Develop Opportunistic Maintenance Models with Considering … Ba_Thesis.pdf · Huy Truong-Ba . Supervisors: Prof Lin Ma (Principal) Dr Michael Cholette (Associate) Submitted in fulfilment

OPPORTUNISTIC MAINTENANCE AND OPTIMISATION

CONSIDERING EXTERNAL OPPORTUNITIES

Huy Truong-Ba

Supervisors:

Prof Lin Ma (Principal)

Dr Michael Cholette (Associate)

Submitted in fulfilment of the requirement for the degree of

Doctor of Philosophy

School of Chemistry, Physics and Mechanical Engineering

Faculty of Science and Engineering

Queensland University of Technology

2018

Opportunistic Maintenance and Optimisation Considering External Opportunities i

Keywords

External opportunity

Maintenance optimisation model

Opportunistic maintenance

Stochastic duration

ii Opportunistic Maintenance and Optimisation Considering External Opportunities

Abstract

Preventive Maintenance (PM) has played an important role in extending the useful life of

industrial assets and reducing the O&M costs of industrial organisations. By taking advantage

of cost saving opportunities, opportunistic maintenance (OM) has significant potential to

further reduce costs and increase the availability of production systems. Most existing studies

have focused on “internal” OM policies where maintenance opportunities result from

maintenance activities performed on another component/system. On the other hand,

opportunities may arrive from sources external to the production system (e.g. low demand, lack

of materials, weather factors) and therefore arrivals and durations are uncertain and

uncontrollable.

In this thesis, mathematical models are developed for OM policies when the opportunities have

the following characteristics: random arrivals with time-varying statistical properties,

stochastic in duration and time-varying maintenance cost savings. Four OM optimisation

models are also developed to investigate and analyse the impact and benefit of OM policies:

• A time-based model where the arrival and duration of external opportunities are

modelled with time-varying characteristics. The time for PM and criteria for accepting

maintenance opportunities are found by minimising the single-cycle total cost.

• The second model adopts the condition-based approach in order to determine the

optimal condition-based OM policy for a finite-horizon mission. The maintenance

optimisation problem is formulated as a finite-horizon Markov Decision Process where

the randomly occurring opportunities are accounted for by augmenting the time-

varying, decision-dependent transition probabilities.

• The third model describes how to integrate predictions of opportunity characteristics

into OM decisions. A Dynamic Programming approach is used to obtain the optimal

policy, consisting of time-varying thresholds on equipment condition, the cost of

conducting maintenance and forecasts of arrived opportunity.

• The fourth model discusses a special OM policy type where the external events can

yield a “free” PM by restoring a part of the current condition (degradation) of

investigated systems with no cost. An important application of this special case is in

Opportunistic Maintenance and Optimisation Considering External Opportunities iii

solar power where the receivers of photovoltaic systems or the mirrors of Concentrated

Solar Power (CSP) systems can be cleaned by rain. The study of this special OM policy

has been conducted as a case study of a condition-based cleaning policy for the mirrors

of a hypothetical CSP system located in Brisbane, Australia.

The proposed models have been illustrated and analysed through numerical examples and real-

world case studies of wind turbines and solar power systems. The numerical results presented

in this thesis have shown the significant savings in maintenance costs compared to the

traditional PM policies. The comparison between the (more traditional) full-opportunity-only

policies and partial opportunity policies has shown that the partial opportunity policies can

yield significant cost savings. The OM policies developed would have a significant benefit for

several production systems affected by external opportunities, e.g. for renewal energy systems

such as wind and solar power and agriculture processing industries.

iv Opportunistic Maintenance and Optimisation Considering External Opportunities

Table of contents

Keywords .................................................................................................................................... i

Abstract ..................................................................................................................................... ii

Table of contents ..................................................................................................................... iv

List of tables ........................................................................................................................... viii

List of figures ............................................................................................................................ ix

Abbreviations .......................................................................................................................... xii

Statement of Original Authorship .......................................................................................... xiii

Acknowledgment ................................................................................................................... xiv

Chapter 1: Introduction ..................................................................................... 1

1.1 Maintenance overview ...................................................................................................2

1.2 Introduction to Opportunistic Maintenance (OM) .........................................................3

1.3 Objectives and main contribution of this thesis .............................................................5

1.3.1 Research problem ................................................................................................5

1.3.2 Research objectives ..............................................................................................5

1.3.3 Research scope .....................................................................................................6

1.3.4 Optimisation approach .........................................................................................6

1.3.5 Contributions and significance .............................................................................8

Chapter 2: Literature review ............................................................................ 10

2.1 Maintenance policy overview ...................................................................................... 11

2.1.1 Time based maintenance .................................................................................. 11

2.1.2 Condition based maintenance (CBM) ............................................................... 13

2.2 Opportunistic Maintenance (OM) ............................................................................... 16

2.2.1 Time based Opportunistic Maintenance ........................................................... 21

2.2.2 Condition based Opportunistic Maintenance ................................................... 23

2.3 OM optimisation models ............................................................................................. 24

2.4 Solution methodology ................................................................................................. 26

2.5 Summary and gaps discussion ..................................................................................... 28

Chapter 3: Joint Opportunistic and Preventive Maintenance policy for an infinite time horizon: Time-based approach ............................................................................ 31

Opportunistic Maintenance and Optimisation Considering External Opportunities v

3.1 Introduction ................................................................................................................. 34

3.2 Model Description ....................................................................................................... 34

3.3 Joint Optimisation Models ........................................................................................... 36

3.3.1 Determining 𝒑𝒑𝑶𝑶𝑶𝑶 .............................................................................................. 39

3.3.2 Model 1: Optimisation when CM is minimal ..................................................... 41

3.3.3 Model 2: Optimisation when CM is perfect ...................................................... 43

3.3.4 Verification of models via simulation ................................................................ 45

3.4 Numerical Examples .................................................................................................... 47

3.4.1 Model 1: OM and PM are perfect; CM is minimal ............................................ 49

3.4.2 Model 2: CM, OM and PM are perfect .............................................................. 55

3.5 Conclusion ................................................................................................................... 57

Chapter 4: Joint Opportunistic and Preventive Maintenance policy for a finite time horizon: Condition-based approach .................................................................... 58

4.1 Introduction ................................................................................................................. 60

4.2 Model description ........................................................................................................ 60

4.2.1 Degradation and maintenance model ............................................................... 63

4.2.2 Opportunity model ............................................................................................ 64

4.2.3 Cost structure .................................................................................................... 65

4.2.4 Optimisation through Markov Decision Process (MDP) .................................... 66

4.3 Policy evaluation via simulation .................................................................................. 71

4.4 Numerical Example ...................................................................................................... 72

4.4.1 Input data .......................................................................................................... 73

4.4.2 Optimal maintenance policy ............................................................................. 80

4.4.3 Sensitivity analysis ............................................................................................. 81

4.5 Conclusion ................................................................................................................... 85

Chapter 5: Joint Opportunistic and Preventive Maintenance policy: Integrating with opportunity forecasts ......................................................................................... 86

5.1 Introduction ................................................................................................................. 88

5.2 Optimisation model ..................................................................................................... 88

5.2.1 Policy description .............................................................................................. 88

5.2.2 Degradation model ............................................................................................ 91

5.2.3 Maintenance actions ......................................................................................... 91

vi Opportunistic Maintenance and Optimisation Considering External Opportunities

5.2.4 Cost structure .................................................................................................... 92

5.2.5 Opportunity model ............................................................................................ 92

5.2.6 Opportunity forecast ......................................................................................... 93

5.2.7 Optimisation model in MDP approach .............................................................. 94

5.3 Age-based versus condition-based opportunistic maintenance policies .................... 96

5.3.1 Age-based approach .......................................................................................... 96

5.3.2 Condition-based approach ................................................................................ 98

5.4 Evaluating policy via simulation................................................................................... 99

5.5 Numerical example .................................................................................................... 100

5.5.1 Input ................................................................................................................ 100

5.5.2 Comparison between the policies with and without forecasts ...................... 104

5.5.3 Benefits of short-term forecasts of opportunities .......................................... 105

5.6 Conclusion ................................................................................................................. 110

Chapter 6: Cost-free opportunistic maintenance: Case study of cleaning strategy for solar power 111

6.1 Introduction ............................................................................................................... 113

6.2 Modelling and Optimisation ...................................................................................... 115

6.2.1 Problem description ........................................................................................ 115

6.2.2 Reflectivity degradation model ....................................................................... 117

6.2.3 Total cleaning cost ........................................................................................... 118

6.2.4 Optimisation of the cleaning policy ................................................................. 119

6.3 Condition-Based Cleaning Using Real Data and the System Advisor Model (SAM) .. 120

6.3.1 Data-based computation of transition probabilities ....................................... 121

6.3.2 Data-based computation of Potential Revenue 𝑯𝑯(𝒕𝒕𝒌𝒌) ................................... 125

6.4 Policy Evaluation Strategy using Monte Carlo Simulation ......................................... 128

6.5 Case Study – Brisbane (Queensland), Australia ......................................................... 130

6.5.1 DNI and electricity prices in Brisbane .............................................................. 130

6.5.2 Solar plant configuration, efficiency and cleaning costs ................................. 133

6.5.3 Evaluation of condition-based cleaning policies vs time-based policy ........... 135

6.6 Conclusion ................................................................................................................. 142

Chapter 7: Conclusions and future work ......................................................... 144

7.1 Findings and contributions ........................................................................................ 145

Opportunistic Maintenance and Optimisation Considering External Opportunities vii

7.2 Potential future studies ............................................................................................. 147

References ....................................................................................................... 149

viii Opportunistic Maintenance and Optimisation Considering External Opportunities

List of tables

Table 2.1 Summary of Types of OM and Modelling Approach .............................................. 18

Table 3.1 The Inputs for Simulation ....................................................................................... 45

Table 3.2 The General Inputs for Model 1 .............................................................................. 49

Table 3.3 PM time 𝒕𝒕𝑷𝑷𝑶𝑶 and cost savings vs no OM for Model 1. Effect of PM/CM cost ....... 51

Table 3.4 PM time 𝒕𝒕𝑷𝑷𝑶𝑶 and Cost Savings vs no OM for Model 1. Effect of 𝒄𝒄𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒄𝒄𝒃𝒃

.................... 52

Table 3.5 Savings with Proposed OM Policy over Traditional PM for Model 1 ...................... 54

Table 3.6 Savings achieved considering Partial Opportunities over full-opportunity-only OM for Model 1 ................................................................................................................... 55

Table 3.7 The General Inputs for Model 2 .............................................................................. 55

Table 3.8 Savings with Proposed OM Policy over Traditional PM for Model 2 ...................... 56

Table 3.9 Savings Achieved Considering Partial Opportunities over Full-opportunity-only OM for Model 2 ................................................................................................................... 57

Table 4.1 Discrete States of Main Bearing Degradation ........................................................ 74

Table 4.2 Failure and Maintenance Parameters .................................................................... 75

Table 4.3 The Scale Parameters of Weibull Distribution for Wind Speed at Mount Emerald, Queensland, Australia ............................................................................................ 76

Table 4.4 Saving Rates of Joint OM and PM Policies when Failure Cost Varies ..................... 82

Table 4.5 Saving Rates of Joint OM and PM Policies when Downtime Cost Varies ............... 83

Table 5.1 Input Parameters .................................................................................................. 103

Table 5.2 Savings of using Short-term Forecasts when Forecasting Accuracy Varies (the smaller the values the more accurate) for two cases: a) TB-OM model, and b) CB-OM model .............................................................................................................................. 109

Table 6.1 Comparisons among Three Cleaning Policies: CBC with/without considering Rain and Fixed Cycle Strategy .............................................................................................. 137

Opportunistic Maintenance and Optimisation Considering External Opportunities ix

List of figures

Figure 1.1. The number of publications relating to OM. ..........................................................4

Figure 1.2. Research objectives. ...............................................................................................6

Figure 1.3. Overall Approach of Developing Optimisation models of OM policy. ....................7

Figure 3.1. Two examples of cycles for scenario 1: Minimal corrective maintenance. In Example a, OM terminates the cycle when an opportunity of duration greater than 𝒅𝒅𝒎𝒎𝒎𝒎𝒎𝒎(𝒕𝒕) occurs. In Example b, PM terminates the cycle since no opportunity occurs with duration greater than 𝒅𝒅𝒎𝒎𝒎𝒎𝒎𝒎(𝒕𝒕). .............................................................................. 35

Figure 3.2. Three examples of cycles for scenario 2: Perfect corrective maintenance. In Example a, CM terminates the cycle (i.e. renews the machine), while in Example b OM terminates the cycle since an opportunity of sufficient duration occurs prior to PM and failure. Finally, Example c illustrates the scenario where PM terminates the cycle, since neither a failure nor opportunity of sufficient duration occur prior to PM. ................................................................................................................................ 35

Figure 3.3. Illustrative plot of penalty cost function. ............................................................. 38

Figure 3.4. Comparison between probabilities determined from simulation and calculation in scenario Minimal CM. ............................................................................................ 46

Figure 3.5. Comparison between probabilities determined from simulation and calculation in scenario perfect CM. .............................................................................................. 47

Figure 3.6. Comparison between cost rates determined from simulation and calculation. . 47

Figure 3.7. Optimisation procedure. ...................................................................................... 49

Figure 3.8. GA convergence. .................................................................................................. 50

Figure 3.9. Minimum duration threshold for Model 1 when PM cost 𝒄𝒄𝑷𝑷𝑶𝑶 varies. ............... 52

Figure 3.10. The minimum duration thresholds for Model 1 when ratios between production loss cost and direct cost 𝑪𝑪𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍

𝑪𝑪𝒃𝒃 are varied. ................................................................ 53

Figure 3.11. The optimal solutions of Model 1 when maximum penalty cost of OM cpmax varies. ................................................................................................................................ 54

Figure 4.1. The mechanism of opportunistic maintenance policy. ........................................ 61

Figure 4.2. OM scenarios: a) Doing PM; b) No PM and doing OM with suitable opportunity; c) No PM and do-nothing for OM due to unsuitable opportunity; and d) Neither PM nor OM due to no opportunity. .................................................................................... 63

Figure 4.3. Optimisation procedure. ...................................................................................... 70

Figure 4.4. Simulation flow chart. .......................................................................................... 72

Figure 4.5. Representative example of weekly degradation in the main bearing of the wind turbine. ................................................................................................................... 74

Figure 4.6. Annually probability of Opportunity occurrences (at least one week day has average wind speed lower than 3.5 m/s). .............................................................. 77

x Opportunistic Maintenance and Optimisation Considering External Opportunities

Figure 4.7. Discrete distributions of opportunity durations. ................................................. 77

Figure 4.8. Weekly average electricity price in Queensland – Australia. .............................. 78

Figure 4.9. Average daily possible electricity generation of wind turbine REpower 3.4-104 WTG at air density of 1.09 kg/m3. .................................................................................. 79

Figure 4.10. Average downtime cost per day. ....................................................................... 79

Figure 4.11. Optimal Maintenance policy. ............................................................................. 80

Figure 4.12. Probability of conducting OM and PM (average for 1 year). ............................. 81

Figure 4.13. Probabilities of maintenance types with full OM and partial OM policies. ....... 83

Figure 4.14. Ratios between probabilities of OM and PM with full OM and partial OM policies. ................................................................................................................................ 84

Figure 4.15. Savings of OM policies with different wind speed thresholds creating maintenance opportunities. ......................................................................................................... 85

Figure 5.1. The maintenance options for each time interval: a) Doing nothing; b) Doing PM; and c) Planning OM. ............................................................................................... 90

Figure 5.2. The optimisation approach for each time interval. ............................................. 91

Figure 5.3. One simulation repetition flow chart. ............................................................... 100

Figure 5.4. Production loss cost per time unit. .................................................................... 101

Figure 5.5. Opportunity Intensity......................................................................................... 102

Figure 5.6. Probability of conducting OM and PM of the policy with and without short-term forecasts for TB-OM approach. ............................................................................ 104

Figure 5.7. Probability of conducting OM and PM of the policy with and without short-term forecasts for CB-OM approach. ............................................................................ 105

Figure 5.8. Savings of using short-term forecasts compared to OM without forecasts. ..... 106

Figure 5.9. Savings in using short-term forecasts compared to OM without forecasts. ..... 107

Figure 5.10. Savings of using short-term forecasts when forecast window varies. ............ 108

Figure 6.1. Reflectivity degradation and cleaning threshold. .............................................. 116

Figure 6.2. The model structure and cleaning policy implementation. ............................... 117

Figure 6.3. Monte Carlo simulation of Cleaning Policy 𝝅𝝅. ................................................... 129

Figure 6.4. Historical DNI in Brisbane from 2005 to 2015 . ................................................. 131

Figure 6.5. Historical price in Brisbane from 2005 to 2015. ................................................ 131

Figure 6.6. Produced hourly average DNI, Price and Price × DNI over one year. ............... 132

Figure 6.7. Produced random hourly samples of Price × DNI over one year. ..................... 132

Figure 6.8. A simulated CSP plant layout in Brisbane, Queensland. .................................... 133

Figure 6.9. Hourly cosine efficiencies for selected mirrors over one year. ......................... 134

Figure 6.10. Average Daily probability of rain in Brisbane, Australia. ................................. 135

Opportunistic Maintenance and Optimisation Considering External Opportunities xi

Figure 6.11. The cleaning reflectivity thresholds of two optimal policies for one year: Considering rain vs not considering rain. These thresholds repeat almost periodically for each year in the horizon. ................................................................................ 136

Figure 6.12. Probability of a clean for each day of the year. ............................................... 138

Figure 6.13. The effect of cleaning cost on savings compared to the optimal fixed cycle strategy. ................................................................................................................ 139

Figure 6.14. The effect of cleaning delay. ............................................................................ 140

Figure 6.15. The effect of rain probabilities. ........................................................................ 141

Figure 6.16. Reflectivity thresholds for one year with different rain probabilities. These thresholds repeat almost periodically for each year in the horizon. ................... 142

xii Opportunistic Maintenance and Optimisation Considering External Opportunities

Abbreviations

AMEO: Australian Energy Market Operator

AREMI: Australian Renewable Energy Mapping Infrastructure

CBC: Condition-based cleaning

CBM: Condition-based maintenance

CB-OM: Condition-based opportunistic maintenance

CM: Corrective maintenance

CSP: Concentrated Solar Power

DNI: Direct Normal Irradiation

DP: Dynamic Programming

HPP: Homogeneous Poisson Process

MDP: Markov Decision Process

NHPP: Non-homogeneous Poisson Process

OM: Opportunistic maintenance

O&M: Operation and Maintenance

PM: Preventive maintenance

PV: Photovoltaic

SAM: System Advisor Model

TBM: Time-based maintenance

TB-OM: Time-based opportunistic maintenance

Opportunistic Maintenance and Optimisation Considering External Opportunities xiii

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet requirements

for an award at this or any other higher education institution. To the best of my knowledge and

belief, the thesis contains no material previously published or written by another person except

where due reference is made.

Signature: QUT Verified Signature

Date: October 2018

xiv Opportunistic Maintenance and Optimisation Considering External Opportunities

Acknowledgment

First and foremost, I would like to express my sincere gratitude to my principal supervisor,

Prof Lin Ma for the continuous support from the very beginning of my PhD application to the

end of this journey. Without her ultimate support, I am afraid that I had no chance to pursue

my PhD study in QUT. I also appreciate her guidance and assistance in conducting my research

and finishing this thesis.

My sincere thanks also go to my associate supervisors, Dr Michael Cholette and Dr Pietro

Borghesani, for the exciting and invaluable time working with them. I am especially grateful

to their knowledge and experience sharing as well as comments and revisions that help me not

only in finishing this thesis but also for my research life in future.

I would also like to thank my family for supporting me spiritually throughout writing this thesis

and my life in general. In particular, I must acknowledge my beloved wife, without whose love,

encouragement and editing assistance, I would not have finished this thesis.

Completing this work would have been all the more difficult were it not for the emotional

support and friendship provided by the other members of Prof Lin Ma’s research team such as

Dr Ruizi Wang (Norman), Dr Hongyang Yu and my friends Sinda, Arif and Giovanni. I am

indebted to them for their help.

Chapter 1: Introduction 1

Chapter 1: Introduction

Overview

This chapter provides an overview of maintenance and the concept of opportunistic

maintenance. Opportunistic maintenance has been widely studied in recent years and has

proven benefits compared to traditional preventive maintenance. The research problem and the

contributions of this thesis are also discussed.

2 Chapter 1: Introduction

1.1 MAINTENANCE OVERVIEW

Maintenance plays an important role in the activities of industrial enterprises. Maintenance

activities improve the productivity and profitability of a company through improving

availability of production systems, maintaining the quality of products and keeping the safety

of working environments (Alsyouf, 2007). Wireman indicated in his book (Wireman, 2010)

that the downtime costs have historically been two to 14 times higher than maintenance costs.

He also said that the maintenance expenditures in United States (US) have risen 10–15% per

year since 1979 and were estimated to be about three trillion USD in 2010 (Wireman, 2010).

However, about one-third of this cost was ineffective (Wireman, 2010). Heng et al. (2009) also

stated that about 30% to 50% of expense of US companies is lost due to ineffective maintenance

policies and it is similar in other countries. Thus, applying efficient maintenance strategies is

the most important consideration for many companies in order to increase the competition

position in market.

Many studies on maintenance have been conducted over many decades. Maintenance can be

classified into two major types: Corrective Maintenance (CM) and Preventive Maintenance

(PM). CM events are unplanned actions on failed components or systems to restore them to

functioning condition; on other hand, PM is defined as scheduled activities to improve or

extend the expected life of working systems (Wang and Pham, 2006). Researchers have

focused more on PM, seeking to optimise PM actions to minimise costs. In the traditional

approach, the plan for PM activities can be set at an optimal period (Barlow and Hunter, 1960).

In the past few decades, with the development of technology in wide-range areas, particularly

in computer and information processing fields, the real-time condition of industrial systems

has become increasingly available. These condition measurements enable better informed PM

decisions for both maintenance activities and timing. The types of PM strategies based on

condition are so-called condition-based maintenance (CBM) (Jardine et al., 2006; Prajapati et

al., 2012). The typical policy of PM in CBM is to conduct PM when the condition of system

reaches an optimal threshold (Jardine et al., 2006; Prajapati et al., 2012). Another approach for

PM derived from CBM strategy is predictive maintenance (Aizpurua et al., 2017; Heng et al.,

2009; Jardine et al., 2006; Nguyen et al., 2015; Prajapati et al., 2012). In this approach, the

information of system conditions is used to forecast the failure time or the remaining life of

these systems. Based on this prediction, decisions of appropriate PM actions will be selected

for an optimal time (i.e. before any failures but as not to waste too much of the remaining life).

According to Wang and Pham (2006), maintenance actions can be classified into five types:


• Minimal repair is a maintenance action that restores the system operating condition

to the condition right before failure.

• Perfect repair or replacement is a maintenance action that restores the system to as

good as new state.

• Imperfect repair is a maintenance action that brings the system’s condition to a better

state than before maintenance. The state of the system is somewhere between

minimal and perfect repair.

• Worse repair makes the system’s operating condition worse than just prior to its

failure, but the system still works after maintenance.

• Worst repair is the maintenance action that cannot restore the normal operation of

the system.

Among these maintenance types, the first three, i.e. minimal, perfect and imperfect repairs, are

considered in most studies.

1.2 INTRODUCTION TO OPPORTUNISTIC MAINTENANCE (OM)

With the increasing complexity of industrial systems and the wider application of PM, the cost

of PM activities at enterprises is escalating (Jardine et al., 2006) and new approaches for PM

have been proposed. A well-known advanced approach is Opportunistic Maintenance (OM).

This kind of maintenance activity can be defined as PM; however, it is not at a fixed point of

time or condition but at an arriving “opportunity” (Ab-Samat and Kamaruddin, 2014; Saranga,

2004). Commonly considered opportunities are the repairs for failures of other components in

the systems (Saranga, 2004; Sherwin, 1999), or the events of conducting PM for a certain

component of considered systems (Abdollahzadeh et al., 2016; Zhou, Huang, et al., 2015;

Zhou, Lin, et al., 2015). OM can also be defined as combined or group maintenance activities

of some components of the systems in order to reduce the maintenance cost and time (Pham

and Wang, 2000). These opportunity types (e.g. conducting maintenance activities for a certain

component) can be classified as internal opportunities in the sense that the opportunity

originates within the system itself. In contrast, external opportunities are external

factors/events that can yield considerable cost savings such as stops due to weather conditions

or production issues (e.g. lack of material), or some periods when production loss costs or

maintenance-related costs are low (low demand, low costs for spare parts, etc.).


The number of published papers relating to OM from year 1963 up to now are presented in

Figure 1.1 according to Scopus database.

Figure 1.1. The number of publications relating to OM.

The interest in OM has been increasing dramatically since approximately 10 years ago. It shows

that the advantages of OM policies have been considered by researchers and also by industrial

organisations. There are two main benefits of OM compared to PM: 1) the savings of set-up

cost when initiating a maintenance action, and 2) reduced downtime cost which is caused by

frequent PM activities if applying traditional PM policies (Ab-Samat and Kamaruddin, 2014).

The most challenging aspect of OM is the nature of uncertainty of opportunity occurrence. The

failures of components, the stoppages of production systems due to economic or environmental

causes (i.e. rain, storm, etc.) can be considered as opportunities, and they have some uncertain

aspects. Thus, the OM policies have decisions that depend on the arrival time of opportunity.

However, the occurrence time is not only stochastic. The benefit of opportunities, especially

the external opportunities, can also be subject to significant randomness. For example, the lost

time necessary for repairing a machine or the duration of a rain or storm is generally uncertain

and thus must produce a random saving in downtime cost. Due to the stochastic properties of

benefits induced by external opportunities, the savings of external opportunities may be

considered as “partial” instead of “full” benefits. For instance, if the PM time is longer than the

stoppage induced by the (random) opportunity duration, only a portion of the maintenance

downtime cost would be saved (e.g. stoppages due to wet weather in sugarcane processing).

This property of external opportunities is a significant difference compared to internal

opportunities, which usually induce the “full” savings (e.g. saving full set-up cost when doing

maintenance on a group of components).

0

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Number of publications in OM over time

Number of publications


1.3 OBJECTIVES AND MAIN CONTRIBUTION OF THIS THESIS

1.3.1 Research problem

OM can improve the efficiency of the maintenance activities for industrial organisations.

Therefore, OM has become a topic in reliability and maintenance research which has attracted

many researchers. However, the external OM policies, which involve unavoidable production

stoppages, have not been widely studied. In particular, the impacts of non-stationary

occurrences and stochastic durations of opportunity and uncertain future opportunity

information have yet to be considered.

The aim of this research is to answer the general research question relating to external OM

strategy: How can OM policies be optimised when considering the following characteristics of

external opportunities:

• stochastic characteristics and partial benefit of opportunities;

• interaction between optimal PM and OM; and

• imperfect forecast information on occurrence and other properties of external

opportunities.

1.3.2 Research objectives

The research aims to obtain achievement of the following objectives:

1. Develop a one-cycle Time-based OM (TB-OM) optimisation model for a production

system where stochastic cost savings of external opportunities are considered.

2. Develop a Condition-based OM (CB-OM) optimisation model for a production

system in a finite mission time where stochastic cost savings of external

opportunities are considered.

3. Develop an optimisation model of OM that integrates the information of prediction

of opportunity occurrence and characteristics.

4. Develop an OM model considering cost-free external maintenance events under the

case study of condition-based cleaning policy for a CSP system with the effects of

natural cleaning events of rains.


Figure 1.2. Research objectives.

1.3.3 Research scope

This study is restricted by the following limitations:

• Studied OM policies consider only external opportunities. Opportunities created by

the failure of components of the system are not investigated in optimisation models.

• Objectives of OM optimisation models are limited to cost-related functions. Other

performances of maintenance policies (e.g. availability) are not considered.

1.3.4 Optimisation approach

The overall approach for developing an OM optimisation model is introduced in Figure 1.3.


Figure 1.3. Overall Approach of Developing Optimisation models of OM policy.

First, the system states must be described with two popular approaches: two-state reliability

and multi-state degradation models. The two-state (i.e. functional and failure) models are

typically developed based on the traditional reliability theory. Whereas the multi-state models,

which are suitable to describe the degradation processes of systems in CBM approaches, are

developed according to some stochastic process (i.e. Gamma, Wiener, etc.) or Markov chain.

The random arrivals and cost savings of opportunities are described in the opportunity model.

Most literature considered stationary processes for opportunity occurrences; however, some

opportunities such as weather-caused ones have time-varying intensities. The cost savings of

opportunities can be achieved through the sharing of maintenance set-up costs or the downtime

cost of doing maintenance during an opportunity period. The state and opportunity models are


combined in an optimisation formula where the cost-related objectives are defined. Some

studies defined the optimisation with some constraints such as limitation of maintenance

resources (e.g. time, spare parts, labour, etc.). The optimisation formula can be solved globally

according to some analytical methods. In some situations, the near-optimal solutions are

obtained by using some heuristic algorithms (e.g. Genetic Algorithm, Particle Swarm

Optimisation, etc.) and simulation. The proposed OM policies are evaluated through real-case

studies. However, when the real data is not always available, the numerical (simulated)

examples can be used to evaluate the policies, and sensitivity analysis is conducted to

understand the effects of varying inputs.

1.3.5 Contributions and significance

Improving the efficiency of maintenance policy is one of the best ways of cutting down the

operation cost of companies. Therefore, OM should be studied and applied in order to increase

the effectiveness of PM activities. In particular, the OM when considering external

opportunities should be focused, especially in industrial organisations where their operations

rely on external factors (e.g. weather conditions, demands, prices, etc.). However, studies of

OM with external opportunities have been limited in literature and that is the main motivation

for this thesis.

For the research community, this work contributes to knowledge of OM field by extending the

study of external opportunities and their characteristics as well as integrating them to the

optimisation. The thesis exploits the mathematical models for joint PM and OM policies in

both time-based and condition-based approaches. Specifically, two practical characteristics of

external opportunities are investigated in this thesis: (1) the non-stationary stochastic

occurrences; and (2) random durations. Benefits (cost savings) of external opportunities are

“partial” where just a part of downtime costs is reduced due to the duration of opportunities

being less than the required time of maintenance activities. The methods developed in this

thesis allow for the exploitation of these “partial” opportunities for the first time. OM is applied

only if the duration of arrived opportunity is long enough (i.e. creates enough cost savings).

The interaction between the decisions of OM and PM is also investigated and analysed in

developed models. This thesis investigates the use of short-term forecasts of external

opportunities in OM decisions by analysing how their benefit varies with forecast accuracy and

availability. A special opportunity is also investigated in this thesis for the first time: the case

of an external, randomly arriving, cost-free restoration of the condition. Such an opportunity

may enable the deferral of an expensive maintenance and was studied in the context of


optimising mirror cleaning policies for CSP systems, where rain events can yield free cleaning

events for mirrors.

The contributions of this research are not only in the scientific body of knowledge but also for

real-world applications. For industry, this study proposes a new maintenance policy that is

promising to achieve more benefits (maintenance cost savings) than traditional ones. These

policies consider external sources that make a significant reduction in maintenance costs such

as the downtime costs. Therefore, implementing an optimal maintenance policy that can take

advantage of these events will contribute significant savings in O&M costs for companies. The

numerical results in this thesis also show that the proposed joint OM and PM policies can yield

significant maintenance cost reductions especially when the production loss cost is significant

in maintenance cost. The case studies for wind turbine maintenance and cleaning policies for

CSP systems demonstrate the possibility and benefit of implementation developed OM policies

in renewal energy systems such as wind turbines and solar power where their operation (and

maintenance) are affected by weather factors.

10 Chapter 2: Literature review

Chapter 2: Literature review

Overview

This chapter gives a review of maintenance studies with emphasis on opportunistic

maintenance (OM) research areas. Firstly, section 2.1 summarises some main properties of

time-based and condition-based maintenance policies. Secondly, the details of OM models and

approaches in previous studies are reviewed in sections 2.2, 2.3 and 2.4. Finally, the discussion

of some research gaps in OM studies are presented in the last section.

Chapter 2: Literature review 11

2.1 MAINTENANCE POLICY OVERVIEW

Maintenance includes planned and unplanned actions conducted to keep or restore a system to

an acceptable operating condition. Optimal maintenance policies aim to ensure best system

reliability and safety performance at the lowest possible cost (Wang and Pham, 2006). The

study of different maintenance policies is a significant research field to prevent the occurrence

of system failures and to reduce maintenance costs (Wang and Pham, 1996). Barlow and

Hunter (1960) first proposed a well-known and basic optimal maintenance policy where a

single-component system is preventively maintained at an optimal life time (age) in order to

minimise the maintenance cost. Since the study of Barlow and Hunter was published, many

maintenance models have been proposed that considered and integrated many aspects of

maintenance such as cost categories, maintenance repair types, inspection and replacement,

system structures, dependencies and opportunistic maintenance, system condition

detection/prediction and information/signal processing, etc. Some researchers (Ab-Samat and

Kamaruddin, 2014; Dekker et al., 1997; Gorjian et al., 2010; Heng et al., 2009; Jardine et al.,

2006; Pham and Wang, 1996; Prajapati et al., 2012; Wang, 2002) tried to summarise the

research and practice of maintenance in different approaches. To date, maintenance models can

be classified into two main approaches, i.e. time (age) based maintenance (TBM) and

condition-based maintenance (CBM).

2.1.1 Time based maintenance

The first and most well-known TBM policy was the model proposed by (Barlow and Hunter,

1960). There were two proposed optimal replacement policies. The first one is that a

component/system is replaced at a time when a failure occurs or at an optimal time for

preventive maintenance. The second one is similar in that systems are replaced at an optimal

time with the assumption that minimal repairs (i.e. the maintenance action that restores systems

from failure to functioning state but without changing the reliability of systems) is applied

when failures happen. Coolen-Schrijner and Coolen (2007) analysed the age replacement

policy in a different way to Barlow and Hunter where the one-cycle criterion was considered

instead of the renewal process. Lu et al. (2012) investigated the TBM policy under a time-

varying environment. De Jonge et al. (2015) studied the benefit of delaying the TBM due to

the uncertainty of lifetime distribution. Some articles integrated the TBM policy with

production factors such as scheduling, inventory, etc. in order to jointly optimise the operation

and maintenance (O&M) costs. Bergeron et al. (2009) developed an optimal integrating model

of lot sizing and maintenance strategy to satisfy the demand of products without backlogging.


Tam et al. (2007) proposed a maintenance schedule considering lead time for spare parts and

ageing due to the operation rate of the system. Gan et al. (2013) analysed the buffer inventory

and maintenance plan for a two-series machine system with intermediate buffer. Nourelfath et

al. (2015) developed a joint optimisation model of maintenance and production plan where the

cost of product quality was accounted. Xiao et al. (2016) considered a joint optimal model for

both production scheduling and machine group preventive maintenance. Van Horenbeek et al.

(2013) summarised and predicted potential future research about the combination of

maintenance and inventory.

Aside from two traditional repair types, i.e. perfect repair (replacement) and minimal repair,

the imperfect repair was a notable aspect of TBM research. Kijima et al. (1988) considered

imperfect maintenance modelled as virtual age approach in their proposed PM policy. Wang

and Pham (1996) also considered the imperfect repairs that were modelled by quasi-renewal

process in their TBM. These authors also summarised the imperfect maintenance in a review

paper (Pham and Wang, 1996). Bartholomew-Biggs et al. (2009) developed a TBM model for

a sequence of imperfect maintenances. Liu et al. (2011) investigated the TBM policy where the

quality of maintenance actions was random. Wang et al. (2011) similarly developed an optimal

model for the sequence of imperfect PM where the quality of maintenance actions is stochastic.

Similar study with stochastic imperfect maintenance was conducted by Khatab and Aghezzaf

(2016).

The research on TBM in the recent decade focused on structures of system and the dependence

among components of systems. Yu et al. (2007) developed a TBM model for a cold-standby

system. Gao et al. (2015) developed a dynamic interval for PM scheduling for a series system.

Zhang et al. (2017) studied the TBM policy for a load-sharing system which consists of two

components. A popular research track of TBM policy for multi-component systems was

selective maintenance. This maintenance policy is to select some components of a certain

system for PM activity under the limitation of resources (time, spare parts, budget, etc.)

(Cassady et al., 2001; Rajagopalan and Cassady, 2006; Richard Cassady et al., 2001). The

series-parallel systems were the most focused structure in selective maintenance (Dao et al.,

2014; Khatab and Aghezzaf, 2016; Khatab et al., 2016; Moghaddam and Usher, 2011; Richard

Cassady et al., 2001; Xu et al., 2016). The maintenance resources could be uncertain and

stochastic (Khatab et al., 2017). Some articles discussed the selective maintenance for multi-

component systems that have some special properties. Dao and Zuo (2016) investigated the

multi-state systems with variable loading condition. Dao et al. (2014) and Xu et al. (2016)


considered the economic dependence among component maintenance in selective maintenance

policy. The economic dependence among components of systems related directly to another

research track called opportunistic maintenance that will be discussed in detail in Section 2.2

of this chapter.

Another approach in TBM research discussed on delay-time models where detectable defects

appear some time duration (delayed time) before actual failures occurrences. The developed

TBM policies for this approach focused on the defect inspection scheduling to reduce the

consequences of failures (Christer, 1999; Murthy and Kobbacy, 2008; Osaki, 2002; Wang,

2012). Christer et al. (1995) developed a delay-time model where arrivals of defects (faults)

followed stochastic processes and delayed time were random. Wang and Syntetos (2011)

presented a forecasting model for maintenance spare parts using for a delay-time maintenance

policy. Berrade et al. (2017) and Cavalcante et al. (2017) considered the delay-time concept in

their postponed maintenance policies where the maintenance is delayed for waiting

maintenance opportunities after defects are recognised in inspection. The maintenance policies

developed by these authors related to the concept of opportunistic maintenance discussed in

Section 2.2 of this chapter. Besides, the concept of delay-time was naturally combined with the

condition-based maintenance (CBM) policies (Wang, 2012) discussed in following section.

2.1.2 Condition based maintenance (CBM)

The rapid development of technology, especially the advancements in information technology,

has enabled wide application of CBM in all industrial areas (Alsyouf, 2007; Kumar, 1996;

Prajapati et al., 2012). CBM policy recommends maintenance actions based on the information

collected through condition monitoring (Jardine et al., 2006). CBM attempts to avoid

unnecessary maintenance tasks by taking maintenance actions only when there is evidence of

abnormal behaviours of a physical asset (Jardine et al., 2006; Prajapati et al., 2012). In

summary, the CBM is the policy that enables the conduct of maintenance activities when

required based on the knowledge about current condition of industrial systems. CBM can be

categorised as a type of PM and consists of three main steps (Jardine et al., 2006): 1) data

acquisition; 2) data processing; and 3) maintenance decision-making. There are two main

processes in CBM that support making maintenance decisions, i.e. diagnostics and prognostics

(Jardine et al., 2006) or current condition evaluation-based (CCEB) and future condition

prediction-based (FCPB) (Ahmad and Kamaruddin, 2012). Diagnostics is a process of finding

the fault after or in the process of the fault occurring in the system, while prognostics is the

process of predicting the future failures by observing and analysing the current and historical


deviation rate of the operation from normal state (Prajapati et al., 2012). According to reviews

by Jardine et al. (2006), the authors state that the prognostics process is superior to diagnostics

because it can prevent the failure or prepare for the problems. However, the prognostics cannot

completely replace the diagnostics because there are some faults and failures that are not

predicted in 100% or even not predictable (Ahmad and Kamaruddin, 2012).

In CBM studies, the degradation of components/systems was a stochastic process. There were

some stochastic processes used to describe the degradation process of systems. The most

popular one was Gamma process (Castanier et al., 2003; Cheng et al., 2017; Do, Voisin, et al.,

2015; Grall, Bérenguer, et al., 2002; Grall, Dieulle, et al., 2002; Hong et al., 2014; Peng and

van Houtum, 2016). In Gamma process, the increment (decrement) of degradation for an

arbitrary time interval ∆𝑡𝑡 follows Gamma distribution and independent. Giorgio et al. extended

the Gamma process where the state after time gap ∆𝑡𝑡 is dependent on current states (Giorgio et

al., 2010) as well as the age of system (Giorgio et al., 2011). The time (age) varying Gamma

processes were an extended approach for degradation modelling which was introduced in some

articles (Fouladirad and Grall, 2012; Guida et al., 2012). The second popular stochastic process

used in CBM studies was Brownian (or Wiener) process which is different to Gamma process

in that the increment (decrement) can be negative (van Noortwijk, 2009; Zhou, Ma, et al.,

2009). Some studies (Liu, Wu, et al., 2017; Zhang et al., 2015) used the Brownian process to

model the degradation path. Multiple Brownian processes were used by Zhang, Huang, et al.

(2016) to describe a degradation process of a mechanical system that changes states according

to shock events (following a Homogeneous Poisson Process – HPP). Another possible

approach for a degradation path was using the Markov chain (Chen et al., 2003; Moustafa et

al., 2004; Zhou, Lin, et al., 2015). Nourelfath et al. (2012) also described the degradation of

each component in a series-parallel system as a Markov chain. Besides, some articles used

other random processes. Li and Pham (2005) developed a CBM model for a two-component

system with competing degraded process and random shocks where stochastic time functions

with random parameters described the degradation processes and Poisson process was used for

random shocks. Chen and Li (2008) extended the CBM model with competing failures and

random shocks with a stochastic process named geometric increasing process. Chen et al.

(2011) and Liu, Liang, et al. (2017) used a linear time function with random parameters for

modelling the degradation path.

As discussed above, the CBM strategies can be implemented only if decision makers have

some knowledge about the condition (degradation) of considered systems. Thus, the


installation of continuously monitoring systems is required and sometimes these systems are

complex and expensive (De Jonge et al., 2015). This requirement is considered as a drawback

of CBM. One approach studied and implemented to overcome the lack of continuously

monitoring systems was developing an inspection schedule. Wang (2003) and Ferreira et al.

(2009) used delay-time concept to determine inspection intervals for CBM policies. Zhang et

al. (2015) considered an inspection and CBM policy with stochastic imperfect repairs where

the inspection time is periodical. A. Grall et al. (Grall, Bérenguer, et al., 2002; Grall, Dieulle,

et al., 2002) determined a dynamic inspection schedule for a CBM maintenance policy where

the next inspection depends on the current condition state of systems. Castanier et al. (2003)

and Do, Voisin, et al. (2015) developed CBM policies with non-periodic inspections where the

imperfect repairs were considered. Wang et al. (2008) considered a combined inspection/CBM

and inventory model where the maintenance and order are decided according to a detected

system condition after inspection. Rasmekomen and Parlikad (2016) used the reliability of a

system as the threshold for inspection decision and the decision of maintenance is decided after

inspection. The inspection sometimes is not perfect. Kallen and van Noortwijk (2005)

considered the case that the inspection gives imperfect information where the error of

inspection was modelled as Normal (Gaussian) distribution. Zhou et al. (2011) developed an

imperfect inspection model based on partially observable Markov Decision Process where the

condition observation was modelled as Gaussian distribution.

Most studies about CBM policies developed optimal decision models to make maintenance

decisions according to the knowledge of current condition (Berenguer et al., 2003; Fouladirad

and Grall, 2012; Grall, Bérenguer, et al., 2002) or predicted condition (Cholette and

Djurdjanovic, 2014; Grall, Dieulle, et al., 2002; Hong et al., 2014; Iung et al., 2008; Ming Tan

and Raghavan, 2008; Mobley, 2002). The maintenance decisions were selected based on the

optimal threshold of monitored system degradation. Moustafa et al. (2004) modelled a CBM

policy with two condition thresholds that divide the state spaces for three possible maintenance

decisions, i.e. doing nothing, minimal repair and replacement. Liu, Wu, et al. (2017) proposed

a CBM strategy where the state (condition) thresholds are varied through time according to the

effects of age-state-dependent operating costs.

Some studies proposed maintenance policies that are the combination of both TBM and CBM.

Chen et al. (2003) developed a PM policy that is dependent on both system state and time.

Saassouh et al. (2007) proposed the policy of two condition thresholds creating the activation

zone where the time for maintenance order (maintenance is conducted after some delayed time)


is decided according to the mode of deteriorating process. Liu, Liang, et al. (2017) proposed a

CBM model where the monitoring condition was the hazard rate which is dependent on both

system age and cumulative degradation. Zhang et al. (2014) proposed a CBM and TBM policy

for heterogeneous populations where the CBM is to identify and replace the defective products

and TBM is applied for the remaining products (where degradation is higher than a threshold)

after inspection.

The studies of CBM policies originally focused on developing maintenance decision strategies

for a single-component system. Some CBM articles mentioned above (Hong et al., 2014;

Nourelfath et al., 2012; Rasmekomen and Parlikad, 2016) considered the CBM for a multi-

component system where there is some dependence among the degradation processes of each

component. The maintenance policies for a multi-component system led to another track of

CBM research named condition-based opportunistic maintenance (CB-OM) which is reviewed

in detail in the following sections.

2.2 OPPORTUNISTIC MAINTENANCE (OM)

The concept and study of OM first appeared a long time ago in the research of Jorgenson and

McCall (1963); McCall (1963); Radner and Jorgenson (1963). In these papers, the authors used

the term “opportunistic replacement” to describe the concepts of OM. Another term of OM

descriptions was “piggyback preventive maintenance” which was introduced by Liang (1985).

This concept mentioned that the PM of a certain component was conducted with an

unscheduled maintenance of another component (Ab-Samat and Kamaruddin, 2014; Liang,

1985). From these early studies, the OM has been a notable topic for many researchers in the

field of Reliability and Maintenance, especially from 2000 until now (Ab-Samat and

Kamaruddin, 2014). Dekker and van Rijn (2003), according to a review in Ab-Samat and

Kamaruddin (2014), gave a general definition of the term “opportunity” in OM approach as

“any event” that can be used for PM of a system which saves the penalty cost of a system

shutdown. Based on this description of “opportunity”, the authors of OM studies considered

and defined the OM in accordance to their own purposes. Some OM studies described the

opportunities as any event that reduces some of the maintenance cost (Berrade et al., 2017;

Cavalcante and Lopes, 2015; Dekker, 1995; Dekker and Smeitink, 1994; Dekker and van Rijn,

2003; Dekker et al., 1997; Dijkstra and Dekker, 1992; Wildeman et al., 1997).


OM is considered as a more efficient maintenance policy than the traditional PM strategy. The

reason is economic dependence as the joint maintenance type of OM may require less cost and

less time (Wang and Pham, 2006). One popular type of the cost saving of OM is the set-up cost

(Wildeman et al., 1997). The savings of set-up cost also include the disassembly/assembly cost

when conducting maintenance activities (Hu and Zhang, 2014; Zhou, Huang, et al., 2015). The

downtime cost is potentially saved by OM. Zhou, Xi, et al. (2009) mentioned downtime cost

savings would be obtained when grouping some items together for maintenance. The system

downtime cost caused by failures and frequent maintenance actions would be reduced by

applying OM (Samhouri and Samhouri, 2009). Koochaki et al. (2012) mentioned that OM not

only saves cost, but also increases the availability of systems. Overall, the OM objective is to

reduce the planned downtime for machines and maximise the lifetime or reliability of

components simultaneously (Ab-Samat and Kamaruddin, 2014).

One of the most common approaches is to consider OM as the PM activity of a certain

component in the time window of the CM of another unit (Wang and Pham, 2006). Pham and

Wang (2000) considered the OM policy of combining the PM of other components with the

CM of failed ones for a k-out-of-n system. Cui and Li (2006) investigated the OM integrating

with CM for a multi-component system with shock models. Another typical type of opportunity

is taking the advantage of a planned preventive maintenance event of another component to do

maintenance for other suitable units. Saranga (2004) considered the OM policy for components

of an aircraft system when a PM is conducted. Zhou, Xi, et al. (2009) developed a dynamic

decision process to select suitable components for a policy of an OM and PM combination of

a series system. Castanier et al. (2005) and Koochaki et al. (2012) investigated the effectiveness

of OM with CBM approach. Lu and Zhou (2017) proposed a grouping of PM policy for serial-

parallel multistage manufacturing systems where the quality and production rates of different

product routes are considered in maintenance policy. Aizpurua et al. (2017) proposed a logical

algorithm for dynamic maintenance planning according to the prognostic information where

the non-critical failures are delayed to be jointly maintained with a critical failure.

OM that combines the PM activities of components with the CM or PM of another component

has been also considered as “grouping maintenance” in many studies (Dekker et al., 1997).

This term came from the nature of OM that other units of a multi-component system are

considered for integrating maintenance with the other required maintenance (CM or PM) of a

certain component for saving some maintenance cost, e.g. set-up cost (Dekker et al., 1997;

Wildeman et al., 1997). Li et al. (2016) developed a grouping strategy for maintenance of a


multi-item system that includes stochastic and economic dependence. Gunn and Diallo (2015)

also proposed the method for opportunistic grouping PM actions for a multi-component system.

Nguyen et al. (2015) proposed a two-level decision-making process that decides whether to or

not to conduct PM and then find an optimal component group for maintenance activity. Hu et

al. (2017) proposed a dynamic PM grouping strategy for a production system during the

opportunistic time between two successive production batches.

Another type of opportunity that has been investigated is production stoppages due to blockage

or starvation of product flow in the system. Gu et al. (2015) defined two types of maintenance

opportunity: active and passive maintenance opportunities. The active type is the time that a

certain machine may stop without affecting the production rate by consuming the products

already in inventory (buffers). The passive opportunity is the time that some machine must be

stopped because of the starvation or blockage caused by the downtime of another machine.

Another study by the same authors Gu et al. (2017) discussed the determination of active

maintenance opportunity window (AMOW) for a series production line with buffers where the

production loss is minimised. Some other studies discussed similar approaches; for example,

Zhou, Yu, et al. (2015) proposed the bottleneck-based OM strategy for a series production

system, while Xia et al. (2015) considered the characteristics of a batch production system to

develop an efficient OM policy. Tao et al. (2014) integrated the production concept of theory

of constraints into OM policy where the lower priority machines are considered for OM when

doing maintenance for the bottleneck one. Cheng et al. (2017) proposed a joint optimisation

model of production lot sizing and grouping maintenance for a production system where the

production lot size and weighted reliability thresholds for components are simultaneously

optimised to minimise the maintenance and production cost rate.

The classification of some literatures corresponding to these types of approaches in OM

research is summarised in Table 2.1.

There are two popular approaches in OM modelling, i.e. time (age) based type and condition-

based type. These two types of OM models are also the popular approaches of maintenance

modelling in general. Further discussion of the two approaches is presented in the following

sections.

Table 2.1

Summary of Types of OM and Modelling Approach


Articles

Opportunity types TB-

OM

CB-

OM PM

activities CM

activities Production

related

Undefined,

external

(L'Ecuyer and Haurie, 1983) x x

(Dijkstra and Dekker, 1992) x x

(Dekker and Smeitink, 1994) x x

(Dekker et al., 1997) x x

(Jhang and Sheu, 1999) x x

(Mohamed-Salah et al., 1999) x x x

(Sherwin, 1999) x x

(Pham and Wang, 2000) x x

(Saranga, 2004) x x x

(Castanier et al., 2005) x x

(Cui and Li, 2006) x x

(Mechefske and Zeng, 2006) x x x

(Chang et al., 2007) x x

(Levrat et al., 2008) x x

(Zequeira et al., 2008) x x

(Bedford and Alkali, 2009) x x x

(Laggoune et al., 2009) x x x

(Samhouri and Samhouri, 2009) x x x x

(Zhou, Xi, et al., 2009) x x

(Laggoune et al., 2010) x x x

(Bedford et al., 2011) x x

(Almgren et al., 2012) x x x

(Ding and Tian, 2012) x x

(Koochaki et al., 2012) x x x x

(Taghipour and Banjevic, 2012) x x

(Do Van et al., 2012) x x


(Tambe et al., 2013) x x x

(Do Van et al., 2013) x x

(Cavalcante and Lopes, 2014) x x

(Hu and Zhang, 2014) x x x

(Tao et al., 2014) x x x

(Gustavsson et al., 2014) x x

(Cavalcante and Lopes, 2015) x x

(Do, Scarf, et al., 2015) x x x

(Gunn and Diallo, 2015) x x x

(Nguyen et al., 2015) x x x x

(Shafiee et al., 2015) x x x

(Zhou, Lin, et al., 2015) x x x

(Zeng and Zhang, 2015) x x

(Zhang and Zeng, 2015) x x

(Zhou, Huang, et al., 2015) x x

(Vu et al., 2015) x x x

(Shafiee and Finkelstein, 2015) x x

(Abdollahzadeh et al., 2016) x x

(Shi and Zeng, 2016) x x x

(Li et al., 2016) x x

(Sarker and Ibn Faiz, 2016) x x

(Vu et al., 2016) x x x

(Zhu et al., 2016) x x x

(Iung et al., 2016) x x x

(Shi and Zeng, 2016) x x x

(Babishin and Taghipour, 2016) x x x

(Chalabi et al., 2016) x x

(Hu and Jiang, 2016) x x

(Zhang, Gao, et al., 2016) x x


(Letot et al., 2016) x x

(Feng et al., 2017) x x

(Kilsby et al., 2017) x x x x

(Peng and Zhu, 2017) x x x x

(Sheikhalishahi et al., 2017) x x

(Zhang and Zeng, 2017) x x x

(Pargar et al., 2017) x x

(Cheng et al., 2017) x x x

(Nguyen et al., 2017) x x x

(Lu and Zhou, 2017) x x

(Berrade et al., 2017) x x

(Aizpurua et al., 2017) x x

2.2.1 Time based Opportunistic Maintenance

The time (age) based studies proposed maintenance models whose failures are determined

according to their “age”. Barlow and Hunter (1960) proposed two basic optimal PM age-based

models corresponding respectively to two assumptions: minimal repairs and perfect repairs

when failures occur. These models were used to determine the optimal age thresholds for

making a decision of conducting PM. In basic approaches of age-based OM models, the age

thresholds are also considered as the criteria of OM decision when opportunities arrive. These

basic age-based OM policies comprise two-age thresholds where the larger one is the limit

control for traditional PM and the smaller one is the signal for OM when a suitable opportunity

arrives (Cavalcante and Lopes, 2014, 2015; Cavalcante et al., 2017; Coolen-Schrijner et al.,

2009; Dijkstra and Dekker, 1992; Jhang and Sheu, 1999; L'Ecuyer and Haurie, 1983; Sherwin,

1999). The general mechanism of two-age threshold policies is described as following. When

an opportunity occurs (usually, a certain component fails, and CM is conducted or PM is

conducted when PM age threshold is reached), other functioning components are selected for

OM if ages are larger than their corresponding OM age thresholds. The age threshold OM

policy was not only limited as two-age policy but also extended to multi-age group policy.

Sarker and Ibn Faiz (2016) proposed an OM policy which includes multiple age groups

associating in varied degrees of maintenance actions when an opportunity arrives.


Besides the basic approach described above, there have been other approaches that did not

consider the age of a component as a direct criterion but used other measurements developed

from ages of components. Mohamed-Salah et al. (1999) compared the difference between the

expected PM time of functioning components and the PM time or instant failure time of a failed

item with a threshold 𝛿𝛿. Ding and Tian (2012) considered an OM policy with two-age

thresholds which were defined as percentages of mean time to failure (MTTF) for the

functioning components of failed and working wind turbines respectively. Pham and Wang

(2000) considered two-control time (𝜏𝜏,𝑇𝑇) policy for a k-out-of-n system where the OM is

conducted if 𝑚𝑚 components fail in interval [𝜏𝜏,𝑇𝑇]. A policy of replacing all operating

components whose failure times fall before a control limit 𝐿𝐿 was proposed by Mechefske and

Zeng (2006). Laggoune et al. (2009, 2010) proposed a grouping OM policy where components

are considered for OM by comparing the expected cost of OM and the expected cost of waiting

for the next scheduled PM. These costs are functions of age of components. Dekker R and

Smeitink E (1995; 1994) proposed the method to set the priorities for selecting components for

OM in a random opportunity with restricted duration. The priorities were determined by

comparing the cost rates between OM and optimal PM of each component. A similar cost

comparison was also suggested by Samhouri and Samhouri (2009) where the remaining

lifetime cost and risk cost of letting the component function to end of life were developed from

the ages and condition of components. Do Van et al. (2013) suggested a procedure to group

components for maintenance when opportunity occurs where the duration of opportunity was

known and considered as a constraint. Abdollahzadeh et al. (2016) and Atashgar and

Abdollahzadeh (2016) developed an OM policy for a windfarm where criteria of OM actions

are defined as reliability thresholds. The reliabilities of components are determined as functions

of age and compared to these corresponding thresholds to decide an appropriate maintenance

action, i.e. perfect or imperfect maintenance actions. Similar age-based reliability thresholds

was used in the OM model for a windfarm (Erguido et al., 2017) where the thresholds are

dynamically changed according to wind speeds (low wind speed is the preferred opportunity

for doing maintenance). The hazard rate thresholds, determined according to the ages of

components, were used to select the group of machines for PM in study of Lu and Zhou (2017).

The age of machines sometimes did not relate directly to criteria for OM decision but was used

to select the “candidates” for OM. Hu and Zhang (2014) proposed a two-stage OM policy

where a certain functioning component is considered as an OM candidate if its age is higher

than an age threshold and is subsequently selected for OM if its determined risk value is higher


than a risk threshold. Gunn and Diallo (2015) investigated an OM policy of grouping machines

for PM where the candidates for grouping are limited to the ones whose lifetimes (ages) locate

in a predefined interval [𝐸𝐸𝑐𝑐,𝐹𝐹𝑐𝑐].

Tambe et al. (2013) approached OM in a different way. They developed a selective

maintenance policy to select appropriate maintenance actions, i.e. imperfect, perfect repairs

and doing nothing for each component of a system within an available duration when an

opportunity occurs. In this model, the effects of failures and maintenance actions were

modelled as age functions. Berrade et al. (2017) proposed a strategy for postponing the PM to

await a suitable opportunity (lower maintenance cost). Cavalcante et al. (2017) proposed a

combined policy of inspection and opportunistic maintenance where a single-unit system can

be preventively replaced when either a defect is detected at a fixed interval inspection, an

opportunity occurred in suitable time or at a planned PM time.

2.2.2 Condition based Opportunistic Maintenance

The CBM is currently expected to be popular in industry application because of the automation

and development in information technologies (Alsyouf (2007) cited to Kumar (1996)).

Therefore, the OM concept can be applied in CBM as well. One of the basic approaches of

applying OM in CBM is setting a threshold of the condition of a machine to make a decision

of OM conduction. Saranga (2004) used a condition threshold policy to select components for

OM. This approach is similar to age-based OM that a threshold of condition is used instead of

an age threshold mentioned in age-based approach. Koochaki et al. (2012) investigated and

compared OM policies of age-based and condition-based approaches for a series system. The

trigger for OM consideration in this condition-based model was defined as a proportion of the

time from the detection of the critical condition to failure time. Zhu et al. (2017) also optimised

the condition threshold for maintenance of a CBM system in unscheduled and scheduled times

caused by the maintenance (CM/PM) of other time-based components. Do, Scarf, et al. (2015)

also proposed a condition OM policy for a two-component system with degradation level

thresholds. At a certain inspection time, if a component fails or is over the preventive threshold,

it will be conducted in CM or PM and the other component will be considered for OM if its

degradation level is higher than the OM threshold. Similar approaches were proposed in Zhang

and Zeng (2015) and Zeng and Zhang (2015). In these studies, the thresholds for OM, PM were

used to make decisions of OM and PM for a certain component. A multi-threshold OM policy


associating with imperfect OM were proposed in Zhou, Lin, et al. (2015) where the decision

of OM relates to the restoration level (imperfect repair) of the components.

Other approaches for condition-based OM policies have been proposed. Nguyen et al. (2017;

2015) developed a model to predict reliabilities of components as system based on their

conditions. PM and OM were decided by comparing these predictive reliabilities to two

thresholds corresponding to PM and OM respectively. Li et al. (2016) suggested a condition-

based grouping OM policy for a stochastic-dependent multi-component system with different

decision criteria. At a moment of inspection, a functioning component is considered for OM

group if it satisfies: 1) the probability, that current deterioration level is greater than preventive

threshold, is higher than a decision control 𝑃𝑃𝑃𝑃; and 2) the probability that the degradation level

is larger than the failure threshold at the next inspection time and is also larger than another

decision control. Bedford et al. (2011) considered an OM policy where the condition

deterioration of a system is observed through signals and these signals are used to inform OM.

2.3 OM OPTIMISATION MODELS

Similar to other maintenance optimisation studies, the objectives of OM optimisation models

are usually related to maintenance cost. The most popular objective used is cost rate defined as

the maintenance cost per time unit. The basic formula determining the cost rate is based on the

renewal theory (Wang and Pham, 2006):

𝐶𝐶𝐶𝐶(𝑇𝑇) =𝐶𝐶(𝑇𝑇)𝐷𝐷(𝑇𝑇)

(2.1)

Where: 𝐶𝐶𝐶𝐶(𝑇𝑇) is cost rate, 𝐶𝐶(𝑇𝑇) is expected cost function, 𝐷𝐷(𝑇𝑇) is expected cycle time and

𝑇𝑇 is decision variable of model.

The renewal theory application is valid if the policy is applied for a long time (long run). In

some situations, the optimisation is just for one cycle. Therefore, the cost rate with one-cycle

criterion should be used (Ansell et al., 1984; Coolen-Schrijner and Coolen, 2007):

𝐶𝐶𝐶𝐶1(𝑇𝑇) = 𝐸𝐸 �𝐶𝐶(𝑋𝑋)𝑋𝑋

� (2.2)

Where: 𝑋𝑋 is the life of system, 𝐶𝐶(𝑋𝑋) is the cost function.

The cost rate has usually been considered as the objective if the OM policies had the property

of “cycle”. It means that in these policies, the applied maintenance activities will restore


systems to a same predefined “state” after a specific interval. Another popular objective of OM

optimisation has been the expected total maintenance cost. This objective was typically used

when the maintenance policies were developed for a fixed-time window or mission (Almgren

et al., 2012; Besnard et al., 2009; Li et al., 2016; Samhouri and Samhouri, 2009; Saranga, 2004;

Taghipour and Banjevic, 2012). Because OM policies can save more cost compared to

traditional PM policies, some authors considered the objective to maximise the cost savings

when applying OM (Mechefske and Zeng, 2006; Tao et al., 2014; Zhou, Xi, et al., 2009). Some

other studies considered multi-objective optimisation models for OM policies. One of the

objectives in multi-objective OM optimisation models is relating to cost. Cavalcante and Lopes

(2014, 2015) combined the availability of systems and total maintenance cost in their

optimisation models. Abdollahzadeh et al. Abdollahzadeh et al. (2016) considered the energy

producing rate and maintenance cost in developing the optimisation model for OM policy of a

windfarm.

One of the interesting issues of OM research is the cost description or cost categories of the

objective functions. The advantage of OM is maintenance cost saving, particularly the set-up

and downtime costs. Many articles describe these savings in their studies. Other studies

(Abdollahzadeh et al., 2016; Gunn and Diallo, 2015; Hu and Zhang, 2014) used a fixed set-up

cost for maintenance conduction. Besnard et al. (2009) considered the transportation cost of

maintenance crews to an offshore wind power system as the set-up cost. When the team is sent

to a wind power system, OM is considered to save this cost. Others (Ding and Tian, 2012;

Sarker and Ibn Faiz, 2016) did not consider only the fixed cost of sending the maintenance

crews to a windfarm, but also the accessing cost when conducting maintenance for any

components of a wind turbine. Laggoune et al. (2009, 2010) and Do, Scarf, et al. (2015)

proposed two different set-up costs relating to CM and PM when conducting either of these

maintenance activities for any item of systems. The approach of different set-up costs

according to each maintenance type was also proposed in Shafiee et al. (2015). Zhou, Xi, et al.

(2009) defined the savings cost as including the downtime cost when grouping some items

together for maintenance and the smaller risk cost of failure if no OM. These authors also

considered the penalty cost of OM when doing maintenance in advance. Shi and Zeng (2016)

also suggested both savings costs of OM as set-up cost and the penalty cost of shifting OM

sooner. Li et al. (2016) developed a cost model that considered both downtime cost and

maintenance set-up cost as saving costs of OM. It was also applied in OM model of Nguyen et

al. (2015). A special set-up cost mentioned by Zhou, Huang, et al. (2015) was the cost for


disassembly process when doing PM or CM. Therefore, this cost depends on the component

which is conducted CM or PM. Nguyen et al. (2017) described the cost categorises of spare

parts inventory (e.g. ordering costs in normal and emergency orders, holding costs, etc.) in their

joint policy of maintenance and inventory.

One type of cost usually mentioned in OM models is the penalty cost of conducting OM.

Saranga (2004) defined the cost of remaining life as penalty cost of OM. A similar remaining

life cost was used in the study of Samhouri and Samhouri (2009). Tao et al. (2014) considered

the penalty cost as a linear function of different time of OM and planned PM. Some studies

defined the OM cost separately to other costs in maintenance policy. Mohamed-Salah et al.

(1999) defined the OM cost as a linear function according to the time of OM and this cost was

higher than PM cost. Pham and Wang (2000) used two separate costs for PM alone and OM

combined with PM and CM. Cavalcante and Lopes (2014, 2015) distinguished the types of

costs of CM, PM and OM separately.

2.4 SOLUTION METHODOLOGY

In OM studies, besides the stochastic nature of failures of components or systems, the

opportunity occurrences are also uncertain. Thus, developing mathematical models for OM

policies were challenging in both modelling and solution. An efficient approach used widely

by many authors was simulation, some coupled with a heuristic optimisation method. Monte

Carlo simulation was the most popular simulation method (Abdollahzadeh et al., 2016; Bedford

and Alkali, 2009; Bedford et al., 2011; Dekker and Smeitink, 1994; Do, Scarf, et al., 2015; Hu

and Zhang, 2014; Laggoune et al., 2009, 2010; Li et al., 2016; Nguyen et al., 2017; Sarker and

Ibn Faiz, 2016; Shafiee et al., 2015; Zhou, Yu, et al., 2015; Zhou, Huang, et al., 2015; Zhu et

al., 2017). Among studies above, some used the simulation – optimisation (Sim – Opt)

approach to obtain a solution. This approach is a mechanism that integrates two parts: the

optimisation module is to generate solution candidates; and the simulation module evaluates

them. Sarker and Ibn Faiz (2016) also applied the mechanism of Sim – Opt to determine the

optimal solution. A similar approach was proposed by Zhou, Huang, et al. (2015) and

Laggoune et al. (2009); (2010) where the generated solution is evaluated through simulation

process. Abdollahzadeh et al. (2016) developed a more complicated solution process where

Particle Swarm Optimisation (PSO) algorithm was used as the core of an optimisation module

in Sim – Opt approach. Some studies used simulation software for evaluating performance of


OM policies. Mohamed-Salah et al. (1999) developed their simulation model by using

Promodel software. Mechefske and Zeng (2006) used the simulation software dedicated to

simulating the wear-out condition of electrode tips. The simulation model proposed by

Koochaki et al. (2012) was developed using the discrete event simulation software tool

Tecnomatix Plant Simulation.

Another approach in solving OM optimisation models used meta-heuristic algorithm. Saranga

(2004) used genetic algorithm (GA) to find the optimal age and condition thresholds of OM

policies. GA was also applied in the study of Samhouri and Samhouri (2009). Tambe et al.

(2013) used three popular heuristic algorithms to solve their optimisation model, i.e. GA,

simulated annealing (SA) and sequence heuristic algorithm, and the results comparison was

conducted to evaluate the efficiency of algorithms. Lung et al. (2007) and Levrat et al. (2008)

proposed the “odds-algorithm” to determine the production stops for OM. Almgren et al.

(2012) developed the solution procedure according to greedy algorithm. Abdollahzadeh et al.

(2016) and Atashgar and Abdollahzadeh (2016) used PSO for the optimisation module in Sim

– Opt approach.

In modelling techniques, Dynamic Programming (DP) was widely used in many OM studies.

L'Ecuyer and Haurie (1983) used DP to determine the optimal strategy for an opportunistic

replacement policy. This approach was also proposed by Do Van et al. (2013); Zhou, Xi, et al.

(2009). Xia et al. (2015) developed a two-level DP model to make a decision on OM according

to the production status where the PM intervals are first determined, and PM actions are

scheduled or postponed according to real-time cost-saving analysis. Some other studies used

renewal theory as their modelling method. Pham and Wang (2000) modelled the OM strategy

for a k-out-of-n system integrating the imperfect repairs by applying renewal theory. These

authors also proposed a quasi-renewal model to determine the optimal OM strategy of doing

maintenance for other subsystems at the time of PM for a main subsystem (Wang and Pham,

2006). The Markov chain was also considered in modelling methods of OM policies by some

studies such as L'Ecuyer and Haurie (1983); Zhou, Lin, et al. (2015).

There are some special methods proposed in OM studies. Derigent et al. (2009) used fuzzy

logic to develop their model. Besnard et al. (2009) developed an Integer Linear Programming

(ILP) model to determine the group of wind turbines maintained when sending the maintenance

crew. Gunn and Diallo (2015) used shortest path algorithm for modelling their OM problem.


2.5 SUMMARY AND GAPS DISCUSSION

OM has been an interesting topic in reliability and maintenance research. The term

“opportunistic” or “opportunity” in OM indicates the main concept of this maintenance type.

That is, the PM for a certain component or system is conducted by chance. The considered

chances for OM can be the maintenance time (PM or CM) of another component/machine, the

production stoppages due to lack of demand, blockage/starvation of upstream or downstream

machines, the redundancy of inventory/buffer, or any other reason other than failure that stops

machines/components. Thus, in general, OM occurrences may be random. The basic benefit of

OM is taking advantage of machine stoppages for doing maintenance. These benefits may

include saving the set-up cost, reducing the downtime cost, or reducing the maintenance stop

frequency by grouping maintenance for many components/machines. However, in some

situations, OM also induces some unwanted costs, such as wasting the remaining life of

components/systems or incurring costs for rescheduling maintenance activities. The

characteristics of OM can be summarised as:

1. A type of preventive maintenance.

2. Attempts to utilise opportunities which occur stochastically.

3. A trade-off between saving maintenance costs such as maintenance set-up cost and

downtime cost, and penalty costs for conducting OM.

Even though many studies have been conducted on OM in the last half century, they have not

mentioned and discussed some issues. One of them relates to mechanism of opportunity

occurrences. The types of OM mentioned mostly in previous studies is the combined

maintenance policy of PM or CM of a certain machine and the OM for other machines. This

kind of OM strategy can be defined as “internal” OM because the opportunities are created by

the system itself. On the other hand, “external” OM considers system stops that are not related

directly to the system under consideration, such as failures of components, but rather are caused

by outside sources. Some examples of external OM are the production stoppages due to low

demand, lack of materials, weather issues, etc.

For external OM, the mechanisms of opportunity occurrences have not been developed in

previous studies. Many papers have assumed that the random opportunity occurs according to

stationary Poisson process (Bedford and Alkali, 2009; Bedford et al., 2011; Cavalcante and

Lopes, 2015; Coolen-Schrijner et al., 2009; Jhang and Sheu, 1999; Zhu et al., 2017). Thus,


another key gap is the development of OM models that use more realistic opportunity

occurrence models.

Another issue of OM that has received little attention is the duration of opportunities. All

maintenance activities require a certain time to conduct and this time window depends on the

types and degrees of maintenance activities. Yet, durations also affect the quality of the

opportunity (even for a single machine); longer opportunities are more desirable since they can

significantly reduce induced downtime costs. However, in most OM studies, the duration of

opportunity was ignored or implicitly assumed to be sufficient for all considered maintenance

actions. This assumption may not be practical. Thus, the duration of opportunities should be

one of the constraints of optimisation models.

Generalising the view of opportunity duration discussed above, the benefit of opportunities,

especially of external opportunities, has not been focused in previous articles. Grouping

maintenance policies (internal OM) as discussed above aims to save the maintenance set-up

costs that are shared among maintenance activities for all components. For external

opportunities, the benefit is various and depends on the definition of opportunities. The benefits

could be downtime cost saving due to the stoppages of production systems or the reduced

maintenance action costs in some special time periods (low-production seasons, promoting a

program of outsourcing maintenance services, etc.). Hence, the external opportunities are not

only limited to discrete events (e.g. production stoppages) which are suitable for conducting

maintenance, but also continuous operative conditions of systems (e.g. production,

maintenance costs, etc.) which can reduce the maintenance costs. Besides, the cost savings of

(external) opportunities sometimes is just a part of some maintenance cost categories. For

example, the duration of a production stoppage is not long enough to do a full maintenance but

still induces some savings in downtime cost. These opportunities can be considered as “partial”

ones that reduce only a part of maintenance cost. Therefore, discussion and development of the

cost savings models for external opportunities, particularly partial opportunities, are notable.

Moreover, the benefit of external opportunities is a challenge in OM modelling. The nature of

opportunity is the uncertainty. Not only is the randomness considered in opportunity

occurrences but the benefit of opportunities is also stochastic, especially the opportunities

coming from “external” sources. For example, in the sugarcane processing industry, the

production depends strictly on the weather during the harvest season. In wet weather, the

harvesting must stop and the starvation of input leads to a production stop as well. Therefore,

the rain or wet weather periods can be considered as the opportunities for maintenance and the


durations of these periods are completely random. Thus, integrating the effects of stochastic

benefits of opportunities to OM decisions is a significant topic that is yet to be studied.

Chapter 3: Joint Opportunistic and Preventive Maintenance policy for an infinite time horizon: Time-based approach 31

Chapter 3: Joint Opportunistic and Preventive Maintenance policy for an infinite time horizon: Time-based approach

Overview

This chapter presents a joint OM and PM policy for a single-unit system using time-based

maintenance. OM is considered when a suitable opportunity occurs. If no opportunity arrives,

the system will be preventively maintained at the end of the cycle. The study develops an

analytical model for opportunistic maintenance considering two critical properties of real-

world opportunities: (i) non-homogeneous opportunity arrivals; and (ii) stochastic opportunity

duration. The model enables exploitation of downtime cost savings from “partial” opportunities

(stoppages shorter than the required maintenance time), thus extending the potential benefit of

OM. This chapter relates to two published papers:

• Ba HT, Cholette ME, Borghesani P, Ma L. A quantitative study on the impact of

opportunistic maintenance in the presence of time-varying costs. 2016 IEEE

International Conference on Industrial Engineering and Engineering Management

(IEEM)2016. p. 1360-4.

• Truong Ba H, Cholette ME, Borghesani P, Zhou Y, Ma L. Opportunistic

maintenance considering non-homogenous opportunity arrivals and stochastic

opportunity durations. Reliability Engineering & System Safety. 2017; 160:151-61.

32 Chapter 3: Joint Opportunistic and Preventive Maintenance policy for an infinite time horizon: Time-based approach

Nomenclature

General variables:

𝑡𝑡 ∈ [0, 𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚]: age of a system.

𝑇𝑇 ∈ [0, 𝑡𝑡𝑃𝑃𝑃𝑃]: length of a renewal cycle.

𝐷𝐷 ∈ [0,∞): duration of opportunity.

Dependent variables:

𝑝𝑝𝑃𝑃𝑃𝑃 : probability of the cycle to stop due to scheduled PM.

𝑝𝑝𝑂𝑂𝑃𝑃 : probability of cycle stop due to OM.

𝑝𝑝𝐶𝐶𝑃𝑃: probability of cycle stop due to CM.

𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃: cost rate in case the cycle stops due to PM.

𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃: cost rate in case the cycle stops due to OM.

𝐶𝐶𝐶𝐶𝐶𝐶𝑃𝑃: cost rate in case the cycle stops due to CM.

𝑁𝑁𝑓𝑓(𝑥𝑥): number of failures during time interval with length 𝑥𝑥.

𝑇𝑇𝐶𝐶: single-cycle total cost.

𝐶𝐶𝐶𝐶: expected cost per unit of time (cost rate).

Decision variables:

𝑡𝑡𝑃𝑃𝑃𝑃 ∈ [0, 𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚]: planned PM maintenance interval.

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡): decision function of minimum opportunity duration that will be accepted.

Parameters:

𝑑𝑑𝑃𝑃𝑃𝑃: required duration of any PM (or OM) activity (fixed).

𝑐𝑐𝐶𝐶𝑃𝑃: cost of one CM instance.

𝑐𝑐𝑏𝑏: direct cost for one PM (or OM) activity.

𝑐𝑐𝑙𝑙: cost rate per unit of downtime of PM (or OM).

Functions:

𝑐𝑐𝑝𝑝(𝑡𝑡): penalty cost when OM is executed at time 𝑡𝑡.

𝑓𝑓(⋅),𝐹𝐹(⋅),𝐹𝐹�(⋅) : system’s PDF, CDF and survival function.


𝑓𝑓𝐷𝐷(⋅), 𝐹𝐹𝐷𝐷(⋅), 𝐹𝐹𝐷𝐷(⋅): PDF, CDF and survival function of opportunity duration.


3.1 INTRODUCTION

Considering the previous literature, it is evident that most existing OM studies utilise simplified

opportunity arrival models (with HPP arrival times), and do not consider stochastic duration of

future opportunities in the maintenance optimisation. Such characterisation is often simplistic

and suboptimal: weather-induced opportunities have seasonal arrival intensities (non-

homogeneous) and highly variable duration; failures of other equipment in the plant may

provide opportunities of different duration depending on the nature and location of the fault;

highly non-stationary market dynamics could trigger low demand-induced production stops

and consequent opportunities.

This study aims to jointly optimise opportunistic and traditional PM considering two critical

properties of real-world opportunities: (i) non-homogeneous opportunity arrivals; and (ii)

stochastic opportunity duration. The optimisation model will determine which opportunities

should be accepted under different maintenance assumptions. This results in a more faithful

and realistic modelling of opportunities and enables the exploitation of “partial” opportunities,

i.e. those having duration shorter than the required maintenance time, but still potentially

constituting a significant downtime cost saving.

The remainder of the chapter is organised as follows: Section 3.2 describes the problem

approach. Section 3.3 derives a closed form expression of the single-cycle total cost and

develops the joint optimisation models for two scenarios: minimal and perfect CM. The

numerical examples and analysis are presented in Section 3.4. The last section summarises the

main conclusions of this study.

3.2 MODEL DESCRIPTION

This study develops a preventive maintenance (PM) policy with opportunistic maintenance

(OM) consideration for a production system. For the purposes of this study, an opportunity is

a forced system stoppage not induced by preventive or corrective maintenance, e.g. a

production line stoppage due to lack of demand or adverse weather conditions. These

unavoidable stoppages do not increase maintenance total cost because they have no effect to

the availability of systems. However, if maintenance activities are conducted in these long

enough opportunities, the maintenance cost is lower due to reducing of downtime cost. In

addition to the traditional preventive and corrective maintenance (PM and CM respectively),

this study considers opportunistic maintenance (OM). OM is conducted when the system is still


functional and an accepted suitable opportunity occurs. On the other hand, CM occurs if a

system fails prior to OM or PM. PM is conducted if the system is still functional at the

predetermined PM interval and no opportunity has been accepted.

The following situation is considered. A machine has been renewed at the beginning of the

cycle and the objective is to optimally select the preventive maintenance interval 𝑡𝑡𝑃𝑃𝑃𝑃 and the

acceptance criteria for an arrived opportunity. This acceptance is modelled using a duration

threshold: an arrived opportunity at time 𝑡𝑡 is accepted if its duration will be greater than a time-

variant threshold 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡).

Figure 3.1. Two examples of cycles for scenario 1: Minimal corrective maintenance. In Example a, OM terminates the cycle when an opportunity of duration greater than 𝒅𝒅𝒎𝒎𝒎𝒎𝒎𝒎(𝒕𝒕) occurs. In Example b, PM terminates

the cycle since no opportunity occurs with duration greater than 𝒅𝒅𝒎𝒎𝒎𝒎𝒎𝒎(𝒕𝒕).

Figure 3.2. Three examples of cycles for scenario 2: Perfect corrective maintenance. In Example a, CM terminates the cycle (i.e. renews the machine), while in Example b OM terminates the cycle since an

opportunity of sufficient duration occurs prior to PM and failure. Finally, Example c illustrates the scenario where PM terminates the cycle, since neither a failure nor opportunity of sufficient duration occur prior to PM.

Two scenarios are investigated in this study:


1. The system can only be renewed by PM or OM, while CM is assumed to be minimal.

This scenario is illustrated in Figure 3.1.

2. The system can be renewed by any of the three possible maintenance types (CM,

OM, PM). This scenario is illustrated in Figure 3.2.

The following additional assumptions are made:

1. The opportunity occurrences follow a NHPP and are independent on the condition

of the system.

2. The duration of maintenance activities and opportunities are small compared to the

cycle length.

3. The duration of opportunity is a random variable with known distribution.

4. OM and PM are assumed perfect maintenance actions.

Without loss of generality, the optimisation is developed for the case where the decision maker

has full knowledge of the opportunity duration, 𝐷𝐷, immediately before its occurrence. Thus,

the decision depends on the minimum acceptable duration, 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡). Thus, 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) = 0 means

accepting any arrived opportunity and 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) = ∞ means rejecting all arrived opportunities.

For the case where the DM has no knowledge of 𝐷𝐷, 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) ∈ {0,∞} (i.e. the opportunity is

accepted or rejected based on 𝑡𝑡 alone). The objective of optimisation will be to minimise the

expected cost rate by manipulating the decision variables 𝑡𝑡𝑃𝑃𝑃𝑃 and 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡).

3.3 JOINT OPTIMISATION MODELS

The probability of failure for the system is considered as a function of system age. Hence, the

PDF, CDF and survival function of system failure are determined based on the age 𝑡𝑡.

In this study, the renewal cost rate cannot be used as the objective function because the

assumption of applying an identical policy in the long-term does not apply when the

opportunity arrivals are non-homogenous. Therefore, the objective is to find the optimal

decision variables 𝑡𝑡𝑃𝑃𝑃𝑃 and 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) minimising the one-cycle cost rate:

𝐶𝐶𝐶𝐶�𝑡𝑡𝑃𝑃𝑃𝑃,𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡)� = 𝐸𝐸 �𝑇𝑇𝐶𝐶𝑇𝑇� (3.1)

Where 𝑇𝑇𝐶𝐶 and 𝑇𝑇 are the total cost and cycle length respectively.


The system is renewed after some maintenance activity (i.e. PM, OM or CM in scenario 2),

resulting in a cycle pattern whose length and cost depends on the type of maintenance that

terminates the cycle. Thus, the expected cost rate may be computed as:

𝐸𝐸 �𝑇𝑇𝐶𝐶𝑇𝑇� = � 𝐸𝐸 �

𝐶𝐶𝑙𝑙𝑇𝑇𝑙𝑙� type ℓ� ⋅ 𝑃𝑃[type ℓ]

ℓ∈{𝑃𝑃𝑃𝑃,𝑂𝑂𝑃𝑃,𝐶𝐶𝑃𝑃}

(3.2)

For the two scenarios under consideration (CM is minimal (scenario 1), CM is perfect (scenario

2)), the expected cost rates are respectively determined as:

𝐸𝐸 �𝑇𝑇𝐶𝐶𝑇𝑇� = 𝐸𝐸 �

𝐶𝐶𝑂𝑂𝑃𝑃(𝑇𝑇𝑂𝑂𝑃𝑃,𝐷𝐷) + 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅ 𝐸𝐸�𝑁𝑁𝑓𝑓(𝑇𝑇𝑂𝑂𝑃𝑃)�𝑇𝑇𝑂𝑂𝑃𝑃

� 𝑝𝑝𝑂𝑂𝑃𝑃

+ �𝑐𝑐𝑃𝑃𝑃𝑃 + 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅ 𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡𝑃𝑃𝑃𝑃 )�

𝑡𝑡𝑃𝑃𝑃𝑃� 𝑝𝑝𝑃𝑃𝑃𝑃

= 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] + 𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃]

(3.3)

𝐸𝐸 �𝑇𝑇𝐶𝐶𝑇𝑇� = 𝐸𝐸 �

𝐶𝐶𝑂𝑂𝑃𝑃(𝑇𝑇𝑂𝑂𝑃𝑃,𝐷𝐷)𝑇𝑇𝑂𝑂𝑃𝑃

�𝑝𝑝𝑂𝑂𝑃𝑃 + �𝑐𝑐𝑃𝑃𝑃𝑃𝑡𝑡𝑃𝑃𝑃𝑃

� 𝑝𝑝𝑃𝑃𝑃𝑃 + 𝐸𝐸 �𝑐𝑐𝐶𝐶𝑃𝑃𝑇𝑇𝐶𝐶𝑃𝑃

� 𝑝𝑝𝐶𝐶𝑃𝑃

= 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] + 𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃] + 𝐸𝐸[𝐶𝐶𝐶𝐶𝐶𝐶𝑃𝑃] (3.4)

where 𝑐𝑐𝐶𝐶𝑃𝑃, 𝑐𝑐𝑃𝑃𝑃𝑃 and 𝐶𝐶𝑂𝑂𝑃𝑃(𝑡𝑡,𝐷𝐷) are the corresponding costs of CM, PM and OM activities. 𝑝𝑝𝐶𝐶𝑃𝑃,

𝑝𝑝𝑃𝑃𝑃𝑃 and 𝑝𝑝𝑂𝑂𝑃𝑃 are the probabilities that the cycle is terminated by CM, PM or OM respectively.

𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡)� denotes the expected number of failures during the cycle in case the CM is assumed

to be minimal (scenario 1).

The CM cost 𝑐𝑐𝐶𝐶𝑃𝑃 is assumed to be a constant value for each failure of machine, while the PM-

related cost is comprised of two parts: the direct cost 𝑐𝑐𝑏𝑏 (e.g. spare parts and labour) and the

production losses due to the PM duration (𝑑𝑑𝑃𝑃𝑃𝑃), which has production loss per unit time 𝑐𝑐𝑙𝑙.

The PM cost 𝑐𝑐𝑃𝑃𝑃𝑃 is then computed as:

𝑐𝑐𝑃𝑃𝑃𝑃 = 𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ 𝑑𝑑𝑃𝑃𝑃𝑃 (3.5)

The OM cost is divided into three parts: 1) the direct cost 𝑐𝑐𝑏𝑏 , which is the same as the direct

cost of PM activity; 2) the production loss 𝑐𝑐𝑙𝑙 ⋅ [𝐷𝐷𝑃𝑃𝑃𝑃 − 𝐷𝐷]+, which in the OM case depends on

the length of opportunity duration, which is a random variable; and 3) A time-variant penalty


cost 𝑐𝑐𝑝𝑝(𝑡𝑡) due to shifting the preventive maintenance to earlier time1. Thus, the OM cost is

computed as:

𝐶𝐶𝑂𝑂𝑃𝑃(𝑡𝑡,𝐷𝐷) = 𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑝𝑝(𝑡𝑡) + 𝑐𝑐𝑙𝑙 ⋅ [𝑑𝑑𝑃𝑃𝑃𝑃 − 𝐷𝐷]+ (3.6)

The penalty cost 𝑐𝑐𝑝𝑝(𝑡𝑡) is the time-variant part of OM cost, which results from shifting

preventive maintenance preparation from the planned time 𝑡𝑡𝑃𝑃𝑃𝑃 to the time of opportunity

occurrence. Therefore, 𝑐𝑐𝑝𝑝(𝑡𝑡) is intuitively higher when 𝑡𝑡 is far to 𝑡𝑡𝑃𝑃𝑃𝑃 and it is close to zero (0)

when 𝑡𝑡 is right before 𝑡𝑡𝑃𝑃𝑃𝑃. In this study, the penalty cost 𝑐𝑐𝑝𝑝(𝑡𝑡) is assumed following the

sigmoid function:

𝑐𝑐𝑝𝑝(𝑡𝑡) = 𝑐𝑐𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚[1 − 𝑆𝑆(𝑡𝑡; [𝑎𝑎 𝑏𝑏])] = 𝑐𝑐𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 �𝑒𝑒−𝑚𝑚(𝑡𝑡−𝑏𝑏)

1 + 𝑒𝑒−𝑚𝑚(𝑡𝑡−𝑏𝑏)� (3.7)

In which, 𝑆𝑆(𝑡𝑡; [𝑎𝑎 𝑏𝑏]) = 11+𝑒𝑒−𝑎𝑎(𝑡𝑡−𝑏𝑏) is a sigmoid function with two parameters 𝑎𝑎 and 𝑏𝑏 and 𝑐𝑐𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚

is the maximum penalty cost. With different values (𝑎𝑎, 𝑏𝑏), the slopes and changing times of

penalty cost are varied. Hence, this function is suitable to illustrate many situation of penalty

cost such as the cost is very high at the early age but significantly reduced when the age is close

to PM time. Figure 3.3 shows some examples of this function with different 𝑎𝑎 and 𝑏𝑏 = 𝑡𝑡𝑃𝑃𝑃𝑃2

.

Figure 3.3. Illustrative plot of penalty cost function.

Considering Eqs. (3.3) and (3.4), to determine expected cost rate given 𝑡𝑡𝑃𝑃𝑃𝑃 and 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡), one

must determine the probabilities 𝑝𝑝𝐶𝐶𝑃𝑃, 𝑝𝑝𝑃𝑃𝑃𝑃 and 𝑝𝑝𝑂𝑂𝑃𝑃 of cycle termination due to CM, PM and

1 Note that: [𝑥𝑥]+ ≜ �𝑥𝑥, 𝑥𝑥 ≥ 00, 𝑥𝑥 < 0


OM respectively. The next three sections discuss the computation of 𝑝𝑝𝑂𝑂𝑃𝑃, 𝑝𝑝𝑃𝑃𝑃𝑃, 𝑝𝑝𝐶𝐶𝑃𝑃 and the

cost rates for the two scenarios.

3.3.1 Determining 𝒑𝒑𝑶𝑶𝑶𝑶

The probability that the cycle ends in OM may be calculated as:

𝑝𝑝𝑂𝑂𝑃𝑃 = �𝑝𝑝𝑂𝑂𝑃𝑃𝑂𝑂𝑘𝑘

∞

𝑘𝑘=1

(3.8)

where 𝑝𝑝𝑂𝑂𝑃𝑃𝑂𝑂𝑘𝑘 (𝑘𝑘 = 1,2, … ) is the probability that the OM is conducted at the 𝑘𝑘𝑡𝑡ℎ arrived

opportunity in time interval [0, 𝑡𝑡𝑃𝑃𝑃𝑃]:

𝑝𝑝𝑂𝑂𝑃𝑃𝑂𝑂𝑘𝑘 = � �� …𝑑𝑑𝑡𝑡𝑘𝑘−2

𝑡𝑡𝑘𝑘−1

0

� 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑘𝑘−1)�𝑓𝑓𝑂𝑂𝑘𝑘−1,0(𝑡𝑡𝑘𝑘−1)𝑑𝑑𝑡𝑡𝑘𝑘−1

𝑡𝑡𝑘𝑘

0

�

𝑡𝑡𝑃𝑃𝑃𝑃

0

⋅ 𝐹𝐹𝐷𝐷��𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑘𝑘)�𝑓𝑓𝑂𝑂𝑘𝑘,0(𝑡𝑡𝑘𝑘)𝑑𝑑𝑡𝑡𝑘𝑘

(3.9)

Eq. (3.9) is a joint event consisting of the 𝑘𝑘𝑡𝑡ℎ opportunity arrival (described by p.d.f 𝑓𝑓𝑂𝑂𝑘𝑘,0(𝑡𝑡𝑘𝑘)),

the rejection of the previous 𝑘𝑘 − 1 opportunities, and the suitability of the 𝑘𝑘𝑡𝑡ℎ opportunity. The

suitability of the 𝑘𝑘𝑡𝑡ℎ opportunity is evaluated according to the survival function at the decision

threshold 𝐹𝐹𝐷𝐷��𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑘𝑘)�, i.e. the probability of opportunity duration exceeding the threshold

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑘𝑘).

In order to describe the distribution 𝑓𝑓𝑂𝑂𝑘𝑘,0(𝑡𝑡𝑘𝑘) of the opportunity arrivals, a hypothesis of NHPP

arrivals with intensity 𝜆𝜆𝑂𝑂(𝑡𝑡) is made. Under this assumption, the 𝑘𝑘𝑡𝑡ℎ opportunity occurrence

at time 𝑇𝑇𝑂𝑂𝑘𝑘 in any arbitrary interval �𝑡𝑡𝑙𝑙, 𝑡𝑡𝑓𝑓� (𝑡𝑡𝑓𝑓 ≥ 𝑡𝑡𝑙𝑙) follows a Gamma distribution, where the

c.d.f 𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑠𝑠(𝑡𝑡) and PDF 𝑓𝑓𝑂𝑂𝑘𝑘,𝑡𝑡𝑠𝑠(𝑡𝑡) are given by Cinlar (2013); Ross (2014):

and 𝑓𝑓𝑂𝑂𝑘𝑘,𝑡𝑡𝑠𝑠(𝑡𝑡) = [Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡)]𝑘𝑘−1

(𝑘𝑘 − 1)!𝜆𝜆𝑂𝑂(𝑡𝑡)𝑒𝑒−Λ(𝑡𝑡𝑠𝑠,𝑡𝑡) (3.11)

where 𝛾𝛾(𝑘𝑘,𝑢𝑢) = ∫ 𝑥𝑥𝑘𝑘−1𝑒𝑒−𝑚𝑚𝑑𝑑𝑥𝑥𝑢𝑢0 is the lower incomplete gamma function and Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡) is the

mean value function of NHPP considered from the start time 𝑡𝑡𝑙𝑙:

𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑠𝑠(𝑡𝑡) = 𝑃𝑃�𝑡𝑡𝑙𝑙 < 𝑇𝑇𝑂𝑂𝑘𝑘 ≤ 𝑡𝑡� =𝛾𝛾�𝑘𝑘,Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡)�

(𝑘𝑘 − 1)! (3.10)


Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡) = �𝜆𝜆𝑂𝑂(𝜏𝜏)𝑑𝑑𝜏𝜏𝑡𝑡

𝑡𝑡𝑠𝑠

𝑡𝑡 ∈ �𝑡𝑡𝑙𝑙, 𝑡𝑡𝑓𝑓� (3.12)

While conceptually straightforward, the integrals of Eq. (3.9) are intractable due to the presence

of the OM decision function 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) in the infinitely nested integrals. To address this, the

timeline will be discretised into 𝑁𝑁 time intervals of length ∆𝑡𝑡, (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚] 𝑖𝑖 = 1,2, … ,𝑁𝑁, where

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) will be considered constant for each interval. Thus, the OM decision function 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡)

is discretised in time to 𝑁𝑁 decision variables, i.e. 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡1),𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡2), … ,𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑁𝑁).

Similarly, 𝑡𝑡𝑃𝑃𝑃𝑃 = 𝑖𝑖𝑃𝑃𝑃𝑃∆𝑡𝑡, where 𝑖𝑖𝑃𝑃𝑃𝑃 ∈ {1,2, … ,𝑁𝑁} is the interval index where PM occurs.

Using this discretisation, the probability of OM may be calculated as:

𝑝𝑝𝑂𝑂𝑃𝑃 = �𝑝𝑝𝑂𝑂𝑃𝑃𝑖𝑖

𝑚𝑚𝑃𝑃𝑃𝑃

𝑚𝑚=1

(3.13)

where 𝑝𝑝𝑂𝑂𝑃𝑃𝑖𝑖 probability of conducting OM in the interval (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚] and 𝑖𝑖𝑃𝑃𝑃𝑃 ∈ {1,2, … ,𝑁𝑁} is the

time index where PM occurs. The probability 𝑝𝑝𝑂𝑂𝑃𝑃𝑖𝑖 may be computed as:

𝑝𝑝𝑂𝑂𝑃𝑃𝑖𝑖 = 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖��1 − 𝑝𝑝𝑆𝑆𝑂𝑂𝑗𝑗�𝑚𝑚−1

𝑗𝑗=0

(3.14)

In Eq. (3.14) above, 𝑝𝑝𝑆𝑆𝑂𝑂𝑖𝑖 is the probability of the arrival of at least one suitable opportunity in

interval 𝑖𝑖:

𝑝𝑝𝑆𝑆𝑂𝑂𝑖𝑖 = �𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)��𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑚𝑚)�

𝑘𝑘−1⋅ 𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡𝑚𝑚)

∞

𝑘𝑘=1

𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� < 1

0 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� = 1

(3.15)

which has the obvious boundary condition 𝑝𝑝𝑆𝑆𝑂𝑂0 = 0. Note that 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� ≤ 1 and

𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡𝑚𝑚) ≤ 1 ∀𝑘𝑘, so this series converges and thus the infinite sum of Eq. (3.15) exists and

is finite2.

The probability 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 in Eq. (3.14) is the probability that an accepted opportunity occurs in

interval (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚]. The expression for 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 depends on the scenario. In case that CM is minimal

2 Practically, the sum is computed by truncating all the terms above some finite 𝑘𝑘 where the product

𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚 (𝑡𝑡𝑚𝑚)�𝑘𝑘−1 ⋅ 𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡𝑚𝑚) is sufficiently small.


(scenario 1), it is easy to see that 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 = 𝑝𝑝𝑆𝑆𝑂𝑂𝑖𝑖 since OM is the only possible cycle termination

event within the interval3. For the scenario 2 where CM is perfect, 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 is the joint probability

that there is a suitable opportunity for OM and the system is still functional at the arrival time

of this opportunity. Then, the probability 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 is computed according to two scenarios:

Scenario 1: 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 = 𝑝𝑝𝑆𝑆𝑂𝑂𝑖𝑖 (3.16)

Scenario 2: 𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖 = �𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)�(𝑘𝑘−1)

⋅ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� � 𝐹𝐹�(𝑡𝑡) ⋅ 𝑓𝑓𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡)𝑑𝑑𝑡𝑡

𝑡𝑡𝑖𝑖

𝑡𝑡𝑖𝑖−1

∞

𝑘𝑘=1

(3.17)

Thus, 𝑝𝑝𝑂𝑂𝑃𝑃 can be computed for both scenarios using Eq. (3.2). It is worth highlighting that the

time discretisation enables defining a time-varying distribution for the opportunity duration

(i.e. 𝐹𝐹𝐷𝐷,𝑡𝑡𝑖𝑖(⋅)) without changing the form of the equations presented in this section. In the rest

of the chapter, a time-invariant 𝐹𝐹𝐷𝐷(⋅) will be considered for the sake of clarity, without any

loss of generality.

3.3.2 Model 1: Optimisation when CM is minimal

For Model 1, the system is renewed by one of two possible causes: OM or PM with

probabilities 𝑝𝑝𝑂𝑂𝑃𝑃 and 𝑝𝑝𝑃𝑃𝑃𝑃 respectively. The probability of OM 𝑝𝑝𝑂𝑂𝑃𝑃 is determined according

to Eq. (3.13), and the probability of PM may be calculated as

𝑝𝑝𝑃𝑃𝑃𝑃 = 1 − 𝑝𝑝𝑂𝑂𝑃𝑃 (3.18)

The expected cost rate is defined as Eq. (3.3) and the term 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅ 𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡)� is the expected cost

of CM (minimal) repairs during the cycle, where 𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡)� is the expected number of failures

up to time 𝑡𝑡. The one-cycle cost rate optimisation problem can be stated as:

min

𝑡𝑡𝑖𝑖𝑃𝑃𝑃𝑃 ,𝑑𝑑𝑚𝑚𝑖𝑖𝑚𝑚(𝑡𝑡𝑖𝑖)𝐶𝐶𝐶𝐶 �𝑡𝑡𝑚𝑚𝑃𝑃𝑃𝑃 ,𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� = 𝐸𝐸 �

𝑇𝑇𝐶𝐶𝑇𝑇� = 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] + 𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃]

Subject to:

𝑡𝑡𝑚𝑚𝑃𝑃𝑃𝑃 ∈ {𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑁𝑁}

𝑡𝑡0 = 0

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) ∈ [0,∞)

(3.19)

3 PM, if it were to occur, occurs at the end of the interval.


𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) = ∞ ∀𝑖𝑖 > 𝑖𝑖𝑃𝑃𝑃𝑃

Where the terms 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] and 𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃] are computed as follows. The PM cost is defined as

in Eq. (3.5) so that the expected PM cost rate is determined as:

𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃] = �𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ 𝑑𝑑𝑃𝑃𝑃𝑃 + 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅ 𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡𝑃𝑃𝑃𝑃)�

𝑡𝑡𝑃𝑃𝑃𝑃� ⋅ 𝑝𝑝𝑃𝑃𝑃𝑃 (3.20)

Next, the expected cost rate of OM will be determined. Consider any interval (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚]

corresponding with fixed decision variable 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚). The expected cost rate given that OM is

conducted at this interval, denoted 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃|𝑂𝑂𝑀𝑀𝑚𝑚], is determined as:

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃|𝑂𝑂𝑀𝑀𝑚𝑚]

= �0 𝐹𝐹�𝐷𝐷(𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)) = 0

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑚𝑚 |𝑂𝑂𝑀𝑀𝑚𝑚] + 𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑝𝑝 |𝑂𝑂𝑀𝑀𝑚𝑚� + 𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃

𝑓𝑓 |𝑂𝑂𝑀𝑀𝑚𝑚� otherwise

(3.21)

where 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑚𝑚 |𝑂𝑂𝑀𝑀𝑚𝑚] is the expected OM cost rate relating to the maintenance action when

OM is conducted, 𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑝𝑝 |𝑂𝑂𝑀𝑀𝑚𝑚� relates to the penalty cost, which depends on the time of

arrival of the accepted opportunity and 𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑓𝑓 |𝑂𝑂𝑀𝑀𝑚𝑚� relates to the minimal repairs until the

time OM occurrence. . These terms have the common formula:

𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑐𝑐𝑙𝑙𝑚𝑚𝑝𝑝|𝑂𝑂𝑀𝑀𝑚𝑚�

=∑ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)�

(𝑘𝑘−1)⋅ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� ⋅ ∫ 𝐶𝐶𝑐𝑐𝑙𝑙𝑚𝑚𝑝𝑝(𝑡𝑡)

𝑓𝑓𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡)𝑡𝑡 𝑑𝑑𝑡𝑡𝑡𝑡𝑖𝑖

𝑡𝑡𝑖𝑖−1∞𝑘𝑘=1

𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� ⋅ ∑ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)�(𝑘𝑘−1)

⋅ 𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡𝑚𝑚)∞𝑘𝑘=1

(3.22)

where: 𝑐𝑐𝑐𝑐𝑚𝑚𝑝𝑝 ∈ {𝑚𝑚,𝑝𝑝, 𝑓𝑓} is the index indicating type of considered costs, i.e. maintenance,

penalty or failure cost; and 𝐶𝐶𝑐𝑐𝑙𝑙𝑚𝑚𝑝𝑝(𝑡𝑡) is the function of relating cost.

In Eq. (3.22), the numerator is component cost rate at an accepted opportunity during time

interval (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚]; and the denominator is the probability that there is at least one suitable

opportunity during this time interval, 𝑝𝑝𝑆𝑆𝑂𝑂𝑖𝑖. Note that the term 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� > 0 since the case

where 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� = 0 is handled separately in Eq. (3.21). The function of 𝐶𝐶𝑐𝑐𝑙𝑙𝑚𝑚𝑝𝑝(𝑡𝑡)

according to maintenance (𝑚𝑚), penalty (𝑝𝑝), and failure (𝑓𝑓) are defined:

𝐶𝐶𝑚𝑚(𝑡𝑡) = �𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ � [𝑑𝑑𝑃𝑃𝑃𝑃 − 𝐷𝐷]𝑓𝑓𝐷𝐷(𝐷𝐷)𝑑𝑑𝐷𝐷

𝑑𝑑𝑃𝑃𝑃𝑃

min(𝑑𝑑𝑚𝑚𝑖𝑖𝑚𝑚(𝑡𝑡𝑖𝑖),𝑑𝑑𝑃𝑃𝑃𝑃)

� (3.23)

𝐶𝐶𝑝𝑝(𝑡𝑡) = 𝑐𝑐𝑝𝑝(𝑡𝑡) (3.24)


𝐶𝐶𝑓𝑓(𝑡𝑡) = 𝑐𝑐𝐶𝐶𝑃𝑃𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡)� (3.25)

The expected cost rate for OM is computed as:

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] = �𝑝𝑝𝑂𝑂𝑃𝑃𝑖𝑖 ⋅ 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃|𝑂𝑂𝑀𝑀𝑚𝑚]𝑚𝑚𝑃𝑃𝑃𝑃

𝑚𝑚=1

(3.26)

In the case where OM is not considered, 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) = ∞ ∀𝑡𝑡, then optimisation model (3.19)

simplifies to the classical PM optimisation proposed by Barlow and Hunter (1960):

min𝑡𝑡𝑃𝑃𝑃𝑃

𝐶𝐶𝐶𝐶(𝑡𝑡𝑃𝑃𝑃𝑃) = 𝐸𝐸 �𝑇𝑇𝐶𝐶𝑇𝑇� =

𝐸𝐸[𝑇𝑇𝐶𝐶]𝑡𝑡𝑃𝑃𝑃𝑃

=𝑐𝑐𝐶𝐶𝑃𝑃 ⋅ 𝐸𝐸�𝑁𝑁𝑓𝑓(𝑡𝑡𝑃𝑃𝑃𝑃)� + 𝑐𝑐𝑃𝑃𝑃𝑃


Subject to:

𝑡𝑡𝑃𝑃𝑃𝑃 ≥ 0

(3.27)

3.3.3 Model 2: Optimisation when CM is perfect

In Model 2, the system is renewed in one of three possible ways: CM, OM or PM with

probabilities 𝑝𝑝𝑃𝑃𝑃𝑃, 𝑝𝑝𝑂𝑂𝑃𝑃 and 𝑝𝑝𝐶𝐶𝑃𝑃 respectively. The probability of OM 𝑝𝑝𝑂𝑂𝑃𝑃 is determined

according to the process presented. In Section 3.3.1. PM occurs when neither CM nor OM

occur. That is, PM is the joint event that the system survives up to 𝑡𝑡𝑃𝑃𝑃𝑃 and no suitable

opportunity has occurred:

𝑝𝑝𝑃𝑃𝑃𝑃 = 𝐹𝐹��𝑡𝑡𝑚𝑚𝑃𝑃𝑃𝑃� ⋅�𝑝𝑝𝑆𝑆𝑂𝑂𝑖𝑖


𝑚𝑚=1

(3.28)

The probability of CM is:

𝑝𝑝𝐶𝐶𝑃𝑃 = 1 − 𝑝𝑝𝑂𝑂𝑃𝑃 − 𝑝𝑝𝑃𝑃𝑃𝑃 (3.29)

The expected cost rate may then be computed using Eq. (3.4). Then the optimisation problem

is described as following:

min

𝑡𝑡𝑖𝑖𝑃𝑃𝑃𝑃 ,𝑑𝑑𝑚𝑚𝑖𝑖𝑚𝑚(𝑡𝑡𝑖𝑖) 𝐶𝐶𝐶𝐶 �𝑡𝑡𝑚𝑚𝑃𝑃𝑃𝑃 ,𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� = 𝐸𝐸 �

𝑇𝑇𝐶𝐶𝑇𝑇�

= 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] + 𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃] + 𝐸𝐸[𝐶𝐶𝐶𝐶𝐶𝐶𝑃𝑃]

Subject to:

𝑡𝑡𝑚𝑚𝑃𝑃𝑃𝑃 ∈ {𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑁𝑁}

(3.30)


𝑡𝑡0 = 0

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) ∈ [0,∞)

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) = ∞ ∀𝑖𝑖 > 𝑖𝑖𝑃𝑃𝑃𝑃

where the terms 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃], 𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃] and 𝐸𝐸[𝐶𝐶𝐶𝐶𝐶𝐶𝑃𝑃] are computed according to the equations

presented as follows. The expected cost rate relating to PM activity is determined:

𝐸𝐸[𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃] = �𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ 𝑑𝑑𝑃𝑃𝑃𝑃

𝑡𝑡𝑃𝑃𝑃𝑃� ⋅ 𝑝𝑝𝑃𝑃𝑃𝑃 (3.31)

The expected OM cost rate 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃] may still be determined using Eq. (3.26), but the term

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃|𝑂𝑂𝑀𝑀𝑚𝑚] is:

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃|𝑂𝑂𝑀𝑀𝑚𝑚] = �0 𝐹𝐹�𝐷𝐷(𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)) = 0

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑚𝑚 |𝑂𝑂𝑀𝑀𝑚𝑚] + 𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑝𝑝 |𝑂𝑂𝑀𝑀𝑚𝑚� 𝑐𝑐𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒𝑒𝑒

(3.32)

where: 𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑚𝑚 |𝑂𝑂𝑀𝑀𝑚𝑚] and 𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑝𝑝 |𝑂𝑂𝑀𝑀𝑚𝑚� are respectively the expected OM cost rate relating

to the maintenance action and the penalty cost given that the OM is conducted at interval

(𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚]. These terms may be computed as:

𝐸𝐸[𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑚𝑚 |𝑂𝑂𝑀𝑀𝑚𝑚] = �𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ � [𝑑𝑑𝑃𝑃𝑃𝑃 − 𝐷𝐷]𝑓𝑓𝐷𝐷(𝐷𝐷)𝑑𝑑𝐷𝐷

𝑑𝑑𝑃𝑃𝑃𝑃

min(𝑑𝑑𝑚𝑚𝑖𝑖𝑚𝑚(𝑡𝑡𝑖𝑖),𝑑𝑑𝑃𝑃𝑃𝑃)

�

⋅∑ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)�

(𝑘𝑘−1)⋅ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� ⋅ ∫

𝐹𝐹(𝑡𝑡) ⋅ 𝑓𝑓𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡)𝑡𝑡 𝑑𝑑𝑡𝑡𝑡𝑡𝑖𝑖


𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖

(3.33)

𝐸𝐸�𝐶𝐶𝐶𝐶𝑂𝑂𝑃𝑃𝑝𝑝 |𝑂𝑂𝑀𝑀𝑚𝑚�

=∑ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)�

(𝑘𝑘−1)𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� ∫ 𝑐𝑐𝑝𝑝(𝑡𝑡)

𝐹𝐹(𝑡𝑡) ⋅ 𝑓𝑓𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡)𝑡𝑡 𝑑𝑑𝑡𝑡𝑡𝑡𝑖𝑖


𝑝𝑝𝐴𝐴𝑂𝑂𝑖𝑖

(3.34)

CM is conducted if a failure happens prior to a suitable opportunity occurrence. Define 𝑃𝑃𝑂𝑂𝑃𝑃(𝑡𝑡)

as the probability that no suitable opportunity occurs prior to time 𝑡𝑡, which may be computed

as:

𝑃𝑃𝑂𝑂𝑃𝑃(𝑡𝑡) = �1 −�𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)�(𝑘𝑘−1)

⋅ 𝐹𝐹𝐷𝐷�𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚)� ⋅ 𝐹𝐹𝑂𝑂𝑘𝑘,𝑡𝑡𝑖𝑖−1(𝑡𝑡)∞

𝑘𝑘=1

��1 − 𝑝𝑝𝑆𝑆𝑂𝑂𝑗𝑗�𝑚𝑚−1

𝑗𝑗=0

(3.35)

where 𝑖𝑖 is the index that satisfies 𝑡𝑡 ∈ (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚]. Hence, the cost rate 𝐸𝐸[𝐶𝐶𝐶𝐶𝐶𝐶𝑃𝑃] relating to CM

activity is defined as:


𝐸𝐸[𝐶𝐶𝐶𝐶𝐶𝐶𝑃𝑃] = � 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅𝑃𝑃𝑂𝑂𝑃𝑃(𝑡𝑡) ⋅ 𝑓𝑓(𝑡𝑡)

𝑡𝑡𝑑𝑑𝑡𝑡

𝑡𝑡𝑖𝑖𝑃𝑃𝑃𝑃

0

= � � 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅𝑃𝑃𝑂𝑂𝑃𝑃(𝑡𝑡) ⋅ 𝑓𝑓(𝑡𝑡)

𝑡𝑡𝑑𝑑𝑡𝑡

𝑡𝑡𝑖𝑖

𝑡𝑡𝑖𝑖−1


𝑚𝑚=1

(3.36)

In the case where OM is not considered, 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) = ∞ ∀𝑡𝑡, the objective function simplifies to

the PM optimisation model as mentioned by Coolen-Schrijner and Coolen (2007).

min𝑡𝑡𝑃𝑃𝑃𝑃

𝐶𝐶𝐶𝐶(𝑡𝑡𝑃𝑃𝑃𝑃) = � 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅𝑓𝑓(𝑡𝑡)𝑡𝑡

𝑑𝑑𝑡𝑡


0

+ �𝑐𝑐𝑃𝑃𝑃𝑃𝑡𝑡𝑃𝑃𝑃𝑃

� ⋅ 𝐹𝐹�(𝑡𝑡𝑃𝑃𝑃𝑃)

Subject to:

𝑡𝑡𝑃𝑃𝑃𝑃 ≥ 0

(3.37)

3.3.4 Verification of models via simulation

This section is to confirm the accuracy and effectiveness of the developed model above through

a Monte Carlo simulation. An example of a certain policy including PM and OM are simulated.

The policy used for two scenarios, i.e. minimal CM and perfect CM, is defined as follows:

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡) = �

1000 𝑡𝑡 ≤ 202.5 20 < 𝑡𝑡 ≤ 352 35 < 𝑡𝑡 ≤ 45

1.5 45 < 𝑡𝑡 ≤ 50

; 𝑡𝑡𝑃𝑃𝑃𝑃 = 50

The penalty cost of OM follows sigmoid function described in Eq. (3.7) with parameters:

𝑎𝑎 =10𝑡𝑡𝑃𝑃𝑃𝑃

; 𝑏𝑏 =𝑡𝑡𝑃𝑃𝑃𝑃

2; 𝑐𝑐𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 200

The initial inputs for both scenarios minimal CM and perfect CM are detailed in Table 3.1.

Table 3.1

The Inputs for Simulation

Input Parameters Notation Values/Descriptions

Minimal CM Perfect CM

Failure time (days) shape

parameter 𝛼𝛼 2 2

Failure time (days) scale

parameter 𝛽𝛽 30 50

Maintenance duration (hours) 𝑑𝑑𝑃𝑃𝑃𝑃 3 3


Direct cost of PM cost 𝑐𝑐𝑏𝑏 500 100

Downtime cost rate 𝑐𝑐𝑙𝑙 500 60

CM cost 𝑐𝑐𝐶𝐶𝑃𝑃 1000

Opportunity intensity rate

(times/day) 𝜆𝜆𝑂𝑂(𝑡𝑡) 𝜆𝜆𝑂𝑂(𝑡𝑡) = �𝜆𝜆𝐻𝐻 = 0.1 𝑡𝑡 ≥ 𝑡𝑡𝑆𝑆

𝜆𝜆𝐿𝐿 = 0.02 𝑡𝑡 < 𝑡𝑡𝑆𝑆

Season change time (days) 𝑡𝑡𝑆𝑆 30

Opportunity duration (hours)

location parameter 𝜇𝜇 0.8

Opportunity duration (hours)

scale parameter 𝜎𝜎 0.6

Time discretisation interval

[days] ∆𝑡𝑡 ∆𝑡𝑡 = 1

The verification of determining probabilities 𝑝𝑝𝑂𝑂𝑃𝑃, 𝑝𝑝𝐶𝐶𝑃𝑃 and 𝑝𝑝𝑃𝑃𝑃𝑃 is presented in Figure 3.4

(minimal CM) and Figure 3.5 (perfect CM). The results show that probabilities computed

through formulas above belong to 95% confidence intervals of simulation outputs. It means

that the developed models are accurate.

Figure 3.4. Comparison between probabilities determined from simulation and calculation in scenario Minimal CM.


Figure 3.5. Comparison between probabilities determined from simulation and calculation in scenario perfect CM.

The verification of cost rate computation is shown in Figure 3.6. The result also confirms the

credibility of calculation process.

Figure 3.6. Comparison between cost rates determined from simulation and calculation.

3.4 NUMERICAL EXAMPLES

To illustrate the use of both optimisation models, the following examples are considered. A

system is assumed to have failure time (in days) following a Weibull distribution with shape

parameter 𝛼𝛼 and scale parameter 𝛽𝛽. The system is as good as new at the starting time of the

cycle. The opportunity is supposed to occur according to NHPP with two intensities associated

with two “seasons”:

𝜆𝜆𝑂𝑂(𝑡𝑡) = �𝜆𝜆𝐻𝐻 = 0.1 𝑡𝑡 ≥ 𝑡𝑡𝑆𝑆𝜆𝜆𝐿𝐿 = 0.02 𝑡𝑡 < 𝑡𝑡𝑆𝑆


where the time of the “season change” is 𝑡𝑡𝑆𝑆 = 30.

The duration of opportunity (in hours) is assumed to follow the Log-Normal Distribution with

parameters (𝜇𝜇 = 0.8;𝜎𝜎 = 0.6). The duration for PM or OM is fixed at 𝑑𝑑𝑃𝑃𝑃𝑃 = 3 hours.

The penalty cost 𝑐𝑐𝑝𝑝(𝑡𝑡) is expressed as sigmoid function in Eq. (3.7). The parameters used for

numerical examples are set at 𝑐𝑐𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 = 0.6 ⋅ 𝑐𝑐𝐶𝐶𝑃𝑃, 𝑎𝑎 = 10𝑡𝑡𝑃𝑃𝑃𝑃

and 𝑏𝑏 = 𝑡𝑡𝑃𝑃𝑃𝑃2

.

In order to solve (3.19) and (3.30), the domain of the decision variable 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) is discretised

to 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) ∈ 𝔻𝔻 = {0,𝑑𝑑1, … ,𝑑𝑑𝑃𝑃𝑃𝑃,∞}, where all the finite values above 𝑑𝑑𝑃𝑃𝑃𝑃 are excluded since

the OM cost is at a minimum constant value for 𝐷𝐷 ≥ 𝑑𝑑𝑃𝑃𝑃𝑃 as shown in Eq. (3.6). The infinite

value is added to allow the option of rejecting all opportunities independently on 𝐷𝐷 because

OM is not beneficial in the time interval (𝑡𝑡𝑚𝑚−1, 𝑡𝑡𝑚𝑚]. For the numerical studies presented here,

∆𝑡𝑡 = 1 and 𝔻𝔻 = �0.25𝑖𝑖 | 𝑖𝑖 = 0,1,2, … , 𝑑𝑑𝑃𝑃𝑃𝑃0.25

, � ∪ {𝑀𝑀} with and 𝑀𝑀 = 1000 is used to represent

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) = ∞.

The optimisation problems are solved approximately using a simple Genetic Algorithm in

MATLAB 2015a on a High Performance Computing (HPC) cluster at Queensland University

of Technology. The parameters used for GA include: crossover rate = 0.8, mutation rate = 0.08,

population = 60. The GA algorithm was run several times to achieve a consistent optimal

solution. The flow chart of optimisation process is described in Figure 3.7. The time for solving

this optimisation problem is dependent on the discretisation of both the time interval and

𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚). If the discretisation is too fine, the computation time increased exponentially but the

outcomes have finer resolution. With the discretisation mentioned above, it takes about 10

hours to get full results.


Figure 3.7. Optimisation procedure.

3.4.1 Model 1: OM and PM are perfect; CM is minimal

The basic data for Model 1 is shown in Table 3.2.

Table 3.2 The General Inputs for Model 1


Failure time (days) shape parameter 𝛼𝛼 2

Failure time (days) scale parameter 𝛽𝛽 20

Maintenance duration (hours) 𝑑𝑑𝑃𝑃𝑃𝑃 3


Opportunity intensity rate (times/day) 𝜆𝜆𝑂𝑂(𝑡𝑡) 𝜆𝜆𝑂𝑂(𝑡𝑡) = �𝜆𝜆𝐻𝐻 = 0.1 𝑡𝑡 ≥ 𝑡𝑡𝑆𝑆𝜆𝜆𝐿𝐿 = 0.02 𝑡𝑡 < 𝑡𝑡𝑆𝑆


Opportunity duration (hours) location parameter 𝜇𝜇 0.8

Opportunity duration (hours) scale parameter 𝜎𝜎 0.6

Time discretisation interval [days] ∆𝑡𝑡 ∆𝑡𝑡 = 1

Maximum number of intervals [-] 𝑁𝑁 𝑁𝑁 = 100

Unless otherwise noted, the costs components are as follows:

• 𝑐𝑐𝑃𝑃𝑃𝑃 = 𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ 𝑑𝑑𝑃𝑃𝑃𝑃 = 2 ⋅ 𝑐𝑐𝐶𝐶𝑃𝑃

• 𝐶𝐶𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝐶𝐶𝑏𝑏

= 𝑐𝑐𝑙𝑙⋅𝑑𝑑𝑃𝑃𝑃𝑃𝐶𝐶𝑏𝑏

= 2

• 𝑐𝑐𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 = 0.6 ⋅ 𝑐𝑐𝐶𝐶𝑃𝑃

Figure 3.8 shows the convergence of GA for 5 optimisation runs according to the inputs

described above.

Figure 3.8. GA convergence.

0 3000 6000 9000 12000 15000

Iteration

120

125

130

135

140

145

150

Cos

t rat

e

Run 1

Run 2

Run 3

Run 4

Run 5


3.4.1.1 Effects of relative costs of PM vs CM

In order to evaluate the effects of the relative cost of PM and CM, the ratio of PM to CM is

altered while other parameters are kept at their original values. Two cases are considered to

evaluate the effect of considering partial opportunities:

• Consider all opportunities, 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡𝑚𝑚) ∈ 𝔻𝔻;

• Consider only full opportunities which have durations at least as large as the required

PM time 𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚 ∈ {𝑑𝑑𝑃𝑃𝑃𝑃,∞}.

Table 3.3 shows the per cent savings of each OM policy when compared to the traditional PM

policy without OM.

Table 3.3

PM time 𝒕𝒕𝑷𝑷𝑶𝑶 and cost savings vs no OM for Model 1. Effect of PM/CM cost

PM cost (% CM cost)

100% 200% 300%

No OM 𝑡𝑡𝑃𝑃𝑃𝑃 20 28 35

Full opportunities only %Savings 2.0% 5.1% 11.2%

𝑡𝑡𝑃𝑃𝑃𝑃 20 36 47

Partial opportunities %Savings 4.0% 12.5% 19.8%


It can be seen that the savings relative to the no-OM baseline increase when the PM cost is

high compared to CM. Intuitively, the PM times are longer when considering OM as well. The

numerical results also indicate that the consideration of partial opportunities can lead to a

significant cost rate saving over the case where only full opportunities are accepted

(approximately two times here).

The optimal minimum duration thresholds of arrived opportunities are shown in Figure 3.9.

Note that OM is not considered at all times. Early in the cycle (when the system is still new),

OM is ignored since the unacceptably short cycle time completely negates the PM savings from

the opportunities. As the cycle time increases, high duration opportunities can be accepted, and

the minimum duration threshold continues to drop as the cycle time increases. This is intuitive


as well; as one approaches the optimal PM time, shorter duration opportunities can be accepted

since the difference in cycle time between OM and PM is small. One also notes that when the

PM cost increases, the time window of OM consideration is extended, indicating that OM is

preferable when PM costs are high.

Figure 3.9. Minimum duration threshold for Model 1 when PM cost (𝒄𝒄𝑷𝑷𝑶𝑶) varies.

3.4.1.2 Effects of relative costs of production loss (𝒄𝒄𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍) vs direct cost (𝒄𝒄𝒃𝒃)

In this example, the ratio between production loss and direct costs is changed to evaluate the

effects of a different contribution of production loss in maintenance cost (again, other

parameters are kept at their default values). The two OM cases described in Section 3.4.1.1 are

again considered here.

Table 3.4 shows the per cent savings of each OM policy when compared to the traditional PM

policy without OM when 𝑐𝑐𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝑐𝑐𝑏𝑏

is varied.

Table 3.4

PM time 𝒕𝒕𝑷𝑷𝑶𝑶 and Cost Savings vs no OM for Model 1. Effect of 𝒄𝒄𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍/𝒄𝒄𝒃𝒃

𝒄𝒄𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍/𝒄𝒄𝒃𝒃

0.5 2 4

No OM 𝑡𝑡𝑃𝑃𝑃𝑃 28 28 28


Full opportunities only %Saving 1.3% 5.1% 7.3%


Partial opportunities %Saving 3.7% 12.5% 16.7%


Since the purpose of OM consideration is saving production loss cost, it is not surprising that

the savings over the no-OM baseline are higher when the production loss cost is a large

component of the PM cost. It can also be seen that considering partial opportunities

approximately doubles the savings compared to considering only full opportunities.

Figure 3.10 presents the best opportunity duration threshold when the ratio between the loss

cost 𝑐𝑐𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 and the direct cost 𝑐𝑐𝑏𝑏 is varied. One can see in this figure that as the production loss

costs increase, the PM time is extended to increase the probability that the less expensive OM

terminates the cycle.

Figure 3.10. The minimum duration thresholds for Model 1 when ratios between production loss cost and direct cost �𝑪𝑪𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍

𝑪𝑪𝒃𝒃� are varied.


3.4.1.3 Effects of penalty cost

In this example, the influence of the penalty cost on the maintenance policy is investigated.

The maximum penalty cost 𝑐𝑐𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 is set to 0%, 20%, 60% and 100% of PM cost while other

parameters are at their default values. The effects of different penalty costs of conducting OM

are shown in Figure 3.11. The highest saving is obtained when there is no penalty of OM and

the savings reduce when the penalty cost increases. However, the saving is still at about 9%

even when the penalty cost is high (100% of PM cost). One also notes that as the penalty cost

increases, the PM time decreases, and the time window of OM consideration shrinks since PM

is more favourable to OM due to the higher penalty costs.

Figure 3.11. The optimal solutions of Model 1 when maximum penalty cost of OM �𝒄𝒄𝒑𝒑𝒎𝒎𝒎𝒎𝒎𝒎� varies.

3.4.1.4 Summary

The numerical results show that combining OM to PM policy can produce a considerable

maintenance cost saving. The results indicate the benefits of considering partial opportunities

and integrating the duration of opportunities to OM decision-making criteria. The most suitable

case for applying combined OM and PM policy is when the PM cost is high and the production

dominates the PM cost (see Table 3.5).

Table 3.5

Savings with Proposed OM Policy over Traditional PM for Model 1



0.5 2.0 4.0

PM C

ost

100% CM cost 1.4% 4.0% 5.3%

200% CM cost 3.7% 12.5% 16.7%

300% CM cost 7.7% 19.8% 25.2%

Table 3.6

Savings achieved considering Partial Opportunities over full-opportunity-only OM for Model 1


0.5 2.0 4.0

PM C

ost

100% CM cost 0.7% 2.1% 2.7%

200% CM cost 2.5% 7.8% 10.1%

300% CM cost 3.7% 9.7% 12.1%

Table 3.6 shows that the consideration of partial opportunities can yield a significant cost

reduction compared to the case of accepting full opportunities only. The saving is also larger

when the PM cost increases and the production loss occupies a major contribution in PM cost.

This result indicates that it is more beneficial to implement the maintenance strategy

considering partial opportunities.

3.4.2 Model 2: CM, OM and PM are perfect

In this case, the PM cost must be less than the CM cost to avoid trivial solutions. If the PM cost

is higher than CM cost, PM is never considered and the system will function until failure when

CM will restore the system as good as new. The basic data for model 2 is shown in Table 3.7.

Table 3.7

The General Inputs for Model 2



Failure time (days) shape parameter 𝛼𝛼 2

Failure time (days) scale parameter 𝛽𝛽 50

Maintenance duration (hours) 𝑑𝑑𝑃𝑃𝑃𝑃 3

Opportunity intensity rate (times/day) 𝜆𝜆𝑂𝑂(𝑡𝑡) 𝜆𝜆𝑂𝑂(𝑡𝑡) = �𝜆𝜆𝐻𝐻 = 0.1 𝑡𝑡 ≥ 𝑡𝑡𝑆𝑆𝜆𝜆𝐿𝐿 = 0.02 𝑡𝑡 < 𝑡𝑡𝑆𝑆


Opportunity duration (hours) location parameter 𝜇𝜇 0.8

Opportunity duration (hours) scale parameter 𝜎𝜎 0.6

Time discretisation interval (days) ∆𝑡𝑡 ∆𝑡𝑡 = 1

Maximum number of intervals 𝑁𝑁 𝑁𝑁 = 100

Unless otherwise noted, the cost components are as follows:

• 𝑐𝑐𝑃𝑃𝑃𝑃 = 𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑙𝑙 ⋅ 𝑑𝑑𝑃𝑃𝑃𝑃 = 0.4 ⋅ 𝑐𝑐𝐶𝐶𝑃𝑃

• 𝐶𝐶𝑙𝑙𝑙𝑙𝑠𝑠𝑠𝑠𝐶𝐶𝑏𝑏

= 𝑐𝑐𝑙𝑙⋅𝑑𝑑𝑃𝑃𝑃𝑃𝐶𝐶𝑏𝑏

= 2

• 𝑐𝑐𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 = 0.6 ⋅ 𝑐𝑐𝐶𝐶𝑃𝑃

The results of Model 2 show a similar trend to Model 1 and the key results are summarised

here.

Table 3.8

Savings with Proposed OM Policy over Traditional PM for Model 2


0.5 2.0 4.0

PM C

ost

20% CM cost 0.6% 1.8% 2.4%

40% CM cost 1.8% 5.7% 7.5%

60% CM cost 3.4% 9.6% 12.2%

Similar to Model 1 (minimal CM), the results of Model 2 show that significant (albeit smaller)

savings can be achieved when the penalty cost is low and the PM cost is high and dominated


by production loss (Table 3.8). Clearly, this is reasonable: the purpose of OM is to reduce the

production loss cost caused by PM activities.

Table 3.9

Savings Achieved Considering Partial Opportunities over Full-opportunity-only OM for Model 2


0.5 2.0 4.0 PM

Cos

t 20% CM cost 0.3% 0.9% 1.2%

40% CM cost 1.1% 3.4% 4.3%

60% CM cost 1.1% 3.6% 4.7%

Table 3.9 also shows the benefit of considering partial opportunities in the OM policy (over

full opportunities only). The benefit is again greater when the PM cost is high and the

production loss is a major component in PM cost.

3.5 CONCLUSION

This study has presented a general model to determine the optimal combined scheduled

preventive maintenance (PM) and opportunistic maintenance (OM) considering two

characteristics of real opportunities: non-homogeneous arrival and uncertain opportunity

duration. In particular, this study considered “partial opportunities” for the first time, i.e. those

that have duration less than the PM time. A time-varying minimum duration threshold for the

acceptance of opportunities was considered and a time discretisation approach was used to

analytically compute the single-cycle cost rate. Subsequently, this cost rate was minimised by

jointly optimising the schedule PM time and time-varying opportunity duration threshold.

Simulation results showed that considering partial opportunities yields a significant saving

compared to both traditional PM policies (no OM) and to OM policies where only full

opportunities are accepted (i.e. only those at least as long as the required PM time). In

particular, the results indicate that the partial opportunity model can yield higher savings when

PM is expensive, primarily due to high downtime costs.

58 Chapter 4: Joint Opportunistic and Preventive Maintenance policy for a finite time horizon: Condition-based approach

Chapter 4: Joint Opportunistic and Preventive Maintenance policy for a finite time horizon: Condition-based approach

Overview

This chapter exploits the partial opportunistic maintenance model for an industrial system in a

finite time horizon where the condition-based maintenance approach is applied. The

maintenance optimisation problem is formulated in this study as a finite-horizon Markov

Decision Process, where the randomly occurring opportunities are accounted for by

augmenting the time-varying, decision-dependent transition probabilities. The numerical

example and analysis is conducted for a hypothetical wind turbine located at Mount Emerald,

Queensland, Australia, to illustrate the benefit of the proposed model.

This chapter relates to papers:

Truong-Ba H, Borghesani P, Cholette ME, Ma L. Optimization of Condition-based

Maintenance Considering Partial Opportunities.” Submitting to IEEE-Transaction on

Reliability.

Chapter 4: Joint Opportunistic and Preventive Maintenance policy for a finite time horizon: Condition-based approach 59

Nomenclature

Indices:

𝑖𝑖, 𝑗𝑗: indices of states, 𝑖𝑖, 𝑗𝑗 = 1,2, … ,𝑁𝑁.

𝑚𝑚: index of maintenance actions, 𝑚𝑚 = 0,1, … ,𝑀𝑀.

𝑘𝑘: index of time period (decision epochs), 𝑘𝑘 = 1,2, … ,𝐾𝐾.

Variables:

𝑆𝑆𝑘𝑘: degradation state of system at time 𝑡𝑡𝑘𝑘.

𝑒𝑒𝑚𝑚, 𝑒𝑒𝑗𝑗: a discrete state of system

𝑎𝑎𝑚𝑚: a maintenance action.

𝐷𝐷𝑘𝑘: opportunity duration, a random variable.

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚�: transition probability of degradation from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 after time interval

(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] given no maintenance action is conducted.

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚�: PM-only transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 after time interval

(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] given maintenance action 𝑎𝑎𝑚𝑚 is conducted.

𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚�: joint PM and OM transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 after time

interval (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] given decision 𝑎𝑎𝑚𝑚 is selected.

Parameters:

𝑑𝑑(𝑎𝑎𝑚𝑚): duration of doing maintenance action 𝑎𝑎𝑚𝑚.

𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚): cost corresponding to state 𝑒𝑒𝑚𝑚 at time epoch 𝑡𝑡𝑘𝑘.

𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚): direct (basic) maintenance cost corresponding to action 𝑎𝑎𝑚𝑚 at time epoch 𝑡𝑡𝑘𝑘.

𝑐𝑐ℓ,𝑘𝑘: downtime cost rate (per unit of stopping time) at time epoch 𝑡𝑡𝑘𝑘.

𝜆𝜆𝑂𝑂(𝑡𝑡): time-varying opportunity occurrence intensity.


4.1 INTRODUCTION

Analogous to what was proposed for TB-OM in the previous chapter and papers (Ba et al.,

2016; Truong Ba, Cholette, Borghesani, et al., 2017), this study aims at developing a CB-OM

methodology that includes non-stationary partial opportunities and time-varying economic

conditions in the decision processes. An optimal CB-OM policy (minimal total maintenance

cost) will be developed for a production system over a finite time horizon considering the time-

varying statistical properties of opportunity arrivals and economic benefits as well as a set of

possible maintenance actions. The optimal maintenance policy proposed in this study is

obtained by jointly optimising: 1) time-varying condition thresholds for selecting PM

maintenance actions among imperfect repair candidates; and 2) rules for the selection of OM

maintenance actions when a suitable opportunity arrives. The OM component of the policy

considers both the asset condition and the economic benefit of the opportunity, which is made

particularly relevant by the “partial” nature of most opportunities. This situation was inspired

by the practical situation of the windfarm where low wind speed (seasonally-dependent arrived

rate) events may stop production of wind turbines and often create partial opportunities for

maintenance and where condition monitoring information is often critical in the optimisation

of maintenance.

The remainder of the chapter is organised as follows: Section 4.2 describes the modelling

approach, formulating the novel CB-OM optimisation problem according to the Dynamic

Programming approach. The evaluation of the identified optimum, based on a Monte Carlo

simulation procedure, is discussed in Section 4.3. The analysis of the benefits provided by the

newly proposed PM-OM policy are discussed through a case study of a hypothetical wind

turbine presented in Section 4.4. Finally, the last section summarises the main conclusions of

this chapter.

4.2 MODEL DESCRIPTION

This study develops a dynamic preventive maintenance (PM) policy with opportunistic

maintenance consideration for a production system in a predefined time horizon 𝑡𝑡0, 𝑡𝑡1, … ,𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑

(which ideally would be the entire life of the system). During the time horizon, PM or OM will

be considered according to the degradation of the system, the economic benefit of each

opportunity as well as the cost of a set of possible (potentially imperfect) maintenance actions.

An opportunity is defined as a forced system stoppage due to external causes (e.g. a production


line stoppage due to lack of demand or adverse weather conditions), which can reduce the

downtime cost if maintenance is conducted. Maintenance costs consist of two components: (1)

fixed basic (direct) maintenance costs (e.g. spare parts, equipment, etc.); and (2) variable costs

(e.g. downtime cost) which can be reduced by OM. For each decision epoch (beginning of time

interval), according to the state of system, a PM action is selected among a set of possible

maintenance actions4. During a certain interval, if no PM action is conducted, it is possible for

OM to occur upon the arrival of an opportunity, the suitability of which is evaluated on the

basis of the maintenance cost savings and the current state of the system. The general concept

of the maintenance strategy is depicted in Figure 4.1.

Figure 4.1. The mechanism of opportunistic maintenance policy.

In this chapter, the total joint PM and OM policy is developed based on the Markov Decision

Process (MDP) approach. The timeline is discretised into 𝐾𝐾 intervals (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1], each with an

equal and sufficiently small length ∆𝑡𝑡 and with 𝑘𝑘 = 1,2, … ,𝐾𝐾 and 𝑡𝑡1 = 0 , 𝑡𝑡𝐾𝐾+1 = 𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑. The

aim of the MDP model is to determine the optimal decision rules for PM at the beginning of

each time epoch as well as the optimal OM policy for the consideration of opportunities

4 Note that the set of maintenance actions includes “do-nothing”.


arriving during each interval. The following assumptions have been considered in the

formulation of the problem:

1. A PM decision 𝑃𝑃𝑀𝑀𝑘𝑘 will be made at the beginning of each time interval, selecting

among a finite set ℳ of possible actions including the “do-nothing” action 𝑎𝑎0.

2. If a maintenance action (other than do-nothing) is selected, OM is not considered in

that interval (Figure 4.2a).

3. If do-nothing is selected for PM at the beginning of the interval, OM is considered

for that interval. An OM decision 𝑂𝑂𝑀𝑀𝑘𝑘 is selected among the same finite set of

possible actions ℳ (including do-nothing) in case an opportunity arrives during the

interval (Figure 4.2b). The other possible outcome (no PM and no OM) is instead

obtained when no suitable opportunity occurs within the interval (Figure 4.2c and

d).

4. It is assumed that the time interval is short enough for at most one opportunity to

occur. This approximation improves if the interval is chosen small enough so that

the probability of two opportunities arriving in an interval is negligible.

5. It is assumed that no maintenance action is implemented at the end of the time

horizon 𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑 (e.g. it is decommissioned).

Assumption (4) is not terribly restrictive and is made largely to simplify presentation. If (4) is

not satisfied, one may either shrink the discretisation interval ∆𝑡𝑡 or employ the methods used

by Truong Ba, Cholette, Borghesani, et al. (2017) to simply extend the methods presented here

to the multiple opportunity case.

According to the structure of MDPs, discrete states (degradation), decisions and corresponding

transition matrices, and reward (cost) functions must be defined (Puterman, 2014). These will

be discussed in the subsequent sections.


Figure 4.2. OM scenarios: a) Doing PM; b) No PM and doing OM with suitable opportunity; c) No PM and do-nothing for OM due to unsuitable opportunity; and d) Neither PM nor OM due to no opportunity.

4.2.1 Degradation and maintenance model

This section discusses on transition probabilities according to system degradation process and

maintenance actions. In reality, these probabilities can be obtained via a data processing

technique according to specific system inspection or monitoring processes. With the

development of data mining techniques and simulation methods, the historical condition data

of systems can be analysed and transformed to appropriate transition probabilities. In the

limitation of this study, the processes to transform monitored/inspected data to transition

probabilities are not discussed in details.

The degradation of the system 𝑆𝑆𝑘𝑘 at time 𝑡𝑡𝑘𝑘 is considered to be a perfectly monitored process

(assumed observed at uniform time intervals 𝑡𝑡𝑘𝑘 = 𝑘𝑘∆𝑡𝑡). The degradation of the system

(naturally continuous in value) is discretised into 𝑁𝑁 states with a constant gap ∆𝑒𝑒. The system

is said to be in state 𝑒𝑒𝑚𝑚, 𝑖𝑖 ∈ [1, … ,𝑁𝑁] at time 𝑡𝑡𝑘𝑘 if the degradation of system 𝑆𝑆𝑘𝑘 satisfies:

𝑒𝑒𝑚𝑚 −∆𝑒𝑒2≤ 𝑆𝑆𝑘𝑘 ≤ 𝑒𝑒𝑚𝑚 +

∆𝑒𝑒2

(4.1)

Thus, in the remainder of the chapter 𝑆𝑆𝑘𝑘 is discretised in:

𝑆𝑆𝑘𝑘 ∈ {𝑒𝑒1, … , 𝑒𝑒𝑁𝑁} = 𝒮𝒮


Note that the last state 𝑒𝑒𝑁𝑁 indicates that the system is completely degraded (e.g. failed). The

one-step transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 is defined in the usual way:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� = Pr�𝑆𝑆𝑘𝑘+1 = 𝑒𝑒𝑗𝑗|𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚 , do − nothing� ∀𝑘𝑘 (4.2)

These probabilities may be constructed from historical data and/or via Monte Carlo simulations

of a continuous degradation model. Because the degradation process is non-decreasing, the

transition probability 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� will satisfy:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� = 0 ∀𝑒𝑒𝑗𝑗 < 𝑒𝑒𝑚𝑚 (4.3)

i.e. the transition matrix is upper triangular.

The degradation state is also affected by maintenance actions. This study considers 𝑀𝑀 + 1

possible maintenance actions with different restoration levels.

ℳ = {𝑎𝑎0,𝑎𝑎1, … ,𝑎𝑎𝑚𝑚, … ,𝑎𝑎𝑃𝑃}

The “do-nothing” action is labelled as 𝑎𝑎0, the action 𝑎𝑎𝑚𝑚 ,𝑚𝑚 = 1 …𝑀𝑀 are imperfect repairs with

the different restoration levels, in which 𝑎𝑎𝑃𝑃 represents for perfect maintenance. The one-step

transition probability of the maintenance action 𝑎𝑎𝑚𝑚 is computed as:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� = Pr�𝑆𝑆𝑘𝑘+1 = 𝑒𝑒𝑗𝑗|𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚,𝑃𝑃𝑘𝑘 = 𝑎𝑎𝑚𝑚 � ∀𝑘𝑘 (4.4)

In case the maintenance action 𝑎𝑎0 (do-nothing) is selected, the transition probability

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚,𝑎𝑎0� is equal to 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚�.

4.2.2 Opportunity model

The opportunity occurrences are assumed to follow a Non-homogenous Poisson process

(NHPP) with intensity 𝜆𝜆(𝑡𝑡). Under this assumption, the 𝑛𝑛𝑡𝑡ℎ opportunity occurrence in any

arbitrary interval �𝑡𝑡𝑙𝑙, 𝑡𝑡𝑓𝑓� (𝑡𝑡𝑓𝑓 ≥ 𝑡𝑡𝑙𝑙) follows a Gamma distribution, where the c.d.f

𝐹𝐹𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙,𝑛𝑛) and p.d.f 𝑓𝑓𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙,𝑛𝑛) are given by (2013); Ross (2014):

and 𝑓𝑓𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙,𝑛𝑛) = [Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡)]𝑚𝑚−1

(𝑛𝑛 − 1)!𝜆𝜆(𝑡𝑡)𝑒𝑒−Λ(𝑡𝑡𝑠𝑠,𝑡𝑡) (4.6)

𝐹𝐹𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙,𝑛𝑛) = 𝑃𝑃�𝑡𝑡𝑙𝑙 < 𝑇𝑇𝑂𝑂𝑚𝑚 ≤ 𝑡𝑡� =𝛾𝛾�𝑛𝑛,Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡)�

(𝑛𝑛 − 1)! (4.5)


In these expressions, 𝛾𝛾(𝑛𝑛,𝑢𝑢) = ∫ 𝑥𝑥𝑚𝑚−1𝑒𝑒−𝑚𝑚𝑑𝑑𝑥𝑥𝑢𝑢0 , is the lower incomplete gamma function and

Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡) is the mean value function of NHPP considering from the start time 𝑡𝑡𝑙𝑙:

Λ(𝑡𝑡𝑙𝑙, 𝑡𝑡) = �𝜆𝜆(𝜏𝜏)𝑑𝑑𝜏𝜏𝑡𝑡

𝑡𝑡𝑠𝑠

𝑡𝑡 ∈ �𝑡𝑡𝑙𝑙, 𝑡𝑡𝑓𝑓� (4.7)

Note that the assumption of only one opportunity arrival within the time interval means that

𝐹𝐹𝑂𝑂(𝑡𝑡𝑙𝑙 + ∆𝑡𝑡; 𝑡𝑡𝑙𝑙, 𝑛𝑛) ≈ 0 for 𝑛𝑛 > 1. Therefore, in this study, only the case of 𝑛𝑛 = 1 is considered.

Hence, for the convenient presentation of the remaining of the chapter, it is assigned that

𝐹𝐹𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙) ≡ 𝐹𝐹𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙, 1) and 𝑓𝑓𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙) ≡ 𝑓𝑓𝑂𝑂(𝑡𝑡; 𝑡𝑡𝑙𝑙, 1).

Each opportunity lasts in random duration 𝐷𝐷𝑘𝑘 which follows a time-varying distribution

𝒟𝒟𝑘𝑘𝑂𝑂 with p.d.f and c.d.f are 𝑓𝑓𝐷𝐷,𝑘𝑘(. ) and 𝐹𝐹𝐷𝐷,𝑘𝑘(. ), respectively. The opportunity duration can yield

some savings in downtime cost if conducting OM.

4.2.3 Cost structure

Each degradation state 𝑒𝑒𝑚𝑚 of system induces an average state cost, denoted as 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚). The

degradation cost 𝑐𝑐𝑙𝑙,𝑘𝑘(⋅) satisfies the condition:

𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒1) ≤ 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒2) ≤ ⋯ ≤ 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑁𝑁) ∀𝑘𝑘 (4.8)

Each maintenance action 𝑎𝑎𝑚𝑚 has its own time-varying direct cost 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) and a variable cost

𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) that can be reduced by an opportunity5. In this study, the time-varying downtime cost

due to production loss is used to represent the variable costs of maintenance action, according

to the following expression:

𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) = 𝑐𝑐ℓ,𝑘𝑘 ⋅ 𝑑𝑑(𝑎𝑎𝑚𝑚) 𝑎𝑎𝑚𝑚 ∈ ℳ (4.9)

where 𝑑𝑑(𝑎𝑎𝑚𝑚) is the duration of maintenance action 𝑎𝑎𝑚𝑚; and 𝑐𝑐ℓ,𝑘𝑘 is the time-dependent

production loss cost per unit of time. Hence, the cost of a maintenance action 𝑎𝑎𝑚𝑚 is defined as:

PM: 𝐶𝐶𝑘𝑘𝑃𝑃𝑃𝑃(𝑎𝑎𝑚𝑚) = 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐ℓ,𝑘𝑘 ⋅ 𝑑𝑑(𝑎𝑎𝑚𝑚) (4.10)

OM: 𝐶𝐶𝑘𝑘𝑂𝑂𝑃𝑃(𝐷𝐷𝑘𝑘,𝑎𝑎𝑚𝑚) = 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐ℓ,𝑘𝑘 ⋅ max(0,𝑑𝑑(𝑎𝑎𝑚𝑚) − 𝐷𝐷𝑘𝑘) (4.11)

5 Note that 𝑐𝑐𝑏𝑏(𝑎𝑎0) = 0 and 𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎0) = 0 ∀𝑘𝑘. The costs of decision doing nothing 𝑐𝑐𝑏𝑏(𝑎𝑎0) and 𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎0) are redefined for case considering OM.


where: 𝐷𝐷𝑘𝑘 is the random duration of opportunity and follows some time-varying distribution

𝒟𝒟𝑘𝑘𝑂𝑂 with p.d.f and c.d.f are 𝑓𝑓𝐷𝐷,𝑘𝑘(⋅) and 𝐹𝐹𝐷𝐷,𝑘𝑘(⋅), respectively.

4.2.4 Optimisation through Markov Decision Process (MDP)

This study aims to develop a PM and OM that minimises the total expected cost over the

specified finite horizon. This means identifying both the degradation states for conducting PM

as well as the policy for accepting an opportunity if it arrives within a time interval. The joint

policy is composed of two components to be optimised together:

• 𝜋𝜋𝑃𝑃𝑃𝑃 = �𝜇𝜇𝑃𝑃𝑃𝑃,1(𝑆𝑆1), 𝜇𝜇𝑃𝑃𝑃𝑃,2(𝑆𝑆2), … , 𝜇𝜇𝑃𝑃𝑃𝑃,𝐾𝐾(𝑆𝑆𝐾𝐾)�, the PM component of the policy,

which is a series of functions 𝜇𝜇𝑃𝑃𝑃𝑃,𝑘𝑘:𝒮𝒮 ⟼ℳ mapping the degradation state into a

preventive maintenance action at the beginning of the epoch 𝑘𝑘;

• 𝜋𝜋𝑂𝑂𝑃𝑃 = �𝜇𝜇𝑂𝑂𝑃𝑃,1(𝑆𝑆1,𝐷𝐷1),𝜇𝜇𝑂𝑂𝑃𝑃,2(𝑆𝑆2,𝐷𝐷2), … , 𝜇𝜇𝑂𝑂𝑃𝑃,𝐾𝐾(𝑆𝑆𝐾𝐾,𝐷𝐷𝐾𝐾)�, the OM component of the

policy, which is a series of functions 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘:𝒮𝒮 × 𝒟𝒟𝑘𝑘𝑂𝑂 ⟼ℳ mapping the degradation

state and opportunity duration into a maintenance action during the time interval 𝑘𝑘.

The solution of this optimisation problem is obtained by backward induction according to the

following procedure:

• Optimise OM. Assuming that 𝑎𝑎0 (do-nothing) is chosen for PM at the beginning of

the interval 𝐾𝐾, the optimal OM decision function 𝜇𝜇𝑂𝑂𝑃𝑃,𝐾𝐾∗ (𝑆𝑆𝐾𝐾,𝐷𝐷𝐾𝐾) is determined. The

expected cost over the interval 𝐾𝐾 incurred following the rule 𝜇𝜇𝑂𝑂𝑃𝑃,𝐾𝐾∗ (𝑆𝑆𝐾𝐾,𝐷𝐷𝑘𝑘) is

therefore equal to the expected cost of selecting PM action 𝑎𝑎0 at the beginning of the

interval 𝐾𝐾;

• Given optimal OM, optimise PM. Once the expected cost of a do-nothing PM

decision is known, the optimal PM decision function 𝜇𝜇𝑃𝑃𝑃𝑃,𝐾𝐾∗ (𝑆𝑆𝐾𝐾) is determined,

selecting among all the possible PM actions ℳ, including the do-nothing 𝑎𝑎0. The

combined OM-PM decision rule will therefore result in a minimal expected cost

𝑉𝑉𝐾𝐾(𝑆𝑆𝐾𝐾) over the last interval;

• Recursion. Knowing 𝑉𝑉𝐾𝐾(𝑆𝑆𝐾𝐾), repeat the above optimisations in a recursive manner

(i.e. backward induction) to compute 𝑉𝑉𝑘𝑘(𝑆𝑆𝑘𝑘), 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘(𝑆𝑆𝑘𝑘,𝐷𝐷𝑘𝑘), and 𝜇𝜇𝑃𝑃𝑃𝑃,𝑘𝑘(𝑆𝑆𝑘𝑘) for 𝑘𝑘 =

𝐾𝐾 − 1,𝐾𝐾 − 2, … ,0.


The following two subsections will discuss the details of the procedure above. The next

sections will discuss the identification of the optimal OM decision function 𝜇𝜇𝑂𝑂𝑃𝑃,𝐾𝐾∗ (𝑆𝑆𝑘𝑘,𝐷𝐷𝑘𝑘), the

subsequent determination of the optimal PM function 𝜇𝜇𝑃𝑃𝑃𝑃,𝐾𝐾∗ (𝑆𝑆𝐾𝐾), and the backward induction

procedure to determine the PM and OM policies (i.e. the decision functions for 𝐾𝐾 − 1,𝐾𝐾 −

2, … ,0).

4.2.4.1 Optimal maintenance policy for OM

When an opportunity occurs, the selection of an OM action will be based on the duration

provided by this opportunity and the current degradation of the system. Consider the situation

where 𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚 and the optimal decision functions are known from time 𝑘𝑘 + 1, … ,𝐾𝐾, i.e. the

expected (optimal) cost-to-go6 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗� is known. If an opportunity arrives with duration 𝐷𝐷𝑘𝑘 =

𝛿𝛿 in the interval (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] and maintenance action 𝑎𝑎𝑚𝑚 is selected upon its arrival, the total

maintenance cost incurred from 𝑘𝑘 to 𝐾𝐾 can then be computed as:

𝑈𝑈𝑂𝑂𝑃𝑃,𝑘𝑘(𝑒𝑒𝑚𝑚, 𝛿𝛿,𝑎𝑎𝑚𝑚) = 𝐶𝐶𝑘𝑘𝑂𝑂𝑃𝑃(𝛿𝛿, 𝑎𝑎𝑚𝑚) + 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚) + �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ⋅ 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�𝑙𝑙𝑗𝑗∈𝒮𝒮

= 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐ℓ,𝑘𝑘 ⋅ max(0,𝑑𝑑(𝑎𝑎𝑚𝑚) − 𝛿𝛿) + 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚)

+ �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ⋅ 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�𝑙𝑙𝑗𝑗∈𝒮𝒮

(4.12)

where 𝐶𝐶𝑘𝑘𝑂𝑂𝑃𝑃(𝛿𝛿,𝑎𝑎𝑚𝑚) is the cost of conducting maintenance action 𝑎𝑎𝑚𝑚 opportunistically and

𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚) is the degradation cost. The optimal decision for OM 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚, 𝛿𝛿) is therefore one that

achieves the minimum cost:

𝑈𝑈𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚, 𝛿𝛿) = min

𝑚𝑚𝑚𝑚∈ℳ𝑈𝑈𝑂𝑂𝑃𝑃,𝑘𝑘(𝑒𝑒𝑚𝑚, 𝛿𝛿,𝑎𝑎𝑚𝑚)

𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚, 𝛿𝛿) = argmin

𝑚𝑚𝑚𝑚∈ℳ𝑈𝑈𝑂𝑂𝑃𝑃,𝑘𝑘(𝑒𝑒𝑚𝑚,𝛿𝛿,𝑎𝑎𝑚𝑚)

(4.13)

Thus, 𝑈𝑈𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚, 𝛿𝛿) represents the optimal cost-to-go conditional upon the arrival of an

opportunity with duration 𝛿𝛿 in the interval (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1].

As discussed above, OM is only considered if maintenance action for PM is do-nothing (𝑎𝑎0).

Therefore, once the OM decision function 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚,𝛿𝛿) is identified for the interval 𝑘𝑘, the

6 The expected optimal cost-to-go is the cost of starting in state 𝑒𝑒𝑗𝑗 at time 𝑘𝑘 + 1 and applying the optimal policy from 𝑘𝑘 + 1 until the end of the horizon [page 83 of reference 30]. The cost 𝑉𝑉𝐾𝐾+1(𝑒𝑒𝑚𝑚) is assumed known (e.g. zero or a fixed plant decommissioning cost).


expected cost 𝔼𝔼[𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎0)] and the effect of OM on the transition probability can be determined.

This will in turn enable the selection of the best PM action (including 𝑎𝑎0) at the beginning of

the same interval.

4.2.4.2 Determining expected cost and transition probability of PM decision 𝒎𝒎𝟎𝟎

In order to determine the expected total cost and transition probabilities under OM, the

probability of conducting maintenance action 𝑎𝑎𝑚𝑚 opportunistically 𝑝𝑝𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃 (𝑒𝑒𝑚𝑚) is first calculated.

Let 𝐼𝐼𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚, 𝛿𝛿) be the indicator function defined as follows:

𝐼𝐼𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚,𝛿𝛿) = �1 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘

∗ (𝑒𝑒𝑚𝑚, 𝛿𝛿) = 𝑎𝑎𝑚𝑚0 otherwise

(𝑚𝑚 = 0,1, … ,𝑀𝑀) (4.14)

An event of selecting maintenance action 𝑎𝑎𝑚𝑚 for OM happens when 1) an opportunity occurs;

and 2) the OM action 𝑎𝑎𝑚𝑚 is optimal (𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚, 𝛿𝛿) = 𝑎𝑎𝑚𝑚). Thus, the probability that an OM

with action type 𝑎𝑎𝑚𝑚 occurs is computed as:

𝑝𝑝𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃 (𝑒𝑒𝑚𝑚) = 𝐹𝐹𝑂𝑂(𝑡𝑡𝑘𝑘+1; 𝑡𝑡𝑘𝑘) ⋅ � 𝐼𝐼𝑚𝑚,𝑘𝑘

𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚,𝛿𝛿) ⋅ 𝑓𝑓𝐷𝐷,𝑘𝑘(𝛿𝛿)𝑑𝑑𝛿𝛿∞

0

𝑚𝑚 = 1, … ,𝑀𝑀 (4.15)

where 𝐹𝐹𝑂𝑂(𝑡𝑡𝑘𝑘+1; 𝑡𝑡𝑘𝑘) is the probability that an opportunity arrives in the interval (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1],

(computed according to Eq. (4.5)). In this study, it is assumed that the length of time interval

is short enough for at most one opportunity to occur. In the case that many opportunities can

arrive in a time interval, the method described in Eq. 14 of previous chapter and paper (Truong

Ba, Cholette, Borghesani, et al., 2017) can be used.

The improper integral in Eq. (4.15) is convergent due to 𝐼𝐼𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚, 𝛿𝛿) ∈ {0,1} and 𝑓𝑓𝐷𝐷,𝑘𝑘(𝛿𝛿) is the

p.d.f. Thus, in order to calculate numerically this integral, it is possible to discretise the duration

𝛿𝛿 and compute up to some big enough value 𝐻𝐻 that make 𝑓𝑓𝐷𝐷,𝑘𝑘(𝐻𝐻) as less than a tolerant value

𝜖𝜖, 𝑓𝑓𝐷𝐷,𝑘𝑘(𝐻𝐻) < 𝜖𝜖; or, to use the numerical integral function of MATLAB (Shampine, 2008).

The probability of conducting OM is determined as follows:

𝑝𝑝𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚) = � 𝑝𝑝𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃 (𝑒𝑒𝑚𝑚)

𝑃𝑃

𝑚𝑚=1

(4.16)

When the PM decision is “do-nothing”, OM is possible during the interval. Therefore, the PM-

only transition probability 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� is not valid anymore and a new transition probability

𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎0� is computed as:


𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎0� = 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚� ⋅ [1 − 𝑝𝑝𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚)] + � 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ⋅ 𝑝𝑝𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃 (𝑒𝑒𝑚𝑚)

𝑃𝑃

𝑚𝑚=1

(4.17)

Finally, when maintenance action 𝑎𝑎0 is selected, the expected maintenance cost incurred in the

interval (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] 𝔼𝔼[𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎0)] is:

𝔼𝔼[𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎0)] = � 𝑝𝑝𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃 (𝑒𝑒𝑚𝑚) ⋅ 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚)

𝑃𝑃

𝑚𝑚=1

+ 𝐹𝐹𝑂𝑂(𝑡𝑡𝑘𝑘+1; 𝑡𝑡𝑘𝑘)

⋅ � � 𝑐𝑐ℓ,𝑘𝑘 ⋅ max(0,𝑑𝑑(𝑎𝑎𝑚𝑚) − 𝛿𝛿) ⋅ 𝐼𝐼𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚,𝛿𝛿) ⋅ 𝑓𝑓𝐷𝐷,𝑘𝑘(𝛿𝛿)𝑑𝑑𝛿𝛿

∞

0

𝑃𝑃

𝑚𝑚=1

(4.18)

The improper integral in Eq. (4.18) is also convergent due to 𝑓𝑓𝐷𝐷,𝑘𝑘(𝛿𝛿) as the p.d.f, 𝐼𝐼𝑚𝑚,𝑘𝑘𝑂𝑂𝑃𝑃(𝑒𝑒𝑚𝑚, 𝛿𝛿) ∈

{0,1} and max(0,𝑑𝑑(𝑎𝑎𝑚𝑚) − 𝛿𝛿) = 0 ∀𝛿𝛿 > 𝑑𝑑(𝑎𝑎𝑚𝑚). Hence, the numerical integral function of

MATLAB is suitable to compute this integral.

In the following subsection, these results will be finally used to determine the optimal PM

decision policy 𝜋𝜋𝑃𝑃𝑃𝑃∗ = �𝜇𝜇𝑃𝑃𝑃𝑃,1∗ (𝑆𝑆1),𝜇𝜇𝑃𝑃𝑃𝑃,2

∗ (𝑆𝑆2), … , 𝜇𝜇𝑃𝑃𝑃𝑃,𝐾𝐾∗ (𝑆𝑆𝐾𝐾)�.

4.2.4.3 Optimal maintenance policy for PM (given optimal OM policy)

As done for the 𝑎𝑎0 case, the transition probabilities and expected maintenance cost of other PM

decisions 𝑎𝑎𝑚𝑚 ≠ 𝑎𝑎0 must be computed. In case 𝑎𝑎𝑚𝑚 ∈ ℳ[𝑎𝑎0] is selected for PM at the beginning

of the interval 𝑘𝑘, the transition probability is not affected by the OM policy (and therefore equal

to that defined in Eq. (4.4)):

𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� = 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� 𝑎𝑎𝑚𝑚 ∈ ℳ[𝑎𝑎0] (4.19)

The expected maintenance cost of selecting maintenance action 𝑎𝑎𝑚𝑚 is the same as PM cost

𝐶𝐶𝑘𝑘𝑃𝑃𝑃𝑃(𝑎𝑎𝑚𝑚) in Eq. (4.10):

𝔼𝔼[𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎𝑚𝑚)] = 𝐶𝐶𝑘𝑘𝑃𝑃𝑃𝑃(𝑎𝑎𝑚𝑚) = 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐ℓ,𝑘𝑘 ⋅ 𝑑𝑑(𝑎𝑎𝑚𝑚) 𝑎𝑎𝑚𝑚 ∈ ℳ[𝑎𝑎0] (4.20)

Finally, the expected total maintenance cost function for time interval 𝑘𝑘, given the system state

𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚, is:

𝔼𝔼[𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚)] = 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚) + 𝔼𝔼[𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎𝑚𝑚)] (4.21)


The optimisation model subject to these Markov chain dynamics falls under the well-

established paradigm of finite-horizon MDPs, which can be easily solved via backward

induction (Puterman, 2014):

𝑉𝑉𝑘𝑘(𝑒𝑒𝑚𝑚) = min𝑚𝑚𝑚𝑚∈ℳ

�𝔼𝔼[𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚)] + �𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ⋅ 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�𝑙𝑙𝑗𝑗∈𝒮𝒮

�

𝜇𝜇𝑃𝑃𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚) = argmin

𝑚𝑚𝑚𝑚∈ℳ�𝔼𝔼[𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚)] + �𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ⋅ 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�

𝑙𝑙𝑗𝑗∈𝒮𝒮

�

(4.22)

The summarised optimisation procedure is presented in following flow chart (Figure 4.3). The

computation time of backward induction algorithm is dependent on the numbers of discretised

states and time intervals. If the condition of systems is finely discretised the computation time

will dramatically increase. However, if the states are roughly discretised, the optimal solution

is not really exact and the benefits of proposed policies is reduced. Therefore, state

discretisation should be considered wisely to compromise both the computation time and the

resolution of outcomes.

Figure 4.3. Optimisation procedure.


4.3 POLICY EVALUATION VIA SIMULATION

Monte Carlo simulation is used to evaluate the maintenance policy in order to give insight into

the variation of OM and PM in the presence of seasonally varying costs. Given:

• OM-PM policy: 𝜋𝜋 = �𝜇𝜇𝑃𝑃𝑃𝑃,1(𝑆𝑆1),𝜇𝜇𝑂𝑂𝑃𝑃,1(𝑆𝑆1,𝐷𝐷1), … , 𝜇𝜇𝑃𝑃𝑃𝑃,𝐾𝐾(𝑆𝑆𝐾𝐾),𝜇𝜇𝑂𝑂𝑃𝑃,1(𝑆𝑆𝐾𝐾,𝐷𝐷𝐾𝐾)�;

• number of time intervals 𝐾𝐾;

• length of interval ∆𝑡𝑡;

• degradation process 𝒟𝒟ℰ;

• opportunity occurrence intensity 𝜆𝜆(𝑡𝑡);

• distribution of opportunity duration 𝒟𝒟𝑘𝑘𝑂𝑂 ∀𝑘𝑘;

• cost of each states 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚) ∀𝑒𝑒𝑚𝑚,𝑘𝑘;

• direct maintenance cost 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) ∀𝑎𝑎𝑚𝑚,𝑘𝑘;

• maintenance duration 𝑑𝑑(𝑎𝑎𝑚𝑚) ∀𝑎𝑎𝑚𝑚 and production loss cost 𝑐𝑐ℓ,𝑘𝑘 ∀𝑘𝑘;

• the optimal cost-to-go 𝑉𝑉𝑘𝑘(𝑒𝑒𝑚𝑚) ∀𝑒𝑒𝑚𝑚,𝑘𝑘.

Figure 4.4 shows the details of the simulation procedure for a single simulation.


Figure 4.4. Simulation flow chart.

After running a series of simulations, the average cost is computed:

𝐶𝐶𝑐𝑐𝑒𝑒𝑡𝑡�� =∑ 𝐶𝐶𝑐𝑐𝑒𝑒𝑡𝑡(𝑒𝑒)𝑟𝑟

Number of repetitions

4.4 NUMERICAL EXAMPLE

This section discusses a case study to demonstrate the effectiveness of the proposed policy. An

optimal CB-OM policy for a finite time horizon of about five years (260 weeks) is developed

to maintain the main bearing of a hypothetical wind turbine located at Mount Emerald

windfarm, Queensland, Australia. The considered external opportunities are days when the

average wind speed is less than the cut-in threshold of the wind turbine. At this time, the wind


turbine is not functioning, and it creates the chance to do preventive maintenance if necessary.

Therefore, the wind data of Mount Emerald and electricity prices of Queensland are used to

develop the opportunity model and production loss cost of the wind turbine. The developed

OM policy for a wind turbine can be considered as a basic maintenance policy to select the

candidates (possible maintained wind turbines) for another grouping maintenance policy where

some wind turbines can be maintained together as mentioned in some literature (Erguido et al.,

2017; Zhang, Gao, et al., 2016; Zhang and Zeng, 2017). It is noted that the wind turbines also

stop when the wind speed is higher than the cut-out threshold and it also creates an opportunity

for maintenance; however, this situation is not considered because it is not safe and convenient

to do maintenance in strong wind conditions.

4.4.1 Input data

The optimal policy is developed for a horizon of 𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑 = 260 weeks (five years) with a time

epoch of ∆𝑡𝑡 = 1 week.

4.4.1.1 Degradation and maintenance models

Degradation of bearings in wind turbines can be monitored based on some condition signals

acquisition via sensors such as vibration and acoustic emission (García Márquez et al., 2012).

Then the collected data is analysed through some signal processing methods (e.g. Fast-Fourier

transform, amplitude demodulation, etc.) to obtain the degradation path of the investigated

main bearing. In this illustrative example, the degradation path of the main bearing proposed

in the study of Zhu (2012) is used to develop the transition matrix. The degradation path in this

study was modelled as the Lundberg–Palmgren formula:

𝑋𝑋(𝑡𝑡,𝜙𝜙1,𝜙𝜙2,𝜃𝜃) = 𝜙𝜙1 + 𝜃𝜃 ⋅ 𝑡𝑡𝜙𝜙2 (4.23)

where the lifetime 𝑡𝑡 represents the number of days of system operation. The positive random

parameter 𝜃𝜃 represents the wind load and is assumed to follow a Weibull distribution with two

parameters, scale 𝜇𝜇 and shape 𝜎𝜎, which can be obtained by data fitting. The constant parameter

𝜙𝜙1 is the initial degradation level, and 𝜙𝜙2 is a given constant that depends on the type of

bearing. The degradation was also limited by a failure threshold 𝐻𝐻.

In the study of Zhu (2012), the parameters were set as following: the wind load 𝜃𝜃 followed a

Weibull distribution with scale 𝜇𝜇 = 2.12 and shape 𝜎𝜎 = 7.9; the initial degradation level 𝜙𝜙1 =

1; the type parameter of the bearing 𝜙𝜙2 = 0.33; and the maximum (failure) threshold of


degradation 𝐻𝐻 = 10. Using the above parameters for the degradation path and Monte Carlo

simulation, a weekly transition matrix is developed for 17 discrete states described in the

following Table 4.1.

Table 4.1

Discrete States of Main Bearing Degradation

State Degradation level (𝑿𝑿)

𝒎𝒎 = 𝟏𝟏 𝑋𝑋 = 1 (original)

𝒎𝒎 = 𝟐𝟐 1 < 𝑋𝑋 ≤ 4

𝒎𝒎 = 𝟑𝟑 4 < 𝑋𝑋 ≤ 5

𝒎𝒎 = 𝟒𝟒…𝟗𝟗 0.5𝑖𝑖 + 3 < 𝑋𝑋 ≤ 0.5𝑖𝑖 + 3.5

𝒎𝒎 = 𝟏𝟏𝟎𝟎…𝟏𝟏𝟏𝟏 0.25𝑖𝑖 + 5.5 < 𝑋𝑋 ≤ 0.5𝑖𝑖 + 5.75

A comparison of the results of the degradation path in the study by Zhu (2012) and those in

this study is shown in Figure 4.5. The results indicate that it can obtain an approximately

consistent degradation path within 95% confident interval.

Figure 4.5. Representative example of weekly degradation in the main bearing of the wind turbine.


In paper of Zhu (2012), the author considered only one perfect PM action with cost 7000 euros

(≈10,862 AUD). In this study, it is proposed that there are two PM actions: minor and major

ones. The major PM, denoted as 𝑎𝑎2, is a perfect maintenance and costs the same amount of

10,862 AUD. It is also assumed it takes three days to conduct this major PM and yield some

production loss cost. The minor PM action, denoted as 𝑎𝑎1, is an imperfect one which restores

the degradation level of system to some value between the best one (e.g. 1) and the failure

threshold7. This study assumes that the restored degradation level is 8; the cost is half of a

major PM; and the duration for conducting minor PM is set at two days. The CM cost of 30,000

euros (≈ 46,723 AUD) in paper of Zhu (2012) is also used in this study as the failure cost.

Hence, the state cost 𝑐𝑐𝑙𝑙,𝑘𝑘 is defined as:

𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒) = �46,723 AUD 𝑒𝑒 = 170 otherwise

The failure cost, maintenance actions and other information is summarised in Table 4.2.

Table 4.2

Failure and Maintenance Parameters

Parameters Maintenance action 𝒎𝒎𝟏𝟏 Maintenance action 𝒎𝒎𝟐𝟐

Restoration level 8 1

Direct cost 𝒄𝒄𝒃𝒃,𝒌𝒌 5431 AUD 10,862 AUD

Maintenance duration 2 3

Failure cost (last states) 46,723 AUD

Number of degradation

states 17

Number of time epochs 𝑲𝑲 260 weeks (5 years)

Time interval length 𝚫𝚫𝒕𝒕 1

4.4.1.2 Wind model

In this case study, the days in the week with average wind speed less than cut-in threshold are

considered as opportunities. If there are 𝑛𝑛 consecutive days with lower cut-in wind speed, it

7 Note that the action 𝑎𝑎0 is doing nothing


means that there is an opportunity with duration of 𝐷𝐷 = 𝑛𝑛 days. Hence, in order to develop the

opportunity model for this windfarm in Mount Emerald, the wind speed model is first studied.

The data on the wind speed distribution at the Mount Emerald site can be found in a report

prepared by Parsons Brinckerhoff, Australia (Mount Emerald Wind Resource and Energy Yield

Assessment, 2012). The wind speed at the case study location is fitted to the Weibull

distribution with shape of 𝛽𝛽 = 2.11 and scale of 𝛼𝛼 = 10.1 (m/s). However, the author also

mentioned that the average wind speed varies monthly within a given year. Therefore, in this

case study, it is supposed that the wind speed follows different Weibull distributions according

to different months of the year. The shape parameter 𝛽𝛽 = 2.11 is maintained and the scale

parameters 𝛼𝛼𝑘𝑘 (𝑘𝑘 = 1 … 12) are determined again according to different mean wind speeds

(Table 4.3).

Table 4.3

The Scale Parameters of Weibull Distribution for Wind Speed at Mount Emerald, Queensland, Australia

Month Average wind

speed (m/s)

Scale

parameters Month

Average wind

speed (m/s)

Scale

parameters

1 6.3 7.113 7 11.6 13.098

2 8.0 9.033 8 9.2 10.388

3 7.2 8.129 9 10.3 11.630

4 11.2 12.646 10 9.8 11.065

5 10.3 11.630 11 9.1 10.275

6 9.4 10.614 12 6.6 7.452

As mentioned above, for each time interval (week), an opportunity occurs if there is at least

one day with average wind speed lower than the cut-in value. According to Mount Emerald

Wind Resource and Energy Yield Assessment 2012), the wind turbines used in this study

produce a small amount of power when it is 4 m/s. Hence, it is reasonable to set the cut-in

threshold of 3.5 m/s. The probability that a week day in a certain week 𝑘𝑘 is an opportunity can

be computed as 𝑝𝑝𝑘𝑘 = Pr(𝑊𝑊𝑆𝑆 ≤ 3.5), where 𝑊𝑊𝑆𝑆 follows the Weibull distribution with

parameters according to the month that week 𝑘𝑘 belongs. From this probability of opportunity

occurrence for each week day, the probabilities of opportunity occurrence for each week can

be determined (i.e. probability that at least one week day has wind speed lower than cut-in

threshold):


𝑝𝑝𝑘𝑘𝑂𝑂 = 1 − (1 − 𝑝𝑝𝑘𝑘)7

The probabilities of opportunity occurrences for 52 weeks are presented in Figure 4.6. The

opportunity is stated lasting for 𝑛𝑛 days if there are 𝑛𝑛 consecutive days with wind speed lower

than the cut-in threshold counted from the first day with low wind speed. With this definition,

the discrete distributions of opportunity durations can be determined by enumeration. Figure

4.7 shows the distribution of possible opportunity durations for each month.

Figure 4.6. Annually probability of Opportunity occurrences (at least one week day has average wind speed lower than 3.5 m/s).

Figure 4.7. Discrete distributions of opportunity durations.

Distributions of opportunity durations

1 2 3 4 5 6 7 8 9 10 11 12

Month

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abilit

y

D = 1 day

D = 2 days

D = 3 days

D = 4 days

D = 5 days

D = 6 days

D = 7 days


It is noted that the wind speed depends on some other weather factors. Hence, using some data

mining techniques to integrate all these factors should be the best way to estimate the wind

speed. However, the limited data cannot support for following this approach.

4.4.1.3 Downtime cost rate

When conducting the required PM activity, the wind turbine must be stopped for the

maintenance duration, which causes some production loss. The downtime (production loss)

cost per time unit (day) depends on the electricity price and average wind speed, which affects

the possible produced electricity output. These factors are stochastic and time-varying.

For the electricity prices, 11 years of prices from Queensland were synchronously averaged

and subsequently smoothed with a sliding window with a width of 31 days (see Section 3.2 in

paper of Truong Ba, Cholette, Wang, et al. (2017)) to determine the average electricity prices

for each week. The resulting weekly average electricity prices are presented in Figure 4.8.

Combining the average wind speed discussed above and the power output of the particular

wind turbine (REpower 3.4-104 WTG at air density 1.09 kg/m3 (Mount Emerald Wind

Resource and Energy Yield Assessment, 2012)) with the electricity efficiency loss of 4%, the

average daily power generation of the wind turbine can be calculated and is shown in Figure

4.9. Finally, the downtime costs are computed as the multiplication between the electricity

price and possible power generation Figure 4.10.

Figure 4.8. Weekly average electricity price in Queensland – Australia.


Figure 4.9. Average daily possible electricity generation of wind turbine REpower 3.4-104 WTG at air density of 1.09 kg/m3.

Figure 4.10. Average downtime cost per day.


4.4.2 Optimal maintenance policy

The computation time for both optimisation and simulation is about 1 hour (computer with

CPU core i7-4790 @3.6GHz, 16 GB Ram). The optimal policy is shown in Figure 4.11.

Figure 4.11. Optimal Maintenance policy.

The white zone represents the doing nothing (decision 𝑎𝑎0) and the blue zone represents the

degradation states where an arrived opportunity is considered. The red zone at the top is

associated with the perfect PM and the green represents the minor (imperfect) PM. In each time

instant, the decision maker will select PM as well as maintenance type or do-nothing according

to the current state of the system. If the do-nothing is chosen for PM (white and blue zones),

maintenance is conducted only if a suitable opportunity occurs (blue zone).

Examining the policy, it can be see that the OM is less preferable in the beginning and the end

of each year. These periods correspond to the time of low-production loss cost (see Figure

4.10). Besides, the OM option is also not favourable in the middle of the year (weeks 29-31

each year) which is caused by the lower chance of suitable opportunities (high wind speed) in

both occurrences and durations on this periods (see Figure 4.6).


Figure 4.11 also shows some indicative time slices (e.g. 25, 34 and 160) to illustrate the OM

consideration zone. For each of these time slices, the criteria for OM acceptance are shown in

a separate plot with the opportunity duration on the horizontal axis and the degradation state

on the vertical axis. It can be seen from these slices that the duration (which creates the cost

savings) influences the selection of the action for low cost savings. It is interesting to note that

certain OM actions are not economical for short opportunity duration but should be considered

when the duration is long (e.g. OM Type 2 at time 25 and 160).

The probabilities with which OM or PM are conducted are presented in Figure 4.12. The figure

shows that probability of PM is especially preferred in week 12, 25 and 34. It is equivalent with

the policy shown in Figure 4.11 that in these weeks, the PM can be conducted with lower

degradation level (9.5). OM is preferable when the downtime cost is low (periods 11-12) and

less favourable in high downtime cost periods (26-28). In the end and beginning of years, the

probability of OM is still high even though the production loss cost is low because the chance

of an opportunity is larger in these periods (Figure 4.6).

Figure 4.12. Probability of conducting OM and PM (average for 1 year).

4.4.3 Sensitivity analysis

4.4.3.1 Effect of failure cost

In this section, a series of different costs of system failure (last state) are simulated to evaluate

the benefit of joint PM-OM policy in comparison to a PM-only policy (other parameters are


kept constant). The policy of joint PM and OM is investigated in two situations, respectively

considering partial opportunities and only full opportunities.

Table 4.4 shows the saving percentages of maintenance policy of joint OM and PM to the

typical CBM policy (policy with PM-only).

Table 4.4

Saving Rates of Joint OM and PM Policies when Failure Cost Varies

Failure Cost (% compared to default value)

25% 50% 100% 200% 300%

Only full

opportunities 1.6% 1.2% 1.2% 0.7% 0.7%

Partial

opportunities 7.9% 8.6% 8.6% 6.0% 5.1%

The joint OM and PM policies yield some savings compared to the policy with PM-only. The

percentage of savings reduces when the failure cost is very high. The reason is when the failure

cost is high, PM is preferable compared to OM. Moreover, the results also show that the OM

policy considering partial opportunities provides a significant benefit in respect to the policy

with full opportunities only. The reason is that considering partial opportunities gives more

chance for maintenance action type 2 (perfect maintenance) as shown in Figure 4.13, which

has a higher restoration (renewal).


Figure 4.13. Probabilities of maintenance types with full OM and partial OM policies.

4.4.3.2 Effect of downtime cost

Similar to the previous section, this section investigates the savings of joint OM and PM

policies to maintenance strategy without OM as a function of downtime cost.

Table 4.5

Saving Rates of Joint OM and PM Policies when Downtime Cost Varies

Production Loss Cost

25% 50% 100% 200% 300%

Only full opportunities 0.0% 0.2% 1.3% 2.2% 3.8%

Partial opportunities 0.1% 2.5% 8.5% 16.2% 21.7%

When the downtime cost 𝑐𝑐ℓ,𝑘𝑘 increases, the benefits of joint OM and PM policies are more

evident. This is reasonable since the ultimate purpose of considering OM is to save downtime

cost. Moreover, the consideration of partial opportunities provides more benefit than the policy

with full opportunities only.

Figure 4.14 shows the preference of OM versus PM when the production loss cost changes.

When the production loss cost is high enough, the percentage of OM events is larger than PM

in case a partial OM policy is applied. OM is more favourable when the downtime cost

increases. Comparing the policies’ partial and only full opportunities, the chances of OM in the

policy with partial opportunities is higher than the other, as expected since more opportunities

are available for consideration.


Figure 4.14. Ratios between probabilities of OM and PM with full OM and partial OM policies.

4.4.3.3 Effect of wind threshold defining opportunity occurrences

In this case study, the cut-in wind speed is considered as the threshold for creating

opportunities, i.e. when the wind turbine is completely stopped. However, the opportunity wind

speed threshold may be set higher to make more opportunities for doing maintenance. There is

certainly some penalty cost for doing OM if the opportunity threshold is higher. This penalty

cost is caused by the production loss when the wind speed is less than opportunity threshold

but higher than cut-in value. Let 𝑒𝑒𝑂𝑂 and 𝑒𝑒𝑐𝑐𝑚𝑚 be the wind speed threshold for opportunities and

cut-in value, respectively. The average penalty cost per maintenance time unit (day) of doing

OM is computed as:

𝑐𝑐𝑒𝑒𝑝𝑝𝑒𝑒𝑚𝑚,𝑘𝑘 = 𝐸𝐸𝑃𝑃𝑘𝑘 ⋅ � 𝑃𝑃𝑂𝑂𝑊𝑊(𝑒𝑒) ⋅ 𝑓𝑓𝑘𝑘(𝑒𝑒)𝑑𝑑𝑒𝑒

𝑤𝑤𝑂𝑂

𝑤𝑤𝑐𝑐𝑖𝑖

(4.24)

where 𝐸𝐸𝑃𝑃𝑘𝑘 is the average electricity price per kWh per day at time epoch 𝑘𝑘; 𝑃𝑃𝑂𝑂𝑊𝑊(𝑒𝑒) is the

power generating function according to wind speed; and 𝑓𝑓𝑘𝑘(𝑒𝑒) is the p.d.f of wind speed

distribution (Weibull distribution) at time epoch 𝑘𝑘.

Figure 4.15 shows the saving percentages of OM policies when the wind speed threshold is

varied. When the threshold is larger, the saving is more significant. This is likely due to the

fact that more opportunities can be created, and their duration is likely longer if the wind speed

threshold is set high. However, if the wind speed threshold is set too high, the penalty cost now


is significant and makes OM less beneficial. Moreover, when the opportunity threshold

increases, the gap between savings of two policies (i.e. full and partial OM) is smaller. It is

because the chance of long opportunity durations is also dramatically increased. The optimal

threshold is about 8 m/s for policy with partial opportunities and 9 m/s for policy with full

opportunities only.

Figure 4.15. Savings of OM policies with different wind speed thresholds creating maintenance opportunities.

4.5 CONCLUSION

This study has proposed a novel optimal condition-based maintenance strategy considering

partial opportunities for maintenance over a finite horizon. In addition to the CBM approach,

this study includes the analysis of a set of imperfect maintenance actions. The mission total

maintenance cost was minimised through a Markov Decision Process where system

degradation thresholds for different PM actions as well as optimal OM selection were jointly

optimised.

A numerical case study was conducted for a hypothetical wind turbine at Mount Emerald wind

farm, Queensland, Australia. Numerical results showed that the maintenance policy

considering partial opportunities would yield significant savings in maintenance cost in

comparison with the traditional CBM policy as well as in respect to the full-opportunity only

CB-OM. In particular, the simulation results confirm the intuitive hypothesis that savings are

more significant when production loss costs are dominant.

86 Chapter 5: Joint Opportunistic and Preventive Maintenance policy: Integrating with opportunity forecasts

Chapter 5: Joint Opportunistic and Preventive Maintenance policy: Integrating with opportunity forecasts

Overview

This chapter discusses an enhancement of opportunistic maintenance where more accurate

short-term forecasts of external opportunities is available. The maintenance optimisation

problem is formulated in this study as a finite-horizon Markov Decision Process, where the

randomly occurring opportunities and their forecasts are accounted for by augmenting the time-

varying, decision-dependent transition probabilities. A Dynamic Programming approach is

used to obtain the optimal CBM policy, consisting of time-varying thresholds on equipment

condition, the cost of conducting maintenance, and predicted opportunity characteristics

(arrival rates and durations). A numerical study is undertaken to evaluate the benefits of the

inclusion of short-term forecasts in OM decisions

Chapter 5: Joint Opportunistic and Preventive Maintenance policy: Integrating with opportunity forecasts 87

Nomenclature

Indices:

𝑖𝑖, 𝑗𝑗: indices of states, 𝑖𝑖, 𝑗𝑗 = 1,2, … ,𝑁𝑁.

𝑚𝑚: index of maintenance actions, 𝑚𝑚 = 0,1, … ,𝑀𝑀.

𝑘𝑘: index of time period (decision epochs), 𝑘𝑘 = 1,2, … ,𝐾𝐾.

Variables:

𝑆𝑆𝑘𝑘: degradation state of system at time 𝑡𝑡𝑘𝑘.

𝑒𝑒𝑚𝑚, 𝑒𝑒𝑗𝑗: a discrete state of system

𝑎𝑎𝑚𝑚: a maintenance action.

𝐷𝐷�𝑘𝑘: forecast opportunity duration.

𝑂𝑂�𝑘𝑘: forecast opportunity occurrence

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚�: transition probability of degradation from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 after time interval

(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] given no maintenance action is conducted.

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚�: PM-only transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 after time interval

(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] given maintenance action 𝑎𝑎𝑚𝑚 is conducted.

𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚�: joint PM and OM transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 after time

interval (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] given decision 𝑎𝑎𝑚𝑚 is selected.

Parameters:

𝑑𝑑(𝑎𝑎𝑚𝑚): duration of doing maintenance action 𝑎𝑎𝑚𝑚.

𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚): cost corresponding to state 𝑒𝑒𝑚𝑚 at time epoch 𝑡𝑡𝑘𝑘.

𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚): direct (basic) maintenance cost corresponding to action 𝑎𝑎𝑚𝑚 at time epoch 𝑡𝑡𝑘𝑘.

𝑐𝑐ℓ,𝑘𝑘: downtime cost rate (per unit of stopping time) at time epoch 𝑡𝑡𝑘𝑘.

𝜆𝜆𝑂𝑂(𝑡𝑡): time-varying opportunity occurrence intensity.


5.1 INTRODUCTION

The previous chapters have developed a joint PM and OM policy considering the more realistic

case of partial opportunities. In that time-based model, decision makers decide to do OM when

an opportunity occurs according to its duration. If the opportunity duration is larger than a time-

varying predetermined threshold, OM will be conducted. This means that decision makers have

perfect information of opportunity duration when it arrives. However, in practice, decision

makers never have perfect knowledge about the opportunities (occurrences, cost benefits, etc.)

and must use forecasts, especially in the short-term, to make decisions.

This chapter aims to extend the previous study in developing a joint OM and PM policy

considering the forecasts of opportunity characteristics for OM decision process in both

approaches: time-based and condition-based. As before, the opportunities are external events

that stop the production systems and create the chance for doing maintenance (e.g. rain days

stop the sugarcane processing system; the low-wind days stop the wind turbines). The benefit

of these opportunities is the saving of the downtime cost of doing maintenance, which depend

on the duration of opportunities. The maintenance decisions, PM and OM, consider both the

asset condition and the estimated economic benefit (duration) of the opportunity.

The remainder of the chapter is organised as follows: Section 5.2 describes the policy and

optimisation model in MDP approach. The detailed discussion of the age-based and condition-

based approaches is presented in section 5.3. The evaluation of the identified optimum, based

on a Monte Carlo simulation procedure, is presented in Section 5.4. The benefits of proposed

policy are discussed through numerical examples presented in section 5.5. Finally, the last

section briefly summarises the key findings of this study.

5.2 OPTIMISATION MODEL

5.2.1 Policy description

The aim of this study is to develop a dynamic planning strategy in a predefined time horizon

mission 𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑 where the decisions are based on the age of system in time-based approach or the

condition of system condition-based approach and the predicted characteristics of opportunities

(i.e. occurrence, duration) in each planning interval. At the beginning of each time interval, the

degradation (or age) of system is detected, and forecasts of potential opportunities are collected.

According the current state of system and information of opportunities, the decision makers


will consider options of doing PM immediately, preparing for OM or doing nothing. Doing

nothing should be selected if the system is still in good condition. In contrast, if the degradation

(or age) of system is over some predefined thresholds, the PM is immediately conducted. In

case, the system condition is degraded enough but not over the PM threshold, OM will be

considered according to the benefit of an arrived opportunity. The opportunities considered in

this chapter are also the external ones that yield some maintenance cost deduction. The

difference of this model to the previous one is that decision makers have no knowledge about

the arrived opportunities but have some short-term predicted information and plan the OM

according to this information.

In this chapter, the total OM policy is also developed based on the MDP approach. The timeline

is discretised into 𝐾𝐾 intervals (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] with an equal and sufficiently small length ∆𝑡𝑡 where

𝑘𝑘 = 1,2, … ,𝐾𝐾 and 𝑡𝑡1 = 0 , 𝑡𝑡𝐾𝐾+1 = 𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑. There are some additional assumptions in the

developed model:

• The imperfect maintenance actions are also considered in this study, which restore

the condition or age of system to a certain point according to restoration levels;

• The state (degradation or age) of system is perfectly detected;

• The occurrences and durations of opportunities are independent.

At the beginning of each time interval (𝒕𝒕𝒌𝒌, 𝒕𝒕𝒌𝒌+𝟏𝟏], decision makers will make maintenance

decisions based on the system state 𝑺𝑺𝒌𝒌 and the opportunity information of occurrence chances

and durations of 𝑳𝑳 periods. A decision of doing nothing, doing PM or planning for OM is

selected for the current time interval 𝒌𝒌 (Figure 5.1):

• Doing nothing (𝑎𝑎0): No maintenance action occurs (even opportunities arrive) and

the states of system are changed according to the degradation process during the time

interval.

• Doing PM: A maintenance action 𝑎𝑎𝑚𝑚 ≠ 𝑎𝑎0 is selected for PM. The states of system

are changed according to the recovering level of maintenance action 𝑎𝑎𝑚𝑚.

• Planning OM: A maintenance action 𝑎𝑎𝑚𝑚 ≠ 𝑎𝑎0 is selected for predicted opportunity.

The states of system are changed according to the recovering level of maintenance

action 𝑎𝑎𝑚𝑚 if opportunity occurs. If opportunity does not arrive, the states of system

are changed according to the degradation process during the time interval.


Figure 5.1. The maintenance options for each time interval: a) Doing nothing; b) Doing PM; and c) Planning OM.

The optimisation model is developed based on MDP approach. The optimisation must

determine the optimal maintenance decision (doing nothing, PM or OM) in current time

interval according to the available short-term forecasts of opportunities. In each time interval

(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1], the policy is re-optimised to determine the maintenance decision 𝑃𝑃𝑘𝑘∗ (𝑆𝑆𝑘𝑘,𝜙𝜙𝑘𝑘) with

updated short-term information 𝜙𝜙𝑘𝑘 and current state 𝑆𝑆𝑘𝑘. The forecasts are available in time

window (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+𝐿𝐿−1] only; however, the optimal policy is developed for whole 𝐾𝐾 time intervals

from time epoch 𝑘𝑘. Therefore, the long-term information (historical data) is used for remaining

time intervals without forecasts (𝑡𝑡𝑘𝑘+𝐿𝐿 , 𝑡𝑡𝐾𝐾] to determine the optimal cost-to-go 𝑉𝑉ℓ𝑙𝑙𝑙𝑙𝑚𝑚𝑙𝑙(𝑆𝑆ℓ),ℓ =

𝑘𝑘 + 𝐿𝐿, … ,𝐾𝐾; and the short-term information is used to compute the cost-to-go of intervals in

the forecast window, 𝑉𝑉ℓ𝑙𝑙ℎ𝑙𝑙𝑟𝑟𝑡𝑡(𝑆𝑆ℓ),ℓ = 𝑘𝑘, … ,𝑘𝑘 + 𝐿𝐿 − 1 (Figure 5.2).


Figure 5.2. The optimisation approach for each time interval.

5.2.2 Degradation model

The discrete states of system are denoted as 𝑆𝑆𝑘𝑘 ∈ 𝒮𝒮 = {𝑒𝑒1, 𝑒𝑒2, … }, which is the age or

degradation of system according to age- or condition- based approaches respectively. The

transition probabilities of states 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� after each time epoch is defined as:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� = Pr�𝑆𝑆𝑘𝑘+1 = 𝑒𝑒𝑗𝑗|𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚 , no maintenance action� ∀𝑘𝑘 (5.1)

Denote ℙ𝑘𝑘,0 = �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚�� as the complete one-step transition matrix of all probabilities

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚�. The details of transition probability determination for age- and condition- based

approach will be discussed in section 5.3.

5.2.3 Maintenance actions

There are 𝑀𝑀 + 1 possible maintenance actions with different restoration levels:

ℳ = {𝑎𝑎0,𝑎𝑎1, … ,𝑎𝑎𝑚𝑚, … ,𝑎𝑎𝑃𝑃}

The one-step transition probability of maintenance type 𝑎𝑎𝑚𝑚 is computed:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� = Pr�𝑆𝑆𝑘𝑘+1 = 𝑒𝑒𝑗𝑗|𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚,𝑃𝑃𝑘𝑘 = 𝑎𝑎𝑚𝑚 � ∀𝑘𝑘 (5.2)

Let ℙ𝑘𝑘,𝑚𝑚 = �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚 ,𝑎𝑎𝑚𝑚�� be the complete one-step transition matrix of all probabilities. In

the case where the maintenance action 𝑎𝑎0 (doing nothing) is selected, the transition probability

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚,𝑎𝑎0� ≡ 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� is defined as Eq. (5.1).


5.2.4 Cost structure

The cost categories also include a state cost 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚), maintenance direct cost 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) and a

variable cost 𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚). The PM cost and OM cost (if OM is conducted) of maintenance action

𝑎𝑎𝑚𝑚 are also defined as:

PM: 𝐶𝐶𝑘𝑘𝑃𝑃𝑃𝑃(𝑎𝑎𝑚𝑚) = 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) (5.3)

OM: 𝐶𝐶𝑘𝑘𝑂𝑂𝑃𝑃(𝑎𝑎𝑚𝑚) = 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + �𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) − 𝐶𝐶𝑘𝑘𝑂𝑂�+

(5.4)

where: 𝐶𝐶𝑘𝑘𝑂𝑂 is the random saving cost due to conducting OM. The term [𝑥𝑥]+ is defined as [𝑥𝑥]+ =

max[0, 𝑥𝑥]. In this study, the variable cost 𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) is defined as the downtime cost of doing

maintenance and the saving cost of OM, 𝐶𝐶𝑘𝑘𝑂𝑂 is caused by the random duration of opportunities:

𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) = 𝑐𝑐ℓ,𝑘𝑘 ⋅ 𝑑𝑑(𝑎𝑎𝑚𝑚) (5.5)

and: �𝑐𝑐𝑣𝑣,𝑘𝑘(𝑎𝑎𝑚𝑚) − 𝐶𝐶𝑘𝑘𝑂𝑂�+

= 𝑐𝑐ℓ,𝑘𝑘 ⋅ [𝑑𝑑(𝑎𝑎𝑚𝑚) − 𝐷𝐷𝑘𝑘𝑂𝑂]+ (5.6)

Where: 𝑐𝑐ℓ,𝑘𝑘 is the production loss cost per unit of time and 𝑑𝑑(𝑎𝑎𝑚𝑚) is the duration of action 𝑎𝑎𝑚𝑚.

𝐷𝐷𝑘𝑘𝑂𝑂 is the random opportunity duration following the time-varying distribution 𝒟𝒟𝑘𝑘𝑙𝑙𝑝𝑝𝑡𝑡 with p.d.f

and c.d.f of 𝑓𝑓𝑙𝑙,𝑘𝑘(𝐷𝐷𝑘𝑘𝑂𝑂) and 𝐹𝐹𝑙𝑙,𝑘𝑘(𝐷𝐷𝑘𝑘𝑂𝑂) respectively. The opportunity duration 𝐷𝐷𝑘𝑘𝑂𝑂 is estimated as

𝐷𝐷�𝑘𝑘 through a forecasting process discussed in Section 5.2.6.

5.2.5 Opportunity model

The opportunity occurrences are also assumed to follow a Non-homogenous Poisson process

(NHPP) with intensity 𝜆𝜆(𝑡𝑡). In this model, it is assumed that there is at most one opportunity

that arrives in every short interval 𝑘𝑘. Thus, the probability of opportunity occurrence (in long-

term) is defined as:

𝑝𝑝𝑘𝑘𝐿𝐿𝑂𝑂 = 1 − 𝑒𝑒−Λ(𝑡𝑡𝑘𝑘,𝑡𝑡𝑘𝑘+1) (5.7)

where: Λ�𝑡𝑡𝑙𝑙, 𝑡𝑡𝑓𝑓� = ∫ 𝜆𝜆(𝜏𝜏)𝑑𝑑𝜏𝜏𝑡𝑡𝑓𝑓𝑡𝑡𝑠𝑠

.

Each opportunity has its random duration 𝐷𝐷𝑘𝑘𝑂𝑂 which follows some time-varying distribution

𝒟𝒟𝑘𝑘𝑙𝑙𝑝𝑝𝑡𝑡 with p.d.f and c.d.f are 𝑓𝑓𝑙𝑙,𝑘𝑘(𝐷𝐷𝑘𝑘𝑂𝑂) and 𝐹𝐹𝑙𝑙,𝑘𝑘(𝐷𝐷𝑘𝑘𝑂𝑂) respectively.


5.2.6 Opportunity forecast

The properties of opportunities (i.e. occurrences and durations) are predicted in a short-term

horizon with some forecasting models. Let 𝑋𝑋𝑘𝑘 and 𝑋𝑋�𝑘𝑘 be the actual and estimated (predicted)

quantities of a certain opportunity measurement respectively. The forecasting model 𝑓𝑓 has the

(potentially autoregressive) form:

𝑋𝑋�𝑘𝑘+1 = 𝑓𝑓([𝑋𝑋𝑘𝑘 , 𝐼𝐼𝑘𝑘], [𝑋𝑋𝑘𝑘−1, 𝐼𝐼𝑘𝑘−1], … )

where 𝐼𝐼𝑘𝑘 is the information (or observations) of other influential factors at time 𝑡𝑡𝑘𝑘. This means

that the predicted value at current time depends on the actual values in past and other necessary

information for the forecasting process. For example, the rain forecasts for near-future are

based on the current (and past) rain events and observation of other meteorological variables

via weather satellite. The forecasting processes of opportunity characteristics are usually

complicated and specific for each type of considered quantities. However, the forecasting

procedure is out of scope of this study because it is not necessary for maintenance decision

makers. Therefore, instead of being concerned with the prediction procedure, this study

considers the accuracy of forecasted quantities which can be described as the conditional

probability distribution Pr�𝑋𝑋�𝑘𝑘|𝑋𝑋𝑘𝑘�. In general, the accuracy can be described as a certain

distribution 𝒟𝒟𝑚𝑚𝑐𝑐𝑐𝑐𝑢𝑢𝑟𝑟𝑚𝑚𝑐𝑐𝑎𝑎 and one can write:

𝑋𝑋�𝑘𝑘~𝒟𝒟𝑚𝑚𝑐𝑐𝑐𝑐𝑢𝑢𝑟𝑟𝑚𝑚𝑐𝑐𝑎𝑎(𝑋𝑋𝑘𝑘,𝛔𝛔 )

where 𝛔𝛔 is a set of distribution parameters. This distribution can be developed easily if

historical data of actual quantities and associated forecasting ones are available. The following

part discusses the accuracy distributions for concerned opportunity characteristics in this study:

durations and occurrences.

The forecast opportunity duration 𝐷𝐷�𝑘𝑘 follows some distribution that the actual value as the

mean (i.e. the forecasts error has mean zero):

𝐷𝐷�𝑘𝑘~𝒟𝒟𝑘𝑘𝑓𝑓𝑐𝑐�𝐷𝐷𝑘𝑘𝑚𝑚𝑐𝑐𝑡𝑡𝑢𝑢𝑚𝑚𝑙𝑙 ,𝛔𝛔𝑑𝑑� (5.8)

where 𝛔𝛔𝑑𝑑 is the set of parameters.

The actual occurrence of opportunity, denoted as 𝑂𝑂𝑘𝑘, is a binary random variable, 𝑂𝑂𝑘𝑘 ∈ {0,1}

where 𝑂𝑂𝑘𝑘 = 1 means opportunity arrives and 𝑂𝑂𝑘𝑘 = 0 otherwise. The forecast 𝑂𝑂�𝑘𝑘 therefore is

not 0 and 1 but randomly between [0,1] where 𝑂𝑂�𝑘𝑘 tends to close to 1 if the opportunity actually

occurs and 0 if it does not. Hence, 𝑂𝑂�𝑘𝑘 possibly follows either two accuracy distributions 𝒪𝒪(𝛔𝛔𝑙𝑙)


or 𝒪𝒪′(𝛔𝛔𝑙𝑙′ ) given the opportunity arrives or not, respectively. Since 𝑂𝑂�𝑘𝑘 has the properties of

probability (i.e. 𝑂𝑂�𝑘𝑘 ∈ [0,1]), it is convenient to consider 𝑂𝑂�𝑘𝑘 as “chance” of opportunity

occurrence:

𝑂𝑂�𝑘𝑘~𝒪𝒪(𝛔𝛔𝑙𝑙) if 𝑂𝑂𝑘𝑘 = 1

𝑂𝑂�𝑘𝑘~𝒪𝒪′(𝛔𝛔𝑙𝑙′ ) if 𝑂𝑂𝑘𝑘 = 0 (5.9)

These distributions 𝒪𝒪(𝛔𝛔𝑙𝑙) and 𝒪𝒪′(𝛔𝛔𝑘𝑘′ ) can be continuous or discrete ones and be determined

with available historical data.

5.2.7 Optimisation model in MDP approach

This model aims to minimise the expected total cost for whole time horizon. Let

min𝜋𝜋

�𝔼𝔼�𝐶𝐶𝑘𝑘�𝑒𝑒𝑚𝑚, 𝜇𝜇𝑘𝑘(𝑒𝑒𝑚𝑚)��𝐾𝐾

𝑘𝑘=1

(5.10)

where 𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚 denotes a discretised degradation state, 𝑒𝑒𝑚𝑚 ∈ 𝒮𝒮; and 𝜋𝜋 = {𝜇𝜇1, 𝜇𝜇2, … , 𝜇𝜇𝐾𝐾} is called

the maintenance policy, which is a series of functions 𝜇𝜇𝑘𝑘:𝒮𝒮 ⟼ℳ at map the degradation state

into a maintenance action at epoch 𝑘𝑘.

The minimisation of (5.10) subject to these Markov chain dynamics falls under the well-


induction (Puterman, 2014).

𝑉𝑉𝑘𝑘(𝑒𝑒𝑚𝑚) = min𝑚𝑚𝑚𝑚∈ℳ


�

𝜇𝜇𝑘𝑘∗(𝑒𝑒𝑚𝑚) = argmin𝑚𝑚𝑚𝑚∈ℳ


�

(5.11)

where 𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� is transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 with decision 𝑎𝑎𝑚𝑚; 𝑉𝑉𝑘𝑘(𝑒𝑒𝑚𝑚)

as the expected “cost-to-go” when the reflectivity loss is 𝑒𝑒𝑚𝑚 and 𝑉𝑉𝐾𝐾+1(𝑒𝑒𝑚𝑚) = 𝑐𝑐𝑙𝑙,1(𝑒𝑒𝑚𝑚) by

definition. The optimal policy is then 𝜋𝜋∗ = (𝜇𝜇1∗, … , 𝜇𝜇𝐾𝐾∗ ).

Therefore, when the transition probabilities are determined and 𝔼𝔼[𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚)] is computed, the

optimisation is clear. These terms depend on the decision of doing nothing, PM or OM. If a

maintenance action 𝑎𝑎𝑚𝑚 ≠ 𝑎𝑎0 is selected for PM, the OM never occurs because the maintenance

action 𝑎𝑎𝑚𝑚 is conducted immediately. Hence, at the beginning of time epoch 𝑘𝑘, the OM


consideration affects only the decision 𝑎𝑎0. The following parts will discuss the separate policy

for OM and its influence on decision 𝑎𝑎0.

5.2.7.1 Optimal maintenance policy for OM

When an opportunity occurs, decision makers are assumed to know the current condition of

system 𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚 but no idea about the opportunity duration 𝐷𝐷𝑘𝑘𝑂𝑂. Hence, the short-term forecast

duration 𝐷𝐷�𝑘𝑘 is used; or, the estimated (mean) values of duration 𝐷𝐷� = constant is used if no

forecast is available. Let 𝑑𝑑𝑘𝑘𝑂𝑂 be the value of opportunity duration which is used for selecting

the optimal OM action:

𝑑𝑑𝑘𝑘𝑂𝑂 = �𝐷𝐷�𝑘𝑘 if forecast is available𝐷𝐷� otherwise

(5.12)

Let optimal OM policy is 𝜋𝜋𝑂𝑂𝑃𝑃∗ = �𝜇𝜇𝑂𝑂𝑃𝑃,1 ∗ , … , 𝜇𝜇𝑂𝑂𝑃𝑃,𝐾𝐾

∗ �. Therefore, the optimal decision of

maintenance action for OM 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂) is defined as:

𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂) = argmin

𝑚𝑚𝑚𝑚∈ℳ𝑉𝑉𝑂𝑂𝑃𝑃,𝑘𝑘(𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂 ,𝑎𝑎𝑚𝑚) (5.13)

Where 𝑉𝑉𝑂𝑂𝑃𝑃,𝑘𝑘(𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂 ,𝑎𝑎𝑚𝑚) is value function of maintenance action 𝑎𝑎𝑚𝑚, degradation state 𝑆𝑆𝑘𝑘 =

𝑒𝑒𝑚𝑚 and duration of opportunity 𝑑𝑑𝑘𝑘𝑂𝑂:

𝑉𝑉𝑂𝑂𝑃𝑃,𝑘𝑘(𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂 ,𝑎𝑎𝑚𝑚)

= �𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐ℓ,𝑘𝑘 ⋅ [𝑑𝑑(𝑎𝑎𝑚𝑚) − 𝑑𝑑𝑘𝑘𝑂𝑂]+ + �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ⋅ 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�𝑙𝑙𝑗𝑗∈𝒮𝒮

�

⋅ 𝑝𝑝𝑘𝑘𝑂𝑂 + �� 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚� ⋅ 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�𝑙𝑙𝑗𝑗∈𝒮𝒮

� ⋅ [1 − 𝑝𝑝𝑘𝑘𝑂𝑂]

(5.14)

where 𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗� is the expected “cost-to-go” of time epoch (𝑘𝑘 + 1) associated with degradation

state 𝑒𝑒𝑗𝑗; and 𝑝𝑝𝑘𝑘𝑂𝑂 is the chance of opportunity occurrence used for the optimisation model. The

probability of opportunity occurrence 𝑝𝑝𝑘𝑘𝑂𝑂 is similar to the opportunity duration in that that

decision makers have no idea about this value in the beginning. Therefore, the short-term

forecast 𝑂𝑂�𝑘𝑘 in Eq. (5.9) is used; or, the historical value 𝑝𝑝𝑘𝑘𝐿𝐿𝑂𝑂 in Eq. (5.7) is used if no forecast is

available:

𝑝𝑝𝑘𝑘𝑂𝑂 = � 𝑂𝑂�𝑘𝑘 if forecast is available1 − 𝑒𝑒−Λ(𝑡𝑡𝑘𝑘,𝑡𝑡𝑘𝑘+1) otherwise

(5.15)


5.2.7.2 Optimal maintenance policy for PM (given optimal OM policy)

As discussed above, the OM is only considered if maintenance action doing nothing (𝑎𝑎0) is

chosen. Therefore, the transition probability 𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎0� that includes OM consideration is

computed as:

𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎0� = 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚, 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂)� ⋅ 𝑝𝑝𝑘𝑘𝑂𝑂 + 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚� ⋅ [1 − 𝑝𝑝𝑘𝑘𝑂𝑂] (5.16)

denotes the maintenance cost where OM is considered for 𝑎𝑎0 as 𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎𝑚𝑚). The maintenance

cost 𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎0) when maintenance action 𝑎𝑎0 is selected is redefined as expected value:

𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎0) = �𝑐𝑐𝑏𝑏,𝑘𝑘 �𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ (𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂)�+ 𝑐𝑐ℓ,𝑘𝑘 ⋅ �𝑑𝑑 �𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘

∗ (𝑒𝑒𝑚𝑚,𝑑𝑑𝑘𝑘𝑂𝑂)� − 𝑑𝑑𝑘𝑘𝑂𝑂�+� ⋅ 𝑝𝑝𝑘𝑘𝑂𝑂 (5.17)

In case a certain maintenance action 𝑎𝑎𝑚𝑚 ∈ ℳ\[𝑎𝑎0] is selected for PM activity, the transition

probability is the same as the definition in the Eq. (5.2), and the maintenance cost of selected

maintenance action 𝑎𝑎𝑚𝑚 is defined in Eq. (5.3).

𝑞𝑞𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� = 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� 𝑎𝑎𝑚𝑚 ∈ ℳ\[𝑎𝑎0] (5.18)

𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎𝑚𝑚) = 𝐶𝐶𝑘𝑘𝑃𝑃𝑃𝑃 = 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) + 𝑐𝑐ℓ,𝑘𝑘 ⋅ 𝑑𝑑(𝑎𝑎𝑚𝑚) 𝑎𝑎𝑚𝑚 ∈ ℳ\[𝑎𝑎0] (5.19)

Finally, the cost function for a maintenance decision 𝑎𝑎𝑚𝑚 and a given state 𝑆𝑆𝑘𝑘 = 𝑒𝑒𝑚𝑚 in Eq. (5.11)

is defined as:

𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚) = 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚) + 𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎𝑚𝑚) (5.20)

5.3 AGE-BASED VERSUS CONDITION-BASED OPPORTUNISTIC MAINTENANCE POLICIES

This section presents the degradation transition probabilities and related cost structures for two

approaches: age-based and condition-based OM models.

5.3.1 Age-based approach

The virtual age of system is used as the state in the MDP model and assumed to be integer

values. In this approach, it is assumed that when the system fails, it is correctively maintained

with minimal repair, which restores the functioning of system but does not change its age. The


failure of system is assumed following the Weibull distribution with shape 𝛾𝛾 and scale 𝜃𝜃.

Hence, the transition probability is defined as follows:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚� = �1 𝑒𝑒𝑗𝑗 = 𝑒𝑒𝑚𝑚 + 10 otherwise

(5.21)

The imperfect maintenance action will restore the age of system to some point between the

current age and original one (0) according to restoration level. The transition probability of

maintenance action 𝑎𝑎𝑚𝑚 is therefore defined as:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚, 𝑎𝑎𝑚𝑚� = �1 𝑒𝑒𝑗𝑗 ≤ [1 − 𝑒𝑒(𝑎𝑎𝑚𝑚)] ⋅ 𝑒𝑒𝑚𝑚 < 𝑒𝑒𝑗𝑗+10 otherwise

(5.22)

where 𝑒𝑒(𝑎𝑎𝑚𝑚) is the restore level of action 𝑎𝑎𝑚𝑚 (𝑒𝑒(𝑎𝑎𝑚𝑚) = 1 is perfect repair). The new

(discretised) state of system after a maintenance action 𝑎𝑎𝑚𝑚 is determined as:

𝑆𝑆𝑑𝑑(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚) = ��1 − 𝑒𝑒(𝑎𝑎𝑚𝑚)�𝑒𝑒𝑚𝑚� (5.23)

Note that ⌊𝑥𝑥⌋ is the lower integer value of 𝑥𝑥.

When failures occur, the system is minimally repaired with an average cost of 𝑐𝑐𝐶𝐶𝑃𝑃. According

to the current age and the maintenance decision at the beginning of each time epoch, the

expected number of failures is varied during the time interval. The expected number of failures

given the system is in age 𝑒𝑒𝑚𝑚 with maintenance action 𝑎𝑎𝑚𝑚 ∈ ℳ\[𝑎𝑎0] at the beginning of interval

(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] which is determined as:

𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓,𝑘𝑘|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� ={[𝑆𝑆𝑑𝑑(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚) + 1] ⋅ ∆𝑡𝑡}𝛾𝛾 − [𝑆𝑆𝑑𝑑(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚) ⋅ ∆𝑡𝑡]𝛾𝛾

𝜃𝜃𝛾𝛾

𝑎𝑎𝑚𝑚 ≠ 𝑎𝑎0 (5.24)

where 𝑁𝑁𝑓𝑓,𝑘𝑘 is the number of failures during the time interval 𝑘𝑘 and ∆𝑡𝑡 is the fixed length of

time interval.

If action 𝑎𝑎0 is selected, the OM will be considered. Therefore, the expected number of failures

is dependent on the occurrence of opportunity and OM action:

𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓,𝑘𝑘|𝑒𝑒𝑚𝑚,𝑎𝑎0� = 𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓,𝑘𝑘|𝑒𝑒𝑚𝑚� ⋅ 𝑒𝑒−𝜆𝜆𝑘𝑘∆𝑡𝑡 + � 𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓(𝑡𝑡)|𝑒𝑒𝑚𝑚, 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ � ⋅ 𝜆𝜆𝑘𝑘𝑡𝑡𝑒𝑒−𝜆𝜆𝑘𝑘𝑡𝑡𝑑𝑑𝑡𝑡

∆𝑡𝑡

0

(5.25)

where 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ is the OM action as discussed above, 𝜆𝜆𝑘𝑘 is the arrival rate of opportunity during

the time interval 𝑘𝑘. 𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓,𝑘𝑘|𝑒𝑒𝑚𝑚� is the expected number of failures during the time interval 𝑘𝑘


without any maintenance action; and 𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓(𝑡𝑡)|𝑒𝑒𝑚𝑚, 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ � is the expected number of failures if

OM action 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ is conducted at time 𝑡𝑡 ∈ (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1]:

𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓,𝑘𝑘|𝑒𝑒𝑚𝑚� =[(𝑒𝑒𝑚𝑚 + 1)∆𝑡𝑡]𝛾𝛾 − (𝑒𝑒𝑚𝑚∆𝑡𝑡)𝛾𝛾

𝜃𝜃𝛾𝛾

𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓(𝑡𝑡)|𝑒𝑒𝑚𝑚, 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘∗ �

=��𝑆𝑆𝑑𝑑�𝑒𝑒𝑚𝑚, 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘

∗ � + 1�∆𝑡𝑡�𝛾𝛾− �𝑆𝑆𝑑𝑑�𝑒𝑒𝑚𝑚, 𝜇𝜇𝑂𝑂𝑃𝑃,𝑘𝑘

∗ �∆𝑡𝑡 + 𝑡𝑡�𝛾𝛾

+ (𝑒𝑒𝑚𝑚∆𝑡𝑡 + 𝑡𝑡)𝛾𝛾 − (𝑒𝑒𝑚𝑚∆𝑡𝑡)𝛾𝛾

𝜃𝜃𝛾𝛾

(5.26)

In this age-based approach, the state cost 𝑐𝑐𝑙𝑙,𝑘𝑘 is set as zero and the maintenance action cost

𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚) is redefined as:

𝐶𝐶𝑘𝑘(𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚) = 𝑐𝑐𝐶𝐶𝑃𝑃 ⋅ 𝔼𝔼𝑘𝑘�𝑁𝑁𝑓𝑓,𝑘𝑘|𝑒𝑒𝑚𝑚,𝑎𝑎𝑚𝑚� + 𝐶𝐶𝑘𝑘𝑃𝑃(𝑎𝑎𝑚𝑚) (5.27)

5.3.2 Condition-based approach

The degradation process of considered systems is specific for each system. Two popular

degradation models used for CBM in literature are Gamma and Wiener process (Gorjian et al.,

2010). In this study, the Gamma process is used for modelling the degradation of system. The

transition matrix ℙ𝑘𝑘,0 can be developed via Monte Carlo simulation.

The system is failed when the degradation is in the last state. In this situation, a failure cost

𝑐𝑐𝑓𝑓𝑚𝑚𝑚𝑚𝑙𝑙𝑢𝑢𝑟𝑟𝑒𝑒 is incurred. Hence, the cost of states 𝑐𝑐𝑙𝑙,𝑘𝑘 is defined as:

𝑐𝑐𝑙𝑙,𝑘𝑘 = �0 𝑒𝑒 ≠ 𝑒𝑒𝑁𝑁

𝑐𝑐𝑓𝑓𝑚𝑚𝑚𝑚𝑙𝑙𝑢𝑢𝑟𝑟𝑒𝑒 𝑒𝑒 = 𝑒𝑒𝑁𝑁 (5.28)

The imperfect maintenance action will restore the state of system to some point between the

current state and original state according to restoration level. The transition probability of

maintenance action 𝑎𝑎𝑚𝑚 is therefore defined as:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚 ,𝑎𝑎𝑚𝑚� = �1 𝑒𝑒𝑗𝑗 ≤ [1 − 𝑒𝑒(𝑎𝑎𝑚𝑚)] ⋅ 𝑒𝑒𝑚𝑚 < 𝑒𝑒𝑗𝑗+10 otherwise

(5.29)

where 𝑒𝑒(𝑎𝑎𝑚𝑚) is the restore level of action 𝑎𝑎𝑚𝑚 (𝑒𝑒(𝑎𝑎𝑚𝑚) = 1 is perfect repair).


5.4 EVALUATING POLICY VIA SIMULATION

The Monte Carlo simulation is used to evaluate the maintenance policy in order to give deeper

views in variation of OM and PM options according to seasonal varying costs. Given:

• number of time intervals 𝐾𝐾;

• length of interval ∆𝑡𝑡;

• degradation process;

• opportunity occurrence intensity (𝑡𝑡);

• distribution of opportunity duration 𝒟𝒟𝑘𝑘𝑙𝑙𝑝𝑝𝑡𝑡 ∀𝑘𝑘;

• distribution of forecast opportunity occurrences given opportunity really occurs 𝒪𝒪𝑘𝑘;

or not occur 𝒪𝒪𝑘𝑘′ ;

• distribution of forecast duration 𝒟𝒟𝑘𝑘𝑓𝑓𝑐𝑐;

• cost of each state 𝑐𝑐𝑙𝑙,𝑘𝑘(𝑒𝑒𝑚𝑚) ∀𝑒𝑒𝑚𝑚,𝑘𝑘;

• direct maintenance cost 𝑐𝑐𝑏𝑏,𝑘𝑘(𝑎𝑎𝑚𝑚) ∀𝑎𝑎𝑚𝑚,𝑘𝑘;

• downtime cost per time unit, 𝑐𝑐ℓ,𝑘𝑘;

• maintenance duration (𝑎𝑎𝑚𝑚).

Figure 5.3 shows the details of the simulation procedure for a single simulation.


Figure 5.3. One simulation repetition flow chart.

After running a series of simulations, the average values are computed:


Number of repetitions

5.5 NUMERICAL EXAMPLE

5.5.1 Input

In order to demonstrate the proposed policy, a production system that has a seasonal-intensive

operation is considered. The considered external opportunities will stop the system and induce

some savings in downtime cost if maintenance is conducted. The opportunity occurrence is


time-varying and the durations are random. The optimal policy is developed for 100 time

intervals (𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑 = 100 ;∆𝑡𝑡 = 1).

In CB-OM approach, the degradation of the system follows a Gamma process with parameters

(𝛼𝛼 = 2.78 ;𝛽𝛽 = 0.018) and it is discretised into 101 intervals with equal width ∆𝑒𝑒 = 0.02.

Each discrete value multiple of ∆𝑒𝑒 corresponds to a state of the system. The transition matrix

of degradation is determined via Monte Carlo simulation. The last state means that the system

has totally failed and it incurs a cost 𝑐𝑐𝑓𝑓𝑚𝑚𝑚𝑚𝑙𝑙𝑢𝑢𝑟𝑟𝑒𝑒. The default value of 𝑐𝑐𝑓𝑓𝑚𝑚𝑚𝑚𝑙𝑙𝑢𝑢𝑟𝑟𝑒𝑒 is $20,000.

In TB-OM approach, the system failure time is assumed following Weibull distribution with

parameters {𝛾𝛾 = 3.304;𝜃𝜃 = 16.722}. When the failure occurs, the minimal repairs with cost

𝑐𝑐𝐶𝐶𝑃𝑃 = $2000 are conducted to restore the function of system but not change its state (age).

Maintenance direct costs are assumed to be fixed in time. The downtime cost due to production

loss when doing maintenance is time-varying (Figure 5.4). The downtime costs of low and

peak seasons are �𝑐𝑐ℓ𝑚𝑚𝑚𝑚𝑚𝑚 = 400 ; 𝑐𝑐ℓ𝑚𝑚𝑚𝑚𝑚𝑚 = 1200�.

Three maintenance types with different restoration levels, direct costs and durations are

considered for OM or PM where the third one is the perfect maintenance. The restoration levels

of maintenance actions are 20%, 60% and 100%, respectively. The direct costs are fixed at

2000, 5000 and 8000 respectively. The durations are 2, 3 and 5 respectively.

Figure 5.4. Production loss cost per time unit.


Opportunity forecasts are available in five time epochs (𝐿𝐿 = 5) from current time. The

opportunity duration is assumed independent on arrival time and following a Log-Normal

Distribution with parameters (𝜇𝜇 = 0.75 ;𝜎𝜎 = 0.5). Due to the duration being a non-negative

value, the forecast duration is assumed to follow truncated Normal distribution

𝐷𝐷�𝑘𝑘𝑓𝑓𝑐𝑐~𝒩𝒩𝑡𝑡𝑟𝑟{𝐷𝐷𝑘𝑘𝑂𝑂,𝜎𝜎𝑑𝑑2, [0,∞)}. The default value of 𝜎𝜎𝑑𝑑 is set at 50% of standard deviation of

duration distribution (i.e. Log-Normal (𝜇𝜇 = 0.75 ;𝜎𝜎 = 0.5)):

𝜎𝜎𝑑𝑑 = 0.5 ⋅ 𝑆𝑆𝐷𝐷(𝐷𝐷𝑘𝑘𝑂𝑂) = 0.5 ⋅ �𝑉𝑉𝑎𝑎𝑒𝑒(𝐷𝐷𝑘𝑘𝑂𝑂) = 0.5 ⋅ �𝑒𝑒𝜇𝜇+12𝜎𝜎

2�𝑒𝑒𝜎𝜎2 − 1� = 0.639

The opportunity arrival intensity is time-varying and season-dependent. The illustration of

opportunity rates is shown in Figure 5.5. The chance of opportunity occurrences forecast 𝑂𝑂�𝑘𝑘 is

assumed to follow the conditional discrete distributions 𝒪𝒪 and 𝒪𝒪′ given opportunity occurs or

not. The distributions 𝒪𝒪 and 𝒪𝒪′ are developed based on the triangular distribution

𝒯𝒯ℛ{𝑎𝑎 = 0, 𝑏𝑏 = 1, 𝑐𝑐 = 1} and 𝒯𝒯ℛ{𝑎𝑎 = 0, 𝑏𝑏 = 0, 𝑐𝑐 = 1} where 𝑎𝑎 is the lower bound, 𝑏𝑏 is the

peak and 𝑐𝑐 is the upper bound.

Figure 5.5. Opportunity Intensity.


The input data is summarised in Table 5.1.

Table 5.1

Input Parameters

Parameters Maintenance

action 𝒎𝒎𝟏𝟏

Maintenance

action 𝒎𝒎𝟐𝟐

Maintenance

action 𝒎𝒎𝟑𝟑

Restoration level 20% 60% 100%

Direct cost 𝒄𝒄𝒃𝒃,𝒌𝒌 2,000 4,000 8,000

Maintenance duration 2 3 5

Other parameters

CB-OM TB-OM

Maximum downtime cost

rate 1200

Minimum downtime cost

rate 400

State cost 20,000 (last state) 2000 (per failure)

Degradation Gamma (α=2.78; β=0.018) n/a

Failure distribution n/a Weibull (𝛾𝛾=3.304;

𝜃𝜃=16.722)

Opportunity duration

distribution (Log-Normal) (μ=0.75; σ=0.5)

Number of time epochs 𝑲𝑲 100

Time interval length 𝚫𝚫𝒕𝒕 1

Number of forecast interval

𝑳𝑳 5

Number of simulation runs 10,000

The computation time for both optimisation and simulation is about 4 hours (computer with

CPU core i7-4790 @3.6GHz, 16 GB Ram).


5.5.2 Comparison between the policies with and without forecasts

This section discusses the benefits of joint OM and PM policy using short-term forecasts of

opportunities compared to the policy without using this information. The simulated results of

the probabilities of OM and PM for the cases of considering short-term forecasts and without

forecasts are presented in Figure 5.6 and Figure 5.7. It shows that the OM is preferred to PM

in most time of horizon duration, especially in peak seasons when the downtime cost is

significant (Figure 5.4). The probability of OM is also large in time 40 to 60 even it belongs to

the low-production time. This is because the chances for opportunity occurrences during this

period is high (Figure 5.5). The probabilities of PM also increase at period (39-43) when the

production loss cost is still low, and the chance of opportunities is lowest at these times. In

comparing the cases of using versus not using short-term forecasts, the probability of doing

maintenance (both PM and OM) of a case without short-term forecasts is higher. It is reasonable

because fewer unnecessary maintenance activities are conducted when using forecasts of

opportunities.

Figure 5.6. Probability of conducting OM and PM of the policy with and without short-term forecasts for TB-OM approach.


Figure 5.7. Probability of conducting OM and PM of the policy with and without short-term forecasts for CB-OM approach.

5.5.3 Benefits of short-term forecasts of opportunities

This section conducts some sensitivity analysis about the benefits of joint OM and PM policy

using short-term forecasts of opportunities versus the policy without using this information.

The benefits are analysed with different inputs of failure cost, downtime cost, forecasting

window and parameters of forecasting distributions.

5.5.3.1 Effect of failure cost

In this section, a series of different costs of system failure in CB-OM approach and CM cost in

TB-OM approach are simulated to evaluate the benefit of policy using short-term forecasts of

opportunities in comparison to the policy with long-term information only. The costs are set as

25%, 50%, 100%, 150% and 250% of default value. The policy of joint PM and OM is


investigated in two approaches, respectively CB-OM and TB-OM. Figure 5.8 shows the saving

percentages of maintenance policy using short-term forecasts.

The savings of joint PM and OM policy using the forecasts is almost unchanged for both cases

of CB-OM and TB-OM when the failure cost varies. Both policies show the reduction in

savings when the failure cost increases. It is explainable that the high failure cost results of the

PM are more favourable in comparison with OM and make the forecast less useful. In TB-OM

approach, the saving is quite low when the failure cost is small. The reason is that the small

failure cost reduces the maintenance necessity and the benefit of forecast therefore decreases.

Figure 5.8. Savings of using short-term forecasts compared to OM without forecasts.

5.5.3.2 Effect of downtime cost

This section investigates the savings of using short-term forecasts when downtime costs are

varied (Figure 5.9). When the downtime cost 𝑐𝑐ℓ,𝑘𝑘 increases, the benefits of joint OM and PM

policies with the short-term information are more significant. That is reasonable because the

OM is intuitively more favourable when downtime cost is higher. That makes using forecasts

more significant because it makes the OM decisions more accurate.


Figure 5.9. Savings in using short-term forecasts compared to OM without forecasts.

5.5.3.3 Effect of forecast window

This section analyses the savings of using versus not using short-term forecasts according to

the length of forecast window. Figure 5.10 shows the savings with different forecast windows.

When the forecasts are available for further into the future, the benefits of joint OM and PM

policies with short-term forecasts’ consideration are more significant. That is reasonable

because the optimal policy is reliable in the longer time with availability of forecasts. The

savings yielded by consideration of TB-OM and CB-OM are almost in the same increasing

pattern. The increasing rate of benefit adapting to the forecast window is reduced when the

forecast is further in the future. This is because the effect of accurate information in the future

is reduced when it is far from current time.


Figure 5.10. Savings of using short-term forecasts when forecast window varies.

5.5.3.4 Effect of forecasting accuracy

The efficiency of using the short-term forecasts is intuitively dependent on its accuracy. Table

5.2 presents the changes of benefits induced by using the short-term forecasts with different

accuracy levels of both predicted opportunity occurrences and durations for a) the TB-OM

approach, and b) the CB-OM policy. In order to change the accuracy levels of predicted

opportunity occurrence and duration, the parameter of occurrence distribution (parameters of

triangle distribution) and the parameter of duration distribution (standard deviation 𝜎𝜎𝑑𝑑 of

truncated Normal distribution) are varied.


Table 5.2 Savings of using Short-term Forecasts when Forecasting Accuracy Varies (the smaller the values the more accurate) for two cases: a) TB-OM model, and b) CB-OM model

(a) TB-OM model Standard deviation 𝝈𝝈𝒅𝒅 (measure accuracy of forecast duration)

𝟎𝟎 𝟎𝟎.𝟐𝟐𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌𝑶𝑶� 𝟎𝟎.𝟓𝟓𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌

𝑶𝑶� 𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌𝑶𝑶� 𝟐𝟐𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌

𝑶𝑶�

accu

racy

of p

redi

cted

occ

urre

nce 𝓞𝓞′ a

nd

𝓞𝓞 (t

il

di

tib

ti)

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎,𝟎𝟎) &

𝓣𝓣𝓣𝓣(𝟏𝟏,𝟏𝟏,𝟏𝟏) 6.2% 6.1% 5.4% 3.2% 0.3%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎,𝟎𝟎.𝟓𝟓) &

𝓣𝓣𝓣𝓣(𝟎𝟎.𝟓𝟓,𝟏𝟏,𝟏𝟏) 6.1% 5.8% 5.3% 3.2% 0.2%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎,𝟏𝟏) &

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟏𝟏,𝟏𝟏) 4.9% 4.5% 3.9% 2.2% -0.6%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟐𝟐𝟓𝟓,𝟏𝟏) &

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟏𝟏𝟓𝟓,𝟏𝟏) 4.2% 4.1% 3.1% 1.9% -0.9%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟓𝟓,𝟏𝟏) &

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟓𝟓,𝟏𝟏) 3.2% 3.2% 2.4% 1.0% -1.3%

(b) CB-OM model Standard deviation 𝝈𝝈𝒅𝒅 (measure accuracy of forecast duration)

𝟎𝟎 𝟎𝟎.𝟐𝟐𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌𝑶𝑶� 𝟎𝟎.𝟓𝟓𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌

𝑶𝑶� 𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌𝑶𝑶� 𝟐𝟐𝑺𝑺𝑺𝑺�𝑺𝑺𝒌𝒌

𝑶𝑶�

accu

racy

of p

redi

cted

occ

urre

nce

𝓞𝓞′

d 𝓞𝓞

(ti

l d

it

ibti

)

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎,𝟎𝟎) &

𝓣𝓣𝓣𝓣(𝟏𝟏,𝟏𝟏,𝟏𝟏) 10.5% 10.4% 8.9% 6.0% 1.5%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎,𝟎𝟎.𝟓𝟓) &

𝓣𝓣𝓣𝓣(𝟎𝟎.𝟓𝟓,𝟏𝟏,𝟏𝟏) 9.1% 8.7% 7.4% 4.5% 0.3%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎,𝟏𝟏) &

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟏𝟏,𝟏𝟏) 6.9% 6.6% 5.5% 2.8% -1.3%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟐𝟐𝟓𝟓,𝟏𝟏) &

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟏𝟏𝟓𝟓,𝟏𝟏) 5.6% 5.6% 4.1% 1.7% -2.2%

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟓𝟓,𝟏𝟏) &

𝓣𝓣𝓣𝓣(𝟎𝟎,𝟎𝟎.𝟓𝟓,𝟏𝟏) 4.5% 4.2% 2.8% 0.8% -3.1%

The result shows that when the forecast is more accurate, the savings are higher. It is reasonable

because the right maintenance options will be selected if the forecasts are precise. The savings

are still significant when the accuracy is quite low (𝒪𝒪′ = 𝒯𝒯𝒯𝒯(0,0,1), 𝒪𝒪 = 𝒯𝒯𝒯𝒯(0,1,1) and 𝜎𝜎𝑑𝑑 =

𝑆𝑆𝐷𝐷(𝐷𝐷𝑘𝑘𝑂𝑂)). It encourages decision makers to consider short-term forecasts to support their

maintenance decision. When the forecasts are very inaccurate, there is no benefit anymore. In

this situation, using the long-term (historical) information is better than using the forecasts.


5.6 CONCLUSION

This chapter investigated the benefit of using short-term forecasts of external opportunities in

implementation of opportunistic maintenance policy. Both time-based and condition-based

OM policies were considered in this study. The mission total maintenance cost was minimised

through a Markov Decision Process where system state thresholds (i.e. degradation in CB-OM

and age in TB-OM) for different PM actions as well as optimal OM selection were jointly

optimised.

Numerical results showed that using short-term forecasts of opportunities could yield a

significant saving in implementation of OM policies. In particular, the simulated results

confirmed the intuitive hypothesis that the forecast is more useful if it is more accurate and its

availability is far in the future.


Chapter 6: Cost-free opportunistic maintenance: Case study of cleaning strategy for solar power

Overview

This chapter discusses condition-based opportunistic maintenance where the effects of cost-

free external maintenance events to maintenance policy are investigated. The proposed policy

is developed as the cleaning policy of mirrors in a CSP system to maintain the reflectivity of

the mirror field. In this study, a condition-based cleaning (CBC) policy is developed for mirrors

whose degradation is stochastic and subject to seasonal variations. The optimal policy is

determined by formulating and solving a finite-horizon Markov Decision Process in which

time-varying transition matrices describe stochastic soiling, rain events and imperfect

cleanings. The optimal cleaning policy is therefore a time-varying reflectivity threshold, below

which cleaning is triggered.

The methodology has been applied to a case study on a hypothetical plant in Brisbane,

Australia. Using publicly available electricity price and weather data, the optimised CBC

policy was found to save 5-30% of total cleaning costs compared with a fixed-time strategy.

Importantly, higher CBC savings are achieved when the direct cleaning costs are high,

indicating that the policy could be particularly significant for countries with high labour or

resource (water etc.) costs (e.g. Australia).

This chapter relates to published paper:

• Truong Ba H, Cholette ME, Wang R, Borghesani P, Ma L, Steinberg TA. Optimal

condition-based cleaning of solar power collectors. Solar Energy. 2017; 157:762-77.

112 Chapter 6: Cost-free opportunistic maintenance: Case study of cleaning strategy for solar power

Nomenclature

𝑁𝑁 number of states.

𝐾𝐾: number of decision epochs.

∆𝑡𝑡: discretisation interval for the decision epoch.

𝑘𝑘: index of time period (decision epochs), 𝑙𝑙 = 1,2, … ,𝐾𝐾.

𝑖𝑖, 𝑗𝑗: indices of states of a certain mirror, 𝑖𝑖, 𝑗𝑗 = 1,2, … ,𝑁𝑁.

∆𝑒𝑒: discretisation interval for the states.

𝑝𝑝𝐷𝐷�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, no rain�: transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 given no cleaning and no rain

at time epoch 𝑘𝑘.

𝑝𝑝𝑅𝑅�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, rain�: transition probability from state 𝑒𝑒𝑚𝑚 to state 𝑒𝑒𝑗𝑗 given rain event in epoch 𝑘𝑘.

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎�: transition matrix in epoch 𝑘𝑘 with action 𝑎𝑎.

𝐻𝐻(𝑡𝑡𝑘𝑘): revenue per square metre with maximum reflectivity (potential revenue).

𝐿𝐿(𝑆𝑆𝑘𝑘, 𝑡𝑡𝑘𝑘): degradation cost per square metre with state 𝑆𝑆𝑘𝑘 at time epoch 𝑡𝑡𝑘𝑘.

𝐶𝐶𝑘𝑘(𝑆𝑆𝑘𝑘,𝑃𝑃𝑘𝑘): cost per square metre when state is 𝑆𝑆𝑘𝑘 and action 𝑃𝑃𝑘𝑘 is taken in epoch 𝑘𝑘.

𝑆𝑆𝑘𝑘: reflectivity loss of the field at time 𝑡𝑡𝑘𝑘.


6.1 INTRODUCTION

Industrial systems degrade through time depending on their working conditions and

environment. In order to maintain and restore their condition, some maintenance activities

should be planed and conducted as discussed in previous chapters. However, there are some

special situations where systems can restore fully or partially their condition by external events

with no cost. One example is that the outside of a building can be cleaned in a heavy rain and/or

strong wind. Another example is the productivity restoration of heliostats for solar power plants

caused by rain events. These external events can do the role of maintenance actions to reduce

the degradation of systems without incurring any cost. Hence, it is possible to call these special

events as “cost-free” maintenance opportunities. Decision makers should consider this type of

external opportunities to enhance maintenance policies and reduce costs. This chapter discusses

on this type of cost-free maintenance opportunities in a case study of cleaning strategy for

mirrors of a concentrating solar power (CSP) system. The impact of these cost-free

maintenance opportunities to cleaning strategies is discussed and analysed.

Solar power systems are important alternatives to fossil fuel energy sources. However, the cost

competitiveness of some of these technologies has been hindered by their high operation and

maintenance (O&M) costs, particularly for CSP systems. A significant contribution to the

O&M cost is the cleaning of the solar collectors that are responsible for focusing the solar

irradiation onto a receiver. Frequent cleanings lead to maintaining high reflectivity and

generation efficiency while infrequent cleaning can save on cleaning costs (Deffenbaugh et al.,

1986). An experimental comparison of solar photovoltaic (PV) technologies has indicated that

though frequent cleaning can improve energy yield and Performance Ratio (PR), the

corresponding Levelized Cost of Energy (LCOE) can actually increase (Fuentealba et al.,

2015). Therefore, an optimal cleaning policy is required to attain the correct balance between

revenue received from generating more electricity (cleaner collectors) and the costs of

conducting cleaning operations (e.g. water, labour). A similar problem has been studied for a

Heat Exchanger Network (HEN) to derive an optimal cleaning schedule with minimum total

operational cost (Sanaye and Niroomand, 2007).

Most studies in cleaning of solar collectors have mainly focused on cleaning technologies

(Cohen et al., 1999; Morris, 1980; Sayyah et al., 2013). The overall cleaning methods can be

classified into four categories: Natural cleaning due to rain; trucks or hosing systems (Cohen

et al., 1999; Trabish, 2013); sprinkler-like systems with fixed nozzles; and robotic cleaning

systems (Schell, 2011). Additionally, a particle-charging based method was proposed with the


benefits of being self-cleaning and water saving (Mazumder et al., 2014). To automate

cleaning, a GPS-based system of mirror washing machines (MWMs) was developed by

BrightSource to optimise cleaning tasks including the location and density of stopping points,

the cleaning order and the heliostats’ orientations (Alon et al., 2014).

Among the few studies focusing on cleaning schedules, most have focused on setting time-

based cleaning intervals considering average degradation rates. Deffenbaugh et al. (1986) and

Bergeron and Freese (1981) suggested optimal cleaning frequencies which balance the daily

reflectivity degradation and the annual performance of solar collectors. Kattke and Vant-Hull

(2012) applied a similar methodology to establish the optimal balance between average field

reflectivity and excess capacity in the design phase of the solar field.

However, such fixed-time interval cleaning strategies neglect the influence of the stochastic

and time-varying factors inherent in cleaning optimisation. The stochastic nature of the soiling

process makes an “average”-based fixed-time approach suboptimal when compared to a

condition-based approach, which can exploit direct on-site measurements of soiling and/or

optical efficiency to adapt the cleaning policy to the current condition. Moreover, the

seasonality of weather and electricity-price statistics play a key role in the economics of solar

collector cleaning. In particular, the soiling process is strongly affected by local weather

conditions (e.g. wind, rain, humidity, (dew)-temperature, airborne dust concentration and type)

and it varies seasonally (Bergeron and Freese, 1981; El-Nashar, 2009; Ghazi et al., 2014; Guan

et al., 2015; Maghami et al., 2016; Sayyah et al., 2013). In addition, rain and events may also

lead to natural cleaning events, which are free of charge (Bethea et al., 1981; Ghazi et al., 2014;

Guan et al., 2015; Sayyah et al., 2013; Vivar et al., 2010). Thus, both the stochastic soiling rate

and the seasonal properties of weather and price should lead to cleaning policies that vary

accordingly. Intuitively, high electricity prices and Direct Normal Irradiation (DNI) mean that

cleaner collectors (high optical efficiency) are preferred to take advantage of the high potential

revenue. In contrast, when prices are low, revenue loss due to soiling will be lower.

Despite the promise of newly developed online measurement technologies (Wolfertstetter et

al., 2014a; Wolfertstetter et al., 2012; Zhu et al., 2014), no cleaning studies have yet considered

soiling/optical efficiency measurements and seasonal variations in weather/electricity price in

their optimisation. In this study, a reflectivity-based cleaning policy is developed for the

particular case of CSP heliostats (but applicable to other solar collectors) under the well-known

paradigm of CBM (Liu et al., 2003; Prajapati et al., 2012), called in this case condition-based

cleaning (CBC). The cleaning optimisation problem is formulated as a finite-horizon MDP


with the aim of minimising the sum of cleaning costs and lost revenue due to reflectivity

degradation. In contrast to the existing work, cleaning decisions are made by comparing the

reflectivity with a time-varying threshold that has been set considering seasonal variation in

the weather factors such as DNI and rain as well as electricity prices. Furthermore, a numerical

study is performed with the aim of assessing the profit impact of the new CBC policy compared

to a traditional time-based cleaning schedule and to test the sensitivity of this benefit to delays

between the cleaning decision and the actual cleaning event.

The remainder of the chapter is structured as follows: Section 6.2 discusses the optimisation

problem set-up and details the objective function and necessary statistical models. Section 6.3

describes how to use commonly available data to estimate the necessary probabilities and

model parameters for the optimisation model, while Section 6.4 discusses how the quality of

the cleaning policy is evaluated using Monte Carlo simulation. Section 6.5 presents a case study

on a hypothetical CSP plant in Brisbane, Australia and the proposed strategy is compared with

a time-based cleaning schedule. Finally, Section 6.6 presents the conclusions of the study and

suggests avenues for future work.

6.2 MODELLING AND OPTIMISATION

6.2.1 Problem description

The aim of this study is to develop a flexible cleaning policy in which the threshold “adapts”

to the time-varying occurrences of natural cleaning (rain) events and electricity price statistics

over a predefined time horizon [0,𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑]. The model concept is depicted in Figure 6.1. It is

assumed that there is an available method for monitoring the mirror reflectivity (via

measurement or estimation) so that it may be used for comparison with a time-varying

threshold to trigger cleaning actions. Cleaning or rain may restore the reflectivity of the

heliostat and the degree of restoration for both is potentially “imperfect” (reflectivity after these

events is described by a probability model). The occurrence of sufficiently intense rain events

and electricity price statistics are considered stochastic with seasonally varying statistical

properties. Thus, the goal of the optimisation process is to identify this threshold optimally so

as to minimise the sum of cleaning costs and lost revenue due to reflectivity degradation. The

optimal time-varying threshold for the CBC policy is determined using the time-varying

average quantities (e.g. DNI, electricity prices, rain occurrences etc.) which are estimated via

historical data (discussed in detail in Section 6.3). To implement the optimal CBC policy, a


decision to clean (or not) at a certain time period is made based on the comparison of the actual

reflectivity loss with the time-varying optimal threshold. Thus, the time-varying threshold is

optimised using statistical models based on historical averages. Once these thresholds are

determined, only current reflectivity measurements are needed to implement the policy.

Figure 6.1. Reflectivity degradation and cleaning threshold.

The remainder of this section describes a framework to identify the optimal thresholds based

on an MDP. In this framework, the timeline is discretised into 𝐾𝐾 decision epochs (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] 𝑘𝑘 =

1,2, … ,𝐾𝐾 and 𝑡𝑡1 = 0 , 𝑡𝑡𝐾𝐾+1 = 𝑇𝑇𝑒𝑒𝑚𝑚𝑑𝑑, with a sufficiently small duration, ∆𝑡𝑡, such that the

reflectivity, 𝜌𝜌(𝑡𝑡𝑘𝑘),can be considered approximately constant. The reflectivity loss, 𝑆𝑆𝑘𝑘 ≝ 𝜌𝜌0 −

𝜌𝜌(𝑡𝑡𝑘𝑘), 0 ≤ 𝜌𝜌(𝑡𝑡𝑘𝑘) ≤ 𝜌𝜌0, is also discretised to the finite set 𝑆𝑆𝑘𝑘𝑑𝑑 ∈ {𝑒𝑒0, … 𝑒𝑒𝑁𝑁−1} = 𝕊𝕊, with 𝑒𝑒𝑚𝑚 =

𝑖𝑖∆𝜌𝜌8. The dynamics of 𝑆𝑆𝑘𝑘𝑑𝑑 will thus be modelled as a discrete-time, finite state Markov chain,

in which transition probabilities will depend on the properties of the random degradation

process, the probability of a rain event, and the cleaning action taken in the epoch. The

computation time of MDP algorithm is correlated to the numbers of discretised states and time

intervals. If the reflectivity is finely discretised and the range of considered reflectivity is too

large, the computation time will dramatically increase. However, if the reflectivity is roughly

discretised, the optimal decision is not really exact. The output of the MDP is an optimised

8 In other words, 𝑆𝑆𝑘𝑘𝑑𝑑 = 𝑒𝑒𝑗𝑗 indicates that 𝑆𝑆𝑘𝑘 ∈ �𝑒𝑒𝑗𝑗 −∆𝜌𝜌2

, 𝑒𝑒𝑗𝑗 + ∆𝜌𝜌2�.


cleaning policy that shall be implemented in the following manner (see Figure 6.2). The

cleaning policy will select an action, 𝑃𝑃𝑘𝑘 (𝑃𝑃𝑘𝑘 = 1 “clean”, or 𝑃𝑃𝑘𝑘 = 0, “do-nothing”), based on

a measurement of the heliostat reflectivity 𝜌𝜌(𝑡𝑡𝑘𝑘). The performance of the policy will be

assessed according to the total cleaning cost, which is the sum of cleaning costs and lost

revenue due to reflectivity loss.

Figure 6.2. The model structure and cleaning policy implementation.

The remainder of this section discusses the technical details of the development and solution

of the MDP. Section 6.2.2 details the development of the Markov transition probabilities.

Section 6.2.3 describes the total cost of the cleaning model and its components, and Section

6.2.4 states the optimisation problem as MDP and the solution procedure.

6.2.2 Reflectivity degradation model

In this study, the dynamics of reflectivity loss are described by a non-homogenous Markov

chain with transition probabilities:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎� ≝ Pr�𝑆𝑆𝑘𝑘+1𝑑𝑑 = 𝑒𝑒𝑗𝑗�𝑆𝑆𝑘𝑘𝑑𝑑 = 𝑒𝑒𝑚𝑚,𝑃𝑃𝑘𝑘 = 𝑎𝑎� (6.1)

which is the probability that the (discretised) reflectivity loss is 𝑒𝑒𝑗𝑗 at epoch 𝑘𝑘 + 1 given that at

𝑡𝑡𝑘𝑘, the reflectivity loss was 𝑒𝑒𝑚𝑚 and action 𝑎𝑎 was selected. These probabilities are influenced by

cleaning decisions and two natural phenomena: Deposition of dust on the mirrors and the

arrival of rain events. In the absence of cleaning (𝑎𝑎 = 0), the transition probabilities can be

written as:


𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎 = 0� = 𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘)𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, rain� + [1 − 𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘)]𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, no rain� (6.2)

where 𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘) is the probability of rain in epoch 𝑘𝑘, and 𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, no rain� describes the

degradation of reflectivity due to soiling:

𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, no rain� = Pr�𝑆𝑆𝑘𝑘𝑑𝑑 = 𝑒𝑒𝑗𝑗 |𝑆𝑆𝑘𝑘−1𝑑𝑑 = 𝑒𝑒𝑚𝑚 , no rain� (6.3)

and 𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, rain� describes the cleaning effect of rain events:

𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, rain� = Pr�𝑆𝑆𝑘𝑘𝑑𝑑 = 𝑒𝑒𝑗𝑗 |𝑆𝑆𝑘𝑘−1𝑑𝑑 = 𝑒𝑒𝑚𝑚 , rain� (6.4)

Methodologies for calculating these probabilities from available data are discussed in detail in

Section 6.3.1.

In this study, cleaning is assumed to occur at the end of the decision epoch. While an imperfect

cleaning model (e.g. similar to rain cleaning) can be used, cleaning actions are considered

perfect in this study and thus reset the reflectivity loss to zero, i.e. 𝑎𝑎𝑘𝑘 = 1 ⟹ 𝑆𝑆𝑘𝑘+1 = 0. Hence,

the transition probability under cleaning is:

𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎 = 1� = �1 𝑒𝑒𝑗𝑗 = 00 𝑒𝑒𝑗𝑗 ≠ 0 (6.5)

6.2.3 Total cleaning cost

In this section, the total cleaning cost optimisation criterion is developed which has two

components: Productivity losses due to mirror reflectivity loss and direct maintenance costs

due to cleaning events. The cost per square metre due to reflectivity loss in a certain time epoch

𝑡𝑡 ∈ (𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1] is defined as:

𝐿𝐿(𝑡𝑡𝑘𝑘) = � 𝑝𝑝(𝜏𝜏) ⋅ 𝐷𝐷𝑁𝑁𝐼𝐼(𝜏𝜏) ⋅ 𝜂𝜂(𝜏𝜏) ⋅ �𝜌𝜌0 − 𝜌𝜌(𝜏𝜏)� ⋅ 𝑑𝑑𝜏𝜏

𝑡𝑡𝑘𝑘+1

𝑡𝑡𝑘𝑘

(6.6)

where 𝜌𝜌(𝑡𝑡) and 𝜌𝜌0 denote the measured and fully-clean reflectivities respectively, 𝑝𝑝(𝑡𝑡) denotes

the electricity price at time 𝑡𝑡, 𝐷𝐷𝑁𝑁𝐼𝐼 denotes the solar direct normal irradiance and 𝜂𝜂 is the total

efficiency (accounts for the effect of optical and thermal losses excluding reflectivity). While

the price, DNI and efficiency vary significantly on an hourly basis, the field reflectivity 𝜌𝜌(𝑡𝑡)

is expected to have slower dynamics, and it is therefore considered constant within each epoch

𝑡𝑡 ∈ [𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1). Using this simplification, Eq. (6.6) can be re-written as:


𝐿𝐿(𝑡𝑡𝑘𝑘) = 𝑆𝑆𝑘𝑘 ⋅ � 𝑝𝑝(𝜏𝜏) ⋅ 𝐷𝐷𝑁𝑁𝐼𝐼(𝜏𝜏) ⋅ 𝜂𝜂(𝜏𝜏) ⋅ 𝑑𝑑𝜏𝜏

𝑡𝑡𝑘𝑘+1

𝑡𝑡𝑘𝑘

. (6.7)

Thus, the cost due to reflectivity loss can be written as the product of the reflectivity loss 𝑆𝑆𝑘𝑘

and the potential revenue per square metre with maximum reflectivity at time epoch 𝑡𝑡𝑘𝑘, 𝐻𝐻(𝑡𝑡𝑘𝑘):

𝐻𝐻(𝑡𝑡𝑘𝑘) = � 𝑝𝑝(𝜏𝜏) ⋅ 𝐷𝐷𝑁𝑁𝐼𝐼(𝜏𝜏) ⋅ 𝜂𝜂(𝜏𝜏) ⋅ 𝑑𝑑𝜏𝜏

𝑡𝑡𝑘𝑘+1

𝑡𝑡𝑘𝑘

(6.8)

The degradation cost per square metre 𝐿𝐿 is then defined as the cost due to lost reflectivity, and

may be computed as:

𝐿𝐿(𝑆𝑆𝑘𝑘, 𝑡𝑡𝑘𝑘) = 𝑆𝑆𝑘𝑘 ⋅ 𝐻𝐻(𝑡𝑡𝑘𝑘). (6.9)

The expected total cost per square metre at time 𝑡𝑡𝑘𝑘 is finally obtained adding this degradation

cost to a cleaning cost per square metre 𝐶𝐶𝑐𝑐𝑙𝑙, itself conditional on the decision of performing

the cleaning action:

𝐶𝐶𝑘𝑘(𝑆𝑆𝑘𝑘,𝑃𝑃𝑘𝑘) = 𝔼𝔼{𝐻𝐻(𝑡𝑡𝑘𝑘)} ⋅ 𝑆𝑆𝑘𝑘 + 𝐶𝐶𝑐𝑐𝑙𝑙 ⋅ 𝑃𝑃𝑘𝑘 (6.10)

where 𝑃𝑃𝑘𝑘 ∈ 𝔸𝔸 = {0,1} is the cleaning action at time 𝑡𝑡𝑘𝑘, with 𝑃𝑃𝑘𝑘 = 0 for the “do-nothing”

action and 𝑃𝑃𝑘𝑘 = 1 if cleaning occurs in time epoch 𝑡𝑡𝑘𝑘 8F

9.

6.2.4 Optimisation of the cleaning policy

The aim of this study is to develop a cleaning policy that minimises 𝐶𝐶𝑘𝑘(𝑆𝑆𝑘𝑘,𝑃𝑃𝑘𝑘) in Eq. (6.10).

However, this cost is subject to several stochastic phenomena: Soiling rates vary with weather

conditions, rain occurrences are random with seasonally varying statistical properties, and the

effectiveness of cleaning operations (particularly rain) may not be deterministic. Thus, it

instead must minimise the expected total maintenance cost denoted as:

min𝜋𝜋

𝔼𝔼 ��𝐶𝐶𝑘𝑘 �𝑆𝑆𝑘𝑘𝑑𝑑 , 𝜇𝜇𝑘𝑘−1�𝑆𝑆𝑘𝑘−1𝑑𝑑 ��𝐾𝐾

𝑘𝑘=0

� (6.11)

9 Note that for this formulation, the cleaning of the entire field is assumed to be instantaneous for optimisation purposes. The impact of the cleaning lead-time on the quality of the optimal cleaning policy will be investigated and discussed in the results section.


where the cleaning policy is denoted as 𝜋𝜋 = (𝜇𝜇0, 𝜇𝜇1, … , 𝜇𝜇𝑘𝑘, … , 𝜇𝜇𝐾𝐾), which is a sequence of

functions 𝜇𝜇𝑘𝑘:𝕊𝕊 ⟼ 𝔸𝔸 that map the reflectivity into an action at each epoch 𝑘𝑘. Thus, it is obvious

that the goal is to develop a cleaning policy that minimises the expected total cleaning cost.

Since the dynamics of mirror reflectivity loss have been described as a non-homogenous

Markov chain (Section 6.2.2), the minimisation is (6.11) subject to these falls under the well-


induction (Puterman, 2014):

𝑉𝑉𝑘𝑘(𝑒𝑒) = min𝑚𝑚∈𝔸𝔸

�𝐶𝐶𝑘𝑘(𝑒𝑒,𝑎𝑎) + �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒,𝑎𝑎�𝑁𝑁−1

𝑗𝑗=0

𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�� (6.12)

𝜇𝜇𝑘𝑘∗(𝑒𝑒) = arg min𝑚𝑚∈𝔸𝔸

�𝐶𝐶𝑘𝑘(𝑒𝑒,𝑎𝑎) + �𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒,𝑎𝑎�𝑁𝑁−1

𝑗𝑗=0

𝑉𝑉𝑘𝑘+1�𝑒𝑒𝑗𝑗�� (6.13)

where the transition probabilities 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒, 𝑎𝑎� have been defined in Eq. (6.2) and 𝑉𝑉𝑘𝑘(𝑒𝑒) is the

expected “cost-to-go” when the reflectivity loss is 𝑒𝑒. It is assumed that there is no cleaning at

the last time epoch 𝐾𝐾; hence, 𝑉𝑉𝐾𝐾(𝑒𝑒) = 𝔼𝔼{𝐻𝐻(𝑡𝑡𝐾𝐾)} ⋅ 𝑒𝑒. However, this assumption will have little

practical effect since 𝐾𝐾 will be large (ideally, the entire life cycle of the plant). The optimal

cleaning policy is clearly 𝜋𝜋∗ = (𝜇𝜇0∗ , 𝜇𝜇1∗, … , 𝜇𝜇𝐾𝐾∗ ) and will repeat almost periodically for each

year. Therefore, once the transition probabilities are established and 𝔼𝔼{𝐻𝐻(𝑡𝑡𝑘𝑘)} is computed, the

optimisation is straightforward.

6.3 CONDITION-BASED CLEANING USING REAL DATA AND THE SYSTEM ADVISOR MODEL (SAM)

The methodology presented in the previous section requires the computation of the transition

probabilities 𝑝𝑝𝑘𝑘�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚,𝑎𝑎� and the expected potential revenue 𝔼𝔼{𝐻𝐻(𝑡𝑡𝑘𝑘)}. This section describes

a methodology to compute these quantities at any specific site using historical weather

conditions and electricity prices and the System Advisor Model (SAM) developed by the

National Renewable Energy Laboratory (NREL) (Blair et al.).

As noted in Section 6.2.2, reflectivity may be described as a Markov process, in which

transition probabilities depend on the soiling rate and rain effects. For any particular site, rain

data may be obtained from historical observations from weather services (e.g. Australian


Bureau of Meteorology). Soiling rate data for a specific site may be obtained from experimental

data or estimated from literature.

The potential revenue of the total cleaning cost model (Section 6.2.3) requires a statistical

characterisation of the electricity prices and DNI (described in Eq. (6.8)). While the expected

values (estimated via historical averages) of these quantities are sufficient for optimisation, a

broader statistical characterisation will enable a more complete validation via Monte Carlo

simulation. This characterisation can be obtained on the basis of widely-available datasets from

such as the Australian Energy Market Operator (Australian Energy Market Operator (AEMO))

in Australia, the New York Independent System Operator (NYISO) (New York Independent

System Operator (NYISO)) and the Electric Reliability Council of Texas (ERCOT) (Electric

Reliability Council of Texas (ERCOT)), Nord Pool (Nord Pool) in northern Europe, and many

publicly available weather data sites (Cooperative Network for Renewable Resource

Measurements (CONFRRM); National Centers for Environmental Information (NCEI)).

In addition to external factors, the characteristics of the CSP plant are key factors influencing

the potential revenue. In the case of an operating plant, the features are already known to the

operator. However, it is also possible to apply the cleaning optimisation strategy in the site

selection and plant design phase by using well-established software to obtain the necessary

plant characteristics. In particular, the freely available System Advisor Model (SAM)

developed by NREL (Blair et al.) allows the quantification of system efficiency based on a

specific plant design.

Combining these data resources, it is clear that the methodology is directly applicable to plants

in a large number of geographical locations, both in the design and operational phase. The main

limitation remains the identification of site-specific soiling rates since one must still rely on the

limited availability of experimental data. Only the development of physical soiling models

could alleviate the need for ad hoc experimental soiling measurement. Unfortunately, currently

available soiling models are not only incomplete, but are also inaccurate and highly dependent

on uncertain parameters. The analysis, improvement and implementation of such models in the

policy is left to future work.

6.3.1 Data-based computation of transition probabilities

6.3.1.1 A statistical reflectivity model due to soiling process

The probability models corresponding to each term in Eq. (6.2) will now be described. The

soiling process model describes the time evolution of soiling leading to the reflectivity loss. In


general, soiling rates possess complex dependencies on many factors, including meteorological

conditions (e.g. humidity, temperature, wind), properties of the local dust in the air, and

properties of the mirrors themselves. Instead of developing a detailed model of the entire

physical phenomenon concerned (many of which are poorly understood), a stochastic process

model may be developed based on the experimental data at the designed plant location or

historical data from an operating plant. The collected data of mirror reflectivity loss over a

certain interval with length ∆𝑡𝑡 can be matched to some distribution 𝒮𝒮, such as Gamma

distribution (Guida et al., 2012; Kallen and van Noortwijk, 2005; van Noortwijk, 2009):

∆𝑆𝑆𝑘𝑘 ∼ 𝒮𝒮 (6.14)

which is widely used in degradation modelling due to its strictly positive increments.

The degradation state is said to be in state 𝑒𝑒𝑚𝑚 if the actual (continuous) reflectivity loss lies in

the interval �𝑒𝑒𝑚𝑚 −∆𝜌𝜌2

, 𝑒𝑒𝑚𝑚 + ∆𝜌𝜌2� as described in Section 6.2.1. Therefore, the transition

probability of reflectivity in the absence of rain and cleaning may then described as the

probability of the incremental degradation ∆𝑆𝑆𝑘𝑘 to be such that it brings the reflectivity from

given state 𝑒𝑒𝑚𝑚 to �𝑒𝑒𝑗𝑗 −∆𝜌𝜌2

, 𝑒𝑒𝑗𝑗 + ∆𝜌𝜌2�, i.e.:

𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗�𝑒𝑒𝑚𝑚, no rain� ≅ Pr �𝑒𝑒𝑗𝑗 −∆𝜌𝜌2≤ 𝑒𝑒𝑚𝑚 + ∆𝑆𝑆𝑘𝑘 < 𝑒𝑒𝑗𝑗 +

∆𝜌𝜌2� (6.15)

which can be computed directly by integrating the appropriate intervals of the distribution 𝒮𝒮 or

indirectly via Monte Carlo simulation. Clearly, seasonal variation in meteorological conditions

could also affect the soiling rate, which would result in a time-varying distribution 𝒮𝒮𝑘𝑘.

However, due to limited reflectivity degradation data, seasonal variation in the soiling rates is

neglected in this study. However, if more detailed historical site-specific soiling data is

available, the transition probabilities can be simply extended to time-varying soiling rates

without altering the optimisation process.

6.3.1.2 Data-based rain model

Two stochastic characteristics of rain events should be considered in cleaning policy: Rain

occurrence and cleaning effect on reflectivity. Rain occurrences are random but have seasonal

distributions according to the climate of the plant location. This seasonal variation of rain

occurrence can be modelled using a time-varying arrival rate 𝜆𝜆(𝑡𝑡) of rain events. This intensity

can be estimated based on daily rainfall data, 𝐶𝐶𝐹𝐹𝑎𝑎,𝑑𝑑, where 𝑑𝑑 = 1, … , 365 denotes the day of

the year (neglecting leap years), and 𝑦𝑦 = 1,2, … ,𝑌𝑌 denotes the year.


The effect of rain has been the subject of several studies. According to a study conducted by

Kimber et al. (2006), the daily rainfalls between 5-10mm result in cleaning of mirrors.

Similarly, Hammond et al. (1997) identified a 5mm rain intensity as the threshold to obtain

almost perfectly clean PV modules, while Mejia and Kleissl identified a threshold of 2.5mm

(Mejia and Kleissl, 2013). On the contrary, Naeem and Tamizhmani observed that a much

higher amount of rain (≥ 13mm) was required to clean the surfaces (Naeem and Tamizhmani,

2015). There are also a number of studies noting that light rainfall can actually enhance soil

deposition (Bergeron and Freese, 1981; Maghami et al., 2016; Sayyah et al., 2013;

Wolfertstetter et al., 2014b). Considering the available quantitative literature on this topic, the

model of rain events on soiling will include: 1) an intensity threshold above which rain events

are considered potentially cleaning; and 2) the possibility that the rain event may result in a net

reflectivity loss.

A rain event is defined as follows:

𝐶𝐶𝑂𝑂𝑎𝑎,𝑑𝑑 = �1 𝐶𝐶𝐹𝐹𝑎𝑎,𝑑𝑑 ≥ 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑐𝑐𝑙𝑙𝑑𝑑0 otherwise

(6.16)

where 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑐𝑐𝑙𝑙𝑑𝑑 is the threshold above which rain is considered sufficiently intense to have a

potential cleaning effect. For most of this work, 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑐𝑐𝑙𝑙𝑑𝑑 = 8mm, which fits well within the

range of the available literature. Altering this threshold has the effect of changing the rain

probabilities, the effect of which on the CBC policy is examined later in this study (Section

6.5.3.3)

The estimation of the statistical properties of rain events (𝜆𝜆(𝑡𝑡)) is based on two assumptions:

• Long-term (yearly) cyclostationarity. The time-varying statistics of rain events

repeat periodically every 365 days. Therefore, the rain occurrence on a specific day

of different years are considered to be samples from the same distribution.

• Short-term almost-stationarity. The statistics of rain vary slowly and are considered

stationary over a week timescale.

Therefore, the arrival rate of rain �̅�𝜆𝑑𝑑 during day 𝑑𝑑 of any year is estimated as the combination

of the moving average of seven days (three days before, three days after and current day) and

the synchronous average (i.e average of the same day over 𝑌𝑌 years):

�̅�𝜆𝑑𝑑 =1

7𝑌𝑌� �𝐶𝐶𝑂𝑂𝑎𝑎,𝑑𝑑+𝑚𝑚

𝑌𝑌

𝑎𝑎=1

3

𝑚𝑚=−3

(6.17)


The rain occurrences are subsequently modelled as a non-homogenous Poisson process with

arrival rate 𝜆𝜆(𝑡𝑡). Thus, the chance of rain in any arbitrary interval [𝑡𝑡𝑘𝑘 , 𝑡𝑡𝑘𝑘+1), 𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘) in Eq.

(6.2), is determined as:

where 𝑁𝑁𝑅𝑅(𝑡𝑡) is the number of rain events up to time 𝑡𝑡, and Λ(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1) is the mean value

function considering the time interval [𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1). Imposing a decision epoch length of ∆𝑡𝑡 =

1 day (which is sufficiently fine to describe the slow degradation process), the mean value

function is simply:

Λ(𝑡𝑡𝑘𝑘, 𝑡𝑡𝑘𝑘+1) = � 𝜆𝜆(𝜏𝜏)𝑑𝑑𝜏𝜏

𝑡𝑡𝑘𝑘+∆𝑡𝑡

𝑡𝑡𝑘𝑘

= 𝜆𝜆𝑑𝑑 ⋅ ∆𝑡𝑡 . (6.19)

The cleaning effect of rain events depend on many factors, including the amount of rainfall.

Long and heavy rain may clean mirrors nearly perfectly while short and light rain events may

actually increase the reflectivity loss (Bergeron and Freese, 1981). Therefore, the cleaning

effect is modelled as a random variable that is supposed to be independent to the current state

of mirrors. Similar to the reflectivity loss model, the distribution of the rain cleaning effect and

its parameters can be obtained via experimental data at plant location or measured data from

an operating plant. The collected data of mirror reflectivity loss after rain events can be fitted

to some distribution ℛ:

𝑆𝑆𝑅𝑅 ~ ℛ (6.20)

The transition probability for the reflectivity loss after a rain event is modelled as a stochastic

restoration of the reflectivity that is independent on the current reflectivity:

𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|𝑒𝑒𝑚𝑚, rain� = 𝑝𝑝𝑆𝑆�𝑒𝑒𝑗𝑗|rain� = Pr �𝑒𝑒𝑗𝑗 −∆𝑒𝑒2≤ 𝑆𝑆𝑅𝑅 ≤ 𝑒𝑒𝑗𝑗 +

∆𝑒𝑒2� ∀𝑖𝑖 (6.21)

which can be computed directly by integrating the appropriate intervals of the distribution ℛ

or indirectly through Monte Carlo simulation. With the rain event transition probabilities

described in Eq. (6.21), the reflectivity of the mirrors after a rain event may be worse than prior

to the rain event, especially when the mirrors are already in a fairly clean state (small 𝑒𝑒𝑚𝑚).

However, the enhanced soiling effects of light rain and other adverse weather conditions (noted

in previous studies) are not modelled directly but are assumed to be absorbed into the

uncertainty in the soiling rate (i.e. contribute to the variance of the soiling rate distribution

𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘) = 1 − Pr[𝑁𝑁𝑅𝑅(𝑡𝑡𝑘𝑘 + ∆𝑡𝑡) −𝑁𝑁𝑅𝑅(𝑡𝑡𝑘𝑘) = 0] = 1 − 𝑒𝑒−Λ(𝑡𝑡𝑘𝑘,𝑡𝑡𝑘𝑘+1) (6.18)


(6.15)). Nevertheless, a similar statistical model of light rain events can be used to describe the

effects of light rain.

It is noted that the effectiveness of rains depends on some other weather factors (e.g. dust

concentration, wind speeds, etc.). Hence, using some data mining techniques to integrate all

these factors should be a better approach to model the cleaning effects of rains. However, the

limited data cannot support for following this approach and this is out of scope of this study.

6.3.2 Data-based computation of Potential Revenue 𝑯𝑯(𝒕𝒕𝒌𝒌)

Given the discrete nature of available price and DNI datasets, the decision epoch ∆𝑡𝑡 = 𝑡𝑡𝑘𝑘+1 −

𝑡𝑡𝑘𝑘 is divided into smaller time intervals ∆𝑡𝑡𝐿𝐿

, corresponding to the sampling interval in the data.

Under the hypotheses that 1) the price and DNI are stationary over the interval; and 2) 𝜂𝜂 = �̅�𝜂

is constant over the interval, the expected maximum revenue can be computed as:

𝔼𝔼{𝐻𝐻(𝑡𝑡𝑘𝑘)} = � 𝔼𝔼{𝑝𝑝(𝜏𝜏) ⋅ 𝐷𝐷𝑁𝑁𝐼𝐼(𝜏𝜏) ⋅ 𝜂𝜂(𝜏𝜏)} ⋅ 𝑑𝑑𝜏𝜏

𝑡𝑡𝑘𝑘+1

𝑡𝑡𝑘𝑘

= �𝔼𝔼�𝐷𝐷𝑁𝑁𝐼𝐼 �𝑡𝑡𝑘𝑘 +ℓ∆𝑡𝑡𝐿𝐿� ⋅ 𝑝𝑝 �𝑡𝑡𝑘𝑘 +

ℓ∆𝑡𝑡𝐿𝐿�� ⋅ �̅�𝜂 �𝑡𝑡𝑘𝑘 +

ℓ∆𝑡𝑡𝐿𝐿�

𝐿𝐿−1

ℓ=0

⋅∆𝑡𝑡𝐿𝐿

(6.22)

The remainder of Section 6.3.2 is devoted to computing the efficiency and the expected values

for price and DNI. For this work, ∆𝑡𝑡 = 1 day, and 𝐿𝐿 = 24, making ∆𝑡𝑡𝐿𝐿

= 1 hour. The decision

epoch is sufficiently fine to describe the (slow) degradation process, while the choice of 𝐿𝐿 is

driven by the (typical) sampling intervals of the available data.

6.3.2.1 DNI and price empirical model

A DNI and electricity price model for the execution of an optimisation study can be based on

empirical data, often available from satellite or ground-station measurements (Dugaria et al.,

2015). The following model is based on an hourly measurement of DNI and electricity price,

collected along a series of years. The evolution of DNI and price in a specific day 𝑑𝑑 =

1, … , 365 of a specific year 𝑦𝑦 = 1, … ,𝑌𝑌 (for simplicity enumerated starting from 1) is

represented by the set of functions 𝐷𝐷𝑁𝑁𝐼𝐼𝑎𝑎,𝑑𝑑(ℎ) and 𝑃𝑃𝑒𝑒𝑖𝑖𝑐𝑐𝑒𝑒𝑎𝑎,𝑑𝑑(ℎ), where ℎ = 0, … , 23 represents

the hour of the day. In order to develop empirical DNI and price models, the following

assumptions are considered:


• A high correlation between hourly samples within the same day was to be considered

to reproduce the common observation of clear days vs cloudy days. In particular, it

motivates the decision to keep a “DNI-day” (i.e. a 24-hour DNI sequence) as the

smallest unit of DNI sampling. Actual DNI-days cannot be split and/or undergo an

hour permutation, since this would violate the hourly correlation assumption. The

same correlation between hourly samples of electricity prices within the same day

(e.g. the high demand and low demand days) was also considered for sampling a

“price-day”.

�𝐷𝐷𝑁𝑁𝐼𝐼𝑎𝑎,𝑑𝑑(ℎ) ∶ ℎ = 1,2, … ,23�~𝒟𝒟(𝑑𝑑,𝑎𝑎)

(𝑃𝑃𝑒𝑒𝑖𝑖𝑐𝑐𝑒𝑒𝑎𝑎,𝑑𝑑(ℎ) ∶ ℎ = 1,2, … ,23)~𝒫𝒫(𝑑𝑑,𝑎𝑎) (6.23)

• Long-term cyclostationarity (yearly cycle). The time-varying statistics of DNI and

price repeat periodically every 365 days (small approximation from the actual 365.25

cycle length). Therefore the 𝑌𝑌 measurements of DNI and price taken in 𝑌𝑌 different

years of data at the same hour and day were considered as samples from the same

distribution 𝒟𝒟(𝑑𝑑) and 𝒫𝒫(𝑑𝑑):

𝒟𝒟(𝑑𝑑,𝑎𝑎) ≅ 𝒟𝒟(𝑑𝑑) for all 𝑦𝑦 = 1, … ,𝑌𝑌.

𝒫𝒫(𝑑𝑑,𝑎𝑎) ≅ 𝒫𝒫(𝑑𝑑) for all 𝑦𝑦 = 1, … ,𝑌𝑌. (6.24)

• Short-term almost-cyclostationarity (daily cycle). The statistics of DNI and price

repeat almost periodically on a daily basis, with the daily DNI and price cycle

varying very slowly with seasons. In particular, the variation of daily DNI and price

statistics was considered small (and therefore neglected) in a 31-day sliding window

(15 days before, 15 days after and current day):

�𝐷𝐷𝑁𝑁𝐼𝐼𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) ∶ ℎ = 1,2, … ,23�~𝒟𝒟(𝑑𝑑) for all 𝑖𝑖 = −15, … ,15 ; 𝑦𝑦 = 1, … ,𝑌𝑌

(𝑃𝑃𝑒𝑒𝑖𝑖𝑐𝑐𝑒𝑒𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) ∶ ℎ = 1,2, … ,23)~𝒫𝒫(𝑑𝑑) for all 𝑖𝑖 = −15, … ,15 ; 𝑦𝑦 = 1, … ,𝑌𝑌

approximately

(6.25)

• To capture the interdependence between price and DNI, instead of sampling DNI

and price separately, a set of hourly products of DNI and price, denoted:

𝑃𝑃𝐷𝐷𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) ≜ 𝐷𝐷𝑁𝑁𝐼𝐼𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) ⋅ 𝑃𝑃𝑒𝑒𝑖𝑖𝑐𝑐𝑒𝑒𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) (6.26)

are described by the “PD-day” distribution 𝒫𝒫𝒟𝒟(𝑑𝑑):


(𝑃𝑃𝐷𝐷𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) ∶ ℎ = 1,2, … ,23)~𝒫𝒫𝒟𝒟(𝑑𝑑) for i = −15, … ,15 ; 𝑦𝑦 = 1, … ,𝑌𝑌

approximately (6.27)

Combining these assumptions, it is possible to build an empirical distribution 𝒫𝒫𝒟𝒟� (𝑑𝑑)for each

“PD-price-day” of the year with 31 ⋅ 𝑌𝑌 daily profiles of price and DNI is similarly constructed:

𝒫𝒫𝒟𝒟 � (𝑑𝑑) = �(𝑃𝑃𝐷𝐷𝑎𝑎,𝑑𝑑+𝑚𝑚(ℎ) ∶ ℎ = 1,2, … ,23) ∶ 𝑦𝑦 = 1, … ,𝑌𝑌 and 𝑖𝑖 = −15, … ,15� (6.28)

These sets are then used to obtain an average yearly history of price-DNI (“average year” of

price-DNI) required for the calculation of the potential revenue of Eq. (6.22):

𝔼𝔼 �𝐷𝐷𝑁𝑁𝐼𝐼 �𝑡𝑡𝑘𝑘 +ℓ∆𝑡𝑡𝐿𝐿� ⋅ 𝑝𝑝 �𝑡𝑡𝑘𝑘 +

ℓ∆𝑡𝑡𝐿𝐿�� =

131𝑌𝑌

� � 𝑃𝑃𝐷𝐷𝑎𝑎,𝑑𝑑𝑚𝑚𝑎𝑎(𝑡𝑡𝑘𝑘)+𝑚𝑚(ℓ)15

𝑚𝑚=−15

𝑌𝑌

𝑎𝑎=1

(6.29)

where 𝑑𝑑𝑎𝑎𝑦𝑦 (𝑡𝑡𝑘𝑘) is the day of the year corresponding to the time 𝑡𝑡𝑘𝑘. The distributions in Eq.

(6.28) are used to simulate the 𝑃𝑃𝐷𝐷 profiles used in the results section of this chapter. To

simulate a PD history over an arbitrary number of years, each “PD-day” [𝑃𝑃𝐷𝐷]𝑎𝑎,𝑑𝑑𝑙𝑙𝑚𝑚𝑚𝑚(ℎ) of the

simulation was sampled randomly among the available samples in the corresponding set 𝒫𝒫𝒟𝒟� (𝑑𝑑)

of available daily profiles for the day 𝑑𝑑 of the year.

This basic empirical distribution approach offers a simple and effective strategy for the actual

implementation of the cleaning optimisation procedure described in this chapter, based on very

general assumptions and commonly available data for a wide range of locations. This level of

modelling accuracy also allows showing the advantages of this technique based on realistic

data. Further improvements in the methodology and results could be achieved by implementing

more complex modelling approaches for DNI and price, which are however, outside the scope

of this study.

6.3.2.2 Efficiency of a CSP plant

Other than DNI and electricity price, the overall plant efficiency, excluding mirror reflectivity,

needs to be identified for the expected revenue per square metre as defined in Eq. (6.8). To this


end, a simulated tower CSP plant is designed using a (SAM) baseline plant (Blair et al.). In

general, the plant efficiency at time 𝑡𝑡𝑘𝑘 can be defined as10:

�̅�𝜂(𝑡𝑡𝑘𝑘) = 𝜂𝜂𝑙𝑙𝑝𝑝𝑡𝑡𝑚𝑚𝑐𝑐𝑚𝑚𝑙𝑙(𝑡𝑡𝑘𝑘) ∙ 𝜂𝜂𝑡𝑡ℎ𝑒𝑒𝑟𝑟𝑚𝑚𝑚𝑚𝑙𝑙 . (6.30)

For a CSP plant without storage, an approximate thermal efficiency can be given by:

𝜂𝜂𝑡𝑡ℎ𝑒𝑒𝑟𝑟𝑚𝑚𝑚𝑚𝑙𝑙 = 𝜂𝜂𝑡𝑡ℎ𝑒𝑒𝑟𝑟𝑚𝑚𝑚𝑚𝑙𝑙 𝑐𝑐𝑎𝑎𝑐𝑐𝑙𝑙𝑒𝑒 ∙ 𝜂𝜂𝑚𝑚𝑒𝑒𝑡𝑡 𝑐𝑐𝑙𝑙𝑚𝑚𝑣𝑣𝑒𝑒𝑟𝑟𝑙𝑙𝑚𝑚𝑙𝑙𝑚𝑚 (6.31)

where thermal cycle loss and net power conversion are considered, and the efficiency is

directly estimated based on the default design parameters from SAM. The optical efficiency

(excluding reflectivity) can be calculated as:

𝜂𝜂𝑙𝑙𝑝𝑝𝑡𝑡𝑚𝑚𝑐𝑐𝑚𝑚𝑙𝑙(𝑡𝑡𝑘𝑘) = 𝜂𝜂𝑐𝑐𝑙𝑙𝑙𝑙(𝑡𝑡𝑘𝑘) ∙ 𝜂𝜂𝑙𝑙ℎ𝑚𝑚𝑑𝑑𝑚𝑚𝑚𝑚𝑙𝑙 ∙ 𝜂𝜂𝑏𝑏𝑙𝑙𝑙𝑙𝑐𝑐𝑘𝑘𝑚𝑚𝑚𝑚𝑙𝑙 ∙ 𝜂𝜂𝑚𝑚𝑡𝑡𝑡𝑡𝑒𝑒𝑚𝑚𝑢𝑢𝑚𝑚𝑡𝑡𝑚𝑚𝑙𝑙𝑚𝑚

∙ 𝜂𝜂𝑙𝑙𝑝𝑝𝑚𝑚𝑙𝑙𝑙𝑙𝑚𝑚𝑙𝑙𝑒𝑒 ∙ 𝜂𝜂𝑚𝑚𝑏𝑏𝑙𝑙𝑙𝑙𝑟𝑟𝑝𝑝𝑡𝑡𝑚𝑚𝑚𝑚𝑐𝑐𝑒𝑒 ∙ 𝜂𝜂𝑒𝑒𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑐𝑐𝑒𝑒 ∙ 𝜂𝜂ℎ𝑒𝑒𝑚𝑚𝑡𝑡 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 (6.32)

including the time-varying mirror cosine loss, air attenuation loss, spillage, blocking and

shading, receiver absorptance and emittance and additional heat loss on receiver. Aside from

cosine losses, all efficiency components in SAM are assumed to be constant (thus neglecting

their potential dependency on time, DNI etc.). Based on the simulated plant, heliostat layout

information can be generated via geometry optimisation so that hourly changing cosine

efficiencies can be calculated for each mirror by using its geometric position and sun angles

for the particular site. Regarding the rest of the efficiency components, upfront default values

obtained from the SAM baseline plant are utilised.

6.4 POLICY EVALUATION STRATEGY USING MONTE CARLO SIMULATION

Monte Carlo simulation is used to evaluate the cleaning policies to give insight into the variance

of the cleaning costs and seasonal variation of the resulting cleaning policy. Given:

• cleaning policy 𝜋𝜋;

• number of decision epochs 𝐾𝐾;

• time and reflectivity discretisation parameters ∆𝑡𝑡 and ∆𝜌𝜌 respectively;

10 All efficiencies are obtained using SAM and represent an “hour-field” average, both within the

hourly discretisation �∆𝑡𝑡𝐿𝐿� and across the field.


• distribution of reflectivity degradation increments 𝒮𝒮;

• rain cleaning effect distribution and rain occurrence probabilities, ℛ and 𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘)

respectively;

• cleaning cost 𝐶𝐶𝑐𝑐𝑙𝑙;

• plant efficiency 𝜂𝜂(𝑡𝑡𝑘𝑘);

• empirical distributions 𝒫𝒫𝒟𝒟� (𝑑𝑑) of product 𝑃𝑃𝐷𝐷 = 𝑃𝑃𝑒𝑒𝑖𝑖𝑐𝑐𝑒𝑒 × 𝐷𝐷𝑁𝑁𝐼𝐼.

The cleaning policy is evaluated by executing the simulation outlined in the flow chart in

Figure 6.3.

Figure 6.3. Monte Carlo simulation of Cleaning Policy 𝝅𝝅.

Each simulation run for one year will result in different investigated outcomes according to the

random input parameters that are generated from developed distribution discussed in Section


6.3. By repeating the simulation 𝑁𝑁𝑟𝑟 times, key quantities such as the average values of cost and

number of cleans may be easily computed:


𝑁𝑁𝑟𝑟 and 𝑁𝑁𝑐𝑐𝑙𝑙 =

∑ 𝑁𝑁𝑐𝑐𝑙𝑙(𝑒𝑒)𝑟𝑟

𝑁𝑁𝑟𝑟

6.5 CASE STUDY – BRISBANE (QUEENSLAND), AUSTRALIA

To evaluate the effectiveness of the proposed cleaning policy, a case study is conducted for a

hypothetical CSP facility located in Brisbane. The empirical models’ DNI and electricity prices

are developed for Brisbane as are the rain occurrence and reflectivity degradation model.

6.5.1 DNI and electricity prices in Brisbane

The historical hourly DNI data (2005 to 2015) for Brisbane, were obtained from Australian

Renewable Energy Mapping Infrastructure (AREMI) and pre-processed with linear

interpolation for missing values as shown in Figure 6.4. The available 30 minutes averaged

electricity price data (between 2005 and 2015) in Queensland were downloaded from

Australian Energy Market Operator (AEMO) and pre-processed for hourly averaged price and

is shown in Figure 6.5.

Based on the historical data, the average price of product and DNI and their empirical

distributions can be constructed. The hourly average PD is shown in Figure 6.6 and is

computed according to Eq. (6.29) (the averages of DNI and price are computed in a similar

way). Figure 6.7 shows the result of sampling the empirical distributions of the “PD-day”

described in Section 6.3.2.1.


Figure 6.4. Historical DNI in Brisbane from 2005 to 2015 .

Figure 6.5. Historical price in Brisbane from 2005 to 2015.

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time (years)

0

2000

4000

6000

8000

10000

12000Prices from 2005 ~ 2015AUD$/MWh

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time (years)

-100

0

100

200

300Zoomed prices (-100 ~ 300 AUD$/MWh)

AUD$/MWh


Figure 6.6. Produced hourly average DNI, Price and Price × DNI over one year.

Figure 6.7. Produced random hourly samples of Price × DNI over one year.


6.5.2 Solar plant configuration, efficiency and cleaning costs

Alongside DNI and electricity prices, the overall plant efficiency must be estimated to compute

the expected potential revenue. In this study, a simulated CSP tower plant in Brisbane was

designed in SAM (Blair et al.) using the default design parameters. The heliostat layout was

optimised as shown in Figure 6.8 (8695 mirrors in total were generated).

The hourly cosine efficiency over one year can be determined based on the geometry

information, receiver tower designing height and local geographic information of Brisbane. For

the mirrors highlighted in Figure 6.8, hourly efficiency over the whole year were presented in

Figure 6.9. By assuming the entire solar field as one single mirror, the obtained hourly

efficiency was arithmetically averaged across the entire field over each hour. Since the

remaining efficiency components are relatively constant, the total efficiency excluding mirror

reflectivity can be directly calculated as the product of cosine efficiency and a number of

default values from SAM for the simulated plant.

Figure 6.8. A simulated CSP plant layout in Brisbane, Queensland.

-2000 -1500 -1000 -500 0 500 1000 1500 2000

East <--------> West (m)

-2000

-1500

-1000

-500

0

500

1000

1500

Nor

th <

------

--> S

outh

(m) 1

23

4

5

6

7

8


Figure 6.9. Hourly cosine efficiencies for selected mirrors over one year.

The unit mirror cleaning cost 𝐶𝐶𝑐𝑐𝑙𝑙 was also approximated based on reference plant costs in SAM

using the “sam-costs-2013-molten-salt-power-tower.xlsx” spreadsheet (System Advisor

Model (SAM); Turchi and Heath, 2013). The SAM default value of 63 washing events per year

were assumed and water usage per wash was taken as 0.7 L m2, aperture⁄ . Using a bulk water

price of approximately 2.3 AUD/m3 for Brisbane (Queensland Department of Energy and

Water Supply) and the labour costs from the spreadsheet of 350,000 AUD (assuming AUD ≈

USD in 2013), the cleaning cost may be obtained as:

𝐶𝐶𝑐𝑐𝑙𝑙 = 𝐶𝐶𝑤𝑤𝑚𝑚𝑡𝑡𝑒𝑒𝑟𝑟 + 𝐶𝐶𝑙𝑙𝑚𝑚𝑏𝑏𝑙𝑙𝑟𝑟 + 𝐶𝐶𝑙𝑙𝑡𝑡ℎ𝑒𝑒𝑟𝑟

≥ 0.7𝐿𝐿

m2 wash× 2300 × 10−6

𝑃𝑃𝑈𝑈𝐷𝐷𝐿𝐿

+ 350,000 AUD

63 washes ⋅ 𝑃𝑃𝑚𝑚𝑝𝑝𝑒𝑒𝑟𝑟𝑡𝑡𝑢𝑢𝑟𝑟𝑒𝑒

≈ 0.0076AUD

m2 wash

(6.33)

where 𝑃𝑃𝑚𝑚𝑝𝑝𝑒𝑒𝑟𝑟𝑡𝑡𝑢𝑢𝑟𝑟𝑒𝑒 = 920,678.1 m2 is the aperture area of the reference plant mirrors.

Considering the uncertainties of price, cleaning technology, labour and other costs (which are

likely higher in the Australian context), a wider range of [0.005, 0.05] AUD/m2 is used in this

study to show the cost impact of the proposed method in the presence of high labour costs.

Time (hours)

Cos

ine

effic

ienc

ies

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror3

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror6

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror7

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0

0.5

1

mirror8


The daily rain probabilities were determined based on the historical data of daily rain in

Brisbane (Bureau of Meteorology (BoM) and Australia) (Figure 6.10). As described in Section

6.3.1.2, a rainy day is the day with rainfalls higher or equal to 8 mm. These daily rain

probabilities are estimated according to procedure discussed in Section 6.3.1.2 with data of

Brisbane daily rainfalls from year 1910 to 2015. The rain is likely to occur in the beginning

and the end of the year (summer) while the dry season occurs in the middle of the year (winter).

Figure 6.10. Average Daily probability of rain in Brisbane, Australia.

6.5.3 Evaluation of condition-based cleaning policies vs time-based policy

In this section, the proposed cleaning policy is compared over a 10 year horizon (i.e. 𝐾𝐾 ≈

3650 days) to the traditional policy where mirrors are cleaned at a fixed predetermined cycle

(Bergeron and Freese, 1981). The fixed cycle is determined according to Bergeron and Freese

(1981):

𝑇𝑇𝑐𝑐𝑎𝑎𝑐𝑐 = �2𝐶𝐶𝑐𝑐𝑙𝑙

𝜂𝜂 ⋅ 𝐷𝐷𝑁𝑁𝐼𝐼 ⋅ 𝑝𝑝 ⋅ ∆𝑆𝑆 (6.34)

where �̅�𝜂, 𝐷𝐷𝑁𝑁𝐼𝐼, 𝑝𝑝 are the daily average values of total efficiency, solar direct normal irradiance

and electric price per square metre respectively; 𝐶𝐶𝑐𝑐𝑙𝑙 is the cleaning cost per square metre of

mirrors; and ∆𝑆𝑆 is the average daily loss of reflectivity.

In order to apply the proposed methodology, distributions of reflectivity degradation and rain

effects are assumed based on values available in literature. The degradation of mirror


reflectivity Gamma process parameters were set to 𝛼𝛼 = 4 ;𝛽𝛽 = 0.001 (which was established

based on a mean reflectivity loss of 0.004 per day (Deffenbaugh et al., 1986; Kattke and Vant-

Hull, 2012)) . The reflectivity of heliostats after a rain event is assumed to follow a Weibull

distribution with parameters scale and shape parameters (𝛼𝛼𝑅𝑅 = 0.471 ;𝛽𝛽𝑅𝑅 = 4.5 × 10−3)

respectively, which correspond to a mean after-rain reflectivity loss of 0.01 and standard

deviation of 0.02.

Three mirror cleaning policies are determined and compared when the cleaning cost is 𝐶𝐶𝑐𝑐𝑙𝑙 =

0.015 (AUD/m2):

1. The proposed strategy, considering rain in the optimisation.

2. The proposed strategy considering 𝑝𝑝𝑅𝑅(𝑡𝑡𝑘𝑘) = 0 ∀𝑡𝑡𝑘𝑘, i.e. not considering rain in the

optimisation.

3. A fixed cycle cleaning scheduled from Eq. (6.34).

The cleaning policies are evaluated via Monte Carlo simulation for one year with 10,000

repetitions. Please note that the reported quantities are annual averages over the ten-year

horizon. The computation time for both optimisation and simulation is about 3 hour (computer

with CPU core i7-4790 @3.6GHz, 16 GB Ram).

Figure 6.11. The cleaning reflectivity thresholds of two optimal policies for one year: Considering rain vs not considering rain. These thresholds repeat almost periodically for each year in the horizon.


Figure 6.11 shows the optimised reflectivity thresholds that trigger cleaning tasks for mirror

cleaning policies 1 and 2 (considering rain in the optimisation and not considering rain). When

the mirror reflectivity drops below the threshold, the cleaning task will be conducted. The

policy with rain consideration results in lower thresholds compared to a policy without rain

consideration, which is intuitive; this policy tends to wait briefly for the zero-cost rain event

cleanings.

Table 6.1

Comparisons among Three Cleaning Policies: CBC with/without considering Rain and Fixed Cycle Strategy

Cleaning Policies

CBC considering

rain

CBC without

considering rain

Fixed

cycle

Annual total cost

per m2

mean 0.7781 0.8155 0.9219

Std. dev 0.0369 0.0318 0.0323

% Savings 15.6% 11.5% n/a

Average number

of cleans per year

mean 18.2253 26.1052 33.1000

Std. dev 0.7910 0.8790 n/a

Table 6.1 shows the cost and number of cleans according to different policies over one year.

The simulated results indicate that the condition-based cleaning policies yield a cost saving up

to 15.6% compared to the traditional fixed-frequency cleaning. In the condition-based policies,

cleaning activities are triggered by reflectivity of mirrors. Therefore, the number of cleans is

reduced by about 21% (policy without considering rain) and 45% (policy considering rain).

Comparing only the two condition-based policies, the policy with rain consideration can

slightly reduce the costs (15.6% vs 11.5%). The number of cleans in the case when considering

rain is also lower than the case of not considering rain.

Figure 6.12 shows the probability of cleaning on each day of the year, computed by averaging

the cleaning actions over all 10 years. The probability of cleaning in the case of not considering

rain is higher than the case of considering rain for the whole year. Additionally, the cleaning

probabilities appear to adapt to the season when rain is considered in the optimisation. Hence,

the policy with considering rain can take advantage of the rain effect (clean mirrors without

cost) to further reduce cost.


Figure 6.12. Probability of a clean for each day of the year.

6.5.3.1 Sensitivity to cleaning cost

Figure 6.13 shows different percentages of cost savings when the cleaning cost varies. It

indicates that the savings of condition-based policies (compared to the fixed cycle policy) is

higher when the cleaning cost is high. This is reasonable since the purpose of condition-based

policy application is to reduce the number of cleans and makes cleaning activities occur at the

“right” time when the reflectivity is below some threshold value.

Comparing two cleaning policies with and without rain consideration, the savings of a policy

with rain increases slightly when the cleaning cost increases. The policy with rain consideration

aims to take cleaning advantage of rain instead of doing cleaning tasks. Therefore, when the

cleaning cost increases, the benefit of this policy is more significant.


Figure 6.13. The effect of cleaning cost on savings compared to the optimal fixed cycle strategy.

6.5.3.2 The effect of cleaning delay

In some situations, the cleaning activities cannot be conducted immediately when the triggering

condition is met. The reason might be due to the temporary unavailability of cleaning resources.

This situation probably occurs when the conditioning cleaning policies are applied because the

cleaning activities cannot be planned and this departure from optimality will decrease the

benefit of policies.

Figure 6.14 shows the changes in total cost due to cleaning delay. Here, it is assumed that no

delay occurs when the fixed cycle cleaning policy is applied since it is planned well in advance.

Clearly, the delay reduces the benefit of condition-based cleaning policies and the cost

increases when the delay is more. When the delay is expected to be high (12+ days), the

traditional policy should be considered because the condition-based cleaning methods are not

beneficial. Besides, the cost increase due to delay of the policy considering rain rises faster

than the method without rain consideration. Hence, when the delay is moderate (~7 days here),

the effect of rain can be ignored11.

11 Alternatively, this delay can be explicitly considered in the optimisation to alter the threshold appropriately. However, this is left to future work.


Figure 6.14. The effect of cleaning delay.

6.5.3.3 The effect of rain probabilities

The probabilities of rain in this illustrative example are estimated based on the historical data

of Brisbane. The results may change dramatically when the chances and seasonality of rains

are varied. This section is dedicated to discussing the benefit of the proposed CBC policy in

case the chances of rain occurrences are different. In order to completely study the effects of

rain probabilities on cost savings, the time-varying (seasonal) rain probabilities of other areas

should be determined with the same procedure for associating historical data. However, it is

challenging and out of the scope of this study. Therefore, to analyse the effect of rain

probabilities, it is assumed that the seasonal patterns of rain probabilities are similar to Figure

6.10, but the investigated probabilities are respectively set at 0% (no rain), 25%, 50%, 100%,

150% and 250% of the default rain probabilities (relevant to Brisbane).

Figure 6.15 shows the changes in cleaning cost per square metre when the likelihoods of rain

are varied. Compared to the fixed-frequency cleaning strategy, both CBC policies, with and

without rain consideration, are more beneficial even when no rain occurs. The cost savings of

both CBC policies increases when the probabilities of rain events are higher. Considering two

CBC polices, the benefits of them are the same in the situation of no rain and the policy with

rain consideration can yield more savings when the rains occur more frequently. It enables one

to make a conclusion that the CBC policies, especially the one considering rain effect, are the

better mirror cleaning strategies for CSP systems.


Figure 6.15. The effect of rain probabilities.

Figure 6.16 presents the different optimal reflectivity thresholds of CBC policies associating

with different probabilities of rain. The reflectivity thresholds of CBC without considering rain,

which are not varied to rain probabilities, is used to make a comparison. When the rains occur

more frequently (higher probabilities), the reflectivity thresholds tend to be lower. In the

situation where rain rarely or never happens, the reflectivity thresholds of policy with rain

consideration is exactly same as the policy not considering rain. It is reasonable because when

the rain occurrences are frequent, decision makers tend to delay (reduce the triggering

thresholds) the cleaning decisions for waiting rains.


Figure 6.16. Reflectivity thresholds for one year with different rain probabilities. These thresholds repeat almost periodically for each year in the horizon.

6.6 CONCLUSION

This study presents a novel approach for condition-based cleaning for CSP solar fields. The

proposed policy accounts for changes in mirror reflectivity, electricity prices and solar

irradiance. This cleaning policy also considers, for the first time, the effects of seasonal rain

events, which restore mirror reflectivity at no cost, in the optimisation model.

The proposed policies are evaluated based on the data of a hypothetical CSP system located in

Brisbane, Australia. Two developed policies, i.e. with and without considering rain, were

compared to traditional cleaning strategy where the cleaning tasks are conducted after a

predetermined fixed cycle time. The results show that the proposed methods in this study can

yield a significant savings compared to traditional approach. Moreover, the policy also gives a

considerable benefit even in the case when cleaning activities are delayed.

Future studies will focus on three issues. First, a sector-specific policy will be developed for

the situation where multiple field sectors are cleaned separately. Second, a deeper study will

focus on modelling the variation of meteorological conditions and electricity prices (especially

spikes) and integrating this model into the decision process of cleaning activities and thermal

storage. Finally, a physical model to estimate the soiling process of mirrors with and without


rain events will be developed and incorporated to provide physical insight into the soiling

process when actual on-site data availability is limited.

144 Chapter 7: Conclusions and future work

Chapter 7: Conclusions and future work

Overview

This chapter summarises the results from previous chapters regarding the opportunistic

maintenance models that consider the partial external opportunities for the first time.

Opportunistic maintenance (OM) policies have been widely studied in recent years and have

demonstrated benefits compared to traditional preventive maintenance. Literature was lacking

in studies on opportunities caused by external events that reduce the maintenance costs. The

savings of such external opportunities are uncertain and partial, which presents some challenge

in modelling and optimisation. Therefore, the primary goal of this research was to develop OM

models that consider the partial and external opportunities for maintenance of industrial

systems. Models were developed for both time- and condition-based approaches and the impact

of using short-term forecasts in OM decision process was studied. The proposed OM policies

have been analysed through simulated numerical example as well as a case study of a

hypothetical wind turbine located in Queensland, Australia.

This thesis has also developed innovative methods for optimising maintenance for the case

where the external opportunity provides a no-cost restoration of the system. This type of

opportunity is prevalent in solar energy systems where rain events can provide free cleaning of

solar collectors. Thus, a reflectivity-based cleaning policy for the mirrors of a hypothetical CSP

System in Brisbane, Australia considering natural rain events was examined. The case study

demonstrates that free opportunities in cleaning policy can make significant savings in the

operation and maintenance cost of CSP systems.

Chapter 7: Conclusions and future work 145

7.1 FINDINGS AND CONTRIBUTIONS

This thesis details research conducted in opportunistic maintenance (OM), with special

attention paid to partial external opportunities and the effects of their stochastic properties (e.g.

occurrences and durations) on OM policies. Four optimisation models for joint opportunistic

and preventive maintenance policies have been developed to fulfil the thesis objectives

mentioned in Chapter 1. The proposed policies in both time-based and condition-based

approaches have shown the benefits of considering the partial external opportunities in

reducing the total maintenance costs of industrial systems. This thesis has also discussed the

development of a fourth model where an opportunistic maintenance is optimised in the

presence of external opportunities that provide a zero-cost restoration. The savings attained

using the developed methods were demonstrated on numerical simulations and two case

studies: one in CSP systems and another in wind turbines. In both cases, natural weather events

(i.e. rain and low-wind events) provide the external opportunities.

Chapter 3 has presented the optimisation models for OM and PM where the time-based

approach is addressed. The optimisation of the maintenance policy decides the optimal time

for doing a perfect PM or OM activity when a suitable external opportunity arrives. Two

characteristics of real opportunities have been considered in the models: non-homogeneous

arrival and uncertain opportunity duration. Partial opportunities, which have durations less than

the PM required time, were also utilised for the first time. A time-varying minimum duration

threshold for the acceptance of opportunities has been considered and a time discretisation

approach was used to analytically compute the single-cycle cost rate. Numerical results have

shown that considering partial opportunities yields a significant saving compared to both

traditional PM policies (no OM) and to OM policies where only full opportunities are accepted

(i.e. only those at least as long as the required PM time). In particular, the results have indicated

that utilising partial opportunities yield higher savings when downtime costs are high.

The condition-based approach for OM policy was discussed in Chapter 4. The optimal

condition-based maintenance strategy has investigated partial opportunities for maintenance

over a finite asset mission. This model has also included the analysis of a set of imperfect

maintenance actions. The mission total maintenance cost has been minimised through an MDP

where both the system degradation thresholds for different PM actions and the selection of OM

were jointly optimised. The proposed OM policy has been illustrated in a case study of a

hypothetic wind turbine at a wind farm at Mount Emerald, Queensland where the external

opportunities are caused by the low-wind speed days with little-to-no production. The


numerical results have shown that the maintenance policy considering partial external

opportunities would yield significant savings in maintenance cost in comparison with the

traditional CBM policy and with respect to considering full opportunities only. The savings of

applying the developed condition-based OM policy are about 8% of maintenance cost and

increase to more than 15% if the wind speed threshold defining opportunities is set higher than

the cut-in value. Moreover, the simulation results reconfirmed the intuitive hypothesis that

savings are more significant when downtime cost is dominant in total maintenance cost.

To enhance the benefit of implementation of the OM policies considering the external partial

opportunities, short-term forecasts were integrated into the maintenance decisions for both the

time- and condition-based approaches in Chapter 5. The total maintenance cost was minimised

through an MDP where system state thresholds (e.g. degradation in CB-OM and age in TB-

OM) for different PM actions as well as optimal OM selection were jointly optimised. The

proposed policies have been analysed in numerical examples through the Monte Carlo

simulation. The results demonstrated that using opportunity forecasts could yield a significant

saving in the implementation of OM policies. Moreover, the simulated results confirmed the

intuitive hypothesis that the benefits of forecasts are greater when they are more accurate and

longer-term.

In Chapter 6, a maintenance optimisation methodology was developed for external

opportunities that provided zero-cost restoration (and must be accepted when they arrive).The

policy was again determined using an MDP with transition probabilities that account for the

possibility of the zero-cost restorations. The methodology was developed using a case study on

reflectivity-based cleaning policy for mirrors of a hypothetical CSP system located in Brisbane,

Queensland, Australia, where rain events could imperfectly restore mirror reflectivity. The

proposed policy accounted for changes in mirror reflectivity, electricity prices and solar

irradiance. This cleaning policy has also considered, for the first time, the effects of seasonal

characteristics of rain events in the optimisation model. Thus, the proposed cleaning strategy

has provided two advantages: 1) cleaning actions are only carried out when required since they

are based on the current (monitored) reflectivity of mirrors; and 2) cleaning decision thresholds

are tuned considering the seasonal characteristics of prices, solar irradiance and rain events for

the specific location. The simulation results confirmed that the proposed methods in this study

yield significant savings compared to traditional fixed-schedule approach.

In summary, this thesis has contributed to the maintenance research community by answering

the research question on how OM policies are optimised when considering external

Chapter 7: Conclusions and future work 147

opportunities and their characteristics through four optimisation models. Two practical

properties of external opportunities have been investigated and modelled thoroughly in

optimisation: (1) the non-homogeneous stochastic arrivals; and (2) random durations. The

impacts and interaction between PM and OM have been analysed in all four policies where the

PM times or conditions are delayed or set higher to take advantage of OM. This thesis has also

investigated the effects and benefits of using short-term forecasts of opportunities when

implementing the OM policies. The partial characteristic of external opportunities has been

considered and compared to the policies with full opportunities only. The results have

demonstrated the benefit of considering partial opportunities. Finally, the developed policies

were illustrated and their possibility of implementation in practical industrial systems shown,

especially the agriculture and renewal energy systems where the operations were affected by

weather factors.

7.2 POTENTIAL FUTURE STUDIES

This thesis has investigated a new approach in OM study considering external partial

opportunities. It also provides an opportunity to extend the OM research in some potential

issues as follows.

The proposed OM policies in this thesis have aimed to provide policies for decision makers to

take maximal advantage of external opportunities, which represent opportunities to do

maintenance at lower cost. At those times (i.e. opportunity occurrences), the grouping

maintenance policies could be applied to select the suitable components of a multi-component

system for maintenance activities and therefore can save some set-up costs of maintenance

activities. Developing optimisation models for OM policies combining the external

opportunities and grouping maintenance is of interest because of its potential benefit in saving

both maintenance set-up and downtime costs. The developed models will integrate both

characteristics of external opportunities (i.e. stochastic arrivals and partial benefits) and

economic dependence of components in a complex system. The modelling is challenging due

to the stochastic occurrences and durations of external opportunity associated with the

complexity of component dependences, i.e. economical, structural and failure (stochastic)

dependences. One possible approach is two-stage decision models where the external

opportunities will be considered in the first stage and the component grouping policy will help


to select the components for OM. These types of OM policies will yield more benefit to total

maintenance costs as well as be more practical and applicable for many industrial systems.

Another issue to consider in OM is other special types of “opportunities”, particularly

continuous opportunities which do not require an “arrival”, but in which the benefit is

uncertain. In order to take advantage of this kind of continuous opportunity, studies about the

mechanism and prediction of opportunity benefits should be conducted. For instance, the

electricity price, which varies significantly and continuously through time, can create some

maintenance opportunities for the power systems (e.g. wind or solar) when the price is low. In

this way, electricity price can be considered as a continuous opportunity. The stochastic

properties and predictability of these external continuous opportunities should be studied and

combined with optimal OM policies for power systems.

One possible extension of the study on external opportunities is considering imperfect

maintenance actions which are appropriate to external opportunity durations. When an

opportunity arrives with not long enough duration, an imperfect repair that is suitable for this

opportunity duration can be conducted and restores a part of system condition.

Another issue worthy of consideration is the joint optimisation of OM and maintenance

logistics. Maintenance resources (labour, spare parts, etc.) are always limited, and therefore,

logistic planning for maintenance is required and has been an engaging topic in maintenance

research. Logistics planning for the OM activities with external opportunities where

occurrences and durations are uncertain may be a worthwhile area for future work.

Some other issues should be studied about the maintenance/cleaning policies for solar power

systems. The degradation of mirror reflectivity in a CSP system and its prediction is a

significant topic. The loss of mirror reflectivity is a stochastic process but dependent on many

random weather factors such as rain, dust concentration, wind, dew-temperature, etc. that make

it difficult to predict. Therefore, if there is a model that can effectively predict the loss of

reflectivity while accounting for weather factors, a prognosis maintenance (cleaning) policy

can be developed for CSP systems. Another issue in solar collector cleaning is the multi-sector

problem where the (usually very large) solar field must be cleaned by a set of finite resources

(equipment, employees). Therefore, the selection of mirror sectors and sequencing of the

cleaning actions should be discussed in future studies.

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Develop Opportunistic Maintenance Models with Considering … Ba_Thesis.pdf · Huy Truong-Ba . Supervisors: Prof Lin Ma (Principal) Dr Michael Cholette (Associate) Submitted in fulfilment