the 10 CFR 50.61a flaw tables, and to update the distribution models in 

13 VFLAW into new flaw distributions that are representative of the Beaver Valley 2 RPV.

14 C.3.1 Description of the NDE Data Used as Evidence to Build the Likelihood Function

15 The EPRI report by Spanner [C-13] provides UT-based measured flaw data near the inner 

16 surface (~2.5 inches) of the Beaver Valley 2 RPV, including small flaw sizes. These data, while 

17 subject to POD and measurement error, provide a more vessel-specific perspective of the 

18 distribution of flaws for Beaver Valley 2 than does the VFLAW distributions used in FAVOR.

19 The observed Beaver Valley 2 NDE data (with detection and sizing uncertainty) are mostly in 

20 the form of interval-censored data as summarized below [C-13]:

21 1. 19 weld flaws were detected by the UT NDE of Beaver Valley 2 in the first inch (the 

22 inspection volume specified in Supplement 4 to Mandatory Appendix VIII to Section XI of 

the ASME Code) of the RPV (~0.616 ft3 23 ), all having a flaw depth less than 0.125".

24 2. 103 weld flaws were detected and reported by the UT NDE of Beaver Valley 2 in the 

25 first 3/8t (2.953") (the inspection volume specified in Supplement 6 to Mandatory 

26 Appendix VIII to Section XI of the ASME Code) of the RPV were reported, all with flaw 

27 depths less than 0.125", except for one flaw that measured 0.260".

28 The lower limit of the detected flaw intervals described above is not stated in [C-13], but is 

29 certainly not zero. Two possible subjective lower limits were assumed, and later the sensitivities 

30 of the final (posterior) flaw-distribution results to these two choices were assessed. The two 

31 lower limits are 0.04" and 0.075".

32 The NDE data described above provide an incomplete depiction of the true flaws in the Beaver 

33 Valley 2 RPV because they contain uncertainties associated with the UT technology used to

34 detect and size flaws. The associated weld bead thicknesses are not reported. However, the 

35 weld region of the observed flaws and flaw length is reported. Such results should be corrected 

36 for detection and sizing capability, particularly for the small flaws.

37 The Beaver Valley 2 RPV weld map is shown in Figure C-3 [C-14]. In the absence of the 

38 Beaver Valley 2 average weld bead thickness as the point of transition between large and small 

39 depths, it is assumed that all flaws reported are Submerged Arc Welds (SAWs), which form over 

90% of welds in the VFLAW data, with the bead thickness of D = t 40 tr = 0.26". Using the Beaver 
C-20

1 Valley 2 RPV information shown in Figure C-3, the following weld characteristics were 

2 computed and used to specialize the VFLAW data to represent the prior distributions of flaw 

3 depth and density for Beaver Valley 2 RPV:

4 Total Weld Length = 816.18"

Total Weld Volume2 = 4.975 ft3 5

Total Weld Fusion Area = 92 ft2 6

Weld Volume of 1" = 0.616 ft3 7

Weld Volume of 3/8t = 1.865 ft3 8

9

10

11 Figure C-3. Beaver Valley 2 weld map [C-14]

12 C.3.2 Description of Flaw-Depth and -Density Information Used in VFLAW as Prior PDFs 

13 in This Example

14 Smaller flaws neighboring the inner surface of the RPV that would be subjected to high thermal 

15 stresses during PTS events are dominant contributors to the risk of vessel failure posed by PTS. 

16 During the PTS re-evaluation effort in the late 1990s and early 2000s, the Marshall distribution 

17 was updated using the results of destructive tests of the Pressure Vessel Research User Facility 

18 (PVRUF) and Shoreham RPVs [C-2]. These analyses assumed that the flaws were near the 

19 inner surface of the RPV and included very small flaws of less than 0.125 inches (3 mm) in 

20 through-wall extent. The purpose of the re-evaluation was to provide an up-to-date estimate of 

21 the distribution of weld flaw density and sizes using modern NDE methods of analysis. These 

22 distributions represent the flaw-depth and flaw-density models in VFLAW [C-3].

 2 The total volume of reactor vessel weld area according to the weld dimensions shown in Figure C-3 =

[1.375 × 99.45 × 2 × (86.531 - 78.656)] + [2 × 1.375 × 61.68 × (86.531 - 78.656)] + [1.25 × 3.14 × 2 × 

86.531 × (86.5312 - 78.6562)] = 8,595.29 in3 = 4.974 ft3

.
C-21

1 Because we need prior distributions in the Bayesian inference described earlier, the VFLAW 

2 distributions many be specialized to Beaver Valley 2. This is done by using the corresponding 

3 bead thickness and other vessel characteristics for Beaver Valley 2 to determine the prior 

4 distributions of the flaw depth and flaw density from VFLAW distributions. The prior distributions 

5 fill the gaps in the limited inspected data and uncertainties associated with the NDE results.

6 Analysis of flaws should consider different vessel regions and welding types. Welding types 

7 include the Submerged Arc Weld (SAW), Shielded Metal Arc Weld (SMAW), and repair weld. 

8 Depending on the known details of fabrication, NUREG/CR-6817 estimated flaw distributions for 

9 each weld type used in a particular region of the vessel. Measured flaw data showed 

10 vessel-to-vessel variability in both flaw density and depth. To supplement the limited data from 

11 the PVRUF and Shoreham flaw measurements, NUREG/CR-6817 used expert elicitation. 

12 Distribution functions to characterize the number and sizes of flaws in the various regions of the 

13 RPVs were then developed based on the measured data and insights from an expert elicitation 

14 exercise and used in VFLAW.

15 The prior distributions used in this example updated the PVRUF flaw models described in 

16 NUREG/CR-6817 and used in VFLAW, including the hyper-distributions that address the 

17 uncertainties associated with the parameters for the distribution functions describing flaw depth 

18 and flaw density. For example, the exponential distribution was used to represent the variability 

19 of large flaw depths. However, the parameter λ of the exponential distribution (i.e., assuming 

f(a) = λe-λa 20 , where a is the flaw depth) was in turn considered to be a random variable and 

21 described by the gamma distribution. The gamma distribution itself has two parameters, 

22 α1 and α2, that are given as constants in NUREG/CR-6817. These parameters were updated in 

23 this example based on the observed data from Beaver Valley 2. The values of the 

24 hyper-distributions used in VFLAW are shown in Table C-2. Note that the flaw-depth data in

25 Table C-2 are in meters, whereas this example calculation is based on inches. The relationship 

26 between the flaw-depth and flaw-density PDFs and the corresponding hyper-PDFs of their 

27 parameters are illustrated in Figure C-4.

28

29 Figure C-4. Flaw-depth and flaw-density distributions used in VFLAW and their corresponding 

30 parameter hyper-PDFs.
C-22

1 Table C-2. Flaw-Depth (a) and Flaw-Density (b) Distributions and Hyper-PDF Parameters Used 

2 in VFLAW for PVRUF

3 (a) Flaw-depth characteristics (in meters) based on data from PVRUF RPV

Case Flaw 

Size 

Category

Welding 

Process

Random 

Variable 

Representing 

Flaw Depth

PDF of 

Flaw 

Depth

Parameters 

of PDF

Distribution 

Describing 

Uncertainty 

of 

Parameters 

of PDF

Parameters 

of 

Uncertainty 

Distribution

1 Small

(a ≤ Δ)

SAW a/Δ Multinomial P1, P2, P3 Dirichlet U1 = 34

U2 = 8

U3 = 1

2 Small

(a ≤ Δ)

SMAW a/Δ Multinomial P1, P2, P3 Dirichlet U1 = 34

U2 = 8

U3 = 1

3 Small

(a ≤ Δ)

Repair 

Weld

a/Δ Multinomial P1, P2, P3 Dirichlet U1 = 34

U2 = 8

U3 = 1

4 Large

(a > Δ)

SAW a/Δ Exponential λ Gamma α1 = 21.68

α2 = 52

5 Large

(a > Δ)

SMAW a/Δ Exponential λ Gamma α1 = 21.68

α2 = 52

6 Large

(a > Δ)

Repair 

Weld

a/Δ Exponential λ Gamma α1 = 17.58

α2 = 13

4

5 (b) Flaw-density characteristics based on data from PVRUF RPV

Case Flaw 

Size 

Category

Welding 

Process

Random 

Variable 

Representing 

Flaw Depth

PDF of 

Flaw 

Depth

Parameters 

of PDF

Distribution 

Describing 

Uncertainty 

of 

Parameters 

of PDF

Parameters 

of 

Uncertainty 

Distribution

1 Small

(a ≤ Δ)

SAW Flaws per 

cubic meter

Poisson λ Gamma α3 = 0.180

α4 = 1419

2 Small

(a ≤ Δ)

SMAW Flaws per 

cubic meter

Poisson λ Gamma α3 = 0.014

α4 = 197

3 Small

(a ≤ Δ)

Repair 

Weld

Flaws per 

cubic meter

Poisson λ Gamma α3 = 0.00123

α4 = 12

4 Large

(a > Δ)

SAW Flaws per 

cubic meter

Poisson λ Gamma α3 = 0.180

α4 = 4

5 Large

(a > Δ)

SMAW Flaws per 

cubic meter

Poisson λ Gamma α3 = 0.014

α4 = 4

6 Large

(a > Δ)

Repair 

Weld

Flaws per 

cubic meter

Poisson λ Gamma α3 = 0.00123

α4 = 7

6

7
C-23

1 C.3.3 Detection and Sizing Error Models

2 Analysis of UT-detected performance data reported by EPRI [C-6] was used to identify a POD 

3 function for the purpose of this example. Currently, EPRI is updating these performance data 

4 and will supply more appropriate POD models for future consideration; the POD and flaw-sizing 

5 error functions used here are therefore intended only to illustrate this numerical method. The 

6 threshold limit of this POD function (Eqn. (C-4)) was taken as 0.00 and the epistemic 

7 uncertainties associated with the parameters of the model were not considered in this example. 

Accordingly, the following mean POD function based on Eqn. (C-4) was used3 8 :

63.2100(a 0.1124) 1 e

1 POD(a) 1 − + = − Eqn. (C-25)

9 where a is the true, unbiased flaw size. Figure C-5 shows a plot of this POD function.

10

11

12

13

14

15

16

17 Figure C-5. POD vs. flaw depth.

18 The measurement error was assumed, based on some examples and measurement errors 

19 reported in EPRI report [C-6]. The measurement error is a linear function that oversizes small 

20 flaws and undersizes large flaws. Figure C-6 depicts a plot of the measurement-error model 

21 used in this example. The model itself can be described as a line with a slope of -0.1905 and 

22 an intercept of 0.0476, with the variability resulting from the model error described by a normal 

distribution. That is, M 0.1905a 0.0476 N(0,0.0298) * 23 ε = − + + , where N(0, 0.0298) is a normal 

24 distribution with a mean of zero and a constant standard deviation of 0.0298. Note that in this 

25 example we are not accounting for the epistemic uncertainties of the measurement-error model. 

26 However, such uncertainties should be considered when new measurement-error models are 

27 developed. Equally, as described by Eqn. (C-10), it is possible to represent the above line by 

the normal PDF: * 2 564.55(M 0.1905a 0.0476) g'(M ) 13.41e− + −

e e 28 = .

29

30

 3 The model error associated with the logistic distribution was not considered.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

POD

True size (inch)

(a)

a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

POD

True size (inch)

(a)

a
C-24

1

2 Figure C-6. Measurement error vs. flaw depth.

3

4 C.3.4 Bayesian Updating of the Parameters of the Flaw-Depth Distributions

5 In this section, the process of updating the prior data specialized to Beaver Valley 2 with the 

6 observed NDE data using Bayesian inference will be discussed. Figure C-7 describes the main 

7 elements of the updating process. The observed UT-based flaw data discussed in 

8 Section C.3.1 is corrected for POD and measurement error. The POD and measurement error 

9 models were discussed in Section C.3.3. In the remainder of this section, development of the 

10 likelihood functions for the two sets of observed flaw data (one for the first one inch and another 

11 for the 3/8t of the inner RPV welds) will be discussed first. Then the Bayesian inference that 

12 combines the observed data with the prior data will be considered. The analysis will be 

13 performed for each of the NDE observed data sets to determine different conclusions.

14 1. The observed data for the first 3/8t of the RPV welds were used to update the prior 

15 models shown in Table C-2 in order to specialize them for the Beaver Valley 2 RPV. 

16 The updated values can be used for PTS calculations specific to the Beaver Valley 2 

17 RPV using VFLAW and FAVOR.

18 2. The observed data for the first inch of the vessel were used to estimate the true number 

19 of flaws in the flaw-depth ranges of the 10 CFR 50.61a tables. This information is used 

20 to assess how the estimated number of flaws of a given size in Beaver Valley 2 

21 (estimated based on inspection data, VFLAW distributions, and Bayesian inference) 

22 compares to the number and size of flaws permitted by the 10 CFR 50.61a tables.
C-25

1

2 Figure C-7. Bayesian updating elements of the Beaver Valley 2 example.

3

4 C.3.4.1 Bayesian Updating of the Parameters of Large-Flaw Depth Distribution Based 

5 on the 3/8t UT Inspection Data

6 According to the prior data used by VFLAW as described in Section C.3.3, small-to-large 

7 flaw-size transition occurs at flaw depths exceeding the SAW weld bead thickness. According 

8 to Table C-2, large flaws will follow an exponential distribution, conditional on exceeding the 

9 transition flaw depth. Thus, the distribution of large-flaw depths corrected for the bias would be:

( ) (a M t )

1

(a t )

l

tr

l

l tr tr * l tr l e e

Pr(a t )

f(a | ) f a | , a t −l − −l − e − = l = l

>

l l > = Eqn. (C-26)

where λl is the large-flaw depth intensity (per inch) and atr 10 is the transition flaw depth (assumed 

11 to be 0.26"). To build the likelihood function, the flaws should be represented in terms of the 

measured flaw depths. Because the NDE data reported only one exact flaw depth (n* 12 = 1, 

a1

* = 0.26) in the large-flaw region (exceeding the transition t 13 tr) for Beaver Valley 2, the likelihood 

14 function based on Eqn. (C-16) would be:

( ) [ ]( ) Π∫ ε

λ

− −

ε ε

ε

 λ



 





+

− λλ

1

-1

- a -M -0.26 -564.55(M +0.1905 a -0.0476)

l 63.2100(a M 0.1124)

1

i=1 1

l

l

*

data e 13.41e dM

1 e

1 1

[POD( )]

1 L(a | ) = * 2 i

*

l i

*

i

Eqn. (C-27)

where ∫ λ λ λ

∞

0.26

- (a-0.26) POD( l) = POD(a) l e da 15 l would be near 1 in this case because the POD(a) reaches unity 

16 when the flaw depth exceeds 0.2 inch in Eqn. (C-25). Note that, for simplicity, the random 

variable Mε 17 is integrated over the range of -1 to 1 because these are the extreme ranges of the 

18 measurement error. The next step is finding the posterior distribution of λl based on Eqn. (C-1). 

19 That is:
C-26

∫

λ

λ π λ λ

λ π λ

π λ

l

l 0 l l

*

data

l 0 l

*

* data

1 l data L(a | ) ( )d

L(a | ) ( ) ( | a ) = Eqn. (C-28)

1 Using Eqn. (C-27) as the likelihood, the Bayesian updating with the gamma PDF as the prior 

hyper-PDF for λl

, ( ) = ( | 1.2, = 4) π0 ll gamma ll α1 = α2 2 (or the alternative form

( ) = ( | = 0.8333, = 4) π0 ll gamma ll β α2 3 ), from the VFLAW (SAW) information specialized to 

4 Beaver Valley 2 by using the bead thickness of 0.26" and converted to per unit inch (using the 

SAW data listed in Table 6.6 of NUREG/CR-68174

), the posterior distribution of λl 5 is calculated 

6 and fitted into another gamma distribution. The posterior gamma PDF representing the Beaver 

Valley 2 RPV would be ( ) = ( | = 1.186, = 5) 7 π1 ll gamma ll α1 α2 and plotted and compared to the 

8 prior PDF in Figure C-8 (see Appendix C-1). The gamma distributions used to describe 

9 flaw-depth intensity and flaw density in Table C-2 are normalized based on the bead thickness. 

10 As evident from Table C-2, VFLAW uses normalized flaw-depth distributions (i.e., based on a/Δ 

11 instead of a). Assuming a bead thickness of 0.26", the normalized gamma prior and posterior 

12 PDFs for large-flaw-depth intensity for Beaver Valley 2 would be 

( ) = ( | 4.615, = 4) π0 ll gamma ll α1 = α2 and ( ) = ( | = 4.563, = 5) π1 ll gamma ll α1 α2 13 . Note that 

14 Figure C-8 is not the normalized version. The normalized posterior would be the new 

15 hyper-distribution that reflects the measured data and can be used in the VFLAW and FAVOR 

16 runs for Beaver Valley 2 RPV.

17 From Figure C-8, it is evident that the prior and posterior values are very similar, because only 

18 one flaw depth data point is reported and used, so the posterior relies primarily on the prior 

19 information. The impact will, however, affect the density PDF more strongly.

20

Figure C-8. Prior and posterior distributions of the flaw-depth intensity of parameter λl 21 .

22 Appendix C-1 describes the MATLAB routine used to generate the posterior distribution and 

23 solutions to Eqn. (C-26) through Eqn. (C-28).

 4 Note that to specialize VFLAW flaw-depth distributions for Beaver Valley 2, only PVRUF large-flaw 

depths reported for SAW in Table 6.6 of the NUREG/CR-6817 were used instead of the distributions 

summarized in Table 2 of this report.
C-27

1 C.3.4.2 Bayesian Updating of the Parameters of Small Flaw Depth Distribution Based 

2 on the 3/8t UT Inspection Data

3 Flaw depths smaller than the bead thickness for PVRUF (from Table 6.4 in NUREG/CR-6817) 

4 were analyzed and appeared to best fit an exponential PDF. That is:

, > 0, a > 0

1 e

e f(a | ) = -0.26 s

- a

s

s s

s

λ λ λ λ

λ

Eqn. (C-29)

5 where λs is the flaw depth (per inch) for small flaws. The hyper-distribution of λs was also 

estimated as the gamma distribution π ( ) = ( | 2.854, 39) 6 0 λs gamma λs α1 = α2 = that was used as the

7 prior PDF for Beaver Valley 2.

8 The procedure from this point on is the same as that for the large flaws with the exception that 

9 the NDE data for small flaws were given in the form of a single flaw depth interval, as discussed 

10 in Section C.3.1. Accordingly, with m = 1 and using Eqn. (C-17), the likelihood function for the 

11 evidence consisting of 103 detected (observed) flaws, all with a depth of less than 0.125" but 

12 larger than 0.04" (or 0.075" as sensitivity value), will be:

( ) 103 0.125

0.075or0.04

*

1

1

-564.55(M +0.1905 a -0.0476)

-0.26

- (a M )

s

63.2100(a M 0.1124) s

s 13.41e dM da

1 e

e 

1 e

1 1

POD( )

1 L(Data | ) * 2

s

* s

*

































λ 



 





+

- λλ = ∫ ∫

e λ

λ - e

e

e

Eqn. (C-30)

13 The Bayesian updating for small flaws is the same as Eqn. (C-28) and the likelihood given by 

14 Eqn. (C-30) results in the posterior distributions of λs as shown in Figure C-9 (see 

15 Appendix C-1). Also, a plot of the POD(λs) as expressed by equation:

( ) ∫ − λ

−λ

− −

λ 



 





+

λ = −

0.26

s

s

0.26 a

a

s s 63.2100(a 0.1124) 1 e

e

1 e

1 1

0

POD( ) da Eqn. (C-31)

16 is shown in Figure C-10. Clearly, because small flaws are associated with small POD and 

17 considerable measurement error, the impact of these would be significant in the posterior 

18 results shown in Figure C-9. The posterior can be expressed as the gamma PDF 

π ( ) = ( | 1.614, 60) 19 1 λs gamma λs α1 = α2 = . The normalized versions of the prior and posterior 

20 can be expressed by dividing α1 by the bead thickness of 0.26". Note that Figure C-9 does not 

21 show the normalized versions of the prior and posterior distributions.

22 The VFLAW data for small flaw depths uses the multinomial distribution, whose parameters are 

23 described by the Dirichlet PDF. Accordingly, it is possible to find the equivalent Dirichlet PDFs 

24 that best describe the posterior PDF of λs expressed in form of the gamma distribution.
C-28

1

2 Figure C-9. Bayesian prior and posterior PDFs of λs (per unit inch).

3

4 Figure C-10. Marginal POD independent of small flaw depth.

5 To update the Dirichlet hyper-distribution that represents uncertainties in the parameters of the 

6 multinomial distribution representing small-flaw-depth PDF used in VFLAW, the posterior PDF 

7 of λs shown in Figure C-9 is used to calculate the number of flaws in each of the ranges 

8 (0" to 0.072", 0.072" to 0.126", and 0.126" to 0.170") used by the multinomial distribution in 

9 VFLAW. Because the Dirichlet distribution is a conjugate distribution, the posterior number of 

10 flaws is found (see Section C.3.4.3.2 for estimating the number of posterior flaws) by adding the 

11 posterior mean number of flaws in each flaw-depth bin (see Table C-3). When added to the 

12 prior number of flaws (see Appendix A of NUREG/CR-6817 for the procedure), the updated 

13 Dirichlet values are viewed as being more representative of the distributions of true flaws in the 

14 Beaver Valley 2 RPV and are summarized in Table C-3.

15 Table C-3. Updated VFLAW Distribution for Small Flaws

16

0

0.04

0.08

0.12

0.16

0.2

0 10 20 30 40 50 60

Posterior

Prior

PDF

Flaw size intensity (per unit inch)

Flaw-Depth 

Bins (in)

Normalized 

Flaw-Depth 

Bins (per 

NUREG/

CR-6817) per 

Unit of Bead 

Thickness

Prior U 

(see 

Table C-2)

Prior 

Multinomial 

Parameter Posterior U

Updated 

Mean of

Multinomial 

Parameter

a1 (0 to 0.072) 0.25 (0.0 to 0.4) U1 = 34 P1 = 0.7907 U'1 = 479.75 P'1 = 0.9639

a2 (0.072 to 0.126) 0.55 (0.4 to 0.7) U2 = 8 P2 = 0.1860 U'2 = 9.71 P'2 = 0.0332

a3 (0.126 to 0.17) 0.85 (0.7 to 0.1) U3 = 1 P3 = 0.0233 U'3 = 5.354 P’3 = 0.0029
C-29

1 A critical observation from this example is that the posterior results of the flaw-depth distribution

2 in general and small depth distribution in particular are most sensitive to the POD model and 

3 measurement error model and their associated uncertainties. The lower tail of the POD model, 

4 in the ranges of small flaw depths (<0.1"), expects small probabilities of detection. When a flaw 

5 of small depth is reported, because of the corresponding small POD, it increases the likelihood 

6 that there are several flaws not detected for the one detected. This will substantially affect the 

7 results.

8 Appendix C-1 describes the MATLAB routine used to solve the integrals in Eqn. (C-29) through 

9 Eqn. (C-31) to estimate the posterior PDF of small flaws.

10 C.3.4.3 Results of Updating Parameters of Flaw Density Distribution Based on the 3/8t 

11 UT Inspection Data

12 In Section C.2.3.2, two methods were presented to update the flaw densities. One was based 

13 on a binomial likelihood function (Eqn. (C-19)) and one used the Poisson density and took 

14 advantage of the conjugate properties of the prior gamma PDF used as the prior distribution of 

15 the intensity, ρ, of the Poisson distribution (Eqn. (C-23)). The later method is used to update the 

16 flaw densities for this example. This method is based on the mean values only and is simpler 

17 than the former.

18 C.3.4.3.1 Posterior Density of Large Flaws 

19 The number of large flaws (size > 0.26 inch), given the single UT-detected flaw, yields the 

posterior flaw depth intensity, λl 20 , which was shown in Figure C-8. The marginal POD 

21 independent of the flaw size was almost 1 (for any value of posterior λl). Accordingly, using 

22 Eqn. (C-20), the number of flaws based on the posterior estimates of flaw depth would be 1.

Using the conjugate features of the prior gamma distribution for ρl, 23 the posterior distribution (see 

24 Appendix A of NUREG/CR-6817) will be:

3

l 1 2

1 l l l 1 2 l

( | 8.5119, 5) flaws/ ft

( | n ) ( | 6.6467 V, 4 n )

ρ a = a =

π ρ = ρ a = + a = + =

gamma

gamma

Eqn. (C-32)

25 Figure C-11 compares prior and posterior distributions of the number of flaws and flaw intensity 

of large flaws for Beaver Valley 2 in a volume of 1.8652 ft3 26 . Note that VFLAW values are in 

27 cubic meter. The equivalent prior and posterior distributions in cubic meter are summarized in

28 Table C-4.
C-30

1

2 (a)

3

4

5 (b)

6

7 Figure C-11. Prior and posterior flaw density (a) and intensity (b) for large size flaw

8 C.3.4.3.2 Posterior Density of Small Flaws

9 Similar to the large flaw density, the prior gamma distribution representing the flaw density per 

10 unit volume, ρs, used in the VFLAW was updated based on the 103 UT-detected small flaws in 

11 the interval of 0.04" to 0.125". This would yield the posterior flaw depth intensity, λs, as was 

12 shown in Figure C-9. The mean value of the flaw depth intensity (i.e., λs = 37.16) estimated by 

13 the posterior shown in Figure C-9 yields a marginal probability of detection independent of flaw 

14 depth of, POD(λs) = 0.0274.

15 The likelihood of observing ns small flaws would be expressed as:

(Table 6.2 SAW PVRUF NUREG/CR - 6817)

( ) ( | 6.6467, 1419) flaws / ft 

( | 8.5119, 1909)

( | n ) ( | 6.6467 V, 1419 n )

3

0 s s 1,s 2,s

pos

2,s

pos

s 1,s

s

pos

2,s

pos

1 s s s 1,s

p ρ = ρ a = a =

= ρ a = a =

p ρ = ρ a = + a = +

gamma

gamma

gamma

Eqn. (C-33)

16 Figure C-12 compares prior and posterior distributions of the number of flaws and volumetric 

17 flaw intensity of small flaws.

0 0.5 1 1.5 2 2.5 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Flaw density (per unit cubic feet)

PDF
C-31

1

2 (a)

3

4

5 (b)

6

7 Figure C-12. Prior and posterior flaw density (a) and intensity (b) for small size flaws

8 Because the posterior flaw density distributions shown in Figure C-11 and Figure C-12 are 

9 based on the mean values of λ for both large and small flaws, a Monte Carlo simulation was 

10 used to select random realizations of λ from the posterior PDFs of λ for large and small flaws to 

11 develop the corresponding posterior flaw density PDFs. Figure C-13 shows the results of the 

12 spread of the number of the flaws of various depths for the Beaver Valley 2 RPV, including the 

13 epistemic uncertainties shown in the form of box plots.

14 Similar to the flaw-depth distribution, the posterior distribution of flaw density will be dominated 

15 by the POD and measurement error models and their uncertainties. Specifically, when a large 

16 number of small flaws are reported during the NDE inspection, the corresponding low POD 

17 value increases the number of expected undetected flaws.

18 Based on the results of this section, Table C-4 summarizes the posterior results to be used for 

19 Beaver Valley 2 VFLAW and FAVOR runs. The posterior results have been changed to metric 

20 units in this table, because VFLAW uses metric values.

21
C-32

Table C-4. Summary of the Posterior Parameter PDFs of Flaw Depth and Flaw Density to be used in VFLAW a1

Beaver Valley 2 2

Case Flaw Size 

Category

Welding 

Process

Random Variable 

Representing Flaw 

Depth

PDF of Flaw 

Depth

Parameters 

of PDF

Distribution 

Describing 

Uncertainty 

of 

Parameters 

of PDF

Original

VFLAW

Parameters of 

Uncertainty 

Distribution

(Flaw Depth 

parameters are 

specialized to 

Beaver Valley 2

1 Small

(a ≤ Δ)

SAW Flaws per cubic meter Poisson ρ Gamma α3 = 0.180

α4 = 1419

2 Large

(a > Δ)

SAW Flaws per cubic meter Poisson ρ Gamma α3 = 0.180

α4 = 4

3 Small

(a ≤ Δ)

SAW a/Δ Multinomial P1, P2, P3 Dirichlet U1 = 34

U2 = 8

U3 = 1

4 Large

(a > Δ)

SAW a/Δ Exponential λ Gamma α1 = 4.615§§

α2 = 4

3

4

 §§ VFLAW combined all SAW and SMAW flaw data to arrive at the gamma distribution with α1 = 21.68 α2 = 52. A

Section C.3.4.1, this example used only PVRUF SAW data for large flaw depths to build the specialized Beaver 
C-33

1

2

3 Figure C-13. Posterior number of flaws in the 3/8t region in various flaw size intervals including 

4 epistemic uncertainties

5 C.3.4.3.3 Sensitivity of the Flaw Density Results to a Lower Bound of 0.075" for the 

6 Interval Data Observed in the 3/8t Region

7 Estimations of the posterior PDFs presented in Sections C.3.4.3.1 and C.3.4.3.2 have been 

8 made by assuming that the observed interval of small flaws was 0.075" to 0.125". To assess 

9 the sensitivity of this assumption, it is assumed that the detection up to 0.075" is possible and all 

10 data would be between 0.075" and 0.125". Repeating the calculations in Section C.3.4 

11 (including subsections) yields the posterior flaw depth and density PDFs that are very similar. 

12 Therefore, it was concluded that the results are relatively insensitive to choice for the lower 

13 bound of the data. More discussions of this topic are presented in Section C.3.5.

14 C.3.5 Updating the 10 CFR 50.61a Tables

15 The NDE data from the inspection volume specified in Supplement 4 to Mandatory 

16 Appendix VIII to Section XI of the ASME Code (the first one-inch of the weld) is used as the 

17 evidence for the Bayesian updating procedure described in the previous sections to obtain the 

18 flaw depth and density characteristics for Beaver Valley 2. The posterior characteristics are 

19 used to develop the corresponding 10 CFR50.61a flaw tables (consisting of only 19 

20 UT-detected flaws in the interval 0.04" to 0.125"). The procedure is the same as the one used 

21 to update the UT data observed in the 3/8t region as discussed in Section C.3.4.

22 If the posterior distributions of the flaw density are used to estimate the mean number of flaws in 

23 the ranges of flaw sizes described in the 10 CFR 50.61a flaw tables, the results should be 

24 prorated to the number of flaws per 1,000" of weld. Because Beaver Valley 2 has a total weld 

25 length of 816.18", the prorated mean number of flaws was calculated and compared with the 

26 allowable number of flaws per length of weld, as shown in Table C-5. If the lower limit of the 

27 observed data interval is changed from 0.04" to 0.075" to line up with the smallest bin size in the 

28 10 CFR 50.61a flaw tables, the results summarized in Table C-5 are calculated. In this case, 

0 - 0.075 0.075 - 0.125 0.125 - 0.175 0.175 - 0.225 0.225 - 0.26 10-1

100

101

102

Actual number of flaws

Flaw size intervals (inch)

Max

75%

25%

Min

Median
C-34

1 the results of the mean number of flaws were also not too sensitive to the choice of the lower 

2 limit of the interval of NDE data.

3 To assess the epistemic uncertainties in the POD and measurement error uncertainties, again 

4 for the case of the Supplement 4 inspection volume (the first one inch), a Monte Carlo 

5 simulation was used to select random realizations of λ from the posterior PDFs of λ for both 

6 large and small flaws to develop the corresponding posterior flaw density PDFs. Repeating this 

7 process will result in the epistemic uncertainty for each flaw depth interval used in Table C-4. 

8 Figure C-14 shows the results of the number of the flaws of various depth intervals, by 

9 employing box and whisker plots. Epistemic uncertainties are larger (than those shown in 

10 Figure C-13), because of the smaller evidence, because there were only 19 observed flaws in 

11 the Supplement 4 inspection volume.

12

13
C-35

1 Table C-5. Alternate PTS Rule Flaw Table Assessment using Mean Number-of-Flaws by Size 

2 Interval for Beaver Valley 2 (Assuming a Lower Limit of 0.04" for Observed Data 

3 Interval)

Flaw Depth

(inch)

Observed 

(Detected) 

Number of Flaws 

(Biased)

Posterior Mean

Number of Flaws 

(Unbiased and 

Corrected for 

POD)

Rule’s Limits 

per 1,000" of 

Weld

0.000 < a ≤ 0.075

19

124.98 No limit

0.075 < a ≤ 0.475 38.21 166.70

0.125 < a ≤ 0.475 0 14.13 90.80

0.175 < a ≤ 0.475 0 4.86 22.82

0.225 < a ≤ 0.475 0 1.30 8.66

0.275 < a ≤ 0.475 0 0.21 4.01

0.325 < a ≤ 0.475 0 0.13 3.01

0.375 < a ≤ 0.475 0 0.09 1.49

0.425 < a ≤ 0.475 0 0.04 1.00

4

5 Table C-6. Alternate PTS Rule Flaw Table Assessment using Mean Number-of-Flaws by Size 

6 Interval for Beaver Valley 2 (Assuming a Lower Limit of 0.075" for Observed Data 

7 Interval)

Flaw Depth

(inch)

Observed 

(Detected) 

Number of Flaws 

(Biased)

Posterior Mean

Number of Flaws 

(Unbiased and 

Corrected for 

POD)

Rule’s Limits 

per 1,000" of 

Weld

0.000 < a ≤ 0.075 0 122.89 No limit

0.075 < a ≤ 0.475 19 39.05 166.70

0.125 < a ≤ 0.475 0 14.69 90.80

0.175 < a ≤ 0.475 0 5.12 22.82

0.225 < a ≤ 0.475 0 1.38 8.66

0.275 < a ≤ 0.475 0 0.21 4.01

0.325 < a ≤ 0.475 0 0.13 3.01

0.375 < a ≤ 0.475 0 0.08 1.49

0.425 < a ≤ 0.475 0 0.04 1.00

8
C-36

1

2 Figure C-14. Posterior number of flaws in the Supplement 4 inspection volume in various flaw 

3 size intervals including epistemic uncertainties

4

0 - 0.075 0.075 - 0.125 0.125 - 0.175 0.175 - 0.225 0.225 - 0.26

100

101

102

Actual number of flaws

Flaw size intervals (inch)

Max

75%

25%

Min

Median
C-37

1 C.4 References

2 [C-1] U.S. Code of Federal Regulations, Title 10, “Energy,” Chapter I, Section 50.61a,

3 “Alternate Fracture Toughness Requirements for Protection Against Pressurized 

4 Thermal Shock Events” (10 CFR 50.61a).

5 [C-2] Williams, P.T., T.L. Dickson, and S. Yin, “Fracture Analysis of Vessels - Oak Ridge 

6 FAVOR, v09.1, Computer Code: Theory and Implementation of Algorithms, Methods, 

7 and Correlations,” ORNL/TM-2010/5, January 2010, Oak Ridge National Laboratory, 

8 Oak Ridge, TN.

9 [C-3] U.S. Nuclear Regulatory Commission, “A Generalized Procedure for Generating 

10 Flaw-Related Inputs for the FAVOR Code,” NUREG/CR-6817, March 2004, Agencywide 

11 Documents Access and Management System (ADAMS) Accession No. ML040830499.

[C-4] Gelman, A., et al., Bayesian Data Analysis, 2nd 12 Ed., Chapman & Hall/CRC, Boca 

13 Raton, FL, July 2003.

14 [C-5] Hamada, M.S., et al., Bayesian Reliability, Springer Science+Business Media, LLC, New 

15 York, NY, July 2008.

16 [C-6] Becker, F.L., “Reactor Pressure Vessel Inspection Reliability Based on Performance 

17 Demonstrations,” EPRI Report 1007984, June 2004, Electric Power Research Institute,

18 Palo Alto, CA.

19 [C-7] Georgiou, G.A., “Probability of Detection (PoD) Curves: Derivation, Applications and 

20 Limitations,” Research Report 454, UK Health & Safety Executive, London, UK, 2006.

21 [C-8] Li, M., and W. Q. Meeker, “A Noise Interference Model for Estimating Probability of 

22 Detection for Nondestructive Evaluations,” Proceedings of the Review of Progress in 

23 Quantitative Nondestructive Evaluation, July 20–25, 2008, Chicago, IL, American 

24 Institute of Physics, College Park, MD, 2009.

25 [C-9] U.S. Department of Defense, “Nondestructive Evaluation System Reliability 

26 Assessment,” MIL-HDBK-1823, April 30, 1999.

27 [C-10] Chatterjee, K., and M. Modarres, “A Bayesian Probabilistic Approach to Improved Health 

28 Management of Steam Generator Tubes,” published online by the Prognostics and 

29 Health Management Society, 2011, available at 

30 https://www.phmsociety.org/sites/phmsociety.org/files/phm_submission/2011/phmc_11_

31 012.pdf (accessed July 21, 2014).

32 [C-11] Celeux, G., et al., “Using Markov Chain Monte Carlo Methods to Solve Full Bayesian 

33 Modeling of PWR Vessel Flaw Distributions,” Reliability Engineering and System Safety, 

34 66(3):243–252, December 1999.

35 [C-12] Chatterjee, K., and M. Modarres, “A Probabilistic Approach for Estimating Defect Size 

36 and Density Considering Detection Uncertainties and Measurement Errors,” Journal of 

37 Risk and Reliability, 227(1):28–40, February 2013.

38 [C-13] Spanner, J., “Materials Reliability Program: Reanalysis of Reactor Vessel Examination 

39 Data from the 1996 Beaver Valley 2 Vessel Examination (MRP-207), RPV Flaw 

40 Distribution,” EPRI Report 1014548, November 2006, Electric Power Research Institute, 

41 Palo Alto, CA.
C-38

1 [C-14] U.S. Nuclear Regulatory Commission, “Probabilistic Fracture Mechanics - Models, 

2 Parameters, and Uncertainty Treatment Used in FAVOR Version 04.1,” NUREG-1807, 

3 June 2007, ADAMS Accession No. ML072010411.

4
C-39

1 Appendix C-1: Sample MATLAB Routine

2

3 clear

4 %%***********************************************************************

5 %%This MATLAB routine estimates posterior values of parameters of flaw- 6 %%depth distributions for small and large flaws in RPVs using NDE data,

7 %%accounting for measurement error, POD, and epistemic uncertainties. The

8 %%routine solves Equations 16, 17 and 18. The specific data used in this

9 %%routine are related to the POD of Eqn. 25 and the measurement error

10 %%discussed in Section C.3.3 The NDE data of Beaver Valley 2 are used. It

11 %%also shows the solution to the likelihood functions of Eqn. 27 and Eqn. 30. 12 %%Finally the routine solves the Bayesian inferences in Eqn. 28 for both

13 %%small and large flaws.

14 %%***********************************************************************

15 % THIS ROUTINE IS FOR DEMONSTRATION OF ONE WAY TO SOLVE THE BAYESIAN

16 % INFERENCE METHODS DISCUSSED IN THIS REPORT. THE RESULTS MAY BE USED TO

17 % CROSS-CHECK THE RESULTS IF OTHER VALUES OR OTHER SOLUTION ROUTINES ARE

18 % USED. THE ROUTINE DOES NOT COVER ALL E ANALYSES PERFORMED AND DISCUSSED

19 % IN THIS REPORT. FOR EXAMPLE, THE ROUTINE TO ESTIMATE THE FLAW-DENSITY

20 % DISTRIBUTIONS HAS NOT BEEN PRESENTED (ONLY FLAW-DEPTH MODEL

21 % DISTRIBUTIONS HAVE BEEN PRESENTED).

22 %%***********************************************************************

23 %%LIMITATIONS

24 % 1. THIS ROUTINE ASSUMES THE SINGLE PARAMETER EXPONENTIAL DISTRIBUTION

25 % FOR MODELING DISTRIBUTION OF BOTH SMALL AND LARGE FLAW DEPTHS.

26 % 2. THE POSTERIOR FOR SMALL FLAWS WILL GO TO INFINITY IF THE RIGHT EXTREME

27 % OF THE LEFT-CENSORED NDE DATA OR LEFT EXTREME OF INTERVAL DATA FOR

28 % SMALL FLAWS IS EXTREMELY SMALL (THAT IS, LESS THAN 0.09 INCH).

29 % 3. THE POSTERIOR FOR SMALL FLAWS APPROACHES INFINITY FOR LARGE SIGMA;

30 % THAT IS, THE RANDOM ERROR FOR THE MEASUREMENT ERROR MODEL (WHEN SIGMA

31 % EXCEEDS 0.035").

32 %%***********************************************************************

33 %%ASSUMPTIONS

34 % 1. THIS ROUTINE ASSUMES LEFT-TRUNCATED SINGLE-PARAMETER EXPONENTIAL

35 % DISTRIBUTION FOR FLAW DEPTH FOR SMALL FLAWS. 36 % 2. THIS ROUTINE ASSUMES RIGHT-TRUNCATED SINGLE-PARAMETER EXPONENTIAL

37 % DISTRIBUTION FOR LARGE FLAWS.

38 % 3. THIS ROUTINE ASSUMES TWO-PARAMETER LOGISTIC DISTRIBUTION EQN. 4 WITH

39 % A POD THRESHOLD OF ZERO.

40 % 4. SIMILARLY TO EQN. 30, INTEGRAL RANGE FOR THE MEASUREMENT IS BETWEEN

41 % -1 AND 1, WHICH COVERS THE EXTREME RANGES OF THE BIAS AND RANDOM ERRORS.

42 %%***********************************************************************

43 %%NOMENCLATURE:

44 %m & c are parameters discussed in Eqn. 8; sigma is standard deviation of

45 %random error associated with the measurement error model of Eqn. 8.

46 %beta1 & beta2 are POD function parameters in Eqn. 4; ath is the detection

47 %threshold of Eqn. 4.

48 %atr is the transition flaw depth between small and large flaws discussed

49 %in Eqn. 9. 50 %lambda1 & lambda2 are flaw-depth distribution parameters (i.e., the flaw51 % depth intensity) for small and large defects respectively.

52 %n1 is the total number of censored NDE data for small flaws. 53 %are is the right extreme of left-censored or interval of small-flaw54 %depth NDE data.

55 %ale is the left extreme of interval of small-flaw-depth data; ale must

56 % be greater than the parameter c of the measurement error model.

57 %a1 is a set, representing values of exact small-flaw-depth NDE data

58 %reported. 59 %n2 is the number of exact small-flaw-depth NDE data reported

60 %n3 is the total number of exact large-flaw-depth NDE data reported

61 %n4 is the total number of interval or left-censored large-flaw-depth NDE

62 %data reported
C-40

1 %a2 is set, representing exact values of large-flaw-depth NDE data

2 %reported

3 %are_large is the right extreme of the left-censored or interval large- 4 %flaw-depth NDE data reported

5 %ale_large is the left extreme of interval data of large-flaw-depth NDE

6 %data reported

7 %%***********************************************************************

8 %%Measurement error parameters

9 m=0.84;c=0.04;sigma=0.0298;

10 %%***********************************************************************

11 %%POD parameters

12 ath=0;beta1=63.21;beta2=0.1124;

13 %%***********************************************************************

14 %%Transition point for large flaws

15 atr=0.26;

16 %%***********************************************************************

17 %%***********************************************************************

18 %%NDE data of small flaws

19 n1=103;are=0.125;ale=c;a1=[0.15];n2=0;

20 %%***********************************************************************

21 %%***********************************************************************

22 %%NDE data of large flaws

23 a2=[0.26];n3=length(a2);n4=0;ale_large=0.27;are_large=0.3;

24 %%***********************************************************************

25 %BAYESIAN INFERENCE FOR ESTIMATING POSTERIOR DISTRIBUTION OF THE PARAMETER

26 %LAMBDA (FLAW-DEPTH INTENSITY) OF THE SMALL-FLAW-DEPTH EXPONENTIAL PDF

27 lambda1=linspace(0,60,100);

28 Norm_const=0;

29 for i=2:length(lambda1)

30 Marginal_Pod=@(a)((1-(1+exp(-beta1*beta2))./(1+exp(beta1*(a-beta2-ath)))).*(la

31 mbda1(i).*exp(-lambda1(i).*a)./(1-exp(-lambda1(i).*atr))));

32 Likeli_int=@(Ea,a)((1-(1+exp(-beta1*beta2))./(1+exp(beta1*(a-Ea-beta2-ath)))).

33 *(lambda1(i).*exp(-lambda1(i).*(a-Ea))./(1-exp(-lambda1(i).*atr))).*((1/sqrt

34 (2*pi*sigma^2)).*exp(-(1/(2*sigma^2)).*((m*Ea+(1-m)*a-c)/m).^2)));

35 Likeli_exact=@(Ea)((1-(1+exp(-beta1*beta2))./(1+exp(beta1*(a1-Ea-beta2-ath))))

36 .*(lambda1(i).*exp(-lambda1(i).*(a1-Ea))./(1-exp(-lambda1(i).*atr))).*((1/sqrt

37 (2*pi*sigma^2)).*exp(-(1/(2*sigma^2)).*((m*Ea+(1-m)*a1-c)/m).^2)));

38 Likelihood_exact(i)=((quad(Likeli_exact,-1,1))./(quad(Marginal_Pod,0,atr)))^n2

39 ;

40 Likelihood_int(i)=((dblquad(Likeli_int,-1,1,ale,are))./(quad(Marginal_Pod,0,at

41 r)))^n1;

42 Prior_small(i)=(gampdf(lambda1(i),39,.3504)); 43 Numerator(i)=Likelihood_int(i).*Likelihood_exact(i).*Prior_small(i);

44 Norm_const=Norm_const+Numerator(i)*(lambda1(i)-lambda1(i-1));

45 end

46 Posterior_small=Numerator/Norm_const;

47 Mean_Posterior_Small_Flaw_Depth_Intensity=sum(lambda1.*Posterior_small)*0.6061

48 Mean_Posterior_Small_Flaw_Depth=1/Mean_Posterior_Small_Flaw_Depth_Intensity

49 POD_of_Posterior_Mean_Small_Flaw_Depth=1-(1+exp(-beta1*beta2))./(1+exp(beta1*(

50 Mean_Posterior_Small_Flaw_Depth-beta2-ath)))

51 subplot(211)

52 plot(lambda1,Posterior_small,lambda1,Prior_small,'-.','LineWidth',2)

53 xlabel('Flaw Depth intensity (per unit 

54 inch)','fontsize',12,'fontweight','b','FontName','Times New Roman')

55 ylabel('Relative Frequency','fontsize',12,'fontweight','b','FontName','Times 

56 New Roman')

57 title('Posterior and prior flaw depth intensity for small 

58 flaws','fontsize',14,'fontweight','b','FontName','Times New Roman')

59 h = legend('Posterior_small','Prior_small');

60 set(h,'Interpreter','none')

61 set(gca,'FontSize',12,'FontWeight','b','FontName','Times New Roman')

62 %%***********************************************************************

63 %BAYESIAN INFERENCE FOR ESTIMATING POSTERIOR DISTRIBUTION OF THE PARAMETER

64 %LAMBDA (FLAW-DEPTH INTENSITY) OF THE LARGE-FLAW-DEPTH EXPONENTIAL PDF
C-41

1 lambda2=linspace(0,10,100);

2 Norm_const1=0;

3 for i=2:length(lambda2)

4 Marginal_Pod1=@(a)((1-(1+exp(-beta1*beta2))./(1+exp(beta1*(a-beta2-ath)))).*(l

5 ambda2(i).*exp(-lambda2(i).*(a-atr))));

6 Likeli_exact1=@(Ea)((1-(1+exp(-beta1*beta2))./(1+exp(beta1*(a2-Ea-beta2-ath)))

7 ).*(lambda2(i).*exp(-lambda2(i).*(a2-Ea-atr))).*((1/sqrt(2*pi*sigma^2)).*exp(-

8 (1/

9 (2*sigma^2)).*((m*Ea+(1-m)*a2-c)/m).^2)));

10 Likelihood_exact1(i)=((quad(Likeli_exact1,-1,1))./(quad(Marginal_Pod1,atr,10))

11 )^n3;

12 Likeli_int1=@(Ea,a)((1-(1+exp(-beta1*beta2))./(1+exp(beta1*(a-Ea-beta2-ath))))

13 .*(lambda2(i).*exp(-lambda2(i).*(a-Ea-atr))).*((1/sqrt(2*pi*sigma^2)).*exp(-(1

14 /

15 (2*sigma^2)).*((m*Ea+(1-m)*a-c)/m).^2)));

16 Likelihood_int1(i)=((dblquad(Likeli_int1,-1,1,ale_large,are_large))./(quad(Mar

17 ginal_Pod1,atr,10)))^n4;

18 Prior_large(i)=(gampdf(lambda2(i),4,0.8333));

19 Numerator1(i)=Likelihood_exact1(i).*Likelihood_int1(i).*Prior_large(i);

20 Norm_const1=Norm_const1+Numerator1(i)*(lambda2(i)-lambda2(i-1));

21 end

22 Posterior_large=Numerator1/Norm_const1;

23 Posterior_small=Numerator/Norm_const;

24 Mean_Posterior_Large_Flaw_Depth_Intensity=sum(lambda2.*Posterior_large)*0.1010 25 Mean_Posterior_Large_Flaw_Depth=1/Mean_Posterior_Large_Flaw_Depth_Intensity

26 POD_of_Posterior_Mean_Large_Flaw_Depth=1-(1+exp(-beta1*beta2))./(1+exp(beta1*(

27 Mean_Posterior_Large_Flaw_Depth-beta2-ath)))

28 subplot(212)

29 f1=plot(lambda2,Posterior_large,lambda2,Prior_large,'-.','LineWidth',2)

30 xlabel('Flaw depth intensity (per unit 

31 inch)','fontsize',12,'fontweight','b','FontName','Times New Roman')

32 ylabel('Relative Frequency','fontsize',12,'fontweight','b','FontName','Times 

33 New Roman')

34 title('Posterior and prior flaw depth intensity for large 

35 flaws','fontsize',14,'fontweight','b','FontName','Times New Roman')

36 h = legend('Posterior_large','Prior_large');

37 set(h,'Interpreter','none')

38 set(gca,'FontSize',12,'FontWeight','b','FontName','Times New Roman')

39 screen_size = get(0, 'ScreenSize');

40 f1 = figure(1);

41 set(f1, 'Position', [0 0 screen_size(3) screen_size(4) ] );

42 %%***********************************************************************

43

44
C-42

1 Output from the above routine is displayed in Figure C-14

2 (also shown in Figure C-8 and Figure C-9)

3

4 Mean_Posterior_Small_Flaw_Depth_Intensity =

5 37.8118

6 Mean_Posterior_Small_Flaw_Depth =

7 0.0264

8 POD_of_Posterior_Mean_Small_Flaw_Depth =

9 0.0035

10 Mean_Posterior_Large_Flaw_Depth_Intensity =

11 5.4042

12 Mean_Posterior_Large_Flaw_Depth =

13 0.1850

14 POD_of_Posterior_Mean_Large_Flaw_Depth =

15 0.9900

16 f1 =

17 230.0087

18 231.0082

19 Published with MATLAB® 7.12
C-43

1

2

3

Figure C-15. MATLAB Routine Output 4

D-1

1

2 APPENDIX D: SENSITIVITY-STUDY RESULTS ON FLAW 

3 DISTRIBUTIONS CONSIDERING VFLAW DATA, POD, AND 

4 MEASUREMENT ERRORS IN NDE DATA

5

6 Prepared for the U.S. Nuclear Regulatory Commission under the Subcontract 4000111626 

7 with UT-Battelle, LLC (Oak Ridge National Laboratory)

8

9

10

11

12

13

14 Mohammad Modarres

15 University of Maryland

16

17 March 2012

18
D-2

1 Table of Contents

2

3 Table of Contents............................................................................................................D-2

4 List of Tables...................................................................................................................D-3

5 List of Figures..................................................................................................................D-3

6 Abbreviations ..................................................................................................................D-4

7 Symbols ....................................................................................................................D-4

8 D.1 INTRODUCTION............................................................................................................D-5

9 D.2 OVERVIEW OF THE BAYESIAN APPROACH TO UPDATE FLAW DEPTH AND FLAW 

10 DENSITY DISTRIBUTIONS ...........................................................................................D-5

11 D.3 APPLICATION TO A BASE CASE EXAMPLE................................................................D-6

12 D.4 APPLICATION TO SENSITIVITY CASES ....................................................................D-11

13 D.5 DISCUSSION AND SUMMARY ...................................................................................D-18

14 D.6 REFERENCES.............................................................................................................D-19

15
D-3

1 List of Tables

2

3 Table D-1. Alternate PTS Rule Compliance Using Mean Number-of-Flaws by Size Interval

4 for Beaver Valley-2 (assuming a Lower Limit of 0.04” vs. 0.075” for Observed

5 Data Interval)......................................................................................................D-10

6 Table D-2. Results of Sensitivity Case 1 ..............................................................................D-12

7 Table D-3. Results of Sensitivity Case 2 ..............................................................................D-12

8 Table D-4. Results of Sensitivity Case 3 ..............................................................................D-13

9 Table D-5. Results of Sensitivity Case 4 ..............................................................................D-13

10 Table D-6. Results of Sensitivity Case 5 ..............................................................................D-14

11 Table D-7. Results of Sensitivity Case 6 ..............................................................................D-14

12 Table D-8. Results of Sensitivity Case 7 ..............................................................................D-15

13 Table D-9. Results of Sensitivity Case 8 ..............................................................................D-15

14 Table D-10. Results of Sensitivity Case 9 ............................................................................D-16

15 Table D-11. Results of Sensitivity Case 10 ..........................................................................D-16

16 Table D-12. Results of Sensitivity Case 11 ..........................................................................D-17

17 Table D-13. Results of Sensitivity Case 12 ..........................................................................D-17

18

19

20

21

22 List of Figures

23

24 Figure D-1. Beaver Valley-2 Weld Map [D-9] .........................................................................D-8

25 Figure D-2. Flaw Depth and Flaw Density Distributions Used in VFLAW and Their 

26 Corresponding Parameter Hyper-PDFs...........................................................................D-9

27

28

29

30
D-4

1 Abbreviations

Abbreviation Definition

PDF Probability Density Function

NDE Nondestructive Evaluation

POD Probability of Detection

RPV Reactor Pressure Vessel

SAW Submerged Arc Weld

UT Ultrasonic Test

2

3 Symbols and Expressions

Symbol Definition

a True flaw depth (inches)

a* NDE measured (or observed) flaw depth (inches)

ath Threshold flaw depth for detection, below which flaw detection is beyond the 

capability of the NDE technology used

D The event that a flaw is detected

D The event that a flaw is not detected

EM Model error of the NDE measurement error, represented by a normal 

distribution with a mean of zero and known standard deviation

L(a|Φ) Likelihood of true flaw depth given the vector of parameters Φ

L(Data|θ) Likelihood of observed NDE data conditioned on (given) the unknown 

parameter θ

L(n | n )j

* Likelihood of observing n* flaws given that there are a total of nj true flaws

Mε NDE measurement error (combined random and systematic errors)

n Number of flaw-depth intervals (reported in NDE)

n� Mean of the PDF of the number of flaws in volume v

n* Number of exact flaw depths observed (reported in NDE)

















*

j

n

n Combination of n* observed flaws out of total flaws nj (observed or detected, 

and unobserved or not detected flaws) 

POD(a) Probability of detection of a flaw of depth a

Pr(.) Probability

Pr(D) POD independent of flaw depth

Pr(D) Probability of no detection regardless of size

Pr(n | n )

*

j Probability of total flaws nj conditioned on observing n* flaws in NDE

βi Parameter i of the POD model

Δ Weld bead thickness

π0(θ) Prior PDF of an unknown parameter θ

π1(θ|Data) Posterior PDF of an unknown parameter given observed NDE data

Φ Vector of parameters of the flaw-depth PDF

θ An unknown parameter of a model

4

5
D-5

1 D.1 Introduction

2 This report discusses the results of a sensitivity analysis by applying a Bayesian updating 

3 methodology described in Appendix C to account for probability of detection (POD) and 

4 measurement (sizing) error in the data for flaws detected through ultrasonic testing (UT). The 

5 methodology estimates vessel-specific flaw depth and density distributions for comparison to 

6 the 10 CFR 50.61a screening tables and updates the VFLAW code’s distributions for further 

7 analysis with the FAVOR code. The sensitivity analysis was performed using assumed 

8 nondestructive examination (NDE) flaw data. Existing VFLAW distributions for flaw depth and 

9 density, representing welds of the Pressure Vessel Research User Facility (PVRUF) vessel, were 

10 specialized to the Beaver Valley 2 reactor pressure vessel (RPV) welds and assumed to 

11 properly describe prior flaw-depth and flaw-density distributions and characteristics.

12 In the remainder of this Appendix, a very brief overview of the methodology is described first, 

13 followed by the results of the sensitivity analysis. The results are compared to the 

14 10 CFR 50.61a screening tables and conclusions are summarized.

15 D.2 Overview of the Bayesian Approach to Update Flaw-Depth and 

16 Flaw-Density Distributions

17 Finding flaws using NDE requires characterization of associated uncertainties. The reliability of 

18 an inspection is customarily expressed in terms of the POD of a given flaw size and expressed 

19 by curves of POD vs. flaw size. Additionally, sizing or measurement errors occur and model 

20 uncertainties exist. Sizes of UT-detected small flaws in RPVs tend to be overestimated while 

21 the sizes of large flaws are underestimated [D-1].

22 POD is generally modeled as a function of true flaw depth. Different forms of POD curves have 

23 been used in the literature (see References [D-2], [D-3], and [D-4]. The logistic POD model for 

24 hit/miss data is a common model and is represented as:





















β β β β

β β

0 otherwise

 for a > a

1+ e

1+ e 1- POD(a | , ,a ) = th (a- -a )

-

1 2 th 1 2 th

1 2

Eqn. (D-1)

25 Measurement error is defined by Eqn. (D-2), where Mε is the measurement error, a* is the 

26 measured value of flaw depth, and a is the true value of flaw depth:

Mε = a* - a Eqn. (D-2)

27 In its simplest form, Mε can be represented by a linear function of true size, as shown in 

28 Eqn. (D-1), where m is the slope and c is the intercept of the line representing measurement 

29 error versus true flaw depth:

N (0, )

M ma c E

M M M

M

where,a trueflawdepthandE = σ

= + +

=

e

Eqn. (D-3)

30 The likelihood of exact flaw measurements reported may be expressed as a function of POD 

31 and Mε (see Appendix C). If, in addition to or instead of the exact flaw depth data, intervals of 

32 flaw depths are reported (such as the number of flaws observed at less than a given depth or 
D-6

1 between an upper and lower limit), the likelihood may also be similarly expressed (see Appendix 

2 C).

3

4 Regardless of the form of the data (exact flaw depth and/or interval data), the Bayesian 

5 inference of the vector of parameters Φ of the flaw-depth probability density function (PDF)

would be obtained from the Bayesian inference ( | Data) L(Data | ) ( ) 6 π1 Φ ∝ Φ π0 Φ , where 

( |Data) π1 Φ is the posterior multivariate PDF of vector Φ and ( ) 7 π0 Φ is the prior multivariate PDF 

8 of Φ.

9 If flaws are detected with a probability Pr(D) (or not detected with a probability of 

10 Pr(D) = 1− Pr(D) ), the probability that a specific total number of flaws n* observed in a given 

volume v out of the true number of flaws nj 11 follows the binomial distribution (see 

12 References [D-5] and [D-6]:

[ ] [ ] * j * n n n n

*

j

j

* Pr(D) 1- Pr(D)

n

n

L(n | n ) = 















Eqn. (D-4)

13 Accordingly, the mean estimate of the true number of flaws n would be found from:

POD(D)

n n

*

= Eqn. (D-5)

The Bayesian updating of nj

, which allows the estimation of the posterior distribution of nj 14 to 

15 describe the true number of flaws, would be:

∑

=

nj

j j

*

j j

*

*

j L(n | n )Pr(n )

L(n | n )Pr(n ) Pr(n | n )

Eqn. (D-6)

16 where the prior Pr(nj) may be expressed by a Poisson distribution with parameter ρ (defined in 

17 VFLAW).

18 D.3 Application to a Base Case Example

19 The EPRI report by Spanner [D-7] provides UT-based measured flaw data near the inner 

20 surface (~2.5 inches) of the Beaver Valley Unit 2 RPV, including small flaw sizes. These data, 

21 while subject to POD and measurement error, provide a more vessel-specific perspective of the 

22 distribution of flaws for weld metals in the Beaver Valley 2 RPV than do the VFLAW distributions 

23 used in FAVOR.

24 The observed Beaver Valley 2 NDE data (with detection and sizing uncertainty) are mostly in 

25 the form of interval data as summarized below [D-7]:

26 19 weld flaws were detected by the UT NDE of Beaver Valley 2 in the first inch (ASME 

27 Code, Section XI, Mandatory Appendix VIII, Supplement 4 inspection volume) of the 
D-7

1 RPV (these data are assumed normalized to 1,000" of weld length), all having a flaw 

depth less than 0.125". *** 2

3 The lower limit of the detected flaw intervals described above was not stated in [D-7], but is not 

4 zero. Two possible subjective lower limits were assumed, and later the sensitivities of the final 

5 (posterior) flaw distributions to these two choices were assessed. The two lower limits selected 

6 were 0.04" and 0.075".

7 The associated weld bead thicknesses are not reported in [D-7]. However, the weld region of 

8 the observed flaws and flaw length is reported. Such results must be corrected for detection 

9 and sizing capability, particularly for the small flaws. The Beaver Valley 2 RPV weld map is 

10 shown in Figure D-1 [D-8]. In the absence of the Beaver Valley average weld bead thickness as 

11 the point of transition between large and small depths, it was assumed that all flaws reported 

12 are in submerged arc welds (SAWs), which form over 90% of welds in the VFLAW data, with the 

13 bead thickness of D = 0.26". Using the Beaver Valley 2 RPV information shown in Figure D-1, 

14 the VFLAW data were specialized to represent the prior distributions of flaw depth and density 

15 for Beaver Valley 2 RPV.

16

17

18 Total Weld Length = 816.18"

Total Weld Volume = 4.975 ft3 19

Total Weld Fusion Area = 92 ft2 20

Weld Volume of 1" = 0.616 ft3 21

Weld Volume of 3/8t = 1.865 ft3 22

 *** In Appendix C, the same analysis is performed, except that the analysis did not assume that the data 

were normalized to 1,000" of linear length of weld. Instead, it assumed that the data came from 816.18" 

of weld, which is the actual length of weld in the Beaver Valley 2 RPV.
D-8

1

2

3 Figure D-1. Beaver Valley 2 Weld Map [D-8]

4

5
D-9

1 The relationship between the flaw depth and flaw density PDFs and the corresponding prior 

2 hyper-PDFs of their parameters are illustrated in Figure D-2.

3

4

5

6

7 Figure D-2. Flaw-Depth and Flaw-Density Distributions Used in VFLAW and Their 

8 Corresponding Parameter Hyper-PDFs

9

10 Analysis of UT-detected performance data reported by the Electric Power Research Institute 

11 (EPRI) [D-7] was used to identify a POD function for the purpose of this example. The threshold 

12 limit of this POD function was taken as 0.00, and the epistemic uncertainties associated with the 

13 parameters of the model were not considered in this study. Accordingly, the following mean 

14 POD function was used:

63.2100(a 0.1124) 1 e

1 POD(a) 1 − + = − Eqn. (D-7)

15 The measurement error was assumed based on some examples and measurement errors 

16 reported in Reference [D-7]. After combination of Eqn. (D-2) and Eqn. (D-3), the measurement 

17 error (a linear function that oversizes small flaws and undersizes large flaws) is:

M 0.1905a 0.0476 N(0,0.0298) *

ε = − + + Eqn. (D-8)

18
D-10

1 where N(0, 0.0298) is a normal distribution with a mean of zero and a constant standard 

2 deviation of 0.0298. The observed data for the first inch of the vessel were used to estimate the 

3 true number of flaws in the flaw-depth ranges of the Alternate PTS Rule flaw tables. This 

4 information was used to assess how the estimated number of flaws of a given size in Beaver 

5 Valley 2 (estimated based on inspection data, VFLAW distributions, and Bayesian inference) 

6 compares to the number and size of flaws permitted by the Alternate PTS Rule flaw tables. The 

7 posterior characteristics were used to develop the corresponding Alternate PTS Rule flaw 

8 tables.

9 If the lower limit of the observed data interval is changed from 0.04" to 0.075" to line up with the 

10 smallest bin size in the Alternate PTS Rule flaw tables, the results summarized in Table D-1 are 

11 calculated. In this case, the results of the mean number of flaws show no sensitivity to the 

12 choice of the lower limit of the interval of NDE data. Given this flaw observation, true estimated 

13 numbers of flaws in all bins were below the Alternate PTS rule flaw limits.

14

15 Table D-1. Alternate PTS Rule Flaw Table Assessment Using Mean Number of Flaws by Size

16 Interval for Beaver Valley 2 (assuming a Lower Limit of 0.04" vs. 0.075" for 

17 Observed Data Interval)

Flaw Depth

(in)

Base Case 1 Base Case 2

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

Number of 

Flaws 

(Biased 

with Lower 

Detection 

Limit of 

0.04")

Posterior 

Mean 

Number of 

Flaws 

(Unbiased 

and 

Corrected 

for POD)

Observed 

(Detected) 

Number of 

Flaws 

(Biased with 

Lower 

Detection 

Limit of 

0.075")

Posterior 

Mean 

Number of 

Flaws 

(Unbiased 

and 

Corrected for 

POD)

1 0.000 < a ≤ 0.075

19

111.41 0 109.09 No limit

2 0.075 < a ≤ 0.475 25.73 19 27.02 166.70

3 0.125 < a ≤ 0.475 0 8.46 0 8.98 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 2.88 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 0.79 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.05 1.00

18

19
D-11

1 D.4 Application to Sensitivity Cases

2 In order to examine the effects of changes in the number of flaws observed through NDE, as 

3 well as the significance of the POD and the VFLAW prior distributions, twelve sensitivity cases 

4 were identified and examined. Methods and tools discussed in Sections D.2 and D.3 were used 

5 to carry out these sensitivity cases. The sensitivity cases examined were as follows:

6 1. No flaws detected, with consideration of the VFLAW prior distributions, POD, and 

7 measurement error.

8 2. 70% of the Alternate PTS Rule’s allowed flaws in Bins 2 and 3 are detected, with 

9 consideration of the VFLAW prior distributions, POD, and measurement error.

10 3. 110% of the Alternate PTS Rule’s allowed flaws in Bin 3 are detected, with consideration 

11 of the VFLAW prior distributions, POD, and measurement error.

12 4. No flaws detected, with no consideration of VFLAW priors (i.e., non-informative priors) 

13 and POD, but with consideration of flaw measurement error only (i.e., no POD or 

14 VFLAW prior).

15 5. 70% of the Alternate PTS Rule’s allowed flaws in Bins 2 and 3 are detected, with no

16 consideration of VFLAW priors and POD, but with consideration of flaw measurement 

17 error only.

18 6. 110% of the Alternate PTS Rule’s allowed flaws in Bin 3 are detected, with no 

19 consideration of VFLAW priors and no consideration of POD, but with consideration of 

20 flaw measurement error only.

21 7. No flaws detected, no consideration of VFLAW priors, but with consideration of POD and 

22 flaw measurement error.

23 8. 70% of the Alternate PTS Rule’s allowed flaws in Bins 2 and 3 are detected, with no 

24 consideration of VFLAW priors, but with consideration of POD and flaw measurement 

25 error.

26 9. 110% of the Alternate PTS Rule’s allowed flaws in Bin 3 are detected, with no 

27 consideration of VFLAW priors, but with consideration of POD and flaw measurement 

28 error.

29 10. No flaws detected, no consideration of POD, but with consideration of VFLAW priors and 

30 flaw measurement error.

31 11. 70% of the Alternate PTS Rule’s allowed flaws in Bins 2 and 3 are detected, with no 

32 consideration of POD, but with consideration of VFLAW priors and flaw measurement 

33 error.

34 12. 110% of the Alternate PTS Rule’s allowed flaws in Bin 3 are detected, with no 

35 consideration of POD, but with consideration of VFLAW priors and flaw measurement 

36 error.

37 The results of the sensitivity studies are listed in Table D-2 through Table D-13.

38

39

40
D-12

1 Table D-2. Results of Sensitivity Case 1

Flaw Depth

(in)

Base Case 1 Sensitivity Case 1

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 80.13 No limit

2 0.075 < a ≤ 0.475 25.73 0 40.76 166.70

3 0.125 < a ≤ 0.475 0 8.46 0 18.79 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 7.81 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 2.71 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.05 1.00

2

3

4 Table D-3. Results of Sensitivity Case 2

Flaw Depth

(in)

Base Case 1 Sensitivity Case 2

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit 

of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 306.12 No limit

2 0.075 < a ≤ 0.475 25.73 116.69 34.16 166.70

3 0.125 < a ≤ 0.475 0 8.46 63.56 7.71 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 1.69 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 0.44 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.04 1.00

5
D-13

1 Table D-4. Results of Sensitivity Case 3

Flaw Depth

(in)

Base Case 1 Sensitivity Case 3

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 155.03 No limit

2 0.075 < a ≤ 0.475 25.73 0 41.33 166.70

3 0.125 < a ≤ 0.475 0 8.46 99.88 15.21 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 4.51 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 1.21 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.05 1.00

2

3

4 Table D-5. Results of Sensitivity Case 4

Flaw Depth

(in)

Base Case 1 Sensitivity Case 4

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 0.00 No limit

2 0.075 < a ≤ 0.475 25.73 0 0.00 166.70

3 0.125 < a ≤ 0.475 0 8.46 0 0.00 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 0.00 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 0.00 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.00 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.00 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.00 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.00 1.00

5
D-14

1 Table D-6. Results of Sensitivity Case 5

Flaw Depth

(in)

Base Case 1 Sensitivity Case 5

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Lower 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 56.47 No limit

2 0.075 < a ≤ 0.475 25.73 116.69 60.22 166.70

3 0.125 < a ≤ 0.475 0 8.46 63.56 36.13 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 18.67 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 5.98 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.00 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.00 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.00 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.00 1.00

2

3

4 Table D-7. Results of Sensitivity Case 6

Flaw Depth

(in)

Base Case 1 Sensitivity Case 6

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 32.85 No limit

2 0.075 < a ≤ 0.475 25.73 0 67.03 166.70

3 0.125 < a ≤ 0.475 0 8.46 99.88 47.02 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 27.19 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 11.17 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.00 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.00 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.00 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.00 1.00

5
D-15

1 Table D-8. Results of Sensitivity Case 7

Flaw Depth

(in)

Base Case 1 Sensitivity Case 7

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 0.00 No limit

2 0.075 < a ≤ 0.475 25.73 0 0.00 166.70

3 0.125 < a ≤ 0.475 0 8.46 0 0.00 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 0.00 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 0.00 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.00 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.00 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.00 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.00 1.00

2

3

4 Table D-9. Results of Sensitivity Case 8

Flaw Depth

(in)

Base Case 1 Sensitivity Case 8

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 5131.91 No limit

2 0.075 < a ≤ 0.475 25.73 116.69 263.80 166.70

3 0.125 < a ≤ 0.475 0 8.46 63.56 35.54 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 4.88 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 0.39 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.00 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.00 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.00 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.00 1.00

5
D-16

1 Table D-10. Results of Sensitivity Case 9

Flaw Depth

(in)

Base Case 1 Sensitivity Case 9

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 1233.10 No limit

2 0.075 < a ≤ 0.475 25.73 0 202.01 166.70

3 0.125 < a ≤ 0.475 0 8.46 99.88 54.12 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 13.13 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 1.95 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.00 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.00 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.00 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.00 1.00

2

3

4 Table D-11. Results of Sensitivity Case 10

Flaw Depth

(in)

Base Case 1 Sensitivity Case 10

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 79.14 No limit

2 0.075 < a ≤ 0.475 25.73 0 41.18 166.70

3 0.125 < a ≤ 0.475 0 8.46 0 19.09 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 7.93 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 2.42 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.05 1.00

5
D-17

1 Table D-12. Results of Sensitivity Case 11

Flaw Depth

(in)

Base Case 1 Sensitivity Case 11

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 73.49 No limit

2 0.075 < a ≤ 0.475 25.73 116.69 57.04 166.70

3 0.125 < a ≤ 0.475 0 8.46 63.56 31.09 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 14.66 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 4.81 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.05 1.00

2

3

4 Table D-13. Results of Sensitivity Case 12

Flaw Depth

(in)

Base Case 1 Sensitivity Case 12

Alternate 

PTS Rule 

Limit (per 

1,000" of 

Weld)

Bin No.

Observed 

(Detected) 

No. of Flaws 

(Biased with 

Detection 

Limit of 0.04")

Posterior 

Mean 

No. of Flaws 

(Unbiased 

and 

Corrected 

for POD)

Assumed No. 

of Detected 

Flaws

(with Bias)

Posterior 

Mean 

No. of Flaws 

(Unbiased)

1 0.000 < a ≤ 0.075

19

111.41 0 63.04 No limit

2 0.075 < a ≤ 0.475 25.73 0 66.84 166.70

3 0.125 < a ≤ 0.475 0 8.46 99.88 39.42 90.80

4 0.175 < a ≤ 0.475 0 2.98 0 20.55 22.82

5 0.225 < a ≤ 0.475 0 0.79 0 7.21 8.66

6 0.275 < a ≤ 0.475 0 0.29 0 0.29 4.01

7 0.325 < a ≤ 0.475 0 0.14 0 0.14 3.01

8 0.375 < a ≤ 0.475 0 0.09 0 0.09 1.49

9 0.425 < a ≤ 0.475 0 0.05 0 0.05 1.00

5
D-18

1 D.5 Discussion and Summary

2 In this study, analysis of a base case of UT-detected weld-flaw data involving measurement 

3 (sizing) error for Beaver Valley 2 was performed. The base case was evaluated for interval flaw 

4 depths detected in the inspection volume specified in Supplement 4 to Mandatory Appendix VIII 

5 to Section XI of the ASME Code. Detected flaw-depth intervals were analyzed assuming lower 

6 UT detection limits of 0.04" and 0.075". Subsequently, twelve sensitivity cases were assessed 

7 based on variations in the assumed detected flaw-depth data (in the form of intervals), choices 

8 of considering VFLAW flaw depth and flaw densities as prior information (as opposed to no prior 

9 information), and choices of considering the POD (as opposed to perfect detection; i.e., no 

10 POD) were evaluated and compared to the base case as well as the Alternate PTS Rule flaw 

11 table limits. No sensitivities to the choice of the lower UT detection limit on the observed data 

12 were found.

13 The results obtained from the twelve sensitivity cases were consistent and showed that small 

14 overpopulations of flaws in Bins 2 and 3 of the Alternate PTS Rule flaw tables resulting from

15 possible oversizing of small flaws would be shifted to Bins 1 and 2 after accounting for the 

16 measurement error in the Bayesian inference. When POD is considered, the effect of the 

17 missed small flaws was clearly seen in Bins 1 and 2 with an additional number of flaws in the 

18 posterior estimates as compared to the observed flaws.

19 The effects of the consideration and choice of the prior distributions of flaw density and depth 

20 were significant. When no prior information was used to describe the flaw-density and 

21 flaw-depth distributions, POD and measurement error were also sensitive and significantly 

22 amplified the number of flaws in Bins 1 and 2. However, when prior VFLAW PDFs were used, 

23 the posteriors were significantly moderated by the existence of the prior PDFs, and the POD 

24 and measurement errors played less significant roles.

25 If the approach documented in this appendix is used to reassess actual NDE flaws, it would be 

26 advisable to use informative or semi-informative prior estimates of the flaw depth and density 

27 distributions.

28

29
D-19

1 D.6 References

2 [D-1] Becker, F.L., “Reactor Pressure Vessel Inspection Reliability Based on Performance 

3 Demonstrations,” EPRI Report 1007984, June 22, 2004, Electric Power Research 

4 Institute, Palo Alto, CA.

5 [D-2] Georgiou, G.A., “Probability of Detection (PoD) Curves: Derivation, Applications and 

6 Limitations,” Research Report 454, UK Health & Safety Executive, London, UK, 2006.

7 [D-3] Li, M., and W. Q. Meeker, “A Noise Interference Model for Estimating Probability of 

8 Detection for Nondestructive Evaluations,” Proceedings of the Review of Progress in 

9 Quantitative Nondestructive Evaluation, July 20–25, 2008, Chicago, IL, American 

10 Institute of Physics, College Park, MD, 2009.

11 [D-4] U.S. Department of Defense, “Nondestructive Evaluation System Reliability 

12 Assessment,” MIL-HDBK-1823, April 30, 1999.

13 [D-5] Celeux, G., et al., “Using Markov Chain Monte Carlo Methods to Solve Full Bayesian 

14 Modeling of PWR Vessel Flaw Distributions,” Reliability Engineering and System Safety, 

15 66(3):243–252, December 1999.

16 [D-6] Chatterjee, K., and M. Modarres, “A Probabilistic Approach for Estimating Defect Size 

17 and Density Considering Detection Uncertainties and Measurement Errors,” Journal of 

18 Risk and Reliability, 227(1):28–40, February 2013.

19 [D-7] Spanner, J., “Materials Reliability Program: Reanalysis of Reactor Vessel Examination 

20 Data from the 1996 Beaver Valley 2 Vessel Examination (MRP-207), RPV Flaw 

21 Distribution,” EPRI Report 1014548, November 2006, Electric Power Research Institute, 

22 Palo Alto, CA.

23 [D-8] U.S. Nuclear Regulatory Commission, “Probabilistic Fracture Mechanics - Models, 

24 Parameters, and Uncertainty Treatment Used in FAVOR Version 04.1,” NUREG-1807, 

25 June 2007, Agencywide Documents Access and Management System (ADAMS) 

26 Accession No. ML072010411.

27





NUREG

-2163

Draft

Technical Basis for Regulatory Guidance on the Alternate Pressurized Thermal Shock Rule