probabilistic methodology for prioritising replacement of ageing power transformers based on

Probabilistic Methodology for Prioritising Replacement of Ageing

Power Transformers Based on Reliability Assessment of

Transmission System

A thesis submitted to The University of Manchester for the Degree of

Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2014

Mrs Selma Khalid Elhaj Awadallah, B.Sc., M.Sc.,

School of Electrical and Electronic Engineering

3

Table of Contents

1 Introduction ............................................................................................................. 21

1.1 Power System Reliability ...................................................................................... 21

1.1.1 Reliability Evaluation Methods...................................................................... 22

1.1.2 Historical Overview ....................................................................................... 23

1.1.3 System Reliability Definition and Attributes ................................................. 23

1.1.4 Hierarchal Levels of System Reliability Evaluation ...................................... 24

1.1.5 Reliability Cost .............................................................................................. 25

1.1.6 Power System Reliability Applications ......................................................... 25

1.1.7 Scope of the Thesis ........................................................................................ 27

1.2 Research Motivation .............................................................................................. 27

1.2.1 Advanced Age of Installed Equipment .......................................................... 28

1.2.2 Privatisation of Electricity Industry ............................................................... 29

1.2.3 Power Transformers ....................................................................................... 30

1.2.4 National Grid UK ........................................................................................... 31

1.3 Review of Past Work ............................................................................................. 32

1.3.1 End-of-life Failure Modelling ........................................................................ 32

1.3.2 Incorporation into System Reliability Assessment ........................................ 35

1.3.3 Applications in Replacement Planning .......................................................... 36

1.3.4 Uncertainty Quantification in System Reliability .......................................... 37

1.4 Summary of Past Work ......................................................................................... 38

1.5 Research Aim and Objectives ............................................................................... 39

1.6 Research Contributions ......................................................................................... 41

1.7 Outline of the Thesis ............................................................................................. 43

2 Composite Power System Reliability Evaluation ....................................... 47

2.1 Introduction ........................................................................................................... 47

2.2 Modelling .............................................................................................................. 48

2.2.1 Component Failure Models ............................................................................ 48

2.2.2 Load Models .................................................................................................. 54

2.2.3 Network Models ............................................................................................. 55

2.3 Evaluation Techniques .......................................................................................... 57

2.3.1 Fundamental Techniques ............................................................................... 57

2.3.2 Methods for Large Systems ........................................................................... 58

2.3.3 State Enumeration .......................................................................................... 60

2.3.4 Non-sequential Monte Carlo (NMC) ............................................................. 61

2.3.5 Sequential Monte Carlo (SMC) ..................................................................... 62

2.3.6 State Enumeration vs. Monte Carlo Simulation ............................................. 64

2.4 Reliability Indices .................................................................................................. 64

2.5 Summary ............................................................................................................... 65

3 Reliability Assessment Considering End-of-life Failure .......................... 67

3.1 Introduction ........................................................................................................... 67

4

3.2 Integration of End-of-life Failure into System Reliability Evaluation ................. 68

3.2.1 State-of-the-art Method ................................................................................. 68

3.3 Reliability Assessment Software .......................................................................... 70

3.3.1 Overview ........................................................................................................ 71

3.3.2 Functional Definition ..................................................................................... 72

3.3.3 Programming Information ............................................................................. 78

3.3.4 Application Information ................................................................................ 79

3.3.5 Validation ...................................................................................................... 80

3.4 Test Networks Description.................................................................................... 82

3.4.1 Test System .................................................................................................... 83

3.4.2 Load and Network Model .............................................................................. 84

3.4.3 Transformers Fleet Data ................................................................................ 84

3.5 Adjustments for Reliability Assessment ............................................................... 86

3.5.1 Generating Unit Reliability ............................................................................ 86

3.5.2 Repairable Failure .......................................................................................... 87

3.5.3 Accuracy of Non-sequential Monte Carlo ..................................................... 89

3.6 Summary ............................................................................................................... 91

4 Reliability-Based Replacement Framework ................................................ 93

4.1 Introduction ........................................................................................................... 93

4.2 Reliability Importance Measures .......................................................................... 95

4.2.1 Structural Importance Measure ...................................................................... 96

4.2.2 Improvement Potential Measure .................................................................... 96

4.2.3 Criticality Importance Measure ..................................................................... 96

4.2.4 Fussell-Vesely Reliability Measure ............................................................... 97

4.2.5 Further Consideration .................................................................................... 97

4.3 Pareto Analysis ..................................................................................................... 98

4.4 Replacement Justification ................................................................................... 100

4.4.1 Unreliability Cost ......................................................................................... 100

4.4.2 Saving on Reinvestment Cost ...................................................................... 101

4.5 Case Study........................................................................................................... 102

4.5.1 Transformers IP Measure ............................................................................ 102

4.5.2 Pareto Analysis ............................................................................................ 106

4.5.3 Replacement Justification ............................................................................ 110

4.6 Summary ............................................................................................................. 113

5 Incorporation of Unconventional Failure Models into Reliability Studies ..................................................................................................... 115

5.1 Introduction ......................................................................................................... 115

5.2 Transformer Failure Model ................................................................................. 116

5.2.1 Life-stress Models ....................................................................................... 117

5.2.2 Transformer Life-thermal stress Relationship ............................................. 118

5.2.3 Arrhenius-Weibull Failure Model ............................................................... 119

5.3 Estimation of Arrhenius-Weibull Parameters ..................................................... 120

5

5.3.1 Weibull Distribution .................................................................................... 120

5.3.2 Arrhenius-Weibull Distribution ................................................................... 121

5.3.3 Unavailability Estimation............................................................................. 124

5.4 Implementation of Arrhenius-Weibull Distribution ............................................ 124

5.5 Comparison between Gaussian and Arrhenius-Weibull...................................... 126

5.5.1 Load Points with Increased ENS.................................................................. 127

5.5.2 Load Points with Decreased ENS ................................................................ 128

5.5.3 Load Points with No Change in ENS ........................................................... 129

5.6 Summary ............................................................................................................. 130

6 Transformer Criticality for Cascading Failure Events ......................... 131

6.1 Introduction ......................................................................................................... 131

6.2 Dependent Failure ............................................................................................... 132


6.2.2 Calculation of Second Dependent Failure.................................................... 134

6.3 Age and Load based Criticality indicators .......................................................... 136

6.3.1 Indicator of Initiating a Cascading Failure (ICF) ......................................... 136

6.3.2 Indicator of Vulnerability to Consequent Failure (VCF) ............................. 137

6.4 Implementation on the Test System .................................................................... 137

6.4.1 Transformer ICF........................................................................................... 138

6.4.2 Transformer VCF ......................................................................................... 139

6.4.3 Transformer Site Criticality ......................................................................... 143

6.5 Effect of Load Uncertainty on ICF and VCF ...................................................... 145

6.6 Summary ............................................................................................................. 147

7 Quantification of Uncertainty in Reliability Assessment ...................... 149

7.1 Introduction ......................................................................................................... 149

7.2 Failure Model Uncertainty .................................................................................. 150

7.3 Epistemic Uncertainty in End-of-life Failure ...................................................... 151

7.3.1 Gaussian Distribution ................................................................................... 151


7.4 Quantification of Aleatory and Epistemic Uncertainty ....................................... 160

7.4.1 Second Order Probability Method ............................................................... 160

7.4.2 Evidence Theory method ............................................................................. 165

7.5 Uncertainty Based Importance Indicator ............................................................. 169

7.5.1 Probabilistic Sensitivity Analysis ................................................................ 169

7.5.2 Case study .................................................................................................... 170

7.5.3 Application to System Indices ..................................................................... 171

7.5.4 Application to Load Points Indices .............................................................. 173

7.6 Summary ............................................................................................................. 180

8 Conclusions and Future Work ........................................................................ 183

8.1 Conclusions ......................................................................................................... 183

8.2 Future work ......................................................................................................... 187

9 References ............................................................................................................. 190

6

Commonly Used Probability Distributions in Composite System Reliability Appendix A.Assessment ................................................................................................. 198

Illustrative Example for the Reliability Assessment Using the Fundamental Appendix B.Methods ...................................................................................................... 201

Programming and Application Information of the Reliability Assessment Appendix C.Software ..................................................................................................... 206

Test System Data ....................................................................................... 213 Appendix D.

10 Years Ambient Temperature Data ........................................................ 219 Appendix E.

Time Value of Money Formulae ................................................................ 222 Appendix F.

Full List of ICF and VCF Values ............................................................... 223 Appendix G.

Author’s Thesis Based Publications .......................................................... 231 Appendix H.

Total word count: 61,681

7

List of Figures

Figure 1-1: Power system reliability hierarchal levels ......................................................... 24

Figure 1-2: Thesis topic area (red boxes) within power system reliability .......................... 27

Figure 1-3: Spiral of system reliability declining due to age related problems, adopted from

[31] .................................................................................................................. 29

Figure 1-4: Transformer in manufacturing process (photos taken at TIRATHAI

transformers factory, Thailand) ...................................................................... 30

Figure 1-5: Age distribution of transformers owned by National Grid Electricity

Transmission ................................................................................................... 31

Figure 1-6: Bathtub Curve ................................................................................................... 33

Figure 2-1: Aspects related to assessment of the composite power system reliability ........ 48

Figure 2-2: Two state model of repairable failure ............................................................... 50

Figure 2-3: Illustrative limiting values of availability and unavailability based on Markov

theory .............................................................................................................. 52

Figure 2-4: Illustrative example of multi-step load model for the load duration curve ....... 55

Figure 2-5: Flowchart of the steps of composite power system reliability assessment ....... 59

Figure 2-6: Illustrative chronological states of four components ........................................ 62

Figure 3-1: The calculation of probability of having end-of-life failure during the

subinterval j ..................................................................................................... 69

Figure 3-2: Main script in the DPL command object .......................................................... 71

Figure 3-3: Failure Effect Analysis actions ......................................................................... 74

Figure 3-4: Explanatory example of a data flow diagram DFD........................................... 78

Figure 3-5: A snapshot of the function pasted in the active Study Case in the Data Manager

........................................................................................................................ 79

Figure 3-6: Setting the number of NMC iterations in the reliability software ..................... 80

Figure 3-7: Annual Load Duration curve represented by the 20-step load model. .............. 82

Figure 3-8: The single line diagram of the test system ........................................................ 83

Figure 3-9: Age distribution of the test system’s transformers ............................................ 85

Figure 3-10: Unavailability due to end-of-life failure using normal distribution (=65,

=15) for a range of ages (1-58) ..................................................................... 86

Figure 3-11: Heat maps for the test system showing the effects from assuming zero

unavailability due to repairable failure ........................................................... 89

Figure 3-12: The convergence of the ENS index against the number of Monte Carlo

iterations .......................................................................................................... 90

Figure 4-1: Risk matrix used to determine replacement candidates. Adopted from [103] .. 94

Figure 4-2: Illustrative example of the use of incremental change in sensitivity analysis

with NMC simulation ..................................................................................... 98

8

Figure 4-3: Cost of system unreliability as a function of ENS based on Great Britain

regulatory incentives/penalties scheme ........................................................ 101

Figure 4-4: Reliability importance measure (IP) for power transformers. ........................ 105

Figure 4-5: ENS for replacement scenarios of transformers ............................................. 106

Figure 4-6: Pareto plot for the replacement scenarios ....................................................... 109

Figure 4-7: Economic comparison of replacement plans .................................................. 112

Figure 4-8: Economic comparison of replacement plans adding unreliability cost calculated

using VoLL ................................................................................................... 113

Figure 5-1: Modelling transformer reliability: (a) traditional constant unavailability. (b)

Age dependant unavailability. (c) Age-load dependant unavailability ........ 117

Figure 5-2: Illustrative example of Arrhenius-Weibull cdf for two different HST, where

HST1>HST2. ................................................................................................ 119

Figure 5-3: Curve fitting of Gaussian and Weibull distributions ...................................... 120

Figure 5-4: Transformer unavailability due to end-of-life failure for a range of ages (1-58)

using Gaussian and Weibull distributions. ................................................... 121

Figure 5-5: Characteristic life relationship with transformer HST based on the estimated

values of A and B. ........................................................................................ 123

Figure 5-6: The unavailability for transformer age range (1-58 years) calculated using

Arrhenius-Weibull distribution for maximum loading level and average

loading level. ................................................................................................ 124

Figure 5-7: Critical load points based on ENS obtained using Gaussian and Arrhenius-

Weibull distributions. ................................................................................... 129

Figure 6-1: Flowchart of second dependent failure calculations due to thermal stress. .... 135

Figure 6-2: The top 25 transformers in ICF ranking for the annual load model ............... 140

Figure 6-3: The top 25 transformers in VCF ranking for the annual load model .............. 142

Figure 6-4: Age and loading for the top 25 transformers in VCF ranking. ....................... 143

Figure 6-5: The top 25 transformers in ICF ranking without considering the local effect on

the transformer sites ..................................................................................... 144

Figure 6-6: Area affected by T9 (marked by X sign in the figure) outage ........................ 145

Figure 6-7: Introduced uncertainty in the 6-step load model ............................................ 146

Figure 6-8: Frequency of coming in 5 top-ranked transformers based on ICF ................. 146

Figure 6-9: Frequency of coming in 5 top-ranked transformers based on VCF ................ 147

Figure 7-1: Illustrative example of mixed aleatory-epistemic uncertainty model for

repairable and end-of-life failure .................................................................. 151

Figure 7-2: Examples of the histogram of the unavailability for the transformers

considering ±10% variation in Gaussian distribution parameters ................ 152

Figure 7-3: Uncertainty in unavailability of transformer age range (1-58 years)

corresponding to ±10% variation in the Gaussian distribution parameters. . 152

Figure 7-4: Critical transformer sites based on ENS obtained using deterministic and

uncertain parameters of Gaussian distribution ............................................. 154

9

Figure 7-5: Projection of the uncertainty on characteristic life and the relationship with

transformer HST ........................................................................................... 155

Figure 7-6: Examples of the histogram of the unavailability for the transformers

considering ±10% variation in Arrhenius-Weibull distribution parameters . 156

Figure 7-7: Uncertainty in unavailability of the age range (1-58 years) corresponding to

±10% variation in the Arrhenius-Weibull distribution parameters for

HST=36.37ºC ................................................................................................ 157

Figure 7-8: Uncertainty in unavailability for a 40 year old transformer for load range (1-

140%) corresponding to ±10% variation in the Arrhenius-Weibull distribution

...................................................................................................................... 157

Figure 7-9: Critical transformer sites based on ENS obtained using deterministic and

uncertain parameters of Arrhenius-Weibull distribution .............................. 159

Figure 7-10: Use of nested sampling process to propagate the mixed aleatory-epistemic

uncertainty to power system reliability indices. ........................................... 161

Figure 7-11: Distinguishing between aleatory and epistemic uncertainty forms using

horsetail plot ................................................................................................. 162

Figure 7-12: The single line diagram of IEEE-RTS .......................................................... 163

Figure 7-13: cdfs of PLC index generated using SOP method and considering the mixed

aleatory-epistemic uncertainty in transformer failure rate ............................ 163

Figure 7-14: The cdf of PLC index mean values ( ), Most probable cdf and horsetail

cdfs bounds ................................................................................................... 164

Figure 7-15: The cdfs of PLC index derived using the typical aleatory uncertainty model

and the mixed aleatory-epistemic uncertainty model ................................... 164

Figure 7-16: The belief structure of obtained by evidence theory and the cdf obtained

by SOP .......................................................................................................... 167

Figure 7-17: Belief structure of obtained in Case study I and Case study II.............. 168

Figure 7-18: The probability distribution function of transformers’ failure rate. .............. 170

Figure 7-19: The ENS histogram with fitted normal distribution calculated from 1000

random values of components failure rate .................................................... 172

Figure 7-20: IEEE-RTS components ranked using correlation coefficient between failure

rate and system’s ENS .................................................................................. 172

Figure 7-21: Scatter plots of Line 3-9 and transformer Tx 3-24 failure rates and system’s

ENS. .............................................................................................................. 173

Figure 7-22: Scatter plot of Bus 4 ENS and Line 15-24 failure rate.................................. 174

Figure 7-23: IEEE-RTS components ranked using correlation coefficient between failure

rates and Bus 6’ ENS .................................................................................... 175

Figure 7-24: IEEE-RTS components ranked using CRI .................................................... 175

Figure 7-25: Line 15-21(1) correlation coefficient and contribution to ENS to load buses

...................................................................................................................... 176

Figure 7-26: Correlation coefficient of IEEE-RTS components with load point Bus 14 .. 178

Figure 7-27: IEEE-RTS reliability map-importance of components ................................. 178

Figure 7-28: IEEE-RTS reliability map: Area of vulnerability for Line 15-21 ................. 179

10

Figure 7-29: IEEE-RTS reliability map: criticality of load points .................................... 179

11

List of Tables

Table 1-1: Life-time estimation and percentage of population within the life-time range in

1998 & 2008 for some of the components, taken from [31] ............................. 28

Table 1-2: Examples of widespread power outage initiated by transformer failure ............ 31

Table 2-1: Comparison between analytical techniques and simulation techniques in

assessing composite system reliability .............................................................. 64

Table 3-1: Selection procedure for the component state using random numbers and the

XOR probability rule ......................................................................................... 73

Table 3-2: Calculation of the power required to overcome the overload of a component (di)

........................................................................................................................... 76

Table 3-3: Comparison between the annualised system reliability indices reported in [24]

and indices produced by the developed software .............................................. 81

Table 3-4: A comparison of annual reliability indices for the three multi-step load models

........................................................................................................................... 82

Table 3-5: The 6-step load model of the test system ........................................................... 84

Table 3-6: The effect of assuming zero unavailability due to repairable failure on ENS

[MWh/year] ....................................................................................................... 88

Table 3-7: Accuracy and computation time of NMC for 10,000, 15,000 and 20,000

iterations ............................................................................................................ 89

Table 3-8: Reliability indices of the test system using the given Gaussian distribution...... 90

Table 4-1: Ranking of the transformers in the test system based on IP importance measure

......................................................................................................................... 104

Table 4-2: The system ENS resulted from the replacement of transformers one by one .. 107

Table 4-3: Reduction and cumulative reduction in ENS due to replacement scenarios

following Pareto’s new ranking ...................................................................... 108

Table 4-4: Cost of unreliability for replacement plans ...................................................... 110

Table 4-5: Saving on reinvestment cost for the replacement scenarios ............................. 111

Table 5-1: Parameters of hot-spot temperature model ....................................................... 122

Table 5-2: The 6-step load model with associated ambient temperature values................ 125

Table 5-3: Examples of the unavailability of transformers calculated using Arrhenius-

Weibull distribution......................................................................................... 126

Table 5-4: ENS for load points and system of the test network using Arrhenius-Weibull

distribution ...................................................................................................... 126

Table 5-5: A comparison between reliability studies using Gaussian and Arrhenius-Weibull

distributions ..................................................................................................... 127

Table 5-6: Load points which gained an increase in the ENS when using Arrhenius-


Table 5-7: Load points which experienced a decrease in the ENS when using Arrhenius-


12

Table 5-8: Example of loading percentage at different levels for two transformers located

at Bus 19 and 28 .............................................................................................. 129

Table 6-1: Top five transformers for load levels ranking based on ICF ........................... 138

Table 6-2: Maximum, average, and median values of transformers’ ICF for load levels in

(%) .................................................................................................................. 139

Table 6-3: Top five transformers for load levels ranked based on VCF ........................... 141

Table 6-4: Maximum, average, and median values of transformers’ VCF for load levels in

(%) .................................................................................................................. 141

Table 6-5: Transformer sites rank using an average ICF .................................................. 144

Table 7-1: ENS and the ranking of load points obtained using deterministic and uncertain

parameters of Gaussian distribution ............................................................... 153

Table 7-2: ENS and the ranking of load points obtained using deterministic and uncertain

parameters of Arrhenius-Weibull distribution ................................................ 158

Table 7-3: PLC for load points obtained using deterministic and uncertainty parameters of

Arrhenius-Weibull distribution ....................................................................... 158

Table 7-4: A comparison between aleatory model and mixed aleatory-epistemic model . 165

Table 7-5: Case Study I: Belief structure of .............................................................. 167

Table 7-6: Assumed epistemic uncertainty in transformer failure rate used for case study II

........................................................................................................................ 168

Table 7-7: IEEE-RTS circuits correlation coefficient with load buses ............................. 174

Table 7-8: IEEE-RTS’ load points ranked using SRI ........................................................ 177

13

List of Abbreviation

A Availability

Bel Belief function in evidence theory

BPA Basic probability assignment

CBF Cumulative belief function

CC Correlation coefficient

CPF Cumulative plausibility function

CPU Central processing unit

CRI component ranking index

DC Direct current

DFD Data flow diagram

DP Degree of polymerisation

DPL DIgSILENT programming language

DSET Dempester-Shafer Evidence theory

EDLC Expected duration of load curtailment

EDNS Expected Demand Not Supplied

EENS Expected Energy Not Supplied

EIC Energy interruption unit

ELC Expected load curtailments

ENLC Expected number of load curtailments

ENS Expected Energy Not Supplied

FV Future value

GDP Gross Domestic Product

HLI Reliability assessment of hierarchal Level 1

HLII Reliability assessment of hierarchal Level 2

HLIII Reliability assessment of hierarchal Level 3

HST Hottest Spot Temperature

IC Criticality importance measure

ICF Indicator for initiation of cascading failure

IEC International Electrotechnical Commission

IEEE Institute of Electrical and Electronic Engineering

IEEE-RTS IEEE-reliability test system

IP Improvement potential measure

IS Structural importance measure

K Degree Kelvin

NERC North American Electric Reliability Corporation

NMC Non-sequential Monte Carlo

OF Oil forced cooling

ON Oil natural cooling

OPF Optimal power flow

PC Personal computer

PCC Point of common coupling

PLC Probability of load curtailment

14

PV Present value

RCM Reliability Centred Maintenance

SMC Sequential Monte Carlo

SOP Second order probability

SRI Substation ranking index

TDCG Total dissolved combustible gas

TOF Time on failed

TTF Time to failure

TTR Time to repair

U Unavailability

UMIST University of Manchester Institute of Science and

Technology

VCF Indicator for vulnerability to consequent failure

VoLL Value of lost load

XOR Exclusive OR

15

Abstract

Title: Probabilistic Methodology for Prioritising Replacement of Ageing Power

Transformers Based on Reliability Assessment of Transmission Systems

Selma Khalid Elhaj Awadallah, The University of Manchester, July 2014

Doctor of Philosophy (PhD)

Customers expect electricity to be not only available but also affordable whenever they

need it. Due to the stochastic nature of power system component failure, the management

of power interruption is challenging. Although the reliability of supply can usually be

increased by employing redundant equipment; this means that affordability is

compromised. At present, many power utilities have a considerable amount of aged

equipment in their networks. Although they have already started replacement planning, the

price control schemes imposed by regulatory authorities constrain their capital expenditure

budget.

This thesis has studied the influence of the end-of-life failure of power transformers on

transmission system reliability in order to make decisions on their replacement. Power

transformers are selected for the analysis because they are technically complex, expensive,

and main feed points of electricity for end users. In addition, surveys on ageing asset show

that 50% of transformer populations, in many utilities, have been classified as old since the

year 2008. The focus of these reliability analyses is to identify the most critical

transformers and to establish a reliability based replacement framework. Modelling of end-

of-life failure was reviewed, and the state-of-the-art method of its incorporation into

system reliability was adopted. A reliability assessment tool within DIgSILENT

PowerFactory package was developed in order to perform reliability studies.

This thesis has four original contributions surrounding transmission system reliability

analysis. The first contribution is the development of a cost-effective framework that

concerns the application of reliability studies on asset replacement decision making. The

developed framework has employed reliability importance measures, the Pareto analysis

and economic comparison based on reliability incentive/penalty schemes. All the three

elements of the framework are original applications to system reliability area. The second

contribution is the integration of unconventional end-of-life failure models into system

reliability. The unconventional model used in this study is Arrhenius-Weibull distribution,

which characterises end-of-life failure under different loading conditions. This study has

evaluated the added value provided by including loading levels in failure models and how

this enhances the understanding of the effect of operational factors on system reliability.

The third contribution is the investigation of dependent failure of power transformers

caused by thermal stress. This investigation has led to the development of two probabilistic

indicators to rank power transformer based on their criticality to multiple failure events.

These new indicators have related the transformer reliability to its age and loading levels.

In the fourth contribution, comprehensive studies of the effect of uncertainty associated

with failure model parameters were performed. The first study has established bases for a

system related approach for refining failure models. The approach is based on assessing the

sensitivity of the system reliability or the system reliability applications to the uncertainty

in failure model parameters. In the second study, two quantification methods were adopted

to propagate the uncertainty in failure model parameters to system reliability indices.

These are the second order probability and evidence theory. The last uncertainty study has

described the use of sampling based sensitivity analysis to identify the most critical

transformers and their area of vulnerability. Studies throughout the thesis have been

performed on a realistic transmission network and the IEEE Reliability Test System.

16

Declaration

No portion of the work referred to in this thesis has been submitted in support of an

application for another degree or qualification of this or any other university or institute of

learning.

17

Copyright Statement

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Theses.

18

Acknowledgements

In the name of Allah the beneficent the merciful. Allah says “I only intend reform as much

as I am able. And my success is not but through Allah. Upon Him I have relied, and to Him

I repent.” (Hŭd: 88). All praise and thank be to Allah. He confers upon us sense of reason

and intellect other than all creatures, and may the peace and blessings of Allah be upon

Mohammed his servant and messenger.

I must express special appreciation and thanks to my supervisor, Prof. Jovica V. Milanović

for giving me the opportunity to be one of his PhD students and for his guidance, advice

and encouragement throughout this research. His dedication has inspired and motivated me

to pursue not only technical research excellence but also professional practice.

I would like to acknowledge the useful and constructive discussions about power

transformers with Prof Zhongdong Wang and Mr. Bevan Patel while working on this

project.

Special thanks go to National Grid plc for the financial support for this PhD research

project. I would particularly like to express my gratitude to Mr. Paul Jarman of National

Grid plc for the valuable discussion and feedback throughout this research project.

I would like to thank everyone at The University of Manchester who has contributed in any

way to the completion of this thesis including IT services, researcher development office,

and staff at school of Electrical and Electronic Engineering. I would particularly like to

thank my colleagues in Power System Quality and Dynamics Group. Their commitment to

highly research standard has been inspirational. Thanks are also due to Dr. Atia Adrees,

Dr. Tuba Gozel and Mr. Sami Abdelrahman for their encouragement and support.

I am also indebted to my friend Sawsan Mubarak Abdallah for taking care of my toddler

daughter while I was working on this project. Her love to my daughter has put my mind at

rest and I was able to focus on my research. I would also like to thank my friends the Taha

for lending me a hand whenever I need it.

My sincerest thanks are due to my dearest husband Amgad and my children Ahmed and

Yousra for understanding, patience and support during the completion of this thesis. This

thesis would not have been possible without their encouragement. I would like to express

my heart-felt gratitude to my parents Khalid and Fatima for their prayers, love and support.

19

To my parents

Chapter 1 Introduction

21

1

Introduction

1.1 Power System Reliability

In modern developed societies, electric energy is one of the main pillars on which all

aspects of life are based. Power systems may be the most complicated large-scale

engineering systems; nevertheless they are expected to have a very high degree of

reliability. In many power systems, the average annual interruption duration does not

exceed 2 hr/year [1], which can be interpreted as 99.977% reliability. Therefore, customers

take electricity for granted and expect it to be not only available but also affordable

whenever they need it.

It is very challenging, however, for power system utilities to meet customers’ expectations.

This is due to the stochastic nature of the failure of power system components, which

makes the prediction and control of power interruption a difficult task. Although the

reliability of supply can be generally increased by employing redundant equipment, the

affordability gets compromised which is against customers’ expectations. Consequently,

maintaining the reliability of supply within budget constraints is important and the

foremost issue for all power system utilities. The system reliability is mentioned in the

strategic objectives of almost all power system utilities and companies, e.g., [2-4]. For all

power utilities, decisions related to power system activities, e.g., operation, development,

reinvestment and maintenance, are made based on system reliability and the continuity of


22

supply for end users. As a result, power system researchers and engineers have developed,

over many decades, techniques and ways to evaluate power system reliability and achieve

the balance between reliability and low cost.

1.1.1 Reliability Evaluation Methods

There are two approaches to assess the reliability of power systems; a deterministic

approach and a probabilistic approach [5, 6]. The deterministic approach is performed by

selecting one or many operational base cases and then subjecting each of them to different

incidents. The number of components out of service in these incidents may be one or more.

Then the reliability of power system can be assessed based on these cases [6]. Examples of

this approach are percentage reserve in generation planning and N-1 transmission system

planning criteria. The deterministic approach does not reflect the probabilistic nature of the

failure of power system equipment nor the operational and parameters uncertainties of the

system. It treats all the incidents and/or operating conditions as if they have the same

likelihood of occurrence.

On the other hand, the probabilistic assessment of system reliability solves this issue by

applying a probabilistic control on the selection of incidents and operating cases [5, 6]. The

probabilistic evaluation of power system reliability, also known as risk assessment of

power system, involves two principles. The first principle is the characterisation of system

states, i.e., components’ outages and operating cases, by probabilistic models. These

probabilistic models are usually derived from historical failures and operation data. The

second principle is the quantification of the consequence of individual system states.

Combining these two principles together gives the probabilistic evaluation of the reliability

of power systems. There are two main categories in probabilistic evaluation techniques,

analytical and simulation techniques. The difference between the two techniques is in the

way that the system states are selected.

At present, the most commonly used approach is the probabilistic one. As a matter of fact,

for the last three decades the term reliability generally refers to the probabilistic evaluation

of power system reliability.


23

1.1.2 Historical Overview

The first significant studies that used probabilistic approach in reliability evaluation

appeared in 1947 [7-10]. The studies applied the probabilistic approach on the generating

reserve capacity. By the early 1960s, many North American utilities adopted this method

to become the primary assessment method of generating reserve capacity [11].

Simultaneously, more complex analytical studies on using the probabilistic approach on

transmission and distribution systems started to appear [12, 13]. After the 1965 blackout,

power system reliability analysis received further attention, which resulted in the

formulation of the National Electric Reliability Council (NERC) whose membership

includes most power utilities in US and Canada. The same interest in reliability studies

appeared in Europe, but most studies employed the Monte Carlo simulation rather than the

analytical method employed in North America [11]. Both techniques, the analytical and

Monte Carlo simulation, adopted the conventional Markov models in the reliability

evaluation [14, 15]. In 1970, Billinton wrote the first book on power system reliability

[16]. Since Billinton’s book, an ever increasing number of publications, which adopted

probabilistic evaluation of power system reliability, has been added to power system

literature [17-22].

1.1.3 System Reliability Definition and Attributes

In general, the term system reliability is broken into two attributes: adequacy and security.

The system adequacy concerns the ability of supplying customers with electricity in all

steady state cases. This includes the existence of generation, transmission and distribution

facilities, which are able to meet customer demand. The system security involves the study

of the transition between these steady states and the ability of the system to survive after

disturbances, specifically, studies surrounding system dynamics and transients [5, 14]. The

system security attribute is not a well-developed research compared to system adequacy

due to the complexity involved in these studies. As a matter of fact, reliability studies are

mostly performed to assess the system adequacy.

The definition of power system reliability as given by NERC is: “Reliability is the ability

to meet the electricity needs of end-use customers, even when unexpected equipment

failures or other factors reduce the amount of available electricity” [23]. From the

definition, it is obvious that the evaluation of reliability must contemplate the stochastic

nature of the failure of power system components. The ability mentioned in the definition


24

is measured by quantitative indicators known as system reliability indices. Reliability

indices are the output of reliability evaluation. It must be appreciated that these indices are

not deterministic; they are statistical expected values. They, however, give a reasonable

measure that includes the probability of success or failure system states and their

consequences [24].The calculations of these indices vary with the evaluation techniques.

That is to say, the indices formulae of analytical methods are different from formulae of

simulation methods.

1.1.4 Hierarchal Levels of System Reliability Evaluation

The hierarchal levels of system reliability evaluation follow, to some extent, the regular

three parts of the power system; generation, transmission and distribution. System

reliability evaluation, however, is not performed for each part separately. It is performed in

three levels explained by Figure 1-1 [5, 16, 24-26]. Level-1 (HLI) is concerned with

generation and how it meets the demand at all times including the peak demand. Level-2

(HLII) assesses the generation and transmission ability to meet the electricity needs at the

grid supply points. At this level, the transmission operational constraints such as thermal

and voltage limits are involved. Finally, level-3 (HLIII) combines all three power system

parts and assesses the ability of generation, transmission and distribution to meet the

customers’ demand. Due to the complexity of modelling and performing such studies, it is

a common practice to use the output from HLII studies as input to the distribution system

reliability studies [5, 25, 26]. There are different reliability indices for different hierarchal

levels.

Figure 1-1: Power system reliability hierarchal levels


25

1.1.5 Reliability Cost

In the applications of reliability analysis in power system activities, reliability studies alone

do not give the full picture. The estimation of the cost involved to achieve the required

reliability level is a necessary complement of reliability studies. As stated earlier, the

balance between required reliability and cost must be achieved in all decision-making

processes. In order to examine this balance, the reliability of the system has to be measured

in monetary units. This is usually accomplished by estimating the Energy Interruption Cost

(EIC). The unreliability cost is then calculated by multiplying this monetary measure by

the Energy Not Supplied index (ENS). The evaluation of the monetary value of power

system reliability is not an easy task. Many studies have been carried out to accomplish

this task [27, 28]. Among these, there are three main methods [26]:

I. Customer damage functions: In this method, EIC is derived from customer

damage functions. These damage functions are obtained by customer surveys and

presented as functions of interruption duration. EIC is the average damage cost of

different types of customers and is region, country or area dependent. Although this

method is complicated and time consuming, it reflects the social cost of

interruption [24, 26, 29].

II. Capital investment vs acquired reliability: EIC is estimated in this method as the

ratio of capital investment spent on reliability reinforcement to the achieved

increase in reliability. As an illustration, for individual reliability reinforcement

projects, the capital investment estimation and reliability assessment are performed.

Then, the average of the ratio of capital investment to the achieved increase in

reliability is calculated and considered as EIC [24, 26].

III. Gross Domestic Product: EIC can be calculated by dividing the Gross Domestic

Product (GDP) by the annual delivered energy. This is the simplest method to

calculate EIC and is suitable for utilities owned by government [26].

1.1.6 Power System Reliability Applications

The main function of power systems is to provide customers with electricity at an

acceptable level of continuity. Consequently, almost all power system activities focus on

system reliability and related decisions are based on the continuity of electricity supply. As

a result, power system reliability has many applications including transmission


26

development planning, transmission operation planning, generation source planning,

selection of substation configurations, reliability-centred maintenance, and probabilistic

spare-equipment analysis [26, 30]. Most reliability applications are employed to make

decisions among alternatives based on risk and cost comparisons. The basic idea is to

estimate for each alternative the capital investment (CI), the operational cost (OC), and the

unreliability or risk cost (RC). The best alternative is the one with the minimum total cost

CT (CT = CI+OC+RC). In the following, examples from these applications are briefly

discussed to reflect on the practical implementations of system reliability. It should be

mentioned that the application of reliability assessment in system activities is a framework

from which decisions on these activities can be taken; it is not a technical process.

1. Transmission Development planning

The application of system reliability in transmission development planning comes after the

design stage. Generally, the design stage produces more than one alternative that pass the

utility design standard criterion. Then, for all alternatives, the reliability of the system over

a future period of time is evaluated to assess the cost of the risk of customer interruption.

Similarly, the capital investment and the equivalent present value of future operation and

maintenance costs are estimated for individual alternatives. Finally, the total cost of

individual alternatives is calculated and compared in order to select the one with the

minimum cost. In this way, the cost of the risk of customer interruption is included in the

decision-making process [30].

2. Reliability-Centred Maintenance

The reliability centred maintenance (RCM) is a maintenance methodology in which

maintenance plans and schedules are enacted so that minimum cost and effort, along with

better reliability, are achieved. There are two elementary tasks in RCM, the comparison

between a component’s maintenance alternatives and the lowest-risk maintenance

scheduling. The first task is performed in a similar way to transmission development

planning, i.e., risk and cost based comparisons among the maintenance alternatives is

performed. The second RCM task can be only performed when using reliability evaluation

methods that are time-dependent, e.g., sequential Monte Carlo simulation. Then, the


27

reliability of the system is assessed by moving maintenance outages over all possible time

intervals. The lowest risk scheduling is selected [26].

1.1.7 Scope of the Thesis

This thesis aims to carry out a probabilistic analysis of the adequacy of the transmission

system for making decisions on the replacement of power transformers. This aim focuses

the thesis research on the adequacy of HLII. The chosen system reliability application of

the thesis is replacement planning. Figure 1-2 shows where the topic of this thesis falls

within the power system reliability assessment area.

1.2 Research Motivation

Power system equipment is designed to have sufficient withstand strength to endure most

power system disturbances without failure. The withstand strength, however, decreases

with the age of the equipment until it may no longer be able to withstand the stresses

caused by power system disturbances. On the other hand, the cost of replacing components

in a transmission system is particularly high and the decisions regarding replacement

should not be taken lightly. These two factors bring asset replacement planning to the area

of system reliability applications, where cost and risk can be balanced. The main

motivation for the research presented in this thesis is the presence of advanced ageing

Figure 1-2: Thesis topic area (red boxes) within power system reliability

Power Systems Reliability

Security Adequacy

Deterministic Probabilistic

HLII HLI HLIII

Replacement Planning


28

assets in power networks and price control schemes imposed by electricity regulatory

authorities, which have the fixed capital expenditure budget.

1.2.1 Advanced Age of Installed Equipment

The building of electricity infrastructure reached its peak between the 1950s and 1960s.

Taking into account that the average design life-time of most equipment is around 45

years, electrical transmission systems are indeed facing the problem of ageing assets.

According to a survey conducted by Cigre Working Group 37-27 in 1998 [31], “the

population peak will enter the window of life-time estimates over the next 10 years for

almost all types of major equipment”, which is the year 2008. For instance, 50% of

transformer populations of the 10 utilities that participated in the survey have been

considered as old since 2008. Table 1-1 shows the percentage of the population within the

life-time range in 1998 and 2008 for some types of equipment according to the survey [31].

Table 1-1: Life-time estimation and percentage of population within the life-time range in 1998 & 2008 for

some of the components, taken from [31]

Type of

Equipment

Life-time range

[years]

% of population within

life-time range in 1998

Expected % of population

within life-time range in 2008

Transformers 34 - 50 20 50

Overhead Lines 44 - 63 24 46

Circuit Breakers 35 - 47 11 38

Apparently, the ageing of power system equipment is one of the major issues facing power

system utilities at present since it has major effects on reliability, performance and the

environment. The reduction in reliability caused by the fact that much of the installed

equipment has exceeded its design life time and will, in time, potentially become less

reliable could lead to an endless collapsing process. The cost of replacing transmission

system components is excessive and usually the replacement is deferred to later years. This

delay may result in more failures in the system, which definitely causes longer repair time,

lower availability, and the cancellation of planned outages. Consequently, this may create

more failure occurrences because of cancellation of maintenance work, and more delays in

replacement [31]. This is well-demonstrated by the “spiral of decline” shown in Figure 1-3,

adopted from [31]. In addition to reliability issues, the existence of ageing assets in a

system could potentially lead to a lower system performance. Since the withstand strength

decreases with the component’s age, the operational limits, e.g., thermal limit, may be set


29

to a lower value in order to leave a safety margin between component withstand strength

and system stress. This leads to de-rating of the system installed capability [31]. Moreover,

some aged components have environmental impact such as: pollution, visual impact and

electromagnetic field interference. When the equipment was installed, these environmental

issues were not considered, but for modern society they are becoming prominent [31].

Figure 1-3: Spiral of system reliability declining due to age related problems, adopted from [31]

1.2.2 Privatisation of Electricity Industry

The classical electricity industry is a monopoly system by nature, which makes the

implementation of competition difficult. Therefore, the way in which it is privatised, is

based on price controls [32]. In price control methods, the amount of money that utilities

can gain is limited to a specific value. As a result, investment budgets are restricted, and

utilities are under economic pressure to reduce their capital expenditure. Decisions

regarding investment are taken with greater caution than before privatisation. Furthermore,

since the year 2000, European national regulatory authorities have started to impose a

reliability regulation scheme in order to ensure that the budget constraints on transmission

system investment do not affect the continuity of supply for the end users [33]. The

reliability regulation schemes are based on incentives/penalties calculated using some of

the reliability indices, commonly Energy Not Supplied (ENS). For example in Great

Britain, the regulator has applied an incentive scheme to National Grid plc [34-36]. The

proposal has set a target of ENS equal to 316 MWh. Achieving an ENS less than this target

will be rewarded at a rate of £16,000 per MWh. Conversely, any values of ENS more than


30

the target will incur penalties with the same rate. All these regulation schemes make

replacement planning of ageing system equipment a challenging task for power utilities.

1.2.3 Power Transformers

In addition to the Cigre survey results given in Table 1-1, there are three reasons for

studying power transformers as this research case equipment. Firstly, transformers are

technically complex. Their design involves electromagnetic, mechanical and chemical

aspects [37]. Therefore, their design and manufacturing processes are very complex (see

Figure 1-4). As a result of this complexity, transformers are usually custom designed, and

manufacturers do not have them in stock. In addition, they have a long manufacturing time,

which increases with the size of the transformer. Hence, replacing a power transformer

should be planned a long time in advance. Even if there is a spare transformer available

when a failure occurs, transformers are very difficult to transport, and their installation and

commissioning need very careful handling. Furthermore, as most of them are oil

immersed, the failure of a transformer may result in environmental and safety issues. The

transformers also have high capital costs, which makes it very difficult for asset managers

to decide whether or not to replace them. For example, the cost of replacing a power

transformer can be around £4 million based on experience of transmission system asset

managers and [38]. Finally, their location in the network at the grid supply points makes

them the feed point of electricity to customers. In fact, the history of power outages across

the world has shown that a transformer/multiple transformer failure was the initiation event

to widespread outage or even a blackout (see Table 1-2). A classic example for these

events is the Moscow 2005 blackout, which was triggered by the failure of ageing

transformers [39].

Figure 1-4: Transformer in manufacturing process (photos taken at TIRATHAI transformers factory,

Thailand)

a)Transformer winding b) Transformer core


31

Table 1-2: Examples of widespread power outage initiated by transformer failure

Date Location Affected

customer

Notes

28.08.2003 London 500,000

Buchholz alarm of a transformer led to disconnection

action by operator for safety reasons. As a result, a line

tripped out.

25.05.2005 Moscow 2 million Explosion of multiple transformers, which were

around 40 years old

08.09.2011 Arizona & Southern

California 2.7 million

A line trip caused multiple transformers to fail (trip or

damage), which initiated cascading failure

14.01.2012 Istanbul 12 million A transformer failure initiated a blackout

19.02.2014 Tulsa, Oklahoma 2,800 Multiple transformers fire

1.2.4 National Grid UK

This research project was sponsored by National Grid, UK. National Grid is the national

system operator of Great Britain, and owns the electricity transmission system in England

and Wales [40]. A large number of transformers on National Grid’s network was installed

between 1955 and 1975 [41]. Since the estimated life-time of power transformers is about

40 years, National Grid’s transformers population contains a considerable number of aged

power transformers. The histogram shown in Figure 1-5 represents the age distribution of

transformers owned by the National Grid as of June 2012. Consequently, it is important for

National Grid to study transformer end-of-life failure and their influence on system

reliability in order to accurately identify critical transformer candidates for replacement.

Figure 1-5: Age distribution of transformers owned by National Grid Electricity Transmission


32

1.3 Review of Past Work

As explained earlier, the main purpose of power system reliability assessment is to provide

probabilistic measures of the system adequacy that can be used in decision-making

processes. The nature of the specific decision making process determines which models or

methods have to be used. Since this research focuses on decisions regarding the

replacement planning of transmission equipment, end-of-life failure is the core element in

the modelling aspect. This literature review addresses issues related to end-of-life failure

modelling, its integration into system reliability, past studies on reliability application in

replacement planning, and finally how the uncertainty associated with failure models have

been addressed.

1.3.1 End-of-life Failure Modelling

The failure of power system components can be classified into two types; repairable failure

and end-of-life failure. For repairable failure, a component transits between in-service state

and in-repair state in the time period of study, whereas for end-of-life failure, when a

component fails it cannot be repaired. Specifically, in end-of-life failure, there is one

transition and it occurs only once.

In engineering reliability, the failure rate function of a component throughout its life time

is usually described by the bathtub curve given by Figure 1-6 [11, 26, 42]. The bathtub

curve consists of three stages; early life, useful life and wear-out. Failures in the early life

stage are caused by design and manufacturing defects, which can be identified and solved

in the early operation period. Hence, the failure rate in this stage decreases with time. In

the useful life period, failures occur randomly and hence the failure rate does not change

with time. In the wear-out stage, the failure rate increases with time since failures at this

stage are age related [11]. The three stages of the bathtub curve can be employed to

describe the failure rate of both types. In other words, end-of-life failure may have early

life, useful life, and wear-out stages in the same way as repairable failure. For example, an

end-of-life failure may occur randomly during the useful life period of a component due to

operational mistakes or bad weather conditions, e.g., lightning strikes and flooding. On the

other hand, a component failure may occur at the wear-out stage, but a repair action brings

it back to service. The issue that should be noticed is that end-of-life failure is not similar

to age related repairable failure. This review concerns end-of-life failure at the wear-out

stage.


33

Time

Fa

ilu

re

rate

Early

lifeUseful life Wear-out

Figure 1-6: Bathtub Curve

At this level, when one talks about modelling component failure events, whether it be,

repairable or end-of-life failure, this means deriving a probabilistic distribution function

that characterises these failure events [11, 25, 26, 42]. For example, the probability

distribution function for repairable failure describes the probability of time to failure

(TTF), whereas the functions for end-of-life failure characterise the life-time of the

component. A lot of research has already been done into the probabilistic modelling of the

end-of-life failure of power system components. Although this area of research is beyond

the scope of this thesis, obtaining knowledge about the available models is necessary to

complete it. The conventional techniques of modelling failure events are statistical

techniques where probabilistic distribution functions are formulated from historical failure

data [26, 42, 43]. The attained probabilistic functions can be parametric or non-parametric

[44]. In the parametric analysis, the probabilistic distribution “model” is chosen first then

its parameters are estimated from the data and a goodness-of-fit test is performed to check

the suitability of the model to represent the data, e.g., selecting Weibull and Log-normal

distributions to characterise the end-of-life failure of power transformers [45] . When no

probability distribution can characterise the data, the non-parametric approach is used, e.g.,

estimating the failure rate function without predefined probabilistic distribution functions

as in the case study reported in [46]. One major drawback of the non-parametric approach

is that it cannot be used to estimate the probability outside the data range [44]. Since power

system components have a relatively long life-time, and hence there is no sufficient failure

data to derive an accurate probabilistic distribution function, further studies in this area

have focused on enhancing the accuracy of the estimation of the parameters of these

probabilistic functions. The use of survival component data besides failure data was

employed to enlarge the data set and consequently improve the accuracy of parameter

estimation [47-49]. Further to previous studies, some research articles used the


34

chronological failure data to estimate the parameters of distribution functions [50, 51]. In

these studies, the distribution parameters were estimated in an earlier year than the current

year. Then, based on the following year’s failure data, the previously estimated parameters

were updated. When looking at all these studies, there are three common probabilistic

distribution functions used to characterise end-of-life failure. These are normal, Weibull

and lognormal distributions [44, 45, 47, 49-51].

Nevertheless, the formulation of these functions based on the failure data or failure and

survival data may not be sufficient to reflect the failure mechanism as operational

conditions have a significant impact on component failure. That is to say, components that

operate under high system operational stress fail at an earlier age than components that

operate under low stress [52]. The probabilistic functions derived from historical data do

not reflect the operational stress. Accordingly, some recent studies refined the conventional

probability functions by integrating operational factors such as, thermal stress due to

loading levels, and voltage surges into failure models [53-58]. In these “unconventional”

failure models, the parameters of the probabilistic distributions were presented as functions

of operational parameters. An example of these unconventional models is the Arrhenius-

Weibull distribution, which describes the effect of loading levels on failure probability.

This is achieved by defining the scale parameter of the Weibull distribution by the

Arrhenius relationship, which is a function of temperature [54, 55]. In this way, part of the

valuable knowledge of the physics of power system components is applied to the failure

models. The parameters of the model are derived from historical operation and failure data

[58].

In addition, with all the data available from condition monitoring, some research studies

have been carried out to formulate failure probability functions based on the component’s

condition [59, 60]. In these studies, a threshold value for each condition diagnostic

measure is set, e.g., for transformers it can be degree of polymerisation (DP) or total

dissolved combustible gas (TDCG). Then, the failure probability function is estimated as

the probability that the condition diagnostic measure exceeds this threshold.

From this review, one can state that there are two main types of end-of-life failure models

available in the literature. The first one is the conventional probability distribution function

derived from failure, or from failure and survival data. The second type is unconventional

probability function in which some operational factors are included in probability function.

The parameters of these unconventional functions are not constant, they change with


35

operational stress. Unconventional failure models also include non-parametric probability

function, which are derived from condition diagnostic measures and a predefined

threshold.

1.3.2 Incorporation into System Reliability Assessment

As stated earlier, when reliability assessment was first introduced, the analytical and

simulation techniques were based on Markov models. In order to apply Markov models to

an element or system, there are some conditions that have to be met [42]. For repairable

failure, these conditions are not a problem and they are all fulfilled. For end-of-life failure,

however, there are two conditions that are not true: the possibility to transit between all

states and lack of memory [42]. The former is not applicable for end-of-life failure because

if the component fails, it stays in this state. The latter is not applicable as the end-of-life

failure probability is conditional on the fact that the component has survived up to the

present time [42] . This conditional probability is known as posteriori failure probability,

i.e., it depends on the behaviour of the component in the previous time period, and it is not

a lack of memory. As a result of these two issues, assessing system reliability considering

end-of-life failure cannot be accomplished using the traditional Markov method.

A method of the incorporation of end-of-life failure into system reliability was first

introduced in 2002 by Dr. W. Li [61, 62]. In this method, the average unavailability is

estimated in order to incorporate end-of-life failure into the most common methods of the

composite power system reliability evaluation, i.e., State Enumeration and Non-sequential

Monte Carlo simulation. Dr. Li defined the average unavailability as “an average

probability that a component is found unavailable due to aging failures during a specified

time period t, given that it has survived for T years” [61]. The definition comprises two

elements of time, a specific age T and a future study period t. Accordingly, a discretisation

method was proposed to estimate this average unavailability. The estimation is based on

the division of the study period t into a number of equal subintervals followed by the

estimation of the average unavailability in all subintervals. The unavailability during a

subinterval is calculated by multiplying the probability of transition to end-of-life failure

by the average unavailable duration. This involves the assumption that the failure

probability within it is approximately constant. The main issues with this method are the

difficulties that arise when an attempt is made to calculate the probability of transition to

end-of-life failure. This occurs because the calculation depends heavily on the probability


36

distribution function that characterises end-of-life failure. Although Dr. Li has given

approximate formulae for normal (Gaussian) and Weibull distributions [61], it is still

difficult to have a general algorithm to assess system reliability for all probability

distribution functions. In spite of these shortcomings, Dr. Li’s method is the state-of-the-art

in the incorporation of end-of-life failure into system reliability.

Most published studies that assessed system reliability with end-of-life failure adopted this

method [30, 63-69]. In all these publications only conventional probability functions were

used. The only attempt to integrate unconventional failure model into system reliability

was presented in [55] using State Enumeration method. However, the method used to

integrate this model was based on probability of end-of-life failure rather than the

unavailability. Since the calculation of reliability indices using State Enumeration method

is totally based on the concept of unavailability [11, 25, 26], use of probability of failure

instead of unavailability was the major shortcoming of this study.

It should be mentioned that there are some publications that discuss the integration of age

related repairable failure, e.g., [70, 71], in system reliability assessment. However, this area

is beyond the scope of the thesis and the examples cited here are for further clarification of

the concept of end-of-life failure and its integration in reliability assessment.

1.3.3 Applications in Replacement Planning

The application of system reliability in asset replacement planning is based completely on

the integration of end-of-life failure in reliability assessment. There are very few known

studies that have been published in this area; actually only two publications: [63] and [72].

In the first study, the reliability assessment was applied so that a decision regarding the

optimal time to replace a transmission component could be taken [63]. This was achieved

by conducting an economic comparison between the unreliability cost and the savings on

capital expenditure when deferring the replacement action to later years. The comparison

started by assessing the unreliability cost for a specific number of future years and the

accumulated savings on costs that resulted from not reinvesting in replacement. Then, the

optimal time to replace the component under study is when the cost of unreliability

exceeds the savings on reinvestment costs. It was the first study that applied system

reliability in replacement planning. The only issue with this study is that it sets a

framework for making decisions on the replacement of a single component rather than the

entire fleet. In the second study, the reliability assessment is used to compare between


37

different replacement plans for a cable population in order to justify the cost of proactive

replacement compared to upon-failure replacement, i.e., active replacement [72].

Nevertheless, the replacement volumes of cables were determined based on age only and

did not consider the criticality of the cables to the system reliability.

1.3.4 Uncertainty Quantification in System Reliability

The first step of the uncertainty quantification procedure is the identification of the forms

of uncertainty [73]. There are two main forms of uncertainty: aleatory and epistemic. The

aleatory uncertainty, also known as irreducible uncertainty and variability, represents the

inherited random behaviour of power systems [73, 74]. The epistemic uncertainty, also

called reducible uncertainty and state of knowledge uncertainty, models the uncertainty in

parameter estimation due to data shortages or model simplifications [73, 74]. Each form of

uncertainty has its representation models and quantification methods.

The aleatory uncertainty originates from the random behaviour of the system. A classic

example of this is the TTF of a component. The common method of modelling the aleatory

uncertainty is by probabilistic distribution functions. Based on this, the aleatory uncertainty

is propagated by using one of the probability based approaches, such as sampling,

analytical reliability method, or Polynomial Chaos Expansion [74]. The epistemic

uncertainty characterises the uncertainty in parameter estimation due to data shortage or

model simplifications. The representation of the epistemic uncertainty depends on the

amount of available data. Where there is enough data, the probability distribution function

can be the potential model. Then, epistemic uncertainty can be quantified using probability

based approaches. In practice, however, it is difficult to get enough data to derive a

probability density function (pdf). Therefore, a subjective pdf is used: typically a uniform

or normal distribution. In order to avoid this subjective assignment of uncertainty,

researchers have developed different representation and quantification methods that can

deal with the deficiency of data. Among those are Fuzzy set, possibility theory and

evidence theory.

In power system reliability, the aleatory uncertainty has been quantified using the sampling

approach in the Monte Carlo method form. The Sequential and Non-sequential Monte

Carlo techniques have been used to propagate the aleatory uncertainty and build

probabilistic distribution for the reliability indices [24, 75]. The epistemic uncertainty in

power system reliability has been studied in many different ways. The earliest papers


38

calculated the mean and variance of reliability indices from the mean and variance of

failure rate and load forecast using analytical methods [16, 76, 77]. The issue of these

analytical studies is that they do not provide distribution functions of the indices.

Accordingly, another group of publications has characterised the uncertainty by pdf and

has propagated the effect on reliability evaluation by sampling values from the pdf or by

dividing the pdf into intervals [25, 78, 79]. Furthermore, [80-82] have introduced the Fuzzy

set theory to quantify the epistemic uncertainty in failure rate and load forecast values. In

all these previous studies, the concept of distinguishing between the forms of uncertainty

were not presented till the publication of studies reported in [83], which was the first article

discussing the different forms of uncertainty in power system reliability. Then, [84] studied

the classification of uncertainty forms in distribution systems and propagated the two

forms of uncertainty in one framework to the reliability indices using evidence theory.

Although it is worth studying the uncertainty in end-of-life failure models more than

repairable failure, there are no existing studies that address this issue. A reasonable

explanation is that the integration of end-of-life failure is a recent area of research.

1.4 Summary of Past Work

From the previous review, it is apparent that the application of reliability assessment in

replacement planning is a new area of research. There are further studies which can be

carried out to develop this area and improve the existing research. These are summarised

as follows:

The reliability application in replacement planning can further be improved by applying

it to the entire fleet rather than to a single component. The economic comparison can

then be performed to determine the optimum number of components to be replaced. The

volume of replacement can be determined by considering the criticality of components

to system reliability rather than by using the age profile.

Modelling of end-of-life failure of power system components has been extensively

researched. As a result, there are many unconventional and more advanced models,

which have not been employed in system reliability assessment. Studies can be

performed to employ unconventional failure model into power system reliability

assessment. This will enhance the assessment of system reliability and at the same time

assess the value of refining these failure models and their effect on system reliability.


39

No studies discussing the probability of component failure as a consequence of other

components failure, i.e., dependent failure, have been introduced. In particular,

dependent failure is a potential source of concern for ageing systems because the

withstand strength of components decreases with age and they become more vulnerable

to consequent failure. Such studies will help in the investigations of cascading failure

events.

Since power system components have a considerably long life-time, the lack of

historical failure data makes the uncertainty associated with end-of-life failure more

questionable than repairable failure. In addition, end-of-life failure is more related to

operational conditions, and hence, it is not appropriate to use models developed by other

utilities or combine the data of different utilities to derive one model. The quantification

of uncertainty associated with end-of-life failure models contributes to existing research

in this area. Studies can be conducted to examine the sensitivity of system reliability to

the uncertainty in the parameters of end-of-life failure models.

The use of Sequential Monte Carlo (SMC) simulation can perhaps deal with the

difficulties of applying the state-of-the-art method as SMC does not involve the

calculations of component unavailability. In SMC, the system state selection is achieved

by producing chronological component state transition that is sampled from the

probability distribution function. Such study will be a valuable contribution since the

SMC method is more accurate in the evaluation of monetary value of unreliability

because it accurately estimates failure duration.

1.5 Research Aim and Objectives

Following the identification of some of the issues that have not been adequately addressed

in past work related to system reliability application in replacement planning and end-of-

life failure, the main aim of the thesis is to undertake a thorough analysis of transmission

system reliability considering end-of-life failure of power transformers in the context of

replacement planning application. The analysis must include the application of reliability

studies in replacement decision making, the integration of more advanced end-of-life

failure models into system reliability assessment, and the quantification of uncertainty

related to failure models and operational conditions. The results of the analysis will be


40

used to assist in developing replacement plans for transformers against system reliability

requirements to ensure an optimum and justifiable prioritisation of transformer

replacement. In order to achieve this aim, the research has the following objectives:

1. To present state-of-the-art in power system reliability assessment considering ageing

asset and the effect of ageing power transformers in particular.

2. To develop research grade software for assessing composite power system reliability.

The reason for setting this objective is that commercial software packages perform

conventional reliability assessment based on Markov models, which is not applicable

for end-of-life failure, and at the same time they do not allow user-defined failure

models. Therefore, it is necessary to develop a reliability assessment tool first, in order

to be able to conduct relevant studies subsequently.

3. To evaluate system reliability considering end-of-life failure of power transformers.

This objective serves as the first step towards the analysis. It involves applying the

state-of-the-art method of incorporating end-of-life failure into system reliability

assessment.

4. To develop a methodology to identify the most critical transformers for power system

reliability and their area of vulnerability.

5. To develop a methodology for cost-effective replacement planning of power

transformers. The methodology must include the advantages of the existing techniques

and overcome the identified limitations.

6. To integrate unconventional failure models of power transformer into system reliability

studies. To this end, only available failure models in the open literature will be

employed.

7. To explore ways of assessing the dependent failure of power transformers, and to

consider it in the identification of transformer criticality. This helps in studying the

effect of end-of-life failure in multiple failure events.

8. To quantify the effect of uncertainty associated with end-of-life failure model

parameters on system reliability.


41

1.6 Research Contributions

The main contributions of this thesis are summarised in the following list. (Note:

References given in parentheses indicate that the related results are published in

international journals or presented at international conferences. The full list of thesis based

publications is given in Appendix H)

The clarification of the end-of-life failure concept. The thesis provides a

comprehensive explanation of the concept of end-of-life failure and how it is different

from the traditional repairable failure. Ambiguities within reliability studies related to

age related failure are clarified. Based on this, the state-of-the-art method in the

integration of end-of-life failure into system reliability assessment is identified and

adopted (H.1 - H.4).

The development of a reliability based framework for the replacement of power

transformers fleet. The framework combines, for the first time, the advantages of the

two existing methods of replacement planning; risk matrix method and the reliability

indices method. The reliability importance measures were adopted to identify the most

critical transformers and to determine the volume of transformers to be replaced. The

use of the Pareto analysis incorporated within the framework (previously not

implemented in system reliability applications) provides an insight into the effect of

equipment replacement volume on system reliability. Furthermore, the framework also

demonstrates how the regulatory reliability incentive/penalty scheme can be included in

replacement planning (H.1).

The integration of unconventional end-of-life failure models of power transformers

into system reliability assessment. This is the first study to integrate available

unconventional end-of-life failure models of power transformer into system reliability

assessment. The failure model employed considers the thermal stress due to loading

conditions, and is characterised by Arrhenius-Weibull distribution. This study

contributed to the system reliability assessment as it evaluates the added value provided

by including loading levels in failure models, and how this enhanced the understanding

of the effect of operational factors on system reliability (H.3).

The development of transformer criticality indicators to cascading failure event.

The research adopted the Arrhenius-Weibull distribution model in order to investigate


42

the probability of occurrence of second dependent failure due to thermal stress. The

second dependent failure might ultimately lead to a multiple failure event or even a

cascading failure. This has led to the development of two new probabilistic indicators,

relating the reliability of transformers to their age and loading levels, to rank power

transformers based on their criticality for multiple failure events. The developed

indicators could be practically useful for asset management, system planning and

operation applications. They are also proven to be robust with respect to load

uncertainty (H.2 and H.4).

The establishment of bases for a system related approach to refining transformer

failure models. The approach facilitates assessing to what extent the reliability of

transmission network is sensitive to transformer failure model parameters with respect

to specific application, e.g., the identification of most critical transformer sites. Once

this is established, the attention and resources can be focused on refining failure models

of key transformers only as the system reliability, or the system reliability application,

are most sensitive to uncertainty in model parameters of these transformers (H.3).

The quantification and propagation of the uncertainty associated with failure

model parameters to transmission system reliability assessment. The thesis has

characterised the uncertainty associated with failure models as mixed aleatory-epistemic

uncertainty and hence adopted two methods to quantify this form of uncertainty: the

second order probability and Evidence theory. The quantification of both forms of

uncertainty in one framework has not been done before this study. In addition,

reliability important measures, based on sampling based sensitivity analysis, were

proposed to identify the components with major impact on system reliability and their

area of vulnerability. This is an original application of uncertainty studies related to

system reliability (H.3, H.5, and H.6).

In addition to the previous original research contributions, research grade software was

developed to assess transmission system reliability considering end-of-life failure. In the

development of the software, DIgSILENT PowerFactory, which is one of the power

system commercial packages, was used as the primary computational environment. The

software can be easily modified to conduct further studies in the area of power system

reliability involving other power system components. The software is proprietary of The

University of Manchester and National Grid.


43

1.7 Outline of the Thesis

This thesis consists of eight chapters including this introductory one. The remaining seven

chapters are outlined below:

Chapter 2 - Composite Power System Reliability Evaluation

This chapter describes the stages of composite power system reliability evaluation. It has

three main parts. Part one discusses the representation of components failure and

operational conditions in probabilistic models. Part two explains the methods and

techniques used to evaluate composite power system reliability. It also provides the

calculations steps of the reliability indices for each method. The final part lists power

system reliability indices and relevant calculation steps for their evaluation. The chapter

presents the fundamentals of composite power system reliability assessment.

Chapter 3 - Reliability Assessment Considering End-of-Life Failure

In this chapter, details of the state-of-the-art method for the incorporation of end-of-life

failure into system reliability assessment are given. The chapter also describes the

reliability software developed within the research. A full documentation of the software is

provided including functional definitions, programming information, and validation. In

addition, the test network used throughout the research is described. The chapter also states

all assumptions and considerations applied to reliability assessment studies presented in

this thesis.

Chapter 4 - Reliability-Based Replacement Framework

A reliability-based framework for cost-effective replacement of power transformers is

proposed in this chapter. The framework is capable of combining the merits of existing

methods and enhancing some of the identified limitations. The framework comprises the

identification of the most critical transformer, the use of Pareto analysis to relate

replacement scenarios to system reliability, and the economic comparison to determine the

optimum number of transformers to be replaced. The application of the framework is

shown by a case study, where its suitability for assisting in replacement decision-making is

demonstrated.


44

Chapter 5 – Incorporation of Unconventional Failure Models into Reliability studies

This chapter illustrates the integration of unconventional failure model of power

transformer into system reliability. The model includes the effect of thermal stress due to

loading levels, and is characterised by the Arrhenius-Weibull distribution. Reliability

assessment is performed using this unconventional model, and the results are compared to

the results obtained using the traditional probability function in order to examine the

impact of the integration of the advanced unconventional model. The results show that

using the Arrhenius-Weibull distribution provides more insight into understanding the

system reliability and identifying the failure conditions of most critical transformers.

Chapter 6 - Transformer Criticality for Cascading Failure Events

This chapter presents a method to assess the probability of the dependent failure of aged

power transformers induced by thermal stress. The Arrhenius-Weibull distribution is also

used in this chapter. The chapter also investigates the criticality of power transformers to

multiple failure events. Based on this investigation, probabilistic indicators for assessing

age and loading based criticality of transformers to cascading failure events are developed.

The indicators measure the probability of initiating a cascading failure event and the

vulnerability to a consequent failure event. In addition, the chapter presents a study of the

robustness of the indicators to uncertainty in transformer loading.

Chapter 7 - Uncertainty Quantification in Reliability Assessment

This chapter firstly discusses the forms of uncertainty in power system reliability

assessment; aleatory and epistemic and the sources from which they are originated. The

chapter solely deals with uncertainty associated with the estimation of failure model

parameters. The chapter addresses the uncertainty issues in three aspects. Firstly, it

introduces a study of the effect of uncertainty in the parameters of two end-of-life failure

models: Gaussian and Arrhenius-Weibull distributions. The purpose of this study is to

assess to what extent the system reliability can be sensitive to the uncertainty in the

parameters of end-of-life failure models. Secondly, it describes the methods of the

uncertainty quantification and propagation to system reliability indices. It gives case

studies on the IEEE-Reliability Test System (IEEE-RTS) on two methods of

quantifications: second order probability and evidence theory. Thirdly, a case study on the

application of sampled based sensitivity analysis in the identification of the most important


45

components and their area of vulnerability is provided. In this way, the chapter covers all

issues and applications of uncertainty quantification related to failure models.

Chapter 8 - Conclusions and Future Work

The main conclusions of the research are given in this chapter. The chapter also presents

suggestions for future development of proposed methodologies and indicates and discusses

the areas for further research.


46

Chapter 2 Composite Power System Reliability Evaluation

47

2

Composite Power System

Reliability Evaluation

2.1 Introduction

Composite power system reliability is a phrase used to describe the assessment of

generation and transmission systems reliability together. This includes many power system

considerations such as load flow, different generation dispatches, available reactive power

sources, and capacity of lines. This chapter provides a detailed background about the

assessment of composite power systems reliability.

Generally speaking, there are three aspects that are involved in composite system

reliability assessment: modelling, evaluation, and applications. The modelling aspect

involves the probabilistic modelling of component failure, annual load and network

operational status. The evaluation aspect refers to the methods and techniques for assessing

and calculating the reliability indices. The last one is the application of the reliability

analysis on power system activities. The application of reliability is the starting point of the

analysis. The purpose of the reliability analysis should be clearly defined prior to

performing any study because the type of the application determines which models are the

most dominant ones, and which evaluation technique is the most appropriate. Figure 2-1


48

shows these three aspects and the relationship between them. This chapter deals with the

modelling and evaluation aspects only. In discussing component modelling, the chapter

focuses on the difference between repairable failure and end-of-life failure and how they

are modelled. The chapter also provides a description of the three main evaluation

techniques: State Enumeration, Non-sequential Monte Carlo simulations and Sequential

Monte Carlo simulations. In addition, a summary of composite system reliability indices

and their calculation formulae are given.

Figure 2-1: Aspects related to assessment of the composite power system reliability

2.2 Modelling

Modelling from the reliability perspective means the probabilistic representation of all

events involved in the calculation of the reliability indices. In composite system reliability,

there are three input models that need to be considered in the evaluation procedure. These

are: components failure model, load model, and network models. This section describes

these models and explains how the data is processed to estimate model parameters.

2.2.1 Component Failure Models

The failure of a power system component is a stochastic event, whose time of occurrence is

a random variable. In other words, in reliability studies the random variable, which must be

modelled, is the time of occurrence of the failure [85]. Commonly, stochastic events are

modelled by probabilistic distributions. There are many forms of functions for interpreting

probabilistic distributions of random variables. The easiest one to explain is the cumulative

distribution function (cdf). Generally, a cdf of any random variable gives the probability of

the random variable being equal to or less than a specific value [11, 42]. For example, if d

is a random variable, the cdf(D) gives the probability of d≤ D. When projecting this

concept to the time of occurrence of the failure, the cdf(T) defines the probability that a


49

component will fail at time ≤ T, which is simply the probability of failure. Therefore, in

engineering reliability studies the cdf is known as the probability of failure function and it

is commonly denoted by Q(t)[42]. The complementary function of the failure function is

the survival or the reliability function R(t). Given that the total probability of any two

complementary events equals one, R(t) can be calculated by (2.1):

)(1)( tQtR . (2.1)

The value of Q(t) at t=0 is zero, while Q(∞) equals one. In the same way, the reliability of

a component R(t) equals one at t=0, whereas it equals zero when t=∞. The third form of

probability distribution functions is the probability density function pdf, which is denoted

as f(t) in reliability engineering. The f(t) is the derivative of the cumulative distribution

function as given by (2.2):

dt

tdQtf

)()( . (2.2)

The integral of pdf over a period of time gives the probability of the failure occurring

during this period. Accordingly, the integral of pdf from zero to infinity equals one.

There is an additional function for interpreting the probabilistic distribution, which is

particularly related to the reliability analysis; this is the hazard rate function. It is also

known as the failure rate function and is designated as (t). This function is introduced to

have a sense of the instantaneous probability of failure at a specific point of time [86]. For

the next period of time t, the hazard rate function is the probability that a component will

fail during t given that it has survived till the beginning of t divided by t [11, 87, 88].

The division by t is the reason for having the word rate in its name. Consequently, the

hazard rate function has units 1/time. The relationship between (t)and other distribution

functions is given by (2.3) [11]:

)(

)()(

tR

tft .

(2.3)

In engineering reliability, the hazard rate function of an element throughout its life-time is

usually described by the bathtub curve, which was given earlier in Figure 1-6.The hazard

rate function of the exponential distribution can be calculated as follows:

The pdf of the exponential distribution is given by (2.4):


50

tetf

)(

(2.4)

where is the parameter of the exponential distribution.

The reliability function of exponential distribution is given by (2.5):

tetR

)( . (2.5)

Then using (2.3), the hazard rate function of the exponential distribution is calculated as

shown in (2.6):

t

t

e

et)( . (2.6)

The constant hazard rate function is a very unique feature of the exponential distribution,

and hence, exponential distribution is used to characterise a failure event in the useful life

stage of the bathtub curve [11]. This feature is also the reason for referring to the parameter

of the exponential distribution as failure rate.

As discussed in Chapter 1, the failure of power system components can be classified into

two types; repairable failure and end-of-life failure. For repairable failure, a component

transits between in-service state and in-repair state in the time period of study, whereas for

end-of-life failure, when a component fails it cannot be repaired. Specifically, in end-of-

life failure, there is one transition and it occurs only once. Consequently, the random

variable in repairable failure is Time To Failure (TTF), whereas it is the life-time for end-

of-life failure.

2.2.1.1 Repairable Failure

The concept behind the repairable failure is that the component can be repaired to the same

condition before the failure. For power system components, the repair duration takes

considerable time, and hence the repair process is also defined as a stochastic process.

Based on this, the component outage can be modelled by two states: the up and down

states. The transition rates between these two states are the failure rate and the repair rate.

This model is depicted in Figure 2-2 [26].

Figure 2-2: Two state model of repairable failure

Up State

Repair Rate

Down State

Failure Rate


51

In typical power system reliability studies, the failure and repair rates are assumed

constant, which means the failure and repair processes have an exponential distribution

[11, 25]. With this assumption, the two state model meets the requirements of the Markov

process, which are the possibility of transitions between all states, lack of memory, and

stationary transition rates between the states [42]. The first requirement is an inherent

feature of the two state model since the component transits between the states. The second

and last requirements are satisfied by the exponential distribution assumption. It has been

shown previously that the exponential distribution has a constant hazard rate or failure rate

functions. In order to demonstrate that it is memory-less, one may assume that a

component has operated for a period of time T and the probability of failure in the next

period of time t has to be evaluated. The main consideration here is that the component

cannot fail in T+t if it has failed in the previous time T. This is a conditional probability

problem because what needs to be assessed is the probability of failure during t given that

it has survived up to T. The conditional probability rule is given by (2.7) [42]:

)(

)()/(

BP

BAPBAP

.

(2.7)

A reflection of this on the stated problem, )( BAP is the probability of surviving up to T

and failing during t. This probability can be estimated by integrating the pdf of exponential

distribution from T to T+t as shown by (2.8):

)()()(

tTTtT

T

ttT

T

eedtedttfBAP

. (2.8)

P(B), which is the probability of survival up to T, is actually one minus the probability of

failure during the previous period T given as in (2.9):

dttfBPT

0

)(1)( . (2.9)

Given that the integration of the pdf from zero to infinity equals one, P(B) can then be

expressed as

T

T

t

T

TeedttfdttfdttfBP

)()()()(00

. (2.10)

Substitution of the previous probabilities in (2.7) gives (2.11) which is the probability of

failure during t given that the component has survived up to T:


52

teBAP

1)/( . (2.11)

From (2.11), it is obvious that the conditional probability calculated for exponential

distribution does not depend on the previous period T, it only depends on the future study

time t. Therefore, the exponential distribution is memory-less.

Markov process theory states that the probability of being found in any state (up or down)

reaches a limiting value that is independent of the initial conditions (up or down). The

probability of being found in the up state is termed as the availability. Likewise, the

probability of being found in the down state is known as the unavailability. The availability

and unavailability are essential measures of component performance in system reliability.

Referring to the Markov process, for the repairable failure, the availability and

unavailability of a component are constant in the long run. Figure 2-3 shows an illustrative

example of the limiting values of availability and unavailability. The up state is denoted in

the figure as 1 and the down state is as 0. As shown in the figure, the availability and the

unavailability reach the same limiting values regardless of the initial state of the

component.

Figure 2-3: Illustrative limiting values of availability and unavailability based on Markov theory

The unavailability (U) and availability (A) are calculated by (2.12) and (2.13) respectively,

which are the Markov limiting state probabilities [11, 26, 42]:

U

(2.12)

UA 1

(2.13)

where λ andµ are the failure rate and repair rate, respectively.


53

It should be noticed that the Markov process is not applicable to repairable failure

occurring in the wear out stage of the bathtub curve because the failure rate is not constant;

it changes with time, hence it cannot be modelled by exponential distribution.

2.2.1.2 End-of-life Failure

The end-of-life failure of a component occurs only once, and there is no repair action. If

the component has failed in the previous period of time, it cannot fail in the future period

of time. This is similar to the conditional probability given in the previous illustration of

lack of memory of the exponential distribution. This conditional probability is known as

posteriori failure probability, i.e., it depends on the behaviour of the component in the

previous period time [42, 89]. Accordingly, the Markov process is inapplicable to the

calculations of probability related to end-of-life failure.

End-of-life failure is usually related to the wear out stage or the aging of the component.

This does not mean that it does not occur during other stages, but it seldom happens. This

thesis focuses on end-of-life failure in the wear out stage. The probability of having end-

of-life failure in the future period of time PEoL(t) is calculated using the same conditional

probability rule given by (2.7). Hence, if end-of-life failure of a component is characterised

by f(t), the probability of the transition to end-of-life failure in the future time t given that

it has survived up to T can be calculated by (2.14):

T

tT

TEoL

dttf

dttf

tP

)(

)(

)( .

(2.14)

It must be pointed out that this probability is not the unavailability due to end-of-life

failure; it is the probability of transition from up state to end-of-life failure state. The

unavailability is the probability of finding the component in the failure state. The review of

available literature has revealed that there is only one method that has been introduced to

estimate the unavailability due to end-of-life failure. This state-of-the-art method has been

adopted in this thesis. The full explanation of the method is given in Chapter 3. By

providing this distinction between types of failure, the thesis enhanced the understanding

of age related failure and stimulated researchers to focus on models and methods that are

related to the applications under consideration. This explanation is the first original

contribution of this thesis.


54

2.2.2 Load Models

The simplest practice of modelling load is to consider it as a single level that remains

constant over a yearly period. The peak load is usually employed for this model in order to

account for the worst case scenario. The reliability indices calculated using this model are

known as annualised indices. The major advantage of this model is that it reduces the

computation time of the reliability assessment; however, it does not reflect the variation in

the load demand throughout the year.

For some system reliability applications, it is essential to consider the load variation during

the study period. Accordingly, the annual load curve has to be modelled and incorporated

in these reliability analyses. There are two approaches for modelling an annual load curve

[24, 26]. The first approach is to consider the chronological annual load curve and to

perform reliability assessment at each hourly load. The annual reliability indices are

calculated using an equal probability of 1/8760 for each hourly load. This approach is the

most accurate one, but it requires excessive computation time and effort. The second

approach is to represent the annual load variation by the load duration curve and then

convert this duration curve into a multi-step load model [24]. An illustrative model is given

in Figure 2-4. The K-means clustering technique is the most common method for obtaining

multi-step load models. The main steps in this technique are:

1. Determine the number of steps.

2. Set an initial value of the load level for individual steps.

3. Calculate the distance between hourly load points and all load levels. Then cluster

the hourly points to the nearest load level.

4. Recalculate each load level by dividing the summation of all hourly load points by

the number of points in that level.

5. Repeat steps 3 and 4 with an acceptable level of accuracy.

The calculated load levels and the number of hourly load points formulate the multi-step

levels with the associated duration. The accuracy of the reliability results is proportional to

the number of steps, and hence, proportional to the computation time. The selection of the

number of steps is a trade-off between the required level of accuracy and the computational

time of the evaluation. Different transmission networks have different sensitivities to the

load levels, and hence to the number of steps in the load model.


55

In order to incorporate this multi-step model into system reliability assessment, one can

either enumerate the levels one by one or randomly sample them within the simulation

iterations. For the former, the reliability indices are assessed at each level, and then the

annual indices are calculated using the indices obtained for individual levels and their

associated probabilities. The latter approach is only applicable for reliability assessment

techniques based on simulations. In this method, the probabilities of load levels are sorted

from smallest to largest, and then accumulated. For each iteration of the reliability

assessment, a random number between 0 and 1 is generated and compared to the

accumulated probabilities to sample the load levels.

Time (hours)

Load

[M

W]

1000 2000 3000 4000 5000 6000 7000 8000

Figure 2-4: Illustrative example of multi-step load model for the load duration curve

2.2.3 Network Models

The network models in composite system reliability studies can be classified into two

categories. The first one is the same as modelling the network for load flow studies. This

category is important for examining whether a system state is a system success event or a

system failure event and to quantify the consequences of system failure events. (Note:

system success events are system states where there is neither violation of system limits

nor load shedding actions, whereas system failure events are system states where system

limits are exceeded or load shedding actions are taken to bring the system back to normal

operation. “System success" and "system failure" terms are used throughout the thesis to

refer to specific system states defined above.)

The second category in network modelling is the probabilistic modelling of operating

conditions and some practical system considerations. This category is associated with

system state selections. In this chapter, some of the major aspects are discussed and they

are: transmission system adjustments, external power injections, and planned outages [14].


56

2.2.3.1 Transmission System Adjustments

This aspect is not related to the probabilistic representation, it refers to the adjustments

made in modelling the transmission system to make the reliability assessment more

manageable. There is no standard way of doing these adjustments, however, they should be

completed with much care so that the reliability results are not significantly affected. An

example of these adjustments is the aggregation of load connected to the distribution

system. In this process, the connections at the distribution level between the grid supply

points must be accounted for. Another example is the simplification of transmission

substations by modelling them as single busbars.

2.2.3.2 External Power Injections

The need to consider the external power injections arises because many systems have

points of common coupling (PCC) with other transmission systems due to contractual

transfer or interconnection. Those PCCs may also result from study zones classification

within large transmission systems. The probabilistic modelling of these power injections is

essential for reliability assessment since they are directly related to the power flows

through the transmission system. A probabilistic model of these injections can be derived

from the historical power flow. The common model is to have multiple levels of injection

with their associated probabilities. This can be formulated using the same K-means

clustering technique used in annual load modelling [14].

2.2.3.3 Planned Outages

Modelling of planned outages (maintenance and operation) can be done in two ways. The

first one is to treat the planned outage as failure outage and model it with the two state

model. The transition rates between the two states are estimated form historical planned

outage. Then, the unavailability due to planned outage can be estimated using the Markov

process [14, 26]. By doing this, planned outages are considered as random events. The

second model is to have the predetermined schedule of planned outages for the period of

study [14, 26]. This model is more realistic because it ensures that maintenance planning

criteria set by the utility is fulfilled. For instance, utilities commonly do not allow for more

than one component in a substation to be out of service for maintenance. This condition is

not granted attainment when considering the planned outages as random events.


57

2.3 Evaluation Techniques

An engineering system consists of a group of components that are connected together and

affect each other in some manner. For assessing the reliability of a system, the individual

component failures and their combination should be represented. There are many

developed methods for the reliability assessment of engineering systems. The most

fundamental ones are parallel/series and the Markov process. Although these methods are

not suitable for large and complex power systems, they are briefly explained here due to

their conceptual importance.

2.3.1 Fundamental Techniques

In the series and parallel method, the system failure due to the failure of its component is

logically represented. For series representation, the failure of any component leads to the

system failure or all the components must be available for the system to be available. The

unavailability of a system consists of two series components, which have unavailability

values Ua and Ub, can be calculated by (2.15) [26]:

babasys UUUUU . (2.15)

For parallel representation, only one available component leads to the system’s success or

all the components must be unavailable to the system to fail [26, 42]. The unavailability of

a system consists of two parallel components (Ua and Ub) is calculated by (2.16) [26]:

basys UUU . (2.16)

A system can be modelled by combination of series and parallel networks. This type of

modelling is suitable for reliability assessment of simple substations and radial distribution

systems.

The Markov process is based on system states and transition between these states. It has

been introduced earlier in this chapter to describe the two state failure model. It is

convenient for modelling failure events of the individual components, but it is difficult to

apply to large systems. This difficulty arises due to the need for developing the state space

diagram of the system using components’ states. For example, for a system with N

components where each has S states the size of the state space diagram will be SN. The


58

next step in the Markov process is to build the transition matrix (T) based on the state

space diagram. The transition matrix (T) is a square matrix with a dimension that is equal

to the number of the states. The value of the element at the ith

row and jth

column, for

example, is the transition rate from state i to state j. If there is no transition between the

two states, the element is zero. The diagonal element is one minus the sum of all other

elements on the same row. The probability of system states (Ps) after M discrete transition

can be calculated using the matrix multiplication given by (2.17) [42]:

Mss TPMP )0()(

(2.17)

where PS(0) is the vector of the probability of initial conditions. The vector PS(0) has

always one element that equals one (the initial state has probability=1) and all remaining

elements are zeroes. According to the Markov principle, once the limiting state

probabilities have been reached, any further multiplication by the transition matrix does

not change the matrix Ps values. Consequently, if P is the state limiting probability vector,

then (2.18) is true:

PPT . (2.18)

Using (2.18) along with the full probability condition which states that the sum of the

probabilities should be one; the limiting state probabilities can be determined. In addition

to state probability, the Markov process enables the calculation of the frequency of

entering a state and the duration of staying in it.

Appendix B gives an illustrative example of using series/parallel and the Markov process

on a simple network.

2.3.2 Methods for Large Systems

As stated earlier the two previous methods are appropriate for evaluating the reliability of a

simple network or part of a large one. For larger and more complex systems, different

approaches are used. The commonly used approaches for composite power system

reliability are State Enumeration, Non-sequential Monte Carlo and Sequential Monte Carlo

simulation. State enumeration is an analytical method, while obviously Non-sequential and

Sequential Monte Carlo are simulation methods. In general, there are two basic stages in

these assessment techniques: system state selection and failure effect analysis [26]. The


59

three assessment techniques are different in selecting the system states while failure effect

analysis is the same for all of them. In other words, the way in which system states are

generated identifies the reliability assessment technique. Consequently, the reliability

indices calculation is different in each technique.

Data

System State selection

Load flow analysis

Any problem

Remedial actions

Any problem

Update indices

Stopping rules?

Results

no

no

no

yes

yes

yes

Load shedding

Figure 2-5: Flowchart of the steps of composite power system reliability assessment

Figure 2-5 shows the general steps of composite system reliability assessment. The first

step is to select system state, which will be discussed in the following sections. The

remaining steps in the flowchart describe the failure effect analysis stage. This stage

involves the analysis of system states to define ones that are considered as failure states.

The first step in this stage, as shown in the flowchart, is to perform a load flow analysis to


60

determine if this system state causes disconnection of loads or any violation of loading and

voltage limits. If this is true, then, remedial actions by the system operator are simulated to

alleviate the problem. The remedial actions include switching reactors or capacitors,

adjusting FACT devices and phase shifters, and rescheduling the generators. If the problem

still exists, an optimisation procedure for load shedding is performed. Since load shedding

action leads to inadequate system state, it is counted as a failure state and the reliability

indices are updated.

2.3.3 State Enumeration

In the State Enumeration method, the system states are generated by enumerating them one

by one according to a predetermined level of contingency, e.g., first independent failure or

second independent failure. Since all the events in a power system are considered as

independent, the system state probability is calculated by multiplying the probabilities of

the combination elements, i.e., components, load level and network. This is explained by

(2.19):

nl

Ccci

pppp

(2.19)

where pi is the probability of the system state i, pc is the probability of component c state

and C is the set of all components in the system, pl is the probability of the load level, and

pn is the probability of the network state. The probability of the component state is

represented by either its availability or unavailability according to the enumerated system

state. As an illustration, if a system contains ten components and the enumerated state has

one failed component, the system states probability will equal the unavailability of the

failed component × the availability of the nine other components × load level probability ×

network state probability. Next, the system state is examined to decide if it is a system

success state or a system failure state. If the latter is the case, the consequence of this

failure state is obtained using failure effect analysis. The consequence can be any of the

risk measures such as: demand not supplied or energy not supplied. Then, the contribution

of this system state to reliability indices is calculated using (2.20):

iii

RpIndex (2.20)


61

where Indexi is the contribution of the failure state i to the reliability index, and Ri is the

risk measure. The total reliability index (Index) is the summation of index contributions

from all failure states (set S) as given by (2.21):

Si

iIndexIndex . (2.21)

The main strength of the State Enumeration method is its simplicity compared to

simulation methods, but it is infeasible to deal with large systems due to long computation

time. This is specifically true in cases where the level of contingency is higher than the

first failure level or N-1. Another drawback of this method is that it cannot handle the

events that are chronologically time dependent [24, 26].

2.3.4 Non-sequential Monte Carlo (NMC)

Generally, the simulation techniques are often used with large networks, and when

complex operation situations have to be considered, e.g., weather impact, as the

consideration of those situations will increase the size of system state space [26]. In these

simulation methods, the system states are randomly selected based on sampling

approaches. When the state sampling approach is used, the technique is termed as Non-

sequential Monte Carlo simulation (NMC), whereas, when the duration sampling approach

is employed it is termed as Sequential Monte Carlo simulation technique (SMC).

In the NMC approach, a system state is selected by randomly sampling components’ states,

a load state and a network state. Components’ states are selected by comparing their

unavailability to random numbers. For each component, if the random number is equal to

or less than the component unavailability (U), the component is unavailable or in a down

state and it is true the other way around. This can be explained by (2.22):

UR

URCs

1

0

(2.22)

where Cs is the component sate, 0 presents the failure state and 1 presents the success state,

and R is a random number. This process is repeated for individual components. The multi-

step load model can be incorporated into NMC in two ways: it can be enumerated one by

one or sampled based on the probabilities of levels. The probability of a system state is

estimated by dividing the number of the occurrence by the total number of NMC iterations

as given by (2.23):


62

MC

inip

)( (2.23)

where n(i) is the number of occurrences of state i and MC is the total number of NMC

iterations. The contribution of a system state to a reliability index and the reliability index

are calculated using (2.20) and (2.21), respectively.

It is apparent that State Enumeration and NMC depend on the calculation of the

unavailability of power system components. In addition, they cannot capture the

chronological time-dependant events in composite power system reliability.

2.3.5 Sequential Monte Carlo (SMC)

In the SMC technique, the system is represented by chronological time states over a period

of time. The difference between SMC and NMC is that each iteration in SMC is not a

system state; it is a period of time, usually a year. The chronological presentation of the

system is composed by combining the chronological time cycle of individual components,

load curve and network operational points. The component chronological time cycle is

formulated by sampling the time to failure (TTF) and time to repair (TTR) from their

distribution functions. An illustrative example of chronological states of four components

is given in Figure 2-6. For the load model, the annual load curve can be easily used.

Similarly, the network states such as generator dispatch and planned outage can be

represented in the time domain. In this way, accurate modelling of the annual load curve

and the operational conditions is achieved. Once all the states are developed, the system

availability is examined by conducting failure effect analysis for each hour. Reliability

indices are the average of indices calculated in all iterations.

Figure 2-6: Illustrative chronological states of four components

Component 1

Component 2

Component 3

Component 4

Up

Down

Up

Up

Up

Down

Down

DownTime


63

Since NMC and SMC are iteration based approaches, the estimated indices fluctuate. The

bound of this fluctuation decreases with the increase of the number of iterations. Therefore,

it is appropriate to set a convergence criterion to stop the simulation. It is common practice

to use the coefficient of variation as an accuracy measure of the Monte Carlo simulation.

The coefficient of variation, which is given by (2.24), measures the dispersion of the

indices [24]:

(2.24)

where is the accuracy measure, is the standard deviation, and is the mean value.

Furthermore, the results from the iterations are used to construct the distribution of the

reliability indices. In this way, NMC and SMC are employed to quantify the aleatory

uncertainty in reliability assessment.

In addition to the representation of chronological time events, SMC has two additional

main strengths. Firstly, since the technique does not employ the calculation of the

component unavailability, it handles all types of probability distributions that are used to

model component outages without any difficulty. Secondly, the frequency and duration

indices are accurately estimated as they are clearly defined in the chronological

representation [26]. The only disadvantage comparable to the NMC method is that it needs

more computational time and memory space.

It should be noted that this chapter provides only the principal concept of State

Enumeration, NMC and SMC. System reliability literature, however, contains enhanced

and computationally more efficient techniques for assessing composite power system

reliability. Some of these studies adopted more efficient sampling techniques than

conventional Monte Carlo analysis. For example, studies reported in [90] adopted Latin

hypercube sampling technique to enhance the evaluation of the reliability indices

distributions. Other studies employed artificial intelligence, e.g., genetic algorithm [91]

and Evolutionary Particle Swarm optimisation [92], to enhance the state selection by

focusing on the state that contributed to system unreliability. There are also studies that

focused on reducing the computational time of failure effect analysis by employing

intelligent system methodologies [93, 94].


64

2.3.6 State Enumeration vs. Monte Carlo Simulation

As can be seen, there are differences between analytical and simulation techniques and it is

extremely beneficial to understand the capability of each method in order to select the

appropriate techniques for the undertaken research. Table 2-1 gives a comparison between

the two methods in some important aspects. As stated earlier, the main difference between

the two is the way in which system states are selected and consequently the calculation of

the indices.

Table 2-1: Comparison between analytical techniques and simulation techniques in assessing composite

system reliability

Criteria State Enumeration Monte Carlo Simulation

System state selection Predetermined contingency level Random generation of contingency

Load model Predetermined load model 8760 hourly load samples

Indices Calculated by analytical laws Average of iteration results

Computational time Depends on predetermined

contingencies level

Depends on total number of iterations

Propriety For small systems and/or systems with

small state probability

Large systems and/or systems with

complex operation consideration

Uncertainty Not included Distribution of the indices

Chronological event Cannot be considered With sequential technique

2.4 Reliability Indices

Reliability indices are the quantitative measures of systems performance from the

perspective of system adequacy. As stated earlier, these indices are expected statistical

values that give a reasonable measure of future system performance. The composite system

reliability indices can generally be classified into: probability, frequency, duration, and

expectation indices [95]. The probability indices measure how likely an event will occur.

Frequency indices measure the expected rate of recurrence of an event per unit of time.

Duration indices measure the expected time that an event will last for. Expected indices are

the average of an expected consequence of an event [95]. Reliability indices are usually

calculated for load points and the overall system. The following indices are the most

commonly used in composite power system reliability [24].


65

a) PLC – Probability of Load Curtailment

Si

ipPLC (2.25)

where pi is the probability of system state i and S is the set of all system states associated

with load curtailment.

b) ENLC – Expected Number of Load Curtailments

Si

iFENLC (2.26)

where Fi is the system state frequency which can be calculated by

Nk

kii pF (2.27)

where δk is the departure rate of the component corresponding to system state i and N is the

set of all possible departure rates corresponding to state i.

c) EDLC – Expected Duration of Load Curtailment (hr/year)

8760 PLCEDLC (2.28)

d) ELC – Expected Load Curtailments (MW/year)

Si

ii FCENLC (2.29)

where Ci is the load curtailment in system state i.

e) EDNS – Expected Demand Not Supplied (MW/year)

Si

ii pCEDNS (2.30)

f) EENS/ENS – Expected Energy not Supplied (MWh/year)

Si

ii pCEENS 8760 (2.31)

2.5 Summary

This chapter described the assessment of the composite power system reliability. It was

shown that there are three aspects involved in the assessment: modelling, evaluation and


66

applications. The chapter gave details of modelling and evaluation aspects. The key models

in reliability assessment are the component failure model, load model and network model.

The component failure is classified into repairable and end-of-life failure. A detailed

description of both types, and how they are modelled, was given. With respect to annual

load models, the chronological hourly load points and multi-step load level are the

frequently used models. In addition, the probabilistic modelling of some of the network

operational conditions was discussed. The chapter also explained the main techniques of

assessing composite power system reliability. These are State Enumeration, Non-

sequential Monte Carlo and Sequential Monte Carlo simulation. There are two common

stages for all three of them: system state selection and failure effect analysis. The three

techniques vary in the way the system states are selected and accordingly in the indices

calculation. A general comparison between the State Enumeration technique and the two

Monte Carlo techniques was given to show the capabilities and limitations of each type.

The chapter also provided a list of the most commonly used indices in composite power

system reliability studies.

Chapter 3 Reliability Assessment Considering End-of-life Failure

67

3

Reliability Assessment Considering

End-of-life Failure

3.1 Introduction

The first objective of this thesis was to develop research grade software for assessing

composite generation and transmission power system reliability. The development of the

reliability assessment software is necessary because commercial software packages

perform conventional reliability assessment based on the Markov process, which is not

applicable for end-of-life failure. In addition, they are not open-source and hence there is

no way to integrate user-defined failure models. This chapter firstly provides details of the

state-of-the-art method in the incorporation of end-of-life failure into system reliability

assessment. Then, the main part of the chapter describes the developed reliability software,

and provides full documentation including functional definitions, programming

information, and validation. In addition, the test network and models used throughout the

research are described. All the assumptions and considerations applied to reliability

assessment studies presented in the thesis are stated and appropriate justifications are

given.


68

3.2 Integration of End-of-life Failure into System Reliability Evaluation

As shown in Chapter 2, both types of failure, repairable and end-of-life failure, are random

or stochastic events, and therefore, they are modelled by probability distribution functions.

The integration of each type into reliability assessment, however, is different. The review

of past work has shown that there are two studies which have addressed this issue. This

research project has adopted the method introduced by Dr W Li as it is the state-of-the-art

method in the integration of end-of-life failure into composite power system reliability [61,

62]. This method is applicable to State Enumeration and Non-sequential Monte Carlo

assessment techniques.

3.2.1 State-of-the-art Method

The system state selection in State Enumeration and Non-sequential Monte Carlo

techniques is based on the estimation of the components’ unavailability. For conventional

repairable failure, unavailability is calculated using the Markov process as the limiting

state probability of the failure state [42]. In contrast, the probability of end-of-life failure is

a posteriori failure probability, and the Markov model cannot be used. In engineering

reliability, the unavailability (U) is estimated as the ratio between the time on failed (TOF)

to the total time of the study as given by (3.1) [42]:

timeoperatingfailedontime

failedontimeU

.

(3.1)

The state-of-the-art method has adopted this concept in order to calculate unavailability

due to end-of-life failure. In order to estimate the unavailability due to end-of-life failure,

the method has defined it as “an average probability that a component is found unavailable

due to aging failures during a specified time period t given that it has survived for T years”

[61]. The definition comprises two elements of time, a specific age T and a future study

period t. Relating this definition to the unavailability concept given by (3.1), the future

study period t is the denominator of (3.1). Hence, the only missing part from (3.1) is the

time on failed (TOF). The state-of-the-art method has introduced a discretisation method in

order to calculate TOF. The study period t has been divided into S equal subintervals,

which have a small value t. The value of t has to be small enough so that the probability

of failure at any point within t is the same. The average TOF is then calculated as the


69

summation of the unavailable duration if a failure occurs within any subinterval j times the

probability of failure during j, as given by (3.2):

)(1

j

S

j

j UDPTOF

(3.2)

where Pj is the probability of having an end-of-life failure during subinterval j and UDj is

the average unavailability duration when end-of-life failure occurs during subinterval j. For

any subinterval j, if the end-of-life failure occurs at the beginning of the interval the

unavailable duration will be t, and it will be zero if it occurs at the end of this interval.

Then, the average unavailable duration within each interval will be t/2. Based on this, the

unavailable duration with respect to the future study period t when the end-of-life failure

occurs in the subinterval j can be calculated by (3.3) [61]:

2/)12( tjtUD j . (3.3)

According to (3.2), each value of unavailable duration has to be scaled by the probability

of having end-of-life failure during the subinterval j, i.e., Pj. Using the conditional

probability given in (2.14), Pj can be estimated as the difference between the probability of

having end-of-life failure in the duration from T to jt and the probability of having end-

of-life failure in the duration from T to (j-1)t as given in (3.4) [61, 62]:

T

tjT

T

tjT

Tj

dttf

dttfdttf

P

)(

)()(

)1(

(3.4)

where f(t) is the probability density function that characterises end-of-life failure. This

equation can be further explained by Figure 3-1, which is adopted from [62]. Since the

only available condition is that the component has survived up to T, Pj can only be

estimated by the subtraction of the probabilities of the two durations: jt and (j-1)t.

Figure 3-1: The calculation of probability of having end-of-life failure during the subinterval j


70

Accordingly, the average unavailability due to end-of-life failure can be estimated by (3.5)

[61, 62]:

)(1

1

j

S

j

jEoL UDPt

U

. (3.5)

3.3 Reliability Assessment Software

The reliability assessment software was developed using DIgSILENT Programming

Language (DPL) and is based on the Non-sequential Monte Carlo (NMC) simulation.

DIgSILENT PowerFactory package was chosen as the primary computational environment

for the software since it is one of the most trusted commercial software packages by power

system utilities, e.g., National Grid/UK, ESKOM/South Africa, and TransGrid/Australia.

The Non-sequential Monte Carlo simulation method (NMC) was adopted because

simulation techniques are more suitable for large systems as the computing time does not

depend on the size of the systems, i.e., number of components. Due to the time constraints,

developing software based on Sequential Monte Carlo in prescribed DIgSILENT

PowerFactory environment and completing the project tasks would not be achievable. It

should be mentioned that the main purpose of the developed tool was to facilitate studies

of the influence of end-of-life failure of power transformers on power system reliability; it

was not intended to introduce more advanced techniques for power system reliability

evaluation.

Although, the developed tool is research grade software, it is necessary to have good

documentation for many reasons. Generally, the documentation ensures that accurate

mathematical models were used in the programming of the software. Secondly, it serves as

a technical manual for the users. Finally, it opens an opportunity for further improvements

and modifications by other researchers. Therefore, in this chapter, full documentation of

the software, which follows ANSI/ANS 10.3-1995 Standard for Documentation of

Computer Software [96], is given. The ANSI/ANS 10.3-1995 Standard is a guideline for

documenting scientific and engineering software. It gives the general categories or sections

in which the information of the software should be given. Accordingly, the documentation

is classified into four categories: overview, functional definition, programming

information, and application information. As the overview and functional definition are the

main categories, they have been explained in detail in this chapter. The other two


71

categories, i.e., programming information and application information, have been

moderately discussed and the details are given in Appendix C.

3.3.1 Overview

The reliability assessment software is a command object of DIgSILENT Programming

Language (DPL). This DPL command contains input files, output files, sub-commands,

and the main script. Figure 3-2 shows the main script in the DPL command object of the

software. The button circled in red is the execution command of the software, while the

one circled in blue opens a new window which contains input files, output files, and sub-

commands. As mentioned before, the main purpose of the function is to assess the

adequacy of transmission networks, and produce system reliability indices along with load

point indices.

Figure 3-2: Main script in the DPL command object

3.3.1.1 Specifications and Capabilities

The software is executed via DIgSILENT software only, and it uses DIgSILENT’s

database and library. Therefore, there are no extra input files except transformer


72

age data, which is added inside the DPL command in a matrix format (see

Appendix C).

It uses the Non-sequential Monte Carlo simulation (NMC) method as the

assessment technique.

It evaluates the transmission reliability assuming perfect generation, i.e., 100%

reliable, throughout the year. This will not affect the analysis because for this

research only the influence of the transmission equipment on system reliability is of

interest [63]. This assumption is further clarified in section 3.5.1 of this chapter.

It evaluates the system reliability considering both types of failure; repairable and

end-of-life failure.

It utilises DC power flow to examine the consequences of system states. Therefore,

the reactive power constraints are not addressed. This is appropriate since for long

term reliability assessment the active power constraints are the crucial aspect. In

addition, many of the reliability tools adopted the DC power flow for their

algorithms [14].

3.3.1.2 Limitations

The software takes a considerable time to be executed (see section 3.5.3) as it is

developed using DIgSILENT programming language, and it uses DIgSILENT in-

built commands.

3.3.2 Functional Definition

The software consists of four main functional steps, which are: 1) load level selection, 2)

component state selection, 3) failure effect analysis, and 4) calculation of reliability

indices. Each of the steps is discussed in the following sections.

3.3.2.1 Load Level Selection

Since the software employs the NMC simulation method, the annual load curve is

modelled by a multi-step annual load model. The load steps are enumerated one by one

from the multi-step load model. For each step, NMC simulation is executed, and reliability

indices are calculated. The reliability indices estimated for each step are annualised indices

which present the reliability indices if a constant load equal to the step load is used

throughout the year.


73

3.3.2.2 Component State Selection

Generally, in the NMC simulation method, the state of each component is determined by

comparing a random number between 0 and 1 with the component’s unavailability. If the

random number is less than or equal to the unavailability value, the components will be

considered as unavailable. When considering both forms of failure, repairable and end-of-

life failure, two random numbers are independently generated and each one is compared to

each unavailability value. The reason for generating two random numbers is that the two

forms of failure are two different events. The state of the component is then determined by

applying the exclusive union (XOR) probability rule. The XOR rule is applied because the

component cannot be unavailable due to a simultaneous occurrence of both types of

failure. Accordingly, the component is considered unavailable if one of the random

numbers is less than or equal to the associated unavailability, but it is considered available

if both random numbers are less than or equal to the unavailability values. Table 3-1

illustrates the selection procedure for the component state, where UEoL is the unavailability

due to end-of-life failure and UR the unavailability due to repairable failure.

Table 3-1: Selection procedure for the component state using random numbers and the XOR probability rule

Random number-1 <= UR Random number-2 <= UEoL Component State

No No Available

Yes No Unavailable

No Yes Unavailable

Yes Yes Available

3.3.2.3 Failure Effect Analysis

The failure effect analysis step involves a series of actions in order to determine if the

system state is a failure state or a success state [24-26]. Since most transmission systems

are designed to meet the demand if there is no forced outage, only system states, which

contain components in down states, are examined by failure effect analysis. The first action

in this series is the examination of load interruption and system limit violations, which is

completed by applying DC load flow, hence, only thermal limits violation is dealt with as

previously mentioned. If there are neither load interruptions nor violations of the system

limits, the system state is a success state. If only the load interruptions condition is found,

failure effect analysis is terminated and reliability indices are updated. If the system limit

violations condition exists, the second action, which is generation re-dispatch, is applied in


74

order to eliminate this condition. If the generation re-dispatch accomplishes the task and

relieves the overload, the state is considered as a success state. If not, the load shedding

action is performed, which results in a failure state. A descriptive sequence of the actions

performed in failure effect analysis is given in Figure 3-3. In the figure, system states

indicated using green coloured text are system success states and ones indicated using red

coloured text are system failure states.

The procedure of the simulation of the generation re-dispatch and load shedding actions to

relieve transmission system overloads are adopted from COMPASS software, which was

developed at UMIST to assess the composite system reliability [14, 97, 98]. The procedure

is based on the sensitivity of the system overload to bus power injections. The generation

re-dispatch and load shedding is performed on buses whose injections affect the

overloaded components. On this basis, a contribution factor to the power flow through the

overloaded components is calculated for each bus. The overload elimination is completed

by performing iterations of generation re-dispatch and load shedding actions.

3.3.2.4 Contribution Factor

The main concept is to assign a contribution factor to each generation and/or load bus in

the system to describe the relationship between its power injection and the system total

overload. This is based on the concept that each bus in the system contributes to the flow in

a component (line, cable, and transformer) by an amount that is proportionally related to

the distribution factor of the bus for the component and the bus net power injection.

Accordingly, the contribution factor of bus n to the system overload is given by (3.6).

knin DC )1( (3.6)

where

Cn is the contribution factor of bus n,

Figure 3-3: Failure Effect Analysis actions

Neither load

interruption nor

limits violation

System limit

violations state

System state with

components in

down state

Generation re-dispatch

relieves the overload

Load shedding is

necessary to relieve

the overload


75

k 0 if Fi >0 or =1 if Fi<0,

Fi is the power flow through component i,

Dni is the distribution factor of bus n for component i.

The summation in (3.6) is extended to overloaded lines only. The distribution factors (Dni)

are obtained by executing a build-in function in DIgSILENT called load flow sensitivity.

This function produces different forms of sensitivity factors; the one that is used in this

function is termed branch sensitivity (dPbranch/dPbus). As can be seen in (3.6), the sign

and the value of the contribution factor depend on the direction of power flow through the

overloaded components.

Buses with a negative contribution factor have an inverse relationship between their

injection and the system overload. This means that any increase in the injection at this bus

reduces the overload and any decrease in the injection increases the system overload. The

opposite is true for buses with a positive contribution factor. Generation and load buses are

sorted according to the sign and the value of the contribution factor. Consequently,

generation re-dispatch and load shedding are carried out on the most sensitive buses.

3.3.2.5 Generation Re-dispatch and Load Shedding

Some overloads in the system might be relieved by generation re-dispatch. This possibility

exists when there is a generation reserve, and this reserve is available on the buses where

increasing the injection reduces the system overload. The generation re-dispatch action

starts by choosing a pair of buses: the increasing bus (Kin) is the bus with the highest

negative contribution and available increasing reserve and the reducing bus (Kre) is the bus

with the highest positive contribution and available positive power injection for reduction.

If the re-dispatch is possible, the amount of generation exchange between the two selected

buses is the minimum of i) the reserve available in the increasing bus, ii) the positive

injection in the reducing bus and iii) the exchange power that will relieve the overload of a

component. This can be represented by (3.7):

),,min( dGRGGE KinKre (3.7)

where GE is the amount of generation to be exchanged, GKre is the mount of generation at

bus Kre, GRKin is the generation reserve available at bus Kin, and d is the minimum power

required to overcome an overload of individual components. The value of d is obtained by

calculating the power required to relieve the overload of individual components without

affecting the non-overloaded ones and selecting the minimum value. For each component,


76

the amount of power required di is calculated using its distribution factors for the selected

pair of buses and the power flow through it. Table 3-2 shows how this value can be

calculated [97]. The power flow sign given in the first column, is determined according to

the power flow towards the terminal, which was used to calculate the distribution factor.

The column labelled as DIF represents the difference between the component distribution

factors of the selected pair of buses, i.e., DIF= Dkin-Dkre.

Table 3-2: Calculation of the power required to overcome the overload of a component (d i)

Flow sign Component flow F DIF di

(+) Limits < F (+) 0

(+) Limits < F (-) (F-Limits)/|DIF|

(+) F < Limits (+) (Limits-F)/DIF

(+) F < Limits (-) (F+Pnom)/|DIF|

(-) Limits < F (+) (F-Limits)/DIF

(-) Limits < F (-) 0

(-) F < Limits (+) (F+Pnom)/DIF

(-) F < Limits (-) (Limits-F)/|DIF|

As an illustration, if the component is overloaded and its flow sign is opposite to the DIF

sign, then di= (F-Limits)/|DIF|. This amount of exchange power ensures that the

component power flow will equal the limit. If the flow and DIF have the same sign, the

amount of power is 0, which means the power exchange between the pair of the buses

increases the overload on the component, and hence, another pair of buses should be

selected. When the component is not overloaded, and its flow and DIF have the same sign,

the value of di= (Limits-F)/DIF. This value of di is the power required to increase the

power flow through this component to the limit, which ensures that there are no overloaded

components due to generation re-dispatch. If the signs of the flow and DIF are different for

non-overloaded components, then di= (F+Pnom)/|DIF|, where Pnom is the component

rated power. In this case, the power exchange causes a change in the direction of the power

flow through the component but it will not exceed the limit.

After each generation exchange is performed, the whole process must be recomputed. This

includes component flows, distribution factors, contribution factors, a new pair of buses,

and generation exchange. This procedure is repeated until the system overload is removed

or the re-dispatch of the generation becomes unfeasible.

The load shedding method is similar to the generation reschedule method. The only

difference is the increasing bus Kin which is now a load bus, i.e., shedding the load in bus


77

Kin gives the same effect as injection increase. The exchange amount between the pair of

buses is calculated by (3.7) except that the increasing reserve GRKin is the total load at bus

Kin available for load shedding. After each load shedding action, the generation re-dispatch

is firstly attempted but if it is not feasible, load shedding is performed. This ensures

optimum load shedding actions. The calculations of generation re-dispatch and load

shedding are repeated until the overload is eliminated.

3.3.2.6 Calculation of Reliability Indices

The calculation of the reliability indices in the software starts by calculating the annualised

indices of each load level. Then, the annual indices for the load model are calculated by

using corresponding indices for each load level and their associated probabilities.

For each loading level, the reliability indices are obtained by using the number of

occurrences of a specific state over the total number of samples as an approximated

probability of occurrence [24]. For example, the Probability of Load Curtailment (PLC)

index is calculated by (3.8):

N

LCnPLC

)(

(3.8)

where n(LC) is the number of occurrences of a load curtailment state, and N is the number

of samples. The software calculates four main composite power system reliability indices,

which are:

PLC Probability of Load Curtailment

EDLC Expected Duration of Load Curtailment [hr/year]

EDNS Expected Demand Not Supplied [MW/year]

EENS Expected Energy Not Supplied [MWh/year].

The formulae for EDLC, EDNS, and EENS are given by (3.9), (3.10), and (3.11),

respectively:

8760 PLCEDLC (3.9)

ii CPEDNS

(3.10)

8760 ii CPENS

(3.11)

where Pi is the probability of occurrence of state i approximated by the number of

occurrence of the state i divided by the total number of NMC iterations and Ci is the

amount of load curtailed in state i.


78

3.3.3 Programming Information

The documentation of programming information is very valuable as it provides future users

with the structure and the logic of the software to allow further development. Moreover, it

gives the opportunity to replicate the function using an alternative programming language.

In this category of the software documentation, the DPL commands and sub-commands

that compose the tool are described and explained. The interrelations and data exchange

between these commands have been depicted by means of data flow diagrams (DFD)

adopted from [97]. A simple example of a DFD is given in Figure 3-4. As seen in the

figure, a DFD consists of three parts, top, main and bottom part. The top part of the chart

states the name of the command, which consists of two fragments; a code and a phrase.

The code indicates the position of the command in the software’s structure. The number of

digits in the code indicates the structural level of the DFD and the last digit shows the

order of the DFD at that level. For SF-031, as an example, there are three digits in the

code, which means it is located in the third structural level, and it is the second command

(the last digit is 1) inside the SF-03 command. The code of the main command is SF-0.The

phrase in the name indicates the main function of the DFD, e.g., it is sensitivity matrix in

the given example. This way of naming the commands helps to follow the structure of the

function. The main part of the DFD contains sub-commands that are executed by the

command and the interrelation between them. In turn, each one of these sub-commands has

its own DFD. Lastly, the details of input data and the results of the command are given in

the bottom part of the DFD.

INPUTS OUTPUTS

SF-031 SENSITIVITY MATRIX

MAIN INPUTS MAIN OUTPUTS

Name of the

command

Command

contents

Inputs/Outputs

Figure 3-4: Explanatory example of a data flow diagram DFD

As the software is basically a DPL command, all its developed commands and sub-

commands have access to the network components, power system analysis functions, study


79

results and library. The complete documentation for programming information including

the DFDs is given in appendix C.

3.3.4 Application Information

This section describes how the software can be applied to a network built in DIgSILENT

to calculate system reliability indices. Before applying the function, it has to be ensured

that the power flows through the transmission elements are within the limits. An optimal

power flow can be implemented to meet this crucial condition. Once this step is done, the

software can be applied following the steps given in application information

documentation in appendix C. In this chapter only the procedure of how to execute the

software is given.

3.3.4.1 Execution of Reliability Software

To execute the software, the user needs to copy the main DPL command object and paste it

in the active Study Case in the Data Manager as shown in Figure 3-5. The only direct input

to the software is the number of Monte Carlo iterations (mont), as shown in Figure 3-6.

Then, the DPL is executed by pressing the execute button on the command.

Figure 3-5: A snapshot of the function pasted in the active Study Case in the Data Manager


80

Figure 3-6: Setting the number of NMC iterations in the reliability software

3.3.5 Validation

The validation of the developed reliability software was performed by comparing the

results of calculations with the results reported in the literature. Two case studies, which

were performed on the well-known IEEE Reliability Test System (IEEE-RTS) [99] and

reported in [24], were chosen for the validation: annualised indices and annual indices

studies. In the former, the peak load is assumed to be constant throughout the year whereas

in the latter, the effect of the annual load curve is considered. In these two case studies, the

reliability assessment involves both generation and transmission adequacy.

3.3.5.1 Annualized System Indices

The description of the annualized indices case study given by [24] did not include

information about the generation dispatch values. Therefore, two sets of generation

dispatch values were employed to reproduce the results using the software. The first

dispatch (D1) was obtained by performing an optimal power flow study (OPF) using

DIgSILENT in-built command, while the second dispatch (D2) was adopted from [99].

The tool was executed for each dispatch set, and power system reliability indices were

calculated. Table 3-3 shows the comparison between the indices reported in [24] and the

indices produced by the software for both dispatch sets. The columns noted as in the


81

table give the absolute relative error in percentage between the reported indices and indices

obtained by the developed software, which is calculated using (3.12):

100

reported

reported

Index

indexIndex.

(3.12)

As can be seen from Table 3-3, the two dispatch sets give different values for the indices.

For generation dispatch D2, the value of the largest difference/error in indices (see column

6) does not exceed 0.88%. The results of this assessment are almost completely identical to

the results reported in [24]. Although the error in the calculation when using generation

dispatch D1 is higher than the error associated with generation dispatch set D2, the values

of error are still reasonable (the largest error in calculated indices is 8.24%) considering

that the reliability assessment is based on random sampling of NMC. Therefore, it can be

said that in the case of annualised indices, the developed software assesses the power

system reliability accurately.

Table 3-3: Comparison between the annualised system reliability indices reported in [24] and indices

produced by the developed software

Index Results

reported in

[24]

Results obtained

using D1 -using

D1 (%)

Results obtained

using D2 -using

D2 (%)

PLC 0.08 0.075 6.25 0.0793 0.88

EDLC (hr/year) 699 655 6.29 695 0.57

EDNS (MW/year) 13.9 12.8 7.91 14 0.72

EENS (MWh/year) 122,046 111,993 8.24 122,825 0.64

3.3.5.2 Annual System Indices

In order to include the effect of the annual load curve, IEEE-RTS load duration curve is

modelled as a multi-step load duration curve. Ref [24] reported case studies for 15-step and

70-step load models, however, there is no information about those levels and their

associated probability. The case study presented here used a 20-step load model adopted

from [24] and compared the estimated results to the two reported studies. Figure 3-7 shows

the 20-step load model of IEEE-RTS load duration curve.


82

Figure 3-7: Annual Load Duration curve represented by the 20-step load model.

The indices for the three multi-step load models were compared and given in Table 3-4.

The indices calculated by the software using the 20-step load model are greater than the

indices calculated using the 70-step load model and are smaller than the indices calculated

using the 15-step load model. This is an expected result for the 20-step model, which

clearly proves that the software estimates the annual system indices correctly.

Table 3-4: A comparison of annual reliability indices for the three multi-step load models

Index Results reported in [24]

using 70 steps

Results obtained by

the tool using 20 steps

Results reported in [24]

using 15 steps

PLC 0.00117 0.00137 0.00178

EDLC (hr/year) 10.23695 12.0173 15.54475

EDNS (MW) 0.13137 0.13419 0.21761

EENS (MWh) 1147.6 1175.53 1901.038

From the results of the previous case studies, it can be concluded, with high confidence,

that the developed reliability software estimates the system reliability indices accurately,

i.e., within an acceptable margin of error. The conclusion is valid for both annualised and

annual indices.

3.4 Test Networks Description

Case studies conducted during this research project were performed on two transmission

networks. The first one was the IEEE Reliability Test System (IEEE-RTS), which is

widely used for system reliability assessment. All the data regarding this test system

network and load models can be found in [24] and [99]. The second network was a realistic

Per

cen

tage

of

the

pea

k l

oad

Time (hours)


83

meshed transmission system, which, from this point forward, will be referred to as the test

system.

3.4.1 Test System

The test system broadly represents the transmission network of a large metropolitan city.

The single line diagram of the network is shown in Figure 3-8. The transmission voltage

levels are 400 and 275kV. The network has 8 equivalent generation buses/in-feed points

and 25 load buses at different voltage levels (132, 66, and 33kV). It has 28 interbus

transmission transformers (400/275 kV tagged in the single line diagram as T1–T28) and

42 transmission lines and cables. Each load bus represents a substation that contains step

down transformers, substation cables, circuit breakers and disconnectors. The total number

of step down transformers is 126 (not shown in the single line diagram). Though the test

network does not represent any existing real network, all of its components are modelled

using the typical parameters of the UK transmission network.

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T2T1

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T6T5 T7T8

T9 T11 T12

T10

T13 T14

T15 T16

T17 T18

T19 T20

T21 T22

T23 T24

T25 T26

T28T27

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45

53

Figure 3-8: The single line diagram of the test system


84

3.4.2 Load and Network Model

The annual demand variation is presented by an optimum 6-step load model. The model is

constructed from historical operating points of the England and Wales network system.

They are chosen to accurately represent assumed periods of time that they cover during the

year. Each loading level has a different demand and different power injections from

generation and in-feed buses. This supply and demand model accounts for various power

flows since the load and generation variation were independently modelled. The total

supply/demand of the system (as a percentage of the peak) for each load level and the

probability of its occurrence are shown in Table 3-5. The demand/supply percentages

shown in the table are not necessarily the same for individual load and/or generation and

in-feed buses since each of them may have different demand/supply patterns. In addition,

the variation of the thermal capability of equipment (higher or lower than the nameplate

thermal rating) with different seasons of the year is also modelled. The planned outages are

modelled using a predetermined maintenance schedule.

Table 3-5: The 6-step load model of the test system

Step No. Load level (%) Covered time period (weeks) Probability

1 98.9 9 0.173

2 92.3 12 0.230

3 88.1 7 0.135

4 80.4 6 0.115

5 74.5 5 0.0962

6 71.4 13 0.250

∑ = 52 ∑ = 1

3.4.3 Transformers Fleet Data

3.4.3.1 Transformer Age Distribution

The age of the transformers in the test system (Figure 3-8) has been assigned in accordance

with the age distribution of transformers in the National Grid transmission network shown

in Figure 1-5. Accordingly, the transformer age range in the test system is between 1-58

years. The age distribution of transformers in the test system is shown in Figure 3-9. It is

apparent that, there are a considerable number of transformers that have exceeded their


85

original design life-time of 40 years; in fact 56.5% of the transformer population is 40

years or older.

Figure 3-9: Age distribution of the test system’s transformers

3.4.3.2 Transformer End-of-life Failure Models

The end-of-life failure of power transformers in the test system is characterised by a

Gaussian or normal distribution with a mean value () equal to 65 years, and a standard

deviation () of 15 years. In order to calculate the unavailability due to end-of-life failure

using a Gaussian distribution function, Pj given in (3.4) is approximated by (3.13), which

is adopted from [61]:

)(

)())1(

(

T

Q

tjTQ

tjTQ

Pj (3.13)

where Q is calculated by

0)(1

0)()(

xifxw

xifxwxQ

(3.14)

))(()(5

54

43

32

21 sbsbsbsbsbxzxw (3.15)

)2

exp(2

1)(

2x

xz

(3.16)

rxxs

11

)( (3.17)

r=0.2316419, b1=0.31938153, b2=-0.35656378, b3=1.781477937, b4=-1.821255978, b5=1.330274429.

Figure 3-10 shows the unavailability calculated using the mentioned Gaussian distribution

for the age range of the transformer in the test system for a one year future study period.


86

Figure 3-10 demonstrates that there is a non-linear relationship between age and

unavailability. It is apparent that the youngest transformers (0-20 years) have small values

of unavailability that do not increase rapidly with age, whereas the unavailability of older

transformers (30-58 years) increases rapidly with age. This shows that a one year age

difference can make the unavailability significantly different when the transformer is old.

Figure 3-10: Unavailability due to end-of-life failure using normal distribution (=65, =15) for a range of

ages (1-58)

3.5 Adjustments for Reliability Assessment

The reliability evaluation of the test system using the developed reliability software

involves some assumptions and adjustments. These are: generating unit reliability,

repairable failure consideration, and convergence of NMC. The main reason for applying

these adjustments is to reduce the computation time of the reliability assessment without

jeopardising the results and/or the applications of the reliability assessment. This section

describes these assumptions, their effect on the estimation of reliability indices, and

justifications for using them.

3.5.1 Generating Unit Reliability

For composite generation and transmission reliability evaluation, i.e., HLII, there are three

different conditions for conducting the reliability evaluation depending on the requirements

and the applications of the evaluation [26]. These conditions are:

1. To consider the failure of generating units and transmission components, which is

the full composite reliability evaluation.


87

2. To consider the failure of transmission components only and assume that the

generating units are 100% reliable. This condition is suitable when the

transmission system reliability evaluation is of interest.

3. To consider the failure of generating units only (transmission components are

ideal). This evaluation is different from generation reliability evaluation, i.e., HLI,

as the transmission system operational constraints are included.

As described earlier, this research focuses on the reliability of the transmission system

only. Hence, the second condition is adopted throughout the research, that is to say, all the

generating units are assumed to be 100% reliable in all the case studies. With respect to the

in-feed points, it is assumed that the maximum power transfer capability can be delivered

all the time. It should be mentioned that the transmission only reliability evaluation has

been previously adopted in reliability studies [63], which are relevant to the studies in this

thesis.

With respect to overhead lines and cables, they were assumed to be ideal in order to

evaluate the contribution of transformers only to reliability indices. In general, such an

assumption may underestimate the contribution of transformers to reliability indices as it

ignores the cases where simultaneous failures of lines and transformers cause a system

failure state. These cases usually have low probability of occurrence. In addition, this

adjustment has been practised by the National Grid in their reliability models [36], and

hence it has been presumed that it is feasible for this particular test system.

3.5.2 Repairable Failure

In this thesis the unavailability due to repairable failure is assumed to be zero as it is much

smaller than the unavailability due to end-of-life failure, which is of primary importance in

this research. It is a fact that the system reliability indices will be affected by this

assumption. However, the assumption is justified by the fact that the average unavailability

due to repairable failure of power transformers is 0.0012 [100-102], which is equivalent to

the unavailability of 30 year old transformers, and the test system has 93 transformers

older than 30 years. In order to quantify the effect of this assumption on the reliability

indices, three reliability studies were performed: 1) to consider both repairable and end-of-

life failures, 2) to consider end-of-life failure only, and 3) to consider repairable failure

only. Table 3-6 shows the ENS index for the load points and the overall system in each


88

study. The load points which are not shown in the table have a zero ENS index. As

expected, the first study, where both types were dealt with, has the largest ENS values.

Also, the ENS of the system, when assuming zero unavailability due to repairable failure,

i.e., end-of-life failure only, is about 64.4% of the full study ENS. Considering that the

contribution of the study using only repairable failure is 42.69 MWh/year which is 10.4%

of the full study ENS, it can be safely argued that the assumption does not have a major

impact on the reliability indices. Specifically, all load points which have non-zero ENS in

the full study also have non-zero ENS in end-of-life failure study except Buses 24, 53, and

36. These buses however have notably smaller ENS values.

Table 3-6: The effect of assuming zero unavailability due to repairable failure on ENS [MWh/year]

Load point Repairable & End-of-life End-of-life only Repairable only

Bus 19 85.51 41.69 8.13

Bus 20 79.23 60.95 2.12

Bus 26 70.23 31.11 10.23

Bus 18 64.44 51.06 0

Bus 12 38.90 29.20 19.37

Bus 16 27.11 27.11 0

Bus 28 17.03 10.76 0

Bus 27 13.85 5.23 0

Bus 17 6.71 6.63 0

Bus 24 3.82 0 2.84

Bus 53 1.41 0 0

Bus 36 1.20 0 0

System 409.4 263.74 42.69

By representing these results using heat maps (see Figure 3-11) of the test system, it can

also be proven that there is no significant change resulting from the assumption. The same

critical buses and areas were highlighted in both studies: considering repairable and End-

of-life failure together and End-of-life failure only. These three studies have also

demonstrated the significant underestimation of the reliability indices when performing

traditional reliability assessment, i.e., repairable failure only, especially for present ageing

systems.


89

Figure 3-11: Heat maps for the test system showing the effects from assuming zero unavailability due to

repairable failure

3.5.3 Accuracy of Non-sequential Monte Carlo

As mentioned in Chapter 2, the accuracy of the indices estimated using simulation

techniques increases with the number of iterations. This, however, results in significant

computation time. Since the reliability software takes a considerable time to execute the

assessment, optimum 10,000 iterations for each load level have been employed. Table 3-7

shows system ENS, accuracy, and computation time of NMC when using 10,000, 15,000,

and 20,000 iterations. Since the accuracy is measured by the coefficient of variation, a

smaller value means better accuracy. It is apparent that the accuracy improved when

performing 20,000 iterations, but this leads to double the computation time. The value of

ENS, however, has not changed significantly. Figure 3-12 shows the convergence of the

ENS index with the number of NMC iterations. As can be seen, the value of ENS when

performing 10,000 iterations falls in the beginning of the relatively converged values.

Therefore, 10,000 iterations are feasible.

Table 3-7: Accuracy and computation time of NMC for 10,000, 15,000 and 20,000 iterations

10,000 iterations 15,000 iteration 20,000 iteration

ENS

[MWh/yr]

Accuracy

[%]

Time

[h:min]

ENS

[MWh/yr]

Accuracy

[%]

Time

[h:min]

ENS

[MWh/yr]

Accuracy

[%]

Time

[h:min]

263.74

42.5 03:45

254.74

35.4 05:50

240.24

31.1 07:57

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T19 T20

T21 T22

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85 76.5 68 59.5 51 42.5 34 25.5 17 8.5 0 a) Consideration of both repairable and end-of-life failures b) Consideration of end-of-life failure only

60 54 48 42 36 30 24 18 12 6 0

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T17 T16

T19 T20

T21 T22

T23 T24

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Figure 3-12: The convergence of the ENS index against the number of Monte Carlo iterations

To sum up, the reliability of the test system was assessed throughout this thesis considering

the following:

a. Ideal generation and in-feed points.

b. Ideal cables and overhead lines.

c. The unavailability due to repairable failure is zero.

d. The number of Monte Carlo iterations is 10,000.

Consequently, the reliability indices for the load points and overall system, which are

calculated using the given Gaussian distribution, are presented in Table 3-8. System

reliability indices, as well as load point indices, were calculated as described previously for

a one year study period. Load buses, which are not shown in the table, have ENS =0. Bus

20 has the largest PLC and ENS indices, and hence it is the most unreliable bus in the

system. It contains six aged transformers, two of them are 53 years old, and four are 48

years old (see Appendix D).

Table 3-8: Reliability indices of the test system using the given Gaussian distribution

Bus ID PLC [%] ENS [MWh/year]

Bus 20 0.428 60.95

Bus 18 0.007 51.06

Bus 19 0.014 41.69

Bus 26 0.005 31.11

Bus 12 0.005 29.20

Bus 16 0.005 27.11

Bus 28 0.002 10.76

Bus 17 0.001 6.63

Bus 27 0.009 5.23

System 0.474 263.74


91

3.6 Summary

This chapter firstly described the state-of-the-art method in the incorporation of end-of-life

failure into system reliability studies. This method has been adopted into the dedicated

reliability assessment software developed during the completion of this thesis. The main

part of the chapter gave a detailed documentation of the reliability software. The

documentation includes a list of the capabilities and limitations, the functional definition,

programming information and application. In the functional definition documentation, the

models and mathematical concepts behind the software were given. A part of the function,

which deals with failure effect analysis, was programmed as a replication of software

(COMPASS) that was developed at UMIST in 1991. The chapter also provided a brief

description of the data flow diagrams, which were used to explain the interrelation between

the function commands and sub-commands. The full details of the programming

information and the application of the software are given in the appendices. In addition, the

validation of the software using two case studies on the IEEE-RTS system was given. It

showed that the developed reliability software estimates annualised and annual reliability

indices within an acceptable margin of error. The chapter also included a full description of

the test system and models used throughout the thesis. All the adjustments that were made

to manage the reliability assessment were given in this chapter along with the reasonable

justifications and quantification of their effect on reliability assessment.


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Chapter 4 Reliability-Based Replacement Framework

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4

Reliability-Based Replacement

Framework

4.1 Introduction

It is commonly accepted that the cost of replacing components in a transmission system is

particularly high. For example, the cost of replacing a power transformer can be around £4

million [38]. Consequently, it is very difficult for transmission system owners to replace all

their aged assets in a short time. Specifically, since the privatisation of the electricity

industry, the allowed level of return on reinvestment is determined by the price control

scheme applied by regulatory authorities. As a result, the replacement of the majority of

assets is deferred to the following years. Having a considerable amount of aged equipment

in a network, however, will increase the risk of customer interruptions, which eventually

could reach a level which is no longer acceptable. Particularly, electricity regulatory

authorities commonly apply a reliability incentive scheme under which a reliability level is

determined. Therefore, making correct asset management decisions is critical, and careful

analysis is required to find the balance between required reliability and reinvestment costs.


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One can consider the preventive replacement of power system asset as a risk mitigation

process. Based on this consideration, methodologies of asset replacement can be classified

into two groups: non-quantitative risk methods and quantitative risk assessment methods.

Methods that are based on non-quantitative risk focus on the asset failure, and they do not

measure its consequences. In these methods, replacement is undertaken to reduce these

consequences without knowing how much they are and by how much they are being

reduced. Examples of these methods are: age limit model, statistic hazard function model

and reliability curve model. Further details of those models can be found in [103]. The

same models can be applied to asset condition instead of age.

In recent years, there has been an increasing amount of literature on using quantitative risk

approaches in replacement decision making [63, 72, 104-110]. One can classify those

approaches into two classes, the risk matrix method [104-110] and the risk indices method

[63, 72]. In the former, a risk matrix is constructed to define the replacement priority of

components using their condition or age as the first axis and their criticality to the system,

environment, and safety as the second axis. An illustrative example of risk matrix is shown

in Figure 4-1 taken from [104]. The risk matrix method is effective in prioritising

replacement candidates into broad categories. The result, however, is sensitive to the

characteristics of the matrix which may not be optimal for all cases as it does not directly

link replacement with system reliability.

Figure 4-1: Risk matrix used to determine replacement candidates. Adopted from [104]


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The risk indices method, which can also be referred to as the reliability indices method,

allows integration of reliability optimisation techniques. The studies that used this method

were discussed in the review of past work [63, 72]. As it was stated, the first study did not

take into account the replacement planning of the whole component fleet, whereas the

second one did not consider the criticality of the individual components for the system

reliability. Hence, it can be argued that the currently applied quantitative risk methods in

the area of power system component replacement planning have some shortfalls which

need to be overcome.

This chapter presents a replacement planning framework for power transformer based on

system reliability. The proposed framework is the second original contribution of this

thesis since it combines, for the first time, the advantages of the risk matrix method and the

reliability indices method. It has three elements. The first one is the identification of the

most critical transformers for system reliability using reliability importance measures. This

identification of critical transformer is equivalent to the risk matrix mapping since it

depends on both transformer age and its effect on system reliability. The second element in

the framework is Pareto analysis, which is performed to determine the effect of

replacement scenarios on system reliability. The use of Pareto analysis in system reliability

studies is a new application in this area and represents additional original contribution of

the developed framework and the thesis in general. After determining this effect, the final

element of the framework is carried out; this is a cost-benefit analysis to determine the

optimum replacement plan. The cost of unreliability in this analysis is calculated using data

from an incentive/penalty scheme which is typical of those commonly applied to regulate

power system reliability.

4.2 Reliability Importance Measures

The reliability of a complex system is certainly built upon the reliability of its individual

components. Yet, some components have greater impact on system reliability than others,

and hence, the importance of individual components has to be identified. The assessment

of reliability importance measures is an integral part in the applications of the reliability of

engineering systems and has been introduced and used by power system reliability

engineers in many studies [111-117]. Four different reliability importance measures have

been used: Structural Importance, Improvement Potential, Criticality Importance, and


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Fussell-Vesely Importance. These importance measures are discussed in the following

sections.

4.2.1 Structural Importance Measure

Structural importance (IS), also known as Birnbaum’s reliability importance, is calculated

using the partial derivative of the system reliability to the component reliability.

Commonly, IS is obtained by performing simple sensitivity analysis instead of the partial

derivative as shown in (4.1) [111, 113, 114]:

rC

rSIS

(4.1)

where ΔSr is the incremental change in system reliability and ΔCr is the incremental change

in component reliability. The system reliability (Sr) can be described by any of the

reliability indices. For example, PLC [111], ENS and customer interruption cost [114,

116]. For representing the component reliability (Cr), the failure rate or unavailability can

be applied. In the calculations of IS, ΔCr is given the same value for all components under

study. As a result, IS assesses the structural importance of the component or in other words

the importance of the location of the component in the network because it represents the

degree of the changes in the system reliability with respect to the changes in the

component reliability without taking into account the level of the component reliability

[118].

4.2.2 Improvement Potential Measure

The Improvement Potential (IP) expresses the effect of improving the reliability of the

component on the system reliability. It is calculated by subtracting the system reliability

estimated considering the component under study as ideal (Sr,i) from the base case system

reliability (Sr) as shown by (4.2):

irr

SSIP,

. (4.2)

4.2.3 Criticality Importance Measure

The criticality importance measure (IC) is calculated using the IS as given by (4.3) [111,

113]:


97

r

r

S

CISIC (4.3)

where Cr and Sr are the base case component reliability and system reliability, respectively.

As can be seen from (4.3), IC considers the reliability of the component in addition to its

structural importance. Therefore, it reflects both the structural importance and the criticality

of the condition of the component. Hence, when using the IC measure, the less reliable

component of the two having equalled IS, will be considered as more critical. This feature

makes IC the most suitable importance measure for maintenance and replacement decision

making as it is coherent to maintaining/replacing the least reliable components.

4.2.4 Fussell-Vesely Reliability Measure

The Fussel-Vesely reliability importance measure is defined as the probability of at least

one cut set of failed components, which leads to a system failure state, containing the

component involved in the study [111, 113]. Fussel-Vesely can be computed in Monte

Carlo simulations, for example, by the ratio between the numbers of occurrence of

component failure (alone and within a cut set) that leads to system failure and the total

number of Monte Carlo iterations [119]. It is apparent from the definition that Fussel-

Vesely describes the importance of a component from the failure probability perspective

and ignores the quantification of the consequences of the failure, e.g., ENS.

4.2.5 Further Consideration

There are two final remarks regarding the application of the reliability importance

measure. Firstly, referring to (4.1), more accurate sensitivity results of ENS would be

obtained by using a small incremental change in component unavailability values. In the

standard Non-sequential Monte Carlo reliability assessment approach, as in this research,

however, the unavailability values of the components are not directly integrated into the

reliability indices. It is included in the calculations via the random sampling process. As

explained in 3.3.2.2, at each iteration of NMC, a component is considered unavailable

when its unavailability is greater than or equal to a generated random number. If a very

small incremental change in the unavailability value is used it may not affect the results of

reliability assessment in some cases as the generated random number may still be smaller

than the component unavailability ± the incremental change. For example, if a component,


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Figure 4-2: Illustrative example of the use of incremental change in sensitivity analysis with NMC

simulation

whose unavailability is 0.02, is considered to be failed because the generated random

number is 0.01, it will also be considered to be failed if an incremental change of ±0.002 is

applied to its unavailability since the unavailability will still be greater than 0.01. This idea

is depicted in Figure 4-2. Therefore, IS and IC measures may give partial importance level

when using NMC as they ignore some cases. As a result, the replacement framework has

implemented IP importance measure.

Secondly, it is apparent that all the importance measures except FV are calculated using

one way sensitivity analysis in which the reliability of a component is changed one by one

and the corresponding change in system reliability indices is recorded. There are some

researchers who argued that this kind of sensitivity analysis underestimates the effect of

uncertainty in the reliability model and correlation between components [120, 121]. In

particular, this is absolutely true for reliability assessment of nonlinear systems such as

power systems. Therefore, this thesis introduced a method for measuring reliability

importance based on probabilistic sensitivity analysis as a more comprehensive approach

to assess the importance of components. This method can be used to jointly perform

uncertainty analysis and sensitivity analysis. The steps of the method and a case study are

included in Chapter 7 as a part of the application of uncertainty quantification in power

system reliability studies.

4.3 Pareto Analysis

Power system reliability is inherently a non-linear function of component reliability,

involving many hidden correlations and interactions between the components. Increased

reliability is usually obtained using a redundant parallel operation, which creates one of the

most important correlations. When calculating criticality measures using traditional

0.02

0.02 ± 0.002

0 1

Random number


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sensitivity analysis, parallel operation may cause misleading results. For example, two

components may have the same effect on system reliability with one way sensitivity

analysis, but replacing one of them can eliminate the effect of the other. Conversely, in

some cases replacing one of them may not improve the reliability at all and both have to be

replaced. Furthermore, most transmission systems were designed based on the N-1

criterion or better, which prevents a load shedding event with one component out of

service. Therefore, most load shedding events are caused by having more than one

component out of service and further analysis is needed to distinguish between individual

component effects on reliability. Ranking of power system components based on

importance measures, therefore, provides only part of the essential information for

replacement decision making. It brings attention to specific critical equipment but does not

reflect the effect on the system reliability if they are replaced or left in service. For this

purpose, the framework includes Pareto analysis as the second step in making decisions on

asset replacement.

Pareto analysis or the 80/20 principle has been applied in many different disciplines since

it was promoted in the 1950s by quality control engineers [122]. The 80/20 principle was,

however, discovered by the Italian economist Vilfredo Pareto (1848–1923), who was

studying the relationship between wealth and income in England in the nineteenth century.

From the study, he found that there is a pattern in the distribution of the wealth among the

population and this pattern is the same for many other data. From Pareto realisation, the

80/20 principle is formulated, and it states that “approximately 20–30 per cent of any

resource accounted for 70–80 percent of the activity related to that resource” [122]. This

means that a large number of achievements can be completed by fewer inputs. It has been

found that this principle can be applied to any kind of resource, and that the linear

conception that 50 percent of the causes will lead to 50 percent of the results is not true for

the vast majority of cases [122].

In the proposed replacement framework, this principle has been applied to determine the

contribution of individual components to system unreliability. This has been completed in

two stages. The first stage is to replace the critical components one after another starting

from the top of the ranking based on the importance measure IP. The results of this stage

give a clearer indication of the importance of the components, and hence, a new ranking is

acquired. Secondly, based on the new ranking, the reduction in system unreliability, i.e.,

increase in system reliability, resulted from individual replacement scenarios and the


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accumulated reduction are computed. Clearly, the Pareto plot can be obtained from this

and the 20/80 principle can be applied. In the view of the previous stages, Pareto analysis

involves taking the most influential components and quantifying the effect of replacing

them on system reliability. By performing this analysis, the relationship between

replacement scenarios and resulting increase in system reliability can be determined.

4.4 Replacement Justification

For transmission system asset managers there is a choice between replacing the equipment

now and delaying the replacement to the following years. The typical decision for the asset

manager would be to postpone the replacement of assets to the following year in order to

achieve maximum utilisation of assets and savings in the reinvestment cost. Postponement

of the replacement, however, increases the risk of having an end-of-life failure, and hence,

customer supply interruptions. The replacement decision should be justified by performing

cost-benefit analysis to compare the cost of unreliability when the replacement is deferred

and the benefit gained by saving on reinvestment costs.

4.4.1 Unreliability Cost

As mentioned earlier, since the year 2000, many regulatory authorities have started to

impose a reliability regulation scheme in order to ensure that the budget constraints on

transmission system investment do not affect the continuity of supply for the end users

[33]. The reliability regulation schemes are based on incentives/penalties calculated using

some of the reliability indices, commonly Energy Not Supplied (ENS). The replacement

framework uses the incentive/penalties scheme set by the regulator in Great Britain as a

measure of the cost of unreliability. This reliability scheme is illustrated in Figure 4-3. The

scheme has set a target of ENS equal to 316 MWh/year and hence the cost of unreliability

when achieving this ENS is zero. Achieving an ENS less than this target will be rewarded

at a rate of £16,000 per MWh as shown by the green dotted line in Figure 4-3. In this case

the cost of unreliability is negative. Comparably, any values of ENS more than 316

MWh/year will incur penalties with the same rate and the cost of unreliability is positive as

shown by the red line in Figure 4-3. It is recognised that this may not represent the full

societal or reputational costs of extended or widespread power failure, and these factors

also need to be taken into account in replacement planning.


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Figure 4-3: Cost of system unreliability as a function of ENS based on Great Britain regulatory

incentives/penalties scheme

4.4.2 Saving on Reinvestment Cost

In order to calculate the saving on reinvestment cost (ΔCsaving), the time value of

replacement cost in the current year and time value of replacement cost in the following

year have to be calculated. The time value of the replacement cost in the current year

(referred to as present value (PV) in economics terminology) equals the current cost of

replacement. The reinvestment is usually carried out as a series of equal instalments at

equal time intervals, i.e., uniform annual payments (AV). The present value (PV) can be

calculated from annual values (AV) adjusted for time value of money. The time value of

money depends on the type (simple or compound) of interest rate considered [123, 124].

For a simple interest rate, PV can be calculated using (4.4):

ir

irAVPV

n)1(1 (4.4)

where ir is the interest rate and n is the number of instalments. The time value of

replacement cost in the following years (future value FV) can be calculated using PV. In

fact, PV, FV and AV are all related by time value of money formulae. If any one of them is

known, the others can be calculated. Standard formulae include calculation of PV from AV

(known as P given A, P/A), calculation of FV from PV (known as F given P, F/P),

calculation of FV from AV (known as F given A, F/A). All time value of money formulae

are given in Appendix F. For a simple interest rate, FV can be calculated from PV by (4.5):


102

)1( irnPVFV (4.5)

where ir is the interest rate and n is the number of future years.

The saving on reinvestment equals the difference between the future value of the

replacement cost and the present value. Then, the saving on reinvestment cost (ΔCsaving)

can be obtained by:

irnPVCsaving . (4.6)

Equation (4.6) shows the interest earned on the money when it is not spent on the

replacement.

In order to economically justify the postponement of reinvestment decision, i.e., the

replacement decision, the cost of unreliability and the saving in reinvestment costs are

calculated for all the replacement scenarios obtained from Pareto analysis. These scenarios

are then compared to determine the maximum number of components whose replacement

can be deferred to the following year without compromising the system reliability. The

optimum replacement decision is the scenario for which the cost of unreliability is less than

the saving in reinvestment costs and has the maximum number of components that can be

left in service for an additional year.

4.5 Case Study

The proposed framework has been applied to the power transformers fleet in the test

system. Although this thesis, and hence this case study, engages power transformers, the

replacement framework can lend itself to other types of assets such as cables, overhead

lines, and switchgear.

4.5.1 Transformers IP Measure

In order to calculate the importance measures, the Energy Not Supplied index (ENS) was

chosen as the system reliability indicator whereas the unavailability due to end-of-life

failure was defined as the component reliability indicator. Accordingly, the IP for

individual transformers was calculated using a simple sensitivity analysis of the ENS to the

change in the component unavailability. In addition, for the calculations of the importance


103

measures, the seed of the random number generator for the NMC simulation is kept

constant to ensure consistent results of reliability indices for different loading levels.

4.5.1.1 Sensitivity Analysis Procedure

Sensitivity analysis of ENS to transformers’ unavailability is performed according to the

following steps:

1. Assess the base case system reliability and estimate base case ENS.

2. Perform sensitivity analysis by considering that the transformers are ideal

(unavailability=0), one by one, and estimate ENS in each case.

3. Repeat steps 1 and 2 for individual loading levels.

4. Calculate annual base case ENS and annual ENS for individual cases in step 2 using

loading level probabilities.

5. Calculate the importance measure IP for each transformer using (4.7):

idealTbase ENSENSIP _ (4.7)

where ENSbase is the ENS of the system for the base case and ENST_ideal is the ENS when

considering the transformer under consideration as ideal.

4.5.1.2 Results

The results of the importance measures show that only 29 out of 154 transformers in the

test system have an influence on system reliability. For the remaining 125 transformers, the

ENS value does not change if they have been considered ideal. Table 4-1 shows the

transformers that have an influence on the system reliability, their importance measure, age

and unavailability data, and their ranking according to IP measures. All the transformers

that appear in the table are step down transformers. They are not shown in the simplified

single line diagram at Figure 3-8 due to the complexity of the network, instead they are

named by the load point number. From this analysis, one can conclude that the reliability

problems, which are linked with transformer failure, originate from load supply points. It is

also apparent from Table 4-1 that all the critical transformers are in the age range of 43-54

years. Compared with the age histogram of the test system’s transformers (see Figure 3-9),

this age range has the highest frequency of occurrence in the test system. Interestingly, the

oldest transformers (55-58 years old) did not appear in the table. This can be explained by


104

the fact that the reliability assessment results depend on network structure as well as the

reliability of individual transformers.

Table 4-1: Ranking of the transformers in the test system based on IP importance measure

No. ID Age

[years]

Unavailability

[%] IP

1 L18-T2 47 0.75 51.06

2 L26-T4 45 0.62 31.11

3 L20-T4 48 0.82 30.73

4 L20-T5 48 0.82 30.73

5 L20-T6 48 0.82 30.73

6 L19-T3 43 0.50 29.26

7 L12-T2 47 0.75 29.20

8 L18-T1 47 0.75 26.75

9 L19-T1 47 0.75 24.47

10 L18-T3 47 0.75 24.32

11 L16-T3 54 1.34 21.10

12 L16-T6 43 0.50 21.10

13 L19-T4 51 1.06 20.56

14 L12-T1 47 0.75 19.47

15 L26-T3 45 0.62 16.06

16 L16-T5 43 0.50 12.01

17 L26-T2 47 0.75 11.37

18 L28-T1 45 0.62 10.76

19 L28-T2 45 0.62 10.76

20 L12-T4 45 0.62 9.73

21 L19-T2 50 0.98 9.097

22 L20-T3 53 1.25 7.94

23 L17-T1 46 0.68 6.63

24 L17-T4 49 0.90 6.63

25 L20-T1 53 1.25 6.31

26 L27-T3 45 0.62 5.23

27 L27-T1 45 0.62 4.22

28 L26-T1 45 0.62 3.68

29 L27-T5 44 0.56 1.01


105

28

37

36

35

29

5251

49

40

43

42

7

33

48

4746

44

15

5

34

32

31

3027

25

24

22

21

20

19

18

17

16

14

13

12

1110

9

4 123

6 8

23

26

38

T2T1

T4T3

T6T5 T7T8

T9 T11 T12

T10

T13 T14

T18 T15

T17 T16

T19 T20

T21 T22

T23 T24

T25 T26

T28T27

39

41

45

5354

Most critical

transformer sites

Uncritical

transformer sites

Figure 4-4: Reliability importance measure (IP) for power transformers.

The alternative representation of the IP measure is given by a ‘heat’ map showing areas in

the system most affected by ageing of the components. This representation is particularly

useful when the network contains areas of particular strategic importance. Figure 4-4

shows the heat map of the test system. As seen in the figure, the most critical transformers

are located in a limited area. The strategic impact of unreliability in this area in terms of

societal, reputational and environmental impacts can be considered in further studies if

required.

By incorporating the end-of-life failure model into importance studies, this step of the

framework identifies the most critical components for system reliability in terms of ageing.

This is a new application for the importance measure in reliability evaluation.

One final remark is that all calculations are performed using a 2.83-GHz quad core CPU

PC with 3.5 GB RAM. The calculation of the importance measures for the test system was

completed in a round one week. The computation time for a large power network, e.g.,


106

England and Wales transmission network, which is approximately 5 times larger than the

test system, would take approximately 6 weeks using the same PC. For the assessment of

large power networks, the computation time can be reduced by using multiple PCs as in the

case study reported in [101].

4.5.2 Pareto Analysis

4.5.2.1 First Stage

In order to perform the first stage in Pareto analysis, transformers shown in Table 4-1 were

replaced one at a time starting with the top ranked transformer. That is to say, the first

scenario is replacing only one transformer (L18-T2), the second scenario is replacing 2

transformers (L18-T2 and L26-T4), and so on until all 29 transformers are replaced. Figure

4-5 presents the calculated ENS against the number of replaced transformers following the

previously explained procedure. As it appears in Figure 4-5, the reduction in ENS has an

inverse exponential relationship with the number of replaced transformers which illustrates

the suitability of the 80/20 principle for replacement planning.

Figure 4-5: ENS for replacement scenarios of transformers

Figure 4-5 also shows that there are 15 transformers whose replacement will not achieve a

further reduction in ENS (red crossed bars and the last 3 transformers). In other words, for

the test system a 0 ENS, or a 100% reliable system, can be achieved by replacing a smaller

number of transformers than might be expected from the IP results that are shown in Table


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4-1. Table 4-2 gives the numerical values of Figure 4-5 beside the total reduction in system

ENS and the reduction caused by individual transformer replacement. In this table, the 15

transformers, whose replacement does not improve the system reliability, are highlighted

in red.

Table 4-2: The system ENS resulted from the replacement of transformers one by one

NO ID IP

Resulted

system ENS

[MWh/year]

Total reduction

in ENS

[MWh/year]

Reduction in ENS

due to transformer

replacement

[MWh/year]

1 L18-T2 51.06 212.69 51.03 51.03

2 L26-T4 31.11 181.58 82.13 31.11

3 L20-T4 30.73 150.85 112.87 30.73

4 L20-T5 30.73 135.49 128.23 15.37

5 L20-T6 30.73 135.49 128.23 0

6 L19-T3 29.26 106.22 157.49 29.26

7 L12-T2 29.20 77.02 186.69 29.20

8 L18-T1 26.75 77.02 186.69 0

9 L19-T1 24.47 64.59 199.13 12.43

10 L18-T3 24.32 64.59 199.13 0

11 L16-T3 21.10 43.49 220.23 21.10

12 L16-T6 21.10 37.48 226.23 6.00

13 L19-T4 20.56 37.48 226.23 0

14 L12-T1 19.47 37.48 226.23 0

15 L26-T3 16.06 37.48 226.23 0

16 L16-T5 12.01 37.48 226.23 0

17 L26-T2 11.37 37.48 226.23 0

18 L28-T1 10.76 26.72 237.00 10.76

19 L28-T2 10.76 26.72 237.00 0

20 L12-T4 9.73 26.72 237.00 0

21 L19-T2 9.097 26.72 237.00 0

22 L20-T3 7.94 18.77 244.94 7.94

23 L17-T1 6.63 12.14 251.57 6.63

24 L17-T4 6.63 12.14 251.57 0

25 L20-T1 6.31 5.23 258.49 6.91

26 L27-T3 5.23 0 263.72 5.23

27 L27-T1 4.22 0 263.72 0

28 L26-T1 3.68 0 263.72 0

29 L27-T5 1.01 0 263.72 0


108

When considering the 15 transformers with zero reduction in ENS, common feature among

them can be noticed, i.e., there are other transformers located at the same buses which have

been replaced at an earlier stage. This replacement has resulted in eliminating the effect of

those 15 transformers on system reliability. For example, replacing transformer L20-T6

does not improve the system reliability because L20-T4 and L20-T5 were replaced at the

previous scenario. One question can be raised here, why does the replacement of these

transformers not eliminate the effect of L20-T3 and L20-T1? The reason is that they are

feeding a load, which is located at a separate low voltage bus from the load fed by other

transformers at Bus 20 (see the detailed single line diagram in Appendix D). Pursuing this

further, the three transformers: L20-T4, L20-T5, L20-T6 had exactly the same criticality

(IP = 30.73 for the three), but it turns out that replacing just two of them eliminates the

apparent effect shown by one way sensitivity analysis. This case is applicable to all other

transformers and is a solid demonstration of the benefit of performing Pareto analysis in

this replacement framework.

4.5.2.2 Second Stage

In the second step of Pareto analysis, transformers are ranked based on the reduction in

ENS, i.e., last column in Table 4-2. Table 4-3 shows the reduction in ENS and the

cumulative reduction in percentage of the total ENS for the replacement scenario excluding

15 transformers, which have no effect on the system reliability.

Table 4-3: Reduction and cumulative reduction in ENS due to replacement scenarios following Pareto’s new

ranking

NO ID

Reduction in ENS due to

individual replacement

[MWh/year]

Cumulative reduction

in ENS

[%]

1 L18-T2 51.03 19.35

2 L26-T4 31.11 31.15

3 L20-T4 30.73 42.80

4 L19-T3 29.26 53.89

5 L12-T2 29.20 64.97

6 L16-T3 21.10 72.97

7 L20-T5 15.37 78.80

8 L19-T1 12.43 83.51

9 L28-T1 10.76 87.59

10 L20-T3 7.94 90.61

11 L20-T1 6.91 93.23

12 L17-T1 6.63 95.74

13 L16-T6 6.00 98.02

14 L27-T3 5.23 100.00


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Figure 4-6 gives the Pareto plot for the replacement scenarios. The figure shows the

reduction in ENS for each replacement scenario (red columns) and the cumulative

reduction in ENS in percentage of the total ENS of the system (blue line). It is apparent

that 80% reduction in the ENS can be achieved by replacing 7 transformers (24.1% of the

most important transformers) as confirmation of the applicability of Pareto analysis to

power system reliability and component replacement. The figure also shows that 100%

reduction in ENS can be achieved by replacing 14 out of the 29 transformers that were

identified as critical. Table 4-3 and Figure 4-6 directly link the reduction in ENS to the

number of replaced transformers.

Figure 4-6: Pareto plot for the replacement scenarios

4.5.2.3 Final Remark

One final remark regarding Pareto analysis is that it is not only applicable to replacement

scenarios but is also valid all the way through the analysis. For example, the importance

measure has defined 29 transformers as critical, which is 18.8% of 154 transformers in

total the test system. Another example is the case of the transformers at Bus 20 (L20-T4,

L20-T5, L20-T6), replacing L20-T4 (33.3%) has resulted in 30 MWh/year reduction in

ENS, which is 66.6% of the total ENS (46.1 MWh/year) caused by the failure of these

transformers. The exact percentage is not the issue; the main point is the pattern of Pareto

analysis and how its applications can lead to “achieve more with less” [122].


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4.5.3 Replacement Justification

4.5.3.1 Unreliability Cost

In order to match the National Grid regulatory incentive/penalty scheme to the test system,

the ratio of the test system demand to the total demand of the England and Wales network

was calculated. The ENS baseline target of the test system was calculated by multiplying

the ratio by 316 MWh (the baseline target for National Grid). The calculation determined

60 MWh as the baseline target for the test system. The same incentives/penalties rate

(£16,000) was used to determine the cost of unreliability.

Table 4-4 shows the cost of unreliability starting from no replacement scenario (0

transformers to be replaced) to replacing all 14 transformers defined by Pareto analysis. As

can be seen, the cost of unreliability calculated using the scheme changes its sign after

replacing 7 transformers. This is due to achieving the baseline target ENS by replacing 8

transformers only. In addition, the maximum incentive for achieving 0 MWh of ENS is

about £1 million (£0.96 million).

Table 4-4: Cost of unreliability for replacement plans

Number of

replaced

transformer

ENS

[MWh]

Difference

from target

[MWh]

Cost of

unreliability

[k£]

0 263.72 203.72 3259.45

1 212.69 152.69 2443.00

2 181.58 121.58 1945.30

3 150.85 90.85 1453.61

4 135.49 75.49 1207.76

5 106.22 46.22 739.58

6 77.02 17.02 272.36

7 64.59 4.59 73.44

8 43.48 -16.52 -264.24

9 37.48 -22.52 -360.30

10 26.72 -33.28 -532.54

11 18.77 -41.23 -659.65

12 12.14 -47.86 -765.69

13 5.23 -54.77 -876.31

14 0.00 -60.00 -960.00


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4.5.3.2 Saving on the Reinvestment Cost

In order to calculate the future value of reinvestment cost for the replacement plans, a

simple annual interest rate of 5.4% is used [125]. The saving on the reinvestment cost is

calculated using (4.6) for a one year postponement. The cost of replacing a power

transformer, which is the present value of reinvestment, is taken as £4 million. It should be

mentioned that the cost of replacing a power transformer varies with the size and the

voltage level of the transformer [126]. The value of £4 million is an approximate average

cost for the size range of transformers identified by Pareto analysis performed in the

previous section and shown in Table 4-3 (120-240 MVA). Table 4-5 shows the saving on

reinvestment cost for the replacement scenarios.

Table 4-5: Saving on reinvestment cost for the replacement scenarios

Number of

replaced

transformers

Present value

(PV) of

deferring the

replacement

(£millions)

Future value

(FV): non spent

money with

interest

(£millions)

Saving on

reinvestment

cost

(£millions)

0 56.00 59.02 3.02

1 52.00 54.81 2.81

2 48.00 50.59 2.59

3 44.00 46.38 2.38

4 40.00 42.16 2.16

5 36.00 37.94 1.94

6 32.00 33.73 1.73

7 28.00 29.51 1.51

8 24.00 25.30 1.30

9 20.00 21.08 1.08

10 16.00 16.86 0.86

11 12.00 12.65 0.65

12 8.00 8.43 0.43

13 4.00 4.22 0.22

14 0 0 0

Considering the first replacement scenario as an example, the cost or the PV of replacing 0

transformers, i.e., deferring the replacement of 14 transformers to the next year, equals £56

million, which is 14×£4M. If this amount is not spent on replacement, its FV calculated

using (4.5) will be £59.024M. Hence, the saving or reinvestment cost is £3.02M. This


112

would be the amount of interest earned by postponing the replacement of 14 transformers

for one year, i.e., the saving on the reinvestment cost. In the same way, when replacing 14

transformers, there is no PV amount to accumulate interest, and hence, the saving on the

reinvestment cost is zero.

4.5.3.3 Economic Comparison

Figure 4-7 shows an economic comparison between the cost of unreliability and saving on

reinvestment cost for different replacement plans. The aim of this comparison is to

estimate the maximum number of transformers which can be left in service for the

following year without jeopardising system reliability. This number is determined when

the saving on the reinvestment cost due to deferring the replacement of transformers

becomes greater than the cost of unreliability caused by leaving them in service. It can be

seen from Figure 4-7 that the cost of unreliability when deferring replacement of 14

transformers to the following year is greater than the saving on the reinvestment cost. After

replacing one transformer, the saving on the reinvestment cost becomes larger than the

unreliability cost. Therefore, the maximum number of transformers to be replaced in the

following years without compromising the system reliability is 13. Hence, the

economically optimum number of transformers to be replaced is 1. (Note: this result is

based on the approximate average cost of replacing power transformers (£4 million). If the

exact cost is available the optimum number of transformers to be replaced might change).

Figure 4-7: Economic comparison of replacement plans


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The previous economic comparison considers only the amount of money paid as a penalty

to the regulator. Any extra cost, however, can be included in the calculation such as

customer compensation and reputational impact. For example, considering the Value of

Lost Load (VoLL) for the test system is £4000 per MWh, an extra unreliability cost can be

calculated and added to the cost shown in Table 4-4. Then, an economic comparison can

be performed. Figure 4-8 shows the comparison after modifying the cost of unreliability

estimated using the assumed VoLL of the test system. It is apparent that the cost of

unreliability is higher than the previous case. Therefore, the optimum number of

transformers to be replaced this year increases to 3, i.e., replacement of 11 transformers

can be postponed to following years.

Figure 4-8: Economic comparison of replacement plans adding unreliability cost calculated using VoLL

4.6 Summary

This chapter introduced a framework for making decisions about the replacement of power

system transformers. The framework is based on three elements. Firstly, the criticality of

the transformers for system reliability is identified using a reliability importance measure.

Secondly, the impact of replacement of the most critical transformers on system reliability

is determined by using Pareto analysis. The final element of the framework is to perform

an economical comparison between the cost of unreliability based on regulatory incentives


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and the saving on reinvestment cost by deferring replacement in order to determine an

optimum replacement plan for transformer fleet.

The first contribution of the framework is that it combines the merits of the two

commonly-used quantitative risk approaches, risk matrix and risk indices. By applying this

merger, the framework introduces a more comprehensive decision-making framework for

transformer replacement. As second contribution, the framework brings out the use of

Pareto analysis in this area of power system studies. The use of Pareto analysis provides an

insight into the effect of equipment replacement volume on system reliability.

Finally, the chapter provides an example of the use of the reliability regulation scheme in

decision making. It must however be recognised that the societal importance of a reliable

transmission network could be significantly greater than the incentive mechanism,

particularly in the case of widespread and extended failures.

The case study presented in this chapter illustrates how the framework can be practically

employed in order to determine an economically optimal number of transformers to be

replaced. This framework can be also applied to other types of assets such as cables,

overhead lines, and switchgear.

Chapter 5 Incorporation of Unconventional Failure Models into Reliability Studies

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5

Incorporation of Unconventional

Failure Models into Reliability

Studies

5.1 Introduction

The review of past work has revealed that despite the availability of some unconventional

failure models of power transformers, they have not been integrated into system reliability

studies. The only available attempt [55] in the open literature to attain this has a notable

weakness. The study employed the failure probability in selecting system states rather than

the unavailability, which is inconsistent with the main concept of State Enumeration

technique used in the study. Hence, it can be argued that the integration of unconventional

failure model into system reliability has not been done, yet.

The unconventional models relate end-of-life failure of a transformer to its age and

operational conditions. At present, in addition to the advancing age of installed

transformers, there is increasing commercial pressure to operate electricity transmission

systems close to their limits. This increases the stress on the equipment and hence

increases the likelihood of failure occurrence. Therefore, the integration of these


116

unconventional failure models would strengthen the accuracy of system reliability

assessment.

This chapter focuses on the integration of one of the available unconventional failure

models, which is Arrhenius-Weibull distribution. The Arrhenius-Weibull distribution

characterises the relationship between the life-time of transformers and the thermal stress

due to loading levels, and it has been employed to characterise end-of-life failure of power

transformers [54, 55]. The chapter also assesses the added value of the integration of such

unconventional failure models into system reliability studies. This was completed by

comparing the reliability indices calculated using the Arrhenius-Weibull distribution to the

indices calculated using the Gaussian (normal) distribution. The study presented in this

chapter is the original contribution of this thesis in the area of the integration of

unconventional failure models.

5.2 Transformer Failure Model

Previous chapters have shown the assessment of system reliability considering end-of-life

failure. In this reliability assessment, the unavailability due to end-of-life failure was

expressed as a function of the age of the transformer. The end-of-life failure of a power

transformer, however, does not only depend on its age but also on the operational

environment. Transformers’ end-of-life failure occurs, as mentioned in [52], when any

operational stress exceeds the withstand strength of the transformer. This is interpreted as

either a low withstand strength of transformer or high stress induced by operational

environment that may lead to the transformer end-of-life failure. The combination of the

low withstand strength and highly stressed operational environment makes the likelihood

of transformer failure occurrence even higher. Therefore, it is the ultimate objective of an

advanced failure model of transformers to include other electrical operational factors.

The thermal stress is one of the major factors of inducing ageing and failure mechanism of

power transformers. Therefore, in this thesis, the loading level of the transformer is

integrated into unavailability calculations so that the thermal stress effect on the reliability

is taken into account. Figure 5-1 gives an illustrative example of the unavailability

estimation when considering additional aspects. The figure shows the improvements in

modelling from the traditional constant unavailability for all ages; through age dependent


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unavailability; to the unavailability estimation when considering loading conditions

besides the age. The age-load dependant unavailability can be calculated using life-stress

models, e.g., Arrhenius-Weibull distribution.

Figure 5-1: Modelling transformer reliability: (a) traditional constant unavailability. (b) Age dependant

unavailability. (c) Age-load dependant unavailability

5.2.1 Life-stress Models

The characterisation of the failure model of engineering components by probabilistic

distribution functions is accomplished by analysing historical failure data without

considering the operational environment and stresses on components. On the other hand, it

is not an easy task to formulate failure models that include these factors using the

knowledge of failure physics; especially, when the components are technically complex

[127]. One way to represent the operational factors in failure models is using life-stress

models. These models are also used in accelerated life testing in order to obtain failure data

for long life components. Life-stress models combine the knowledge of the component

physics with the statistical analysis of failure data. In general, there are two elements in a

life-stress model: (1) a life-time probability distribution function and (2) a relationship that

describes the change in the parameter of the probability distribution function with different

levels of stress [128]. The life-time probability distribution function has been previously

discussed and employed in this thesis (Chapters 3 and 4). The second element of life-stress

model is a life-stress relationship, which relates the parameter of the probability

distribution function to operational stress. There are different life-stress relationships for

different types of stress, which are usually measured by appropriate relevant factors such

as, temperature, humidity, vibration, and pressure [129].


118

5.2.2 Transformer Life-thermal stress Relationship

There are two common life-stress relationships used in engineering reliability to describe

the acceleration/deceleration of component failure resulting from a change in temperature

(thermal stress). These are Arrhenius and Eyring relationships [130, 131]. These life-stress

relationships relate any life measure, e.g., mean life, median, or characteristic life, to the

temperature. Clearly, these relationships can be merged into probability distribution

functions containing any quantifiable life measure. This thesis has adopted the Arrhenius

relationship since it has been used previously in many power transformer studies [54, 55,

132-135]. Arrhenius relationship is given by (5.1):

)exp(

BAL

(5.1)

where L is a quantifiable life measure, e.g., characteristic life, Θ is the temperature in

Kelvin, and A and B are empirical constants mostly estimated from historical data.

The thermal effect of loading on the power transformer is usually represented by the Hot-

Spot Temperature (HST), i.e., the temperature of the hottest part of the winding. Therefore,

Θ in the Arrhenius relationship is typically substituted by HST [54, 55, 132-134].

Considering that most power transformers are oil-immersed, the IEC 60076-7 loading

guide [136] is used to calculate the HST as given by (5.2):

ykrHg

x

R

kRRTOaHST

1

21

,

(5.2)

where ΘHST is the HST [ºC], Θa is the ambient temperature [ºC]; ΔΘTO,R is the top-oil

temperature rise at the steady state at rated losses [K]; R is the ratio of load losses at the

rated current to the no-load losses; k is the load factor (load current/rated current); x is the

oil exponent; H is the hot-spot factor; gr is the average-winding-to-average-oil temperature

gradient at the rated current [K]; y is the winding exponent. The parameters in the HST

model (5.2) are transformer specific and should be determined by a heat-run test [136]. If,

however, the measured values are not available, recommended values from the IEC

loading guide can be used [136]. Equation (5.2) calculates HST for steady state loading,

i.e., the loading level has been applied on the transformer for more than 8 hours [136].

There are different formulae to calculate dynamic values of HST corresponding to dynamic

loading level, e.g., daily load curve and step change load.


119

5.2.3 Arrhenius-Weibull Failure Model

As the Gaussian distribution has two life measures; and , it is not commonly used with

the Arrhenius relationship. Probability distribution functions, which have one life measure,

such as Weibull and exponential distributions, are typically used [137]. Since Weibull

distribution has been used often in the past to model a power transformer’s failure [54, 55],

it has been used in this study as well. The failure distribution function of the Weibull

distribution, which is, in fact, the cdf, is given by (5.3):

))((exp1)(

ttcdf (5.3)

where and are the characteristic life and shape parameters of Weibull distribution,

respectively. When the transformer enters the ageing period of the bathtub curve (the

exponentially increasing right-hand part of the curve) [42], the shape parameter > 1

indicates an increasing failure rate. The scale parameter represents the age at which

63.2% of the population will fail. Since in (5.3) is a life measure, it is substituted by L

from the Arrhenius relationship given by (5.1) [54, 55]. Hence, the cdf of the Arrhenius-

Weibull failure model can be represented by (5.4):

))((

)273

exp(exp1)(

HST

BA

ttcdf .

(5.4)

Figure 5-2 shows an illustrative example of the cdf of Arrhenius-Weibull failure model for

two different values of HST (HST1>HST2). As the HST increases, the characteristic life

( reduces, which means that a transformer with HST1 may experience end-of-life failure

earlier than a transformer with HST2.

Figure 5-2: Illustrative example of Arrhenius-Weibull cdf for two different HST, where HST1>HST2.


120

5.3 Estimation of Arrhenius-Weibull Distribution Parameters

5.3.1 Weibull Distribution

The first step of implementing Arrhenius-Weibull distribution to the test system is to

estimate a Weibull distribution which is equivalent to the Gaussian distribution used in the

previous chapters. This has been completed using a method that is based on the gamma

function [47]. The equivalent values were found to be =70.79 years and =5. In addition,

validation of these values using curve fitting method has been performed. One thousand

random samples were generated from the Gaussian distribution, and then were fitted to a

Weibull distribution using Matlab dfittool. The values of and were found to be the

same as the ones estimated using the gamma function. Figure 5-3 shows the two

distributions which were fitted to the one thousand random samples.

Figure 5-3: Curve fitting of Gaussian and Weibull distributions

In order to calculate the unavailability due to the end-of-life failure from the Weibull

distribution, the integral used to calculate the probability of having end-of-life failure in

(3.4) can be approximated by (5.5) which is adopted from [61]:

)exp(

)exp())1(

exp(

T

tjTtjT

Pj

(5.5)

where Pj, T, and Δt are as in (3.4) and andare as in(5.4). Figure 5-4 shows the

unavailability for the two distributions: Gaussian and Weibull calculated for the age range


121

between 1 and 58 years. It is apparent that the unavailability values calculated from the two

distributions are not significantly different, which is an additional confirmation of the

accurate estimation of Weibull distribution parameters from the Gaussian distribution.

Figure 5-4: Transformer unavailability due to end-of-life failure for a range of ages (1-58) using Gaussian

and Weibull distributions.

5.3.2 Arrhenius-Weibull Distribution

As can be seen from (5.4), Arrhenius-Weibull failure model has three parameters; A, B,

and In accelerated testing data, Arrhenius-Weibull parameters are commonly estimated

in two steps [138]. Firstly, is estimated by fitting the failure data from each temperature

test to a Weibull distribution. These Weibull distributions are forced to have the same

value of [138, 139]. Secondly, A and B are estimated using and temperature values. As

there are no available data to estimate these parameters, the equivalent Weibull

distribution, historic loading data, and data from IEC Standard 60076-7 are used to

estimate those parameters.

5.3.2.1 Parameter

Based on the common practice in the calculation of the explained earlier, the value of

from the Weibull distribution (=5) was kept the same for Arrhenius-Weibull failure

model.


122

5.3.2.2 Parameters A and B

Parameters A and B are estimated based on two assumptions with respect to Firstly it is

assumed that an average HST (HSTa) of transformers, which is related to the historical

loading level, leads to =70.79 years. The value of HSTa is calculated by the following

steps:

a. Calculate, for each transformer, an annual equivalent loading level using loading

data from the 6-step load model.

b. Estimate a yearly weighted ambient temperature that causes the same ageing as the

variable ambient temperature during one year (a,E) using (5.6) given in [136]. The

value of a,E was calculated from 10 years of historical ambient temperature data

taken from [140] (see Appendix E). It was found to be 11.38ºC.

85.1

max,, ))(2(01.0 yamyaEa (5.6)

where ya is the yearly average temperature [˚C] and m,max is the average

temperature of the hottest month in the year [˚C].

c. Calculate an equivalent yearly HST (HSTb), for each transformer, using its annual

equivalent loading and the yearly weighted ambient temperature. The parameters

used to calculate HST are taken from [141], which contains results of heat-run tests

carried out on transformers owned by the National Grid. Table 5-1 shows the

values of those parameters. The values of HSTb for individual transformer are

given in Appendix D.

d. Obtain the average value of the calculated HSTb of transformers.

Table 5-1: Parameters of hot-spot temperature model

Parameter Value

ΔΘTO,R [K] 37.7

H 2.1

gr [K] 11.2

R 1.6

x 0.8

y 1.3


123

The value of HSTa was found to be 36.38ºC, which was then used as the HST

corresponding to =70.79 years. This value of HST is equivalent to 31.5% loading when

using the yearly weighted ambient temperature. The previous steps give one pair of and

HST values, while the calculation of A and B needs another pair of values. To obtain the

second pair, it is assumed that at HST=80 ºC, equals 40 years. This assumption is made

based on the IEC Standard 60076-7 [136] and IEEE Std C57.91-1995 [133]. Both

standards state that at HST equals 110˚C, a transformer may survive for about 20 years.

Having known that the HST of transformers in the test system at the full load is 72.6 ºC

(calculated using the yearly equivalent ambient temperature and the parameters given in

Table 5-1), the assumption of equals 40 years at HST=80ºC seems reasonable. The

values of A and B are estimated to be 0.56 and 1500, respectively. Figure 5-5 shows the

relationship between and the transformer HST based on the estimated values of A and B.

Figure 5-5: Characteristic life relationship with transformer HST based on the estimated values of A and B.

The assumptions made in the derivation of the relevant parameters were necessary since no

data is available in the open literature. Ideally, parameters A, B and should be calculated

from historical transformer failure data. The reliability evaluation, however, is not affected

by the numerical values of the parameters and will work equally well with any parameter

values. The numerical results though will depend on the values of the parameters used.


124

5.3.3 Unavailability Estimation

Using the Arrhenius-Weibull distribution, the unavailability due to end-of-life failure can

be calculated using (3.2), (3.3) and (5.5). For each transformer, the calculated

unavailability from Arrhenius-Weibull distribution depends not only on its age but also on

its loading level and the ambient temperature, i.e., HST. To illustrate this relationship,

Figure 5-6 shows the unavailability for transformers age range (1-58 years) under two

loading levels: the maximum historical loading of transformers (loading=59%,

HST=47.9ºC calculated for Θa=11.38ºC) and the average historical loading

(loading=31.5%, HST=36.38ºC). The unavailability curve of the average historical loading

(red dashed curve) is the same as the curve shown in Figure 3-10 because the average

historical loading data is assumed to be equivalent to the Gaussian distribution. When the

loading level is higher than the average, as shown by the unavailability curve of maximum

loading (black solid curve), the unavailability values for a transformer aged 20 years or

more will be higher. The figure also illustrates that, for young transformers (0-20 years),

the loading stress has no effect on their unavailability values and the effect of loading level

on transformer unavailability increases with the age of the transformer.

Figure 5-6: The unavailability for transformer age range (1-58 years) calculated using Arrhenius-Weibull

distribution for maximum loading level and average loading level.

5.4 Implementation of Arrhenius-Weibull Distribution

In order to conduct system reliability assessment using Arrhenius-Weibull distribution as

the end-of-life failure model, average values for the ambient temperature for the 6-step


125

load model have to be determined. Table 5-2 shows the average values for individual load

levels, which are calculated from the temperatures in Central England recorded in 2012

[140] (see Appendix E).

Table 5-2: The 6-step load model with associated ambient temperature values

Step

no.

Supply/Demand

level [%]

Probability Ambient

temperature [ºC]

1 98.9 0.173 5.8

2 92.3 0.230 5.8

3 88.1 0.135 8.25

4 80.4 0.115 13.0

5 74.5 0.0962 16.6

6 71.4 0.250 11.98

∑ = 1

Table 5-3 shows examples of transformers’ loading and unavailability, which is calculated

using Arrhenius-Weibull distribution, for different load levels. The table demonstrates the

effect of the main three factors of Arrhenius-Weibull distribution: age, loading, and the

ambient temperature, on the unavailability. The effect of loading level can be observed by

comparing the unavailability values of the three transformers, which are of the same age:

L30-T1, L31-T2, and L26-T2. For example, examining the unavailability values for load

level 1 for these three transformers, the highest loading percentage (68.7), leads to

significantly high value of unavailability (1.7%). The effect of age on the unavailability is

clearly apparent when comparing the unavailability of L13-T6 to the remaining

transformers. Although this transformer is highly loaded compared to the others, its

unavailability values are considerably small. The effect of the ambient temperature can be

appreciated by observing the unavailability values of L31-T2 at load level 1 and level 5.

The loading of this transformer at level 1 (33.2%) is greater than the loading at level 5

(21.8%), but the unavailability value at the lower loading (0.79%) is higher than the value

at the greater loading (0.47%). This is because the ambient temperature at load level 1 is

smaller than load level 5 (see Table 5-2). Furthermore, despite the fact that loading at level

1 is double the loading at level 6 for this transformer, the unavailability values are nearly

the same (0.47% for level 1 and 0.48% for level 6) due to of the effect of ambient

temperature.


126

Table 5-3: Examples of the unavailability of transformers calculated using Arrhenius-Weibull distribution

TX

age

Loading [%] Unavailability [%]

1 2 3 4 5 6 1 2 3 4 5 6

L30-T1 47 7.9 27.5 25.7 20.4 10.8 27.2 0.25 0.39 0.45 0.58 0.64 0.62

L31-T2 47 33.2 21.4 24.7 22.1 21.8 14.7 0.47 0.33 0.44 0.60 0.79 0.48

L26-T2 47 68.7 47.8 62.3 42.3 43.6 49.7 1.70 0.72 1.47 1.05 1.42 1.23

L13-T6 6 49.0 65.0 42.4 46.7 44.7 48.2 2E-4 4E-4 2E-4 4E-4 5E-4 4E-4

System reliability was assessed using Arrhenius-Weibull distribution. Table 5-4 shows the

ENS for the load points and the overall system. As shown in Table 5-4, Bus 20 has the

highest value of ENS. A closer investigation in the loading data shows that all the

transformers at Bus 20 operate at an equivalent annual loading level ranging between 32.3

– 57.9%, which is larger than the average transformers loading level.

Table 5-4: ENS for load points and system of the test network using Arrhenius-Weibull distribution

PLC [%] ENS [MWh/year]

Bus 20 0.436 91.9

Bus 12 0.016 87.5

Bus 26 0.010 68.8

Bus 18 0.008 54.1

Bus 19 0.012 31.4

Bus 16 0.005 27.1

Bus 28 0.001 0.1

System 0.468

360.8

5.5 Comparison between Gaussian and Arrhenius-Weibull Distributions

In order to evaluate the added value on incorporation Arrhenius-Weibull distribution into

reliability assessment, a comparison between the reliability studies conducted using

Gaussian and Arrhenius-Weibull distributions was performed. The ENS index was selected

as the primary reliability measure for the comparison. Table 5-5 shows the ENS values

calculated using the two models: Gaussian and Arrhenius-Weibull distributions. The main

observation based on the comparison is that the same unreliable buses detected based on

Gaussian distribution have been listed based on Arrhenius-Weibull distribution. The only

difference is that Buses 17 and 27 have a zero ENS when using Arrhenius-Weibull

distribution. Despite this, the overall system ENS increased from 263.8 MWh/year when

using Gaussian distribution to 360.8 MWh/year when using Arrhenius-Weibull


127

distribution. Table 5-5 also shows that for some of the load points ENS increased when

using Arrhenius-Weibull distribution, while for the others ENS decreased in the ENS value

(see last column in Table 5-5). Bus 16 kept the same ENS value in the two studies.

Table 5-5: A comparison between reliability studies using Gaussian and Arrhenius-Weibull distributions

ENS [MWh/year] Change

Gaussian Arrhenius-Weibull

Bus 20 61.0 91.9 increase


Bus 19 41.7 31.4 decrease



Bus 16 27.1 27.1 no change

Bus 28 10.8 0.1 decrease

Bus 17 6.6 0 decrease

Bus 27 5.2 0 decrease

System 263.8 360.8

increase

5.5.1 Load Points with Increased ENS

Load points for which ENS increased are Buses 12, 26, 20 and 18. Table 5-6 shows these

buses ranked based on the amount of the increase and as a percentage of the ENS

calculated using Gaussian distribution. The table also gives the age and loading ranges of

the transformers at each load point. The general observation from Table 5-6 is that the

loading ranges of transformers located at these buses are higher than the average loading of

31.5%, which is equivalent to the Gaussian distribution. Consequently, there is an increase

in ENS. It is apparent that the most significant increase in ENS is for Bus 12. The amount

of the increase (58.3 MWh/year) is double the value of ENS when using Gaussian

distribution (29.2 MWh/year). Although the transformers located at Bus 12 are in the 4 -

47 years age range, they operate at equivalent annual loading level between 44.8% and

58.4% which is higher than the average loading of 31.5%. The loading range of

transformers at Bus 12 is the highest among all other buses shown in Table 5-6. Similarly,

Bus 26 has two times higher ENS value than before since the equivalent annual loading

level of its transformers is higher than the average, 36.3 – 52.8%. For Bus 20 the top

ranked among the unreliable buses ranking when using Gaussian distribution, ENS

increased by 51%. Even though the increase in ENS was not the highest in Table 5-6, Bus

20 remains the most unreliable bus when using Arrhenius-Weibull distribution. Bus 18 has


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the lowest increase in ENS (6% of ENS calculated from Gaussian distribution) as the

loading range of transformers located at it is relatively low (32.2 – 37.7%).

Table 5-6: Load points which gained an increase in the ENS when using Arrhenius-Weibull distribution

Amount of the

increase in ENS

[MWh/year]

Percentage of

the increase in

ENS [%]

Transformers

age range

[years]

Transformers

loading range

[%]

Bus 12 58.3 200 4 – 47 44.8 – 58.4

Bus 26 37.7 121 32 – 47 36.3 – 52.8

Bus 20 30.9 51 48 – 53 32.3 – 57.9

Bus 18 3 6 47 32.2 – 37.7

5.5.2 Load Points with Decreased ENS

Load points for which ENS decreased are Buses 17, 27, 28 and 19. Table 5-7 shows these

buses ranked based on the percentage of the decrease in ENS. The table also gives the age

and loading ranges of the transformers at each load point.

Table 5-7: Load points which experienced a decrease in the ENS when using Arrhenius-Weibull distribution

Amount of the

decrease in ENS

[MWh/year]

Percentage of

the decrease in

ENS [%]

Transformers

age range

[years]

Transformers

loading range

[%]

Bus 17 6.6 100 46 – 49 25.3 – 38.9

Bus 27 5.2 100 43 - 45 23.5 – 37.8

Bus 28 10.7 99 8 – 45 31.3 – 36.1

Bus 19 10.3 25 43 – 51 35.5 – 40.6

The first two buses in the table are the buses which have zero ENS when using Arrhenius-

Weibull distribution to characterise end-of-life failure of transformers. When looking at the

loading range of these two buses, it is clear that there are transformers located at these

buses which operate at much lower loading level than the average. This leads to lower

values of transformers unavailability, and hence a zero ENS. Bus 28 and 19, on the other

hand, have lower ENS values although the historical loading data shows that the annual

equivalent loading of transformers located at these buses is between 31.3 – 40.6%, which

is higher than the average transformer loading. The detailed seasonal loading data,

however, shows that the transformers have high loading during the low temperature

seasons (levels 1 and 2; see Table 5-2). This leads to HST values that are less than 36.37ºC

and consequently lower unavailability values than the values obtained from the Gaussian


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distribution. Table 5-8 shows an example of the loading of two transformers located at Bus

19 and Bus 28 for individual load levels.

Table 5-8: Example of loading percentage at different levels for two transformers located at Bus 19 and 28

Level1 Level2 Level3 Level4 Level5 Level6

L19-T4 54.7% 57.3% 26.6% 31.3% 34.2% 29.4%

L28-T3 44.8% 37.3% 32.5% 33.0% 33.2% 25.3%

5.5.3 Load Points with No Change in ENS

Bus 16 is the only bus that did not show any change in the ENS value. The loading range

of the transformers located at this bus is 19.4 – 29.1%, which is the lowest loading range

among all buses. One may expect that due to this low level of loading, the ENS of Bus 16

would decrease, but this did not occur. When looking at the PLC index of this bus obtained

from the two studies, it is found to be the same (0.005%). A closer investigation of the

failure states at Bus 16 revealed that there were planned outages at this bus, which

weakened it. Hence, even the low unavailability values resulted from Arrhenius-Weibull

distribution they did not enhance bus reliability.

Figure 5-7: Critical load points based on ENS obtained using Gaussian and Arrhenius-Weibull distributions.

Figure 5-7 shows the heat maps of the test system for the two reliability studies: Gaussian

and Arrhenius-Weibull distributions. The heat maps illustrate the effect of using

a) Gaussian distribution b) Arrhenius-Weibull distribution

60 54 48 42 36 30 24 18 12 6 0

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unconventional failure model in system reliability studies. These heat maps and the

previous tables illustrated that refining the transformer failure model by including

influential operational agents can contribute significantly to the assessment of system

reliability. The reliability indices calculated when using unconventional models involve the

age, the operation condition, and the ambient temperature effect.

5.6 Summary

The chapter presented the integration of unconventional failure model into system

reliability studies. A previously introduced unconventional failure model of power

transformers, the Arrhenius-Weibull distribution, was used in the study. The Arrhenius-

Weibull distribution characterises end-of-life failure of power transformers based on the

age and the loading conditions. The chapter described in details how this model was

implemented to the test system. The parameters of the Arrhenius-Weibull distribution were

estimated based on an equivalent Weibull distribution and historical loading data. The

assumptions made during the estimation were necessary since there is no available data in

the open literature.

In addition, the chapter demonstrated the effect of the three factors related to Arrhenius-

Weibull distribution on the unavailability values. These are age, loading level and the

ambient temperature. The chapter also assessed the added value of the incorporation of

such unconventional models into system reliability studies. It was shown that the

integration of thermal stress due to loading conditions into end-of-life failure model

(Arrhenius-Weibull distribution) provides more insight into understanding the system

reliability and identifying the failure conditions of most critical load points. The findings of

this chapter contributed to the understanding of the effect of refining the failure model of

power transformers on power system reliability studies.

Chapter 6 Transformer Criticality for Cascading Failure Events

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6

Transformer Criticality for

Cascading Failure Events

6.1 Introduction

Electricity transmission systems typically avoid the serious societal, environmental and

economic impacts of blackouts by employing design and operational planning practices

which consider security constraints. They are, however, still vulnerable to multiple failure

events caused by hidden failures or errors in decisions taken by operators. It is, therefore,

necessary to investigate thoroughly the causes and consequences of such events.

Much work has been already done in order to understand and simulate cascading failure

events, including papers [142-146] and case study presentations [147-150]. In these

studies, many important factors have been considered, including generator instability,

sympathetic tripping, protection failure, and weather conditions. Age related failure,

however, has never been included in cascading failure investigations. With respect to the

identification of critical loading level and components to cascading failure events, there are

some further studies in the open literature [151-153]. The critical loading level was

defined as the level at which a distinct increase in the effect of cascading failures measured

by ENS occurs [151, 152]. For the identification of most crucial components, a criticality


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indicator was developed by employing cascading failure simulation method. In this

cascading simulation, the component was considered out-of-service if its loading level

exceeds a specific threshold. The indicator is then calculated as the ratio between the

number of times that a component appears in cascading failure event to the total number of

cascading events. It is apparent that the introduced criticality indicator did not consider the

effect of the age of the components. As the equipment ages and its capability to withstand

stress reduces there is a need to consider the effect of ageing related failure in cascading

failure studies.

The work presented in this chapter integrates a power transformer reliability model that is

age and loading level dependent into multiple and cascading failure investigations. It

focuses on estimating the probability of a second dependent failure of a transformer, which

might ultimately lead to a multiple failure event or even a cascading failure, occurring due

to thermal stress. Two new probabilistic indicators relating the reliability of transformers to

their age and loading levels are developed to rank power transformers based on their

criticality for multiple failure events. The first indicator (ICF) identifies which

transformers can initiate a sequence of multiple failures when they fail, while the second

(VCF) identifies transformers which are the most vulnerable to a consequential failure. The

indicators are calculated for individual transformers and transformer sites, and their

robustness to load uncertainty is assessed. The results of this work can be used to inform

asset managers about the criticality of transformers and to assist them with replacement

decision making. The investigation of second dependent failure and the development of

these probabilistic indicators represent the original contributions of this thesis to

cascading failure research area.

6.2 Dependent Failure

Generally speaking, there are three types of dependency in failure of power system

components: physical dependency, operational dependency, and environmental

dependency. The physical dependency pertains to the structure and the connections of

components. For example, a common cause failure occurs to double circuits that are on the

same tower or group of components are out of service because of substation

configurations. In operational dependency, the dependent failure occurs because the failure

of the triggering component changes the operational conditions of other components.


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Cascading failure events belong to this type of dependency. The environmental

dependency is related to weather conditions such as snowstorms, floods, and tornados,

which cause the failure of a group of components. This chapter focuses on the operational

dependency, specifically loading condition dependency failure. The development of

Arrhenius-Weibull distribution based model facilitates the assessment of the effect of

loading on the unavailability of power transformers. Accordingly, the effect of the change

in loading conditions on the unavailability can be assessed, which results in evaluating the

probability of second dependent failure due to thermal stress.


The first step for employing Arrhenius-Weibull distribution to model varying load is to

calculate the transformer HST at varying loading level. According to IEC 60076-7 loading

guide, HST can be calculated by (6.1) [136]:

wTOaHST (6.1)

where ΘHST is the hot-spot temperature (ºC), Θa is the ambient temperature (ºC), ΔΘTO is

the top-oil temperature rise (K), and ΔΘw is the hot-spot-to-top-oil gradient (K). According

to [136], ΔΘTO and ΔΘw at varying load can be calculated in two alternative ways; the

exponential equation method (suitable for a step load variation) and the differential

equation method (suitable for time-varying load curves, e.g., a daily load curve). As this

study focuses on the sudden change in loading conditions due to a transformer outage, the

exponential equation method is used. For the exponential equation method, ΔΘTO and ΔΘw

are calculated differently for step load increase and decrease. For step load increase, ΔΘTO

and ΔΘw are given by (6.2) and (6.3), respectively:

)(1

11,

2

,, tfR

kRiTO

x

RTOiTOTO

(6.2)

)(2, tfkHg iwy

rw . (6.3)

For step load decrease, they are calculated by (6.4) and (6.5):

)(1

1

1

13

2

,,

2

, tfR

kRR

kRx

RTOiTO

x

RTOTO

(6.4)

y

rw kHg . (6.5)


134

The functions, f1(t), f2(t) and f3(t), are calculated by (6.6), (6.7) and (6.8):

)

)((

1111)( ok

t

etf

(6.6)

)((

21

))(

(

21222/22 1)1(1)(

k

t

k

t

ow ekektf (6.7)

)

)((

311)( ok

t

etf

(6.8)

where ΔΘTO,i is the top-oil temperature rise at the start (K); ΔΘTO,R is the top-oil

temperature rise at steady state at rated losses (K); ΔΘw,i is the hot-spot-to-top-oil gradient

at the start (K); R is the ratio of load losses at rated current to no-load losses; k is the load

factor (load current/rated current); x is the oil exponent; H is the hot-spot factor; gr is the

average-winding-to-average-oil temperature gradient at rated current (K); y is the winding

exponent; t is time (min); k11, k21, and k22 are thermal model constants; τO is the average oil

time constant (min); τW is the winding time constant (min) [136]. For the HST model

parameters, the recommended values from the IEC loading guide were used except for

ΔΘTO,R, H, gr and R. Those parameters were the same as the ones given in Table 5-1.

(Note: All the parameters used in (6.2) to (6.8) are for the natural oil flow (ON)

transformer cooling mode. For transformers with dual ON and oil forced (OF) cooling

modes, the cooler will operate if the HST exceeds the threshold and different parameters,

corresponding to the OF cooling mode, should be used in HST calculations.)

6.2.2 Calculation of Second Dependent Failure

The calculation of second dependent failure was accomplished by the steps depicted in

Figure 6-1. Because it is impossible to predict when a transformer will fail, the model

starts by enumerating all possible load levels from the annual load curve. Then, for each

load level, load flow calculation is performed to define the initial loading and consequently

the initial hot-spot temperature (HST) of individual transformers. The next step of the

model is to perform a loop for taking transformers out of service one by one and

calculating the new loading conditions to assess thermal stress (represented by HST) on the

remaining transformers. If the outage of a transformer leads to an unfeasible system, i.e.,

load flow does not converge; the load flow calculation algorithm will be changed to DC

load flow, which always converges. This will not affect the results of the calculation of

HST as it is mainly based on real power flow. In practice, however, it is unusual that a


135

single component outage leads to system violations since most power systems are designed

to meet N-1 security criterion.

Run load flow to obtain initial loading of components

Take one transformer out of service

Run load flow to obtain new loading of components

Obtain the corresponding failure model

Enumerate load level from the annual load curve

All load levels

Finish

All transformers

Calculate Hot-Spot Temperature

Estimate the average unavailability

Load flow

Converge?

yes

no Switch to DC

load flow

yes

yes

no

no

Figure 6-1: Flowchart of second dependent failure calculations due to thermal stress.

After calculating HST due to new loading conditions, the parameters of the Arrhenius-

Weibull model of the remaining transformers are updated as described in the previous

section. The unavailability is then estimated using the updated model. It must be

appreciated that the updated unavailability value is a function of age and the new loading

condition. Specifically, the estimated unavailability represents the probability of finding a

transformer, of a specific age under a specific increase in loading, in a failed state.

Although in the vast majority of cases the outage of a transformer will not lead to system

overload, the change in loading will change the stress on transformers and consequently


136

change the probability of having end-of-life failure. The accuracy of the estimated change

in unavailability due to the transformer outage depends on the accuracy of Arrhenius-

Weibull model parameters, which are typically derived using historical loading and failure

data. When this loop completes for all transformers, the unavailability of each transformer

due to another transformer failure is calculated. The execution of the transformers loop is

repeated for all load levels.

6.3 Age and Load based Criticality indicators

6.3.1 Indicator of Initiating a Cascading Failure (ICF)

Using the results produced from the second dependent failure model, a probabilistic

indicator is formulated in order to rank transformers according to probability of initiating a

cascading failure (ICF). ICF measures the effect transformer unplanned outage on the

failure probability of the other transformers, and hence, the probability of a transformer to

initiate a cascading failure. There are two factors in the calculation of a transformer’s ICF,

its initial unavailability and the change in the values of unavailability of other transformers.

Both factors are functions of the age and the loading level of transformers. The formula for

calculating the ICF of a transformer n is given by (6.9):

),,(,

1binibnn UUUICF

N

i

(6.9)

where Un,b is the unavailability of transformer n at the initial case (all transformers are in

service), Ui,n is the unavailability of transformer i when transformer n is out of service, Ui,b

is the unavailability of transformer i at the initial case and N is the total number of

transformers.

The summation in (6.9) describes the effect of the transformer outage on the remaining

transformers’ unavailability. It should be mentioned that for the same loading level, Ui,n

(the unavailability of transformer i when transformer n is out of service) is different for

different transformer ages. For the same loading, older transformers experience higher

value of Ui,n than younger ones. The initial case unavailability of a transformer n (Un,b) is

used in the calculation in order to weigh the transformer’s probability of failure. If Un,b is

not integrated in (6.9), the ICF will represent the criticality of the transformer from a


137

system structure perspective only, without considering the condition of the transformer,

i.e., age and loading.

An important issue to be highlighted is that the ranking of transformers using the ICF

indicator is the main aim of this study, rather than the value of the indicator itself.

6.3.2 Indicator of Vulnerability to Consequent Failure (VCF)

The indicator of vulnerability to a consequent failure (VCF) is formulated by observing the

unavailability of each transformer when other transformers fail. Accordingly, VCF of a

transformer shows how much the failure probability of the transformer will be affected by

the outage of other transformers. This effect on the failure probability of the transformer

depends on its age and the change in its loading level due to the outage of other

transformers. The value of the indicator for a transformer n is calculated by (6.10):

bibninn UUUVCF

N

i,),,(

1

(6.10)

where Un,b is the unavailability of transformer n in the initial case (all transformers are on

service), Un,i is the unavailability of transformer n when transformer i is out of service, Ui,b

is the unavailability of transformer i in the initial case, and N is the total number of

transformers. As can be seen form (6.10), the calculation of VCF involves assessing the

effect on the failure probability of the transformer, represented by Un,i-Un,b, and weighing it

the probability of the occurrence of the outage, represented by Ui,b. Therefore, VCF

captures the two main influential factors, the change in the transformer failure probability

and the probability of occurrence of this change. As mentioned previously, the ranking of

transformers is the primary goal of this study rather than the value of the VCF itself.

The ICF and VCF can be calculated for the annual load curve to inform long term planning

decisions. Furthermore, ICF and VCF can be calculated for specific load level/operating

state to inform short term operational decision making.

6.4 Implementation on the Test System

The proposed steps were applied to the test system using the parameters discussed in the

previous sections. In order to calculate the HST after a transformer outage, the time in (6.2)


138

to (6.8) is set as 30 minutes. The 30 minute value was chosen for illustrative purposes only

as the electricity market of Great Britain is a half hourly market. The initial values of top-

oil temperature rise (ΔΘTO,i) and hot-spot-to-top-oil gradient (ΔΘw,i) were calculated from

the pre-outage loading levels.

6.4.1 Transformer ICF

The ICF indicator was calculated for individual transformers for each load level as well as

for the annual load model. The ICF for the annual load model is calculated using the

individual load levels values and their associated probabilities. The results of the

calculations for either individual load levels or the annual load model show that ICF can

have a value of 0 for some transformers. Those transformers are young (1 – 24 years old)

and operate under low loading conditions. The full list of ICF value is given in appendix G,

in this chapter only the top ranked transformers are represented.

6.4.1.1 Load Levels

The top five transformers for each load level ranked by ICF are shown in Table 6-1. The

top five transformers are all step down transformers. It is obvious that some transformers

appear in the top five at more than one level. For example, transformer L10-T4 comes in

the top five for five loading levels. While transformer L20-T1 appears in four levels, but it

is always top ranked. Transformers located at Bus 20 and Bus10 are frequently in the top

ranks.

Table 6-1: Top five transformers for load levels ranking based on ICF

Level 1 Level 2 Level 3 Level 4 Level 5 Level 6

L20-T1 L20-T1 L20-T5 L20-T1 L20-T1 T18

L20-T3 L19-T1 L20-T6 L10-T4 L20-T3 L10-T4

L10-T4 L20-T3 L20-T4 L20-T3 L16-T3 L10-T2

L12-T1 L10-T4 L16-T3 L16-T3 L10-T4 L19-T1

L12-T2 L10-T2 L54-T3 L10-T2 L16-T2 L19-T3

In order to compare the influence of the loading level on the occurrence of multiple failure

events, the maximum, average, and median values of the ICF of all transformers are

calculated for each load level and shown in Table 6-2. It can be seen that there is a non-

linear relationship between maximum ICF values and load level. Though the ICF depends

on the loading of individual transformers, the loading of transformers varies not only with

the demand level but also with the topology of the network and the planned outages. From


139

the results shown in Table 6-2, level 5 has the largest average and median values of ICF

which makes it the most critical level as there are many transformers, which can initiate a

multiple failure event. Although level 6 has the largest maximum ICF, its criticality is

lower than level 5 because the median ICF is significantly smaller. Conversely, level 3 has

the smallest values of maximum, average and median ICF, and therefore it is the least

critical level. The results in Table 6-2 illustrate the advantage of using ICF, and its

dependency on transformer condition (both age and loading) which makes it suitable as a

criticality indicator to identify critical transformers for short term multiple failure

mitigation. By considering the maximum and the median ICF values for each load level,

one can identify a high criticality group of transformers and by prioritising their

maintenance or replacement, reduce the probability of disruptive events in the network.

Table 6-2: Maximum, average, and median values of transformers’ ICF for load levels in (%)


Max 0.083

0.076

0.010

0.024

0.108

0.114

Average 0.005 0.003

0.001 0.003 0.006 0.005

Median

0.0005 0.0004 0.0001 0.0009 0.0014 0.0002

6.4.1.2 Annual load Model

Figure 6-2 shows the location and the ICF values of the 25 most critical transformers for

the annual load model. As can be seen from the ICF values, there are only a few

transformers with a significant ICF. The figure also shows that there is a very small

difference in the, already small, ICF values between the 12th

(L20-T5) and the 25th

transformer (L20-T6), and that there are effectively only 12 or 13 transformers in the

network critical for the initiating a cascading failure. Four out of six of the top ranked

transformers are located at Bus 20 and Bus 10, as in the case of the individual load level

ranking discussed previously.

6.4.2 Transformer VCF

The VCF indicator was calculated for individual transformers for each load level and for

the annual load model. Similar to the ICF, the results of the calculations for either

individual load levels or the annual load model show that VCF can have a value of 0 for

some transformers, which are generally young and operate under low loading level.


140

6.4.2.1 Load Level

The top five transformers for each load level ranked by VCF are shown in Table 6-3.

Generally, the results shown in this table are comparable to results of ICF ranking given in

Table 6-1. Transformers located at Buses 10 and 20 frequently appear in the top five

transformers for individual load levels.

Figure 6-2: The top 25 transformers in ICF ranking for the annual load model

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T2T1

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T9 T11 T12T10

T13 T14

T18 T15

T17 T16

T19 T20

T21 T22

T23 T24

T25 T26

T28T27

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Table 6-3: Top five transformers for load levels ranked based on VCF


L20-T3 L20-T3 L20-T5 L20-T3 L20-T3 L17-T4

L20-T1 L19-T4 L20-T6 L10-T2 L20-T1 L10-T2

L26-T2 L20-T1 L26-T2 L16-T1 L16-T1 L10-T4

L10-T2 L19-T2 L20-T4 L20-T1 L16-T4 L19-T4

L12-T1 L10-T2 L16-T1 L10-T4 L10-T2 L19-T2

The maximum, average, and median values of the VCF of all transformers are calculated

for each load level and shown in Table 6-4 to assess the criticality of load levels. Similar to

ranking of the transformers, the maximum, average, and median values of VCF are

comparable to the ICF values. This demonstrates that the criticality of load level is the

same for ICF and VCF. This is a predictable conclusion since both ICF and VCF

calculations involve mutual effect of transformer outages.

Table 6-4: Maximum, average, and median values of transformers’ VCF for load levels in (%)


Max 0.083 0.075

0.011 0.024

0.108 0.112

Average 0.005

0.003

0.001

0.003

0.006

0.005

Median 0.0006

0.0005 0.0001 0.001

0.0016 0.0002

6.4.2.2 Annual Load Model

The location and the VCF values of the 25 transformers which are most affected by

cascading failure, for the annual load model, are given in Figure 6-3. The general trend of

the VCF ranking is similar to the ICF ranking. There are few transformers (seven) whose

VCF is large while the values in the tail of the ranking (after the 11th

transformer) do not

vary notably.

From the heat maps Figure 6-2 and Figure 6-3 showing the location of the top-ranked

transformers for both ICF and VCF rankings, it is apparent that they are located at the

same buses (Bus 20 and Bus 10). When looking at the age of those transformers, it is found

that L10-T2 and L10-T4 are 58 years old, and L20-T1 and L20-T3 are 53 years old.

Furthermore, the pairs of these transformers are at the same site, so their age and mutual

influence (due to connection in the substation) together contribute to this high ranking.

In order to study the correlation between VCF, age and loading, the loading and age of the

top 25 ranked transformers based on VCF, are shown in Figure 6-4. The loading values


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shown are the annual equivalent loading. This study is necessary for investigating whether

the transformer is vulnerable to consequent failure because it is old or because it operates

under high loading level. It can be seen that although all of the top 25 ranked transformers

are older than 40 years they do not appear in the figure in age order. The figure clearly

shows the transformer loading contribution to the overall ranking. Generally, the younger

transformers are more loaded and the opposite is true for old transformer. The loading

together with the age of transformer; therefore, influences the final VCF ranking. This is

one of the advantages and crucial feature of VCF, it indicates that the criticality of

transformer does not depend on only one obvious factor.

Figure 6-3: The top 25 transformers in VCF ranking for the annual load model

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Figure 6-4: Age and loading for the top 25 transformers in VCF ranking.

6.4.3 Transformer Site Criticality

The fact that the transformers L10-T2, L10-T4, L20-T1 and L20-T3 are found to be the

most critical according to both indicators leads to further study to examine the nature of

this effect and whether it is limited to the local site or whether it spreads across the

network. An average ICF for each transformer site is therefore calculated, to establish the

extent of its influence on the network. This is completed by excluding transformers located

at the same site from (6.9) and then, calculating the average of the ICF of the transformers

located at each site. It was found that 8 (Buses 18, 19, 24, 25, 26, 34, 38 and 53) out of 26

transformer sites do not affect another site, and are not affected by the outage of

transformers located at other sites, i.e., have zero ICF. Referring to Figure 6-2 and Figure

6-3, it can be seen that transformers located at Bus 18 and 19 are among the top 25

transformers based on ICF and VCF ranking. From this further study, it is apparent that the

effect of these two buses/sites is local and that they do not influence the rest of the

network.

Table 6-5 shows the transformer site rank using the average ICF along with the number of

affected sites. The table shows only buses with non-zero ICF. As can be seen, the top two

buses do not have the largest number of affected sites. This is because the ICF rank mainly

reflects the severity of the outage, i.e., the increase in the unavailability value, rather than

the number of affected sites. Table 6-5 also shows that the most critical transformer site is

Bus 10, which reasonably matches the results given in Figure 6-2 and Figure 6-3.

However, Bus 20, which has two transformers that come in the top 5 in the ICF and VCF


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ranking, comes the 9th

in sites ranking. The change in the ranking is attributed to the fact

that the effect of the outage of transformers located at Bus 20 is more severe on

transformers located at the bus itself. The effect of the outage of transformers at Bus 20 on

the other sites is negligible.

Table 6-5: Transformer sites rank using an average ICF

Site ID ICF (%) Number of

affected sites

Site ID ICF (%) Number of

affected sites

Bus 10 0.0033 5 Bus 30 0.0001 6

Bus 17 0.0027 2 Bus 54 0.0001 6

Bus 12 0.0008 11 Bus 32 6E-05 7

Bus 14 0.0008 6 Bus 35 5E-05 5

Bus 16 0.0005 6 Bus 27 4E-05 1

Bus 15 0.0004 5 Bus 23 6E-06 7

Bus 31 0.0002 6 Bus 28 5E-06 2

Bus 36 0.0002 6 Bus 29 3E-07 6

Bus 20 0.0002 7 Bus 13 2E-08 1

In order to identify the area affected by an individual transformer outage, the transformers

are ranked using the ICF values calculated using (6.9) and disregarding the effect on

transformers located at the same bus. Figure 6-5 shows the 25 most critical transformers

for this ICF ranking. The first fact to notice is that the inter-bus transformers are the top-

ranked ones. They connect transmission level buses, therefore, their outage will affect the

loading of other buses. Once more, L10-T4 appears in the 5 most critical transformers.

Ranking transformers using this ICF value gives a clearer idea about the criticality of

transformer to cascading failure events as it indicates the effect on other sites only.

Figure 6-5: The top 25 transformers in ICF ranking without considering the local effect on the transformer

sites


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As T9 is the most critical transformer in this ranking, the sites affected by its outage are

shown in Figure 6-6, excluding the local bus, i.e., Bus 14, the dark red areas are the most

affected buses.

Figure 6-6: Area affected by T9 (marked by X sign in the figure) outage

It is obvious that the outage of T9 will lead to increase of the loading level at Bus 54 as all

the power injection form Bus 6 will go to this bus. The reason for the increase in loading

level of Bus 16 due to T9 outage is that loads at Buses 17 and 18 are fed by T9 and T8,

when T9 is out, loads would be fed by T8 only and hence Bus 16 gets affected. This

identification of the area of vulnerability due to transformer outage is essential for

weighing the strategic criticality of the transformer. For example, if the most critical area

shown in Figure 6-6 (the area around bus 54) does not contain any crucial loads, then the

attention may be shifted to the second most critical transformer on the list, i.e., T18, etc.

6.5 Effect of Load Uncertainty on ICF and VCF

The variability of the demand due to consumer random behaviour results in load

uncertainty. The effect of this uncertainty on ICF and VCF values, and hence the ranking

has to be studied. In order to examine the ICF and VCF ranking robustness with respect to

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load uncertainty, the uncertainty in individual load levels (see 6-step load model in Table

3-5) is modelled by a normal distribution with a mean value () equal to the load level and

a standard deviation () = 0.033×This is equivalent to ±10% variation in for 99.7% of

the values (3=10% of see Appendix A). Accordingly, the 6-step load model is

probabilistically represented by uncertainty areas of individual load levels as shown in

Figure 6-7. Then, 100 random values of loads values were sampled from this uncertainty

area, and ICF and VCF values for individual transformers were calculated for each sample.

Figure 6-7: Introduced uncertainty in the 6-step load model

The top five ranked transformers only were observed to study the uncertainty effect. The

results for ICF and VCF rankings are given in Figure 6-8 and Figure 6-9, respectively.

Both figures show that the 5 most critical transformers are the same transformers as were

shown in Figure 6-2 and Figure 6-3. Although transformers change ranking order

compared to previous ranking this only occurs for adjacent places, e.g., the forth and the

fifth places in ICF ranking (see Figure 6-8) and the third, the forth and the fifth places in

VCF ranking (see Figure 6-9). A closer inspection of Figure 6-2 and Figure 6-3 reveals that

the values of ICF and VCF at these ranks are very similar and so this is why these

transformers swap places. It can be concluded therefore, that the effect of load uncertainty

does not greatly change the base transformer ranking.

Figure 6-8: Frequency of coming in 5 top-ranked transformers based on ICF


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Figure 6-9: Frequency of coming in 5 top-ranked transformers based on VCF

6.6 Summary

This chapter presented a methodology to identify the criticality of power transformers for

multiple failure events. The proposed methodology is based on the assessment of the

probability of a second dependent failure of an aged power transformer due to the thermal

stress. The probability of a second dependent failure is calculated by relating the

transformer unavailability to its age and loading condition, which is accomplished by

employing Arrhenius-Weibull distribution. The criticality of transformers is then

quantified by introducing two newly defined probabilistic indicators, an indicator of

initiating a cascading failure (ICF) and an indicator of vulnerability to a consequent failure

(VCF). The proposed methodology, for the first time, integrates simultaneously the age

and the loading level of transformers into multiple and cascading failure studies and

facilitates the identification of the most critical transformers for cascading failure events

and hence represents another original contribution of this thesis.

The probabilistic indicators, ICF and VCF, provide a valuable insight, not possible before,

into understanding the probabilities and the consequences of a second dependent failure. It

is also demonstrated, by the use of these indicators, that load uncertainty does not have a

significant effect on the relative criticality ranking of the transformers.

The proposed indicators can be used for ranking transformers based on their significance

for, and vulnerability to, cascading failure and such represent a useful tool for short and

long term power system asset replacement planning.


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Chapter 7 Quantification of Uncertainty in Reliability Assessment

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7

Quantification of Uncertainty in

Reliability Assessment

7.1 Introduction

The uncertainty in power system reliability assessment is inevitable. Component failure

events are random and hence they are the main source of aleatory uncertainty. The aleatory

uncertainty is neither irreducible nor controllable. The common method of modelling this

form of uncertainty is probabilistic distribution function. In power system reliability, the

aleatory uncertainty has been quantified using the sampling approach in the Monte Carlo

simulation methods. The Sequential and Non-sequential Monte Carlo techniques have been

used to propagate the aleatory uncertainty and build probabilistic distribution for the

reliability indices [24, 75, 83].

System reliability assessment also involves epistemic uncertainty in parameters estimation

due to data shortage or model simplifications. The representation of the epistemic

uncertainty depends on the available amount of data. For example, when there is enough

data, pdfs can be used to model the epistemic uncertainty. Another example is Fuzzy set

theory which is used when the available data is fuzzy, e.g., weather condition described in

words like normal or adverse [30]. Evidence theory is another way of modelling and


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quantifying epistemic uncertainty in which different sources of data can be combined

together.

This chapter presents the thesis contribution in the area of uncertainty quantification in

reliability assessment. The chapter focuses on the uncertainty associated with transformers

failure models: repairable and end-of-life failure. It contains three different uncertainty

studies. The first study involves the assessment of the effect of epistemic uncertainty in

end-of-life failure models on the system reliability. In this study, the uncertainty of

transformer unavailability values was derived from the uncertainty of parameters of

transformer end-of-life failure models. By quantifying the effect of uncertainty, the study

established bases for a “system related approach” to refine transformer failure models,

which is the fifth contribution of this thesis. The second study focuses on the quantification

of both forms of uncertainty: aleatory and epistemic. The study employed two

quantification methods: second order probability and evidence theory. The quantification

of both forms of uncertainty in one framework is done for the first time in this thesis. The

last study presented in this chapter is an application of uncertainty studies. Sampling based

sensitivity analysis (also known as probabilistic sensitivity analysis) was used to identify

the most critical components to system reliability. The identification of the most critical

components using probabilistic sensitivity analysis is a new study in the area of system

reliability importance measure.

7.2 Failure Model Uncertainty

The failure models of power system components involve two forms of uncertainty. The

aleatory uncertainty due to the stochastic nature of failure is characterised by pdfs. For

examples, the repairable failure is commonly described by exponential distribution and

end-of-life failure is described by Gaussian and Weibull distributions. The parameters of

these distributions are estimated from historical statistics, and the mean value of

parameters is used for the whole population [26]. This estimation leads to epistemic

uncertainty in these parameters. The epistemic uncertainty is knowledge based, and can be

reduced by collecting more data or by a refining statistical process. By combining these

two forms, the failure models uncertainty can be characterised by a mixed aleatory-

epistemic uncertainty model, which is represented by a group of pdfs describes the

reliability of a component rather than just one pdf. Figure 7-1 is an illustrative example of


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mixed aleatory-epistemic uncertainty model for repairable failure and end-of-life failure.

The random variable in repairable failure is Time to Failure (TTF), whereas it is the life-

time for end-of-life failure.

a) Repairable failure (TTF is the random variable) b) End-of-life failure (life-time is the random variable)

Figure 7-1: Illustrative example of mixed aleatory-epistemic uncertainty model for repairable and end-of-life

failure

7.3 Epistemic Uncertainty in End-of-life Failure

Throughout this thesis, two end-of-life failure models were used: Gaussian and Arrhenius-

Weibull distributions. The studies presented in previous chapter employed deterministic

values of the parameters of these two distributions. This study is set to assess the effect of

uncertainty in the parameters of these two models, and how it influences the identification

of the most critical transformer sites.

7.3.1 Gaussian Distribution

In order to study the uncertainty in Gaussian distribution it is assumed that the parameters

and vary within ±10% of the given values (=65 and =15 years) following uniform

distribution. The corresponding uncertainty in transformer unavailability is calculated from

100 random samples (for illustrative purposes only, though higher number of samples can

be used) of and within the introduced variation. The histograms in Figure 7-2 represent

the distribution of the unavailability of transformers L30-T1 (47 years old) and L24-T1 (6

years old) obtained using the random samples. Despite the fact that the uncertainty in the

parameters was assumed to be uniformly distributed, the unavailability histograms are left

skewed. The unavailability values calculated using the deterministic parameters (Ud) for

L30-T1 and L24-T1 are 0.0025 and 6. 35×10-6

, respectively, which fall in the peaks of the


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two histograms. Comparing the ranges of unavailability values, it is apparent that the range

becomes larger when the transformer is older. The area of uncertainty of the unavailability

of the transformers at different age in the test system (1-58 years) is shown in Figure 7-3.

This figure shows that the uncertainty in Gaussian distribution parameters has minor effect

on the unavailability of the transformers when they are younger than 20 years.

Figure 7-3: Uncertainty in unavailability of transformer age range (1-58 years) corresponding to ±10%

variation in the Gaussian distribution parameters.

Figure 7-2: Examples of the histogram of the unavailability for the transformers considering ±10% variation

in Gaussian distribution parameters


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System reliability is assessed by randomly sampling the unavailability of individual

transformers from the shaded area shown in Figure 7-3. This random sampling process

may result in having transformers with similar age, but not necessarily having the same

unavailability value. A comparison between ENS obtained using deterministic parameters

and ENS obtained considering the uncertainty is given in Table 7-1. The same buses were

identified as critical in the two studies, but with a different order of importance. In general,

the ENS values and the ranking of the buses did not change significantly, except for Buses

12 and 20, which experienced a decrease in ENS by 67% and 27%, respectively. The

ranking of these two buses, however, was not significantly affected. Bus 12 was the 5th

in

the deterministic study, whereas it came 7th

when considering uncertainty. Bus 20 was the

1st in the deterministic study while it came 2

nd in this study.

Table 7-1: ENS and the ranking of load points obtained using deterministic and uncertain parameters of

Gaussian distribution

ENS [MWh/year] Ranking Change in

ENS Deterministic Uncertain Deterministic Uncertain

Bus 20 61.0 43.8 1st 2

nd decrease

Bus 18 51.1 56.6 2nd

1st increase

Bus 19 41.7 31.4 3rd

3rd

decrease

Bus 26 31.1 22.6 4th

5th

decrease

Bus 12 29.2 9.7 5th

7th

decrease

Bus 16 27.1 27.1 6th

4th

no change

Bus 28 10.8 10.8 7th

6th

no change

Bus 17 6.6 6.6 8th

8th

no change

Bus 27 5.2 5.2 9th

9th

no change

System 263.8 213.8 decrease

The heat maps shown in Figure 7-4 illustrate the critical transformer sites identified using

deterministic and uncertain parameters of Gaussian distribution. It can be seen that the

critical areas identified by the heat maps are similar. The only difference is the level of

criticality for some of the transformer sites (Bus 20 and Bus 12). For example, Bus 20 was

in the dark red area (high ENS value) when using deterministic parameters, while it is in

the orange area when using uncertain parameters.


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a) Deterministic parameters b) Uncertain parameters

Figure 7-4: Critical transformer sites based on ENS obtained using deterministic and uncertain parameters of

Gaussian distribution

These results of this study suggest that when using a probability distribution function to

characterise the lifetime of power transformers for the identification of the most critical

transformer sites, the exact values of the probability distribution parameters are not

important, and a reasonably rough estimate can be used instead (for the test system, ±10%

is still acceptable). It should be pointed out though this conclusion should not be

generalised to other systems, at least not in terms of the level of uncertainty that can be

tolerated. Individual systems with a different network configuration, a different

transformer age distribution, and a different probability distribution failure model may

show different sensitivity of results to model parameter uncertainty. The method presented,

however, can be applied to any system to assess the effect of uncertainty and determine the

acceptable level of uncertainty in model parameters.


Assuming that the shape of Weibull distribution does not change with uncertainty, the

uncertainty of parameters A and B of Arrhenius-Weibull distribution (see (5.4)) was

considered. It was assumed that these parameters vary within ±10% of the given values

(A=0.56 and B=1500) following uniform distribution. By projecting this uncertainty on the

characteristic life (), the introduced uncertainty is equivalent to 37% decrease and 80%

increase of the deterministic value (=70.79 years). That is to say, varies in the range

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from 47 to 130 years. Figure 7-5 shows the projection of the introduced uncertainty on the

characteristic life and the relationship with transformer HST. The same assumption that the

average annual HST (HSTa=36.38ºC) is corresponding to the range of uncertain values of

is used in projecting the uncertainty in A and B. As explained in Chapter 5, the

Arrhenius-Weibull distribution parameters were estimated based on the Gaussian

distribution and historical loading data. In addition, some other necessary assumptions

were made in the derivation of the parameters as discussed in Chapter 5. Therefore, the

fact that the resulting range of uncertainty in the Arrhenius-Weibull distribution is bigger

than Gaussian distribution range is an acceptable assumption.

Figure 7-5: Projection of the uncertainty on characteristic life and the relationship with transformer HST

Following the same procedure as for Gaussian distribution, the uncertainty in transformer

unavailability is calculated from 100 random samples of A and B from the introduced

specified range. The distributions of the unavailability of transformers L30-T1, L26-T2

and L24-T1 obtained using the random samples under loading level 1 (the peak load level)

are given in Figure 7-6.

The area of uncertainty in the unavailability of the age range of the transformers in the test

system (1-58 years) is shown in Figure 7-7. Figure 7-7 illustrates the effect of the

uncertainty in these parameters on the unavailability of transformers aged 1-58 years

calculated for the average HST (36.37ºC), i.e., the same HST as for the Gaussian

distribution. This figure shows the increase of the area of uncertainty with transformer age.


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Similar to Figure 7-3, there is no effect of uncertainty in Arrhenius-Weibull distribution

parameters for transformers younger than 20 years. Figure 7-8 shows the area of

uncertainty in unavailability of a 40 year old transformer when it operates in the loading

range from 1 to 140%. It is apparent that the area of uncertainty increases at loading levels

above 40% loading.

Figure 7-6: Examples of the histogram of the unavailability for the transformers considering ±10% variation

in Arrhenius-Weibull distribution parameters


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Figure 7-7: Uncertainty in unavailability of the age range (1-58 years) corresponding to ±10% variation in the

Arrhenius-Weibull distribution parameters for HST=36.37ºC

Figure 7-8: Uncertainty in unavailability for a 40 year old transformer for load range (1-140%) corresponding

to ±10% variation in the Arrhenius-Weibull distribution

System reliability indices for the test system were also calculated using random samples of

the transformer unavailability considering the uncertainty in Arrhenius-Weibull

parameters. The indices and the ranking of the buses for deterministic and uncertain

parameters are given in Table 7-2. Although the uncertainty area is much wider than the

area in Gaussian distribution, the reliability results have not changed much. The top

transformer sites given in Table 7-2 were indicated critical in previous studies as well. The

top four buses (Bus 20, 12, 26, 18) from the two studies based on the Arrhenius-Weibull

model are in the same order except for Bus 20. Bus 20 experienced the largest change in

ENS as it decreased from 91.9 to 30.8 MWh/year. The ENS of Bus 12, 26 and 18 obtained

with uncertain parameters are comparable to the ENS obtained using deterministic

parameters. Since PLC index is directly related to the unavailability of transformers, these

results can be explained by observing the load points’ PLC values of the two case studies

(with deterministic and uncertain parameters). The values of PLC for load points of the test

system are given in Table 7-3. It can be seen that the PLC for Bus 20 is extremely high


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when using deterministic parameters. This means that Bus 20 is more sensitive to the

change in the unavailability values of transformers than the other buses. Buses 23, 36, 27,

and 17 were highlighted as critical when considering the uncertainty, but their ENS is

notably low. Out of 4 case studies (2 studies using Gaussian distribution and 2 studies

using Arrhenius-Weibull distribution) Bus 23 and Bus 36 have only been highlighted as

critical in this case study.

Table 7-2: ENS and the ranking of load points obtained using deterministic and uncertain parameters of

Arrhenius-Weibull distribution

ENS [MWh/year] Ranking Change in

ENS Deterministic Uncertain Deterministic Uncertain

Bus 20 91.9 30.8 1st 4

th decrease

Bus 12 87.5 76.3 2nd

1st decrease

Bus 26 68.8 67.0 3rd

2nd

decrease

Bus 18 54.1 58.4 4th

3rd

increase

Bus 19 31.4 10.2 5th

6th

decrease

Bus 16 27.1 15.1 6th

5th

decrease

Bus 28 0.1 0 7th

N/A decrease

Bus 23 0 3.8 N/A 7th

increase

Bus 36 0 1.2 N/A 8th

increase

Bus 27 0 1 N/A 9th

increase

Bus 17 0 0.3 N/A 10th increase

System 360.8 264.2 decrease

Table 7-3: PLC for load points obtained using deterministic and uncertainty parameters of Arrhenius-Weibull

distribution

PLC [%]

Deterministic Uncertain

Bus 20 0.436 0.004

Bus 12 0.016 0.008

Bus 19 0.010 0.004

Bus 26 0.008 0.022

Bus 18 0.012 0.039

Bus 16 0.005 0.002

Bus 28 0.001 0

Bus 17 0 0.009

Bus 27 0 0.002

Bus 36 0 0.002

Bus 23 0 0.001


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The heat maps shown in Figure 7-9 illustrate the critical transformer sites identified using

deterministic and uncertain parameters of Arrhenius-Weibull distribution. It can be seen

that the most critical transformers sites were the same for both studies, deterministic and

uncertain parameters studies. The difference is in the ranking only. As can be seen, since

Bus 20 experienced the highest change, this resulted in changing it is criticality level from

the dark red (around 90 MWh/year) to green (around 36 MWh/year).

Figure 7-9: Critical transformer sites based on ENS obtained using deterministic and uncertain parameters of

Arrhenius-Weibull distribution

By comparing results given in Table 7-1 and Table 7-2, it can be seen that there are

common buses identified as critical by all studies: Bus 20, 12, 16, 18, 19 and 26. The order

of importance is different, though. The buses, which are not common, i.e., Bus 28, 17, 27,

23, 36, are all at the bottom of ranking lists with remarkably low ENS. Although this study

was carried out using a particular test system, the results suggest that using different end-

of-life failure models with roughly estimated parameters identifies the same critical buses,

i.e., the value of model parameters are not critical for these types of studies. The only issue

is the actual ranking of the important buses. For system reliability studies that require

accurate values of reliability indices, e.g., optimisation application, more accurate model

parameters are required and these can be obtained by focussing modelling and monitoring

effort (i.e., financial and human resources) to identified critical sites rather than spreading

them across all sites in the network. This study established a system based approach for

a) Deterministic parameters b) Uncertain parameters

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refining failure models of transformers. In this approach, the sensitivity of system

reliability to parameters of transformer failure model is assessed first to determine if

refining the failure model is necessary or not. This is completed by assessing the effect of

uncertainty associated with the transformer failure model on the system reliability

application, i.e., the purpose of performing the analysis. In this case study reliability

analysis was carried out to identify the most critical transformer sites, but other types of

reliability studies can be performed in the same way.

7.4 Quantification of Aleatory and Epistemic Uncertainty

This study explores two methods for representing and quantifying the mixed aleatory-

epistemic uncertainty. The first method is second order probability (SOP) and the second

one is evidence theory. Case studies were carried out on the IEEE-RTS to demonstrate the

feasibility of the methods. The quantification of failure model uncertainty characterised by

mixed aleatory-epistemic uncertainty in reliability assessment represents an advancement

of the current state-of-the-art of uncertainty representation and quantification [25, 76-81,

83, 84]. Consequently, deeper understanding of the nature of uncertainty in system

reliability assessment is gained.

7.4.1 Second Order Probability Method

The propagation of the mixed aleatory-epistemic uncertainty using second order

probability (SOP) method is accomplished by establishing two nested sampling processes

[73]. The SOP method, after all, was named like this because of these two nested loops

[73, 74]. The first sampling loop associated with the epistemic uncertainty. The sampled

values are the parameters of distribution functions, which characterise the aleatory

uncertainty. In the second loop, for each sampled value in the previous loop, another

sampling procedure is formulated to propagate the aleatory uncertainty. The propagation of

mixed aleatory-epistemic uncertainty to the reliability indices results in obtaining a group

of probabilistic distribution functions for every reliability index. The commonly used

probabilistic function in the uncertainty propagation is the cdf. Figure 7-10 illustrates the

concept of the nested sampling loops. In the figure, the epistemic uncertainty in the failure

rate () of repairable failure is characterised by a normal distribution pdf(). Each

generated value from this pdf has its associated exponential pdf(TTF), which describes the


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aleatory uncertainty in Time to Failure. The propagation of each exponential distribution

formulates a distribution function for each reliability index. Due to the epistemic

uncertainty there are more than one cdf. In reliability assessment the inner loop is

simulated using the standard Monte Carlo iterations. The output of the SOP method, i.e.,

the group of cdfs, is called the horsetail plot. The importance of the horsetail plot comes

from the fact that it clearly distinguishes between the aleatory and epistemic uncertainty. It

indicates which form of uncertainty is the dominant one. For example, if the horsetail plot

has a small acute angle with the x axis (see Figure 7-11(a)), this means the aleatory

uncertainty is dominant. If the angle of horsetail plot with x axis is close to 90˚, the

epistemic uncertainty is dominant (see Figure 7-11(b)). Furthermore, the horsetail plot

assists in capturing the upper and lower bounds of the cdfs. The SOP is the most

appropriate method when there is enough data to model the epistemic uncertainty by a

probability distribution function. However, in practice, due to a lack of information, an

arbitrary distribution function is used to represent the epistemic uncertainty, commonly the

uniform distribution. p

df

()

pd

f(T

TF

)

Time

cd

f (I

nd

ex)

Figure 7-10: Use of nested sampling process to propagate the mixed aleatory-epistemic uncertainty to power

system reliability indices.


162

Figure 7-11: Distinguishing between aleatory and epistemic uncertainty forms using horsetail plot

7.4.1.1 Case Study

This case study examines the propagation of the failure rate uncertainty of power

transformers to the transmission systems reliability indices. Uncertainty in other

components outage model is not considered. The test network used is the IEEE-RTS [99].

As shown in Figure 7-12, the system has five power transformers that have the same

failure rate of repairable failure.

Bus 1

Bus 3

Bus 19 Bus 20

Bus 22Bus 21

Bus 18

Bus 17

Bus 23

Bus 14

Bus 15

Bus 24Bus 12

Bus 11

Bus 13

Bus 10Bus 9

Bus 6

Bus 8

Bus 5Bus 4

Bus 7Bus 2

Bus 16

Synch.

Cond.


163

Figure 7-12: The single line diagram of IEEE-RTS

In order to introduce epistemic uncertainty, it is assumed that the failure rate varies within

±20% of the given value (0.02 1/year). Accordingly, a uniform distribution within the

range [0.016, 0.024] is used to characterise this epistemic uncertainty. The SOP method is

applied to propagate this uncertainty to PLC index. Figure 7-13 shows the cdfs horsetail

plot obtained by performing 50 samples in the outer loop and 10,000 Monte Carlo

iterations in the inner loop. The horsetail plot shows that the epistemic uncertainty has

greater effect on the PLC index because all but a few of the cdfs, which represent the

aleatory uncertainty, are steep and do not spread over a wide range of values. By

examining Figure 7-13, the cdf which forms the lower bound seems to be an outlier.

Hence, by excluding it, more accurate lower bound can be obtained.

0 0.005 0.01 0.015 0.02 0.0250

0.2

0.4

0.6

0.8

1

CD

F(P

LC

)

Probability of Load Curtailment Index (PLC)

Figure 7-13: cdfs of PLC index generated using SOP method and considering the mixed aleatory-epistemic

uncertainty in transformer failure rate

In power system reliability studies, for most practical purposes only one distribution

function is needed. Therefore, the horsetail cdfs have to be consolidated into one. This is

can be achieved using two methods. The first method is to calculate the mean value of PLC

index ( ) for each cdf, then obtain the cumulative distribution function of those mean

values (cdf ( )). The second method is to obtain the most probable cdf. This is

completed by considering the sampled values of the all cdfs as one set of data and

obtaining the cumulative distribution function of this set. Figure 7-14 shows the plot of cdf

( ) and the most probable cdf along with upper and lower bounds of the horsetail plot.

Both cdf ( ) and the most probable cdf are between the bounds of the horsetail cdfs as

expected. The two methods produced overlapped cdfs. Therefore, either method can be

used for the consolidation of the horsetail plot. In this study, the cdf ( ) was selected to


164

represent the consolidated horsetail plots and it will be referred to as aleatory-epistemic cdf

in the rest of this chapter. cd

f(P

LC

)


Lower bound

Lower bound excluding outlier

Upper

bound

Most Probable cdf cdf (PLC)

Figure 7-14: The cdf of PLC index mean values ( ), Most probable cdf and horsetail cdfs bounds

In order to compare the introduced aleatory-epistemic model represented by cdf ( ) with

the conventional aleatory model, a cdf is produced by using the average value of the failure

rate of power transformers (0.02 1/year). This cdf represents the effect of the conventional

aleatory uncertainty. Both cdfs are shown in Figure 7-15. It can be seen in Figure 7-15 is

that aleatory-epistemic cdf covers a wider range of values than the conventional aleatory

cdf. This means that the conventional model underestimates the uncertainty in the outage

model. A detailed comparison between the statistical parameters of the two models is given

in Table 7-4.

cdf(

PL

C)


Aleatory-epistemic

Aleatory only

Median = 0.0049

Median = 0.0066

Figure 7-15: The cdfs of PLC index derived using the typical aleatory uncertainty model and the mixed

aleatory-epistemic uncertainty model


165

Table 7-4: A comparison between aleatory model and mixed aleatory-epistemic model

Model Mean

()

Standard

Deviation ()

3

(-) (+)

Aleatory only 0.0047 0.0008 0.0023 0.0071

Aleatory-epistemic 0.0069 0.0021 0.0006 0.0132

7.4.2 Evidence Theory method

Evidence theory method, also known as Dempster-Shafer evidence theory (DSET), is an

alternative method of the probability theory to represent and quantify the epistemic

uncertainty. In the probability theory, the lack of information leads to a subjective

probability distribution for representing the uncertainty. To overcome this subjectivity,

evidence theory represents the uncertainty in each input by intervals or sets with their

associated degree of belief [154-156]. Those intervals can be overlapping, adjacent or have

gaps. The degree of belief for individual intervals is indicated by what is known as basic

probability assignment (BPA). There are two main rules that control the derivation of

BPAs of intervals. The first rule is that the BPA of each interval has to be greater than zero

(BPA>0). The second rule is that the summation of BPAs of intervals, which describe an

uncertain variable, equals one (∑ The intervals and their BPAs are derived from

different information sources such as a statistical process, an expert elicitation, and

experimental data. Actually, the capability of combining those different sources is the key

advantage of evidence theory. This feature is the main reason to use evidence theory in this

study. Besides, some power system researchers have already utilised it to combine the

available evidence from different condition monitoring results to predict transformer

failure [157-159].

To quantify the epistemic uncertainty using evidence theory, the BPAs are used as inputs

to construct a belief structure of the uncertainty in the output. This belief structure is

measured by two functions: Belief (Bel) and Plausibility (Pl), which are equivalent to the

lower and upper bounds of the cdf, respectively. The difference between the Bel and Pl and

lower and upper bounds is that the Bel and Pl are consistent with the evidence and do not

involve any subjective assumptions. The steps for the propagation of epistemic uncertainty

to the output can be summarised as follows [154, 155]:

Enumerate all the possible combinations of the intervals which describe the uncertain

inputs. The total number of combinations equals the multiplication of the number of all

intervals.


166

Calculate the BPA of each combination using (7.1):

)(VBPABPASV

C

(7.1)

where V is an interval in the combination and S is a set which contains all the intervals of

that combination.

Sample many values from each combination and propagate through the system to

obtain the minimum and maximum output responses.

Accumulate all the minimum and maximum responses of each combination along with

its associated BPAC to construct a belief structure of the output uncertainty. This belief

structure consists of the cumulative belief function (CBF) and the cumulative

plausibility function (CPF).

To propagate the mixed aleatory-epistemic uncertainty using evidence theory, two loops,

that are similar to the two loops in the SOP method, are established. The outer loop

propagates the epistemic uncertainty and the inner loop propagates the aleatory one.

However, in the evidence theory method, the aleatory uncertainty is not represented by full

probability distribution. Instead, statistics such as mean, variance and standard deviation

are used to act as the aleatory uncertainty quantity. For epistemic uncertainty

quantification, the minimum and maximum responses of these statistics are calculated, and

the belief structure is built form these responses [154, 155].

7.4.2.1 Case Studies

To demonstrate the capability of evidence theory, two case studies were conducted. The

first case study uses the same uncertainty assumption used in SOP case study. That is to

say, each uncertain failure rate is represented by only one interval. This case study is

established to compare the two methods; SOP and evidence theory. In the second case

study, more modifications were added to the uncertainty assumption in order to test the

ability of evidence theory to handle different information sources.

7.4.2.2 Case Study I

The main assumption in this case study is that the transformer failure rate uncertainty is

represented by the interval [0.016 – 0.024]. The BPA of this interval equals 1 because there

are no other intervals. The total number of combinations is 1 (for five transformers

1×1×1×1×1). Consequently, the belief structure of the statistics of a reliability index is an


167

interval showing the minimum and maximum responses. In this case study, the same

number of samples, which is used with SOP (50 samples in the outer loop and 10,000 in

the inner loop), is also used. The uncertainty propagation to PLC is observed by recording

the mean value ( ) for each set generated by the inner loop. The total number of the

recorded mean values is equal to number of iterations in the outer loop. Then, the

minimum and maximum values among these 50 values are selected to obtain the Bel and

Pl interval. Table 7-5 shows the values of the belief interval. Figure 7-16 shows a

comparison between the belief structure obtained by evidence theory and the cdf obtained

from SOP. It can be seen that the cdf obtained from SOP falls within the belief structure

boundaries as expected since the belief structure is equivalent to upper and lower bounds.

Table 7-5: Case Study I: Belief structure of

Reliability Index Pl Bel

0.0011 0.0239

cd

f(P

LC

)


CBF(PLC)

CPF(PLC)

cdf(PLC)

Figure 7-16: The belief structure of obtained by evidence theory and the cdf obtained by SOP

7.4.2.3 Case Study II

In this case study, it is assumed that there are some additional sources of failure rate

estimation, which allows the introduction of another interval for each transformer failure

rate besides the main interval. The assumptions for each transformer failure rate and the

BPA are given in Table 7-6. As each transformer has two intervals for characterising the


168

uncertainty in failure rate and there are five transformers, the total number of the possible

combinations is 32 combinations (2×2×2×2×2).

Table 7-6: Assumed epistemic uncertainty in transformer failure rate used for case study II

Transformer Interval [BPA, min – max]

Tx 3-24 [0.75,0.02 - 0.024] [0.25,0.016 - 0.024]

Tx 9-12 [0.75,0.02 - 0.024] [0.25,0.016 - 0.024]

Tx 10-11 [0.75,0.02 - 0.024] [0.25,0.016 - 0.024]

Tx 10-12 [0.25,0.016 - 0.02] [0.75,0.016 - 0.024]

Tx 9-11 [0.25,0.016 - 0.02] [0.75,0.016 - 0.024]

For each combination, 25 samples were taken for propagating the epistemic uncertainty

and 10,000 iterations were performed for the aleatory uncertainty in the inner loop. The

mean value of PLC index is recorded at each sample of the outer loop (25 values). Then,

the minimum and maximum values among these 25 mean values are selected. The number

of pairs of minimum and maximum values (Bel and Pl) is equal to the number of

combinations, which are 32 pairs. The BPA of each pair is calculated using (7.1). By

accumulating the Bel and Pl values with the BPA values, the cumulative belief function

(CBF) and the cumulative plausibility function (CPF) are obtained.

CPF(PLC)

cdf(PLC)

CBF(PLC)

PLC: Mean value of PLC Index

CB

F(P

LC

) o

r cd

f(P

LC

) o

r C

PF

(PL

C)

Case study I

Case study II

Figure 7-17: Belief structure of obtained in Case study I and Case study II.

The belief structures obtained in case study I and II are shown in Figure 7-17 with the cdf

( ) obtained by applying SOP method. Figure 7-17 illustrates two points. Firstly, the

cdf( ) falls within the belief boundaries in both cases. This shows how the belief


169

structure obtained in evidence theory is related to the upper and lower pounds of horsetail

plot. Secondly, When comparing the belief structures of case study I and II, it is obvious

that introducing the additional intervals in case study II resulted in reduction in the area

between CBF( ) and CPL( ), which means reducing uncertainty. Therefore;

evidence theory shows more capabilities in quantification of uncertainty in reliability

assessment. The only limitation with evidence theory is that engineers are not familiar

with the presentation of the results compared with the SOP method which has been applied

in many engineering areas and its results can be easily understood by wider engineering

community. In addition, some difficulties may arise when trying to interpret and use

evidence theory results in decision making process.

7.5 Uncertainty Based Importance Indicator

The main aim of this study is to identify the most important components in power systems

and providing a ranking list showing the effect of the uncertainty in components failure

rates on system reliability indices. This list can be used to refine the outage models

parameters of the components having major influence on system reliability in order to

reduce the uncertainty. As explained in Chapter 4, the existing reliability importance

measures are calculated by changing the failure rate of components one by one and

evaluating the corresponding change in system reliability indices. This kind of sensitivity

analysis under-estimates the effect of uncertainty of failure rates because it measures the

influence of one component at a time [160]. For nonlinear systems like power system

reliability this may not reflect the correlation between the components failure. The study

has adopted probabilistic sensitivity analysis (also known as sampling based sensitivity

analysis) in order to rank power system components and produce a reliability map that

describes the most important components and their area of vulnerability. In this study, all

types of equipment were considered including transformers, lines and cables. DIgSILENT

reliability calculation function was used, which considers only repairable failure [161].

7.5.1 Probabilistic Sensitivity Analysis

Sampling based sensitivity analysis is one approach developed to determine the

contributions of individual uncertain input parameters to the uncertainty of the system


170

output. A detailed description of the technique is given in [162]. The general steps of the

approach are:

1. Assign a probability density function for each input under consideration.

2. Generate sampled values for the inputs.

3. Evaluate the output results using the values in the previous step.

4. Analyse the effect of each input on the output.

There are many different ways to analyse and measure the effect of each input on the

output. Some of these are qualitative, for example scatter plots, and some of them give

quantitative measures such as correlation coefficients, and standardized regression

coefficients [162].

7.5.2 Case study

This technique has been applied to the IEEE reliability test system (IEEE-RTS). The system

and load point reliability indices were calculated using the in-built reliability function. For

assigning a pdf for each component’s failure rate, a normal distribution with a standard

deviation that enables 99.7% of the data to be within a 20% increase or decrease of the

mean (3 = 0.2mean) was used. Figure 7-18 shows the pdf of the failure rate of

transformers. All transformers in IEEE-RTS have the same average failure rate ( = 0.02

outage/year).

Figure 7-18: The probability distribution function of transformers’ failure rate.

The matric of component’s failure rate was generated using random sampling form the

assumed pdfs. Then, using DIgSILENT PowerFactory reliability calculation function, the

ENS for the system and load buses was evaluated. To measure the sensitivity of the ENS to


171

the failure rates of the components (λ), the correlation coefficient (CC) for each component

was calculated using (7.2):

N

i

N

i

iij

N

i

iij

jj

ENSENS

ENSENS

ENSCC2

122

12

])([])([

))((

),(

(7.2)

where

N

N

i

ij

(7.3)

N

ENS

ENS

N

i

i

(7.4)

where CCj is the correlation coefficient for component i and N is the number of samples.

The CC measures the linear relationship between λ and ENS and has a value between -1

and 1. The positive sign indicates a positive slope and a negative sign indicates a negative

slope in the linear relationship. The absolute value (0–1) shows the strength of the linear

relationship, with 0 representing no linear relationship and 1 representing an exact linear

relationship [162]. Using the CC, the power system components were ranked based on the

idea that the components that have the largest correlation coefficient are the most important

ones.

7.5.3 Application to System Indices

To apply the method, one thousand sample values were generated randomly from each

individual pdfs of the component failure rates. In the sampling process, the lines on a

common tower were assigned the same sample value. Then, ENS for the system was

calculated using these random combinations of failure rate values. The histogram and fitted

normal distribution of system ENS is shown in Figure 7-19. The figure shows that the ENS

calculated from normal probability distribution function of failure rates has a normal

distribution with 3σ equals 4.5% of the mean.


172

Figure 7-19: The ENS histogram with fitted normal distribution calculated from 1000 random values of

components failure rate

The results of the technique have been analysed by calculating the correlation coefficients

between individual components and ENS of the system. Then using the correlation

coefficients as ranking index, the components were ranked based on their effect on system

ENS. A graphical presentation of this ranking is given in Figure 7-20. Using the correlation

coefficient, the components can be grouped into levels and then each group can be studied

differently according to the application of ranking, e.g., refining the failure rate.

Figure 7-20: IEEE-RTS components ranked using correlation coefficient between failure rate and system’s

ENS

Figure 7-20 also shows that transformer Tx3-24, which is connected to Bus 3 and Bus 24,

has the largest correlation coefficient (CC) equal to 0.48. It is obvious that there is no

component which has an exact linear relationship with system ENS (CC=1). Line 3-9,

which connects bus 3 to bus 9, has the smallest correlation coefficient equal to -0.003. The

negative sign which represents falling or inverse relationship with the system reliability

described by ENS is not a reasonable relationship. However, the absolute value is very


173

small and for a 0.95 confidence interval for this correlation coefficient, the upper limit was

(0.059) and the lower limit was (-0.064). The conclusion is that Line 3-9 has no major

effect on the system ENS. Figure 7-21 shows the scatter plot of Line 3-9 and Tx3-24

failure rate and the system ENS. The scatter plot reveals that the correlation coefficient is a

suitable measure of the relationship between the failure rate and ENS as the values speared

in linear trend and there is no other nonlinear trend of the scatter. Figure 7-21 also shows

that there are some outliers or extreme values that might be the reason for the negative

correlation coefficient between Line 3-9 and the system reliability as the correlation

coefficient is sensitive to these values.

a) Line 3-9 b) Tx 3-24

Figure 7-21: Scatter plots of Line 3-9 and transformer Tx 3-24 failure rates and system’s ENS.

7.5.4 Application to Load Points Indices

When the reliability evaluation was applied to calculate the load points’ reliability index,

Bus 1, Bus 2, Bus 3, Bus 13, and Bus 18 had zero ENS (MWh/year) because they are

connected to a generation that is greater than their local demand. Furthermore, Bus 9 and

Bus 10 are connected to power sources with a redundant transmission system which leads

to zero ENS (MWh/year) for both of them. This result occurred because the generation was

assumed to be perfect, as mentioned earlier. Analysing the sensitivity of the ENS of load

buses to power system components’ failure rate is not only useful for defining which

components are the most important ones for each bus but it can also give guidance on

Ranking the components taking into account their relationship with load buses.

Defining the component’s area of vulnerability.

Ranking the load buses based on vulnerability to components failure.


174

7.5.4.1 Ranking of the components

The importance of components for load buses appears in the case of load priority or when

regulatory standards are set on load supply points. In these two cases ranking the

components using the load points’ indices is necessary. Table 7-7 shows examples of the

correlation coefficients between some components and load points. The table shows that

some of the load points have negative correlation coefficients with some components.

However, since the absolute values of correlation coefficient in these cases were very small

compared to the positive ones, it was considered that there is no linear relationship

between these components and load points. Furthermore, the scatter plots reveal many

extreme values that may affect the correlation coefficient. Figure 7-22 shows the scatter

plot of Bus 4’s ENS and Line 15-24 failure rate as these two have the largest negative

correlation coefficient, and it is obvious that there are some outliers and Line 15-24 has no

effect on ENS of Bus 4.

Table 7-7: IEEE-RTS circuits correlation coefficient with load buses

Name Bus 3 Bus 4 Bus 5 Bus 6 Bus 8 Bus 14 Bus 15 Bus 19 Bus 20

Cable 1-2 0.00 -0.02 0.03 0.08 -0.01 -0.06 -0.03 0.02 0.03

Line 1-3 0.02 0.04 -0.04 0.10 0.10 0.00 0.00 -0.02 0.05

Line 1-5 0.00 0.52 0.11 0.00 0.03 0.04 -0.03 0.01 0.01

Line 2-4 -0.06 0.68 0.39 0.00 -0.02 -0.04 0.00 0.03 0.00

Line 2-6 0.01 -0.01 0.00 0.17 0.01 -0.01 -0.02 -0.01 0.06

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

Line 17-22 -0.01 -0.01 0.02 0.11 -0.01 -0.01 0.01 -0.03 0.21

Line 18-21 0.00 -0.01 -0.02 0.18 -0.02 0.05 -0.02 0.04 0.09

Line 19-20 -0.01 0.02 -0.01 0.09 0.02 0.00 -0.02 0.01 -0.01

Line 20-23 -0.03 -0.02 0.00 0.07 0.01 -0.03 0.01 -0.04 0.48

Line 21-22 0.03 -0.01 0.00 0.12 0.00 -0.03 -0.03 0.03 0.07

Figure 7-22: Scatter plot of Bus 4 ENS and Line 15-24 failure rate.


175

The same procedure that is used to rank the components using system ENS can be applied

to each individual bus to study the effect of the component failure rate on them. This gives

a ranking list for each of the load buses. An example of this is given in Figure 7-23. It

shows the ranking of the components that have influence on Bus 6.

Figure 7-23: IEEE-RTS components ranked using correlation coefficient between failure rates and Bus 6’

ENS

In addition to each load point ranking list, a similar approach can be used to include the

effects of components on load points and give only one ranking list. This can be achieved

by summing the correlation coefficients of each component to produce a ranking index

which considers the relationship between components and load points. This component

ranking index (CRI) is defined by (7.5).

B

i

ijj CCCRI (7.5)

where B is the number of load buses. The index is equal to the summation of each row in

Table 7-7 and it ranks the power system components considering their linear relationship

described by correlation coefficient with load buses ENS. Figure 7-24 shows the

components of IEEE-RTS and their CRI.

Figure 7-24: IEEE-RTS components ranked using CRI


176

7.5.4.2 Area of Vulnerability of a Component

In addition to identifying the most influential components of the network, it is also

important to identify the area or buses that are affected by a particular component failure

rate. The sampling based results can be used to define this area. By considering each

component correlation coefficient with all load buses, it can be determined which buses are

affected by component failure rates. As an example, the correlation coefficients of Line 15-

21(1) with load buses were presented in Figure 7-25. The figure shows that the line has a

strong linear relationship with Bus 20, Bus 15, and Bus 6 and it has no effect on the rest of

the buses. From this figure, it can be said that the area of vulnerability of this line involves

these three buses. The figure also shows the contribution of Line 15-21(1) to each bus ENS

calculated using the average failure rate. It can be seen from Figure 7-25 that the

correlation coefficient of Bus 15 is greater than correlation coefficient of Bus 6. However,

the contribution of Line 15-21(1) to ENS of Bus 6 is approximately double of that for Bus

15. Using these two criteria, the area of vulnerability of each component can be clearly

defined.

Figure 7-25: Line 15-21(1) correlation coefficient and contribution to ENS to load buses

7.5.4.3 Ranking of Load Points

The load points or load substations have been ranked in power systems according to their

historical performance [163]. This type of ranking considers the substation as a separate

unit and does not include its interaction with the network and other power system

components. An additional ranking index (SRI substation ranking index) was developed in

this study using the sensitivity analysis results without any additional computational effort.

This index gives a second dimension to the previous measures to characterise substations.

The main idea is to assess how a substation or a load point is affected by the components’


177

outage. Using the same results for the correlation coefficient of an individual component

and load points given in Table 7-7, the SRI for each substation was calculated using (7.6):

C

i

ijj CCSRI (7.6)

where C is the number of components. The SRI is equal to the summation of each column

in Table 7-7. Table 7-8 gives the load points of IEEE-RTS ranked using the SRI. The table

also gives the ENS of each bus calculated using the mean failure rate of the components.

Obviously, Bus6 comes as the most vulnerable bus because it has a linear relationship with

most of the network’s components. This is depicted in Figure 7-23, which represents the

correlation coefficients of failure rates of all components with Bus 6’ ENS.

Table 7-8: IEEE-RTS’ load points ranked using SRI

Ranking

Load point ENS (MWh/y) SRI Based on ENS Based on SRI

Bus 3 1527 1.51 Bus 6 Bus 6

Bus 4 154 1.94 Bus 20 Bus 20

Bus 5 940 2.02 Bus 3 Bus 5

Bus 6 7338 4.16 Bus 5 Bus 4

Bus 8 183 1.50 Bus 15 Bus 15

Bus 14 189 1.28 Bus 19 Bus 19

Bus 15 568 1.90 Bus 14 Bus 3

Bus 19 434 1.54 Bus 8 Bus 8

Bus 20 5617 3.20 Bus 4 Bus 14

Figure 7-26 shows the information for Bus14 which comes last in the ranking process. The

SRI gives an indication on how much certain load buses can be affected by system

components. Bus 6 is a very weak load point because most of components’ outage directly

affects the energy not supplied at this bus. On the other hand, Bus 14 is a very strong bus

because there is no strong relationship between it and system components except Line 14-

16.

The reliability map of IEEE-RTS is given in Figure 7-27, Figure 7-28, and Figure 7-29. In

these three figures, the red colour indicate high importance, the yellow indicates medium

importance and the green indicate low importance. Figure 7-27 shows the most important


178

components according to the classification shown in Figure 7-20. Figure 7-28 provides the

area of vulnerability of Line 15-21 which includes Bus 6, Bus 15 and Bus 20. Finally, the

weakest load points as identified by SRI ranking are given in Figure 7-29.

Figure 7-26: Correlation coefficient of IEEE-RTS components with load point Bus 14

Figure 7-27: IEEE-RTS reliability map-importance of components


179

Figure 7-28: IEEE-RTS reliability map: Area of vulnerability for Line 15-21

Figure 7-29: IEEE-RTS reliability map: criticality of load points


180

7.6 Summary

This chapter focused on studying the uncertainty associated with failure model in power

system reliability assessment. It focused on three studies: 1) the influence of epistemic

uncertainty in end-of-life failure models on system reliability, 2) quantification of aleatory

and epistemic uncertainty, 3) development of uncertainty based importance indicator.

The first study investigated the influence of the uncertainty in the two end-of-life failure

models used in this thesis: Gaussian and Arrhenius-Weibull distributions. The study

assessed the effect of the uncertainty on the identification of the most critical transformer

sites. It has also been demonstrated that for this identification, from the point of view of the

overall system reliability measured by ENS, a rough estimate of the transformer failure

model parameters can be used and such significant transformer failure model parameter

estimation and monitoring effort can be spared. Using the proposed approach it can be

established to what extent the reliability of transmission network is sensitive to transformer

failure model and/or model parameters and focus the attention and resource to important

transformers and/or model parameters only. Though the findings of the analysis may be

different for different systems with different age distribution of transformers and/or

network topologies, the study establishes a framework to assess the effort and resource

needed to obtain reliability based criticality of transformer sites.

The second study proposed the mixed aleatory-epistemic model to characterise and to a

certain extent bound the uncertainty in the component failure models in power system

reliability assessment. It is demonstrated that the proposed aleatory-epistemic model gives

more accurate estimate of the uncertainty than the conventional aleatory model. By

introducing this characterization, the study contributes to the understanding and

classification of uncertainty in power system reliability studies. In the light of this

characterisation, the study explored two methods to represent and quantify the proposed

uncertainty model. First, the epistemic uncertainty is represented by a probability

distribution function and quantified using the second order probability method. Second,

evidence theory is used where the epistemic uncertainty is interpreted by intervals and

degrees of belief. The study validates the proposed uncertainty model and the two methods

by analysing the uncertainty in the failure model of power transformers in the IEEE-RTS.

The propagation of this uncertainty to PLC was analysed. A comparison between the

results produced by the two methods shows that they are compatible despite the different


181

representation of the uncertainty. However, based on this analysis it seems that evidence

theory has the potential to provide better quantification of uncertainty in reliability studies.

Therefore, further study of evidence theory applications is needed.

The third study resulted in a reliability map of power system networks. It identified the

critical areas in the network that have major effect on system reliability indices. The map

was produced using probabilistic sensitivity analysis. The results of sensitivity analysis

were then used to assess the interaction between component failure rates and power system

reliability indices. As a case study, the reliability map of the IEEE-RTS network was

identified using the proposed method. The correlation coefficients between the component

failure rates and the ENS were evaluated as a measure of sensitivity. Using the correlation

coefficient between the failure rate of components and the load points’ reliability indices,

the area of vulnerability of each component was defined. Furthermore, the load points

(substations) were ranked based on their vulnerability to component outage. This ranking

of load point identifies substations that are strongly connected to/influenced by the system.


182

Chapter 8 Conclusions and Future Work

183

8

Conclusions and Future Work

8.1 Conclusions

This thesis studied the influence of the end-of-life failure of power transformers on

transmission system reliability in order to make number of optimal decisions on their

replacement. The research was driven by the presence of high ageing assets in power

networks and price control schemes imposed by electricity regulatory authorities. The

ageing assets reduce the expected level of reliability of power systems, whereas price

control schemes constrain the capital expenditure budget. This means that the balance

between reliability and cost should be managed carefully. Power transformers are selected

for the analysis because they are technically complex, expensive, and main feed points of

electricity for end users. The analyses conducted throughout the research are particularly

beneficial for asset managers. They assist in identifying the most critical transformers for

system reliability and in setting cost-effective replacement plans.

The thesis focused on end-of-life failure of power transformers, and provided

comprehensive explanation of its concept and highlighted differences with the traditional

repairable failure of power transformers. Based on this, the state-of-the-art method in the

integration of end-of-life failure into system reliability assessment is identified and

adopted. By providing this distinction between types of failure, the thesis enhanced the

understanding of age related failure and stimulated researchers to focus on models and


184

methods that are related to the applications under consideration. This explanation is the

first original contribution of this thesis.

During the completion of this thesis, a dedicated tool was developed to assess transmission

system reliability. The DIgSILENT PowerFactory package was chosen as the primary

computational environment for the tool since it is one of the most trusted commercial

software packages for power system utilities. Accordingly, DIgSILENT Programming

Language (DPL) is used to develop the tool based on the Non-sequential Monte Carlo

(NMC) simulation. The four steps of the system reliability assessment procedure are: 1)

load level selection, 2) component state selection, 3) failure effect analysis and remedial

action by system operator, 4) calculation of reliability indices. Full documentation of the

tool is provided in Chapter 3 including functional definitions, programming information,

and validation. This documentation creates an opportunity for further improvement by

other researchers and serves as a manual and a reference for users.

The review of past work showed that system reliability has been applied when making

decisions on replacement planning of a single component and component fleet. The

replacement volume, however, was determined based on the age distribution of the fleet. It

was also found that the risk matrix method is widely used by power utilities to identify

replacement candidates based on their condition and criticality. Following this, in Chapter

4 of this thesis, a reliability based framework for cost-effective replacement of power

transformers was developed. The framework combines, for the first time, the advantages of

the two existing methods of replacement planning: risk matrix method and the reliability

indices method. To this end, the framework consists of three elements. Firstly, the

framework adopted reliability importance measures to identify the most critical

transformers and to determine the volume of transformers to be replaced. This is a new

application of the reliability importance measures and is equivalent to the risk matrix

results. Secondly, it uses Pareto analysis, which is not previously implemented in system

reliability applications, to provide an insight into the effect of equipment replacement

volume on system reliability. Finally, the framework used a reliability incentive/penalty

scheme to estimate the cost of unreliability in order to select the optimum replacement

scenario based on economic comparison. By using this, the framework demonstrated how

the regulatory reliability incentive/penalty scheme can be included in replacement

planning. This developed framework as a whole is the second original contribution of this


185

thesis. Additionally, the three elements of the framework represent original applications in

the area, and such original contribution of the thesis, as well.

Despite the considerable amount of publications in the area of transformer life-time

modelling, they are not employed in system reliability assessment and applications. The

review revealed that only the conventional probability functions have been used in system

reliability and the valuable knowledge applied to unconventional models have not been

utilised yet. The only attempt to achieve this was not correctly executed. In this thesis, a

study on the integration of available unconventional end-of-life failure models was

introduced. The Arrhenius-Weibull distribution was employed as the failure model in

which the thermal stress due to loading conditions was considered. The reliability

assessment results were compared to the results calculated using the conventional Gaussian

model. The comparison illustrated that the integration of the thermal stress into reliability

assessment provides more insight into understanding the system reliability and identifying

the failure conditions of most critical transformers. This study presented in Chapter 5,

contributed to the system reliability assessment as it evaluates the added value provided by

including loading levels in failure models, and how this enhanced the understanding of the

effect of operational factors on system reliability. The integration of unconventional failure

model in system reliability studies is the third original contribution of this thesis.

It is a fact that the problem of ageing assets creates a concern and a threat of multiple

failure events. This is because component reliability deteriorates with age, and hence they

become more susceptible to consequent failure. The literature on power system reliability

did not contain studies that addressed dependent failure or multiple failure events that are

related to ageing assets or end-of-life failure. Hence, the thesis adopted the unconventional

end-of-life failure model in order to investigate the probability of occurrence of second

dependent failure due to thermal stress. The second dependent failure might ultimately lead

to a multiple failure event or even a cascading failure. This investigation led to the

development of two new probabilistic indicators, which relate the reliability of

transformers to their age and loading levels. The first indicator identifies which

transformers can initiate a sequence of multiple failures when they fail, while the second

identifies transformers which are the most vulnerable to a consequent failure. The main

purpose of these indicators is to rank power transformers based on their criticality for

multiple failure events. In addition, further study was performed to assess the effect of

transformer loading uncertainty on the indicators, and proved indicators’ robustness to


186

loading uncertainty. The proposed indicators can be used for ranking transformers based

on their significance for, and vulnerability to, cascading failure and as such represent a

useful tool for short and long term power system planning. Development of these

indicators is the fourth original contribution to system reliability studies. The approach for

developing the indicators and a case study was given in Chapter 6.

The review of modelling end-of-life failure revealed that a considerable effort was spent on

refining the failure model. The uncertainty in the parameters of these failure models

originated from the insufficient failure data due to the long life time of power system

components. However, the effect of this uncertainty on the overall system reliability was

not assessed. In Chapter 7, a study on assessing the effect of uncertainty in the parameters

of the Gaussian and Arrhenius-Weibull distribution on system reliability was described. An

area of uncertainty in the component unavailability was constructed by varying the

parameters of the two models within a specific range. Then, the system reliability is

assessed by randomly sampling this constructed area of uncertainty. The most critical

transformer sites were identified with and without considering the uncertainty. It was

demonstrated that for the identification of critical transformer sites, from the point of view

of the overall system reliability measured by ENS, a rough estimate of the transformer

failure model parameters can be used. The study established bases for an effective

approach for refining transformer failure models. The proposed approach assists in

measuring the extent to which the reliability of transmission network is sensitive to

transformer failure model parameters and focuses the attention and resources only when

the system or the system reliability applications are sensitive to the uncertainty in

parameters. This approach, which is the fifth original contribution of the thesis, introduced

a new application of reliability assessment in the area of refining failure models.

In addition, the thesis also discussed in Chapter 7 the two forms of uncertainty associated

with power system reliability studies: aleatory and epistemic uncertainty. The importance

of distinguishing between them is clearly demonstrated. Then, it used two methods to

quantify the uncertainty in failure model and gave case studies on the IEEE-RTS. The

methods used are the second order probability and Evidence theory. Furthermore, as a part

of the investigation on reliability importance measures, the chapter provided an example of

the use of sampling based sensitivity analysis to identify the most critical components and

their area of vulnerability. The quantification of failure model uncertainty and all aspects

related to it is the final original contribution of this thesis.


187

8.2 Future work

There are several research directions, which can be followed based on the work presented

in the thesis, in order to further develop this area of research.

Reduction of the computational time of the reliability assessment tool. The

reliability assessment tool can be further improved in terms of execution time in

order to enhance its capability of handling large size power networks. This can be

completed by exploring parallel computational techniques. In these techniques, the

program is split into several parts that are not dependent on each other, and then

each part is executed by a different processor. All the parts are executed

simultaneously and hence the computation time is reduced.

The integration of condition based end-of-life failure models into system

reliability studies. The area of integration of unconventional failure model can be

further developed by using models that are formulated from condition monitoring

data. These failure models are different from conventional distribution functions

because the condition data describes the historical operational stress on the

component. Therefore, data from different utilities can be used, which would

enlarge the data set and reduce the uncertainty in model parameters. These models,

however, are non-parametric, and hence incorporating them into system reliability

studies would not be an easy task. Such a study would result in more accurate

system reliability assessment and overcome the problem of insufficient data about

components end-of-life failure. In this way, online data on reliability of individual

components obtained from condition monitoring can be directly employed in

system reliability studies. In particular, this can be valuable in operational system

planning.

The integration of the criticality indictors for cascading failure events into the

replacement framework. A further area for the extension of the developed

replacement framework is to investigate the integration of the two developed

probabilistic indicator ICF and VCF into the framework. This integration would

merge the criticality of transformer for ENS and for cascading failure events.

Replacement plans developed from this future study would cover all the aspect of

criticality of transformers.


188

The use of Sequential Monte Carlo simulation method in end-of-life failure

studies. The use SMC simulation can be further development of the incorporation

of end-of-life failure into system reliability assessment as SMC does not involve

the calculations of component unavailability. In SMC, the system state selection is

achieved by producing a chronological component state transition that is sampled

from the probability distribution function. Such study will be a valuable

contribution, especially, as the Sequential Monte Carlo method is more accurate in

the evaluation of monetary value of unreliability as it accurately estimates failure

duration. The main difficulty with this study would be sampling the posteriori

probability function from the priori failure distribution functions. A second

research issue regarding this study would be the modelling of replacement time and

assumptions with respect to the replacement process itself, e.g., replace the

component with a new component that has the same electrical capability and

parameters (like for like replacement) or with a different one.

The incorporation of operational plans into the replacement framework. The

reliability based replacement framework can be extended by including the effect of

operational plans when mitigating customer interruption along with the

replacement of components. This can be achieved by adopting unconventional

failure models in the calculations. In this way the planning scenarios can involve

not only replacement of components but also operational plans to prevent the

failure of components. The greatest challenge in this research would be the network

modelling.

Development of optimal schedule for components replacement. Further

development of the framework that uses system reliability assessment for

component replacement planning could lead to the proposal of the optimal schedule

for components replacement. This would be an important area for further research.

It could be conducted in the same way as the reliability based maintenance

schedule is currently commonly performed. A system reliability evaluation tool

using Sequential Monte Carlo could be used for this purpose. Then, the reliability

of the system can be assessed by moving replacement outages over all possible

time intervals. The lowest risk scheduling can then be selected as the optimum

schedule.

System reliability studies with ageing models of other types of power system

equipment. A final area for the extension of work following this thesis is to apply


189

all the analyses carried out in this thesis to other types of power system equipment

including lines, cables and switchgear. The results of each individual study and

considering all components together would lead to the most accurate assessment of

system reliability to date.

Chapter 9 References

190

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Appendix A. Commonly Used Probability Distributions in Composite System Reliability Assessment

198

Appendix A.

Commonly Used Probability Distributions in

Composite System Reliability Assessment

A.1. Exponential Distribution

A.1.1 General Background

The exponential distribution is the most widely used distribution in reliability studies. The

two main features of exponential distribution are the constant hazard rate and the

memorylessness. Hence, it is employed in Markov models. The exponential distribution is

described by one parameter which is the hazard rate or the failure rate.

A.1.2 Distribution Functions

Table A.1: The distribution functions, the expected value and the variance of the exponential distribution

Probability

density function

pdf(t)

Cumulative

distribution

function cdf(t)

Hazard

function λ(t)

Expected value Variance

te

t

e

1

2

1

A.2. Gaussian Distribution


The Gaussian distribution or normal distribution characterises many stochastic or random

events, and hence it commonly used to model random variables when their probability

distributions are not known. The main feature of Gaussian distribution is that it symmetric

about the mean value of the random variables. Figure A.1 shows the tolerance intervals of

Gaussian distribution and their relationship with the standard deviation. As can be seen, the

range of ± one standard deviation results in the 68.2% of the population, ± two standard


199

deviations results in the 95.4% of the population, and ± three standard deviations results in

the 99.6% of the population.

Figure A.1: Gaussian distribution tolerance interval


Table A.2: The distribution functions, the expected value and the variance of the Gaussian distribution

Probability

density function

pdf(t)

Cumulative

distribution

function cdf(t)

Hazard

function λ(t)

Expected value Variance

]22

2)([

2

1

t

e

)

2(1

2

1

terf

)(1

)(

tcdf

tpdf

2

where erf is

x

x

texerf

21)(

A.3. Weibull Distribution


Weibull distribution offers a broad range of different distributions with different parameter

values which assist in fitting historical life data. The two main parameters of Weibull

distribution are the scale parameter and the shape parameter. The parameter is


200

known as the characteristic life as it gives the age at which 63.2% of population fails. The

value of is directly related to failure rate of Weibull distribution as follows:

is less than 1, the failure rate decreases with time, i.e., early life period of bathtub

curve.

equal to 1, the failure rate is constant, i.e., useful life period of bathtub curve.

is greater than 1, the failure rate increase with time, i.e., wear-out period of

bathtub curve.


Table A.3: The distribution functions, the expected value and the variance of the Weibull distribution

Probability

density function

pdf(t)

Cumulative

distribution

function

cdf(t)

Hazard

function

λ(t)

Expected

value Variance

t

et

1

t

e

1

t

1

1

1

12

1 22

where is gamma function.

Appendix B. Example for Reliability Assessment Using Fundamental Methods

201

Appendix B.

Example for Reliability Assessment Using

Fundamental Methods

In this appendix, the reliability of simple network will be evaluated using parallel/series

model, Markov model and State Enumeration methods. The network is given in Figure B-

1. The data of the network is given in Table B-1. The total demand at the terminal of the

network is 100 MW. The supply is assumed to be a perfect supply (reliability = 1).

SupplyLine 1

T2

T1

Line 2

Figure B-1: Simple network used to illustrate the methods of reliability evaluation

Table B-1: Equipment Data for the simple network

Component Capacity (MW) Failure rate ()

(failure/year)

Repair duration (MTTR)

(hours)

Line 1 100 0.3 40

Line 2 100 0.3 40

T1 75 0.02 800

T2 75 0.02 800

To evaluate the reliability of this network the probability of failure or the unavailability (Q)

and probability of success or availability (P) for each component were calculated using

equations (B-1) and (B-2) respectively. These two equations are driven by using Markov

method for repairable component.

)1(

BQ


202

)2(

BP

The repair rate () is calculated from the mean time to repair by annualizing it.

)3(8760

BMTTR

Table B-2 gives the availability and unavailability of the example network’s components.

Table B-2: The repair rate, availability and unavailability of the small network components

Component (repair/year) P Q

Line 1 219 0.998632 0.001368

Line 2 219 0.998632 0.001368

T1 10.95 0.998177 0.001823

T2 10.95 0.998177 0.001823

B.1 Series/Parallel method

To model the example network into series and parallel reliability block diagram, the

mechanism of the system failure should be analyzed. The failure of this network can be

defined as system inability to meet the demand or partial of the demand at the terminal of

the system. By looking at each component capacity, the two lines can be modelled as

parallel component because only one is needed to be up to supply the 100 MW demand.

On the other hand, as the two transformer capacity is less than the 100 MW, the two can be

modelled as series component because both of them are needed for system success. This

illustrates that the physical topology of components is not necessary the same reliability

presentation in series/parallel method. The reliability block diagram of the network is

shown in Figure B-2.

0.001368

0.001368

0.0018230.001823L1

T2T1

L2


203

Figure B-2: The series and parallel reliability model for the example network

The unavailability of two components in parallel, from the definition of parallel network, is

given by:

From the definition of the series network, the unavailability is calculated using:

Then, the total system unavailability can be calculated as

Qsystem = (0.001368*0.001368) + (0.001823+0.001823-0.001823*0.001823) -

(0.001368*0.001368) * (0.001823+0.001823-0.001823*0.001823)

Qsystem = 0.003644849

The system unavailability is in fact the PLC, hence:

PLC = 0.00364485.

B.2 Markov Model

To apply Markov model to the example network, the state space of the network is

generated by considering only first independent failure so that there is no state in the

system where more than one component is down. The state space diagram is shown in

Figure B-3.

)4(21 BQQQparallel

)5(2121 BQQQQQseries

seriesparallelseriesparallelsystem QQQQQ


204

All component Up

(1)

L1

Down

(2)

L2

Down

(3)

T1

Down

(4)

T2

Down

(5)

L1T2T1

L2

L1

T2

T1L2

Figure B-3: The state space of the example network considering first failure.

Having generated that, the transition matrix can be formulated

95.900095.10

095.90095.10

002180219

000218219

02.002.03.03.036.0

1000

0100

0010

0001

)(1

22

11

22

11

21212121

TT

TT

LL

LL

TTLLTTLL

T

0

0

0

0

0

95.1000002.0

095.100002.0

0021903.0

0002193.0

95.1095.1021921964.0

5

4

3

2

1

P

P

P

P

P

Solving the above algebraic equations along with full probability condition,

P1+P2+P3+P4+P5 =1.

001815.0

001815.0

001361.0

001361.0

993648.0

5

4

3

2

1

P

P

P

P

P

For this system only states 4 and 5 are considered as failure states. Therefore, the

probability of failure for this system is PLC = P4+P5 = 0.00363, which is approximately

equal the index calculated using series/parallel network.


205

B.3 State Enumeration Method

The State Enumeration method is the one among all previous used method that is suitable

for large system. The unavailability of the overall system equals the sum of the

unavailability of transformers T1 and T2, given in Table B-1.

PLC = 0.001823 + 0.001823= 0.003646

The all three methods give approximately the same value for PLC. The network used in

this example is very simple network. For larger systems, series/parallel and Markov

methods are very irritating and time consuming. Therefore, state enumeration is used in

complex system.

Appendix C. Programming and Application Information of the Reliability Assessment Software

206

Appendix C.

Programming and Application Information of

the Developed Reliability Assessment Software

C.1 Programming Information

SF-0 RELIABILITY FUNCTION

This is the main command of the function. The only input of this command is the number

of Monte Carlo iterations (mont), and the main outputs are the reliability indices. It

contains all the subcommands inside it. It has five direct subcommands; SF-00, SF-01, SF-

02, SF-03, and SF-04. All these subcommands do not have any other subcommands except

SF-03 which will be described in a zoom DFD in the next section. The descriptions of the

other are given in the following subsections. The DFD of this command is given in Figure

C-1. The main feature of this function is the loop of Monte Carlo iteration. SF-0 and SF-

02-SF-04 are repeated till the number of iteration reached the limit set by the user (mont).

SF-0 RELIABILTY FUNCTION


N: number of Monte Carlo Iteration

SF-01

COMPONENT

UNAVAILABILITY

SF-0

UPDATE

INDICES

SF-03

FAILURE EFFECT

ANALYSIS

SF-02

OPEN

OUTAGES

SF-00

Pnom

CALCULATIONS

DIgSILENT

DATA

BASE

mont

If n<mont

SF-04

CLOSE

OUTAGES

SF-0

STATE

SELECTION

ex

ex

C

n+1

INDICES

pn

yes ex

no

yes

no

Figure C-1: The DFD of the main command of the reliability function.


207

SF-00 Pnom CALCULATION

The main task of SF-00 is to calculate the nominal power (Pnom) of transmission branches

(cables and overhead lines) from the basic data (nominal voltage and current). This Pnom

is needed in the failure effect calculations. The command does not need inputs, and its

outputs are the nominal power of the overhead lines and cables.

SF-01 COMPONENT UNAVAILABILITY

The outage data of each component in DIgSILENT is given by the outage frequency and

duration. The unavailability is required to be calculated because it is the main input in non-

sequential Monte Carlo method. This subcommand calculates the unavailability of

transmission elements from the basic data of outage (outage frequency and repair

duration).

SF-0 STATE SELECTION

This function selects the state of each transmission element. This is done by generating a

random number (in-build routine in DPL) and comparing it to the element unavailability

given by the previous command. Then, equation 2-5 is used to determine the state of the

element. If there are any down states, the command sends its output along with execution

trigger to SF-02, SF-03 and SF-04. The output is a set that contains the elements which are

selected to be in the down state.

SF-02 OPEN OUTAGES

This command applies the contingency state on the network. This is done by opening the

switches/circuit breakers to isolate the elements.

SF-04 CLOSE OUTAGES

This command restore the normal topology of the network by closing all the switches that

opened by the SF-02. This command is executed after SF-03 FAILURE EFFECT

ANALYSIS command finishes.


208

SF-0 UPDATE INDICES

After each system state is examined by failure effect analysis (SF-03), this command

updates the reliability indices. It collects the indices in a result object which is the output of

the SF-0 Reliability function.

SF-03 FAILURE EFFECT ANALYSIS

This command performs the failure effect analysis. The inputs are; execution trigger and

nominal power of transmission elements. This command is executed only when there is a

down state. The key feature of this function is the loop to eliminate the overload that is

resulted from the contingency state. The DFD is given by Figure C-2. The sub-commands

are explained in the following subsections.

SF-030

OVERLOAD

TEST

SF-03

FEASIBILITY

CHECK

SF-03 FAILURE EFFECT ANALYSIS


DIgSILENT

DATA

BASE

ex

If i<it

yes

i+1

it

SF-031

SENSITIVITY

MATIX

SF-032

DISPATCH

SF-033

SHEDDING

SF-034

UPDATE

CONSEQ

cexnofe

yesno

ex:execution command

it: Number of overload relief iterations

pn: nominal Power of transmission component

c: consequences of the system state.

pn

Figure C-2: DFD of Failure Effect Analysis command

SF-03 OVERLOAD TEST

The main task of this command is to analyse the state of the system for any overloaded

elements. If there is no overload, the failure effect analysis will be terminated. If there are

overloaded elements, the command will trigger other subcommands to relief this overload.


209


This command calculates the contribution factors of all generation and load buses to

branches’ power flows. The output is a matrix that contains the contribution factors,

reserve to be increased, and reserve to be decreased. The DFD is shown in Figure C-3.



DIgSILENT

DATA

BASE

ex

If b<bu

b+1

SF-031

SENS

EXECUTE

SF-0311

UPDATE

MATRIX

ma: matrix of contribution factor bu: Number of buses

SF-0310

CALCULATE

P_RESERVE

SF-0312

SORT

MATRIX

ma

Figure C-3: DFD of SF-031 Sensitivity Matrix

SF-032 DISPATCH

This command attempts to relief the overload by re-dispatch the generators. This is

achieved by selecting pair to keep the balance of the generation and demand. The DFD of

this command is depicted in Figure C-4. The main input is the contribution matrix from

SF-031. The outputs are: fe is a variable to tell if the overload can be relieved by re-

dispatch and ec is the exchange amount of power injection between the selected buses. The

command SF-0320 calculates ec value based on concept given in Chapter 3.

SF-032 DISPATCH


DIgSILENT

DATA

BASE

ex

If c<co

c+1

SF-032

PAIR

SELECTION

SF-0320

EXCHANGE

CALCULATION

ecma

SF-032

CHECK

FEASIBILITYfe

no

yes

fe: if =0 the dispatch can relief

overload, if ≠0 need load shedding

ec: exchange amount between pair

buses.

co: number of combinations of pair

ma: matrix of contribution factors

Figure C-4: DFD of SF-032 Dispatch


210

SF-03 FEASIBILITY CHECK

This function receives the outputs of SF-032 DISPATCH fe and ec. If there is a feasible

relaxation of the overload by re-dispatch, this command will perform it. If there is no

solution, the command will trigger load shedding SF-033 SHEDDING.

SF-033 SHEDDING

The load shedding command SF-033 works in the same way as SF-032 DISPATCH.

Expect that, there is always a relaxation of the overload with load shedding action. The

technique for minimize the amount of load shed is explained in Chapter 3. The DFD of this

command is shown in Figure C-5. The main input is the contribution factors matrix and the

output is the amount of load shed.

SF-033 SHEDDING


DIgSILENT

DATA

BASE

ex

If c<co

c+1

SF-033

PAIR

SELECTION

SF-0330

L_SHEDDING

CALCULATION

lsma

SF-033

CHECK

LIMITS

no

yes

ls: load shedding amount. co: number of combinations of pair

ma: matrix of contribution factors

Figure C-5: DFD of SF-033 SHEDDING

SF-034 UPDATE CONSEQ

This is the last sub-command of SF-03. It is responsible for updating the amount of load

shed after each iteration in the process of overload relief. When the process finishes, it

sends this accumulated load shed to the reliability function SF-0 to calculate the indices.


211

C.2 Application Information

Age matrix

This matrix contains information about individual transformers in the test system and their

ages. The data in the matrix is used to calculate the unavailability due to end-of-life failure.

Figure C-6 shows the age matrix inside the DPL command.

Figure C-6: Age matrix contains the age data about individual transformers

Indices

The reliability indices of the system and load points are stored in Results Object. Figure C-

7 shows example of the results object. In order to get them one can export the data in the

results object to the output window of DIgSILENT or can export it as Windows Clipboard.

Figure C-8 shows all the options for exporting data stored in results object.


212

Figure C-7: snapshot of results object (system_indices)

Figure C-8: Options for exporting data stored in results object

Appendix D. Test System Data

213

Appendix D.

Test System Data

D.1 Failure Model Data

Tx Age Unavailability-

Gaussian

Unavailability-

Weibull

Equivalent

loading [%]

HSTb [˚C]

L10-T1 51 0.01064722 0.009703 29.2 40.55

L10-T2 58 0.01772474 0.016109 33.2 44.63

L10-T3 16 0.00006899 0.0001 25.5 36.83

L10-T4 58 0.01772474 0.016109 35.7 47.07

L10-T5 51 0.01064722 0.009703 30.7 42.14

L10-T6 49 0.00899425 0.008285 34.4 45.77

L10-T7 3 0.00000285 1.79E-07 23.9 35.27

L12-T1 47 0.00751072 0.007027 58.4 69.79

L12-T10 30 0.00092997 4.09E-06 44.9 56.25

L12-T11 7 0.00000823 5.05E-07 26.9 38.31

L12-T2 47 0.00751072 0.007027 57.9 69.28

L12-T3 32 0.00125951 0.001536 45.5 56.93

L12-T4 45 0.00619563 0.005917 58.1 69.52

L12-T5 45 0.00619563 0.005917 57.0 68.40

L12-T6 19 0.00012943 0.000197 50.9 62.30

L12-T7 4 0.00000374 4.09E-06 49.9 61.27

L12-T8 7 0.00000823 4.09E-06 46.3 57.68

L12-T9 7 0.00000823 4.09E-06 47.7 59.11

L13-T1 28 0.00067531 0.000906 49.5 60.83

L13-T2 28 0.00067531 0.000906 49.5 60.83

L13-T3 23 0.00028167 0.000417 48.3 59.67

L13-T4 29 0.00079414 0.001041 48.8 60.18

L13-T5 9 0.00001361 1.07E-05 48.5 59.89

L13-T6 6 0.00000635 2.28E-06 50.9 62.30

L14-T1 52 0.01153687 0.010476 17.5 28.92

L14-T2 6 0.00000635 2.28E-06 20.6 31.97


214

L14-T3 55 0.01445303 0.013069 17.5 28.92

L14-T4 3 0.00000285 1.79E-07 21.0 32.34

L14-T5 55 0.01445303 0.013069 17.6 28.95

L14-T6 6 0.00000635 2.28E-06 34.6 46.02

L14-T7 50 0.00979959 0.008973 30.7 42.04

L15-T1 46 0.00683229 0.006454 18.7 30.11

L15-T2 42 0.00452987 0.004504 24.0 35.42

L15-T3 56 0.01550524 0.01403 32.9 44.26

L15-T4 57 0.01659614 0.015043 32.7 44.05

L15-T5 45 0.00619563 0.005917 26.9 38.28

L16-T1 54 0.01344038 0.012157 29.1 40.46

L16-T2 48 0.00823130 0.007637 26.8 38.19

L16-T3 54 0.01344038 0.012157 25.5 36.92

L16-T4 49 0.00899425 0.008285 22.6 34.00

L16-T5 43 0.00504523 0.004943 28.7 40.13

L16-T6 43 0.00504523 0.004943 19.4 30.74

L17-T1 46 0.00683229 0.006454 26.1 37.51

L17-T2 47 0.00751072 0.007027 25.3 36.64

L17-T3 48 0.00823130 0.007637 33.9 45.25

L17-T4 49 0.00899425 0.008285 38.9 50.33

L18-T1 47 0.00751072 0.007027 34.9 46.26

L18-T2 47 0.00751072 0.007027 37.7 49.12

L18-T3 47 0.00751072 0.007027 37.7 49.12

L18-T4 47 0.00751072 0.007027 32.2 43.58

L19-T1 47 0.00751072 0.007027 37.5 48.85

L19-T2 50 0.00979959 0.008973 45.8 57.21

L19-T3 43 0.00504523 0.004943 35.5 46.92

L19-T4 51 0.01064722 0.009703 45.8 57.21

L20-T1 53 0.01246811 0.011293 49.8 61.23

L20-T2 48 0.00823130 0.007637 37.8 49.18

L20-T3 53 0.01246811 0.011293 43.2 54.60

L20-T4 48 0.00823130 0.007637 32.3 43.70

L20-T5 48 0.00823130 0.007637 47.1 58.45

L20-T6 48 0.00823130 0.007637 33.8 45.17

L23-T1 32 0.00125951 0.001536 22.5 33.86

L23-T2 18 0.00010541 0.000159 22.8 34.21

L23-T3 24 0.00033840 0.000493 19.9 31.30

L24-T1 6 0.00000635 2.28E-06 31.8 43.16


215

L24-T2 41 0.00405306 0.004094 29.8 41.18

L24-T3 42 0.00452987 0.004504 28.2 39.58

L24-T4 7 0.00000823 4.09E-06 20.6 31.97

L25-T1 26 0.00048215 0.000676 31.2 42.60

L25-T2 24 0.00033840 0.000493 26.2 37.59

L25-T3 7 0.00000823 4.09E-06 18.5 29.88

L25-T4 1 0.00000163 5.34E-09 34.6 45.96

L25-T5 34 0.00167818 0.001952 33.2 44.55

L25-T6 7 0.00000823 4.09E-06 24.3 35.70

L26-T1 45 0.00619563 0.005917 40.4 51.78

L26-T2 47 0.00751072 0.007027 52.8 64.20

L26-T3 45 0.00619563 0.005917 40.4 51.78

L26-T4 45 0.00619563 0.005917 42.0 53.35

L26-T5 32 0.00125951 0.001536 50.3 61.64

L26-T6 44 0.00560018 0.005414 36.3 47.69

L27-T1 45 0.00619563 0.005917 34.0 45.39

L27-T2 45 0.00619563 0.005917 26.6 37.96

L27-T3 45 0.00619563 0.005917 31.6 43.03

L27-T4 45 0.00619563 0.005917 23.5 34.91

L27-T5 44 0.00560018 0.005414 37.8 49.19

L27-T6 43 0.00504523 0.004943 31.4 42.74

L28-T1 45 0.00619563 0.005917 38.7 50.06

L28-T2 45 0.00619563 0.005917 34.4 45.75

L28-T3 8 0.00001061 6.81E-06 40.9 52.24

L30-T1 47 0.00751072 0.007027 21.4 32.75

L30-T2 48 0.00823130 0.007637 22.0 33.39

L30-T3 46 0.00683229 0.006454 20.4 31.81

L30-T4 47 0.00751072 0.007027 27.5 38.86

L31-T1 46 0.00683229 0.006454 24.7 36.10

L31-T2 47 0.00751072 0.007027 22.3 33.73

L31-T3 47 0.00751072 0.007027 34.8 46.22

L31-T4 47 0.00751072 0.007027 19.5 30.84

L32-T1 43 0.00504523 0.004943 46.4 57.74

L32-T2 43 0.00504523 0.004943 40.8 52.17

L33-T1 2 0.00000216 4.43E-08 15.4 26.83

L33-T2 2 0.00000216 4.43E-08 15.4 26.83

L33-T3 2 0.00000216 4.43E-08 15.4 26.74

L34-T1 20 0.00015824 0.000241 36.4 47.76


216

L34-T2 20 0.00015824 0.000241 36.4 47.76

L34-T3 16 0.00006899 0.0001 35.6 46.96

L34-T4 9 0.00001361 1.07E-05 36.5 47.90

L35-T1 44 0.00560018 0.005414 33.0 44.40

L35-T2 44 0.00560018 0.005414 32.9 44.32

L35-T3 5 0.00000489 1.15E-06 27.0 38.36

L36-T1 43 0.00504523 0.004943 40.3 51.69

L36-T2 22 0.00023343 0.00035 15.8 27.15

L36-T3 43 0.00504523 0.004943 41.2 52.62

L36-T4 50 0.00979959 0.008973 22.7 34.04

L36-T5 43 0.00504523 0.004943 40.5 51.86

L36-T6 22 0.00023343 0.00035 15.8 27.15

L36-T7 43 0.00504523 0.004943 41.2 52.62

L36-T8 50 0.00979959 0.008973 22.5 33.93

L36-T9 1 5.34E-09 5.34E-09 17.4 28.79

L38-T1 2 0.00000216 4.43E-08 14.6 26.01

L38-T2 2 0.00000216 4.43E-08 13.5 24.85

L53-T1 49 0.00899425 0.008285 23.5 34.88

L53-T2 32 0.00125951 0.001536 14.2 25.57

L53-T3 47 0.00751072 0.007027 16.1 27.47

L53-T4 49 0.00899425 0.008285 22.2 33.56

L54-T1 44 0.00560018 0.005414 21.8 33.15

L54-T2 3 0.00000285 1.79E-07 24.4 35.74

L54-T3 44 0.00560018 0.005414 34.1 45.50

T1 32 0.00125951 0.001536 36.0 47.42

T10 46 0.00683229 0.006454 33.5 44.91

T11 14 0.00004436 5.95E-05 27.1 38.48

T12 46 0.00683229 0.006454 15.6 26.97

T13 8 0.00001061 6.81E-06 13.7 25.09

T14 14 0.00004436 5.95E-05 46.1 57.46

T15 7 0.00000823 0.001536 28.3 39.66

T16 32 0.00125951 0.00119 8.9 20.28

T17 30 0.00092997 0.000241 8.9 20.27

T18 20 0.00015824 0.00119 25.0 36.39

T19 2 0.00000216 4.43E-08 6.3 17.71

T2 2 0.00000216 4.43E-08 40.4 51.74

T20 2 0.00000216 4.43E-08 7.3 18.71

T21 14 0.00004436 5.95E-05 40.1 51.47


217

T22 13 0.00003534 4.45E-05 43.8 55.16

T23 14 0.00004436 5.95E-05 37.9 49.29

T24 14 0.00004436 5.95E-05 41.1 52.47

T25 18 0.00010541 0.000159 6.8 18.13

T26 18 0.00010541 0.000159 8.7 20.06

T27 6 0.00000635 2.28E-06 43.3 54.71

T28 7 0.00000823 4.09E-06 43.3 54.71

T3 20 0.00015824 0.000241 57.6 68.96

T4 20 0.00015824 0.000241 57.9 69.30

T5 43 0.00504523 0.004943 14.7 26.06

T6 43 0.00504523 0.004943 12.3 23.73

T7 44 0.00560018 0.005414 13.8 25.23

T8 44 0.00560018 0.005414 11.8 23.21

T9 46 0.00683229 0.006454 41.4 52.82


218

D.2 Detailed Single line diagram

21

20

17

10

27

30

31

26

25

24

23

38

37

29

32

33

18

19

4 3 1

5

36

35

28

34

13

11

12

9

15

14

7

8

2

6

1

16

22

3940

41 42

43

44

4546

47

48

49

54

53

Appendix E. 10 Years Ambient Temperature Data

219

Appendix E.

10 Years Ambient Temperature Data

E. Average monthly temperature for 10 years in ˚C

year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Av

2003 4.5 3.9 7.5 9.6 12.1 16.1 17.6 18.3 14.3 9.2 8.1 4.8 10.5

2004 5.2 5.4 6.5 9.4 12.1 15.3 15.8 17.6 14.9 10.5 7.7 5.4 10.48

2005 6 4.3 7.2 8.9 11.4 15.5 16.9 16.2 15.2 13.1 6.2 4.4 10.44

2006 4.3 3.7 4.9 8.6 12.3 15.9 19.7 16.1 16.8 13 8.1 6.5 10.82

2007 7 5.8 7.2 11.2 11.9 15.1 15.2 15.4 13.8 10.9 7.3 4.9 10.48

2008 6.6 5.4 6.1 7.9 13.4 13.9 16.2 16.2 13.5 9.7 7 3.5 9.96

2009 3 4.1 7 10 12.1 14.8 16.1 16.6 14.2 11.6 8.7 3.1 10.11

2010 1.4 2.8 6.1 8.8 10.7 15.2 17.1 15.3 13.8 10.3 5.2 -0.7 8.83

2011 3.7 6.4 6.7 11.8 12.2 13.8 15.2 15.4 15.1 12.6 9.6 6 10.7

2012 5.4 3.8 8.3 7.2 11.7 13.5 15.5 16.6 13 9.7 6.8 4.8 9.7

E.2 Calculation on m,max

Daily temperature for the hottest month in the last 10 years in ˚C

Day of

Month

2012 (July) 2011(July) 2010(July) 2009(August)

Max Min Max Min Max Min Max Min

1 20.8 13.4 24.6 15.5 23.8 12.8 18.6 12.4

2 20.1 12.9 24.4 15.4 22.7 14.7 19.3 10.6

3 20.7 11.1 25.9 13.2 21.9 12 20.6 11.1

4 20.8 13.3 20 14.4 21.4 10.1 22.1 15.8

5 19.6 11.5 21.1 13.5 20 12.1 23.1 16.1

6 19.6 11.6 18.1 11.3 21.3 8.8 20 12.4

7 18.7 9.9 17.9 10.4 21.1 14.2 21.3 12.5

8 21.6 12.3 18.7 9.4 22.2 13.5 21.7 9.6

9 23.2 10.3 19 9.6 25.5 12.7 22.9 11.5

10 24.9 10.4 19.7 9.1 25.8 14 18.9 14

11 22.1 12.1 20.8 13.6 21.5 14.8 22.3 14.1

12 23.1 13.5 19.8 14.3 19.4 13.4 20.5 14.6

13 21.2 14.7 19.4 14.6 18.4 11.3 20.4 12.9


220

14 23.6 14.5 20.1 12.4 20.9 13.3 20.8 10.4

15 21.4 15.8 19.7 9.3 19.6 13.6 21.8 15.5

16 20.9 14.2 20.2 12.5 19.9 11.6 20.4 12.7

17 23.4 15.9 18.4 9.2 18.9 11.2 20.8 12.4

18 24.6 15.9 16.4 7.8 20.9 10.1 21.7 11.3

19 24.5 14.9 20.2 6.9 25.6 13.7 25.7 13.9

20 22.6 13.7 20.7 12.3 23.6 14.9 20.7 15.6

21 20.7 12.9 21.7 12.9 21.7 13.6 18.2 10.9

22 19.6 11.8 21.6 8.2 19.6 12.3 20.9 8.8

23 19.1 11.3 18 12.8 19.6 12 23.3 13.9

24 19 11.7 20.7 11.4 20.8 10.4 19.9 13.3

25 19.7 13.6 19 10.4 22.7 14.6 19.2 9.7

26 19.4 11.8 15 10.9 21.2 15.4 19.5 11.9

27 18.2 10.6 17.5 9.6 21.4 15.7 21.2 12.3

28 20.4 12.5 17.2 10.3 20.4 11.4 17.7 10.2

29 16.9 13 15.2 8.9 18.7 12.4 17.6 9.8

30 15.9 11.3 15.2 11 19.9 11.4 19.1 9.3

31 16.9 4.9 16.4 10.3 20.5 14.9 22.7 14.9

average 20.7 12.5 19.4 11.3 21.3 12.8 20.7 12.4

(max+min)/2 16.62097 15.3871 17.06129 16.56935

Day of

month

2008(July) 2007(August) 2006(July) 2005(August)

Max Min Max Min Max Min Max Min

1 25.2 10.7 19.4 8.7 28.1 12.5 20.1 12.8

2 18.4 11.2 19.1 11.3 29.2 16.1 23.1 10.3

3 19.7 11.2 17.8 9.7 28.9 14.9 20.4 12.9

4 20.6 7.7 18.2 15.3 28.1 14.6 20.2 11.8

5 20 13.7 21.6 12.5 24.5 15.5 19.6 12.5

6 18.4 12.8 22.3 13.4 23.5 16.1 18.6 11

7 17.7 11.4 20.4 8.9 20.1 15 20.6 8.4

8 17.1 10.9 18.7 7.9 19.5 11.3 21.9 7.2

9 17.1 10.5 19 9.3 20.5 13.4 23.6 8.7

10 19.6 12.7 18.6 8.1 20.6 11.6 22.2 13

11 17.7 12.1 18.6 11 21.1 13.6 23.8 12.4

12 16.2 9.1 20.6 12 22.9 9.6 20.4 12.6

13 19.2 7.8 21.3 10.8 21.5 12 18.6 9.8

14 19.2 10.6 18.4 13.2 22.5 8.7 19.7 12.5

15 22 14.9 19.1 14.1 24.8 9.2 20.9 10.1

16 17.5 12.9 19 10.1 27.8 10.8 23.3 12.1

17 17.1 11.8 14.3 9 29.6 12.1 25.4 11.7

18 17.7 12.9 14 12.6 30.9 13.3 24.9 11.2

19 18.3 13.8 16.9 14.1 32.9 16 18.8 13.1

20 17.1 8.4 18.6 10.5 27.6 17.7 20.7 10.4

21 18.8 7.5 17.2 11.4 27.7 15.8 23.3 9.6

22 21.5 9.8 19.8 13.1 26.9 17 18.8 13.7

23 23.9 13.9 19.7 11.3 24.3 15 20.3 9.1

24 24.2 13.8 15.8 11.3 27.6 12.4 17.2 13

25 24.9 14.7 15.7 11.7 30.2 15.2 17 9.9


221

26 24.4 13.5 11.7 12.1 28.6 17.8 17.2 9.3

27 26.7 12.9 12.5 9.1 26.4 16 18.8 11.3

28 26.4 15.7 13.9 9.7 26.2 13.4 21.9 10.4

29 22.2 15.2 16.2 8.8 28.8 14.1 22.5 13

30 24.1 13 15 10.1 23.5 14.6 25.7 10.7

31 22.9 14.5 18 12 21.5 14.1 28 13.8

average 20.5 12.0 17.8 11.1 25.7 13.9 21.2 11.2

(max+min)/2 16.24839

14.42742

19.76935

16.22258

Day of month 2004(August) 2003(August)

Max Min Max Min

1 24.6 10.7 20.9 14.8

2 25.7 14.7 21.3 9.5

3 21.7 16.6 24.7 10.1

4 22.3 15.6 29.4 12.4

5 24.6 14.5 29.7 18.2

6 23.9 14.1 26.1 16.2

7 27.5 13.1 25.8 15.2

8 28.4 15.8 26.7 16.2

9 21.2 18.2 31.5 16.2

10 23.3 15.9 26.7 17.3

11 23 13.7 24.3 16.1

12 22.7 15.4 24.5 16.3

13 20 12.7 22.8 12.8

14 23 13.5 21.7 10.5

15 22.7 14.2 22.5 9.4

16 21.5 15.3 22.9 12

17 22.3 14.3 23.4 12.3

18 21.6 15.6 20.7 14.9

19 21.5 13 19.4 11.3

20 17.4 12.9 20.5 10

21 18.3 9.1 21.7 12.6

22 20.2 9.7 22.8 16.6

23 19.4 13 24.7 16.5

24 19.5 13.4 24.1 16.4

25 19.3 12.2 19.8 13

26 18.9 12.5 20.1 14.2

27 19.3 13.8 20.5 12.8

28 18 9.2 15.5 12.7

29 18.3 11.1 18.2 9.7

30 18.1 11.8 18 6.6

31 18.4 8.8 17.1 6.7

average 21.5 13.4 22.8 13.2

(max+min)/2 17.43548

18.02419

Appendix F. Time Value of Money Formulae

222

Appendix F.

Time Value of Money Formulae

Name Symbols Formula

Single Payment-Compound Amount Factor (F/P, i%, n) i n1

Single Payment-Present Worth Factor (P/F, i%, n) )1(

1

in

Sinking Fund Factor (A/F, i%, n) 1)1( i

in

Capital Recovery Factor (A/P, i%, n) 1)1(

)1(

i

iin

n

Uniform Series-Compound Amount Factor (F/A, i%, n) i

in 1)1(

Uniform Series-Present Worth Factor (P/A, i%, n) )1(

1)1(

ii

in

n

i is the compound interest rate per interest period

n is the number of periods, commonly years

P is the present value

F is the future value

A the amount of a single payment in a uniform series

Appendix G. Full List of ICF and VCF Values

223

Appendix G.

Full List of ICF and VCF Values

G.1 ICF Values

Tx Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Annual

L10-T1 0.0068 0.0051 0.0004 0.0078 0.0021 0.0041 0.004535

L10-T2 0.0001 0 0 0 0.0001 0.0001 5.19E-05

L10-T3 0.0081 0.0055 0.0005 0.0051 0.0023 0.0069 0.005273

L10-T4 0 0.0009 0.0003 0.0001 0.0014 0.0017 0.000819

L10-T5 0.0068 0.0051 0.0004 0.0079 0.0022 0.0042 0.004581

L10-T6 0.0001 0 0 0 0.0001 0.0001 5.19E-05

L10-T7 0.0081 0.0055 0.0005 0.0051 0.0023 0.0069 0.005273

L12-T1 0.0007 0.0009 0.0003 0.0001 0.0014 0 0.000515

L12-T10 0 0 0 0 0 0 0

L12-T11 0 0 0 0 0 0 0

L12-T2 0 0 0 0.0001 0 0 1.15E-05

L12-T3 0.083 0.076 0 0.0236 0.1075 0.0004 0.045063

L12-T4 0.0076 0.0103 0 0.0073 0.0101 0.0066 0.007156

L12-T5 0.0816 0.0289 -0.0007 0.0171 0.0961 0 0.031912

L12-T6 0.0079 0.0031 0.0084 0.0054 0.0096 0.0049 0.005985

L12-T7 0.018 0.0115 0.0099 0.0081 0.0185 0.0066 0.011465

L12-T8 0.0087 0.0034 0.0088 0.0058 0.0104 0.0053 0.006469

L12-T9 0.0002 -0.0001 0 0.0004 0.0001 0.0002 0.000117

L13-T1 0 0 0 0 0 0 0

L13-T2 0.0002 0 0 0 0.0003 0.0001 8.85E-05

L13-T3 0.0001 0 0 0 0 0 1.73E-05

L13-T4 0.0001 0.0013 0.001 0.002 0.0011 0.0039 0.001763

L13-T5 0.0001 0.0014 0.0011 0.0023 0.0012 0.0044 0.001969

L13-T6 0.0001 0.0015 0.0011 0.0027 0.001 0.0016 0.001319

L14-T1 0 0.0016 0.0011 0.0027 0 0 0.000829

L14-T2 0.0234 0.0298 0.0016 0.0075 0.0168 0.0382 0.023173

L14-T3 0.013 0.0167 0.0008 0.0041 0.0091 0.0211 0.012835

L14-T4 0.0144 0.0181 0.0011 0.005 0.0113 0.0239 0.014456

L14-T5 0.0135 0.0172 0.0008 0.0042 0.0095 0.0218 0.013262

L14-T6 0.0001 0.0004 0 0.0002 0.0003 0.0002 0.000212

L14-T7 0.0001 0.0004 0 0.0002 0.0003 0.0002 0.000212

L15-T1 0 0.0002 0 0.0001 0.0001 0.0001 9.23E-05


224

L15-T2 0.0001 0.0003 0 0.0002 0.0003 0.0002 0.000188

L15-T3 0 0 0 0 0 0 0

L15-T4 0 0 0 0 0 0 0

L15-T5 0.0205 0.0033 0.0027 0.0053 0.0095 0.0044 0.007298

L16-T1 0.0217 0.0034 0.0025 0.0055 0.0098 0.0045 0.007579

L16-T2 0.0217 0.0034 0.0025 0.0055 0.0098 0.0045 0.007579

L16-T3 0 0.0035 0 0.0056 0.0101 0.0047 0.0036

L16-T4 0.0038 0.0021 0.0029 0.0027 0.0053 0.0007 0.002529

L16-T5 0.0198 0.0208 -0.0008 0.0121 0.0256 0.0744 0.030577

L16-T6 0 0 0 0 0.0001 0 9.62E-06

L17-T1 0.0294 0.0228 0 0.0236 0.0338 0.0947 0.039998

L17-T2 0.006 0.0025 0 0.0058 0.0074 0.0013 0.003321

L17-T3 0.0056 0.0023 0.0031 0.0054 0.0069 0.0012 0.003504

L17-T4 0 0 0 0 0 0 0

L18-T1 0.0001 0 0.0001 0 0.0001 0.0001 6.54E-05

L18-T2 0 0 0 0 0 0 0

L18-T3 0.0034 0.0037 0.0035 0.0065 0.0142 0.0016 0.004429

L18-T4 0.0043 0.0043 0.0037 0.0085 0.0157 0 0.004725

L19-T1 0 0 0 0 0 0 0

L19-T2 0 0 0 0 0 0 0

L19-T3 0 0 0 0 0 0 0

L19-T4 0 0 0 0 0 0 0

L20-T1 0 0 0 0 0 0 0

L20-T2 0 0 0 0 0 0 0

L20-T3 0.0006 0.0003 0.0003 0.0005 0.0009 0.0006 0.000508

L20-T4 0.0006 0.0003 0.0004 0.0005 0.0008 0.0005 0.000487

L20-T5 0 0 0 0 0 0 0

L20-T6 0.0018 0.0014 0.0012 0.0027 0.0061 0.0021 0.002219

L23-T1 0 0 0 0 0 0 0

L23-T2 0.0022 0.0017 0.0014 0.0032 0.007 0.0025 0.002629

L23-T3 0 0 0 0 0 0 0

L24-T1 0.0022 0.0017 0.0014 0.0032 0.0071 0.0025 0.002638

L24-T2 0 0 0 0 0 0 0

L24-T3 0.0067 0.0039 0.0015 0.0091 0.0053 0.0069 0.005546

L24-T4 0.0006 0.0166 0.0031 0.0056 0.0027 0 0.005258

L25-T1 0.0004 0.0023 0.0011 0.0005 0.0017 0 0.000969

L25-T2 0 0.0001 0.0001 0 0.0001 0.0002 9.62E-05

L25-T3 0.0006 0.0007 0.0015 0.0001 0.0036 0.0001 0.00085

L25-T4 0.0025 0.0017 0.0044 0.0043 0.0064 0.0053 0.003854

L25-T5 0.0027 0.0018 0 0.0046 0.0069 0.0056 0.003477

L25-T6 0.0035 0.0022 0.0039 0.0055 0.0081 0 0.003052

L26-T1 0.0034 0.0022 0.004 0.0054 0.0081 0.0108 0.005737

L26-T2 0 0 0 0.0001 0 0 1.15E-05

L26-T3 0 0 0 0 0 0 0

L26-T4 0 0 0 0 0 0 0


225

L26-T5 0 0 0 0 0 0 0

L26-T6 0 0 0 0.0001 0 0 1.15E-05

L27-T1 0.0032 0.0004 0.0006 0.0009 0.0017 0.0004 0.001094

L27-T2 0 0.0001 0.0002 0.0002 0.0004 0.0006 0.000262

L27-T3 0.0025 0.0004 0 0.0009 0.0017 0 0.000792

L27-T4 0.0028 0.0004 0.0006 0.0009 0.0017 0 0.000925

L27-T5 0 0 0 0 0.0001 0 9.62E-06

L27-T6 0 0 0 0 0 0 0

L28-T1 0 0 0 0 0 0 0

L28-T2 0 0 0 0 0 0 0

L28-T3 0.0001 0 0 0.0001 0.0001 0 3.85E-05

L30-T1 0 0 0 0 0 0 0

L30-T2 0.0057 0.0002 0.005 0.0005 0.0014 0.0002 0.001948

L30-T3 0 0 0 0 0 0 0

L30-T4 0.0161 0.0005 0.0051 0.0013 0.0008 0.0005 0.00394

L31-T1 0.0016 0.001 0 0.0012 0.0022 0.0011 0.001133

L31-T2 0.0021 0.0012 0 0.0008 0.0025 0.0022 0.001523

L31-T3 0 0 0 0 0 0 0

L31-T4 0.0027 0.0012 0.0006 0.0023 0.0039 0.0036 0.002365

L32-T1 0.0028 0.0013 0.0006 0.0024 0.0042 0.0027 0.002221

L32-T2 0 0 0 0 0 0 0

L33-T1 0.0262 0.0088 0 0.0101 0.0147 0.0234 0.014994

L33-T2 0.0237 0.0081 0.0049 0.0095 0.0138 0 0.009054

L33-T3 0.0001 0 0.0001 0.0003 0.0004 0 0.000104

L34-T1 0.0234 0.0079 0.0045 0.009 0.0131 0.0216 0.014177

L34-T2 0.0221 0.0075 0.0045 0.0087 0.0126 0.0207 0.013552

L34-T3 0.0001 0 0 0.0001 0.0001 0.0001 6.35E-05

L34-T4 0 0 0 0 0 0 0

L35-T1 0 0 0 0 0 0 0

L35-T2 0 0 0 0 0 0 0

L35-T3 0.0002 0.0015 0 0.001 0.0002 0.001 0.000765

L36-T1 0 0 0 0.0001 0 0.0001 3.65E-05

L36-T2 0 0 0 0 0 0 0

L36-T3 0.0001 0.0001 0 0.0008 0.0001 0.1135 0.028517

L36-T4 0 0 0 0 0 0 0

L36-T5 0 0 0 0 0 0 0

L36-T6 0.0028 0.0008 0.0014 0.0025 0.0049 0.0015 0.001992

L36-T7 0.0032 0.0008 0.0013 0.0029 0.0046 0.0012 0.00199

L36-T8 0.0032 0.0009 0.0011 0.0032 0.0046 0.0039 0.002696

L36-T9 0.0032 0.0008 0.0013 0.003 0.0044 0.0003 0.001758

L38-T1 0 0 0 0 0 0 0

L38-T2 0 0 0 0 0 0.0001 0.000025

L53-T1 0 0 0 0 0 0 0

L53-T2 0 0 0 0 0 0.0001 0.000025

L53-T3 0.0024 0.005 0.0018 0.0016 0.0051 0.0023 0.003062


226

L53-T4 0.0032 0.0046 0.0028 0.0026 0.0056 0.0027 0.003506

L54-T1 0.0051 0.0064 0.0033 0.0038 0.0084 0.004 0.00505

L54-T2 0.0052 0.0065 0.0034 0.0039 0.0086 0.0041 0.00516

L54-T3 0.004 0 0.0021 0.0021 0.0071 0.0032 0.0027

T1 0 0 0 0 0 0 0

T10 0 0 0 0 0 0 0

T11 0 0 0 0 0 0 0

T12 0 0 0 0 0 0 0

T13 0.0168 0.0026 0.0046 0.0049 0.0092 0.0076 0.007477

T14 0.0224 0.0033 0.0034 0.0059 0.011 0.0096 0.009235

T15 0.0168 0.0026 0.0046 0.0049 0.0092 0.0076 0.007477

T16 0.0188 0.0029 0.0047 0.0053 0.0098 0.0083 0.008185

T17 0.0068 0.0009 0.0011 0.0016 0.003 0.0026 0.002656

T18 0 0.0025 0 0.0047 0.0087 0.0072 0.003756

T19 0.0026 0.0016 0.0012 0.0034 0.0059 0.0178 0.00639

T2 0.0011 0.0006 0.0006 0.0014 0.0034 0.0075 0.002773

T20 0.0026 0.0016 0.0012 0.0035 0.006 0 0.001962

T21 0.0011 0.0006 0.0006 0.0014 0.0034 0 0.000898

T22 0.0025 0.0036 0.0012 0.0033 0.0057 0.0182 0.006904

T23 0.001 0.0021 0.0005 0.0013 0.0031 0.0076 0.003073

T24 0 0 0 0 0 0 0

T25 0 0 0 0 0 0 0

T26 0 0.0016 0.0025 0.0085 0.0199 0.0015 0.003975

T27 0 0.0023 0.0034 0.0118 0.0258 0.0018 0.005281

T28 0.012 0.0058 0.0055 0.0146 0.0439 0.0026 0.010712

T3 0 0 0.0019 0.0014 0.0195 0.0008 0.002492

T4 0.0073 0 0.0024 0.0016 0 0.0009 0.001996

T5 0.0051 0.0013 0.0023 0.0015 0.0128 0.0008 0.003096

T6 0.0005 0 0.0006 0.0045 0.0026 0.0001 0.000962

T7 0.0011 0.0001 0.0007 -0.0001 0.0053 0 0.000806

T8 0.0005 0 0.0006 0.0029 0.0028 0 0.000771

T9 0.0011 0.0001 0.0006 0.0001 0.0041 0.0004 0.0008

G.2 VCF Values

Tx Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Annual

L10-T1 0.007 0.0052 0.0004 0.0072 0.0022 0.0047 0.004683

L10-T2 0.0001 0 0 0 0 0.0001 4.23E-05

L10-T3 0.0079 0.0054 0.0005 0.0058 0.0023 0.0063 0.005146

L10-T4 0 0.0009 0.0003 0.0001 0.0015 0.0014 0.000754

L10-T5 0.007 0.0052 0.0004 0.0072 0.0022 0.0048 0.004708

L10-T6 0.0001 0 0 0 0 0.0001 4.23E-05

L10-T7 0.0079 0.0054 0.0005 0.0058 0.0023 0.0063 0.005146


227

L12-T1 0.0007 0.0009 0.0003 0.0001 0.0015 0 0.000525

L12-T10 0 0 0 0 0 0 0

L12-T11 0 0 0 0 0 0 0

L12-T2 0 0 0 0 0 0 0

L12-T3 0.0817 0.0307 0 0.017 0.0959 0 0.032408

L12-T4 0.0096 0.009 0 0.0066 0.0113 0.0067 0.007262

L12-T5 0.0831 0.0748 0 0.0238 0.1082 0 0.044794

L12-T6 0.0103 0.0047 0.0075 0.0062 0.0118 0.0058 0.007177

L12-T7 0.0112 0.0095 0.0112 0.0072 0.0127 0.0054 0.00904

L12-T8 0.0113 0.0051 0.0083 0.0066 0.0127 0.0062 0.007783

L12-T9 0 0 0 0 0 0.0002 0.00005

L13-T1 0 0 0 0 0 0 0

L13-T2 0.0001 0 0 0 0.0001 0 2.69E-05

L13-T3 0.0001 0 0 0 0 0 1.73E-05

L13-T4 0.0002 0.0013 0.001 0.0022 0.0011 0.0035 0.001704

L13-T5 0.0001 0.0015 0.0011 0.0024 0.0011 0.0037 0.001819

L13-T6 0.0001 0.0014 0.001 0.0024 0.001 0.0025 0.001473

L14-T1 0 0.0016 0.0011 0.0026 0 0 0.000817

L14-T2 0.0088 0.011 0.0007 0.0032 0.0071 0.0148 0.008908

L14-T3 0.024 0.0307 0.0015 0.0074 0.0165 0.0388 0.023581

L14-T4 0.0062 0.0078 0.0005 0.0024 0.0059 0.0107 0.00646

L14-T5 0.0253 0.0323 0.0016 0.0078 0.0174 0.0408 0.024821

L14-T6 0.0001 0.0003 0 0.0002 0.0003 0.0002 0.000188

L14-T7 0.0001 0.0003 0 0.0002 0.0003 0.0002 0.000188

L15-T1 0 0.0002 0 0.0001 0.0001 0.0001 9.23E-05

L15-T2 0.0001 0.0003 0 0.0002 0.0003 0.0002 0.000188

L15-T3 0 0 0 0 0 0 0

L15-T4 0 0 0 0 0 0 0

L15-T5 0.0186 0.0031 0.0022 0.005 0.0089 0.0041 0.006688

L16-T1 0.0226 0.0036 0.0027 0.0058 0.0105 0.0048 0.007985

L16-T2 0.0226 0.0036 0.0027 0.0058 0.0105 0.0048 0.007985

L16-T3 0 0.0033 0 0.0053 0.0093 0.0044 0.003367

L16-T4 0.005 0.0022 0.0029 0.0038 0.0062 0.0012 0.003098

L16-T5 0.0289 0.0223 0 0.0223 0.0327 0.0936 0.039265

L16-T6 0.0001 0 0 0 0.0001 0 2.69E-05

L17-T1 0.0203 0.0213 0 0.0133 0.0266 0.075 0.031271

L17-T2 0.0053 0.0025 0 0.0051 0.0069 0.0012 0.003046

L17-T3 0.005 0.0023 0.0032 0.0048 0.0064 0.0011 0.003271

L17-T4 0 0 0 0 0 0 0

L18-T1 0 0 0 0 0 0 0

L18-T2 0 0 0 0 0 0 0

L18-T3 0.0039 0.004 0.0036 0.0082 0.015 0 0.004471

L18-T4 0.0037 0.0038 0.0036 0.0066 0.0145 0 0.004158

L19-T1 0 0 0 0 0 0 0

L19-T2 0 0 0 0 0 0 0


228

L19-T3 0 0 0 0 0 0 0

L19-T4 0 0 0 0 0 0 0

L20-T1 0 0 0 0 0 0 0

L20-T2 0 0 0 0 0 0 0

L20-T3 0.0006 0.0003 0.0004 0.0005 0.0008 0.0005 0.000487

L20-T4 0.0006 0.0003 0.0004 0.0005 0.0009 0.0006 0.000521

L20-T5 0 0 0 0 0 0 0

L20-T6 0.0029 0.0019 0.0014 0.0041 0.0067 0.0032 0.003046

L23-T1 0 0 0 0 0 0 0

L23-T2 0.0035 0.0023 0.0017 0.0049 0.0078 0.0039 0.003656

L23-T3 0 0 0 0 0 0 0

L24-T1 0.0035 0.0023 0.0017 0.0049 0.0078 0.0039 0.003656

L24-T2 0 0 0 0 0 0 0

L24-T3 0.0041 0.0032 0.0015 0.0047 0.0048 0.0052 0.003954

L24-T4 -0.0001 0.0031 0.0008 0.0013 0.0025 0 0.001196

L25-T1 0.0009 0.0095 0.002 0.0015 0.0022 0 0.003002

L25-T2 0 0 0 0 0 0 0

L25-T3 0.0009 0.0006 0.0005 0.0005 0.0016 0.0015 0.000948

L25-T4 0.0018 0.0012 0.0019 0.0031 0.0046 0.0089 0.003869

L25-T5 0.0019 0.0013 0 0.0033 0.005 0.0095 0.003865

L25-T6 0.0042 0.0027 0.0053 0.0067 0.0099 0 0.003788

L26-T1 0.0042 0.0027 0.0052 0.0067 0.0099 0.1122 0.031825

L26-T2 0 0 0 0.0001 0 0 1.15E-05

L26-T3 0 0 0 0 0 0 0

L26-T4 0 0 0 0 0 0 0

L26-T5 0 0 0 0 0 0 0

L26-T6 0 0 0 0.0001 0 0 1.15E-05

L27-T1 0.0033 0.0004 0.0007 0.0011 0.002 0.0006 0.001227

L27-T2 0 0.0001 0.0001 0.0002 0.0003 0.0004 0.000188

L27-T3 0.0017 0.0003 0 0.0006 0.0012 0 0.000548

L27-T4 0.0035 0.0005 0.0006 0.0011 0.002 0 0.001121

L27-T5 0 0 0 0 0.0001 0 9.62E-06

L27-T6 0 0 0 0 0 0 0

L28-T1 0 0 0 0 0 0 0

L28-T2 0 0 0 0 0 0 0

L28-T3 0.0001 0 0 0.0001 0.0001 0 3.85E-05

L30-T1 0 0 0 0 0 0 0

L30-T2 0.0147 0.0004 0.0054 0.001 0.0018 0.0006 0.003802

L30-T3 0 0 0 0 0 0 0

L30-T4 0.0061 0.0005 0.0049 0.0008 0.0009 0.0006 0.00216

L31-T1 0.0018 0.0011 0 0.0009 0.0023 0.0015 0.001265

L31-T2 0.002 0.0012 0 0.0011 0.0024 0.0018 0.001431

L31-T3 0 0 0 0 0 0 0

L31-T4 0.0028 0.0013 0.0006 0.0024 0.0042 0.0026 0.002196

L32-T1 0.0027 0.0012 0.0006 0.0023 0.0039 0.0037 0.00239


229

L32-T2 0 0 0 0 0 0 0

L33-T1 0.026 0.0088 0 0.0101 0.0147 0.0237 0.015035

L33-T2 0.0239 0.0082 0.0046 0.0095 0.0138 0 0.009071

L33-T3 0.0001 0 0.0001 0.0003 0.0003 0 9.42E-05

L34-T1 0.0233 0.0079 0.0047 0.0091 0.0131 0.0219 0.014273

L34-T2 0.0222 0.0075 0.0045 0.0087 0.0127 0.0212 0.013704

L34-T3 0.0001 0 0 0.0001 0.0001 0.0001 6.35E-05

L34-T4 0 0 0 0 0 0 0

L35-T1 0 0 0 0 0 0 0

L35-T2 0 0 0 0 0 0 0

L35-T3 0.0002 0.0004 0.0001 0.0004 0.0001 0.0003 0.000271

L36-T1 0 0 0 0.0001 0 0.0001 3.65E-05

L36-T2 0 0 0 0 0 0 0

L36-T3 0.0002 0.0001 0 0.0004 0.0001 0.001 0.000363

L36-T4 0 0 0 0 0 0 0

L36-T5 0 0 0 0 0 0 0

L36-T6 0.0024 0.0007 0.001 0.0024 0.0039 0.0017 0.001788

L36-T7 0.0027 0.0008 0.0011 0.0026 0.0042 0.0014 0.001854

L36-T8 0.0047 0.0012 0.0019 0.0042 0.0064 0.0033 0.003271

L36-T9 0.0026 0.0008 0.001 0.0026 0.0041 0.0009 0.001688

L38-T1 0 0 0 0 0 0 0

L38-T2 0 0 0 0 0 0 0

L53-T1 0 0 0 0 0 0 0

L53-T2 0 0 0 0 0 0 0

L53-T3 0.0013 0.0018 0.0007 0.0009 0.0031 0.0013 0.001462

L53-T4 0.0013 0.0018 0.0009 0.0012 0.0026 0.0012 0.00145

L54-T1 0.007 0.0093 0.0049 0.0053 0.0119 0.0057 0.007198

L54-T2 0.0072 0.0095 0.005 0.0055 0.0123 0.0059 0.007404

L54-T3 0.0032 0 0.0017 0.0019 0.0054 0.0026 0.002171

T1 0 0 0 0 0 0 0

T10 0 0 0 0 0 0 0

T11 0 0 0 0 0 0 0

T12 0 0 0 0 0 0 0

T13 0.0132 0.0022 0.0028 0.0041 0.0077 0.0064 0.005983

T14 0.0321 0.0048 0.0079 0.0085 0.0159 0.0138 0.013687

T15 0.0132 0.0022 0.0028 0.0041 0.0077 0.0064 0.005983

T16 0.0145 0.0024 0.0031 0.0044 0.0083 0.0069 0.006512

T17 0.0087 0.0012 0.0019 0.0021 0.004 0.0034 0.003515

T18 0 0.0021 0 0.0039 0.0073 0.0061 0.003162

T19 0.0026 0.0019 0.0012 0.0034 0.0059 0.0182 0.00656

T2 0.0011 0.0008 0.0006 0.0014 0.0034 0.0076 0.002844

T20 0.0026 0.002 0.0012 0.0035 0.006 0 0.002054

T21 0.0011 0.0008 0.0006 0.0014 0.0034 0 0.000944

T22 0.0025 0.0039 0.0012 0.0033 0.0057 0.0178 0.006873

T23 0.001 0.0031 0.0005 0.0013 0.0031 0.0075 0.003279


230

T24 0 0 0 0 0 0 0

T25 0 0 0 0 0 0 0

T26 0 0.0047 0.0063 0.0205 0.0483 0.0023 0.009517

T27 0 0.0025 0.0034 0.0112 0.0174 0.0013 0.004325

T28 0.0083 0.0034 0.0022 0.0082 0.0149 0.0011 0.005171

T3 0 0 0.0037 0.0033 0.0393 0.0019 0.005133

T4 0.0121 0 0.0021 0.0019 0 0.0011 0.002871

T5 0.0042 0.0009 0.001 0.0011 0.0061 0.0007 0.001958

T6 0.0007 0 0.0006 0.0017 0.0029 0.0001 0.000702

T7 0.0007 0.0006 0.0004 0.0005 0.0026 0.0004 0.000721

T8 0.0006 0.0003 0.0004 0.0012 0.0024 0.0008 0.000796

T9 0.0006 0.0018 0.0005 0.0006 0.0022 0.0004 0.000967

Appendix H. Thesis Based Publications

231

Appendix H.

Author’s Thesis Based Publications

Journal Papers

H.1. S. Awadallah, J. V. Milanovic and P. N. Jarman, “Reliability based framework for

cost-effective replacement of power transmission equipment”, IEEE Trans. on Power

Systems, vol.29, no.5, pp.2549-2557, Sept. 2014.

H.2. S. Awadallah, J. V. Milanovic, P. N. Jarman, and Z. D. Wang “Probabilistic

indicators for assessing age and loading based criticality of transformers to cascading

failure events”, IEEE Trans. on Power System, vol.29, no.5, pp.2558-2566, Sept.

2014.

H.3. S. Awadallah, J. V. Milanovic and P. N. Jarman, “The Influence of Modelling

Transformer Age Related Failures on System Reliability”, IEEE Trans. on Power

System, DOI: 10.1109/TPWRS.2014.2331103, Jun. 2014, to be published.

Conference Papers

H.4. S. Awadallah, J. V. Milanovic and P. N. Jarman, “Assessment of probability of

thermal stress induced dependent failure of aged power transformer”, presented at

Cigre SC A2 & C4 Joint Colloquium Zurich, Sep. 2013.

H.5. S. Awadallah and J. V. Milanovic, “Quantification of aleatory and epistemic

uncertainty in bulk power system reliability evaluation”, presented at IEEE PES

Grenoble PowerTech 2013, Grenoble, France. 2013.

H.6. S. Awadallah, J. V. Milanovic and Z.D. Wang, “Probabilistic identification of power

system reliability map” presented at 12th

international conference on Probabilistic

Method Applied to Power systems PMAPS, Istanbul, Turkey. 2012.

probabilistic methodology for prioritising replacement of ageing power transformers based on

Documents