A contribution to risk-based inspection and maintenance

University of Calgary

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Graduate Studies The Vault: Electronic Theses and Dissertations

2017

A contribution to risk-based inspection and

maintenance planning for deteriorating pipelines and

pressure vessels

Haladuick, Shane

Haladuick, S. (2017). A contribution to risk-based inspection and maintenance planning for

deteriorating pipelines and pressure vessels (Unpublished doctoral thesis). University of Calgary,

Calgary, AB. doi:10.11575/PRISM/24682

http://hdl.handle.net/11023/4180

doctoral thesis

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UNIVERSITY OF CALGARY

A contribution to risk-based inspection and maintenance planning for deteriorating pipelines

and pressure vessels

by

Shane Haladuick

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN CIVIL ENGINEERING

CALGARY, ALBERTA

SEPTEMBER, 2017

© Shane Haladuick 2017

ii

Abstract

Engineering systems are subject to deterioration processes, such as corrosion and fatigue,

which reduce the resistance to failure. If failure occurs, it can have large social, economic, and

environmental consequences. To mitigate this risk, regular inspections and maintenance are

performed. To minimize the overall cost of operating the system, it is important to optimize the

inspection and maintenance plan. Lifecycle risk-based inspection and maintenance planning,

which involves determining the expected total cost of operating a system over its lifecycle, is the

most established method of determining the optimal inspection and maintenance plan. However,

lifecycle risk-based inspection and maintenance planning for complex engineering systems with

many components requires a detailed analysis that can be computationally demanding; therefore,

simplifications and assumptions are commonly used in the literature.

The objective of this dissertation is to expand the state of knowledge in risk-based inspection

and maintenance planning for pipelines and pressure vessels, removing many common

assumptions and simplifications. Some aspects of the research specifically target pipelines or

pressure vessels, while others are generic to any engineering system. Specifically regarding

pressure vessels, a simple methodology is presented to determine the optimal maintenance time

for a pressure vessel with an unexpectedly severe defect. This methodology is then expanded to

account for the dependent failure events in pressure vessels with multiple defects and failure

modes. For pipelines, a risk-based maintenance methodology is developed to decide whether it is

better to continuously repair defects in a pipeline or to replace entire pipeline sections. This

methodology also examines the impact of an uncertain lifecycle on risk-based maintenance

planning. For general engineering systems, the use of heuristic algorithms in improving the

computational efficiency of solving risk-based inspection and maintenance optimization

iii

problems is examined. Finally, a methodology is developed to perform risk-based inspection

planning for the next inspection type, without requiring a lifecycle analysis.

This study aids system operators in determining the optimal inspection and maintenance

plan. It also provides methodology to perform risk-based inspection and maintenance planning in

a computationally efficient or simpler manner, to make the techniques more practically

applicable.

iv

Acknowledgements

Thank you to Dr. Markus Dann for helping develop this project, and for teaching me the

intricacies of risk and decision analysis. Our discussions always left me with many new ideas to

ponder. Thank you as well to my fellow graduate students, Chiara Belvederesi and Carlos Melo,

for your help in proofreading papers and for interesting conversation. Also, thank you to my

committee members for your help in improving this dissertation, especially Dr. Marc Maes and

Dr. Gopal Achari for acting on my supervisory committee.

I would like to give a special thanks to my fiancée Maggie Maxwell for her support

throughout this project. You patiently listened to me ramble about mathematics on many chair

lifts and road trips. Also, thank you to my parents for their tireless support throughout my last

half decade as a grad student; I could not have completed this journey without you. Finally,

thank you to my brother for always being happy to discuss engineering over a beer.

I am grateful for the financial support from the National Sciences and Engineering Research

Council (NSERC), the Alberta Scholarship program, and the Pipeline Engineering Centre at the

University of Calgary. I am also thankful to the Department of Civil Engineering for entrusting

me to teach young engineers, and to the Internship Centre at the University of Calgary for

smoothing my transition into industry.

v

Table of Contents

Abstract ......................................................................................................................... ii Acknowledgements ...................................................................................................... iv Table of Contents ...........................................................................................................v

List of Tables ............................................................................................................. viii List of Symbols, Abbreviations, and Nomenclature ................................................... xii

1 INTRODUCTION ......................................................................................................1 1.1 Background / introduction .......................................................................................1 1.2 Problem statement ....................................................................................................2

1.3 Research objectives ..................................................................................................4 1.4 Statement on the author’s contribution to the research ............................................5 1.5 Thesis overview .......................................................................................................6

2 BACKGROUND ........................................................................................................7 2.1 Decision analysis .....................................................................................................8

2.1.1 Bayesian decision analysis .............................................................................10

2.1.2 Risk-based inspection (RBI) and maintenance (RBM) planning ...................13 2.2 Consequence analysis ............................................................................................15

2.3 Structural reliability analysis .................................................................................16 2.4 Structural deterioration modelling .........................................................................18

2.4.1 Corrosion growth modelling ..........................................................................19

2.4.2 Model calibration / parameter estimation ......................................................21 2.5 Inspection and maintenance ...................................................................................24

2.5.1 Inspections .....................................................................................................24 2.5.2 Maintenance ...................................................................................................25

2.6 Overview of the applied methodologies ................................................................26

3 RISK-BASED INSPECTION PLANNING FOR DETERIORATING PRESSURE

VESSELS ..................................................................................................................28 3.1 Abstract ..................................................................................................................29 3.2 Introduction ............................................................................................................30

3.3 Literature review ....................................................................................................32 3.4 Decision analysis framework .................................................................................34 3.5 Corrosion growth modelling ..................................................................................39 3.6 Numerical example ................................................................................................43

3.7 Conclusion .............................................................................................................48

4 RISK-BASED PLANNING FOR DETERIORATING PRESSURE VESSELS WITH

MULTIPLE DEFECTS ............................................................................................50 4.1 Abstract ..................................................................................................................52 4.2 Introduction ............................................................................................................52 4.3 Literature review ....................................................................................................54

4.3.1 Decision analysis ............................................................................................54

4.3.2 RBM planning ................................................................................................55 4.4 RBM framework ....................................................................................................58 4.5 Reliability analysis .................................................................................................62

vi

4.5.1 Corrosion growth model ................................................................................62 4.5.2 Limit state functions for leak and burst failures ............................................66 4.5.3 System reliability analysis .............................................................................68

4.6 Numerical example of a corroding pressure vessel ...............................................71

4.7 Conclusion .............................................................................................................80

5 DECISION MAKING FOR LONG TERM PIPELINE SYSTEM REPAIR OR

REPLACEMENT .....................................................................................................81 5.1 Abstract ..................................................................................................................82 5.2 Introduction ............................................................................................................82

5.3 Background on pipeline RBM ...............................................................................85 5.4 RBM framework for the decision of whether to repair or replace .........................88 5.5 Implementation of the RBM framework ................................................................91

5.5.1 Corrosion growth modelling ..........................................................................91 5.5.2 Determining the time to failure and the expected number of failures ...........97 5.5.3 Expected costs for the RBM framework ........................................................99

5.6 Numerical example of a corroding upstream oil pipeline ....................................102 5.7 Conclusion ...........................................................................................................114

6 GENETIC ALGORITHM FOR INSPECTION AND REPAIR PLANNING OF

DETERIORATING STRUCTURAL SYSTEMS: APPLICATION TO PRESSURE

VESSELS ................................................................................................................116

6.1 Abstract ................................................................................................................117 6.2 Introduction ..........................................................................................................117

6.3 Developing the objective function .......................................................................121 6.3.1 Generic objective function ...........................................................................121

6.3.2 Objective function for RBM of a corroding pressure vessel .......................122 6.4 Optimization with a genetic algorithm ................................................................125

6.5 Numerical example of a corroding pressure vessel .............................................128 6.5.1 Small solution space example ......................................................................129 6.5.2 Large solution space example ......................................................................132

6.6 Conclusion ...........................................................................................................135

7 AN EFFICIENT RISK-BASED DECISION ANALYSIS OF THE OPTIMAL NEXT

INSPECTION TYPE FOR A DETERIORATING STRUCTURAL SYSTEM ....137 7.1 Abstract ................................................................................................................138

7.2 Introduction ..........................................................................................................138 7.3 Methodology ........................................................................................................143

7.3.1 Value of information ....................................................................................143 7.3.2 Isolating the decision of the inspection type for the next inspection ...........145 7.3.3 Implementation of the methodology ............................................................153

7.4 Numerical example of a corroding pipeline.........................................................156 7.5 Conclusion ...........................................................................................................162

8 CONCLUSION .......................................................................................................164 8.1 Results ..................................................................................................................164 8.2 Contributions........................................................................................................166

vii

8.2.1 Scientific knowledge ....................................................................................167 8.2.2 Applications in practice ...............................................................................169

8.3 Limitations and future work.................................................................................171

REFERENCES ................................................................................................................173

COPYRIGHT PERMISSION ..........................................................................................185

viii

List of Tables

Table 2.1. Overview of the applied methodologies. Brackets show the method used in the

example problem. .................................................................................................................. 27

Table 3.1. Posterior mean and COV of the unknown model variables. ........................................ 44

Table 4.1. Inspection results of remaining wall thickness and length of corrosion defects. ........ 72

Table 4.2. Input variables for the corrosion growth model, reliability analysis, and decision

making analysis. .................................................................................................................... 73

Table 4.3. Posterior mean and confidence interval of the corrosion growth model variables. ..... 75

Table 4.4. Optimal repair plans for each analysis method; NR is ‘no repair.’ ............................. 77

Table 5.1. Results of the corrosion growth model and reliability analysis. ................................ 106

Table 5.2. Top three maintenance strategies for each of the three cases of fixed service life as

well as the uncertain service life. Gray segments are replaced and white segments are

repaired. Analysis is performed with the base case of input parameters. ........................... 111

Table 6.1. Input variables for the decision analysis. ................................................................... 130

Table 7.1. Boundaries of the absolute VoI from the next inspection for different inspection

types. ................................................................................................................................... 160

Table 7.2. Boundaries of the relative VoI from the next inspection for different inspection

types. ................................................................................................................................... 161

ix

List of Figures

Figure 1.1. Optimization problem for inspection and maintenance planning (modified from

Straub, 2004). .......................................................................................................................... 2

Figure 2.1. Flow of risk-based inspection and maintenance planning. The section numbers

correspond to sections in Chapter two, and the shaded area shows the focus areas of this

study. ....................................................................................................................................... 7

Figure 2.2. Decision tree to visualize decision problem. p is the probability of each outcome

occurring as a function of the state of nature X, given the decision d. ................................... 9

Figure 2.3. Generic decision tree for RBI and RBM analysis (modified from Raiffa and

Schlaifer, 1961). .................................................................................................................... 14

Figure 2.4. Categorized examples of model calibration techniques. The methods used in this

study are shaded. ................................................................................................................... 22

Figure 3.1. Decision tree for the pressure vessel repair decision through the service life. t is

the scheduled shutdown times, n is the number of shutdowns, tsl is the service life, CR is

the repair indexed by the repair time, CF is the failure cost. ................................................. 35

Figure 3.2. Decision tree for the pressure vessel repair decision when a repair is required

either during the current shutdown or the next shutdown. .................................................... 37

Figure 3.3. Graphical Hierarchical Model of the corrosion growth process. ............................... 42

Figure 3.4. Prior and posterior cdf of actual corrosion depth X2. ................................................. 44

Figure 3.5. Expected cost of repairing next against the expected cost of repairing now, both

factored by the base cost of repairing now. The point shows the example case. .................. 46

Figure 3.6. Cost of failure against cost of repairing now, both factored by the cost of

repairing at next shutdown. The point shows the example case. .......................................... 47

Figure 3.7. Measured remaining wall thickness at t1 against the inspection cost ratio. The

point shows the example case. .............................................................................................. 48

Figure 4.1. Hierarchical graphical model of the corrosion growth process. ................................. 64

Figure 4.2. Probability of failure using three analysis methods. .................................................. 76

Figure 4.3. Expected lifecycle cost comparison of the three analysis methods to assess the

impact of system reliability. .................................................................................................. 78

Figure 4.4. Expected lifecycle cost of the top 4 ranked repair plans for the system reliability

method; NR is ‘no repair.’ .................................................................................................... 79

x

Figure 5.1. Decision tree for the decision of whether to repair the defects or replace the

segment. ................................................................................................................................ 89

Figure 5.2. Hierarchical graphical model of the segment based corrosion growth process. ........ 93

Figure 5.3. Results of ILI 1 and 2. ILI1 is blue and ILI2 is red. J is the number of defects. ...... 105

Figure 5.4. Exceedance probability plot of the corrosion process for segment 1 at time t2. ...... 106

Figure 5.5. pdfs of the corrosion process for segment 1. ............................................................ 107

Figure 5.6. cdfs of the time to failure and repair for segment 1. ................................................ 108

Figure 5.7. Risk profile for fixed and uncertain service lives. .................................................... 108

Figure 5.8. Sensitivity of the expected costs to the fixed service life. The optimal

maintenance decision for each section of the fixed service life is shown, gray segments

are replaced, white segments are repaired. ......................................................................... 110

Figure 5.9. Sensitivity of the expected costs to the cost of failure. ............................................ 113

Figure 5.10. Sensitivity of the expected costs to the interest rate. .............................................. 113

Figure 6.1. cdfs of failure times for each defect, with the defect numbers shown beside the

lines. .................................................................................................................................... 129

Figure 6.2. Progress of the genetic algorithm and exhaustive search solutions. ........................ 131

Figure 6.3. cdfs of failure time for each defect, with the defect numbers shown beside the

lines. .................................................................................................................................... 132

Figure 6.4. Progress of the genetic algorithm with different crossover rates. ............................ 134

Figure 6.5. Optimal solution for the case with crossover rate of 0.9 and population of 2J.

Repair time of 0 years is the current time, and a repair time of 100 years actually means

never, as 100 years is the end of the service life. ................................................................ 135

Figure 7.1. Time to failure for the prior case and for different inspections plans. VoI areas

shown for inspection and maintenance plan e1. .................................................................. 148

Figure 7.2. Absolute and relative VoI from the next inspection. VoI boundaries shown for

inspection and maintenance plan e1. ................................................................................... 152

Figure 7.3. Time to failure for the prior case and for different inspection accuracies. ............... 158

Figure 7.4. Absolute VoI for the three possible inspection types, with the upper and lower

boundaries shown. ............................................................................................................... 159

xi

Figure 7.5. Relative VoI comparing each of the inspection types to each other, with the upper

and lower boundaries shown. .............................................................................................. 161

Figure 8.1. Contributions of each chapter of this dissertation to the different areas of risk-

based inspection and maintenance planning. ...................................................................... 167

xii

List of Symbols, Abbreviations, and Nomenclature

An effort was made to maintain consistent notation throughout this dissertation; however,

because the study is a collection of distinct papers, and because of the sheer number of variables,

the notation is not always consistent between chapters, but is always consistent within a chapter.

Below are lists of the general and chapter specific symbols, abbreviations, and nomenclature. All

variables are redefined in the text of each chapter.

General symbols

E[] expectation operator

f() probability density function

F() cumulative distribution function

Pr() probability

SD[] standard deviation operator

⋃ union operator

∩ intersect operator

VAR[] variance operator

σ standard deviation

Σ summation operator

μ mean

∫ integration operator

Γ gamma function

* denotes the optimal decision

Symbols for Chapter 2

d decision option

xiii

d maintenance decision plan

e inspection plan for RBI

M total number of decision options

o outcome index

O total number of outcomes

p probability of a specific outcome

u utility of a specific outcome

X state of nature

Y random inspection result


C total cost

CF failure cost

CR repair cost

d1 decision option 1: repair now

d2 decision option 2: repair at next shutdown

gleak limit state function for leak

i inspection index

n total number of shutdowns

pF probability of failure

R actual remaining wall thickness

t scheduled shutdown times

tinit corrosion initiation time

tsl service life

xiv

W initial wall thickness

xcrit critical defect depth

X actual corrosion depth

Y measured remaining wall thickness

β scale parameter of the gamma process

Δα shape parameter of the gamma process

ΔpF probability of failure for a specific time increment

ΔX corrosion depth growth

ε measurement error

ϴ1 corrosion model multiplier parameter

ϴ2 corrosion model exponential parameter

λ hyper parameters of the hierarchical model


c consequences of a specific outcome

C total cost

CB cost of burst failure

CF cost of failure

CL cost of leak failure

CR cost of repair

d decision option

dcrit critical corrosion depth

D actual corrosion depth

e base of the natural logarithm

xv

gB limit state function for burst failure

gL limit state function for leak failure

i inspection index

I total number of inspections

ID internal diameter of the pressure vessel

j corrosion defect index

J number of corrosion defects

k corrosion length growth factor

L actual corrosion length

m total number of future shutdowns

M Folias factor

n work hardening coefficient

nL number of leaks

o outcome index

O total number of outcomes

p probability of a specific outcome

Papp applied pressure

Pbc burst pressure capacity

pB probability of system burst

Pbu undamaged burst pressure


pL probability of leak for a specific defect

r interest rate for discounting

xvi

R actual remaining wall thickness

Rt remaining thickness ratio

RSF remaining strength factor

t time

t0 corrosion initiation time

tsl service life

tR repair time for a specific defect

tR {tR1, …, tRJ

}, maintenance decision (vector of repair times)

W0 initial wall thickness


X actual corrosion depth

YD measured remaining wall thickness

YL measured corrosion length

β scale parameter of the gamma process

ΔD corrosion depth growth

ΔL corrosion length growth

Δα shape parameter of the gamma process for a specific time increment

ΔpF probability of failure for a specific time increment

εD measurement error in depth

εL measurement error in length

ϴ1 corrosion model multiplier parameter

ϴ2 corrosion model exponential parameter

λ shell parameter

xvii

σu tensile strength


C total cost

CF cost of failure

CL cost of replacement

CR cost of repair

d decision index

D total number of decision options

i inspection index

I total number of inspections

j defect index

J total number of defects

Js total number of defects for segment s

k {sL, sR, tL}, maintenance strategy

L likelihood function

nF number of failures

ntotal total number of trials

o outcome index

O total number of outcomes O


po probability of outcome o

q number of segments to be replaced

r interest rate

xviii

Rs risk of failure for segment s

s segment index

sL set of segments to replace

sR set of segments to repair

S total number of segments

t future time

ti time of inspection i

tL replacement time

tsl fixed service life

TF time to failure

TR time to repair

Tsl uncertain service life

uo utility of outcome o


Xi actual corrosion depth for inspection i

X(t) actual corrosion depth at future time t

Yi measured corrosion depth for inspection i

Yi,j measured corrosion depth for inspection i and defect j

β location specific scale parameter

�̂� point estimate of β

Δαi location and time specific shape parameter for inspection i

Δ�̂�I point estimate of αi for inspection i

ΔXi incremental corrosion growth for inspection i

xix

εi measurement error for inspection i

θ1 multiplier parameter

θ2 exponential parameter

λ {θ1, θ2, β}, hyper-parameters

Ω proportional cost of an additional segment replacement


B benefit provided by the system

C total cost

Cc cost of construction

CF cost of failure

CI cost of inspection

CM cost of maintenance

CP cost of pollution

CR cost of repair

e inspection and maintenance plan

J number of defects

k number of defects requiring repair

K number of defects scheduled for repair at time t

L set of defects scheduled for repair at time t

m number of potential repair times

nF number of failures

pR probability of repair

r interest rate

xx

S subset of k defects selected from the set L of defects scheduled for repair at time t

t time

T failure time

tR repair time

tsl service life

U utility of a system

Ω factor of the cost of each additional repair


C total cost

CF cost of failure

CI cost of inspection

CM cost of maintenance

CR corrosion rate

e inspection and maintenance plan

g limit state function for pipeline leak

i inspection index

k constant

m set of maintenance rules

n total number of inspections

pM probability of maintenance

r interest rate

t time

T failure time

xxi

t0 corrosion initiation time

tconst constraint time on upper boundary time tupper

te elapsed time between corrosion initiation time and now

tint intersection time of the pdfs of VoI1 curves for different plans e

tmax maximum time at which to evaluate the upper boundary of the VoI

tsl service life

tupper min(tsl, tmax, tconst), time at which to evaluate the upper boundary of the VoI

VoI absolute value of information

VoI1U upper boundary on the amount of VoI1 obtained from only the next inspection at t1

VoI1L lower boundary on the amount of VoI1 obtained from only the next inspection at t1

ΔVoI relative value of information

ΔVoI1U upper boundary on the amount of VoI1 obtained from only the next inspection at t1

ΔVoI1L lower boundary on the amount of VoI1 obtained from only the next inspection at t1

X actual deterioration

X0 initial corrosion depth

Xcrit critical deterioration depth

Y measured deterioration

ε measurement error

Abbreviations

API American Petroleum Institute

ASME American Society of Mechanical Engineers

cdf cumulative distribution function

CSA Canadian Standards Association

xxii

DOT Department of Transportation

DT destructive testing

FORM first order reliability method

FPSO floating production, storage, and offloading facility

HBM hierarchical Bayesian model

ILI in-line inspection

pdf probability density function

pmf probability mass function

MFL magnetic flux leakage

MC Monte Carlo

MCMC Markov Chain Monte Carlo

NDT non-destructive testing

PHMSA Pipeline and Hazardous Materials Safety Administration

RBI risk-based inspection

RBM risk-based maintenance

SORM second order reliability method

US United States

UT ultrasonic testing

VoI value of information

wt wall thickness

1

1 INTRODUCTION

1.1 Background / introduction

Structural and infrastructure systems are subject to deterioration processes, such as corrosion

and fatigue, which can lead to failure. To reduce the probability of failure, risk reduction

measures are taken, such as design changes, and inspection and maintenance actions within the

integrity management system. For systems already in operation, design changes are typically not

possible, and inspections and maintenance is the only option.

The goal of the system operator is to minimize the expected total cost of operating the

system. This requires a balance between the benefit of risk reduction measures, and the cost of

these measures. This balance can be viewed as an optimization problem (Figure 1.1). On the

horizontal axis is the system reliability, which is defined as 1 minus the probability of failure. As

the reliability of the system increases, the expected cost of failure decreases, because failures are

less likely. Concurrently, the expected cost of the inspection and maintenance actions increases,

because it is more costly to maintain higher system reliability. This trade-off produces a

minimum in the expected total cost, which corresponds to an optimal inspection and maintenance

plan for the system. Inspection and maintenance planning, whereby the expected cost is

optimized, is termed risk-based inspection (RBI) and risk-based maintenance (RBM) planning

respectively.

2

Figure 1.1. Optimization problem for inspection and maintenance planning (modified from

Straub, 2004).

1.2 Problem statement

The most widely accepted method of determining the optimal inspection and maintenance

plan is lifecycle risk-based inspection and maintenance. In lifecycle RBI and RBM, the optimal

inspection and maintenance plan is the one that minimizes the expected total cost of operating

the system over its entire lifecycle. While lifecycle RBI and RBM are well established

methodologies, the literature detailing how to perform lifecycle RBI and RBM for complex

systems, meaning systems with multiple components that may interact with each other, such as

pipelines and pressure vessels, is still limited. This is because lifecycle RBI and RBM for

complex systems is, well, complicated. As the complexity of the system increases, the difficulty

and computational demand of the analysis increases as well, often exponentially. This can lead to

the case where the lifecycle RBI and RBM methodology is understood in general, but it is not

actually feasible or sometimes even possible to determine the optimal plan. To address this

problem, many studies in the literature make assumptions and simplifications to reduce the

3

difficulty and computational demand of the analysis. However, these assumptions and

simplifications are often unrealistic and can lead to inaccuracy in the analysis.

One common assumption is to restrict the analysis to only one component (or defect) and one

failure mode. Considering only one component and / or failure mode simplifies the analysis in

two ways. First, there are fewer decision options to consider. There is no longer the decision of

which components to inspect or maintain (the extent of the inspection or maintenance). This is a

major simplification, because for systems with multiple components, every combination of

inspection and / or maintenance of every component must be analyzed. Second, the reliability

analysis is simplified, because with only one component and failure mode there is no need to

consider the potential for dependency between the failure events. However, most engineering

systems have multiple components and failure modes, so an analysis of only one component and

failure mode is often unrealistic and impractical for industry use.

Another common assumption is that the lifecycle of the system is deterministic. This

assumption means that the system operator knows exactly when the system will be retired,

without any uncertainty. This assumption is unrealistic, especially when considering structural

systems, which can have lifecycles of 100 years or more. The lifecycle of a structural system

depends on many factors, including the economic feasibility of a project, global economics, the

structural condition of the system, etc., which are out of the control of the system operator. In

order for the decision analysis to be accurate, all uncertainties need to be accounted for,

including the uncertainty in the lifecycle of the system.

Expanding the analysis to include multiple components and failure modes increases the

difficulty of the analysis. For complex systems with many components, solving the lifecycle RBI

or RBM optimization problem becomes computationally demanding, to the point where it

4

becomes impractical. Therefore, the brute force method of analyzing all possible decision

combinations is no longer possible, and more sophisticated techniques must be used.

1.3 Research objectives

The overall objective of this dissertation is to improve RBI and RBM planning to more

realistically, accurately, and efficiently assess deteriorating engineering systems, specifically

pipelines and pressure vessels. The methods all center around risk-based decision making, but

involve different decision questions, engineering systems, and different aspects of the problem.

More specifically, the objectives are grouped into three areas:

1. RBM planning for systems with multiple components and failure modes

The first research area is RBM analysis for systems with multiple components and failure

modes. When considering multiple components and failure modes, the analysis becomes

more difficult, because 1) the number of possible combinations of decision options increases,

and 2) the dependency in the failure events must be accounted for in the reliability analysis.

Specifically, the following research objectives are addressed:

Considering (almost) all of the possible combinations of decision options in RBM

planning for a complex system

Considering the dependency in the failure events between the multiple components and

failure modes

2. Uncertain system lifecycle

The second research area is treating the uncertainty in the lifecycle of the system in the

decision analysis. The lifecycle of a structural system can be long, and depends on many

macro and micro factors that are out of the control of the decision maker, making it

5

uncertain. In the area of uncertain system lifecycles, the following research objectives are

addressed:

Incorporating the uncertainty in the lifecycle of a system into RBI and RBM planning

Assessing the impact of the uncertainty in the lifecycle on the decision analysis

3. Efficient RBI and RBM solutions

This study explores two avenues for improving the efficiency of RBI and RBM planning.

The first avenue is to perform the standard lifecycle RBI or RBM analysis, and then use a

heuristic algorithm to solve the optimization problem in a more computationally efficient

manner. The second avenue is to abandon the lifecycle analysis, and instead restrict the

decision analysis to just the section of the lifecycle decision sequence that is of interest to the

decision maker. In the research area the following objectives are addressed:

Using heuristic algorithms to more efficiently solve the decision optimization problem in

RBI and RBM planning

RBI and RBM planning without a lifecycle analysis

1.4 Statement on the author’s contribution to the research

To address the research objectives, the body of this dissertation comprises five papers: four

journal articles and one conference paper. For each paper, Shane Haladuick was the first author,

and Dr. Markus R Dann was the second author. Shane Haladuick developed the theoretical

methodology, performed the analysis, and wrote each paper. Dr. Dann contributed significantly,

especially in helping develop the methodology, fine tuning the analysis, proof reading the

manuscripts, and through financial and logistical support.

6

1.5 Thesis overview

This dissertation comprises eight chapters. The first chapter provides an introduction to the

topic of risk-based inspection and maintenance planning, and outlines the objectives of the

research. The second chapter presents background information on the entire decision analysis

process, including risk-based decision making, consequence analysis, reliability analysis,

structural deterioration modelling, and pipeline and pressure vessel inspection and maintenance.

Chapters three through seven form the body of the dissertation, where each chapter is an

individual research paper, towards addressing the research questions posed in section 1.3. The

titles of the papers are:

Chapter 3: Risk-based inspection planning for deteriorating pressure vessels

Chapter 4: Risk-based maintenance planning for deteriorating pressure vessels with

multiple defects

Chapter 5: Decision making for long term pipeline system repair or replacement

Chapter 6: Genetic algorithm for inspection and repair planning of deteriorating structural

systems: Application to pressure vessels

Chapter 7: An efficient risk-based decision analysis of the optimal next inspection type

for a deteriorating structural system

Finally, Chapter eight, the conclusion, presents the results, contributions, and limitations of the

research.

7

2 BACKGROUND

This section provides general background information on the overall process of risk-based

decision making. In addition to this section, Chapters three through seven each contain a

background / literature review relevant to the specific methodology in each chapter. Figure 2.1 is

a flow chart of the risk-based decision making process, including the section numbers

corresponding to where each element of the flow chart is found within Chapter two. The shaded

section of the flow chart shows the main areas of research focus in this study.

Figure 2.1. Flow of risk-based inspection and maintenance planning. The section numbers

correspond to sections in Chapter two, and the shaded area shows the focus areas of this study.

8

The risk-based decision making process starts with information on the engineering system,

which is obtained from previous inspections. This information provides knowledge of the past

and present states of the system. This information is then input into the structural deterioration

model (section 2.4) to determine the future states of the system. The future states of the system

are inputs into the structural reliability analysis (section 2.3) to determine the probability of

system failure as a function of time. Simultaneously, the consequences (section 2.2) of system

failure are analyzed. The reliability and consequences are inputs into the decision analysis

(section 2.1) which determines the optimal inspection and / or maintenance plan for the system.

The inspection and maintenance plan is then enacted forming a set of inspection and

maintenance actions (section 2.5). The inspection actions yield new information, which then

updates the current state of the system, and the process repeats.

2.1 Decision analysis

This section provides an overview of decision analysis, also known as utility theory. This

concept was first introduced by von Neuman and Morgenstern (1947), and has been described in

many texts since (Luce and Raiffa, 1957; Jordaan, 2005; JCSS, 2008; Gelman et al., 2014).

In society, decision makers are tasked with making decisions governing societal

infrastructure. These decisions involve all phases of the lifecycle of the infrastructure; including

planning, design, manufacture, inspection, maintenance, operations, etc. The decision maker

must select the “best” choice from the available options. If all aspects of the system were

deterministic, then the decision process would simply involve selecting the option that provided

the highest utility for the decision maker, where the utility is a quantification of the preferences

of the decision maker.

9

However, in reality, all systems operate under uncertainty (JCSS, 2008; Faber, 2005;

Jordaan, 2005). This uncertainty means that when a particular decision option is selected and

some action is taken accordingly, the outcome of the decision and corresponding action is not

known deterministically. In decision analysis, this uncertainty is addressed by using a

probabilistic analysis, where the variables are treated as random with associated probability

density functions (pdfs), or mass density functions (pmfs) in the discrete case. This concept is

most easily visualized in the form of a decision tree (Figure 2.2), showing the decision options d,

the possible outcomes o of each decision option, and the corresponding probability p and utility u

of each outcome. The probability p is a function of the state of nature X. Here the decision tree

for the discrete d and o is shown; however, the same framework applies for continuous decision

options and outcomes. There is some helpful terminology when navigating a tree: a node is an

element of the tree that is connected by branches. Parent nodes are directly superior in the tree

hierarchy to their child nodes. A node that does not have a child node is called a leaf node

(Breiman et al., 1998).

Figure 2.2. Decision tree to visualize decision problem. p is the probability of each outcome

occurring as a function of the state of nature X, given the decision d.

10

The best decision option 𝑑∗ = (E[𝑢(𝑑)])𝑑𝑚𝑎𝑥 is the one that maximizes the expected utility.

The expected utility of decision d3 is computed as per Equation (2.1). This process is repeated to

yield an expected utility for d1 to dM, and the decision with the highest expected utility is

selected. The expected utility of decision option d is given by the following:

E[𝑢(𝑑)] = ∑ 𝑝𝑜(𝑋|𝑑)𝑢𝑜(𝑑)𝑂𝑜=1 (2.1)

where there are o (o = 1, …, O) outcomes of decision option d, and po and uo are the respective

probability and consequences of outcome o. The probability po is a function of the state of nature

X given the decision d.

Utility is quantified in terms of attributes, which are criteria used to assess the benefits and

consequences of an outcome (JCSS, 2008). The simplest example of an attribute is economic

benefit and cost, and an analysis based solely on maximizing the economic gain would be a uni-

attribute decision analysis. Multi-attribute decision analysis considers additional attributes, for

example time to project completion, environmental or societal consequences, project aesthetics,

etc. The standard approach in decision analysis literature (Rackwitz, 2006; Xu, 2015) is to

monetize the attributes to yield a uni-attrbitute analysis. A utility function is used to transform

the consequences of an outcome into the utility specific to the decision maker. In the literature it

is also standard practice to assume a risk-neutral decision maker, allowing a 1:1 transformation

from consequences directly to utility.

2.1.1 Bayesian decision analysis

Bayesian decision analysis (Gelman et al., 2014) is an extension of decision analysis,

providing the ability to update the probabilities of each outcome based on new information that

becomes available. There are three types of Bayesian decision analysis: prior, posterior, and pre-

11

posterior. Prior decision analysis is the straightforward case of decision analysis as shown in

Figure 2.2. A decision d must be made, and all relevant information on the true state of nature X

is known, i.e. probabilities p(X|d) are available. As noted in Straub (2004), this does not mean

that the true state of nature is known, just that it is not possible or reasonable to learn more about

the state of nature before decision d is made. The decision d* that maximizes the utility for the

decision maker is then selected.

Examples of prior decision analysis can be found in the literature of essentially any field.

Classically, von Neuman and Morgenstern (1947) developed decision analysis with applications

to game playing and economics. This methodology became the cornerstone of game theory

(Nash, 1950; Schelling, 1960) with applications to business strategy (Camerer, 1991), politics

and social sciences (Shubik, 1984), and biology (Brown, 1987) among others. Another example

of common application is health care, for instance Weinstien and Stason (1977) used prior

decision analysis to assess the type of medical intervention to perform on a patient.

Posterior decision analysis assesses the expected utility of each decision; however, in this

case additional information is available, such as a test result. Posterior decision analysis uses

Bayes rule to update the true state of nature X given the new information y:

𝑓𝑋|𝑌(𝑥|𝑦) =𝑓𝑌|𝑋(𝑦|𝑥)𝑓𝑋(𝑥)

𝑓𝑌(𝑦) (2.2)

where fX|Y(x|y) is the posterior pdf of X, given Y = y, a new piece of information such as a certain

test result; fY|X(y|x) is the pdf of that information occurring given that X = x, and is known as the

likelihood; fX(x) and fY(y) are the pdfs of X and Y respectively. fX(x) is known as the prior as it is

the pdf of X prior to the new information. The posterior probabilities p(X|d,y) are then

determined from the updated true state of nature, and the optimal decision is determined

following the same method as prior decision analysis.

12

Posterior decision analysis has application whenever it is necessary to make a decision

incorporating new information as well as the existing prior knowledge. In the literature, posterior

decision analysis has been used in many fields. In the health care field, Donner (1982) used

posterior decision analysis to determine how to interpret results of subgroups in clinical trials, by

combining the observed results with the prior belief of the impact of a treatment or variable. In

biology, Aukema et al., (2011) used posterior decision analysis to incorporate both expert

opinion and observations to inform policy decisions regarding invasive insect species. In

resource management, Punt and Hilborn (1997) analyzed the optimal management actions based

on a posterior analysis combining the general prior knowledge of fishery stock and specific

observations of an individual stock in question. In the political science field, Cameron et al.

(2000) use posterior decision analysis to assess how the Supreme Court chooses which cases to

review, based on updating the probabilities with the decisions of the lower courts.

Pre-posterior decision analysis (Raiffa and Schlaifer, 1961) examines whether the decision

maker should use the resources available to obtain additional information. For instance, the

decision maker could choose to pay money to perform a test to gather additional information.

The information gained from the test would then be used to update the probability of different

outcomes occurring, and the decision maker may arrive at a different decision. If the cost of the

test is low enough, the pre-posterior analysis will determine that the added value of the new

information is worth the cost of the test. In this way, pre-posterior decision analysis is ideally

suited to determine whether an inspection should be undertaken or not. This framework can then

be expanded in many ways, for instance to compare different potential inspections, by treating

each inspection as a decision option and determining which option yields the lowest expected

cost.

13

In the literature, pre-posterior analysis is commonly applied in studies that optimize a plan

for gathering information, of which RBI is a prime example. Aside from RBI, pre-posterior

analysis was used by Marin et al. (1999) to develop a water sampling plan to determine the

effects of waste on groundwater quality. Dittes et al. (2017) used pre-posterior analysis to plan

flood infrastructure design, assessing the value of flexibility in the design to accommodate the

uncertainty in future learning and climate change. Hobbs (1997) used pre-posterior analysis to

determine the value of information in assessing the impact of climate change on water resources.

2.1.2 Risk-based inspection (RBI) and maintenance (RBM) planning

RBI and RBM involve the application of decision analysis to the fields of inspection and

maintenance planning respectively. The goal of RBI is typically to optimize the timing, type, and

extent of inspections, and likewise the goal of RBM is typically to optimize the timing, type, and

extent of the maintenance actions. With respect to RBI, Straub (2004), Rackwitz et al. (2005),

and Straub and Faber (2005) provide detailed descriptions of the application of lifecycle RBI

analysis to structural systems. Specifically for RBM, Kahn and Haddara (2003) provide one of

the earlier applications to a structural system, with an example for a heating, ventilation, and air

conditioning (HVAC) system. This section provides a high level review of RBI and RBM. A

more detailed review of the specific literature relevant to each RBI and RBM methodology

developed in this study is provided in each of Chapters three through seven.

Figure 2.3 is a generic decision tree for RBI and RBM. As shown, RBI uses pre-posterior

decision analysis to determine the optimal inspection plan e, and RBM uses prior or posterior

decision analysis to determine the optimal maintenance plan d. As shown in Figure 2.3, prior or

posterior RBM analysis is a subsection of the larger pre-posterior RBI analysis.

14

Figure 2.3. Generic decision tree for RBI and RBM analysis (modified from Raiffa and

Schlaifer, 1961).

RBM analysis (Chapters three through six) uses prior or posterior decision analysis to

determine the optimal maintenance decision. This study uses posterior decision analysis instead

of prior decision analysis because it allows the analysis to incorporate new information in the

form of inspection data. For example, for pipeline corrosion this allows the prior probability that

a defect will fail to be updated to reflect the new information in the form of an inspection result

of the size of the defect. The maintenance decision d can include all combinations of

maintenance time, type, and extent, where type can differentiate between maintenance options

that differ in performance and cost. The optimal maintenance strategy d* is the strategy that

maximizes the expected utility.

RBI analysis (Chapters six and seven, but mainly Chapter seven) uses pre-posterior analysis

to determine the optimal inspection plan. Pre-posterior analysis is used because RBI is concerned

with deciding whether the information gained from a certain inspection plan warrants the cost of

the plan, which is the ideal application of pre-posterior analysis. A generic decision tree for RBI

is shown in Figure 2.3, where e is the entire set of inspection possibilities, including all

15

combinations of inspection number, time, extent, and type; Y is the inspection result, e.g. for

pipeline corrosion this includes indication or not of a defect, and if so the measured defect size; d

is the maintenance decision, which can include all combinations of maintenance time, extent, and

type, and ufail and usurvive are the utility of failure and survival respectively. The optimal

inspection strategy e* is the one that maximizes the expected utility for the pre-posterior tree.

The optimal inspection strategy e* has a corresponding optimal maintenance strategy d*.

A special case of utility maximization is when all of the aspects of an outcome are

consequences. In this case, instead of viewing the decision as maximizing the expected utility, it

can also be viewed as minimizing the expected consequences, also known as the risk. This

typically applies for RBI or RBM, where the benefits of each decision option are independent of

the decision option, e.g. the benefits of a pipeline are constant regardless of the inspection and

maintenance plan.

2.2 Consequence analysis

Consequence analysis (JCSS, 2008) is an important aspect of decision analysis. In RBI and

RBM planning the consequences include inspection and maintenance cost, and failure cost

(Straub, 2004; Rackwitz et al., 2005). Determining the inspection and maintenance cost is

typically straightforward, and most system operators will have a relatively accurate estimate of

these costs. Determining the cost of failure is more difficult (Zhou and Nessim, 2011). The cost

of failure includes many attributes, for instance the economic cost of restoring the failed system,

environmental costs, and human casualties. The failure cost can also include less obvious

indirect costs, such as damage to a company’s reputation, or resulting changes in government

regulation or policies (JCSS, 2008). Modelling the consequences of system failure is a large

16

research topic unto itself and is beyond the scope of this research. Instead, this study assumes

monetary costs for the consequences, and performs sensitivity analyses to determine the impact

of variation in these costs on the decision analysis.

2.3 Structural reliability analysis

Reliability analysis is concerned with determining the reliability of a system, where

reliability is defined as 1 minus the probability of failure. Reliability analysis is divided into two

main streams: classical reliability analysis and structural reliability analysis. Classical reliability

analysis is used when there is empirical failure data available for a given system. This is the case

for a system with many identical replicas, where it is reasonable to test the system to failure, e.g.

a lightbulb. For the example of a lightbulb, classical reliability theory would test many lightbulbs

until failure, yielding a distribution of the failure time of a lightbulb. Structural reliability

analysis (Melchers, 1999) is used to determine the probability of failure for a structural system.

Unlike for a lightbulb, structural systems are unique (no two buildings are identical), and testing

the system until failure is impractical. Additionally, structural system failure generally occurs

under extreme loads. Consequently, there is no empirical failure data available. Instead,

structural reliability analysis uses limit state functions to model the resistance and load of the

system and predict the probability that failure will occur.

The limit state function for any system has the general form of the resistance to failure minus

the load on the system. When the load exceeds the resistance, the limit state function has a value

of less than zero, and the system is in failure. For a deterministic analysis, the values of the

system are input into the limit state function, and the result is either failure or safety. For a

17

stochastic analysis, at least one of the input variables is random, and the probability that the limit

state function is below zero is the probability of system failure.

Depending on the limit state function, there are several possible approaches to determining

the probability of failure. In certain, and typically more simplistic (fewer variables with specific

distribution combinations) cases, there may be a closed form solution available, where the

probability of failure can be determined by mathematical derivation. For more complicated

functions this is likely not the case, and a typical solution requires a numerical method, for

instance approximation with first or second order reliability methods (FORM or SORM

respectively) (Rackwtiz and Flessler, 1976; Hasofer and Lind, 1974; Der Kiureghian et al.,

1987), or Monte Carlo (MC) simulation (Metropolis and Ulam, 1949). For the more involved

reliability analysis (Chapters three and four) this study uses MC simulation. While MC

simulation is not as efficient as an approximation method such as FORM or SORM, it is

straightforward, and accurate if the sample size of the simulation is large enough. Since the focus

of this study is on risk-based decision making and not on reliability analysis techniques, the

greater simplicity of the MC approach was deemed advantageous. For the simpler reliability

analysis (Chapters five and seven) this study uses numerical approximation of the integrals by

discretization.

Reliability analysis can also be grouped into either time independent or time dependent

problems (Melchers, 1999). Time independent problems are more straightforward, and occur

when the resistance to failure and the load are independent of time. Time dependent problems,

where the resistance and / or load are time dependent, are more difficult. In this case the limit

state function, and consequently the reliability, need to be expressed as functions of time. This is

18

often difficult for continuous time, and a common simplification is to discretize time and

evaluate the limit state function for a set of discrete time intervals.

This study is concerned with deterioration (specifically corrosion) failure of oil and gas

systems, such as pipelines and pressure vessels. The CSA (2012) defines two main failure modes

for corroding pipelines: leak and burst. A leak occurs when the maximum depth of the corrosion

defect exceeds the wall thickness. Thus, a leak occurs if the defect is short enough to corrode

through the wall without bursting. For leaks, the resistance to failure is the initial wall thickness,

and the load is the depth of the corrosion defect. Therefore, the resistance to failure is time

independent, but the load is monotonically increasing with time. A burst occurs when the pipe

wall undergoes plastic collapse due to the internal pressure from the fluid in the pipeline. Thus,

bursts occur for defects that are long enough to violate the burst criterion before the defect

corrodes through the wall. For bursts, the resistance to failure is the remaining steel structure that

is resisting the burst, and the load is the internal applied pressure. Therefore, the resistance to

failure is monotonically decreasing with time, and the load is a random function of time.

Depending on the application, this study considers one or both of these failure modes.

2.4 Structural deterioration modelling

To evaluate the time dependent limit state function for deteriorating oil and gas systems, the

state of the deterioration as a function of time is required. This study is specifically concerned

with deterioration due to corrosion; therefore, the state of corrosion as a function of time is

desired. A corrosion growth model is used to model the state of corrosion through time. This

section comprises two parts. The first part provides a general overview of the important

19

considerations in corrosion growth modelling and the main model types. The second part

describes several methods of calibrating the corrosion growth model.

2.4.1 Corrosion growth modelling

The goal of corrosion growth modelling is to gain insight into the future state of corrosion. In

general, corrosion growth models can be divided into three broad categories (Nešić, 2007):

mechanistic (or physical) models, semi-empirical models, and empirical models. Mechanistic

models are based on electro-chemical theory, and the variables in these models typically have

clear physical meaning. Mechanistic models are based on equations describing the chemical

reactions, movement of electrons, and mass transport in the corrosion process. To further

improve the accuracy of the mechanistic models it is also possible to use input data from fluid

flow models. These equations can then be combined with models of the fluid flow rate (Deng et

al., 2006). The goal of mechanistic models is to obtain the rate of corrosion growth for a specific

set of operating conditions (including pressure; temperature; and flow rates of oil, gas, and

water). While mechanistic models are still more accurate when calibrated with empirical data

from a specific system, they do not necessarily require large amounts of data to be accurate.

Empirical models are on the other end of the spectrum from mechanistic models, having very

little or no basis in electro-chemical theory. Instead, these models rely on empirical data to

calibrate the model, and the more data available, the more accurate the model. However, the

accuracy of these models is typically reserved to the set of operating conditions that the input

data is from; outside of these conditions these models are less accurate. The variables in these

models typically do not have a clear meaning, and instead are based on fitting the model to the

20

available data. As the name implies, semi-empirical models are a hybrid of these two, having

some basis in theory, but also have some variables based purely on data.

This study uses a subset of empirical corrosion growth models known as probabilistic

corrosion growth models. A probabilistic corrosion growth model uses previous knowledge of

the corrosion, combined with information obtained during inspections, to describe the corrosion

process over time (Maes et al., 2009). Structural deterioration processes, such as corrosion, are

stochastic in nature (Dann, 2011; Straub, 2004), meaning the future extent of the deterioration is

probabilistic as opposed to deterministic, and involves a high degree of uncertainty. These

uncertainties are grouped into four categories: 1) spatial, 2) temporal, 3) inspection, and 4)

model. Spatial uncertainty is due to the difference between the corrosion at different defect

locations at any given time. This is due to many factors, including material and environmental

heterogeneity, as well as randomness. Spatial uncertainty can be addressed by spatially

discretizing the corrosion model to represent the spatial heterogeneity of the corrosion. Temporal

uncertainty is the uncertainty in predicting the future deterioration based on the present

deterioration, and is due to the stochastic nature of the corrosion process. Inspection uncertainty

is due to several sources, including measurement error, not detecting an existing defect (misses),

detecting a fictitious defect (false call), and truncation due to reportability. Model uncertainty is

due to the inability of a model to accurately represent the actual phenomena.

There are many corrosion growth models suggested throughout the literature. The reader is

referred to Pandey et al. (2009) and Brazán and Beck (2013) who perform in depth comparisons

of corrosion growth models. These sources divide the models into two main categories: random

variable (or corrosion rate) and stochastic process models. Stochastic process models have been

shown to be superior to random variable models because they more adequately account for the

21

temporal uncertainty (Pandey and van Noortwijk, 2004), are less conservative (Zhou et al., 2012;

Brazán and Beck, 2013), and are more accurate at fitting actual corrosion data (Brazán and Beck,

2013).

Depending on the application and the objective, different chapters in this dissertation use

different corrosion growth models. Chapters three, four, and five use a stochastic process to

model the corrosion growth. Chapter six does not use a corrosion growth model, instead

assuming a distribution for the failure time. For simplicity, Chapter seven uses a corrosion rate

model. A more detailed review of corrosion growth modelling specific to each methodology is

provided in Chapters three, four, five, and seven.

2.4.2 Model calibration / parameter estimation

Once the corrosion growth model is developed, the next step is model calibration, also

known as parameter estimation. Treating the model parameters as uncertain addresses the

epistemic uncertainty due to model error. Figure 2.4 shows some examples of model calibration

techniques, with the techniques used in this study highlighted. In general, model calibration can

either be performed numerically or analytically. Examples of analytical techniques include

closed form solutions, maximum likelihood estimation, and the method of moments. Examples

of numerical techniques include expectation maximization algorithm, variational inference, and

Markov Chain Monte Carlo (MCMC).

22

Figure 2.4. Categorized examples of model calibration techniques. The methods used in this

study are shaded.

For the analytical techniques, a closed form solution is when there is a mathematical solution

to determine the parameters of the model. When there is no closed form solution available

another technique must be used. In maximum likelihood estimation (Walpole et al., 2007), the

point estimates of the model parameters are the parameters that maximize the likelihood

function. The likelihood function is a function of the parameters of the distribution given the

observed corrosion data, and mathematically is given by the product of the probability of

observing each instance of the corrosion data. The likelihood function is maximized by taking

the derivative of the likelihood function with respect to each of the unknown parameters, and

setting the derivative to zero. The method of moments (Bowman and Shenton, 2006) makes

point estimates of the model parameters by determining the moments of the observed corrosion

data (the first moment is the mean, the second moment is the variance) and setting the model

23

parameters to satisfy these moments. The number of moments required is the same as the number

of parameters being estimated.

There are two main reasons to use numerical techniques, 1) when analytical techniques are

unable to provide a solution, or 2) when the entire distribution of the parameters is desired, and

not just a point estimate. As an example of the first reason, when the maximum likelihood

method is used to derive equations for a set of unknown parameters, it can be the case that the

resulting system of equations has no analytical solution. In this case expectation maximization

(Moon, 1996) can be used to solve the system of equations by assuming starting values and

iteratively updating one value each iteration. In general it is more accurate to determine the entire

distribution of the model parameters and not just point estimates. Thus, for the second reason, to

determine the entire distribution of the parameters, a more sophisticated method, such as

variational inference or Markov Chain Monte Carlo (MCMC), is required. Variational inference

(Hoffman et al., 2013) is an extension of expectation maximization, whereby the distribution of

the entire likelihood function is estimated and not just the maximum point. MCMC (Gilks,

2005), as with standard Monte Carlo, uses sampling to determine the distribution of the

parameters. The difference is that in MCMC the distributions of the parameters are unknown;

therefore, MCMC uses a Markov Chain to determine the sample values.

Chapters three and four use MCMC simulation to determine the entire distributions of the

parameters of the stochastic process, using a software package called OpenBUGS (Spiegelhalter

et al., 2006). Chapter five uses the method of moments to determine the initial point estimates of

the parameters of the stochastic process, then uses sum of squares optimization to adjust the

parameters to yield a better fit of the upper tail of the corrosion distribution. However, Chapter

five assumes fixed model parameters, and thus does not account for model error as in Chapters

24

three and four. Chapter six does not use a corrosion growth model, instead assuming a

distribution of the failure time. Chapter seven uses a corrosion rate model, but does not perform

parameter estimation, instead assuming fixed parameters for the model.

2.5 Inspection and maintenance

Depending on the engineering system, there are different types of inspection and

maintenance methods available. This section specifically describes the methods for pipelines and

pressure vessels, first for inspections and then for maintenance.

2.5.1 Inspections

Testing techniques are divided into two groups: destructive testing (DT) and non-destructive

testing (NDT). As the name implies, the difference is that DT damages the system during the test

and NDT does not. For pipelines and pressure vessels, common DT techniques are pressure and

hardness tests, and common NDT techniques are magnetic flux leakage (MFL), ultrasonic (UT),

and local inspection. For safety reasons, only NDT is used on pipelines and pressure vessels in

operation, and because this study is concerned with optimizing system operation, only NDT is

considered in this study.

MFL tools, as described in Mandal and Atherton (1998), detect defects using an array of

permanent magnets, which magnetize the wall of the pipe or pressure vessel to near saturation

flux (the magnetization of the wall is nearly maximized). A defect in the wall, such as a

corrosion pit, cause magnetic flux leakage, and this flux leakage is then measured by the tool.

Consequently, these tools can only be used in steel pipes and pressure vessels where the wall

25

thickness is not too thick so that it can be magnetized to saturation. Post processing converts the

flux leakage into a relative measure of defect depth (percentage of wall thickness).

UT tools, as described in Reber et al. (2002), generate an ultrasonic pulse, which are sound

waves at a frequency above human hearing. Part of the pulse is reflected by the inner surface of

the wall, while the rest of the pulse travels through the body of the wall and is reflected by the

outer surface. The reflected pulses are then received by the tool, and the transmission times are

recorded. These times are then used to determine an absolute measure of the wall thickness at all

points, providing an absolute measure of defect depth. For internal pipeline corrosion, UT tools

require a coupling medium between the tool and the wall, so they can only be used in liquid

pipelines, or in a batch of liquid within a gas pipeline. Because UT tools do not rely on

magnetization, they can be used for many types of material, and with much thicker walls than

MFL tools.

The other common method of inspection is local inspection, which is simply a hands on

measurement of the defect size using calipers or a hand held UT tool. This measurement can be

very accurate; however, for systems with many defects this approach is impractical, and it is

typically necessary to target only the severe defects for local inspection, meaning the defects that

are expected to fail first.

2.5.2 Maintenance

Most engineering systems have several different maintenance options available. In some

cases a certain technique must be used for a certain type of defect, but in other cases the decision

maker is free to choose between a set of options. These options can vary in terms of the quality,

extent, and cost. The maintenance options need to be assessed for the specific system being

26

analyzed, and this section provides a review of the maintenance options for pipelines and

pressure vessels.

A thorough review of pipeline repair methodologies is already provided in Chapter five;

however, this section provides a brief overview based on ASME B31.4 (ASME, 2012). A single

defect in a pipeline can either be repaired locally, or an entire section of the pipe can be cut out

and replaced with a new section. Repairing a single defect is less expensive than replacing a

section of pipe; however, the replacement will likely last longer (higher quality) and have a

greater extent (a larger area of the pipe is maintained). Local repairs typically involve either

welding a sleeve or bolting a clamp over the pipe at the location of the defect to reinforce the

pipe wall. Thus the repair reinforces the single defect, but does not affect the rest of the pipe

segment. The costs and performance of welded sleeves and bolted clamps are relatively similar.

Pressure vessels (API, 2007) can either be repaired, or the entire vessel can be replaced. For

repairs, either a plate can be welded on top of the defect, or a small section of the wall can be cut

out, and a new section can be welded in. The cost and effectiveness of welding a plate or

replacing a small section of the wall are relatively similar.

2.6 Overview of the applied methodologies

The thesis comprises a collection of five papers that address different problems within the

field of risk-based inspection and maintenance planning. Consequently, some aspects of the

methodologies are used in multiple chapters, and other aspects are unique to each chapter. For

clarity, Table 2.1 provides a summary of the important aspects of the methodologies and which

chapter uses each.

27

Table 2.1. Overview of the applied methodologies. Brackets show the method used in the

example problem.

Chapter 3 4 5 6 7

RBI or RBM RBM RBM RBM either RBI

Decision maintenance

time

maintenance

time

maintenance

type

generic

(maintenance

time)

inspection

type

System pressure

vessel

pressure

vessel

pipeline generic

(pressure

vessel)

generic

(pipeline)

Fluid gas gas liquid gas liquid

Components single multiple multiple multiple generic (leak)

Failure mode leak leak and burst leak generic

(leak)

generic (leak)

Corrosion

growth model

stochastic

process

stochastic

process

stochastic

process

N/A generic

(corrosion

rate)

Parameter

estimation

MCMC MCMC method of

moments

N/A assumed

Reliability

analysis

MC MC numerical

approximation

N/A numerical

approximation

28

3 RISK-BASED INSPECTION PLANNING FOR DETERIORATING PRESSURE

VESSELS

This chapter presents an RBM methodology for determining the optimal repair time for a

deteriorating pressure vessel. This chapter contributes to the research area of efficient RBI and

RBM planning. Specifically, the methodology addresses the research objective: RBI and RBM

planning a without a lifecycle analysis.

The novel contribution of this chapter is the methodology to restrict the sequential lifecycle

decisions to only the decision to either repair at the current time, or the time of the next

inspection. Because of this restriction, the analysis is only applicable to a severe defect, which is

defined as a defect that requires repair in the near future. However, the restriction on the set of

repair times shrinks the lifecycle decision sequence down to only the decision of whether to

repair now, or at the next scheduled shutdown, greatly simplifying the analysis. Thus, this paper

presents a method that is simple for practical application by pressure vessel operators.

This chapter is a conference paper (Haladuick and Dann, 2016a), which was published in the

proceedings of the ASME Pressure Vessels and Piping Conference, in Vancouver, BC, July

2016, in the Non-destructive Evaluation, Diagnosis, and Prognosis Division. It is included with

the permission of the copyright holder ASME, and the published version can be accessed here:

http://proceedings.asmedigitalcollection.asme.org/proceeding.aspx?articleid=2590409

29

3.1 Abstract

Pressure vessels are subject to deterioration processes, such as corrosion and fatigue. If left

unchecked these deterioration processes can lead to failure; therefore, inspections and repairs are

performed to mitigate this risk. Oil and gas facilities often have regular scheduled shutdown

periods during which many components, including the pressure vessels, are disassembled,

inspected, and repaired or replaced if necessary.

The objective of this paper is to perform a decision analysis to determine the best course of

action for an operator to follow after a pressure vessel is inspected during a shutdown period. If

the pressure vessel is inspected and an unexpectedly deep corrosion defect is detected an

operator has two options: schedule a repair for the next shutdown period, or perform an

immediate unscheduled repair. A scheduled repair is the preferred option as it gives the decision

maker lead time to accommodate the added labour and budgetary requirements. This preference

is accounted for by a higher cost of immediate unscheduled repairs relative to the cost of a

scheduled repair at the next shutdown. Depending on the severity of deterioration either option

could present the optimal course of action. In this framework the decision that leads to the

minimum expected cost is selected. A stochastic gamma process was used to model the future

deterioration growth using the historical inspection data, considering the measurement error and

uncertain initial wall thickness, to determine the probability of pressure vessel failure. The

decision analysis framework can be used to aid decision makers in deciding when a repair or

replacement action should be performed. This method can be used in real time decision making

to inform the decision maker immediately post inspection. A numerical example of a corroding

pressure vessel illustrates the method.

30

3.2 Introduction

Pressure vessels are used in refineries; offshore platforms; floating production, storage, and

offloading facilities (FPSO’s); and other miscellaneous oil and gas facilities to store gas. These

pressure vessels are subject to structural deterioration due to corrosion, which gradually reduces

the wall thickness until failure occurs. The failure of a pressure vessel in an oil and gas facility

can have large social, economic, and environmental consequences. To mitigate this risk the

operators perform regular inspections and repairs of the pressure vessels, along with all of the

other subsystems of the oil and gas facility. These inspections and repair actions require the

facility to be shutdown, and to reduce offline time they are usually coordinated during one

facility wide shutdown. Thus, the times of future inspections and potential repairs are often fixed

and cannot be adjusted.

If a pressure vessel is inspected and an unexpectedly deep corrosion defect is detected the

decision maker has two options: schedule a repair for the next shutdown period, or perform an

immediate unscheduled repair. In the ideal case the repair is scheduled for the next shutdown

period, giving the operator time to allocate labor and materials for the repair. However, if the

defect is severe enough an emergency repair may be required immediately during the current

shutdown period. An immediate emergency repair is more expensive than a repair at the next

shutdown period because the decision maker does not have lead time to mobilize labor forces

and free funding to perform the repair. Also, future repairs are discounted due to the time value

of money. The objective of this paper is to develop a framework to determine the optimal repair

decision for a corroding pressure vessel.

Unfortunately, the decision of whether to perform the repair immediately or at the next

shutdown is complicated by the uncertainty present in the corrosion growth process. For

31

instance, there is uncertainty due to the temporal variability of corrosion growth, inaccuracies in

the measurement tool, and inaccuracies in the corrosion growth model. This uncertainty makes it

difficult for the decision maker to determine their best course of action, as in whether to repair

immediately or delay the repair. Decision analysis is ideally suited for such uncertain scenarios

to aid decision makers in determining the optimal inspection and repair plan. In decision analysis

a decision tree represents the discrete decision options and the outcomes of the decisions. A

probabilistic approach is used to determine the probability of each outcome occurring for a given

decision, and the expected cost of each option is subsequently determined. The option with the

lowest expected cost is selected as the optimal choice. The use of decision analysis to determine

the inspection and repair plan that minimizes the expected consequences (the risk) is termed risk-

based inspection (RBI) and repair planning. This paper uses RBI to determine the optimal repair

decision for the facility operator.

This paper comprises five sections. The second section presents a review of the recent

literature in RBI, especially related to the oil and gas industry. The third section formulates the

decision problem and presents the methodology for addressing this problem with decision

analysis. The fourth section presents the methodology for using the inspection data to model the

corrosion growth process through time to determine the probability of pressure vessel failure.

The fifth section presents a numerical example using a hypothetical pressure vessel to illustrate

the decision making process. The example assumes consequence values for the risk analysis and

performs a sensitivity analysis to assess the impact of these values. Finally, the paper ends with

some concluding remarks.

32

3.3 Literature review

A thorough overview of the history of RBI is provided in (Straub, 2004). More recently,

there have been many studies developing RBI frameworks for structural engineering systems.

RBI was applied to the column supports of a bridge in (Goulet et al., 2015) to determine the

optimal time of the bridge inspections, and whether bridge components should be strengthened.

The optimal type of inspection of a deteriorating steel highway bridge was assessed in (Corotis et

al., 2005). RBI was used to determine the optimal inspection timing and quality for structural

engineering systems subject to fatigue in (Fujimoto et al., 1997). Tang et al. (2015) investigated

a slowly deteriorating system and used RBI to determine the optimal periodic inspection interval

time.

There has also been a lot of progress in the use of RBI in the oil and gas industry. Recently,

several design codes (API, 2009; API, 2016; ASME, 2007) have been developed to govern the

implementation of RBI within the refinery environment. These codes provide a framework for a

decision maker to perform a RBI for their specific refinery system, which includes pressure

vessels. These codes aid in deciding the type of inspection to perform, the extent of the area that

needs to be inspection (the quality of the inspection), and the interval between inspections. While

these codes provide a strong foundation for applying RBI in industry, they have limitations. One

of the main limitations is that the RBI codes use risk limits to aid in decision making, whereby

an action is taken once the risk violates the limit. In theory this is a valid approach; however, in

practice this risk limit is difficult to define with accuracy. A better approach is to determine the

optimal decision as the one that minimizes the overall cost, as this avoids the need for a pre-

defined risk limit. Another limitation is that the RBI codes typically use simplistic models of

corrosion growth which do not account for all of the uncertainties present in the corrosion

33

growth process. The uncertainties in the corrosion growth process are discussed in detail in the

corrosion growth modelling section.

Several studies have overcome this limitation by performing RBI based on minimizing the

lifetime cost for structural systems in the oil and gas industry. For instance, pipeline inspection

and repair has been extensively studied (Hellevik et al., 1999; Pandey, 1998; Sahraoui et al.,

2013; Hong, 1997, 1999; Nessim et al., 2000; Gomes et al., 203; Gomes and Beck, 2014; Zhang

and Zhou, 2014; Goyet et al., 2002; Garbatov and Soares, 2001). Hellevik et al. (1999)

performed RBI for pipelines subject to corrosion, optimizing the timing of inspections, type of

inspection, number of inspections, and the pipe wall thickness during the design phase. After an

inspection the pipe could be replaced, but not repaired, which avoided having to repair multiple

corrosion defects. Replacement was made when the measured defect was greater than the

threshold, which was an optimized parameter. Pandey (1998) performed RBI on pipelines to

determine the optimal inspection time and whether the pipeline should be repaired post

inspection. To simplify the analysis he constrained the problem to only allow one inspection

within the service life of the pipeline. Sahraoui et al. (2013) performed RBI for pipelines subject

to corrosion, and provided a numerical example for a natural gas pipeline. They considered

uncertainties due to the probability of detection, and probability of false detection of defects;

however, they did not account for measurement uncertainty. They optimized the periodic time

between inspections, but the decision of whether to repair the defect was only based on a

reliability threshold instead of optimization. There have also been several studies applying RBI

to FPSO’s. Fatigue deterioration in the welded connections of the hull structure of an FPSO was

examined, to determine the optimal inspection and repair plan (Goyet et al., 2002), and the

optimal period between inspections (Garbatov and Soares, 2001). Within the refinery

34

environment, RBI was used to optimize the periodic inspection frequency of refinery piping

(Chang et al., 2005). With regards to pressure vessels, RBI was used to assess the impact of

increasing the periodic inspection interval for spring operated relief values on pressure vessels at

the Savannah River Site nuclear reservation (Gross et al., 2012).

3.4 Decision analysis framework

Consider a pressure vessel in an oil and gas facility. The pressure vessel was inspected during

the previous shutdown period and no corrosion defects were detected. The vessel was then

inspected during the current shutdown and an unexpectedly deep corrosion defect was detected.

The decision maker is faced with the decision of whether to repair the defect immediately during

the current shutdown, or to not repair the defect and risk that it fails before the next scheduled

shutdown. If the defect is not repaired immediately then at the next scheduled shutdown the

decision maker must again decide whether to repair the defect or to risk it failing before the

subsequent scheduled shutdown. This process repeats indefinitely until the defect is repaired, the

defect fails, or the pressure vessel reaches the end of its service life. This problem can be

modelled with a decision tree (Figure 3.1).

35

Figure 3.1. Decision tree for the pressure vessel repair decision through the service life. t is the

scheduled shutdown times, n is the number of shutdowns, tsl is the service life, CR is the repair

indexed by the repair time, CF is the failure cost.

Following the branches of the tree, if the defect is repaired at t1 the outcome is that the

pressure vessel survives with certainty until the end of the service life tsl. This is based on the

assumption of perfect repair, which means that post repair the defect ceases to exist and cannot

fail. The cost CR,1 of this decision is only the cost of repairing the defect at t1. The repair cost is

dependent on the repair time to account for the potential for the repair cost to vary with time.

There are two potential reasons the cost may vary with time. First, the time value of money

discounts future repairs by the interest rate. Second, if an emergency repair is required

36

immediately at t1, the cost of the repair could be greater because added pressure to mobilize

resources, such as labor and material, to perform the repair. An example of such a scenario is a

facility with many pressure vessels in need of repair. If the facility has a limited budget or labor

force then a delayed repair is desirable. The greater cost of emergency repairs could be viewed as

a penalty on repairs that are required without prior warning.

If the defect is not repaired at t1 there are two possible outcomes; the defect will either

survive until the next scheduled shutdown at t2, or it will fail before then. The probability of each

of these outcomes is required to determine the expected cost of the decision not to repair, and is

discussed in the next section. If the defect fails the branch terminates and the cost is the cost of

failure CF. If the defect survives then at the next shutdown period the decision maker must again

decide whether or not to repair the defect, and the process repeats itself until the end of the

service life of the pressure vessel. While this framework provides a means of determining the

optimal time to repair the defect, the disadvantage is that it requires the analysis to be carried

forward into the future for the entire service life of the pressure vessel. This requires the times of

every shutdown (and corresponding potential repair opportunity), and the service life of the

pressure vessel, to be known years in advance. It is likely that these values are not known with

certainty years in advance, and the uncertainty in these values adds inaccuracy to the analysis.

This paper avoids this issue by considering the case of a corrosion defect severe enough that

it is known the defect will need to be repaired in the near future. The decision then becomes

whether the repair can wait until the next shutdown period or if it needs to be repaired

immediately. In the ideal case the repair can wait until the next scheduled shutdown, giving the

operator time to mobilize their resources, such as labor and material, to perform the repair.

However, if the defect is very severe immediate repair may be required, typically at a higher cost

37

due to the lack of preparation. The advantage of this framework is that only the next shutdown

time is required for the analysis, and none of the subsequent shutdown times or the service life

are required. It is reasonable to expect the operator to know the time of their next facility

shutdown, so the uncertainty is less. Again a decision tree is used to model this scenario (Figure

3.2). The decision tree (Figure 3.2) for this specific case is actually a clipped version of the tree

for the more general case (Figure 3.1), where the option of no repair at time t2 is eliminated and

the decision maker is forced to repair at this node. This in turn eliminates all subsequent no

repair options as well, greatly simplifying the tree. While this is a simplification, for the case of a

severe defect this simplification is minor, as in practice a severe defect will be repaired fairly

quickly.

Figure 3.2. Decision tree for the pressure vessel repair decision when a repair is required either

during the current shutdown or the next shutdown.

38

Again, following the branches of the tree, for decision d1 the defect is repaired now and the

outcome is that the pressure vessel survives with certainty until the next shutdown period. The

expected cost of repairing now is then entirely due to the cost of the repair:

E[𝐶|𝑑1] = 𝐶𝑅,1 (3.1)

For decision d2 the repair is delayed until the next shut down period. If the pressure vessel

survives until the next shutdown period then the defect is repaired at the cost CR,2 of repair at t2.

If the pressure vessel fails before the next scheduled shutdown then the failure cost CF is

incurred. To fully capture the cost of failure, CF includes not only the expected economic cost,

but also a monetization of the expected social cost, such as human casualties, and the

environmental cost. Thus, the expected cost of repairing at the next shutdown is given by:

E[𝐶|𝑑2] = (1 − 𝛥𝑝𝐹,2)𝐶𝑅,2 + 𝛥𝑝𝐹,2𝐶𝐹 (3.2)

where ΔpF,2 is the probability of the pressure vessel failing in the interval from t1 to t2. The

optimal decision is the one that minimizes the expected cost, therefore, if E[C|d1] < E[C|d2] then

decision d1 is the optimal choice, and the defect should be repaired immediately, otherwise the

repair should be postponed until the next inspection period. By populating Equations (3.1) and

(3.2) the optimal repair decision can be determined. These equations show that the values driving

the decision are the costs of repairing immediately, repairing at the next shutdown, and failure, as

well as the probability of failure. Depending on the ratio of these costs the optimal decision will

switch between d1 and d2.

39

3.5 Corrosion growth modelling

There are typically two modes for corrosion failure: leak and burst. A leak occurs when the

maximum depth of the corrosion defect exceeds the wall thickness. Thus, leaks occur for defects

that are short enough to corrode through the wall without bursting. A burst occurs when the wall

undergoes plastic collapse due to internal pressure before the defect grows through the wall. This

paper only considers leak failure, with a limit state function given by:

𝑔𝑙𝑒𝑎𝑘(𝑥𝑐𝑟𝑖𝑡, 𝑋𝑖) = 𝑥𝑐𝑟𝑖𝑡 − 𝑋𝑖 (3.3)

where xcrit is the critical depth and Xi is the defect depth at inspection i, both defined as a

percentage of the wall thickness (% wt). In other words, failure occurs when the defect depth is

greater than the critical depth, and the probability of failure pF = Pr(Xi > xcrit) is the probability

that Xi is greater than the critical depth. For a probabilistic analysis xcrit is usually defined as 100

% wt. Note that since only leak failure is considered, the corrosion growth model is only

concerned with defect depth.

In the scenario considered, the pressure vessel was inspected during the previous shutdown

period at t0, and no corrosion defects were detected. Then the vessel was inspected again during

the current shutdown at t1 and a severe defect was detected. For the decision analysis the

probability of failure ΔpF,2 for the interval from t1 to t2 is required. To determine this value the

depth of the corrosion defect at the next shutdown time t2 is needed. A corrosion growth model is

used to provide insight into the future corrosion depth based on the results of the two inspections.

Structural deterioration processes, such as corrosion, are stochastic in nature, meaning that the

future depth of the corrosion is probabilistic and contains a high degree of uncertainty (Straub,

2004; Dann, 2011). This uncertainty can be grouped into three categories, 1) spatial, 2) temporal,

and 3) inspection. Since this scenario only considers one corrosion defect, the spatial uncertainty

40

is not relevant. Temporal uncertainty is due to the stochastic nature of the corrosion process,

making the prediction of future corrosion based on current corrosion uncertain. Inspection

uncertainty is uncertainty in how well the measured depth of corrosion describes the actual

depth. In the case of UT inspection of a pressure vessel there are two main components of

inspection uncertainty. The first is the measurement error of the inspection tool, which is

relatively low as UT tools are quite accurate. The second is the variability in the wall thickness

of the pressure vessel. The tool measures the remaining thickness of the wall, which is then

subtracted from the original wall thickness to determine the depth of the defect, and variability in

the wall thickness causes uncertainty in the calculated defect depth.

The corrosion growth model needs to appropriately account for these uncertainties. Another

important physical attribute of corrosion growth is that it is always positive and monotonically

increasing with time and the corrosion model should reflect this. Based on these requirements a

stochastic process model based on the homogeneous gamma process (Figure 3.3) has been

shown to be suitable to describe the corrosion growth in steel structures (Pandey and van

Noortwijk, 2004; Pandey et al., 2009; van Noortwijk, 2009). This model is represented by a

hierarchical Bayesian model (HBM) (Koller and Friedman, 2009) (Figure 3.3).

This model has two simplifications; first, the corrosion growth increments are assumed to be

conditionally independent, and second, the initial corrosion X0 from the previous inspection at t0

is deterministically 0. The measured remaining wall thickness Y1 is subject to measurement error:

𝑌1 = 𝑅1 + 𝜀1 (3.4)

where ε1 is the measurement error and R1 is the actual remaining wall thickness. The

measurement error is assumed to be normally distributed where 𝜀𝑖|𝜎𝜀~normal(0, 𝜎𝜀). The actual

41

remaining wall thickness R1 at time t1 at each inspection time is the difference between the initial

wall thickness W0 and the actual corrosion X1:

𝑅1 = 𝑊0 − 𝑋1 (3.5)

The initial wall thickness is assumed to be normally distributed where

𝑊0|𝜇𝑊0, 𝜎𝑊0

~normal(𝜇𝑊0, 𝜎𝑊0

). The actual corrosion is the sum of the corrosion at the previous

inspection and the incremental growth:

𝑋𝑖 = 𝑋𝑖−1 + ∆𝑋𝑖 (3.6)

where i = 1, 2 is the inspection index. In the gamma process the corrosion growth increment ΔXi

is gamma distributed:

∆𝑋𝑖|∆𝛼𝑖, 𝛽~gamma(∆𝛼𝑖, 𝛽) (3.7)

where Δαi is the unknown time specific shape parameter, and β is the unknown scale parameter,

and the gamma pdf is defined as 𝑓∆𝑋(∆𝑥|∆𝛼, 𝛽) = ∆𝑥∆𝛼−1𝑒−∆𝑥 𝛽⁄ (Γ(∆𝛼)𝛽∆𝛼)⁄ . To satisfy the

monotonic deterioration condition the shape parameter Δαi is defined as a function of time:

Δ𝛼𝑖 = 𝜃1 {(𝑡𝑖 − 𝑡𝑖𝑛𝑖𝑡)𝜃2 − (𝑡𝑖−1 − 𝑡𝑖𝑛𝑖𝑡)𝜃2} (3.8)

where ti is the time of inspection i, θ1 > 0 and θ2 > 0 are unknown time invariant corrosion

model parameters, and tinit is the corrosion initiation time. Thus, to define the process the hyper

parameter λ consists of four parameters θ1, θ2, β, and tinit. To fully define the model prior

distributions are required for the hyper parameters. Exponential priors are assumed for θ1 and β.

The two inspections do not provide enough information to establish the exponential parameter θ2

so the growth process is assumed to be linear (θ2 = 1). The corrosion initiation time must be

42

between the time of the previous inspection t0 when no defects where detected and now t1, and is

assumed to have a truncated exponential prior distribution between these values.

Figure 3.3. Graphical Hierarchical Model of the corrosion growth process.

In the reliability analysis, the resistance to failure (the wall thickness) is constant, while the

load (the defect depth) is monotonically increasing with time. As per Melchers (1999), in this

case the probability of failure for a time interval is determined by using the maximum load for

the interval (the greatest defect depth occurs at the end of the interval) and the constant

resistance. Therefore, for the interval from tinit to t2 the posterior probability of failure pF,2|y1 is

given by:

𝑝𝐹,2|𝑦1 = Pr(𝑋2|𝑌1 ≥ 𝑥𝑐𝑟𝑖𝑡) = ∫ 𝑓𝑋2|𝑌1(𝑥2|𝑦1)𝑑𝑥

∞

𝑥𝑐𝑟𝑖𝑡 (3.9)

43

For the gamma distribution this integral is given by:

𝑝𝐹,2|𝑦1 = ∫ 𝑓𝑋2|𝑌1(𝑥2|𝑦1)𝑑𝑥

∞

𝑥𝑐𝑟𝑖𝑡=

𝛤(𝛼2, 𝑥𝑐𝑟𝑖𝑡 𝛽⁄ )

𝛤(𝛼2) (3.10)

where Γ(α2) is the gamma function and Γ(α2, xcritβ) is the upper incomplete gamma function. The

decision analysis requires the posterior probability of failure ΔpF,2|y1 for the interval from t1 to t2,

not tinit to t2. This probability is conditional on the pressure vessel surviving until t1, given by:

∆𝑝𝐹,2|𝑦1 =𝑝𝐹,2|𝑦1−𝑝𝐹,1|𝑦1

1−𝑝𝐹,1|𝑦1 (3.11)

where pF,1|y1 is the posterior probability of failure from tinit to t1, which is determined in the same

way as pF,2|y1 but using the time interval tinit to t1 instead of tinit to t2.

3.6 Numerical example

A pressure vessel in a refinery that contains gas is subject to structural deterioration due to

corrosion. The initial wall thickness is normally distributed with mean of 12 mm and standard

deviation of 0.25 mm (Jiao et al., 1997). The pressure vessel was inspected during the previous

shutdown 5.2 years ago and no corrosion defects were found. Then the pressure vessel was

inspected during the current shutdown and a defect was detected with a remaining wall thickness

of 9.86 mm. Both of these inspections used an ultrasonic tool, which is subject to normally

distributed measurement error in depth with a mean of 0 and a standard deviation of 0.39 mm

(POF, 2009). The decision maker wants to know whether it is necessary to rush to immediately

repair this defect, or if they can wait until the next scheduled shutdown period to repair the

defect. The next shutdown is scheduled for 4.8 years from now. From this information, the time

values are: t0 = 0, t1 = 5.2 yrs, and t2 = 10 yrs. The posterior distributions of all the variables are

summarized in Table 3.1.

44

Table 3.1. Posterior mean and COV of the unknown model variables.

Variable Symbol Posterior

Mean

Posterior

COV

Corrosion initiation time tinit 0.52 0.98

Multiplier parameter θ1 3.30 0.99

Scale parameter β 0.31 1.03

Actual corrosion at t1 X1 2.07 0.23

Actual corrosion at t2 X2 4.46 0.37

Probability of failure from t1 to t2 ΔpF,2 3.4x10-3

0.017

The prior and posterior distributions of the actual corrosion depth X2 are shown in Figure 3.4.

Updating with the measured remaining wall thickness Y1 causes the posterior distribution to have

a lower mean and standard deviation than the prior.

Figure 3.4. Prior and posterior cdf of actual corrosion depth X2.

45

The probability of failure ΔpF,2|y1 for the time interval from t1 to t2 is used in the decision tree

(Figure 3.2) and as an input for (3.2). To solve (3.1) and (3.2) and determine the expected cost,

the costs CR,1, CR,2, and CF are required; however, to perform the decision analysis and determine

the optimal decision only the ratios of these costs to each other are required and not the absolute

values. Taking the cost of repairing at the next shutdown as the base cost, two ratios can be

defined: CR,1 / CR,2 and CF / CR,2. The expected costs can then be plotted against each other to

visualize the decision situation (Figure 3.5), with the expected costs normalized by the base cost.

In this figure the unity line is the decision neutral line where the expected values of each decision

are the same. For points on this line the decision of repair now or next is neutral, points above

the line should be repaired now, and points below the line should be repaired at the next

shutdown. The point in Figure 3.5 corresponds the example case using cost ratios CR,1 / CR,2 = 2

and CF / CR,2 = 500. The arrows show that for different input values the point moves around the

plot while the plot itself and the unit line for decision neutrality remain the same. For instance, as

the cost of repairing immediately increases the point will move along the arrow pointing right,

and the optimal decision will shift to repair next. Conversely, as the cost of repairing at the next

shutdown increases, the cost of failure increases, or the probability of failure increases, the point

will move along the arrow pointing upwards, and the decision to repair immediately becomes

more favored. While this plot shows the x-axis range down to 0, the typical range is 1 and above.

In this region the cost of immediate repair is greater than the cost of repairing at the next

scheduled shutdown, reflecting the penalty on having to rush an immediate and unscheduled

repair.

46

Figure 3.5. Expected cost of repairing next against the expected cost of repairing now, both

factored by the base cost of repairing now. The point shows the example case.

Another way to visualize the decision analysis is to plot the cost ratios against each other

(Figure 3.6). Again the line is decision neutral; however, the line is no longer unity because the

axes are cost ratios instead of expected costs. The equation of the line is determined by setting

the expected cost of each decision given by Equations (3.1) and (3.2) equal to each other and

isolating the cost ratios CR,1 / CR,2 and CF / CR,2:

𝐶𝑅,1 = (1 − ∆𝑝𝐹,2|𝑦1)𝐶𝑅,2 + ∆𝑝𝐹,2|𝑦1𝐶𝐹 (3.12)

𝐶𝐹

𝐶𝑅,2=

1

∆𝑝𝐹,2|𝑦1

𝐶𝑅,1

𝐶𝑅,2+ 1 −

1

∆𝑝𝐹,2|𝑦1 (3.13)

This proves the decision neutral line is linear with slope 1 / ΔpF,2|y1 and y-intercept 1 - 1 /

ΔpF,2|y1. Thus, the decision neutral line is entirely defined by the probability of failure, and once

47

the probability of failure is known the figure can be created. The decision neutral lines for four

probabilities of failure are shown, including the probability of failure for the example case. By

plotting cost ratio point the figure shows the optimal decision; for instance, the point shown

corresponds to the example case with CR,1 / CR,2 = 2 and CF / CR,2 = 500. An advantage of this

figure is it quickly shows which cost ratios will result in which optimal decision. Also, if the cost

ratios are uncertain it allows several ratios to be easily assessed by locating the point

corresponding to each ratio and checking which decision section the point falls within.

Figure 3.6. Cost of failure against cost of repairing now, both factored by the cost of repairing at

next shutdown. The point shows the example case.

The measured remaining wall thickness Y1 can also be plotted against the inspection cost

ratio CR,1 / CR,2 (Figure 3.7). Again the lines are decision neutral, this time for different fixed

failure cost ratios CF / CR,2. Increasing the measured remaining wall thickness moves the point

upwards, and the decision to repair at the next shutdown becomes more favored. Conversely,

48

increasing the cost of repairing now with respect to repairing later moves the point to the right

and the decision to repair later is favored. However, the decision is mostly sensitive to the cost of

repairing now for ratios close to one, at higher ratios the remaining wall thickness and failure

cost dominate. The non-linear decision neutral line reinforces the sensitivity of the decision to

the measured remaining wall thickness.

Figure 3.7. Measured remaining wall thickness at t1 against the inspection cost ratio. The point

shows the example case.

3.7 Conclusion

This paper presents the decision analysis framework for a corroding pressure vessel inside an

oil and gas facility. The analysis examines the decision of whether a corrosion defect needs to be

repaired immediately upon detecting the defect, or if the repair can wait until the next scheduled

facility shutdown. Waiting to repair the defect at the next shutdown is expected to have a cost

49

benefit as the operator can schedule their resource allocation in advance. The method uses a

decision tree to represent the scenario, with all discrete decisions and their outcomes represented.

The optimal decision is the one that minimizes the expected cost. A probabilistic analysis is used

to determine the probability of each outcome occurring, namely the probability that the pressure

vessel fails or survives. To determine this probability a homogeneous gamma process is used to

model the corrosion growth through time. It is acknowledged that the analysis is dependent on

the corrosion growth model used, and the use of a different model will impact the results. The

optimal decision switches between repairing at the next shutdown or repairing now depending on

the severity of the corrosion, as well as the cost of repairing now, repairing at the next shutdown,

and failure. The analysis showed that the decision is very sensitive to changes in the measured

remaining wall thickness, and moderately sensitive to changes in the ratios between the various

costs. The decision analysis can aid facility operators in decision making regarding corroding

pressure vessel repair. In cases of less severe corrosion, the analysis can be useful in supporting

the decision to delay the repair until the next scheduled shutdown when it is more convenient and

economic. Future work in this study will expand the analysis by solving the corrosion model

with Bayesian inference as well as maximum likelihood. Also, burst failure and multiple defects

will be considered, and the additional option of performing an intermediate inspection of the

vessel between facility shutdown times will be added to the decision tree.

50

4 RISK-BASED PLANNING FOR DETERIORATING PRESSURE VESSELS WITH

MULTIPLE DEFECTS

This chapter contributes to the research area: RBM planning for systems with multiple

components and failure modes, specifically addressing the research objectives:



Considering the dependency in the failure events between the multiple components and

failure modes

The novel contribution of this paper is the RBM framework for pressure vessels that

considers multiple corrosion defects and failure modes, and accounts for the dependent failure

events. System reliability analysis is difficult, and many RBM studies avoid the difficulty by

restricting the analysis to only one defect, or ignoring the dependency in the failure events.

However, these simplifications impact the decision making, and can lead to suboptimal results.

There are three novel contributions of this paper. The first contribution is the methodology to

determine the optimal set of repair times for a pressure vessel with multiple defects, considering

multiple defects and failure modes. The second contribution is the comparison of the decision

results with and without considering the dependency in the failure events, to examine the impact

of considering the dependency. The third contribution is a novel corrosion growth model that

predicts both the corrosion length and depth. The majority of probabilistic corrosion growth

models only predict depth; however, to perform accurate burst analysis the defect length is also

required.

51

This chapter is a journal paper (Haladuick and Dann, 2017a) that was published in the ASME

Journal of Pressure Vessel Technology in August 2017. It is included with the permission of the

copyright holder ASME, and the published version can be accessed here:

https://pressurevesseltech.asmedigitalcollection.asme.org/article.aspx?articleID=2618464

52

4.1 Abstract

Pressure vessels are subject to deterioration processes, such as corrosion and fatigue, which

can lead to failure. Inspections and repairs are performed to mitigate this risk. Large industrial

facilities (e.g. oil and gas refineries) often have regularly scheduled shutdown periods during

which many components, including the pressure vessels, are disassembled, inspected, and

repaired if necessary. This paper presents a decision analysis framework for the risk-based

maintenance planning of corroding pressure vessels containing gas. After a vessel has been

inspected, this framework determines the optimal maintenance time of each defect, where the

optimal time is the one that minimizes the total expected cost over the lifecycle of the vessel. The

framework allows for multiple defects and two failure modes (leak and burst), and accounts for

the dependent failure events. A stochastic gamma process is used to model the future

deterioration growth to determine the probability of vessel failure. The novel growth model

presents a simple method to predict both the depth and length of each corrosion defect to enable

burst analysis. The decision analysis framework can aid decision makers in deciding when a

repair or replacement should be performed. This method can be used to immediately inform the

decision maker of the optimal decision post inspection. A numerical example of a corroding

pressure vessel illustrates the method.

4.2 Introduction

Pressure vessels are used in many industrial facilities including offshore platforms, refineries,

factories, power plants, and floating production, storage, and offloading facilities (FPSOs). These

pressure vessels are subject to structural deterioration due to corrosion and fatigue, which

gradually reduce the resistance to a failure event. Should this deterioration lead to failure, there

53

can be large social, economic, and environmental consequences. To mitigate this risk, operators

perform regular inspections and repairs of the subsystems in the facility, including the pressure

vessels. The inspections and repairs often require the facility to be shut down, and are usually

coordinated during one facility wide shutdown to reduce offline time. Consequently, the future

inspection and potential repair times are often fixed in advance and cannot be adjusted.

Upon inspection of a pressure vessel, the operator must decide when to repair each of the

defects in the vessel. Delaying the repair is advantageous because the present cost of the repair is

discounted for the time value of money, and because the repair may never be required. However,

delaying the repair also increases the risk of failure if the deterioration increases. Therefore, it is

important to optimize the decision of when to repair. Unfortunately, due to the high degree of

uncertainty, determining the optimal decision is not straightforward. The uncertainty is due to

several sources, including spatial and temporal variability of corrosion growth, temporal

variability of the applied pressure, and inaccuracies in the measurement tool and corrosion

growth and limit state models. Decision analysis (Luce and Raiffa, 1957) can be used to

determine the optimal maintenance decision under uncertainty. This method uses a probabilistic

approach to quantify the risk due to corrosion over the lifecycle of a pressure vessel, and the

maintenance plan with the lowest risk is the optimal plan. This process is termed risk-based

maintenance (RBM) planning (Kahn and Haddara, 2003).

The objective of this paper is to use RBM to determine the optimal repair time for a

deteriorating pressure vessel. The key contribution of this paper is a holistic RBM framework for

pressure vessels, which considers multiple corrosion defects and failure modes and accounts for

the dependent failure events. Many RBM studies avoid system reliability analysis by considering

only one defect or ignoring the dependency in the failure events. However, in reality pressure

54

vessels can have multiple defects and failure modes with dependent failure events that can

impact decision making and should be considered in RBM. In addition, this paper introduces a

novel corrosion growth model that predicts the corrosion length and depth. The majority of

probabilistic corrosion growth models only predict depth; however, to perform accurate burst

analysis the defect length is also required.

This paper comprises six sections. The second section presents a review of the recent

literature in RBM, especially related to the oil and gas industry. The third section presents the

RBM framework, which determines the optimal pressure vessel repair time. The fourth section

details the reliability analysis to support the RBM framework. The fifth section presents a

numerical example of a corroding pressure vessel to illustrate the decision making process. The

example demonstrates the impact of dependent failure events on the decision analysis. Finally,

the conclusion discusses the impact and limitations of the method.

4.3 Literature review

4.3.1 Decision analysis

Decision analysis was first introduced by von Neuman and Morgenstern (1947) and has been

described in many texts since (Luce and Raiffa, 1957; Pratt et al., 1995; Jordaan, 2005; JCSS,

2008; Parmigiani and Inoue, 2009). Decision analysis is used to inform engineering decision

makers responsible for societal infrastructure. The decision maker is charged with selecting the

‘best’ choice from the available options. If the outcomes of the decision were deterministic, the

decision maker would simply select the option with the highest utility (or equivalently the lowest

consequences), where utility is a quantification of the decision maker’s preferences. However,

corrosion growth and pressure vessel failure have many sources of uncertainty, causing the

55

outcome of the decision to be uncertain. Decision analysis uses a probabilistic analysis to address

this uncertainty where the variables are treated as random. The optimal decision is the one that

minimizes the expected value of the consequences where the expected consequences are also

known as the risk. The risk of decision option d (d = 1, …, D) is given by the following:

E[𝑐(𝑑)] = ∑ 𝑝𝑜(𝑑)𝑐𝑜(𝑑)𝑂

𝑜=1 (4.1)

where there are o (o = 1, …, O) outcomes of decision option d, and po and co are the respective

probability and consequences of outcome o. The best decision option d* is the one that

minimizes the expected consequences min(E[c(d)]). In RBM, the outcomes of a specific

maintenance plan are the cost of the maintenance and the cost of failure. The consequences of

system failure include non-monetary attributes such as human casualties and environmental

damage; however, these attributes can be monetized for comparison with other costs (Xu, 2015).

4.3.2 RBM planning

RBM uses decision analysis to find the best maintenance plan by determining the total

expected cost of operating a pressure vessel over its lifecycle for a set of possible maintenance

plans. The costs are determined for the set of possible plans, and the plan with the lowest total

expected cost is the optimal plan. There has been a lot of progress in the use of RBM in the oil

and gas industry. Recently, several guidelines (ASME, 2007; API, 2009; API, 2016) have been

developed to govern the implementation of RBM within the refinery environment. These

guidelines provide a strong foundation for how to perform RBM; however, they are limited in

some of their methods. For instance, the use of a probabilistic analysis is discussed, but

guidelines on how to develop corrosion growth or limit state models that account for all of the

uncertainties are not provided.

56

Several studies have incorporated probabilistic analysis in RBM for structural systems in the

oil and gas industry. However, most of these studies employ other simplifications to reduce the

difficulty of determining the optimal maintenance plan. This paper removes these common

simplifications to present as complete a framework for pressure vessel RBM as possible. Some

typical simplifications are:

restricting the deterioration to a single defect, negating system analysis e.g. (Hellevik et

al., 1999; Goyet et al., 2002; Straub, 2004; Gross et al., 2012; Sahraoui et al., 2013;

Gomes et al., 2013; Gomes and Beck, 2014; Haladuick and Dann, 2016a)

assuming defect failure is independent for multiple defects e.g. (Hong, 1997, 1999;

Pandey, 1998; Garbatov and Soares, 2001)

assuming deterministic applied pressure for burst analysis e.g. (Hellevik et al., 1999;

Hong, 1997, 1999; Pandey, 1998)

restricting the inspection result to binary detection or no detection of a defect and not the

defect size e.g. (Hellevik et al., 1999; Straub, 2004; Pandey, 1998)

ignoring uncertainties in the variables in the corrosion growth of limit state models e.g.

(Sahraoui et al., 2013; Gomes et al., 2013; Gomes and Beck, 2014)

Many studies developed RBM frameworks for pipelines. Pandey (1998) performed RBM to

decide whether to repair a defect and to optimize the timing of a single inspection over the

lifecycle. Sahraoui et al. (2013) performed RBM to determine the optimal maintenance plan and

the optimal periodic time between inspections. They accounted for uncertainties due to detection

and false detection but not measurement error. Nessim et al. (2000) optimized the timing of the

first and second inspections considering the hazards of corrosion and mechanical damage. After

an inspection, they made the decision to repair based on whether the defect exceeded a threshold,

57

and they performed RBM to determine the optimal repair threshold. Hong (1997, 1999)

optimized the timing and type of inspection and performed RBM to assess the sensitivity to the

repair threshold. Haladuick and Dann (2016a) developed an RBM methodology to determine the

optimal repair time for a severe defect in a pressure vessel, where the repair needed to be

completed in the near future.

Other studies focused on risk-based inspection (RBI) planning. Straub (2004) performed RBI

for structures subject to fatigue deterioration. The model optimized inspection time by using

either a reliability constraint or a constraint requiring periodic inspections. Gomes et al. (2013)

and Gomes and Beck (2014) performed RBI for onshore buried pipelines subject to external

corrosion. Uncertainty in whether an inspection tool detected a defect was accounted for, but

measurement error was not. Hellevik et al. (1999) performed RBI for pipelines subject to

corrosion, optimizing the timing of inspections, type of inspection, and number of inspections.

Zhang and Zhou (2014) performed RBI to determine the optimal timing of a single inspection of

a natural gas pipeline. They constrained the analysis to only one segment, but they allowed

multiple defects in the segment and the generation of new defects; they also considered

measurement error. There are also several studies applying RBI to FPSOs. Fatigue deterioration

in the welded connections of the hull structure of an FPSO was examined to determine the

optimal inspection and repair plan (Goyet et al., 2002) and the optimal period between

inspections (Garbatov and Soares, 2001). Within the refinery environment, RBI was used to

optimize the periodic inspection frequency of refinery piping (Chang et al., 2005). With regards

to pressure vessels, RBI was used to assess the impact of increasing the periodic inspection

interval for spring operated relief valves on pressure vessels at the Savannah River Site nuclear

58

reservation (Gross et al., 2012). The analysis was restricted to a single component system, which

was reasonable as pressure vessels typically only have one relief valve.

4.4 RBM framework

The objective of the RBM for pressure vessels is to minimize the expected lifecycle cost of

operating a pressure vessel. For a given pressure vessel with J ≥ 1 corrosion defects, the decision

maker must decide which of the defects needs to be repaired and when. The defects can either be

defects newly detected in the current inspection or existing defects that were originally detected

in previous inspections. The maintenance strategy is defined by a vector of repair times tR = {tR1,

…, tRJ}, where there is a repair time for each of the J defects. There are m ≥ 1 future shutdown

times scheduled over the lifecycle of a vessel, and these times form the set of possible times from

which to choose tR, as well as the option to not repair. Thus, there are (m+1)J possible repair

plans, which are all the combinations of repairing each defect at any possible time. The objective

of the RBM framework is to determine the optimal repair plan tR* that minimizes the total

expected cost E[𝐶(𝒕𝑹∗)] = min𝒕𝑹

(E[𝐶(𝒕𝑹)]).

Once the decision maker decides which maintenance strategy tR to apply, the outcome is

either vessel survival or failure. If the vessel survives, there is only the cost of repairing the

vessel according to the maintenance strategy tR. If the vessel fails, then there is the cost of repair

and the cost of failure. This framework makes two main assumptions: first, the repairs are

perfect, meaning that a repaired defect cannot fail; second, there are no additional defects

generated for the lifetime of the vessel. Both of these assumptions are expected to have a minor

impact on the analysis; repairs rarely fail, and it is rare that newly generated defects fail before

existing defects. API (2007) defines two failure modes for pressure vessels, leak and burst, while

59

API (2016) defines four modes, small, medium, and large leak, as well as rupture (similar to

burst), providing more gradation in the consequences of failure. This study uses the API (2007)

failures modes of leak and burst. Since there are multiple defects and two failure modes, there is

dependency in the failure events requiring system reliability analysis. Leak and burst failure

modes are treated differently to reflect the physical nature of a pressure vessel. If a leak occurs, it

is repaired, the vessel is placed back in service after the repair, and the cost of the leak CL is

incurred. Thus, multiple leak failures can occur. However, if a burst occurs, then the vessel is

assumed to be damaged beyond repair and needs to be replaced with a new vessel, incurring the

cost of a burst CB. Thus, the cost of leak and burst failures, CL and CB respectively, need to

include all aspects of both direct and indirect costs. The direct costs include repairing or

replacing the vessel and its contents, cleaning up the failure site, losses of having the vessel

offline (especially in the case of burst), environmental costs, and societal costs of potential

human casualties. The indirect costs include any far reaching costs of the failure such as inability

to deliver the hydrocarbon product to market or damage to the reputation of the company,

government, or industry involved.

The requirement that the vessel is replaced after a burst and the assumption of no defect

generation means that the vessel will survive until the end of its lifecycle. Therefore, the

inspections will be performed regardless of the repair plan tR, so their cost is constant and can be

ignored in the optimization problem. The total expected cost E[C(tR)] for each repair plan is the

sum of the expected cost of repair E[CR(tR)] and the expected cost of failure E[CF(tR)] (Rackwitz

et al., 2005):

E[𝐶(𝒕𝑹)] = E[𝐶𝑅(𝒕𝑹)] + E[𝐶𝐹(𝒕𝑹)] (4.2)

The expected cost of failure is given by the following:

60

E[𝐶𝐹(𝒕𝑹)] = ∫ (E[𝑛𝐿(𝒕𝑹, 𝑡)]𝐶𝐿

(1 + 𝑟)𝑡+

𝑝𝐵(𝒕𝑹, 𝑡)𝐶𝐵

(1 + 𝑟)𝑡)

𝑡𝑠𝑙

0

𝑑𝑡 (4.3)

where nL(tR,t) is the time dependent instantaneous number of system leaks for the repair plan tR,

pB(tR,t) is the time dependent instantaneous probability of system burst failure for the given

repair plan tR, and CL and CB are the cost of a single leak and burst, respectively. The integral is

performed from the time t = 0 (now) to the end of the service life tsl. The cost of failure is

discounted by the interest rate r based on the time t when the failure cost was incurred. The

effective interest rate is taken as the difference between the interest rate and the inflation rate.

The expected cost of repair is given by the following:

E[𝐶𝑅(𝒕𝑹)] = ∑𝐶𝑅

(1 + 𝑟)𝑡𝑅𝑗

(1 − ∫ 𝑝𝐿𝑗(𝒕𝑹, 𝑡)𝑑𝑡

𝑡𝑅𝑗

0

) (1 − ∫ 𝑝𝐵(𝒕𝑹, 𝑡)𝑑𝑡𝑡𝑅𝑗

0

)𝐽

𝑗=1

(4.4)

where CR is the cost of a single repair. The term 1 − ∫ 𝑝𝐿𝑗(𝒕𝑹, 𝑡)𝑑𝑡

𝑡𝑅𝑗

0 is the probability that

defect j does not leak before time tRj, and similarly 1 − ∫ 𝑝𝐵(𝒕𝑹, 𝑡)𝑑𝑡𝑡𝑅𝑗

0 is the probability that the

pressure vessel system does not burst before tRj. In other words, the product of these two

probabilities is the probability that defect j survives until repair j is performed. The summation is

performed over all defects yielding the total repair cost. The optimal repair time for the pressure

vessel can now be determined by populating Equations (4.3) and (4.4) with the costs and

probabilities. Deriving the time dependent failure probabilities is described in the subsequent

section.

The expected cost over the lifecycle of the vessel needs to be computed for each potential

repair plan to solve the optimization problem for the optimal repair plan tR*. As noted

previously, the number of potential repair plans is given by (m+1)J, where m is the number of

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future repair opportunities, and J is the number of defects. For relatively small values of m and J,

the optimization problem can be solved exhaustively by evaluating the objective function

E[C(tR)] for each repair plan tR. However, as m and J increase, the computational demand of this

approach grows exponentially with the number of repair plans. Thus, the exhaustive optimization

approach is limited to only relatively small values of m and J.

There are several possible solutions that allow larger values of m and J; three are briefly

discussed. The first solution is to increase the computational power. Parallel processing allows

many simple computations to be performed simultaneously, decreasing the computation time and

thus allowing solutions to larger problems. However, because the number of repair plans is

increasing exponentially, the scale of the problem can still exceed even advanced computational

resources. The second solution is to limit the number of defects in the analysis to only the most

severe defects (meaning the defects expected to fail first) and ignore the rest. This solution

reduces J, allowing the exhaustive optimization technique to be used. However, ignoring the less

severe defects is a simplification that will make the decision analysis less accurate. The third

solution is to use a heuristic algorithm, such as a genetic algorithm (Holland, 1976; Goldberg,

1989) to solve the optimization problem. A heuristic algorithm does not exhaustively evaluate

the objective function for every repair plan. Instead it uses an algorithm to iteratively evaluate

the objective function for different repair plans, with each iteration adjusting the repair plan to

move towards the optimal plan. This solution allows for much larger values of m and J. The

genetic algorithm has the potential limitation of finding only a locally optimal solution and not

the global optimal; however, a well formulated genetic algorithm has been shown to successfully

solve optimization problems.

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4.5 Reliability analysis

This section details the reliability analysis for the RBM framework for pressure vessels,

where reliability is defined as 1 minus the probability of failure. This section is composed of

three parts. The first part describes the probabilistic corrosion growth model to predict the future

state of deterioration based on the historical inspection data. The second part presents the limit

state functions that define leak and burst failures. The third part describes the method to perform

failure analysis of a pressure vessel as a system with multiple defects and two failure modes.

4.5.1 Corrosion growth model

The leak and burst limit state functions require the depth and length, which are determined

from the corrosion growth model, which describes the growth of the defects over time. Many

corrosion growth models only predict the corrosion depth, and either assume a length or use a

fixed length to depth ratio. This section introduces a simple and novel model to predict both the

defect depth and length, facilitating a more accurate burst analysis. Miran et al. (2016) also

proposed a model for both the depth and length of a corrosion defect, by modelling the

correlation between the dimensions of the defect. They used a power law to model the growth of

a defect and a Poisson process to model the generation of new defects.

The RBM framework functions with any corrosion growth model; however, the corrosion

growth model impacts the results of the RBM, and therefore, it needs to properly address the

uncertainties present. There are many types of probabilistic corrosion growth models suggested

in the literature (Pandey et al., 2009; Brazán and Beck, 2013). A satisfactory corrosion growth

model should account for both the aleatory and epistemic uncertainties (JCSS, 2008) present in

the corrosion growth process. Aleatory uncertainty is due to the natural variability of a process

63

and cannot be reduced. For corrosion growth, this includes spatial and temporal uncertainty.

Spatial uncertainty is due to the difference in corrosion growth at different defect locations. This

model accounts for this uncertainty by modelling the corrosion growth as an exchangeable

system of defects. An exchangeable system allows the corrosion to vary between defects while

still sharing the corrosion information globally (Maes, 2006). Temporal uncertainty is due to the

stochastic nature of the corrosion process, making the prediction of future corrosion based on

current corrosion uncertain (Pandey et al., 2009). This paper uses a stochastic process model to

account for temporal uncertainty. For corrosion growth, epistemic uncertainty includes

inspection uncertainty, statistical uncertainty, and model uncertainty. Unlike aleatory

uncertainty, epistemic uncertainty can be reduced. For instance, measurement uncertainty can be

reduced with better measurement tools, statistical uncertainty can be reduced by performing

more inspections, and model uncertainty can be reduced by developing more accurate

probabilistic models. Additionally, corrosion growth monotonically increases with time, and the

corrosion model should reflect this. Any stochastic process that satisfies these requirements is

acceptable for the decision making methodology presented. The pressure vessel operator should

use corrosion data from comparable pressure vessels to select the growth process that is the best

fit for their specific pressure vessel. This paper assumes the corrosion growth follows the gamma

process, as it has been shown to satisfactorily describe the corrosion growth in steel structures

(Pandey et al., 2009; Pandey and van Noortwijk, 2004; van Noortwijk, 2009). This model is

represented by a hierarchical graphical model (Koller and Friedman, 2009) (Figure 4.1).

64

Figure 4.1. Hierarchical graphical model of the corrosion growth process.

The measured remaining wall thickness YDi,j and defect length YLi,j at each inspection i (i = 1,

…, I) at time ti, for each defect j (j = 1, …, J) are subject to measurement error. For simplicity,

the number of defects J is defined as constant for each inspection i; however, it could be defined

as time dependent if necessary.

𝑌𝐷𝑖,𝑗= 𝑅𝑖,𝑗 + 𝜀𝐷𝑖,𝑗

for i = 1, …, I; j = 1, …, J (4.5)

𝑌𝐿𝑖,𝑗= 𝐿𝑖,𝑗 + 𝜀𝐿𝑖,𝑗

for i = 1, …, I; j = 1, …, J (4.6)

where εDi,j and εLi,j are the measurement errors in depth and length, respectively, and Ri,j and Li,j

are the actual remaining wall thickness and actual defect length, respectively. The actual

65

remaining wall thickness Ri,j is the difference between the uncertain initial wall thickness W0j and

the actual corrosion depth Di,j:

𝑅𝑖,𝑗 = 𝑊0𝑗− 𝐷𝑖,𝑗 for i = 1, …, I; j = 1, …, J (4.7)

The actual corrosion in both depth and length is the sum of the corrosion at the previous

inspection and the incremental depth and length growth ΔDi,j and ΔLi,j, respectively:

𝐷𝑖,𝑗 = 𝐷𝑖−1,𝑗 + ∆𝐷𝑖,𝑗 for i = 1, …, I; j = 1, …, J (4.8)

𝐿𝑖,𝑗 = 𝐿𝑖−1,𝑗 + ∆𝐿𝑖,𝑗 for i = 1, …, I; j = 1, …, J (4.9)

This study assumes that the initial corrosion depth D0j and length L0j for each defect j at the

initial time t0j are deterministically 0. In the gamma process, the corrosion depth growth

increment ΔDi,j is gamma distributed (van Noortwijk, 2009):

∆𝐷𝑖,𝑗|∆𝛼𝑖,𝑗, 𝛽𝑗~gamma(∆𝛼𝑖,𝑗, 𝛽𝑗) for i = 1, …, I; j = 1, …, J (4.10)

where Δαi,j is the unknown time and defect specific shape parameter, and βj is the unknown

defect specific scale parameter. The gamma pdf is defined as 𝑓∆𝐷𝑖,𝑗(∆𝐷𝑖,𝑗|∆𝛼𝑖,𝑗 , 𝛽𝑗) =

∆𝐷𝑖,𝑗∆𝛼𝑖,𝑗−1𝑒−∆𝐷𝑖,𝑗 𝛽𝑗⁄ (Γ(∆𝛼𝑖,𝑗)𝛽𝑗

∆𝛼𝑖,𝑗)⁄ . The shape parameter Δαi,j is defined as a function of

time to satisfy the monotonically increasing deterioration condition:

Δ𝛼𝑖,𝑗 = 𝜃1 {(𝑡𝑖 − 𝑡0𝑗)𝜃2 − (𝑡𝑖−1 − 𝑡0𝑗

)𝜃2} for i = 1, …, I; j = 1, …, J (4.11)

where ti > t0j is the time of inspection i, θ1 > 0 and θ2 > 0 are unknown time invariant corrosion

model parameters, and t0j is the defect specific corrosion initiation time. The novel aspect of this

model is the link between the corrosion depth growth increment ΔDi,j and the length growth

increment ΔLi,j. This model assumes that the length growth increment is proportional to the depth

growth increment and related by the defect specific proportionality constant kj:

66

Δ𝐿𝑖,𝑗 = 𝑘𝑗∆𝐷𝑖,𝑗 for i = 1, …, I; j = 1, …, J (4.12)

This model assumes that the corrosion growth increments are conditionally independent from

previous growth increments in both the depth and length. Prior probability distributions are

required for the hyper-parameters to fully define the model. Non-informative priors are used for

θ1, θ2, βj, and kj. The corrosion initiation time t0j must be after the pressure vessel was built but

before the defect was detected, and it is assumed to have a truncated exponential prior

distribution between these values.

The corrosion growth model is iteratively solved using Markov Chain Monte Carlo (MCMC)

(Gamerman and Lopes, 2006) simulation. There are several software packages available to

facilitate such an analysis, and this study uses OpenBUGS (Spiegelhalter et al., 2006). Each

simulation run samples the corrosion growth over time to determine the posterior distribution of

each variable. The posterior distribution of the corrosion depth for each defect at each time is

then used as an input to the limit state functions.

4.5.2 Limit state functions for leak and burst failures

This section presents the limit state functions for leak and burst failures. The failure analysis

is a prediction into the future, so the time dependent variables are defined as continuous

functions of t, where t = 0 is the time of the most recent inspection tI. A leak occurs when the

maximum depth of the corrosion defect exceeds the wall thickness, and the leak limit state

function of an individual defect is given by the following:

𝑔𝐿𝑗(𝑡) = 𝑑𝑐𝑟𝑖𝑡 − 𝐷𝑗(𝑡) (4.13)

where dcrit is the critical depth and Dj(t) is the actual depth. In other words, failure occurs when

the defect depth is greater than the critical depth, and the instantaneous probability of individual

67

leak failure pLj(tR,t) = Pr(Dj(t) > dcrit) is the probability that Dj(t) is greater than the critical depth

for the given repair plan tR. For a probabilistic analysis, dcrit is usually defined as 100 % of the

wall thickness.

A burst occurs when the wall undergoes plastic collapse due to internal pressure before the

defect grows through the wall. The limit state function for burst failure used in this study was

adapted from Kaida et al. (2013), which is based on API (2007) for cylindrical pressure vessels.

The limit state function for the burst of an individual defect is given by the following:

𝑔𝐵𝑗(𝑡) = 𝑃𝑏𝑐𝑗

(𝑡) − 𝑃𝑎𝑝𝑝(𝑡) (4.14)

where Papp(t) is the applied pressure and Pbcj(t) is the instantaneous burst pressure capacity. Thus,

the instantaneous probability of individual burst failure pBj(tR,t) = Pr(Papp(t) > Pbcj(t)) is the

probability that the instantaneous applied pressure is greater than the burst pressure. The

instantaneous burst pressure capacity is given by the equation below:

𝑃𝑏𝑐𝑗(𝑡) = 𝑅𝑆𝐹𝑗(𝑡)𝑃𝑏𝑢𝑗

(4.15)

where Pbuj is the undamaged burst pressure:

𝑃𝑏𝑢𝑗= (

𝑒

𝑛)

𝑛

(0.25

𝑛 + 0.227) 𝑙𝑛 (1 +

2𝑊0𝑗

𝐼𝐷) 𝜎𝑢 (4.16)

where e is the base of the natural logarithm, n is the work hardening coefficient, W0j is the initial

wall thickness, ID is the internal diameter of the pressure vessel, and σu is the tensile strength.

RSFj(t) is the remaining strength factor, given by the following:

𝑅𝑆𝐹𝑗(𝑡) =𝑅𝑡𝑗

(𝑡)

1 − (1 − 𝑅𝑡𝑗(𝑡)) 𝑀𝑡𝑗

(𝑡)⁄ (4.17)

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where Rtj(t) is the remaining thickness ratio and is defined as Rtj(t) = Rj(t) / W0j, where Rj(t) = W0j

– Dj(t) is the measured remaining wall thickness, and Mtj(t) is the Folias factor. There are several

possible values for the depth Dj(t) (e.g. maximum depth, average depth, etc), and this study uses

the maximum depth as per Kaida et al. (2013). The Folias factor is given by the following:

𝑀𝑡𝑗(𝑡) = 1.001 − 0.0014195𝜆𝑗(𝑡) + 0.2909𝜆𝑗(𝑡)2 − 0.09642𝜆𝑗(𝑡)3

+0.02089𝜆𝑗(𝑡)4 − 0.003054𝜆𝑗(𝑡)5 + 2.957𝑥10−4𝜆𝑗(𝑡)6 − 1.8462𝑥10−5𝜆𝑗(𝑡)7

+7.1553𝑥10−7𝜆𝑗(𝑡) 8 − 1.531𝑥10−8𝜆𝑗(𝑡) 8 + 1.4656𝑥10−10𝜆𝑗(𝑡) 9

(4.18)

where λj(t) is the shell parameter, 𝜆𝑗(𝑡) = 1.285𝐿𝑗(𝑡) √𝐷𝑗(𝑡) 𝑊0𝑗⁄ , and Lj(t) is the longitudinal

defect length.

4.5.3 System reliability analysis

The RBM framework integrates the time dependent expected number of system leaks and the

time dependent probability of system burst over the lifecycle of a vessel to determine the

expected cost of failure and repair [Equations (4.3) and (4.4)]. For burst failure, the system can

only burst once; therefore, the limit state function for system burst is the union of each of the

individual burst limit state functions:

𝑔𝐵(𝒕𝑹, 𝑡) = ⋃ 𝑔𝐵𝑗(𝒕𝑹, 𝑡)

𝐽

𝑗=1 (4.19)

The instantaneous probability of system burst failure pB(tR,t) = Pr(gB(tR,t) < 0) is the

probability that the limit state function for system burst is less than 0. Leak failures are

independent of each other, so a system limit state function for leaks is not required. However,

69

leaks cannot occur simultaneous to or after a burst; therefore, the limit state function for

individual leaks needs to be revised to remove the intersection with system burst:

𝑔𝐿𝑗(𝒕𝑹, 𝑡) = 𝑔𝐿𝑗

(𝒕𝑹, 𝑡) ⋂ 𝑔𝐵̅̅̅̅ (𝒕𝑹, 𝑡) (4.20)

where 𝑔𝐵̅̅̅̅ (𝒕𝑹, 𝑡) is the complement of the limit state function of system burst. The probability of

an individual leak failure pLj(tR,t) = Pr(gLj(tR,t) < 0) is the probability that the limit state function

for an individual leak failure is less than 0. The expected number of leaks E[nL(e,t)] for the

system is given by the sum of the probabilities of individual leaks:

E[𝑛𝐿(𝒕𝑹, 𝑡)] = ∑ 𝑝𝐿𝑗(𝒕𝑹, 𝑡)

𝐽

𝑗=1 (4.21)

Implementation of the continuous time dependent probabilities of leak and burst failures is a

field of study unto itself, and so several simplifications are used in this study. The first is to

discretize time, which allows the integrals to be transformed into the equivalent summations. The

instantaneous reliability formulas previously derived still apply in the case of discrete time as

long as the time increments are short enough to accurately assess the failure intersects. Next, the

reliability for each time increment is required. For leak failure, this problem is relatively simple;

the resistance to failure (the critical wall thickness dcrit) is time independent, and the load (the

corrosion depth Dj(t)) monotonically increases. Therefore, the probability of failure for any time

increment is determined by using the maximum load in the increment (occuring at the end of the

increment) and the load (Melchers, 1999). However, for burst failure, the problem is not trivial.

For burst failure, the resistance to failure (the burst pressure Pbj(t)) monotonically decreases, and

the load (the applied pressure Papp(t)) is time dependent. The random variation in the applied

pressure occurs even though vessels have pressure relief valves, because the relief valve only

maintains a constant maximum operating pressure, and not a constant applied pressure. The

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applied pressure is a function of the operating conditions, for example changes in the

temperature and volume of the fluid, and thus is treated as a random variable. The time

dependent reliability problem is transformed into a time independent problem to allow a

numerical solution. As an upper bound, if the distribution of the largest defect during a time

interval is applied along with the distribution of the maximum pressure during the interval, then

an overly conservative estimate of the probability of failure is obtained. Leira et al. (2014)

suggests a reasonable simplification by using extreme value statistics to model the distribution of

the maximum pressure during the time increment along with the average distribution of the

deterioration during the same time increment. Extreme value statistics are applicable for

relatively longer time increments. Thus, there are competing factors driving the length of the

discrete time increments: the increments need to be short enough for accurate failure intersection,

but long enough for extreme value statics to apply. Increments of 1 year were found to satisfy

both. Additionally, 1 year increments then correspond to the annual distribution of the applied

pressure and the annual probability of failure, which are commonly used in industry. The

deterioration at the midpoint of the increment is taken as the average distribution. Note that the

use of time independent extreme value statistics to model the pressure is an approximation to

facilitate a simpler analysis, and it would be more accurate to model the applied pressure as a

stochastic process.

The limit state functions of leak and burst are evaluated for each MCMC simulation run from

the corrosion growth to determine whether leak and burst failures occurred, and if so, the time of

the failure. The future probability of failure is conditional on pF(t=0) = 0; therefore, simulation

runs that fail before t = 0 are ignored. The system reliability is then determined by treating the

multiple defect failures in leak and burst as per Equations (4.19) to (4.21).

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4.6 Numerical example of a corroding pressure vessel

A cylindrical, carbon steel pressure vessel in a refinery, containing gas, is subject to

structural deterioration due to corrosion. The pressure vessel was constructed 15 years ago and

has a remaining service life tsl = 20 years. The vessel was inspected 3 times during facility wide

shutdowns at 5 year intervals, i.e. after 5, 10, and 15 years. Each of the inspections detected 6

defects (Table 4.1) using an ultrasonic inspection tool. The facility will continue to be shut down

every 5 years for inspections and possible repairs. With 20 years remaining in the service life of

the vessel, the set of possible future repair times are 0, 5, 10, and 15 years from now, as well as

to not repair. The facility operator must decide when to repair each of the defects.

72

Table 4.1. Inspection results of remaining wall thickness and length of corrosion defects.

Defect size [mm] Inspection time [years]

t1 = 5 t2 = 10 t3 = 15

Ri,1 12.7 12.4 11.2

Li,1 8 31 65

Ri,2 12.6 10.9 10.5

Li,2 6 23 29

Ri,3 12.2 11.7 10.9

Li,3 14 12 47

Ri,4 12.5 11.6 11.0

Li,4 4 20 43

Ri,5 12.9 11.5 11.1

Li,5 13 33 43

Ri,6 12.0 11.1 10.8

Li,6 11 28 54

The results of the decision analysis are conditional on the input variables for the corrosion

growth model and limit state functions. The pressure vessel operator should use available data to

determine distributions and parameters that are representative of their pressure vessel. The inputs

need to reflect the state of knowledge and uncertainty at the time of the analysis. If the

distributions or parameters are uncertain, then this uncertainty needs to be accounted for in the

analysis. This example assumes the input values as detailed in Table 4.2. MCMC with 106

simulations was used to solve the corrosion growth model to predict the future corrosion. The

73

future corrosion in turn is used as input for the limit state functions to determine the probability

of failure.

Table 4.2. Input variables for the corrosion growth model, reliability analysis, and decision

making analysis.

Variable Symbol Mean COV Distribution Reference

Critical depth dcrit 100 % wt - fixed -

Depth error εD 0 *0.39 mm Normal POF (2016)

Length error εL 0 *7.8 mm Normal POF (2016)

Wall thickness W0 13 mm 0.02 Normal Kaida et al. (2013)

Applied pressure Papp 1.1 MPa 0.03 Gumbel Leira et al. (2014)

Work hardening

coefficient

n 0.2 0.06 Normal Kaida et al. (2013)

Tensile strength σu 400 MPa 0.06 Normal Kaida et al. (2013)

Inside diameter ID 2400 mm 0.03 Normal Kaida et al. (2013)

Leak cost ratio CL / CR 50 - fixed -

Burst cost ratio CB / CR 500 - fixed -

Interest rate r 4 % - fixed -

*Standard deviation is reported

The posterior distributions of the model variables in the corrosion growth model are

summarized in Table 4.3. The mean of the exponential parameter θ2 > 1 indicates that the gamma

process is non-stationary with the corrosion rate increasing with time. The mean of the length

growth proportionality constant kj shows that the defect lengths are growing 11-42 times faster

74

than the their depths. Defect 1 is growing faster in length than the others and defect 2 is growing

slower, which corresponds to the measured corrosion in Table 4.1.

The results are compared with two simpler analysis methods to assess the impact of the

system reliability on the RBM analysis. The first method assumes independent defect failures

and failure modes. The second method considers only the least reliable single defect and ignores

the others, again assuming independent failure modes. The probability of failure in the

unrepaired case (Figure 4.2) demonstrates the difference between the three methods. The system

reliability method is bounded by the independent failure method (more conservative) and the

worst case method (less conservative). The probability of failure for the three analysis methods

ranges approximately half an order of magnitude. At low probability of failure, the system

reliability method approaches the independent failure method; because failures are rare so there

is minimal intersect between the failure events. As the probability of failure increases, the

failures intersect more frequently, and the system reliability has more of an impact, increasing

the difference between the system reliability and independent failure methods. The dependency

in the failure events between the defects comes from the dependency in the defect size due to the

exchangeable HBM and the applied pressure.

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Table 4.3. Posterior mean and confidence interval of the corrosion growth model

variables.

Variable Symbol Posterior Mean Posterior 90% CI

Multiplier parameter θ1 4.7 0.58 – 13.9

Exponential parameter θ2 1.1 0.77 – 1.4

Scale parameter [mm] β1 0.064 0.0051 – 0.21

β2 0.10 0.0091 – 0.34

β3 0.083 0.0071 – 0.27

β4 0.079 0.0070 – 0.26

β5 0.073 0.0064 – 0.24

β6 0.094 0.0082 – 0.30

Corrosion initiation time [years] t0,1 1.69 0.14 – 3.6

t0,2 1.27 0.081 – 3.2

t0,3 1.12 0.070 – 3.0

t0,4 1.29 0.083 – 3.3

t0,5 1.23 0.079 – 3.2

t0,6 1.03 0.063 – 2.8

Length growth factor k1 42.2 26.9 – 65.3

k2 11.2 6.5– 16.5

k3 19.3 12.4 – 28.1

k4 15.6 9.0 – 24.1

k5 24.3 15.9 – 35.5

k6 20.7 14.6 – 28.4

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Figure 4.2. Probability of failure using three analysis methods.

The failure probabilities are combined with the consequences to determine the expected cost

of each repair plan. For the decision process, only the relative costs are required, not the actual

costs. Taking the cost of repair as the base cost, two ratios are defined: CL / CR and CB / CR, the

leak cost and the burst cost ratios, respectively. The costs assumed for this example are shown in

Table 4.2. The repair plans outlined by each analysis method are ranked by lowest expected

lifecycle cost to determine the optimal plan (Table 4.4). The optimal repair plan differs for the

three analysis methods. The difference in the optimal plans demonstrates the impact of

considering the system reliability in the RBM. Without considering the system reliability, a sub-

optimal plan is reached where either some defects are repaired unnecessarily early as in the

independent failure method, or they are being neglected until too late as in the single worst

defect method. Further, the optimal repair plan for the worst case is not very meaningful; in this

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case, defect two is the most severe so it is repaired, and the risk of all the other defects is ignored

so they are not repaired. A key consideration in determining whether system reliability needs to

be considered is the number of defects. As the number of defects increases, system reliability

analysis becomes more important, because there is more frequent intersection between the failure

events. The opposite is also true, with a decreasing number of defects, the three analysis methods

become increasingly similar until they converge when there is only a single defect. The expected

lifecycle cost of the optimal plans (Figure 4.3) reflects the relationship in the probability of

failure, where the system reliability method is bounded by the independent failure and the worst

single defect methods.

Table 4.4. Optimal repair plans for each analysis method; NR is ‘no repair.’

Method tR,1

[years]

tR,2

[years]

tR,3

[years]

tR,4

[years]

tR,5

[years]

tR,6

[years]

System reliability NR 10 10 10 NR 10

Independent failures NR 5 10 10 10 5

Worst defect NR 5 NR NR NR NR

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Figure 4.3. Expected lifecycle cost comparison of the three analysis methods to assess the impact

of system reliability.

Focusing on the system reliability method, the top four repair plans are shown in Figure 4.4.

The objective of the RBM is to minimize the expected lifecycle cost of operating the pressure

vessel; therefore, the expected costs at t = tsl = 20 years are compared.

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Figure 4.4. Expected lifecycle cost of the top 4 ranked repair plans for the system reliability

method; NR is ‘no repair.’

As time progresses, the cumulative expected cost increases stepwise as the repair time for

each defect is reached. Simultaneously, the slope of the cost temporarily decreases as each defect

is repaired, and the risk of failure posed by the defect is eliminated. The slope of the expected

cost gradually increases again until the next repair occurs as the other defects become more

critical. The expected costs of the top 4 repair plans are similar, ranging from 4.42 to 4.45 times

the cost of repair. Consequently, in the event that the optimal repair plan violates some constraint

of the decision analysis (e.g. a reliability or failure risk constraint), one of the other top repair

plans could be substituted with a minimal cost increase.

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4.7 Conclusion

This paper presents a risk-based maintenance (RBM) framework for a deteriorating pressure

vessel inside an industrial facility. A pressure vessel is inspected and multiple defects are

detected. The pressure vessel operator needs to determine the optimal repair time for each of the

defects. For each defect, delaying repair has the benefit of reducing the repair cost due to

discounting for the time value of money; however, a delay also increases the risk of failure. The

proposed RBM framework considers the dependency in the failure events due to the multiple

defects and failure modes to determine the system reliability. The impact of the system reliability

is assessed by comparison to analysis without considering the dependency in the failure events.

The results show that the system reliability impacts the decision process, and an analysis without

considering system reliability can potentially lead to a suboptimal repair plan. The effect of the

dependent failure events is especially important in vessels with many defects, because the

intersection between the failures increases with an increasing number of defects. Further, the

analysis is dependent on the corrosion growth model, as the use of a different model affects the

results; however, the framework itself is generalized and can be used with any corrosion growth

model. The corrosion growth model presented in this study is a novel and simple method of

modelling both the depth and length of corrosion defects, facilitating a probabilistic reliability

analysis that considers both leak and burst failures. The RBM framework can aid facility

operators to make decisions regarding deteriorating pressure vessel repair. In cases of less severe

deterioration, the analysis can be useful in supporting the decision to delay repairs until the next

scheduled shutdown when it is more convenient and economical.

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5 DECISION MAKING FOR LONG TERM PIPELINE SYSTEM REPAIR

OR REPLACEMENT

This chapter presents a framework for long term maintenance decision making for a

deteriorating pipeline system. This chapter contributes to two research areas: RBM planning for

systems with multiple components, and uncertain system lifecycle. This chapter specifically

addresses the research objectives:



Incorporating the uncertainty in the lifecycle of a system into RBI and RBM planning

Assessing the impact of the uncertainty in the lifecycle on the decision analysis

There are two main contributions of this paper. The first contribution is the scale of the

analysis, which entails the full pipeline system over the long term. This allows the analysis of the

question of whether it is better to continuously repair pipeline defects as they become critical, or

to just replace sections of the pipeline. The methodology is applicable to full pipelines, with

many defects. The second contribution is the methodology to consider the uncertainty in the

lifecycle of the pipeline in the decision analysis, and the impact of the uncertainty on the RBM

decision.

This chapter is a journal paper (Haladuick and Dann, 2016b) that was submitted to the ASCE

Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering in May 2016,

and the second revision of the paper is under review as of September 2017.

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5.1 Abstract

Corrosion is one of the main causes of pipeline failure, which can have large social,

economic, and environmental consequences. To mitigate this risk, pipeline operators perform

regular inspections and repairs. The results of the inspections aid decision makers in determining

the optimal maintenance strategy. However, there are many possible maintenance strategies, and

a large degree of uncertainty, leading to difficult decision making. This paper develops a

framework to inform the decision of whether it is better over the long term to continuously repair

defects as they become critical, or to just replace entire segments of the pipeline. The method

uses a probabilistic analysis to determine the expected number of failures for each pipeline

segment. The expected number of failures informs the optimal decision. The proposed

framework is tailored toward mass amounts of in-line inspection data and multiple pipeline

segments. A numerical example of a corroding upstream pipeline illustrates the method.

5.2 Introduction

Corrosion is a process of structural deterioration, gradually reducing the wall thickness of

pipelines until failure occurs. These failures can have large social, economic, and environmental

consequences. In the United States corrosion accounts for 21 % of recent oil pipeline failures

(US DOT PHMSA incident database). To mitigate this risk, pipeline operators perform regular

inspections and repairs. For a large pipeline with many defects, these repair actions can be very

expensive, so it is important to optimize the maintenance plan.

Once a pipeline is inspected, the operator must decide which of the detected defects need to

be repaired before the next inspection. These defects are then repaired, the subsequent inspection

is performed, and the pipeline operator must again decide which defects should be repaired. This

83

process is repeated until the end of the service life of the pipeline. Once the pipeline is retired the

operator can look back over the maintenance history and sum up the total maintenance cost for

the pipeline. If only a few repairs were required the total maintenance cost is low; however, if

many repairs were required the maintenance cost is high. If many repairs were required then in

hindsight it would have been less expensive to cut out the severely corroded section of pipe and

make one replacement instead of continuously repairing the pipeline. While the optimal

maintenance plan is clear in hindsight, the number of repairs that will be required throughout the

service life cannot be known until the pipeline is retired, due to several factors, such as the

spatiotemporal uncertainty in the corrosion growth and the pipeline operating parameters. This

means that the decision of whether to repair multiple defects or to just replace a pipe segment is

subject to uncertainty.

Decision analysis (von Neuman and Morgenstern, 1947; Luce and Raiffe, 1957) is ideally

suited to decision making under uncertainty. Decision analysis uses a probabilistic approach to

quantify the expected cost of following each decision option, and the option with the lowest

expected cost is selected. The application of decision analysis to maintenance planning is termed

risk based maintenance (RBM) planning (Kahn and Haddara, 2003).

Many pipeline RBM studies (Pandey, 1998; Sahraoui et al., 2013; Nessim et al., 2000; Hong,

1997; 1999) focus on methods to determine the optimal next action for a decision maker. For

instance, they address questions like whether a corrosion defect should be repaired (e.g. Pandey,

1998). These questions are important, as in practice a decision maker is typically concerned with

what to do next. However, the optimal maintenance strategy is conditional on the scale of the

analysis. For instance, the optimal strategy for an analysis of a single defect may be to repair the

defect. But, if there are many other defects in that segment, the optimal strategy at the segment

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scale may be to replace the entire segment instead. And, if other adjacent segments are high risk,

then the optimal strategy at the pipeline scale may be to replace multiple adjacent segments since

one is already being replaced. Thus, the maintenance decision for each defect is dependent on the

maintenance decisions of the other defects. A small scale analysis considering only one defect or

segment will miss the global optimum strategy.

The objective of this paper is to inform the decision of whether it is better in the long term,

and over the scale of the entire pipeline, to continuously repair defects in an oil pipeline as they

become critical, or to just replace entire segments of the pipeline. The key contribution of this

paper is the long term perspective of pipeline system maintenance planning, which has not been

addressed in the pipeline RBM field. The analysis uses RBM, where the possible maintenance

plans include all combinations of either repairing or replacing segments of the pipeline, and all

possible replacement times for each segment. Since the actual number of future failures cannot

be known in advance, the method is centered on the expected number of failures for each

pipeline segment through the service life. Plotting the expected number of failures for each

pipeline segment creates a spatial profile of the failure risk across the pipeline. This profile

provides a long term estimation of the maintenance requirements for the pipeline throughout the

entire service life, and is used to inform the decision of whether to continuously repair defects or

to replace entire segments.

This paper comprises six sections. The second section provides a review of RBM and

pipelines. The third section presents the RBM framework for repair or replacement decision

making for a pipeline. The fourth section describes the implementation of the RBM framework

and is subdivided into three parts: corrosion growth modelling, remaining lifetime analysis, and

the expected costs of the RBM framework. The fifth section presents a numerical example of the

85

method, using a corroding upstream pipeline with some historical inspection data available, and

finally, the paper ends with some concluding remarks.

5.3 Background on pipeline RBM

RBM uses decision analysis to determine the best maintenance plan by selecting the plan

with the lowest expected cost of operating the pipeline over its service life. RBM differs from

risk based inspection (RBI) planning, which is concerned with optimizing the timing and type of

inspections. In an RBM analysis that is strictly concerned with optimizing the maintenance plan,

the goal is to determine the optimal maintenance plan given the current state of deterioration

knowledge. The current state of knowledge is a function of the previous inspection results, which

could have been scheduled based on a pipeline code, company policy, or RBI analysis. The

current state of knowledge changes each time an inspection is performed and new information is

becomes available; therefore, the RBM analysis should be undertaken after each subsequent

inspection to determine the new optimal maintenance plan.

There have been recent RBM studies for pipelines. Pandey (1998) performed RBM to decide

whether to repair a defect, as well as optimize the timing of a single inspection within the service

life. The analysis was simplified with a binary inspection result (detection or no detection of a

defect). Sahraoui et al. (2013) performed RBM to determine the optimal maintenance plan and

the optimal periodic time between inspections, accounting for uncertainties due to detection and

false detection, but not measurement error. Nessim et al. (2000) optimized the timing of the first

and second inspections considering the hazards of corrosion and mechanical damage. After the

inspection they made the decision to repair based on whether the defect exceeded a threshold,

and they performed RBM to determine the optimal repair threshold. Hong (1997, 1999)

86

optimized the timing and type of inspection time and performed RBM to assess the sensitivity to

the repair threshold. Other studies have focused on RBI planning, making the repair decision

based on a non-optimized threshold. Gomes et al. (2013) and Gomes and Beck (2014) performed

RBI for onshore buried pipelines subject to external corrosion. They considered only one

pipeline segment that contained at most one defect at a time. Uncertainty in whether an

inspection tool detected a defect was accounted for; however, measurement error was not.

Hellevik et al. (1999) performed RBI for pipelines subject to corrosion, optimizing the timing of

inspections, type of inspection, and number of inspections. The analysis was simplified with a

binary inspection result. Zhang and Zhou (2014) performed RBI to determine the optimal timing

of a single inspection of a natural gas pipeline. They constrained the analysis to only one

segment but allowed multiple defects in the segment and the generation of new defects, and

considered measurement error.

Pipeline failure can generally be defined as a loss of containment of the pipeline fluid. CSA

(2012) defines three failure modes for pipeline corrosion: small leak, burst, and rupture. A small

leak occurs when the maximum depth of the corrosion defect exceeds the wall thickness. A burst

occurs when the weakened pipe wall undergoes plastic collapse due to the internal pressure from

the fluid in the pipeline. Bursts are further categorized as ruptures if the length of the burst is

long enough that unstable axial growth occurs. Because of the catastrophic nature of burst and

ruptures, they are associated with greater consequences than leaks; however, they occur much

less frequently, with leaks accounting for 96.7 % of corrosion caused oil pipeline failure in the

United States (US DOT PHMSA incident database).

To mitigate the risk of pipeline failure, inspections and maintenance actions are undertaken.

Pipelines are typically inspected by in-line inspection (ILI), which involves passing a

87

measurement device, called a smart pig, through the pipeline. The result of the ILI is a dataset of

the location and size for all of the measured corrosion defects along the length of the pipeline.

Once an ILI is performed the pipeline operator has two main maintenance options, they can

either perform a local repair of a single defect, or they can cut out a section of pipe and replace it

(ASME, 2012). There are different types of repairs available, such as a welded sleeve or bolted

clamp, but the costs and performance of the methods are fairly similar, so for the purposes of this

study they are grouped under the “repair” umbrella. In a repair a single defect is located, and a

sleeve or clamp is placed over the pipe at that location to reinforce the pipe wall. The repair does

not affect the rest of the pipe segment. Replacement is larger scale, a section of pipe of some

length with multiple defects inside is located, and that section is cut out of the pipeline, and a

brand new section of pipe is welded in place. So replacement removes all of the defects in the

replaced section. The cost of repairing a single defect is less than the cost of replacing a section;

however, the replacement has the advantage of fixing many defects with one replacement action.

Both repairs and replacements are susceptible to future corrosion and can potentially fail;

however, the probability of failed repairs or replacements is low (Zhang and Zhou, 2014).

The service life of a pipeline can be difficult to predict, as it depends on many factors, e.g.

the economic feasibility of a project, the global economics of the oil and gas industry, the

structural condition of the pipeline etc. These factors are uncertain; therefore, it is more realistic

to treat the service life of a pipeline as uncertain. This study considers both a fixed and uncertain

service life and assesses the impact of treating the service life as uncertain. This is an advantage

over many other RBM frameworks (Gomes et al., 2013; Gomes and Beck, 2014; Hellevik et al.,

1999; Hong, 1997, 1999; Pandey, 1998; Sahraoui et al., 2013; Zhang and Zhou, 2014) that do not

consider the impact of an uncertain service life.

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5.4 RBM framework for the decision of whether to repair or replace

The goal of this study is long term repair or replacement decision making for a pipeline

system. The analysis begins after an ILI has been performed, and uses the inspection data to

decide whether single defects in segments of a pipeline should be repaired as they become

critical, or whether entire segments should be replaced. Ideally, the set of possible maintenance

plans would include all possible combinations of pipeline maintenance, including repairing any

combination of individual defects at any time, and / or replacing any number of sections of pipe

of any length at any time. However, for a pipeline with lots of defects and a long service life, the

number of possible strategies is extremely large, and the optimization problem is nearly

impossible to solve. To address this problem this study discretizes both space and time, and

makes two assumptions to restrict the set of possible maintenance strategies. First, the pipeline is

discretized into segments, and the corrosion in each segment is analyzed as an individual

population of defects (Dann and Maes, 2015a and 2015b). The discretization of the pipeline

impacts the decision analysis by restricting the decision options, meaning that the optimal

maintenance option is not necessarily in the restricted set. However, the impact of this

assumption is minimized by selecting the segments to capture the nature of the corrosion growth

(see the corrosion growth modeling section). Second, only one replacement action is allowed,

during which any number of adjacent segments can be replaced. We acknowledge that for large

pipelines, such as transmission lines, this assumption could be limiting, as multiple replacements

could be warranted. However, this assumption is suitable for shorter lines, such as subsea and

upstream, where multiple replacements at many locations is not common practice. This

assumption shifts the optimal maintenance decision towards repairing more segments than

otherwise would be if more than one replacement action was allowed. The maintenance decision

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then becomes whether to replace a set of adjacent segments or not, and if so, when. Looking at

the entire pipeline, a maintenance strategy k = (sL, sR, tL) is defined by the set of segments to

replace sL, the set to repair sR, and the time of replacement tL. The set of segments to replace sL

and repair sR combine to form the total set of segments s (s = 1, …, S) in the pipeline. Thus, each

segment must follow a strategy of either replacement or repair, including segments with very

minimal corrosion, as the corrosion could become more severe in the future. A decision tree

(Figure 5.1) is used to model this scenario.

Figure 5.1. Decision tree for the decision of whether to repair the defects or replace the segment.

The costs of decision option k are the replacement cost, which is a function of the set of

segments sL to be replaced and the time of replacement tL; the repair cost, which is a function of

the set of segments sR to be repaired; and the failure cost. The consequences of system failure

include non-monetary attributes, e.g. human casualties and permanent environmental damage;

however, it is possible to monetize these attributes for comparison with other costs, or to use

multi-attribute decision making (Xu, 2015). The total expected cost E[C(k)] of maintenance

strategy k is the sum of these costs. The objective of the RBM framework for repair or

replacement decision making is to minimize the total expected cost of maintaining the pipeline,

and the objective function is given by:

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𝐸[𝐶(𝒌∗)] = min(E[𝐶(𝒌)] (5.1)

In this way the RBM framework is formulated as an optimization problem: find the optimal

maintenance strategy k* that minimizes the total expected cost E[C].

Along with the decision of the type of maintenance to perform, the decision of when to

perform the maintenance is also required. Delaying maintenance has the benefit of decreasing the

expected maintenance cost by discounting the cost by the interest rate r, to account for the time

value of money. However, it also impacts the cost of failure. Consider the decision to replace a

segment. Performing the replacement earlier decreases the probability of failure prior to the

replacement. However, if the segment is replaced very early, there is still a lot of time remaining

in the service life and the new segment may begin to corrode and potentially fail. Thus, there is a

tradeoff in delaying replacement, and the decision framework optimizes the replacement time tL.

Also, in this way the risk of potential failure of the replacement is incorporated in the analysis.

The same tradeoff in delaying maintenance exists for the segments that are repaired, except

in this case there is an optimal repair time for each defect in the segment. To determine the

optimal repair time for each defect the analysis must be performed on each defect individually.

However, as previously stated, the analysis in this paper is performed for the population of

defects in each segment to allow analysis of an entire pipeline of many defects. Therefore, the

repair time of each individual defect cannot be determined. Instead, it is assumed that a repair is

performed when a defect exceeds a depth of 80 % wall thickness (wt) which is the repair criteria

as per code B31G (ASME, 1991). Therefore, the repair time is not optimized and is not part of

the maintenance strategy k. Assuming the repair time shifts the results of the decision analysis

slightly towards replacement because the assumed repair times will not necessarily be the

optimal repair times. However, the assumption reflects actual pipeline operation, where the

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repairs must be performed according to regulations. While this assumption allows the repair cost

to be determined, the expected failure cost both before and after repair still cannot be determined

because it is not known when each individual repair is performed. Therefore, this paper assumes

the repaired defects do not fail. Neglecting the cost of failure for the repaired segments slightly

biases the optimal decision towards repairing segments instead of replacing them; however, the

risk of failure is deemed to be low because the defects are repaired at 80 % wt, and repairs rarely

fail (Zhang and Zhou, 2014). The analysis still clearly identifies situations where replacement is

the optimal decision because of this bias towards repair.

5.5 Implementation of the RBM framework

The implementation section is composed of three parts. The first part presents a probabilistic

corrosion growth model to predict the future state of corrosion based on the historical ILI data.

The second part describes the method to determine the time to failure for the population of

defects in each segment. The time to failure is then combined with the finite service life of the

pipeline to determine the expected number of failures for each segment, which forms the long

term risk profile of the pipeline. The third part derives the equations of the expected costs for the

RBM framework for repair or replacement decision making.

5.5.1 Corrosion growth modelling

The RBM framework for repair or replacement decision making requires the expected

number of failures for each segment as an input, which in turn requires a model of the corrosion

growth through time. There are many probabilistic corrosion growth models suggested in the

literature and thorough reviews are provided in Pandey et al. (2009) and Brazán and Beck

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(2013). For the purposes of the RBM framework any suitable corrosion growth model can be

used. The objective of this study is maintenance decision making for the entire pipeline system,

requiring the analysis of mass ILI data. The population based corrosion growth model (Dann and

Maes, 2015a and 2015b) is used as it is tailored towards mass amounts of data. In the population

based model the ILI data are pooled as a single population, and the growth is assessed between

inspection populations at different times.

A satisfactory model must account for both the aleatory and epistemic uncertainty (JCSS,

2008) present in the corrosion growth process. Aleatory uncertainty is due to the natural

variability of a process and cannot be reduced. For corrosion growth this includes spatial and

temporal uncertainty. Spatial uncertainty is due to the difference in corrosion growth at different

defect locations. This paper addresses the spatial uncertainty by applying the population based

approach on a segment wide basis, which is termed the segment based approach. In this approach

the pipeline is divided into segments and each segment is modelled as an independent

population. An example of a possible method is to use the girth welds to separate the segments.

Another possible method is to divide the pipeline into segments of similar geographic regions

and / or elevation profiles. Determining how to divide the pipeline is beyond the scope of this

study, but the method should consider the potential spatial variation in corrosion. For example, if

the pipeline is to be divided using the girth welds, any potential correlation between the heat

affected zones at each girth weld and the corrosion needs to be accounted for. Temporal

uncertainty is the uncertainty in predicting the future deterioration based on the present

deterioration, due to the stochastic nature of the corrosion process. This paper uses a stochastic

process model to account for the temporal uncertainty by treating the corrosion at any time ti as a

random variable. For corrosion growth, epistemic uncertainty includes inspection uncertainty,

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statistical uncertainty, and model uncertainty. Unlike aleatory uncertainty, epistemic uncertainty

can be reduced, for instance by using better measurement tools, performing more inspections,

and developing better models. Figure 5.2 is a graphical representation of the segment based

hierarchical corrosion growth model (Koller and Friedman, 2009).

Figure 5.2. Hierarchical graphical model of the segment based corrosion growth process.

The measured corrosion depth Yi at each inspection i (i = 1, …, I) at time ti is processed to

account for inspection uncertainty, yielding the actual corrosion Xi at each inspection time ti. As

a reference point, the time of the most recent inspection tI = 0, so all previous inspections

occurred at negative times, and all future times t are positive. Inspection uncertainty is due to

several sources, including measurement error, not detecting an existing defect (misses), detecting

a fictitious defect (false call), and a minimum threshold for reported defects. This paper only

considers measurement error; refer to Dann and Maes (2015a and 2015b) for the full model. The

measurement error εi is the difference between the observed corrosion and the actual corrosion:

𝑌𝑖 = 𝑋𝑖 + 𝜀𝑖 for i = 1, …, I (5.2)

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where εi is the measurement error representing the random scatter in the corrosion depth sizing

model. The probability density function (pdf) 𝑓𝑌𝑖(𝑦𝑖) of the population of measured corrosion

depth is obtained by pooling the results of the ILI at time ti. The posterior pdf 𝑓𝑋𝑖|𝑌𝑖(𝑥𝑖|𝑦𝑖) of the

population of the actual corrosion depths is given by Bayes rule:

𝑓𝑋𝑖|𝑌𝑖(𝑥𝑖|𝑦𝑖) ∝ 𝐿𝑋𝑖|𝑌𝑖

(𝑥𝑖|𝑦𝑖)𝑓𝑋𝑖(𝑥𝑖) for i = 1, …, I (5.3)

where 𝐿𝑋𝑖|𝑌𝑖(𝑥𝑖|𝑦𝑖) is the likelihood function of the actual corrosion Xi conditioned on the

measured corrosion yi, and 𝑓𝑋𝑖(𝑥𝑖) is the prior pdf of Xi. The likelihood function is given by:

𝐿𝑋𝑖|𝑌𝑖(𝑥𝑖|𝑦𝑖) = ∑ 𝐿𝑋𝑖|𝑌𝑖,𝑗

(𝑥𝑖|𝑦𝑖,𝑗)𝑓𝑌𝑖(𝑦𝑖,𝑗)𝐽

𝑗=1 for i = 1, …, I (5.4)

where 𝐿𝑋𝑖|𝑌𝑖,𝑗(𝑥𝑖|𝑦𝑖,𝑗) is the likelihood function for a given defect depth yi,j, and the summation is

performed across the set of defects j (j= 1, …, J).The population of the actual corrosion depths

for each inspection is the sum of the corrosion at the previous inspection and the incremental

growth:

𝑋𝑖 = 𝑋𝑖−1 + ∆𝑋𝑖 for i = 1, …, I (5.5)

where ΔXi is the incremental corrosion growth from ti-1 to ti. As a starting point for the process,

the initial population of actual corrosion depths X0 at the corrosion initiation time t0 is assumed to

be deterministically 0. The relationship between the pdfs of Xi-1 and Xi is given by the

convolution integral:

𝑓𝑋𝑖(𝑥𝑖) = ∫ 𝑓𝑋𝑖−1

(𝑥𝑖−1)𝑓∆𝑋𝑖(𝑥𝑖 − 𝑥𝑖−1)𝑑𝑥𝑖−1 for i = 1, …, I (5.6)

where 𝑓𝑋𝑖(𝑥𝑖) is the pdf of Xi, 𝑓𝑋𝑖−1

(𝑥𝑖−1) is the pdf of Xi-1, and 𝑓∆𝑋𝑖(𝑥𝑖 − 𝑥𝑖−1) is the pdf of the

corrosion growth increment ΔXi. Since the pdfs 𝑓𝑋𝑖−1(𝑥𝑖−1) and 𝑓𝑋𝑖

(𝑥𝑖) are known, the objective

is to determine the pdf of the growth. A closed form de-convolution of (5.6) is not available.

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Instead, this paper follows the approach of Dann and Maes (2015b) and assumes a distribution

for the growth increment ΔXi, and performs the convolution to set the parameters. An important

characteristic of physical corrosion growth is that it is always positive and monotonically

increasing with time, and the corrosion model should reflect this. To satisfy the positive growth

requirement the growth increments of the process must only have positive support. While many

processes satisfy this requirement, the gamma process has been shown to be a satisfactory

stochastic process to model the corrosion growth in structures and pipelines (Pandey and van

Noortwijk, 2004; Pandey et al., 2009; van Noortwijk, 2009). In the gamma process the corrosion

growth increment ΔXi is gamma distributed:

∆𝑋𝑖|Δ𝛼𝑖 , 𝛽~gamma(∆𝛼𝑖 , 𝛽) for i = 1, …, I (5.7)

where Δαi is the unknown location and time specific shape parameter, β is the unknown location

specific scale parameter, and the gamma pdf is defined as

𝑓∆𝑋𝑖(∆𝑥𝑖|∆𝛼𝑖, 𝛽) = ∆𝑥𝑖

∆𝛼𝑖−1𝑒−∆𝑥𝑖 𝛽⁄ (Γ(∆𝛼𝑖)𝛽∆𝛼𝑖)⁄ . To satisfy the monotonic deterioration

condition the shape parameter Δαi is defined as a function of time:

Δ𝛼𝑖 = 𝜃1 {(𝑡𝑖 − 𝑡0)𝜃2 − (𝑡𝑖−1 − 𝑡0)𝜃2} for i = 1, …, I (5.8)

where θ1 > 0 and θ2 > 0 are unknown corrosion model parameters. Thus, to define the process

there are three unknown hyper-parameters λ = (θ1, θ2, β).

Since corrosion growth modelling is not the focus of this paper an approximate solution to

the model is provided using the following simplifications; for a full Bayesian solution refer to

Dann and Maes (2015b). First, λ is treated as deterministic; second, the model is solved in a

stepwise manner from the bottom to the top; third, the corrosion growth increments are assumed

to be conditionally independent; and fourth, the defects are grouped by segment. To solve the

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model, pdfs of the population of measured corrosion 𝑓𝑌1(𝑦1) and 𝑓𝑌2

(𝑦2) for each segment are

created by discretizing the inspection results into bins of 1 % wt. The methodology in this study

is compatible with any assumed distribution of the measurement error, and for simplicity this

study follows the common practice of assuming a normal distribution with a mean of 0 and

standard deviation of σε. The likelihood function 𝐿𝑋𝑖|𝑌𝑖,𝑗(𝑥𝑖|𝑦𝑖,𝑗) for each defect is then a

truncated normal with a mean of yi,j and a standard deviation of σε. There is no information

available for the prior distribution of the actual corrosion Xi; therefore, the Jeffrey’s prior

(Gelman et al., 2014) is used as a non-informative prior, which for the normal distribution with a

fixed variance is the un-normalized uniform distribution.

The time increment ti-1 to ti in Equation (5.8) can be adjusted to define the growth for any

increment, and the convolution can be performed to determine the pdf of the corrosion depth

predicted by the growth model at any inspection time. This means there are I-1 pdfs of the

corrosion depth predicted from the growth model to compare with the pdfs of the actual

corrosion obtained from the sizing error model. For the general case of I ≥ 3 previous

inspections, optimization is used to determine the hyper-parameters that produce the best fit of

this series of pdfs. For the special case of I = 2 previous inspections this paper proposes the

following solution. In this case there is not enough information to establish the exponential factor

θ2, therefore the growth process is assumed to be linear (θ2 = 1). Initial point estimates of θ1 and

β are determined by setting the mean and variance of the gamma distributed growth equal to the

first and second moments of the actual growth increment:

∆�̂�𝐼 = {E[𝑋𝐼] − E[𝑋𝐼−1]} 2 {VAR[𝑋𝐼] − VAR[𝑋𝐼−1]}⁄ (5.9)

�̂� = {VAR[𝑋𝐼] − VAR[𝑋𝐼−1]} {E[𝑋𝐼] − E[𝑋𝐼−1]}⁄ (5.10)

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Fitting the mean ensures the process is a good overall fit to the actual corrosion depth from

the sizing error model. Fitting the variance ensures that the process accurately represents the

uncertainty. Next the fit of the upper tail of the corrosion is assessed. The fit of the upper tail is

important because the extreme defects cause failures. If the fit of the upper tail is unsatisfactory

the variance of the gamma distributed growth is adjusted to create a better fit. For the case when

I = 0 or 1 previous inspections there is not enough information to establish either the multiplier

parameter θ1 or the exponential parameter θ2 and prior information must be used to determine θ1

and θ2 if possible. Once the hyper-parameters are known they are used to define the gamma

distributed corrosion growth increment as per Equation (5.8) for the interval from the time of the

most recent inspection tI to any future time t. The growth increment is then convoluted with the

most recent actual corrosion depth XI obtained from the sizing error model as per Equation (5.6)

to determine the growth model predicted depth X(t) at any future time t.

5.5.2 Determining the time to failure and the expected number of failures

The objective of this study is maintenance decision making for an entire pipeline system, so a

segment based approach to corrosion growth modelling is used. In the segment based approach

the defect size data is pooled. Burst failure analysis is not possible with pooled data, so it is

neglected. This simplification is expected to have minor impact because of the much higher

frequency of leak failure. The RBM methodology presented in this study would support other

reliability methods, such as the enhanced Monte Carlo method (Leira et al., 2016), which allow

the analysis of both burst and leak failure. However, other reliability methods probably will not

be as well suited to large amounts of data as the population based method, and their performance

for a large pipeline of many defects would have to be examined.

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For leak failure, the time to failure TF is when the defect grows to a critical depth xcrit (van

Noortwijk et al., 2009). The cumulative distribution function (cdf) of the time to failure 𝐹𝑇𝐹(𝑡)

of the population is the probability that the time to failure TF does not exceed the future time t:

𝐹𝑇𝐹(𝑡) = Pr(𝑇𝐹 < 𝑡) = Pr (𝑋(𝑡) ≥ 𝑥𝑐𝑟𝑖𝑡) = ∫ 𝑓𝑋(𝑡)(𝑥(𝑡))𝑑𝑥

∞

𝑥𝑐𝑟𝑖𝑡

(5.11)

where TF is the time to failure, X(t) is growth model predicted corrosion depth in the segment at

some future time t, and 𝑓𝑋(𝑡)(𝑥(𝑡)) is the pdf of X(t). For a probabilistic analysis CSA (2012)

defines xcrit as 100 % wt, as defects fail once they grow through the full wall thickness. The same

process can be performed with xcrit = 80 % wt, which is the critical level at which repairs are

required as per code B31G (ASME, 1991). This results in the cdf 𝐹𝑇𝑅(𝑡), where TR is the time to

repair.

For a fixed service life tsl, (5.11) is evaluated at t = tsl to determine the probability of defects

out of the population that fail within the service life 𝑝𝐹|𝑡𝑠𝑙 = 𝐹𝑇𝐹(𝑡𝑠𝑙). The probability of failure

at future times is conditional on the pipeline surviving until the most recent inspection with

certainty, thus pF,I = 0. The RBM framework for repair or replacement decision making requires

the number of failures nF. From the classical definition of probability, the probability of failure

event F is given by pF = nF / ntotal, where nF is the number of failures F and ntotal is the total

number of trials. The actual number of failures nF is uncertain, so the expected number of

failures E[nF] is used. Rearranging the equation yields the expected number of failures E[nF] = pF

J, where the total number of trials ntotal is taken as the number of defects J in the population.

Ideally J is the number of defects in the population at tsl; which is uncertain due to the generation

of new defects, as well as detection and false call uncertainties. For simplicity this paper ignores

defect generation and assumes J is the number of defects reported in the most recent ILI. There is

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typically a high degree of uncertainty in the service life of a pipeline. This uncertainty is

accounted for by treating the service life as a random variable Tsl, in which case pF is given by

the integral:

𝑝𝐹 = ∫ 𝐹𝑇𝐹(𝑡)𝑓𝑇𝑠𝑙

(𝑡)𝑑𝑡∞

0

(5.12)

where 𝑓𝑇𝑠𝑙(𝑡) is the pdf of the service life of the pipeline.

The expected number of failures is informative for long term decision making for the

pipeline. A graphical representation of the expected number of failures for each segment s (s = 1,

…, S) shows the segments with the highest number of expected failures within the service life

and can be thought of as a long term spatial risk profile of the pipeline. If the cost of failure CF

varies throughout the pipeline, for instance for a natural gas pipeline with different classes

(sensitivity to failure), this can be incorporated by plotting the risk Rs for each segment s:

𝑅𝑠 = E[𝑛𝑓]𝑠𝐶𝐹 (5.13)

where Rs is the risk of failure for segment s if no mitigation action is taken between now and tsl

and E[nF]s is the expected number of failures for segment s.

5.5.3 Expected costs for the RBM framework

The expected cost E[C(k)] of maintenance strategy k is given by the following:

E[C(k)] = E[CL(k)] + E[CR(k)] (5.14)

where E[CL(k)] is the expected cost associated with the set of replaced segments sL, and E[CR(k)]

is the expected cost associated with the set of repaired segments sR. E[C(k)] is determined for

each maintenance strategy k = (sL, sR, tL) which is made up of the combinations of the set of

segments to replace sL, the set of segments to repair sR, and the replacement time tL. The optimal

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maintenance decision k* corresponds to the minimum expected cost as per (5.1). As mentioned

previously, the set of maintenance strategies is made up of all combinations of one single

replacement of any number of adjacent segments at any time.

The cost of replacing an additional adjacent segment is not twice the cost of a single

replacement because there is a savings in only mobilizing resources once to perform the

replacement. The savings is quantified by defining the total replacement cost as a function of the

number of segments q to be replaced. For instance, the replacement cost can be defined as

linearly increasing at a rate of ωCL where ω is the proportional cost of an additional replacement.

In this case, the expected pipeline wide cost for the replaced segments sL conditional on the fixed

service life tsl is given by:

E[𝐶𝐿(𝒌)|𝑡𝑠𝑙] =𝐶𝐿(1 + 𝜔𝑞 − 𝜔)

(1 + 𝑟)𝑡𝐿

+ ∑ {𝐶𝐹𝐽𝑠 ∫𝑓𝑇𝐹,𝑠

(𝑡)

(1+𝑟)𝑡 𝑑𝑡 + 𝐶𝐹𝐽𝑠 ∫𝑓𝑇𝐹,𝑠

(𝑡)

(1+𝑟)𝑡 𝑑𝑡𝑡𝑠𝑙−𝑡𝐿−𝑡𝐼

0

𝑡𝐿

0}𝑠∈s𝐿 (5.15)

where s is the segment index, tL is the replacement time, CL is the cost of a single replacement,

CF is the cost of a single failure, ω is the proportional cost of an additional segment replacement,

Js is the number of defects in segment s, r is the interest rate, t is the time from now proceeding

into the future, and 𝑓𝑇𝐹,𝑠(𝑡) is the pdf of the time to failure for segment s, which is the derivative

of the cdf 𝐹𝑇𝐹,𝑠(𝑡). The first term is the cost of replacing q segments. The second and third terms

are the cost of failure. The second term is the expected cost of failure prior to the replacement, so

the integral is performed from now until the replacement time tL. The third term is the expected

cost of failure post replacement, which considers the potential for the new segment to fail. The

new segment is assumed to be identical to the original segment; therefore the same pdf of the

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time to failure 𝑓𝑇𝐹,𝑠(𝑡) is used. For post replacement failure the integral bounds are adjusted to

remove the service life of the original segment, which was up to tL, and the time from when the

original pipeline was built until the time tI of the most recent inspection. The summation is

performed over the set of segments sL to be replaced and each term is discounted by the interest

rate.

For repair, the expected pipeline wide cost for the set of repaired segments sR conditional on

the fixed service life tsl is given by:

E[𝐶𝑅(𝒌)|𝑡𝑠𝑙] = ∑ 𝐶𝑅𝐽𝑠 ∫𝑓𝑇𝑅,𝑠

(𝑡)

(1 + 𝑟)𝑡𝑑𝑡

𝑡𝑠𝑙

0𝑠∈s𝑅

(5.16)

where CR is the cost of a single repair and 𝑓𝑇𝑅(𝑡) is the pdf of the time to repair for segment s.

This equation is the cost of performing all of the repairs required in segment s. The repair timing

follows the pdf of the time to repair 𝑓𝑇𝑅(𝑡), which is based on repairing the defects when they

reach the critical repair threshold of 80 % wt. The time to repair is integrated across the service

life tsl and multiplied by the number of defects Js, yielding the expected number of repairs. The

expected number of repairs is then multiplied by the cost of a single repair yielding the total cost

of repairing the segment. The integral is discounted to account for the time value of money. The

summation is performed across the set of segments to be repaired sR yielding the total cost of

repairs for the pipeline. As stated previously, the equation for the expected cost of repair (5.16)

neglects the cost of failure for the repaired segments.

For an uncertain service life the expected costs are given by:

E[𝐶𝐿(𝒌)] = ∫ E[𝐶𝐿(𝒌)|𝑡𝑠𝑙]𝑓𝑇𝑠𝑙

∞

0

(𝑡𝑠𝑙)𝑑𝑡𝑠𝑙 (5.17)

E[𝐶𝑅(𝒌)] = ∫ E[𝐶𝑅(𝒌)|𝑡𝑠𝑙]𝑓𝑇𝑠𝑙

∞

0

(𝑡𝑠𝑙)𝑑𝑡𝑠𝑙 (5.18)

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where 𝑓𝑇𝑠𝑙(𝑡) is the pdf of the uncertain service life Tsl. These equations weigh the expected costs

for a given service life by the probability of that service life occurring.

To implement the RBM framework for repair or replacement decision making, first the

expected cost of replacement is determined as per Equation (5.15) and repair as per Equation

(5.16). There are no closed form solutions to the integrals in these equations, but they can be

approximated numerically by discretizing time and performing the summation. A sensitivity

analysis was performed on the size of the time increment and it was found that discretizing time

into 1 year increments was sufficient to determine the optimal decision k*. The expected costs

are evaluated for each maintenance strategy k and the total expected cost of each maintenance

strategy is determined as per Equation (5.14). The minimum total expected cost corresponds to

the optimal maintenance strategy k*. For the case of an uncertain service life there is no closed

form solution for the integrals in Equations (5.17) and (5.18), but again they can be

approximated by discretizing time and determining the expected cost for the set of discrete

service lives. Again time increments of 1 year were found to be sufficient, and the increments

should extend from tsl = 0 up to a time with a low exceedance probability of Tsl; an exceedance

probability of 10-5

was used in this study.

5.6 Numerical example of a corroding upstream oil pipeline

Consider an upstream oil pipeline undergoing structural deterioration due to corrosion. For

example purposes a short 10 segment pipeline is assessed. The pipeline was built 10 years ago,

and has had 2 ILI’s to date, denoted ILI1 and ILI2. ILI1 was performed 5.4 years ago, and ILI2

was just performed. The ILI’s were performed with a magnetic flux leakage tool, with a

confidence interval of ±10 % wt at an 80 % confidence level (POF, 2009), corresponding to a

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normally distributed measurement error with a mean of 0 and a standard deviation of σε = 7.8 %

wt. The results of ILI’s 1 and 2 are shown in Figure 5.3. ILI1 detected 1528 defects, with an

overall mean defect depth of 4.0 % wt, and a standard deviation of 3.0 % wt. ILI2 detected 1878

defects, with an overall mean depth of 13.9 % wt and a standard deviation of 5.0 % wt. Also,

there is spatial variability in the corrosion between the segments, with some of the segments (e.g.

segment 1) having many more defects than others (e.g. segment 8). The distribution of the

uncertain pipeline service life should reflect the state of knowledge and uncertainty at the time of

the analysis. In this example, the uncertain service life Tsl is assumed to follow a normal

distribution with a mean of 30 years after ILI2 and a standard deviation of 5 years. For

comparison, three cases of fixed service life are also considered: tsl = 25, 30, and 35 years from

now, which correspond to the mean and one standard deviation away from the mean of the

uncertain service life. All calculations in this example are performed for each of the 10 segments

since the pipeline was spatially discretized by segment.

The pdfs of the measured corrosion are processed to determine the pdfs of the actual

corrosion X1 and X2 from the sizing error model. The corrosion growth is assumed to be gamma

distributed, and the growth increment from t1 to t2 is convoluted with X1 yielding the depth

predicted by the growth model, which is then fit to the actual depth X2 from the sizing error

model X2, to determine 𝜃1 and �̂� assuming θ2 = 1 for linear growth. The upper tail of the growth

model predicted corrosion at t2 needs to be a good fit to the actual corrosion from the sizing error

model to yield an accurate reliability estimate. To fit the upper tail of the growth model predicted

corrosion, the variance of the gamma distributed growth increment ΔX12 is adjusted, while the

mean remains fixed. The exceedance plot (Figure 5.4 for segment 1) compares the upper tail of

the growth model predicted corrosion at t2 to the actual corrosion from the sizing error model,

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showing the accuracy of the fit. The measured corrosion Y2 is also shown in Figure 5.4 for

comparison. The measured corrosion Y1 and Y2 are technically discrete since they are binned;

however, they are shown as approximations of continuous pdfs since there are 100 bins. The

final parameters of the gamma growth are shown in Table 5.1 and the pdfs 𝑓𝑌(𝑦), 𝑓𝑋(𝑥), and the

growth model predicted fit pdf 𝑓𝑋(𝑥) are shown in Figure 5.5 for segment 1.

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Figure 5.3. Results of ILI 1 and 2. ILI1 is blue and ILI2 is red. J is the number of defects.

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Figure 5.4. Exceedance probability plot of the corrosion process for segment 1 at time t2.

Table 5.1. Results of the corrosion growth model and reliability analysis.

Segment 𝜃1 �̂� 0.1 percentile

TF (yrs)

0.1 percentile

TR (yrs)

1 0.38 3.2 29.1 17.6

2 0.60 2.5 26.3 16.2

3 0.44 3.0 27.8 16.9

4 0.68 1.8 35.7 22.8

5 0.41 2.8 33.0 20.3

6 0.37 2.2 55.2 36.4

7 0.34 2.1 62.7 41.5

8 0.38 2.1 58.7 39.5

9 0.47 2.3 37.7 23.9

10 0.35 2.6 44.7 28.6

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Figure 5.5. pdfs of the corrosion process for segment 1.

The cdfs of the time of failure 𝐹𝑇𝐹(𝑡) and the time to repair 𝐹𝑇𝑅

(𝑡) (Figure 5.6) are obtained

by convoluting the gamma increments with the current actual corrosion X2 from the sizing error

model, and integrating above the critical levels of 100 % wt and 80 % wt respectively. The 0.1

percentiles of the time to failure TF and the time to repair TR are shown in Table 1. It can be seen

that segment 2 is the quickest to reach a probability of failure of 0.1 % after only 26.3 years,

whereas segment 7 is the slowest after 62.7 years. Similarly, segment 2 will require 0.1 % of the

defects to be repaired after only 16.2 years, whereas segment 7 will require 0.1 % of the defects

to be repaired after 41.5 years. The area in Figure 5.6 where the time to failure TF is less than the

service life tsl or Tsl is related to (but not equal to) the probability of failure pF. The probability of

failure pF for each segment is used to determine the expected number of failures E[nF]. The plot

of the expected number of failures (Figure 5.7) is the reliability profile of the pipeline, revealing

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that segments 2, 3, and 1 respectively are the greatest concern. Cross referencing these segments

with the ILI data (Figure 5.3) shows these segments have a relatively higher combination of the

number of defects and defect growth between ILI’s. The expected number of failures increases

with increasing fixed service life, and the uncertain service life lies between a fixed service life

equal to the mean and one standard deviation above the mean.

Figure 5.6. cdfs of the time to failure and repair for segment 1.

Figure 5.7. Risk profile for fixed and uncertain service lives.

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Defects that are expected to fail need to be maintained during the service life of the pipeline.

The RBM framework for repair or replacement decision making informs the decision maker of

whether these defects should be repaired one at a time or whether entire segments should be

replaced. The optimal maintenance strategy is dependent on the interest rate r, the cost of failure

CF, the cost of a single repair CR, the cost of a single segment replacement CL, and the cost factor

ω which is the proportion of CL that it costs to replace additional adjacent segments. Only the

ratios and not the absolute values of the costs are relevant, and since the cost of repair is the

lowest it is used as the reference. Base values of CF / CR = 1000, CL / CR = 10, r = 3 %, and ω =

0.2 are used for the analysis. The sensitivity of the optimal repair strategy to the fixed service life

is shown in Figure 5.8. The optimal maintenance decision switches as the fixed service life

changes. For fixed service lives of tsl < 29 years the optimal decision is to repair all of the

segments. Then as the service life increases the optimal decision shifts towards replacing more

and more segments, until for fixed service lives tsl > 42 years the optimal decision is to replace

all of the segments. This is because as the fixed service life increases more and more repairs are

required, and so it becomes more economic to replace segments instead. The total expected cost

increases with increasing service life as expected. The expected cost of replacement also

increases with increasing fixed service life, as more and more segments are replaced. The

expected cost of repair is the opposite; decreasing with increasing fixed service life as less and

less segments are repaired. The expected cost curves for repair and replacement change abruptly

as the decision switches; however, the total expected cost curve remains smooth.

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Figure 5.8. Sensitivity of the expected costs to the fixed service life. The optimal maintenance

decision for each section of the fixed service life is shown, gray segments are replaced, white

segments are repaired.

The top three strategies for the cases of fixed tsl = 25, 30, and 35 years, and the uncertain

service life are shown in Table 5.2. In general, the longer and more uncertain the service life, the

more the optimal strategy shifts towards replacing segments. The optimal strategy for the

uncertain service life is to replace the first five segments, which is different than the optimal

strategy for a fixed service life equal to the mean (tsl = 30 years) of the uncertain service life. The

difference in strategy demonstrates the importance of considering the uncertainty in the service

life when making long term decisions, as discussed in de Jonge et al. (2015). Comparing the

strategies to the reliability profile (Figure 5.7) shows that in general the decision switches from

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replacement to repair at abrupt changes in the risk of the pipeline, for instance between segments

3 and 4, and between segments 5 and 6. The optimal replacement time tL has a very minor

sensitivity to the service life, in general shifting slightly later as the service life increases.

Table 5.2. Top three maintenance strategies for each of the three cases of fixed service life as

well as the uncertain service life. Gray segments are replaced and white segments are repaired.

Analysis is performed with the base case of input parameters.

Service life Strategy

Rank

Maintenance strategy k E[𝐶(𝒌)]

𝐶𝑅

sL, sR tL (yrs)

Fixed tsl = 25 yrs 1 NA 7.5

2 22 11.2

3 23 11.5

Fixed tsl = 30 yrs 1 22 13.4

2 22 13.9

3 22 14.6

Fixed tsl = 35 yrs 1 23 15.5

2 23 16.9

3 23 18.2

Uncertain Tsl ~

normal(30 yrs, 5 yrs)

1 22.5 14.4

2 22.3 14.6

3 22.4 15.6

The impact of the cost of failure CF and the interest rate r is also assessed with a sensitivity

analysis (Figure 5.9 and Figure 5.10). The total cost increases non-linearly with increasing cost

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of failure. For a typical range of failure cost CF / CR = 500 to 2000 there is only a moderate linear

increase in the expected cost, showing that the expected cost is only moderately sensitive to the

failure cost. Throughout the range of failure cost the optimal maintenance decision for each

service life switches. As the cost of failure increases the optimal decision shifts towards repairing

more segments, because the cost of failure is not considered in the case of repair. This is the

reason the expected cost for the fixed service life tsl = 25 years is constant, in this case the

optimal decision is always to repair all segments, so the cost does not increase with increasing

cost of failure.

The expected cost decreases non-linearly with increasing interest rate, and again the

sensitivity is moderate. As the interest rate increases the optimal decision shifts towards

replacing more and more segments at a later replacement time tL, as later replacements are less

expensive. However, this increases the cost of failure for these segments, so the optimal decision

at high interest rates is to repair the highest risk segments and replace all the others. Comparing

the different service lives, the total cost increases as the fixed service life increases, and the cost

for the uncertain service life always lies between the cost for the fixed service lives tsl = 30 and

35 years, which is the mean and one standard deviation above the mean of the uncertain service

life.

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Figure 5.9. Sensitivity of the expected costs to the cost of failure.

Figure 5.10. Sensitivity of the expected costs to the interest rate.

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5.7 Conclusion

This paper presents a RBM framework for the decision of whether it is better over the long

term to continuously repair corrosion defects as they become critical or to replace entire pipeline

segments. If replacement is required, the optimal replacement time is also determined. The

optimal maintenance decision, when considering the entire pipeline over its entire service life, is

not the same as the optimal decision when only considering a single defect, or even a single

pipeline segment. The decision of whether to continuously repair defects as they become critical,

or to replace entire pipeline segments, is driven by the expected number of failures for the

pipeline throughout the service life.

Plotting the expected number of failures provides a spatial profile of the long term pipeline

risk. The plot aids operators in anticipating the most critical segments of the pipeline, and the

future maintenance requirements. As the expected number of failures increases, the maintenance

decision switches from repair to replacement, and the proposed methodology determines the

decision switch point. The analysis also examined the impact of the uncertainty in the service life

on the maintenance decision. The longer and more uncertain the service life the more the optimal

maintenance decision shifts towards replacing segments instead of repairing them. The shift

towards replacement is because longer and more uncertain service lives have a higher number of

expected failures. A sensitivity analysis was also performed for the cost of failure and the interest

rate. The results were found to be very sensitive to the cost of failure when the cost of failure is

low. Also, the sensitivity to the cost of failure increased with increasing service life, again

because the expected number of failures is increasing. The results were less sensitive to the

interest rate, and varying the service life had minimal effect on the sensitivity to the interest rate.

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The RBM framework presented in this paper aids pipeline operators in making maintenance

decisions for corroding pipelines. In the case of more severe corrosion, or of a long or uncertain

pipeline service life, this method can demonstrate that it is less expensive for the operator to

replace a segment (or segments) instead of continuously repairing defects. This aids the pipeline

operator in real time maintenance decision making for their pipeline.

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6 GENETIC ALGORITHM FOR INSPECTION AND REPAIR PLANNING OF

DETERIORATING STRUCTURAL SYSTEMS: APPLICATION TO PRESSURE

VESSELS

This chapter contributes to the research area efficient RBI and RBM solutions, specifically

addressing the research objective: Using heuristic algorithms to more efficiently solve the

decision optimization problem in RBI and RBM planning.

This chapter develops a generic objective function for a structural system, and then applies

the generic objective function to the specific case of lifecycle RBM planning of a pressure vessel

with many defects. To solve the RBM optimization problem this chapter presents the

methodology to use a heuristic algorithm, specifically a genetic algorithm, to improve the

efficiency of the solution. The results of the heuristic optimization are compared with exhaustive

optimization (checking all possible solutions). There are two main contributions of this paper.

First is the development of a relatively straightforward methodology to apply a genetic algorithm

to RBI or RBM problems. Second is the assessment of the performance of genetic algorithms in

solving RBI and RBM problems.

This chapter is a journal paper (Haladuick and Dann, 2017b) that was submitted to the

International Journal of Pressure Vessels and Piping in January 2017 and is under review as of

September 2017.

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6.1 Abstract

For engineering systems, decision analysis can be used to determine the optimal decision

from a set of options, through utility maximization. Applied to inspection and maintenance

planning, decision analysis can determine the best inspection and maintenance plan to follow.

For simple systems, decision analysis is relatively straightforward; however, for more complex

systems with many components or defects, the set of all possible inspection and maintenance

plans can be very large. This paper presents the use of a genetic algorithm to perform inspection

and maintenance plan optimization for complex systems. The performance of the genetic

algorithm is compared to optimization by exhaustive search. A numerical example of lifecycle

maintenance planning for a corroding pressure vessel is used to illustrate the method. Genetic

algorithms are found to successfully reduce the computational demand of solving large

inspection and maintenance optimizations.

6.2 Introduction

In lifecycle engineering (Rackwitz et al., 2005), the optimal engineering solution is the one

that maximizes the utility provided by the system, where utility is a measure of the preference of

the relevant stakeholders. Applied to inspection and maintenance planning, lifecycle engineering

can be used to determine the optimal inspection and maintenance plan for a structural system.

Due to the uncertainties involved in predicting structural system failure, a probabilistic risk based

approach is typically taken, where the expected value of the utility is optimized. This approach is

termed “decision analysis,” and was first introduced by von Neuman and Morgenstern (1947),

and expanded upon in many texts since (Luce and Raiffa, 1957; JCSS, 2008; Jordaan, 2005;

Parmigiani and Inoue, 2009; Pratt et al., 1995). The application of decision analysis to the fields

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of inspection planning and maintenance planning are called “risk based inspection” (RBI)

(Straub, 2004) and “risk based maintenance” (RBM) planning (Kahn and Haddara, 2003),

respectively.

The goal of RBI is typically to optimize the timing and type of inspections, and likewise the

maintenance actions for RBM. RBI and RBM have been extensively applied to deteriorating

structures, such as offshore structures (Faber et al., 2003), steel structures (Straub, 2004;

Fujimoto et al., 1997), floating production, storage, and offloading facilities (FPSO’s) (Goyet et

al., 2002; Garbatov and Soares, 2001), pipelines (Pandey, 1998; Hellvetik et al., 1999; Nessim et

al., 2000; Sahraoui et al., 2013; Gomes et al., 2013; Zhang and Zhou, 2014; Haladuick and Dann,

2016b), refinery piping (Chang et al., 2005), bridges (Stewart, 2001; Barone and Frangopol,

2014), nuclear power plants (Martorell et al., 2000), processing plants (Marseguerra and Zio,

2000), and pressure vessels (Haladuick and Dann, 2016a, 2017a).

For structural systems with independent failure, inspection, and maintenance events,

performing RBI or RBM is relatively straightforward, as each component or defect can be

assessed individually to determine the optimal plan. However, most structural systems (e.g.

pressure vessels, bridges, and power plants) do not have independent failure, inspection, and

maintenance events. Failure of one component is typically related to failure of other components

and the whole system (e.g. a bridge support failing leading to bridge failure). Inspection and

maintenance costs for one component is typically not independent of other inspections or

maintenance, as there is often a cost savings in inspecting or maintaining multiple components

simultaneously. For systems with dependent failure, inspection, or maintenance events, the

optimal time and type of inspection and maintenance cannot be determined independently for

each component or defect in the system. Instead, one optimal plan must be determined for the

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entire system. The set of all candidate inspection and maintenance plans is known as the solution

space, and for systems with many inspection and maintenance times or types and components or

defects, the solution space can be very large. The most obvious approach to determine the

optimal plan from a solution space is an exhaustive search of the entire solution space to find the

plan with the greatest utility. However, for sufficiently complex systems, the solution space of all

combinations of inspection and maintenance times and types becomes very large, and the

exhaustive approach becomes computationally demanding.

As a workaround, many RBI and RBM studies propose simplifications to restrict the size of

the solution space, such as the following:

Restricting the set of inspections to allow only one or two inspections over the

lifetime of a system (e.g. Pandey, 1998; Nessim et al., 2000; Zhang and Zhou, 2014)

Restricting inspections to a fixed time interval (e.g. Sahraoui et al., 2013; Straub,

2004; Gomes et al., 2013; Barone and Frangopol, 2014; Haladuick and Dann, 2016a,

2017a)

Using a constraint, such as a reliability or risk constraint, to reduce the set of

inspection times (e.g. Straub, 2004)

Restricting the number of components or defects in the system (e.g. Sahraoui et al.,

2013; Straub, 2004; Gomes et al., 2013; Gomes and Beck, 2014; Hellevik et al.,

1999; Goyet et al., 2002; Haladuick and Dann 2016a)

Treating the inspection, maintenance, and failure events for each component or

defect as independent, so the optimal inspection and maintenance plan can be

determined for each defect individually (e.g. Pandey, 1998; Hong 1997, 1999;

Garbatov and Soares, 2001)

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Restricting the solution space reduces the computation time for an analysis; however, it also

reduces the quality of the final solution since the true optimal solution may not be in the

restricted solution space. An alternative to restricting the solution space is to use a heuristic

algorithm. Heuristic algorithms, such as genetic algorithms (Holland, 1976; Goldberg, 1989) or

simulated annealing (Kirkpatrick et al., 1983), have been well established in computational

engineering, but their adoption in RBI and RBM has been rare. Barone and Frangopol (2014)

used a genetic algorithm to determine the optimal inspection frequency for a deteriorating bridge.

Fujimoto et al. (1997) used a genetic algorithm to determine the optimal inspection times and

inspection type of fatiguing structures with single or multiple components. They simplified the

framework by ignoring the failure cost and instead using a reliability constraint. Martorell et al.

(2000) used a genetic algorithm to optimize the inspection interval in a nuclear power plant.

Marseguerra and Zio (2000) used a genetic algorithm to optimize both the inspection interval and

the type of maintenance for several series-parallel systems, including a chemical processing

plant.

The objective of this study is to develop a framework for, and examine the performance of, a

genetic algorithm used to solve an optimization problem in RBI or RBM. Both the computational

efficiency and the accuracy of the final solution are assessed to examine the value of using a

genetic algorithm for RBI or RBM of complex structural systems. This paper comprises five

sections. The second section formulates the generic RBI and RBM objective function to be

optimized. A corroding pressure vessel is used as an example to demonstrate the derivation of a

specific objective function. The third section introduces the genetic algorithm used to solve the

optimization problem. The fourth section presents the optimization for the corroding pressure

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vessel system and examines the results for examples with both a small and large solution space.

Finally, the impact and limitations of the methodology are discussed.

6.3 Developing the objective function

6.3.1 Generic objective function

The objective function of the expected utility U of a system is equal to the expected benefit B

provided by the system minus all of the expected costs of the system over its lifecycle (Rackwitz

et al., 2005). The cost typically includes the cost of initial construction CC, inspection CI,

maintenance CM, failure CF, and environmental pollution CP. The failure cost includes both the

direct and indirect costs of failure. The direct costs include the cost of repairing or replacing the

system, cleaning up the failure, losses of having the system offline, environmental costs, and

societal costs of potential human casualties. The indirect costs include any far reaching costs of

the failure; for example, for a bridge it could be failure of the greater transportation network and

also damage to the reputation of the companies, government, or industry involved. Some of these

costs are non-monetary, such as human casualties and environmental damage; however, it is

assumed that these attributes can be monetized for comparison with other costs (Xu, 2015). The

expected utility E[U(e)] is given by the following:

E[𝑈(𝒆)] = E[𝐵(𝒆)] − 𝐸[𝐶𝐶(𝒆)] − E[𝐶𝐼(𝒆)] − E[𝐶𝑀(𝒆)] − E[𝐶𝐹(𝒆)] − E[𝐶𝑃(𝒆)]

(6.1)

where e is the vector of all inspection and maintenance plan parameters to be optimized. The

objective of the optimization problem is to determine the inspection and maintenance plan e that

maximizes the objective function over the system lifecycle.

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For a generic structural system, each element in Equation (6.1) must be treated as dependent

on the inspection and maintenance plan e. For instance, more frequent or higher quality

inspections or maintenance increases the expected inspection cost E[CI(e)] and the expected

maintenance cost E[CM(e)], and decreases the expected failure cost E[CF(e)]. However, it can

also potentially affect the expected pollution cost E[CP(e)], because better maintained

components may pollute less or more. The construction cost E[CC(e)] may also be dependent on

the inspection and maintenance plan; for example, more advanced construction techniques may

be required for more advanced inspections and maintenance. Finally, the expected benefit

E[B(e)] is dependent because some systems produce a greater benefit when they are better

maintained. Also, when maintenance or failure occurs, the system will likely be taken offline so

it can be renewed, resulting in a decrease in the benefit. For a generic system, these dependencies

cannot be relaxed; however, for many systems they are not necessary, so they need to be

examined on a case by case basis.

6.3.2 Objective function for RBM of a corroding pressure vessel

To demonstrate the methodology, this paper applies RBM to optimize the repair time for a

set of corrosion defects in a pressure vessel containing gas. Since the objective of the RBM is to

optimize the repair time for each defect, the maintenance plan e is defined as a vector of repair

times e = {tR1, …, tRJ}, where J is the number of defects in the pressure vessel. In reality the

number of defects J will increase due to the generation of new defects; however, this example

ignores defect generation for simplicity. The objective function for a specific system is derived

by starting with the generic objective function Equation (6.1) and relaxing the dependencies on

the inspection and maintenance plan as applicable. For pressure vessel RBM, several costs in

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Equation (6.1) are assumed to be independent of the maintenance plan e. First, the benefit of a

pressure vessel is independent of its state of deterioration, given that failure has not occurred.

There is still the lost benefit due to system offline time during maintenance and failure; however,

this loss can be incorporated into the cost of maintenance and failure. Consequently, the benefit

is independent of the inspection and maintenance plan e. Second, the construction cost of a

pressure vessel is typically independent of the inspection and maintenance plan. Third, pressure

vessels do not typically pollute, again given that failure has not occurred, so the pollution is

independent of the state of deterioration. Finally, because RBM is performed to optimize repair

times, the cost of inspection is independent of the maintenance plan, because regardless of the

repair times, the vessel will still be inspected according to a separate inspection plan. Costs that

are independent of the inspection and maintenance plan are constant for all inspection and

maintenance plans, and therefore can be ignored when comparing the different plans. The

objective function for the pressure vessel system can then be written in terms of the relative

expected cost E[C(e)]:

E[𝐶(𝒆)] = E[𝐶𝑀(𝒆)] + E[𝐶𝐹(𝒆)] + 𝐶 (6.2)

where C is a constant summarizing all of the costs that are independent of e. The objective now

becomes finding the optimal maintenance plan e that minimizes the expected cost over the

lifecycle of the pressure vessel.

Equations for the expected costs of maintenance and failure are required to perform the

RBM. As discussed previously, system RBM optimization is only warranted when there is

dependency in either the maintenance or failure events between the defects. The dependency of

the maintenance and failure events must reflect the physical nature of the system. In this

example, the maintenance events are treated as dependent and the failure events as independent.

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When repairing a pressure vessel, it is often the case that the bulk of the cost is due to the

pressure vessel being offline, emptying it, and opening it up for repair. This cost is a one-time

cost, regardless of the number of defects being repaired. Then there is an additional cost of the

actual repair for each defect. The dependent maintenance events allow the repair cost to account

for this cost relationship by defining the repair cost as a function of the number of simultaneous

repairs. Any function can define the repair cost; for example, a non-linear function of the number

of repairs allows the cost of the first repair to be greater than subsequent repairs. The expected

cost of maintenance is given by the following:

E[𝐶𝑀(𝒆)] = ∫ ∑𝐶𝑅(𝑘)

(1 + 𝑟)𝑡𝑝𝑅(𝒆, 𝑡, 𝑘)

𝐾(𝑡)

𝑘=1𝑑𝑡

𝑡𝑠𝑙

0

(6.3)

where tsl is the service life of the pressure vessel, CR(k) is the cost of repairing k defects, K(t) is

the number of simultaneous repairs scheduled at time t, r is the discount rate, and pR(e,t,k) is the

probability of repairing k defects at time t for maintenance plan e. The summation is performed

over the number of possible repairs k at time t, which is from 1 to the number of repairs K(t) that

are scheduled at time t. All K(t) repairs that are scheduled at time t will not necessarily be

undertaken, because there is the potential for any number of the K(t) defects to fail before the

repair time. The integral is performed over the service life tsl to consider any continuous potential

repair time. The repair cost is discounted by r yielding the present value of the cost. The

probability of repairing k defects at time t for maintenance plan e is given by the following:

𝑝𝑅(𝒆, 𝑡, 𝑘) = ∑ (∏(1 − 𝐹𝑇𝑗(𝒆, 𝑡)) ∏ 𝐹𝑇𝑗

𝑗∊𝐿,𝑗∉𝑆𝑗∊𝑆

(𝒆, 𝑡))

𝑆⊆𝐿

(6.4)

The summation is performed over all possible combinations of defect subsets S of L, where

each S is a subset of k defects selected from the set L of K(t) defects that are scheduled for repair

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at time t. The first product is performed over the subset S of defects that are repaired, and the

term 1 – Ftj(e,t) is the probability that defect j survives until the repair at time t. The second

product is performed over the defects that are in the set L that are scheduled to be repaired at t,

but not in the subset S that are actually repaired, and the term Ftj(e,t) is the probability that defect

j fails before it is repaired at time t.

The failure events are assumed to be independent to reflect a pressure vessel susceptible to

leak failure with no risk of burst failure. A leak at one defect does not affect the other defects,

and thus the failures can be treated as independent. The expected cost of failure is then given by

the following:

E[𝐶𝐹(𝒆)] = ∫E[𝑛𝐹(𝒆, 𝑡)]𝐶𝐹

(1 + 𝑟)𝑡

𝑡𝑠𝑙

0

𝑑𝑡 (6.5)

where E[nF(e,t)] is the time dependent expected number of failures for the repair plan e, and CF

is the cost of a single leak failure. The expected number of failures E[nF(tR,t)] for the system, as a

function of time, is given by the sum of the cumulative probability of failure time for each

defect:

E[𝑛𝐹(𝒆, 𝑡)] = ∑ 𝐹𝑇𝑗(𝒆, 𝑡)

𝐽

𝑗=1 (6.6)

where FTj(e,t) is the cumulative probability of failure time for defect j for maintenance plan e.

The objective function Equation (6.2) is evaluated for each repair plan e by populating Equations

(6.3) and (6.5).

6.4 Optimization with a genetic algorithm

In general, repair times can be selected from any continuous time throughout the lifecycle of

a system. However, allowing continuous repair times creates a continuous optimization problem

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with an infinitely large solution space of candidate maintenance plans e. As a simplification,

repair times are restricted to a discrete set of potential repair times throughout a system lifecycle.

A short time interval between the potential repair times minimizes the impact of the

simplification and is in line with common practice as repair times are not optimized to the

nearest day. Letting m denote the number of potential repair times over a lifecycle, plus the

option to not repair, the solution space contains (m+1)J candidate repair plans, which are the

combinations of repairing each defect at any potential time. Determining the optimal inspection

plan from the set of (m+1)J candidate inspection plans is a discontinuous or integer optimization

problem. Many optimization techniques are not applicable to integer optimization problems;

however, genetic algorithms have been shown to be successful. This section describes the use of

a genetic algorithm to perform the RBM optimization for a corroding pressure vessel.

A genetic algorithm is a heuristic based on biological evolution. Evolution relies on the

processes of natural selection and mutation to evolve a child population that is better adapted to

its environment than the parent generation. Evolution begins with an initial population

possessing variation in their genetic traits. Some members of the population are better adapted

than others, and these members are more likely to survive and reproduce. When members

reproduce, there is heredity in the reproduction process, meaning that the genetic traits of the

parents are passed to their children. The genetic traits of a child are composed of a random

crossover of the parents’ traits. Preferential reproduction means that the genetic traits of the fitter

members are preferentially passed on to the next generation. The traits of a child can also

randomly mutate, allowing the child to possess traits that were not present in either parent.

Through an iteration of the processes of natural selection and mutation, a fitter population

evolves over many generations.

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Genetic algorithms replicate the evolutionary process. This study uses the genetic algorithm

method described in Deep et al. (2009) for solving integer optimization problems. In the pressure

vessel example, the members of the population are the candidate solutions e, and the genetic

traits of each member are the repair times for each defect. The generic algorithm iterates

successive generations of the same population size, where the best member of each generation

progresses towards the optimal solution, based on the applied objective function. Starting with a

randomly selected population of candidate solutions, the objective function is evaluated to rank

the fitness of each solution. The best candidates from the population are termed the ‘elites’ and

are passed onto the child generation unevolved. To produce the remaining members of the child

population, the genetic algorithm replicates the processes of crossover and mutation. Crossover

children are created by combining the genes of two parent members. A crossover function

specifies which traits are inherited from each parent. In the pressure vessel problem, a crossover

child has repair times for some of the defects based on one parent, and the repair times for the

remaining defects based on the other parent. Crossover instills heredity, allowing the genes of the

fitter members of the parent generation to be passed onto the child generation. Mutation children

are created by randomly mutating the genes of a single parent member. A mutation function

defines which parent traits are mutated and the degree of mutation.

In the pressure vessel problem, a mutation is a random change of the repair time for a defect.

Mutation allows the genetic algorithm to search candidate solutions that are not part of the initial

population, promoting a wider search for the globally optimal solution. However, excessive

mutation makes the algorithm less efficient, because it randomly changes already well adapted

solutions. A crossover rate is used to determine the proportion of the remaining children that are

produced by crossover and mutation. At one extreme, when all children are produced from

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crossover, the algorithm risks becoming trapped in a locally optimal solution, because it cannot

mutate new genes that were not in the original population. At the other extreme, when all

children are mutated, the algorithm performs an exhaustive search, because there is no heredity.

A selection function determines which of the members of the parent population are used for

crossover and mutation based on the objective function score for each parent member. By

iteratively repeating this process for many generations the genetic algorithm evolves the

population towards fitter solutions, and the best solution from each generation progresses

towards the optimal solution. A criterion is used to terminate the algorithm, which is typically

either a maximum number of generations or computation time, or a maximum number of

generations without improvement in the score of the best solution. Obtaining the optimal solution

is not guaranteed.

6.5 Numerical example of a corroding pressure vessel

To demonstrate the methodology, two numerical examples are presented: one with a small

and one with a large solution space. The example with the small solution space allows the

optimization to be solved in two ways: using an exhaustive search and using a genetic algorithm.

The solution obtained from an exhaustive search is the overall global optimum, whereas the

solution from a genetic algorithm is not necessarily the global optimum, but the solutions and the

computational demand can be compared. The example with the large solution space is not

solvable with the exhaustive approach because the computational demand is too great; therefore,

this example is used to demonstrate the scalability of the genetic algorithm approach.

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6.5.1 Small solution space example

In this example, a pressure vessel containing gas is subject to structural deterioration due to

corrosion is considered. The pressure vessel was inspected several times in the past and was

recently inspected again, and J = 4 defects were detected. From the inspection data, the

cumulative probability of the failure time for each defect can be determined from reliability

theory (Melchers, 1999). For the purposes of this example, the cumulative distribution functions

(cdfs) of the failure time for each defect are assumed (Figure 6.1). The vessel has a remaining

service life of 100 years and can be repaired every 5 years; therefore, there are m = 20 potential

repair times and the solution space contains (m + 1)J = 194,481 candidate solutions.

Figure 6.1. cdfs of failure times for each defect, with the defect numbers shown beside the lines.

The repair cost can be assumed as any function of the number of defects j to be repaired. This

example defines the repair cost as CR(j) = CR + ω(j - 1)CR, where CR is the cost of the first repair

and ω is a factor governing the cost of each additional repair. The cost of repairing 0 defects

CR(0) = 0. Only the relative ratio of the costs of failure CF and first repair CR are required to

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populate the objective function, not the absolute costs. The values of the input variables are

detailed in Table 6.1.

Table 6.1. Input variables for the decision analysis.

Variable Symbol Value

Failure cost ratio CF / CR 500

Discount rate r 4 %

Repair factor ω 0.5

For the exhaustive approach, the objective function Equation (6.2) is evaluated for each of

the 194,481 candidate maintenance plans e. The optimal solution is e = {60, 15, 60, 40}, with a

corresponding objective function score of the relative expected cost E[C] / CR = 0.95. Comparing

the optimal repair plan with the cdfs of the failure time (Figure 6.1), it can be seen that each

defect is repaired before the failure risk increases too drastically. The repair time for defect 3

would have been later if it was assessed independently, but because defect 1 was repaired after

60 years, it was less costly to repair defect 3 at the same time due to the decreased marginal cost

of the second repair. The genetic algorithm was able to reach the same optimal solution with a

population of 50 members and a crossover rate of 0.8. The progress of the genetic algorithm and

exhaustive search methods are shown in Figure 6.2.

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Figure 6.2. Progress of the genetic algorithm and exhaustive search solutions.

Figure 6.2 shows the normalized objective function score of the current best solution for each

iteration. The genetic algorithm ran for 63 generations, each with a population of 50 members,

for a total of 3150 evaluations of the objective function. The globally optimal solution was

reached in the 13th

generation, after 650 evaluations of the objective function. The algorithm then

ran until reaching the termination criteria of 50 generations without improvement in the solution.

The computational efficiency of the genetic algorithm can be assessed by comparing it with the

exhaustive search method. The exhaustive search evaluated the objective function 194,481 times

before it could confirm the optimal solution. Thus, the exhaustive search required 61 times the

number of evaluations of the objective function, leading to a factor 7 decrease in computation

time for the genetic algorithm.

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6.5.2 Large solution space example

This example considers the same pressure vessel but with a vastly expanded solution space.

Instead of 4 defects, the most recent inspection detected J = 40 defects, and instead of the

opportunity to repair every 5 years, the vessel can now be repaired annually. Annual repairs for

the same 100 year service life yields m = 100 potential repair times and (m + 1)J = 1.5 x10

80

candidate solutions. An exhaustive search of 1080

candidate solutions is not possible, so a

heuristic, such as the genetic algorithm, is the only way to solve an RBM problem of this scale.

The assumed cdfs of the failure time for each of the 40 defects are shown in Figure 6.3 (note the

cdfs of some defects are beyond 100 years).

Figure 6.3. cdfs of failure time for each defect, with the defect numbers shown beside the lines.

In a problem of this scale, with 40 variables and 1080

candidate solutions, it is important to

select appropriate parameters for the genetic algorithm. Three main parameters to select are the

population size for each generation, the crossover rate, and the stopping criteria. A sensitivity

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analysis was performed to assess the appropriate ranges of the genetic algorithm parameters for

RBI or RBM. The population size and crossover rate were varied while the stopping criteria was

kept constant at 105 evaluations of the objective function. A population of 2 times the number of

variables (2J = 80) consistently produced the best results, along with a crossover rate ranging

from 0.6 to 0.9, with 0.9 being the overall best choice. As a general rule, this study recommends

a population size that is the greater of 2 times the number of optimization variables or 50.

The progress of the genetic algorithm is shown in Figure 6.4. Three runs of the genetic

algorithm are shown to demonstrate the impact of crossover and mutation on the solution

progress, with extreme crossover rates of 0 and 1, and with an optimal crossover rate of 0.9. All

three runs have a fixed population of the optimal 2J and a stopping criterion of 105 evaluations.

At the extremes, with a crossover rate of 0, all children are produced by mutation, and with a

crossover rate of 1 all children are produced by crossover. With a crossover rate of 0 the

algorithm does not possess any heredity, so the best genetic combinations are not passed on from

one generation to the next. The lack of heredity essentially disables the learning aspect of the

algorithm, and as can be seen, the algorithm is unable to greatly improve the solution over 105

iterations and would require many more iterations to find the optimum solution. With a crossover

rate of 1 the algorithm does not use any randomly mutated children. Without mutation the

algorithm is at risk of becoming trapped in a local optimum. The objective function score for

crossover rates of 0, 0.9, and 1 respectively is E[C] / CR = 379.7, 6.5, 6.9.

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Figure 6.4. Progress of the genetic algorithm with different crossover rates.

The optimal solution from the run with a crossover rate of 0.9 and a population of 2J is

shown in Figure 6.5. Again, comparing the repair times for each defect to the failure cdfs (Figure

6.3) shows the defects are repaired before the failure probability rapidly increases, with a

preference towards simultaneous repair. The algorithm was unable to produce a better solution

for over 2.2 x104 evaluations, meaning that the optimal solution is at least close to the global

optimal. As a check, the repair time for each defect can be adjusted up and down by one year

from the optimal solution. If this adjustment produces improvements in the optimal solution for

many of the defects, then the algorithm is not producing an acceptable solution. In this example

none of the adjusted permutations are improvements, demonstrating the accuracy of the genetic

algorithm.

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Figure 6.5. Optimal solution for the case with crossover rate of 0.9 and population of 2J. Repair

time of 0 years is the current time, and a repair time of 100 years actually means never, as 100

years is the end of the service life.

This example demonstrates the value of genetic algorithms in solving the RBI and RBM

optimization problem. It is not practical to solve an optimization problem of this scale using the

exhaustive search method because the computational requirement of 1080

evaluations of the

objective function is too great. In contrast, the genetic algorithm reached a solution that

approaches the global optimal within 105 evaluations of the objective function. For reference, the

elapsed time for this computation on a standard computer was approximately 8 minutes, and a

more lenient stopping criterion would further reduce this time.

6.6 Conclusion

This paper presents a method for efficiently determining optimal risk based inspections and

maintenance plans using a genetic algorithm. In risk based inspection and maintenance planning,

the optimal plan is the one that maximizes the utility provided by the system over its lifecycle.

For simple systems, determining the optimal plan is straightforward. However, for more complex

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systems, the solution space of all possible inspection and maintenance plans can be very large,

and searching the entire solution space is not feasible. This paper demonstrates that genetic

algorithms can be successfully used as a heuristic to more efficiently determine the optimal

inspection and maintenance plan. Two examples were used to illustrate the method. First, a risk

based maintenance optimization problem for a pressure vessel was presented, with a relatively

small solution space of 105 candidate solutions. Optimization with a genetic algorithm was

compared to an exhaustive search, and it was found that the genetic algorithm yielded the same

optimal plan as the exhaustive method but was 7 times faster in terms of computation time.

Second, the example problem was expanded to entail a much larger solution space of 1080

candidate solutions. This problem was too large to solve with an exhaustive search, but the

genetic algorithm was still able to determine a solution with a relatively short computation time

on a standard computer. The larger optimization problem is more realistic in scale to a problem

that could be faced by the operator of an engineering system. Additionally, genetic algorithms

are relatively simple to implement and are supported by many software packages. Thus, this

paper shows that genetic algorithms are a practical and efficient method of solving risk based

inspection and maintenance planning problems for real world engineering systems.

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7 AN EFFICIENT RISK-BASED DECISION ANALYSIS OF THE OPTIMAL NEXT

INSPECTION TYPE FOR A DETERIORATING STRUCTURAL SYSTEM

This chapter contributes to the research area: efficient RBI and RBM solutions. Specifically,

this chapter addresses the research objective: RBI and RBM planning without a lifecycle analysis.

This chapter presents a methodology to examine the decision of what inspection type should

be chosen next, from a set of possible inspection types with varying accuracy and cost. Typically

this question would be addressed by performing a lifecycle RBM analysis and assessing all of

the decisions over the lifecycle of the system. The key contribution of this chapter is the

methodology to assess the optimal decision without performing a lifecycle analysis. This greatly

simplifies the analysis, because only one decision is assessed instead of all of the decisions

throughout the lifecycle. Similar to Chapter three, this methodology shrinks the lifecycle

decision sequence down to only the relevant decision. However, unlike Chapter three, the

methodology presented here can be generally applied to determine the optimal inspection type of

any system with any state of deterioration.

This chapter is a journal paper (Haladuick and Dann, 2017c) that was submitted to Structure

and Infrastructure Engineering in May 2017 and is under review as of September 2017.

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7.1 Abstract

Deteriorating infrastructure systems require inspections and maintenance to ensure safe

operation. System operators are often required to decide the optimal type of inspection to

perform, where some inspections are of higher accuracy, and correspondingly higher cost.

Lifecycle analysis is typically used to determine the optimal inspection type. While lifecycle

analysis is able to determining the optimal inspection type, it can be difficult and

computationally demanding, requiring analysis of the entire decision sequence throughout the

system lifecycle. This paper presents an alternative methodology to approximate the decision of

the optimal next inspection type without performing a lifecycle analysis. This methodology

determines the range of the value of information provided by only the next inspection. When the

inspection cost is outside the range of the value of information then this method yields the

decision of which inspection type to choose, negating the need for lifecycle analysis. When the

inspection cost for some inspection types lies within the bounds then a subsequent lifecycle

analysis is required, but perhaps some inspection types can be eliminated, simplifying the

lifecycle analysis. Thus, this method is complimentary to lifecycle analysis, functioning as a

quick preliminary assessment. The methodology is demonstrated through a numerical example of

a corroding pipeline.

7.2 Introduction

Structural and infrastructure systems are subject to deterioration processes, such as corrosion

and fatigue. These deterioration processes gradually reduce the resistance of the system until

failure occurs. To mitigate the risk of failure, many systems, for example bridges, buildings,

pipelines, and power plants, are subject to regular non-destructive inspections and maintenance.

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To minimize the total cost of operating the system, it is necessary to optimize the inspection and

maintenance actions. A risk-based decision analysis (Joint Committee on Structural Safety,

2008; Jordaan, 2005; Luce & Raiffa, 1957; Parmigiani & Inoue, 2009; Pratt, Raiffa, & Schlaifer,

1995; von Neumann & Morgenstern, 1947) is often used to determine the optimal inspection and

maintenance plan because of its ability to account for the uncertainties present, for instance the

spatiotemporal uncertainty in the deterioration process, inspection errors, and model errors. In

lifecycle risk-based inspection (RBI) planning (Rackwitz, Lentz, & Faber, 2005), the expected

total cost of operating the system over its entire lifecycle is determined for all possible inspection

and maintenance plans. The inspection and maintenance plan with the minimum expected total

cost is the optimal plan. Lifecycle RBI has been applied to deteriorating structures, for example

offshore structures (Faber, Straub, & Goyet, 2003), steel structures (Fujimoto, Kim, & Hamada,

2009; Straub, 2004), floating production, storage, and offloading facilities (FSPO’s) (Garbatov &

Soares, 2001; Goyet , Straub, & Faber, 2002), pipelines (Gomes & Beck, 2014; Gomes, Beck, &

Haukaas, 2013; Hellevik, Langen, & Sørensen, 1999; Nessim, Stephens, & Zimmerman, 2000;

Pandey, 1998; Sahraoui, Khelif, & Chateauneuf, 2013; Zhang & Zhou, 2014), refinery piping

(Chang, Chang, Shu, & Lin, 2005), bridges (Barone & Frangopol, 2014; Stewart, 2001), nuclear

power plants (Martorell, Carlos, Sánchez, & Serradell, 2000), and processing plants

(Marseguerra & Zio, 2000).

This paper examines the decision of what inspection type to use for the next inspection. In

many systems there are several options available for the type of inspection to perform, and there

is typically a tradeoff between the accuracy and the cost of the different types. In this case it can

be difficult for the decision maker to determine whether the additional cost of the higher

accuracy inspection is justified. Lifecycle RBI planning is determines the optimal type of the

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next inspection by determining the optimal inspection and maintenance plan for all inspection

and maintenance decisions over the lifecycle if the system. However, in most cases the system

operator is not interested in the inspection or maintenance actions at some point in the distant

future of the system; instead they are interested in the next inspection or maintenance action. It is

not ideal to have to analyze the cost of all possible plans over the lifecycle of the system in order

to determine the next inspection type, and this is especially true for large systems with long

lifecycles and multiple possible inspection types. As an example, consider a pipeline with a

remaining lifecycle of 60 years that can be inspected using two inspection types, magnetic flux

leakage or ultrasonic testing. The pipeline needs to be inspected every three years, and at each

inspection the operator needs to decide which inspection type to choose. With two possible

inspection types for a series of 20 inspections, there are there are 220

= 106 permutations of

different inspection plans that need to be assessed over the lifecycle of the system to find the best

plan, in order to simply decide which of these two inspection types to use for the next inspection.

While not all systems will have RBI planning problems of this size, this example shows that for

large systems it is advantageous whenever possible to only assess the next inspection type,

instead of the set of all inspections types.

The objective of this paper is to present a simple methodology for determining the optimal

type of the next inspection of a structural system, where there are several inspection types to

choose from, with varying accuracy and cost. The methodology is complimentary to lifecycle

analysis, ideally suited as an preliminary approximate decision analysis to be performed before a

full lifecycle analysis. Depending on the results, the preliminary analysis may simplify the

subsequent lifecycle analysis by eliminating potential inspection types, or negate it entirely.

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The methodology centers on assessing the value of information (VoI) of only the next

inspection, for a set of different inspection types. VoI is an economics concept (Hirshleifer,

1971) that has more recently been applied to risk-based analysis of structures (Pozzi & Der

Kiureghian, 2011; Straub, 2014). VoI is the difference between the expected outcome of a

decision that is made with and without said information. If the VoI is greater than the added cost

of obtaining the information (the cost of the inspection), then the inspection is justified.

Similarly, VoI can be used to compare different inspection types. If the additional VoI of the

higher accuracy inspection is greater than the added cost, then the higher accuracy inspection is

justified. Assessing only the VoI of the next inspection avoids the problem of having to assess all

of the subsequent actions in the lifecycle. Of course, it is not this simple. The value of the

information gained from the next inspection is dependent the information gained from the

inspection after it, and the one after that, continuing in perpetuity until the end of the system

lifecycle. Thus, the only way to determine the VoI of an alternative inspection type is to assess

the impact of the inspection on every subsequent event, hence lifecycle analysis. However, it is

possible to examine the bounds of the VoI of an alternative inspection type for the next

inspection, without a full lifecycle analysis.

Depending on the results of the preliminary analysis, lifecycle analysis may be simplified or

even avoided. If the additional cost of an inspection type lies outside the bounds of the VoI,

meaning that either the cost is lower than the lowest expected VoI, or higher than the highest

expected VoI, then one of the inspection types can be eliminated. If this occurs for all of the

potential inspection types then the optimal inspection type is clear, and lifecycle analysis can be

avoided. If there are still several inspection types left, then a simplified lifecycle analysis can be

performed on the remaining inspection types to determine the optimal type.

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Another advantage of only considering the decision of the next inspection type instead of the

full lifecycle decision sequence is that it avoids the inaccuracy introduced by analyzing decisions

over a long time horizon. The long time horizon in lifecycle RBI requires the analyst to make

assumptions of the long-term behavior of the system over its lifecycle. For example, consider a

highway bridge with a remaining lifecycle of 50 years. Lifecycle RBI analysis could be used to

determine the optimal inspection plan for the bridge over the next 50 years. However, what if

within the next 50 years drones become the primary method of bridge inspection (Minnesota

Department of Transportation, 2015)? Then the optimal inspection plan that was determined

based on the assumption of indefinite local inspections will no longer be accurate. Or, what if

within the next 50 years an alternative transportation system (e.g. ride-sharing, autonomous cars,

autonomous flying vehicles etc.) becomes widely adopted? Then the use of the local

transportation network, and potentially the entire structure the city, will change. This will cause

the loading of the bridge to change, resulting in inaccuracy in the fatigue deterioration model and

a suboptimal inspection plan. An additional source of uncertainty is that the lifetime of the

system is also unknown and must be assumed (de Jonge, Klingenberg, Teunter, & Tinga, 2015).

These examples demonstrate that embedded within the lifecycle RBI methodology are numerous

long-term assumptions concerning all aspects of the system. While the impact of these long-term

assumptions is diminished by the discounting of future costs, it is still not possible to determine

this impact, and consequently inaccuracy is introduced into the lifecycle RBI analysis. Assessing

only the decision of the next inspection type removes the need for these long-term assumptions.

However, the present methodology does introduce other assumptions and inaccuracies, such as

inaccuracy in determining how much information is due to only the next inspection. Thus, the

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use of both approaches as complimentary is valuable in offsetting the weaknesses of either

individual approach.

The paper comprises four sections. The second section presents the methodology, the third

section presents an example of the methodology applied to a corroding pipeline, and the fourth

section discusses some limitations and conclusions.

7.3 Methodology

This section presents the methodology for determining the optimal inspection type from a set

of inspection types with varying accuracy and cost, without having to perform a full lifecycle

decision analysis. This section comprises three parts. The first part provides background in

lifecycle decision analysis and the VoI. The second part presents the methodology to isolate the

decision of the next inspection type from the rest of the decision sequence. Finally, the third part

presents the implementation of the methodology.

7.3.1 Value of information

In lifecycle RBI, the expected total cost E[C(t,e)] of following inspection and maintenance

plan e is determined by analyzing the entire set of sequential decisions up to time t. Evaluating

the expected total cost at the end of the lifecycle t = tsl gives the expected total cost of operating

the system over its lifecycle. The expected total cost of operating the system over its lifecycle is

composed of many elements, for instance, the expected cost of system failure, inspection,

maintenance, pollution (not associated with failure, e.g. from a power plant), and dismantling. To

determine the absolute expected total cost of operating the system, all of the elements of the cost

need to be considered. However, when comparing costs between two inspection and maintenance

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plans, only the costs that are dependent on the inspection and maintenance plan impact the

decision. Costs, such as the expected cost of pollution, are likely to be independent of the

inspection and maintenance plan, and therefore remain constant for each plan. Consequently, this

study assumes that the expected total cost is composed of three main elements: failure,

inspection, and maintenance. Some of these costs are non-monetary, for example human

casualties and environmental impact; however, it is assumed that these attributes can be

monetized for comparison with other costs (Xu, 2015). The expected total cost is given by:

E[𝐶(𝑡, 𝒆)] = E[𝐶𝐹(𝑡, 𝒆)] + E[𝐶𝐼(𝑡, 𝒆)] + E[𝐶𝑀(𝒆)] + 𝑘(𝑡) (7.1)

where E[C(t,e)], E[CF(t,e)], E[CI(t,e)], and E[CM(e)] are the system wide expected total cost,

failure cost, inspection cost, and maintenance cost respectively, all for inspection and

maintenance plan e, and k(t) is a constant summarizing all of the costs that are assumed to be

independent of e. The costs are a function of the time t over which the system operates. For a

system with one failure mode, the expected cost of system failure is given by:

E[𝐶𝐹(𝑡, 𝒆)] = 𝐶𝐹 ∫ 𝑓𝑇(𝜏, 𝒆) (1 + 𝑟)𝜏𝑑𝜏⁄𝑡

0

(7.2)

where CF is the cost of system failure, fT(t,e) is the probability density function (pdf) of the

failure time T, and r is the interest rate to account for the time value of money. Similarly,

assuming that if maintenance is system wide and occurs immediately following an inspection,

the expected cost of maintenance is given by:

E[𝐶𝑀(𝑛, 𝒆)] = 𝐶𝑀 ∑ 𝑝𝑀𝑖(𝒆) (1 + 𝑟)𝑡𝑖⁄

𝑛

𝑖=1 (7.3)

where inspection i is performed at time ti (i = 1,…,n), pMi(e) is the probability of maintenance

after each inspection i, CM is the cost of system maintenance, and r is the interest rate to account

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for the time value of money. If the system has multiple failure modes and components, then

reliability analysis (Melchers, 1999) can be used to derive the system reliability and probability

of maintenance.

The value of information (VoI) (Straub, 2014) can be used to quantify the benefit provided

by an inspection. VoI(t,e) denotes the absolute VoI of inspection and maintenance plan e. VoI(t,e)

is the difference between the expected lifecycle cost of operating under inspection and

maintenance plan e, and the baseline cost, denoted the prior, which is to operate without any

inspections and thus no new information. Note that Straub (2014) develops the VoI concept from

lifecycle RBI, whereas the present study adapts the VoI concept to assess the value of only the

next inspection, without performing the full lifecycle analysis. The VoI does not contain the

inspection cost, because the object is to compare the VoI with the inspection cost. The absolute

VoI of inspection and maintenance plan e is given by:

𝑉𝑜𝐼(𝑡, 𝒆) = E[𝐶𝐹(𝑡)] − {E[𝐶𝐹(𝑡, 𝒆)] + E[𝐶𝑀(𝑛, 𝒆)]} (7.4)

where E[CF(t)] is the prior expected cost of failure as a function of time.

The relative ΔVoI(t,e1,e2) of inspection and maintenance plan e2 over e1 is given by the

difference between the absolute VoI’s for plan e1 and e2:

Δ𝑉𝑜𝐼(𝑡, 𝒆1, 𝒆2) = 𝑉𝑜𝐼(𝑡, 𝒆1) − 𝑉𝑜𝐼(𝑡, 𝒆2) (7.5)

where a negative ΔVoI(t,e1,e2) means there is more VoI in plan e1 than e2.

7.3.2 Isolating the decision of the inspection type for the next inspection

The lifecycle inspection and maintenance plan e for a structural system is composed of a

sequence of decisions. In general, e is typically composed of the following decisions: e =

{inspection time, inspection type, inspection extent, maintenance time, maintenance type,

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maintenance extent}, which are repeated at each inspection time over the lifecycle of the system.

This paper is only concerned with one decision from this sequence, the decision of the inspection

type at the next inspection time. The goal of the methodology is to isolate this decision from the

rest of the decision sequence. However, all subsequent decisions are dependent on the decision

of the next inspection type. To only assess the decision of what inspection type to use for the

next inspection, this decision must be isolated from the decision sequence using careful

assumptions.

First, it is assumed that the time t1 and extent of the next inspection have been previously

determined, and are constant for the different inspection and maintenance plans being compared.

This removes the impact of these decisions on the analysis. The method of determining the time

of the next inspection is not relevant to this analysis, it could be set based on an engineering

code, company policy, or by a separate reliability or risk analysis.

Second, the decision of the next inspection type must be isolated from the subsequent

decision of what maintenance actions to perform following the inspections. As explained in

Straub (2014), the information obtained from the inspection does not have value in and of itself,

but instead value is derived if the information results in a change in the subsequent maintenance

action. Therefore, the decision of which inspection to choose cannot be decoupled from the

subsequent maintenance decision. A common solution (e.g. Hong 1997, 1999; Nessim et al.,

2000; Straub and Faber, 2005; Castanier and Rausand, 2006; Zhou and Nessim, 2011; Sahraoui

et al., 2013; Gomes et al., 2013; Gomes and Beck, 2014; Zhang and Zhou, 2014), which is

employed herein, is to assume a set of maintenance rules m that govern the maintenance decision

for a given inspection result. The maintenance rules m do not have to be the same as the optimal

rules obtained from the lifecycle analysis, and thus the analysis with the assumed rules will not

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be as accurate as a full lifecycle analysis. However, many systems are subject to strict

regulations regarding maintenance, and if the applied maintenance rules are such that they

replicate the regulations, the analysis will be accurate for practical application.Third, ideally the

decision of the next inspection type needs to be isolated from the subsequent sequence of

decisions of the time, type, and extent of all future inspections. However, all future inspection

decisions are dependent on the results of the next inspection. It is not possible to determine how

much of the VoI of inspection and maintenance plan e, denoted VoI(t,e), is provided by only the

next inspection, denoted VoI1(t,e), and how much is from all of the subsequent n-1 inspections,

denoted VoI2:n(t,e). Instead of determining the precise value, this paper proposes that it is

possible to determine upper 𝑉𝑜𝐼1𝑈(𝑡, 𝒆) and lower 𝑉𝑜𝐼1

𝐿(𝑡, 𝒆) boundaries on the amount of

VoI1(t,e) obtained from only the next inspection.

To determine the boundaries of VoI1(t,e), the first step is to model the future VoI as if only

the next inspection will be undertaken, and all subsequent inspections are ignored. This is shown

in Figure 7.1 for inspection plans e1 and e2, as well as the prior with no inspections. The only

difference between plans e1 and e2 is that the next inspection in e2 has a higher accuracy. After

the inspection is complete the maintenance rules m are applied.

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Figure 7.1. Time to failure for the prior case and for different inspections plans. VoI areas shown

for inspection and maintenance plan e1.

The areas between the different inspection plans in Figure 7.1 represent the VoI of the

different plans, which is derived from decreasing the expected cost of failure. In Figure 7.1, the

analysis begins at the time of the next inspection t1, with the conditional probability of failure of

zero for each plan. This is because all decisions previous to t1 are kept constant to facilitate

comparison of only the inspection type; therefore, the system performance prior to t1 is

inconsequential. As time progresses into the future, the probability of failure increases. Initially,

probability of failure under plan e2 is lower than under e1, which in turn is lower than the prior.

This is necessarily the case, and reflects the VoI of the inspections relative to each other and to

the prior. This can be illustrated by examining the extreme cases of a small (lower prior failure

probability) and large (higher prior failure probability) defect. The next inspection in e2 provides

higher accuracy information than the next inspection in e1, which in turn provides more

information than the prior (which has no new information). The higher accuracy information

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leads to a narrower probability distribution of the defect size. For the case of a small defect, the

higher accuracy information will confirm the small defect size with less uncertainty, leading to a

lower posterior marginal failure probability. For the case of a large defect, the higher accuracy

inspection will confirm the large size with less uncertainty, which actually increases the posterior

marginal failure probability. But, in this case the decreased uncertainty in the large defect size

enables a better informed maintenance decision, again leading to a lower posterior marginal

failure probability.

As stated, Figure 7.1 assumes that the system is only inspected once at t1, and thus neglects

the value VoI2:n(t,e) that would be provided by the subsequent inspections. Every future

inspection from t2 to tn provides additional information, reducing the residual VoI1(t,e) that was

provided by the inspection at t1. To determine the boundaries of VoI1(t,e) the goal is determine

the times at which to evaluate VoI(t,e) to determine the upper most and lower most possible VoI

that could be provided by only the next inspection. Towards this goal, it is helpful to examine the

extremes of the accuracy of the subsequent inspection at t2. If the inspection at t2 is perfect,

meaning that the inspection precisely measures the actual deterioration without uncertainty, then

the information obtained from the inspection at t2 is complete. This means that, at the time t2, the

residual value gained from the previous inspection at t1 goes to zero, because all information is

now available. This is the case of the least possible VoI to be extracted from the next inspection,

and defines the lower boundary. This is exemplified in Figure 7.1 for the absolute 𝑉𝑜𝐼1𝐿(𝑡, 𝒆1),

where the lower boundary is proportional to the polka dotted area between t1 and t2. The lower

boundary of the absolute 𝑉𝑜𝐼1𝐿(𝑡, 𝒆2) can be determined similarly. In general, the lower boundary

of the absolute VoI of inspection and maintenance plan e is given by:

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𝑉𝑜𝐼1𝐿(𝑡 = 𝑡2, 𝒆) = 𝐶𝐹 ∫

𝑓𝑇(𝜏) − 𝑓𝑇(𝜏, 𝒆)

(1 + 𝑟)𝜏𝑑𝜏 −

𝐶𝑀𝑝𝑀,1(𝒆)

(1 + 𝑟)𝑡1

𝑡2

𝑡1

(7.6)

where E[𝐶𝑀(𝑡, 𝒆)] = 𝐶𝑀𝑝𝑀,1(𝒆) (1 + 𝑟)𝑡1⁄ is the expected cost of maintenance, which is a

simplified version of Equation (7.3), since only the next inspection is considered. Similarly, the

lower boundary of the relative VoI of e2 over e1 is proportional to the area between these lines

for e1 and e2 in Figure 7.1, from t1 to t2, and is given by:

𝛥𝑉𝑜𝐼1𝐿(𝑡 = 𝑡2, 𝒆1, 𝒆2) = ∫

𝑓𝑇(𝜏, 𝒆1) − 𝑓𝑇(𝜏, 𝒆2)

(1 + 𝑟)𝜏𝑑𝜏 +

𝐶𝑀𝑝𝑀,1(𝒆1) − 𝐶𝑀𝑝𝑀,1(𝒆2)

(1 + 𝑟)𝑡1

𝑡2

0

(7.7)

It is possible that time t2 of the subsequent inspection is uncertain. In this case this uncertainty is

accounted for by treating t2 as a random variable with some pdf. Then the lower bound of the

VoI is determined for a given inspection time t2, and the lower bound of the VoI is integrated

across the uncertain t2. In general, uncertainty in any variable can be treated in this way.

Conversely, if the inspection at time t2 and all subsequent inspections are of the worst

possible accuracy, meaning that the inspection results are random, then no information is

obtained from any of the subsequent inspections at t2 to tn, and from a reliability standpoint it is

as if the inspections are not undertaken. Therefore, the residual VoI from the inspection at t1 is

not diminished at all. This is the case of the maximum possible VoI to be extracted from the next

inspection, and defines the upper boundary. This is exemplified in Figure 7.1 for the absolute

𝑉𝑜𝐼1𝑈(𝑡, 𝒆1), where the upper boundary is proportional to the polka dotted area between t1 and

tint. Note that if the pdfs did not intersect at tint the maximum VoI would be determined by

evaluating the VoI at the end of the service life tsl. However, if tsl is far in the future, the pdfs can

intersect before this point (there is one intersection point for each set of 2 pdfs). The intersection

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occurs because for each case, as time progresses into the future, the defect size increases and

more of the pdf lies in the failure domain. This continues until eventually the bulk of the pdf lies

in the failure domain, and only the lower tail lies in the safe domain. At this point, the additional

information in plan e2 over e1, or in e1 over the prior, gained from the inspection at t1, becomes

counterproductive, as the narrower distribution of the defect size leads to a lower probability that

the lower tail of the distribution lies in the safe domain. After this point, the added VoI for

instance of the next inspection in e2 over e1, begins to decrease. For this phenomenon to occur, it

is necessary for the bulk of the pdf of the defect size to lie in the failure domain. While this can

occur in theory, in practice this phenomenon is unrealistic for structural applications, as these

systems would be inspected and maintained far before this point. Therefore, when determining

the upper bound of the VoI, a practical upper boundary is also given by the maximum point of

the VoI, termed tmax. Note that, if the time value of money was ignored, tmax would equal tint;

however, when considering the time value of money, tmax may be shifted earlier. Therefore, the

upper boundary of the VoI is evaluated at time tupper = min(tsl, tmax). As an additional refinement,

if the system has constraints on the maximum reliability or maximum inspection interval, these

restrictions can be incorporated into the analysis. The constraints can be incorporated by

evaluating the upper boundary at tupper = min(tsl, tmax, tconst), where tconst is either a constraint on

the inspection interval, or the time at which the reliability constraint is violated. In Figure 7.1, the

upper boundary of absolute VoI1(t,e1) is proportional to the polka dotted area between t1 and tint,

and in general is given by:

𝑉𝑜𝐼1𝑈(𝑡 = 𝑡𝑢𝑝𝑝𝑒𝑟 , 𝒆) = 𝐶𝐹 ∫

𝑓𝑇(𝜏) − 𝑓𝑇(𝜏, 𝒆)

(1 + 𝑟)𝜏𝑑𝜏 −

𝐶𝑀𝑝𝑀,1(𝒆)

(1 + 𝑟)𝑡1

𝑡𝑢𝑝𝑝𝑒𝑟

𝑡1

(7.8)

Similarly, the upper boundary of the relative VoI of e2 over e1 is given by:

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∆𝑉𝑜𝐼1𝑈(𝑡 = 𝑡𝑢𝑝𝑝𝑒𝑟 , 𝒆1, 𝒆2) = 𝐶𝐹 ∫

𝑓𝑇(𝜏, 𝒆1) − 𝑓𝑇(𝜏, 𝒆2)

(1 + 𝑟)𝜏𝑑𝜏

𝑡𝑢𝑝𝑝𝑒𝑟

𝑡1

+𝐶𝑀𝑝𝑀,1(𝒆1) − 𝐶𝑀𝑝𝑀,1(𝒆2)

(1 + 𝑟)𝑡1 (7.9)

Figure 7.2 illustrates the upper and lower boundaries of the absolute VoI1(t,e1). The

boundaries of the absolute VoI1(t,e2) and the relative ΔVoI1(t,e1,e2) can be determined similarly.

The lower boundaries are all evaluated at t2, and the upper boundaries are all evaluated at tupper =

tmax, because tmax is less than tsl in each case, and there is no constraint. As can be seen in each

case, at times beyond tmax the VoI begins to decrease, which is unrealistic in practice; therefore

tmax is taken as the upper bound.

Figure 7.2. Absolute and relative VoI from the next inspection. VoI boundaries shown for

inspection and maintenance plan e1.

The absolute and relative boundaries of the VoI from the next inspection provide a range of

the VoI of the next inspection, and not the precise VoI. For the absolute VoI, if the cost of the

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inspection at t1 is less than the lower boundary 𝑉𝑜𝐼1𝐿(𝑡, 𝒆), then the inspection is justified.

Conversely, if the cost of the inspection at time t1 is greater than the upper boundary 𝑉𝑜𝐼1𝑈(𝑡, 𝒆),

then the inspection is not justified. Similarly, for the relative VoI, if the additional cost of the

next inspection type in plan e2 over the next inspection type in plan e1 is less than the lower

boundary of the relative 𝛥𝑉𝑜𝐼1𝐿(𝑡2, 𝒆1, 𝒆2), then the next inspection type in e2 is justified, and

conversely, if the additional cost of the next inspection type in e2 is greater than the upper

boundary 𝛥𝑉𝑜𝐼1𝑈(𝑡𝑠𝑙, 𝒆1, 𝒆2), then it is not justified. If the cost lies within the bounds of the VoI

then a more detailed analysis, such as lifecycle RBI analysis, is required to determine the optimal

plan. However, in this case there is still value in this analysis as a first pass, for instance to

eliminate relatively less desirable inspection types, or in aiding semi-quantitative decision

analysis.

7.3.3 Implementation of the methodology

To determine the VoI, the pdf of the failure time fT(t,e), and the probability of maintenance

pM1(e) after the next inspection, are required. The pdf of the failure time is determined from

reliability theory (Melchers, 1999), as described in this section. Inspections of structural systems

are generally imperfect, meaning that the measured deterioration Y differs from the actual

deterioration X, due to the measurement error ε. The measured deterioration is given by:

𝑌 = 𝑋 + 𝜀 (7.10)

The measurement error is a random variable with a standard deviation σε that reflects the

accuracy of a given inspection type. To describe the measurement error a normal distribution

with a mean of zero is commonly used, describing a symmetrical and non-biased measurement

error. The value of an inspection lies in its ability to reduce the uncertainty in the actual

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deterioration. A more accurate inspection type has a lower standard deviation of the

measurement error, and in general will produce measurements closer to the actual deterioration,

providing a greater reduction in uncertainty.

Before the inspection at t1, the prior state of knowledge of the actual deterioration of the

system is represented by the prior pdf fX(x). The prior state of knowledge is based on any

information that is available prior to the inspection, including previous inspection results from

the same system, deterioration (growth) information from similar system, and expert opinion.

Bayes theorem is used to update the prior state of knowledge to the posterior state by

incorporating the result y from the next inspection:

𝑓𝑋|𝑌(𝑥|𝑦) =𝑓𝑌|𝑋(𝑦|𝑥) 𝑓𝑋(𝑥)

𝑓𝑌(𝑦) (7.11)

where fX|Y(x|y) is the posterior pdf of the actual deterioration, fY|X(y|x) is the pdf of the likelihood

of the actual deterioration X given the measured deterioration Y, fX(x) is the prior pdf of the

actual deterioration, and fY(y) is the prior predictive pdf of Y. For a normally distributed

measurement error with a mean value of zero, the likelihood function is given by:

𝑓𝑌|𝑋(𝑦|𝑥) ~ normal(𝑥, 𝜎𝜀) (7.12)

In this way, the prior pdf of the actual deterioration X is updated with the inspection result y

yielding the posterior pdf fX|Y(x|y) of the actual deterioration. However, when deciding which

type of inspection to choose, the inspection result y is not yet known; therefore, it is not possible

to update the prior pdf of the actual deterioration with the measurement result to determine the

posterior distribution. To solve this problem, the inspection result Y is discretized, and the prior

pdf of the actual deterioration X is updated for each discrete value of Y yielding a set of posterior

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pdfs fX|Y(x|y) of the actual deterioration. The probability that each of the posterior pdfs occurs is

given by the prior predictive pdf fY(y):

𝑓𝑌(𝑦) = ∫ 𝑓𝑌|𝑋(𝑦|𝑥)𝑓𝑋(𝑥)𝑑𝑥𝑋

(7.13)

Updating the deterioration for all possible values of Y does not yield any additional

information, because a priori it was already known that the inspection result would follow the

prior predictive distribution. However, applying the maintenance rules m allows the VoI of the

inspection to be harnessed, yielding the posterior pdf fX|Y(x|y,m) given the inspection result y and

the maintenance rules m.

Once the set of posterior pdfs fX|Y(x|y,m) are obtained, the deterioration growth is modelled

through time yielding the set of posterior pdfs fX(t)|Y(x(t)|y,m) as a function of the future time t.

The future deterioration is used as an input for the limit state function describing the failure state

of the system. From the limit state function a probability of failure is determined at each future

time t, which forms the posterior set of time dependent pdfs of the failure time fT(t,e|y). The set

of the posterior pdfs of the failure time are then marginalized according to the prior predictive

pdf fY(y), which is the probability of each inspection value occurring, to determine the posterior

marginal pdf of the failure time fT(t,e) for maintenance plan e, which incorporates the

maintenance rules m:

𝑓𝑇(𝑡, 𝒆) = ∫ 𝑓𝑇(𝑡, 𝒆|𝑦)𝑓𝑌(𝑦)𝑑𝑦 (7.14)

The probability of maintenance pM1(e) at the time t1 of the next inspection is given by the

integral of the prior predictive pdf across all inspection results that lead to maintenance under the

maintenance rules m:

𝑝𝑀1(𝒆) = ∫ 𝑓𝑌(𝑡1)(𝑦(𝑡1))𝑑𝑦

𝑚 (7.15)

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7.4 Numerical example of a corroding pipeline

An example of a corroding oil pipeline system is used to demonstrate the methodology. For

simplicity, this example assumes the pipeline has only one defect, and the defect can only fail by

leaking, not bursting; however, the methodology is applicable for more complex systems with

multiple defects and failure modes. A pipeline leaks when the depth of the corrosion defect

exceeds the wall thickness (wt) of the pipeline (CSA, 2012); therefore, the limit state function for

pipeline leak is given by g(t) = xcrit – X(t), where xcrit is the critical wall thickness, typically

assumed to be 100 % wt. As a reference point for the decision making, it is assumed that the next

inspection is required now at t1 = 0, the subsequent inspection is scheduled for t2 = 4 years, and

the end of the service life is tsl = 10 years. Because pipelines are highly regulated, a reliability

constraint is assumed, corresponding to a maximum allowable failure probability of 10-3

(1/yr).

Without loss of generality, a corrosion rate model (Dann and Huyse, 2016) is used to model

the corrosion growth through time. In a corrosion rate model, the actual corrosion depth X at any

time t is given by X(t) = X0 + CR(t + te), where X0 is the initial corrosion depth at the initiation

time t0, CR is the corrosion rate, and te is the elapsed time between the corrosion initiation time

and now. From the previous inspection results the corrosion rate CR was found to follow a

gamma distribution, with an expected value of 5 % wt / yr, and a standard deviation of 0.75 % wt

/ yr. The elapsed time te between the corrosion initiation and now is 15 years, and the initial

depth is X0 = 0.

The prior corrosion 𝑓𝑋(𝑡1)(𝑥(𝑡1)) at t1 is updated to the posterior corrosion 𝑓𝑋(𝑡1)|𝑌(𝑥(𝑡1)|𝑦)

for each possible inspection result y. To facilitate computation, the actual corrosion X and the

corrosion rate CR are discretized into bins of 0.01 % wt, the inspection results y are discretized

into bins of 1 % wt, and the future time t is discretized into half year increments. The next stage

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of the decision process is the maintenance decision, which is assumed to follow the maintenance

rule m. The maintenance rule m is that the defect is repaired at the inspection time t1 if the

probability of actual depth at the time of the inspection X(t1) exceeding 80 % wt is greater than

10-3

, where the 80 % wt threshold is based on the repair criteria as per B31G (ASME, 1991).

This rule is applied for each posterior pdfs 𝑓𝑋(𝑡1)|𝑌(𝑥(𝑡1)|𝑦), yielding the posterior pdfs

𝑓𝑋(𝑡1)|𝑌(𝑥(𝑡1)|𝑦, 𝑚) given the inspection result and maintenance rule m. Note that if the

probability of failure before time t1 exceeds the maintenance rule m then the defect would be

maintained before the analysis is undertaken; therefore, the analysis is only valid when the prior

does not exceed the maintenance rule m. In this example the probability of failure before t1 is 1.2

x 10-8

.

The prior CR is used to predict the prior distribution of the corrosion at any future time t > t1.

The posterior distributions of the corrosion at any future time t are determined by first updating

the corrosion rate CR using Bayes theorem in the same way as to update the corrosion, and then

modelling the posterior corrosion into the future. Finally, the prior and posterior distributions of

the corrosion at future time t are used to determine the prior pdf fT(t) and the posterior marginal

pdf fT(t,e) of the failure time. This entire process is repeated for different inspection accuracies

σε, and the resulting pdfs of failure time T are shown in Figure 7.3. The inspection accuracy σε

corresponds to 80 % confidence of ± 5, 10, and 15 % wt, where ± 10 % wt is the accuracy of the

standard magnetic flux leakage in-line pipeline inspection tools (POF, 2016). The inspection

types are termed high, medium, and low accuracy respectively. Because the probability of failure

is constrained at 10-3

(1/yr), the plot is cut off in the vertical direction, which imposes a

corresponding maximum constraint of tconst = 6.5 years for each of the inspection types. The time

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constraint could be different for each inspection type; however, as shown in Figure 7.3, in this

example all pdfs of the time to failure intersect the constraint at tconst = 6.5.

Figure 7.3. Time to failure for the prior case and for different inspection accuracies.

Once the pdfs of the failure time are determined, the probability of maintenance pM1(e) must

be determined. The probability of maintenance pM1(e) is the probability that the defect is

maintained subsequent to the next inspection. The value of pM1(e) for the high, medium, and low

accuracy inspections respectively are 0.0012, 0.0045, and 0.0073. Because the inspection occurs

at t1 = 0, the expected cost of maintenance reduces to E[CM(e)] = CMpM1(e), where the cost of

repairing the defect is assumed as CM = $10,000.

From the pdfs of the failure time, the expected cost of failure is obtained by integrating

across the failure time as per Equation (7.2). To determine the expected cost of failure, the cost

of failure CF and the interest rate r are assumed to be $1 MM and 3 % respectively. The absolute

VoI (Figure 7.4) of each inspection type is obtained from the expected cost of failure and

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maintenance, by comparing to the prior baseline with no inspection, as per Equations (7.6) and

(7.8). The lower boundaries of the VoI are obtained by evaluating at t2 = 4 years, and the upper

boundaries are obtained by evaluating at tconst = 6.5 years, because tconst is less than tsl for each

inspection type and thus governs. Note that immediately post inspection the VoI is negative,

because the cost of maintenance temporarily outweighs the value gained in reducing the

expected cost of failure.

Figure 7.4. Absolute VoI for the three possible inspection types, with the upper and lower

boundaries shown.

The boundaries of the absolute VoI are shown in Table 7.1. If the expected cost of inspection

is less than the lower boundary then the inspection is justified. Conversely, if the expected cost

of inspection is greater than the upper boundary then the inspection is not justified. Note that this

example is a simple case of a pipeline with only a single defect; however, in reality pipelines can

have thousands of defects. This means that in reality the VoI will be much higher. For example,

for a pipeline with independent defects, the VoI values from this single defect example would be

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greater by a factor of the number of defects in the pipeline. For reference, medium accuracy in-

line pipeline inspection currently costs in the range of $2000-3300 per km (Koch, Brongers,

Thompson, Virmani, & Payer, 2002). So, in this case, if the pipeline had 30-50 defects per

kilometer then the inspection is justified.

Table 7.1. Boundaries of the absolute VoI from the next inspection for different inspection types.

Inspection type Lower bound

𝑉𝑜𝐼1𝐿(𝑡2, 𝒆) ($)

Upper bound

𝑉𝑜𝐼1𝑈(𝑡𝑠𝑙 , 𝒆) ($)

high accuracy (± 5 % wt) 122.6 1344.6

medium accuracy (± 10 % wt) 67.4 1293.3

low accuracy (± 15 % wt) 2.9 834.3

The relative VoI (Figure 7.5) is determined by comparing the inspection types to each other,

using the lower accuracy inspection type as the baseline in each comparison. The lower

boundary for all three relative curves is defined at the time of the next inspection t2. The upper

boundary of each curve occurs at min(tsl, tmax, tconst). For the high relative to low accuracy and

medium relative to low accuracy curves the governing time is tconst = 6.5 years; however, for the

high relative to medium accuracy curve tmax = 5.5 years governs.

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Figure 7.5. Relative VoI comparing each of the inspection types to each other, with the upper

and lower boundaries shown.

The boundaries of the relative VoI are shown in Table 7.2. To interpret the results, if the

additional cost of the high over the low accuracy inspection is less than $119.8 per defect, then

the high accuracy inspection should be selected over the low. Conversely, if the additional cost

of the high over the low accuracy inspection is greater than $510.3 per defect then the high

accuracy inspection is not justified. The area in between is a grey area, requiring further analysis.

Table 7.2. Boundaries of the relative VoI from the next inspection for different inspection types.

Inspection type Lower bound

∆𝑉𝑜𝐼1𝐿 ($)

Upper bound

∆𝑉𝑜𝐼1𝑈 ($)

high relative to low 119.8 510.3

medium relative to low 64.5 459.0

high relative to medium 55.2 170.3

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A qualitative observation from Table 7.2, especially when considering the upper bounds of

ΔVoI, is that the high and medium accuracy inspections provide a lot of value over the low

accuracy inspection, but the high accuracy inspection only provides moderate value over the

medium accuracy inspection. This suggests that, if there is even a moderate cost increase of the

high accuracy inspection over the medium, then the medium accuracy inspection type is likely a

better decision. Additionally, because the medium accuracy inspection provides a strong relative

improvement over the low accuracy inspection, from this preliminary analysis the medium

accuracy inspection type seems very promising. Qualitative insight of this kind is another benefit

of this methodology. From a quick and simple analysis of only the next inspection, this method

provides insight that can potentially eliminate unsatisfactory inspection types, or, in some cases,

determine the optimal inspection type outright.

7.5 Conclusion

This paper presents a method for determining the optimal next inspection type for a

deteriorating structural system, without performing a lifecycle RBI analysis. This method is a

simple approximation of the decision analysis that is ideally suited to be used in conjunction with

lifecycle analysis to determine the optimal next inspection type. The advantage of this method is

that it is simpler than lifecycle analysis, because only the next inspection is analyzed. The

method centers on the isolation of the decision of the next inspection type from the rest of the

lifecycle decision sequence. Once the decision of the next inspection type is isolated from the

lifecycle decision sequence, it is analyzed independently, and the value of the information

obtained from the potential inspection types are compared. Although the method does not

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determine the precise value of a given inspection, it provides upper and lower bounds on the

value of information from each inspection.

The main limitation of this method is that it only determines the boundaries of the value of

the next inspection, and not the precise value. This can lead to cases where the method cannot

determine the optimal inspection type, because the cost of the inspection lies in the range

between the boundaries of the value of information, and thus it is unclear if the inspection is

justified. However, even in these cases, the method is valuable as a preliminary analysis to

determine if any options can be eliminated, simplifying the subsequent lifecycle analysis. Also,

the methodology could be extended to enable a quick rough estimate of the decision analysis by

using the midpoint of the range as the decision point, and checking if the cost lies above or

below. In cases where the cost of the inspection lies outside of the boundaries of the value of

information this method is invaluable, providing a simple alternative to lifecycle analysis for

determining the optimal next inspection type.

This paper demonstrates the methodology with an example of a corroding pipeline system.

The value of a high, medium, and low accuracy inspection is assessed, and the method identifies

the scenarios where each inspection type is justified. The pipeline example shows a specific

application of the methodology; however, the method is generic and can be applied to any

deteriorating structural system, to aid the decision maker in determining the optimal next

inspection type.

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8 CONCLUSION

This dissertation presents methods for RBI and RBM decision making for deteriorating

engineering systems. The objective is to expand upon the existing body of knowledge in RBI and

RBM to improve the scope, accuracy, and / or efficiency of solutions. This section summarizes

the results and contributions of this research, as well as the limitations and the opportunities for

future work.

8.1 Results

Chapters three and four develop methodologies for RBM of pressure vessels. Chapter three

developed a methodology to determine whether maintenance of an unexpectedly severe defect

can wait until the next scheduled shutdown period, or whether it needs to occur immediately.

The analysis showed that the decision is very sensitive to changes in the measured remaining

wall thickness, and moderately sensitive to changes in the ratios between the various costs.

Chapter four developed a methodology for RBM of pressure vessels with multiple defects and

failure modes. The results show that the system reliability impacts the decision process, and an

analysis without considering system reliability can potentially lead to a suboptimal maintenance

plan. The effect of the dependent failure events is especially important in vessels with many

defects, because the intersection between the failures increases with an increasing number of

defects. Chapter four also showed that the stochastic corrosion growth model linking the depth

and length of the corrosion defects was able to model the defect growth through time.

Chapter five developed a methodology for RBM of pipelines, specifically examining the

decision of whether it is better over the long term to repeatedly repair defects as they become

critical, or to just replace entire pipeline sections. This study found that with increasing number

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of expected failures in adjacent pipeline segments, the optimal decision shifts towards replacing

segments of the pipeline. This is as expected, since an increasing number of expected failures

means an increasing number of expected repairs will be required, shifting the optimal decision

towards replacing pipeline sections instead of performing such a large number of repairs. The

analysis also examined the impact of the service life. It was found that the longer and more

uncertain the service life, the more the optimal maintenance decision shifts towards replacement

instead of repair. The shift towards replacement is because a longer and more uncertain service

life leads to a higher number of expected failures.

Chapter six presented a methodology for using a genetic algorithm to solve the optimization

problem in RBI and RBM. The results show that genetic algorithms can be successfully used as a

heuristic to more efficiently determine the optimal inspection and maintenance plan. For a simple

example with a relatively small solution space of 105 candidate solutions, the genetic algorithm

method determined the optimal plan 7 times faster than an exhaustive search. For a larger

example with of 1080

candidate solutions, exhaustive search was unable to determine the optimal

solution, but the genetic algorithm was still able to determine a solution within a relatively short

computation time on a standard computer. This methodology demonstrated the power of

heuristic algorithms in solving large optimization problems in RBI and RBM, without having to

make assumptions and simplifications to restrict the solution space.

Chapter seven presented a methodology for determining the optimal type of the next

inspection for an engineering system by determining the upper and lower bounds on the expected

value to be extracted from the inspection. The methodology can be used to determine the optimal

next inspection type in cases where the cost of the inspection lies outside of a range of the

expected value, without requiring a lifecycle analysis. For a complex system, this saves a lot of

166

computation. The methodology can also be used to eliminate unsatisfactory inspection types to

decrease the computational requirements in cases where a subsequent lifecycle analysis is still

required. As an approximate approach, the methodology can also be used to determine the

optimal inspection type by determining whether the inspection cost lies within the upper or lower

half of the range of expected value of the inspection.

8.2 Contributions

This section presents the contributions of this dissertation, which are subdivided into two

categories: scientific knowledge and applications in practice. Furthermore, the contributions are

divided into the different aspects of risk-based inspection and maintenance planning that they

apply to, as shown in Figure 8.1.

167

Figure 8.1. Contributions of each chapter of this dissertation to the different areas of risk-based

inspection and maintenance planning.

All of the main body chapters (three through seven) contribute to decision analysis /

inspection and maintenance planning. Chapters four and five also contribute to structural

deterioration modelling and structural reliability analysis.

8.2.1 Scientific knowledge

This dissertation makes contributions to the scientific knowledge in the areas of structural

deterioration modeling, structural reliability analysis, and decision analysis. For structural

deterioration modeling, Chapter four presents a relatively simple stochastic corrosion growth

168

model that predicts both the depth and length of the corrosion defects. Chapter five presents a

population based stochastic corrosion growth model of the depth of a large population of

corrosion defects.

For scientific contributions to structural reliability analysis, Chapter four makes a novel

contribution of a methodology to determine the structural reliability of a pressure vessel,

considering the dependency in the failure events for multiple corrosion defects with two failure

modes (leak and burst). This study demonstrates the importance of considering the dependency

in the failure events, as it was shown that a different optimal plan can be reached when the

dependency is ignored. Chapter five also contributes to structural reliability analysis with a

method to determine the distribution of the failure time for a population of corrosion defects in a

pipeline, and following from the failure time to determine the expected number of failures in the

pipeline for a given service life.

All of the main body chapters (three through seven) make scientific contributions to the field

of decision analysis. Chapter three contributes a simple decision analysis methodology to

determine whether a severe defect in a pressure vessel needs to be repaired immediately, or if the

repair can wait until the next scheduled shutdown period. The methodology considers only a

single defect that can fail by leak. Chapter four contributes the methodology to account for

multiple defects and failure modes when determining the optimal maintenance plan for a

pressure vessel. The methodology expands upon the methodology in Chapter three to consider

multiple defects and two failure modes (leak and burst), and the dependency in the failure events

that this creates. Chapter five contributes the methodology to examine the question of whether it

is better to continuously repair defects as they become critical, or to just replace entire pipeline

sections. This study shows that a sub optimal maintenance plan is achieved if the decision is not

169

analyzed for the entire pipeline and for a long time frame. Chapter five also contributes the

methodology to consider the uncertain service life of the pipeline, and the analysis of the impact

of this uncertainty on the decision analysis. Chapter six contributes the development of the

methodology to implement a genetic algorithm to solve the RBI and RBM optimization problem.

Chapter six also contributes an analysis of the performance of the genetic algorithm in improving

the efficiency of the optimization. An example of a corroding pressure vessel with many defects

that can fail by leaking is used to illustrate the method. The greatest scientific contribution in this

dissertation is Chapter seven, which presents the methodology to determine the optimal next

inspection type, without the need to perform a lifecycle analysis. Much of the research in the area

of RBI relies on a lifecycle analysis to determine the optimal plan. While lifecycle analysis is

able to determine the optimal plan, the analysis can be difficult and computationally demanding.

This is because in order to assess one decision, such as the next inspection type, every decision

throughout the lifecycle needs to be analyzed. This study provides a method to determine the

value extracted from only the next inspection, without having to assess the value of all

inspections in the lifecycle. This methodology is novel, as there has not been any research to date

in unpacking the sequence of decisions throughout the lifecycle of a system to target one specific

decision for RBI. This methodology is a unique way of approaching RBI decision making, and it

could be used as a base from which many other RBI decisions could be simplified, towards the

goal of minimizing the current reliance on lifecycle analysis.

8.2.2 Applications in practice

This study also contributed practically to the oil and gas industry. For pressure vessel

operators, Chapter three provides a simple methodology to determine whether an unexpectedly

170

severe defect needs to be maintained immediately, or if the maintenance can wait until the next

scheduled shutdown period. This methodology was presented at the Pressure Vessel and Piping

Conference in Vancouver, BC, in 2016, and received a positive reception from industry

professionals, who are actively looking to implement practical and simple RBM methodologies.

This study also contributed to the pressure vessel industry by developing a complete

methodology for performing RBM of a pressure vessel with multiple defects and failure modes.

Chapter four of this dissertation contains this methodology, which can be used as a standalone

roadmap to enable a pressure vessel operator to perform RBM for their vessel. This methodology

removed many assumptions that had been used in other studies, but which were perhaps

unrealistic when analyzing an actual system. For pipeline operators, Chapter five provides a

methodology for determining whether it is better to continuously repair defects as they become

critical, or to replace sections of the pipeline. This methodology can be used by pipeline

operators in real time decision making for their pipeline.

For general engineering systems, this study provides methodology to more efficiently or

simply perform RBI and RBM analysis. Chapter six shows system operators how to use heuristic

algorithms, such as the genetic algorithm, to reduce the computational demand in solving the

optimization problem in RBI and RBM. This approach allows system operators to perform RBI

or RBM analysis for a complex system in a much shorter amount of time, hopefully allowing the

analysis to be performed in time to inform a time sensitive decision. Chapter seven provides

system operators with the methodology to determine the optimal type of the next inspection

without requiring a lifecycle analysis, and instead using a quick and comparatively simple

analysis. RBI analysis can be very difficult, and this can be a deterrent to its application by

system operators in the field. This simple preliminary analysis is much easier to perform than

171

RBI analysis, allowing system operators to incorporate RBI into their decision making more

readily.

8.3 Limitations and future work

There are several limitations to this research. One limitation is the reliability analysis for

pressure vessels (Chapters three and four) was performed with Monte Carlo simulation.

Simulation is accurate and simple to implement, which was the motivation for using it; however,

it is not computationally efficient, and there could be savings in using a numerical

approximation, such as FORM or SORM. Regarding pipelines (Chapter five), this study only

considers leak failure, ignoring burst failure. While this assumption is reasonable considering

that the vast majority of failures are leaks, it is still more complete to consider both failure

modes. To consider burst failure, the population based corrosion growth model would need to be

abandoned, because it does not allow the defect size to be modelled in multiple dimensions, as is

required in burst analysis. However, abandoning the population based growth model would in

turn create additional problems in handling large populations of data, as the motivation for using

the population based model its efficiency in modelling many defects. Developing methodology

for long term pipeline system maintenance decision making that includes burst analysis is a

novel avenue for future work in expanding this study. One possible solution is to use the

Enhanced Monte Carlo method (Leira et al., 2014) to determine the failure probabilities, as it

reduces the sample size required for the Monte Carlo simulation.

Another limitation of this study is that it does not present methodology for all aspects of RBI

and RBM decision making. The common aspects of RBI and RBM are the optimization of the

timing, type, and extent of the inspection and maintenance actions. This study thoroughly

172

addresses the common aspects of RBM decision making. Chapters three and four address the

timing of maintenance actions for pressure vessels, and Chapter four also addresses the extent of

maintenance for pressure vessels. Chapter five addresses the timing and type of maintenance for

pipelines. Chapter six addresses the timing and extent of maintenance for a generic engineering

system. However, it was beyond the scope of this study to address all of the aspects of RBI.

Chapter six discusses general concepts of RBI and Chapter seven addresses the optimization of

the type of inspection. This leaves the optimization of the timing and extent of inspections as

avenues for future work. Chapter seven presents a methodology for determining the optimal next

inspection type, without requiring a lifecycle analysis. The expansion of this methodology to the

optimization of inspection timing and extent, also without requiring lifecycle analysis, are the

most exciting avenues for future expansion of this research. However, these questions may prove

more difficult. Consider the decision of when to perform the next inspection. If one inspection is

later than the other, the benefit of that inspection will persist longer into the future, and this

effect needs to be accounted for. Thus, in order to compare two different inspection times, the

timeframe of the analysis has to be normalized, and the methodology for this normalization

needs to be developed. Or, consider the decision of the extent of the inspection. Expanding the

extent of the inspection introduces multiple components or defects to the analysis, and the

dependency between the failure and inspection events will need to be considered, as per the

methodology developed in Chapter four.

173

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COPYRIGHT PERMISSION

This section includes the copyright permissions from ASME for the following two papers

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Haladuick, S., Dann, M. R. 2017a. Risk-based maintenance planning for pressure vessels

with multiple defects. Journal of Pressure Vessel Technology, 139(4), pp. 041602 1-8.

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September 26, 2017

This letter certifies that I, Dr. Markus R. Dann, give permission for Shane Haladuick to use the

following papers, of which I am a co-author, in his manuscript based thesis:



Haladuick, S., Dann, M. R. 2016b. Decision making for long term pipeline system repair or

replacement. ASCE Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil


Haladuick, S., Dann, M. R. 2017a. Risk-based maintenance planning for pressure vessels

with multiple defects. Journal of Pressure Vessel Technology, 139(4), pp. 041602 1-8.

Haladuick, S., Dann, M. R. 2017b. Genetic algorithm for inspection and repair planning of

deteriorating structural systems: Application to pressure vessels. International Journal of

Pressure Vessels and Piping. Manuscript submitted for publication.

Haladuick, S., Dann, M. R. 2017c. An efficient risk-based decision analysis of the optimal

next inspection type for a deteriorating structural system. Structure and Infrastructure


Dr. Markus R. Dann

Documents

A contribution to risk-based inspection and maintenance