Ghavameddin Nourbakhsh PhDThesis begining pages Feb 1-2011€¦ · system reliability of bulk...

RELIABILITY ANALYSIS AND ECONOMIC

EQUIPMENT REPLACEMENT APPRAISAL FOR

SUBSTATION AND SUB-TRANSMISSION SYSTEMS

WITH EXPLICIT INCLUSION OF NON-REPAIRABLE

FAILURES

A Thesis submitted in

Partial Fulfilment of the Requirement for the

Degree of

Doctor of Philosophy

Ghavameddin Nourbakhsh

M.Sc & B.Sc (Electrical Engineering)

Faculty of Built and Environment Engineering

School of Engineering Systems

Queensland University of Technology

Queensland, Australia

January 2011

Key Words

Power System Reliability

Substation Reliability

Sub-transmission Reliability

Distribution Reliability

Failure rate

Outage

Useful Life

Risk

Aging

Degradation

Mean life

Life Cycle

Bathtub Curve

Non-repairable

Replacement

Refurbishment

Markov Model

Enumeration Method

Maximum Flow

Minimum Cut

Contingency Enumeration

Frequency and Duration

Correlation

Clustering

Expected Interruption Cost

ABSTRACT

The modern society has come to expect the electrical energy on demand, while many

of the facilities in power systems are aging beyond repair and maintenance. The risk

of failure is increasing with the aging equipments and can pose serious consequences

for continuity of electricity supply. As the equipments used in high voltage power

networks are very expensive, economically it may not be feasible to purchase and

store spares in a warehouse for extended periods of time. On the other hand, there is

normally a significant time before receiving equipment once it is ordered. This

situation has created a considerable interest in the evaluation and application of

probability methods for aging plant and provisions of spares in bulk supply networks,

and can be of particular importance for substations.

Quantitative adequacy assessment of substation and sub-transmission power systems

is generally done using a contingency enumeration approach which includes the

evaluation of contingencies, classification of the contingencies based on selected

failure criteria. The problem is very complex because of the need to include detailed

modelling and operation of substation and sub-transmission equipment using network

flow evaluation and to consider multiple levels of component failures. In this thesis a

new model associated with aging equipment is developed to combine the standard

tools of random failures, as well as specific model for aging failures. This technique

is applied in this thesis to include and examine the impact of aging equipments on

system reliability of bulk supply loads and consumers in distribution network for

defined range of planning years. The power system risk indices depend on many

factors such as the actual physical network configuration and operation, aging

conditions of the equipment, and the relevant constraints. The impact and importance

of equipment reliability on power system risk indices in a network with aging

facilities contains valuable information for utilities to better understand network

performance and the weak links in the system. In this thesis, algorithms are

developed to measure the contribution of individual equipment to the power system

risk indices, as part of the novel risk analysis tool. A new cost worth approach was

developed in this thesis that can make an early decision in planning for replacement

activities concerning non-repairable aging components, in order to maintain a system

reliability performance which economically is acceptable.

The concepts, techniques and procedures developed in this thesis are illustrated

numerically using published test systems. It is believed that the methods and

approaches presented, substantially improve the accuracy of risk predictions by

explicit consideration of the effect of equipment entering a period of increased risk of

a non-repairable failure.

CONTENTS 1. Introduction

1.1 Motivation and Overview 1

1.2 Aims and Objectives 2

1.3 Original Contributions 4

1.4 Literature Review 5

1.4.1 Power System Reliability 6

1.4.2 Functional Zones and Hierarchical Levels 7

1.5 An Overview of Probabilistic Reliability Evaluation

Developments of High Voltage Network 9

1.6 Development of Scope of the Thesis 12

1.7 Review of Outline of the Thesis 14

2. Contribution of Equipment to Power System Reliability

Performance using the Enumeration Method

2.1 Introduction 21

2.2 State Space Model for Substation and sub-transmission

Components 22

2.2.1 Markov State Probabilities 24

2.2.2 Enumeration of Selected System States 26

2.2.3 Flow Methods 28

3..1 Connectivity 28

3..2 Maximum Flow 30

2.3 Substation and Sub-transmission Indices 31

2.4 Percentage Contribution of Components to Probability and

Frequency of Contingencies 35

2.4.1 Percentage Contribution of Components to Reliability Indices

in a Power System 36

2.4.2 Load Weighted Attributes to Probability and Frequency

Indices 38

Page

2.5 Application Applied to Substation and Sub-transmission

Configurations 40

2.5.1 Application Applied to Small Sub-transmissions System 40

2.5.2 Application Applied to Medium Substation System 49

2.5.3 Application of Limited Capacity in System Reliability

Evaluations 57

2.6 Summary 62

3. A Markov Model for Non-Repairable Aging Equipment and its

Contribution to Substation/Sub-transmission Reliability

Evaluations

3.1 Introduction 64

3.2 Non-Repairable Aging Failure Model 65

3.3 Including Non-Repairable Aging in Markov State Model 69

3.3.1 Non-Repairable Aging using State Space Model 71

3.3.1.1 Aging State with Two States 72

3.3.1.2 Aging State with Three States 74

3.3.1.3 Non-Repairable Aging State with Four States 77

3.4 Procedures in Evaluating Load Point and System Indices,

Including Non-Repairable Aging States 89

3.5 Application to Substation and Sub-transmission Networks 92

3.5.1 Application to Small Networks 93

3.5.1.1 Application to Case “a” 93

3.5.1.2 Application to Case “d” 102

3.5.1.3 Application to Medium Network 118

3.6 Summary 134

4. Classification and Decomposition of Distribution Feeder Loads

4.1 Introduction 136

4.2 Data Collection 138

4.3 Model Formulation 139

4.4 Sub – Sector / Customer Profiles 145

4.5 Decomposing a Load Feeder 147

4.6 Cross Validation and Testing 150

4.7 Load Information for Reliability and Cost / Benefit

Assessments 154

4.8 Summary 155

5. Impact of Non-Repairable Failures of Aging

Transmission/Sub-transmission Network on Distribution Load

Point Indices


5.2 Distribution System Reliability 157

5.2.1 Electric Power Distribution Reliability Indices 158

5.2.1.1 Load Point Indices 158

5.2.1.2 System Indices 160

5.3 Analytical Simulation 161

5.3.1 Switching Operations 162

5.3.1.1 Tripping of Automatic Protection Devices 163

5.3.1.2 Manual Isolation of Components 164

5.3.1.3 Reclosing Alternative Supply 165

5.3.1.4 Isolation of Insufficiently Supplied Load Points 165

5.3.2 Outage Causing Modes 167

5.3.2.1 Active Failure 167

5.3.2.2 Passive Failure 168

5.3.2.3 Temporary Failure 168

5.3.2.4 Maintenance Outage 169

5.3.3 Load Connectivity / Shedding 169

5.3.3.1 Minimum Cut Set 170

5.3.3.2 Maximum Flow 170

5.3.3.2.1 Shortest Path Maximum Flow 171

5.3.3.2.2 Priority Maximum Flow 173

5.4 Simulation Tests 176

5.4.1 RBTS Bus 2 Distribution Network 176



5.4.2 RBTS Bus 4 Distribution Network 183



5.5 Effect of Substation Sub-transmission

Aging on Distribution Network 191

5.5.1 Application of Substation/Sub-transmission

Non-repairable Aging on Distribution System Reliability 193

5.5.2 Effects of Aging Equipment’s Mean Life on Distribution

System Reliability 197

5.6 Summary 202

6. Economic Consideration for Substation and Sub-transmission

Equipment Replacement


6.2 Customer Interruption Costs 206

6.3 Expected Interruption Cost (EIC) 208

6.4 Application to Substation and Sub-transmission Systems 210

6.4.1 Economic Evaluations at Substation

and Sub-transmission Systems 210

6.4.2 Economic Evaluations at Distribution System 218

6.5 Summary 221

7. Software for Substation, Sub–transmission and Distribution

Reliability Evaluations


7.2 General Program Flow Chart 224

7.3 Software application guide 225

7.4 Software Output Result Files 233

7.5 Summary 235

8. Conclusions


8.2 Summary 237

8.3 Future Research 240

References 242

Appendix A 251

A.1 Labelling Algorithm 251

A.2 Augmentation Routine 252

Appendix B 254

List of Figures

1.1 Reliability worth, with investment cost. 6

1.2 Subdivision of Power System Reliability. 7

1.3 Hierarchical levels for power system facilities. 8

2.1 General four sates model representing substation equipment modes. 23

2.2 Example of connectivity flow with an active failure at component 9. 29

2.3 An example of connectivity flow with passive failure at component 3. 29

2.4 An example of maximal flow with active failure at component 9. 30

2.5 An example of maximal flow with passive failure at component 3. 31

2.6 Small Substation Test Cases. 42

2.7 Medium substation test network. 50

3.1 Aging failure or, hazard rate functions. 66

3.2 Aging Failure, using Weibull distribution with α = 45 and β = 7. 67

3.3 Aging Failure, using Normal distribution with µ = 45 and σ = 8. 68

3.4 Aging, Failure Rate, using Weibull distribution with α = 45 and β = 7. 68

3.5 Aging, Failure Rate, using Normal distribution with µ = 45 and σ = 8. 69

3.6 A three state Markov model, including an aging state. 72

3.7 A four state Markov model, including an aging state. 75

3.8 A five state Markov model, including an aging state. 78

3.9 Four states, conventional Markov model for power system equipment. 81

3.10 Operating state probability in Markov models; with and without an

aging state. 83

3.11 Active/switching state probability in Markov models; with and without an aging

state. 84

3.12 Repair state probability in Markov models; with and without an aging state. 84

3.13 Maintenance state probability in Markov models; with and without an aging

state. 85

3.14 Operating state frequency in Markov models; with and without an aging

state. 85

3.15 Active/switching state frequency in Markov models; with and without an aging

state. 86

3.16 Repair state frequency in Markov models; with and without an aging state. 86

3.17 Maintenance state frequency in Markov models; with and without an

aging state. 87

3.18 Non-repairable aging state probability. 87

3.19 Non-repairable aging state frequency. 88

3.20 Non-repairable aging state probability, with replacement. 88

3.21 Non-repairable aging state frequency, with replacement. 89

3.22 System load unavailability due to component aging for network case “a”. 95

3.23 System load EFLC due to component aging for network case “a”. 96

3.24 Percentage contribution of components to System load unavailability, as TX 1 is

aging for network case “a”. 97

3.25 Percentage contribution of components to System load EFLC, as TX 1 is aging

for network case “a”. 98

3.26 Percentage contribution of components to System load unavailability, as TX 1

and TX 2 are aging for network case “a”. 99

3.27 Percentage contribution of components to System load EFLC, as TX 1 and TX 2

are aging for network case “a”. 100

3.28 Percentage contribution of components to System load unavailability, as TXs

and CB1 are aging for network case “a”. 101

3.29 Percentage contribution of components to System load EFLC, as TXs and CB1

are aging for network case “a”. 101

3.30 System load unavailability due to component aging for network case “d”. 103

3.31 System load EFLC due to component aging for network case “d”. 105

3.32 Load unavailability due to component aging for network case “d”. 106

3.33 Load EFLC due to component aging for network case “d”. 108

3.34 Percentage contribution of components to load 1 unavailability, as TX1 is aging

for network case “d”. 109

3.35 Percentage contribution of components to load 1 EFLC, as TX1 is aging for

network case “d”. 110

3.36 Percentage contribution of components to load 2 unavailability, as TX1 is aging


3.37 Percentage contribution of components to load 2 EFLC, as TX1 is aging for


3.38 Percentage contribution of components to load 1 unavailability, as TX1 and CB

are aging for network case “d”. 112

3.39 Percentage contribution of components to load 1 EFLC, as TX1 and CB are

aging for network case “d”. 113

3.40 Percentage contribution of components to load 2 unavailability, as TX1 and CB

are aging for network case “d”. 113

3.41 Percentage contribution of components to load 2 EFLC, as TX1 and CB are


3.42 Percentage contribution of components to load 1 unavailability, as CB is aging


3.43 Percentage contribution of components to load 1 EFLC, as CB is aging for


3.44 Percentage contribution of components to load 1 PLC/unavailability, as TXs and

CB are aging for network case “d”. 116

3.45 Percentage contribution of components to load 1 EFLC, as TXs and CB are


3.46 Percentage contribution of components to all loads unavailability, as CB are


3.47 Percentage contribution of components to all loads EFLC, as CB are aging for


3.48 Overall and single load unavailability due to transformers aging for medium

network. 122

3.49 Overall and single load EFLC due to transformers aging for medium

network. 123

3.50 Overall and single load unavailability due to circuit breakers aging for medium

network. 123

3.51 Overall and single load EFLC due to circuit breakers aging for medium

network. 124

3.52 Percentage contribution of components to load 2 unavailability, as source circuit

breakers are aging for medium network. 125

3.53 Percentage contribution of components to load 2 EFLC, as source circuit

breakers are aging for medium network. 126

3.54 Percentage contribution of components to load 2 unavailability, as circuit

breakers 1 and 2 are aging for medium network. 126

3.55 Percentage contribution of components to load 2 EFLC, as circuit breakers 1 and

2 are aging for medium network. 127

3.56 Percentage contribution of components to load 2 unavailability, as circuit

breakers 1, 2 and 4 are aging for medium network. 128

3.57 Percentage contribution of components to load 2 EFLC, as circuit breakers 1, 2

and 4 are aging for medium network. 128

3.58 Percentage contribution of components to load 3 unavailability, as all

transformers and circuit breakers are aging for medium network. 131

3.59 Percentage contribution of components to load 3 EFLC, as all transformers and

circuit breakers are aging for medium network. 132

3.60 Percentage contribution of components to all loads unavailability, as all

transformers and circuit breakers are aging for medium network. 132

3.61 Percentage contribution of components to all loads EFLC, as all transformers and

circuit breakers are aging for medium network. 133

4.1 Industrial feeder plot for 1 week. 142

4.2 Residential feeder plot for 1 week. 142

4.3 Predicted results versus actual results showing two anomalous peaks. 143

4.4 Weekday, Saturday and Sunday load plot for Industrial loads. 144

4.5 Industrial loads comparison. 145

4.6 (Sub)-Sectors within commercial sector. 147

4.7 Contribution values of Sub-Sectors for an unknown feeder. 149

4.8 Cross Validation Results with an Industrial feeder: weights o = industrial sector,

∆ = commercial sector, + = residential sector. 151

4.9 Cross Validation Results with a Commercial feeder: weights o = industrial sector,

∆ = commercial sector, + = residential sector. 152

4.10 Cross Validation Results for Residential feeder: weights o = industrial sector, ∆

= commercial sector, + = residential sector. 153

5.1 User and internal representations of an analytical simulation network. 162

5.2 Tripping automatic protection devices. 163

5.3 Manual isolation of components. 164

5.4 Closing of normally open switches. 165

5.5 Isolation of insufficiently supplied load points. 166

5.6 Active failure in a network. 167

5.7 Passive failure in a network. 168

5.8 Temporary failure in a network. 169

5.9 Maintenance outage in a network. 169

5.10 Solution for minimum cut set of a simple network. 170

5.11 Solution for shortest path maximum flow of a simple network. 172

5.12 Solution for priority maximum flow of a simple network. 174

5.13 Distribution network at RBTS Bus 2. 176

5.14 Distribution network at RBTS Bus 4. 183

5.15 Bulk Load 2 of medium sub in Figure 2.7, failure probability as TX2 aging with

two different mean lives. 198

5.16 Bulk Load 2 of medium sub in Figure 2.7, failure frequency as TX2 aging with

two different mean lives. 198

5.17 SAIFI index of RBTS Bus 2 distribution system – supplied from medium sub

load Bus 2 of Figure 2.7 as TX2 aging with mean life a) 45, b) 30 years. 200

5.18 CAIDI index of RBTS Bus 2 distribution system – supplied from medium sub

load Bus 2 of Figure 2.7 as TX2 aging with mean life a) 45, b) 30 years. 200

6.1 Sector customer damage function. 206

6.2 Composite customer damage function. 207

6.3 Small sub-transmission, Case “d”. 211

6.4 Medium substation. 214

6.5 Advance breaker and a half substation. 217

7 .1 High level flow chart gives the overview calculating process of reliability

indices. 224

7.2 Dialogue box for creating a new substation or distribution case study. 225

7.3 Screen view of substation software program, with component menu and

tool bar. 227

7.4 Screen view of distribution software program, with component menu and

tool bar. 227

7.5 Screen-view of substation components and the data input dialogue boxes, using

graphical users interface. 228

7.6 Screen-view of distribution components and the data input dialogue boxes, using

graphical users interface. 229

7.7 Screen-view of load point for both distribution substation, and the load data input

dialogue box, using graphical users interface. 230

7.8 Screen-view of the program pull-down menu bar and functions. 231

7.9 Screen-view of system wide component data entry for a) substation and b)

distribution, programs. 232

7.10 Screen-view of simulation settings for a) substation and b) substation &

distribution, programs. 232

B.1 Single Line Diagram of the RBTS 254

B.2 Single Line Diagram of the Distribution System of the RBTS Bus 2 255

B.3 Single Line Diagram of the Distribution System of the RBTS Bus 4 256

List of Tables 2.1 Substation Component Test Data. 43

2.2 Cases “a-e”, Percentage contribution of components to contingency

probabilities. 45

2.3 Cases “a-e”, percentage contribution of components to contingency

frequencies. 47

2.4 Overall, system load indices, for cases “a-e”. 48

2.5 Load point indices, for cases “a-b”, and each load in “c-e”. 49

2.6 Medium Substation Component Test Data. 50

2.7 Overall, system load indices, for the medium substation. 50

2.8 Load point indices, for the medium substation. 51

2.9 Percentage contribution of components to contingency probabilities. 52

2.10 Percentage contribution of components to contingency frequencies. 53

2.11 Percentage contribution of components to contingency probabilities. 54

2.12 Breakdown of load point indices, for the medium substation. 55

2.13 The number of system probability contingency events for loads and sources

in the medium substation. 56

2.14 The number of system frequency contingency events for loads and sources in

the medium substation. 57

2.15 Overall, system load indices, for the medium substation including capacity

limitation from the sources. 58

2.16 Each load point indices, for the medium substation with capacity limited

sources included. 58

2.17 Percentage contribution of components to contingency probabilities,

including capacity limitation of the sources. 59

2.18 Percentage contribution of components to contingency frequencies, including

capacity limitation of the sources. 60

2.19 Breakdown of load point indices, for the medium substation including

capacity limitation of the sources. 61

3.1 Reliability data used for Markov models. 83

3.2 System load unavailability of network case “a”, as components age. 94

3.3 System load EFLC of network case “a”, as components age. 96

3.4 System load unavailability of network case “d”, as components age. 102

3.5 System load EFLC of network case “d”, as components age. 104

3.6 Load unavailability of network case “d”, as components age. 105

3.7 Load EFLC of network “case d”, as components age. 107

3.8 System load unavailability of medium network, as components age. 119

3.9 System load EFLC of medium network, as components age. 120

3.10 Load unavailability of medium network, as components age. 120

3.11 Load EFLC of medium network, as components age. 121

3.12 Overall System Load Outage Indices, as all transformers and breakers are

aging in the medium sub. 129

3.13 Load 1/3 indices, as all transformers and breakers are aging in the medium

substation. 130

3.14 Load 2 indices, as all transformers and breakers are aging in the

medium sub. 131

4.1 Contribution of (Sub)-Sector Profiles for Industrial (Ind), Commercial (Com)

and Residential (Res) loads types. 148

5.1 Component failure rate, repair time and switching time data for distribution

network at RBTS Bus 2. 177

5.2 Feeder lengths in the distribution network at RBTS Bus 2. 177

5.3 Load point data of the distribution network at RBTS Bus 2. 177

5.4 Distribution line capacities in the distribution network at RBTS Bus 2. 178

5.5 RBTS Bus 2 sub-transmission line capacities. 178

5.6 Distribution system indices at RBTS Bus 2, using minimum cut set. 178

5.7 Distribution load point indices at RBTS Bus 2, using minimum cut set. 179

5.8 Distribution system indices at RBTS Bus 2, using shortest path maximum

flow. 180

5.9 Distribution load point indices at RBTS Bus 2, using shortest path maximum

flow. 181

5.10 Distribution system indices at RBTS Bus 2, using priority maximum flow.

182

5.11 Distribution load point indices at RBTS Bus 2, using priority maximum flow.

182

5.12 Component failure rate, repair time and switching time data for distribution

network at RBTS Bus 4. 183

5.13 Feeder lengths in the distribution network at RBTS Bus 4. 184

5.14 Load point data of the distribution network at RBTS Bus 4. 184

5.15 Distribution line capacities in the distribution network at RBTS Bus 4. 184

5.16 RBTS Bus 4 sub-transmission transformer capacities. 184

5.17 Distribution system indices at RBTS Bus 4, using minimum cut set. 185

5.18 Distribution load point indices at RBTS Bus 4, using minimum cut set. 186

5.19 Distribution system indices at RBTS Bus 4, using shortest path maximum

flow. 188

5.20 Distribution load point indices at RBTS Bus 4, using shortest path maximum

flow. 189

5.21 Distribution system indices at RBTS Bus 4, using priority maximum flow.

189

5.22 Distribution load point indices at RBTS Bus 4, using priority maximum flow.

191

5.23 Bus 1 load indices of Figure 2.6(d). 193

5.24 Bus 2 load indices of Figure 2.7. 194

5.25 RBTS Bus 2 distribution system – supplied from aging load Bus 1 of Figure

2.6(d), using minimum cut set. 195


2.7, using minimum cut set. 195


2.6(d), using minimum cut set. 196


2.7, using minimum cut set. 197

5.29 RBTS Bus 2 distribution system indices – supplied from medium sub load

Bus 2 of Figure 2.7 TX2 aging with mean life a) 45, b) 30 years. 199

6.1 Sector customer damage functions for the seven customer categories. 207

6.2 Composite customer damage function for the customer mixture. 208

6.3 Case “d” EIC evaluation for replacement target time, TX1 & CB aging. 212

6.4 Case “d” major reliability indices, TX1 & CB aging. 212

6.5 Case “d” EIC evaluation for replacement target time, TX1 & TX2 aging. 213

6.6 Case “d” EIC and replacement target time, TX1 & TX2 aging with 10% yearly

load increase. 213

6.7 Med Bus EIC and replacement target time, TX2 aging with 12MW load. 215

6.8 Med Bus EIC and replacement target time, TX2 aging with 24 MW load. 215

6.9 Medium Bus major reliability indices, TX2 aging. 216

6.10 Breaker and a half substation, EIC and replacement target time, with circuit

breakers aging. 218

6.11 Medium Bus major reliability indices at load 2, as TX2 aging. 219

6.12 Replacement time by application to RBTS Bus 2 distribution. 219

6.13 Replacement time by application to RBTS Bus 4 distribution. 220

B.1 Peak Loads in the RBTS 257

B.2 Feeder Lengths 257

B.3 Customer Data 258

B.4 Loading Data 259

B.5 Reliability and System Data 260

List of Principle Symbols & Acronyms

λ Failure rate (occ/yr)

λa Active failure rate (occ/yr)

λp Passive failure rate (occ/yr)

λm Maintenance outage rate (occ/yr)

λag Aging failure rate (non-repairable) (occ/yr)

µ Repair rate (occ/yr)

µm Maintenance repair rate (occ/yr)

µSW Switching rate (occ/yr)

µag Aging outage rate (non-repairable) (occ/yr)

f Frequency

f(t) Failure density function

P Probability

Pi Probability of sate i

r Repair time (hours)

TSW Average switching time (hours)

Tm Average maintenance time (hours)

TR Average replacement/refurbishment time (hours)

sw Switching action

L Average load (MW) or; (kW)

E Average energy consumed by load

Cindex Percentage contribution to an index

CWindex Percentage contribution to an index, load weighted

TX Transformer

CB Circuit Breaker

S Source

LDC Load Duration Curve

PLC Probability of Load Curtailments

PSD Probability of Source Disconnection

EDNS Expected Demand Not Supplied

EFLC Expected Frequency of Load Curtailments

U Unavailability

ADLC Average Duration of Load Curtailments

EENS Expected Energy Not Supplied

SAIFI System Average Interruption Frequency Index

SAIDI System Average Interruption Duration Index

CAIDI Customer Average Interruption Frequency Index

ASAI Average Availability Index

SCDF Sector Customer Damage Function

CCDF Composite Customer Damage Function

EIC Expected Interruption Cost

CRF Capital Return Factor

RBTS Roy Billinton Test System

GUI Graphical Users Interface

C# C Sharp, computing language code

Statement of Original Authorship

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

requirements for an award at this or any other higher education institution. To the

best of my knowledge and belief, the thesis contains no material previously

published or written except where due reference is made.

Signature ________________________________

Date______________________________

Acknowledgement

I would firstly like to thank my supervisors, Professor Gerard Ledwich, Professor

Lin Ma and Professor Andy Tan for their support and guidance throughout my

PhD. Thanks also to Dr. Keith Hoffman for his assistance.

I would also like to thank Professor Doug Hargreaves and my friend and

colleague Associate Professor Firuz Zare for their sincere support and

encouragement, especially since last year.

Last but not least, I particularly would like to thank my dear wife Sousan who

endeavoured many hardships in this journey and my boys for their support and

understanding.

Ghavameddin Nourbakhsh

Queensland University of Technology

January 2011

1

Chapter 1

Introduction

1.1 Motivation and Overview

This thesis presents an investigation into the effects of non-repairable aging

equipment on the reliability of substation and sub-transmission power supply systems.

These networks are normally composed of many components including power

transformers, high voltage high breakers, bus bars, manual isolators, normally open

switches and transmission lines. Many of these facilities in existing power supply

industries are aged, degrading and close to their mean functional life cycle. Given

this situation, the increase of failures can lead to wide spread outages, the loss of

supply to many customers, and a high investment cost associated with equipment

replacement and/or refurbishment. This places further pressure on the decision

making organisations.

Recently, power system reliability research has begun to address ways to deal with

this problem more objectively. Various areas of research have addressed the aging in

their own right, such as; condition monitoring, preventive maintenance, proactive

maintenance, reliability centre maintenance and power system reliability, etc. The

task of estimating risk criteria in conjunction with a calculated economic decision is

the centre piece of most approaches.

Many existing older electricity networks have substations and sub-transmission

systems that may have facilities that are aged and non-repairable and these can pose a

high risk of causing wide spread black out. This problem can even pose an even

higher risk if the substation and the networks that it is serving do not enjoy an

2

adequate level of redundancy. Most equipment used in power systems, like power

transformers, are expensive and normally a spare is not readily available and may take

a long time before it is ordered or delivered. Refurbishment is also costly and time

consuming, and therefore systematic and advance planning is required to minimise the

risk of financial losses and maximise reliability.

In the remainder of this chapter the aims and objectives of this thesis will be

described. The scope of the thesis will be defined, and an outline of the thesis will be

presented. Finally, the contributions made in this thesis will be stated.

1.2 Aims and Objectives

The aims and objectives presented in this thesis are to:

1. Improve a non-repairable aging failure methodology and include it as part of a

Markov model which is used to predict the probability of failures in electrical

power substations and sub-transmission systems The inclusion of this new

methodology, which incorporates the ‘useful life’ of a power system

component (known as random failure mode) with an aging model, allows both

the frequency and duration related indices to be determined.

2. Include non-repairable aging contingencies as part of the enumeration method,

including maintenance, switching and repair.

3. Revise the contingency selection so as to reduce computational intensity while

still retaining accuracy.

4. Evaluate the contribution of the non-repairable/aging equipment to bulk load

indices.

5. Identify the distribution feeder customer load types and classifying customer

and sub-customers load types, classifying unknown feeder customer loads into

similar types using correlation between the known and the unknown loads then

decompose the unknown load profiles to known types.

3

6. Evaluate the effect of substation/sub-transmission non-repairable aging to

distribution customer load points. After the evaluation corrective action can

be considered upon exceeding a certain limit of risk indices.

7. Include the reliability methods and approaches developed in this thesis in an

economic appraisal to determine the most advantageous replacement time of

non-repairable components, in a power system.

8. Develop software tools as part of the thesis research work.

The methods proposed within this thesis are designed to work together in fulfilling the

ultimate aims and objectives of the thesis, as noted earlier. The non-repairable aging

components are modelled using explicit states by Markov’s state space model which

is incorporated as part of power system reliability evaluation. This study is brings

together the system configuration and individual equipment reliabilities to form a

combined solution for power network reliability problems.

Some of the components of a power network may be in their ‘early life’ stage but the

probability and frequency of entering a ‘non-repairable’ state increases with age. This

model is also designed to fit such transitions.

In addition, the performance of a power network is measured against the degree in

which it is able to fulfil its intended goal, which is to supply loads as reliably and

economically as possible. The trade off between reliability and economy is best

measured at the distribution customer’s load points.

4

1.3 Original Contributions

The original contributions made in this thesis include:

1. The development of algorithms for a failure frequency and duration approach

to power system reliability, to measure the contribution of equipment

reliability to system load and source disruption as indicators. The advantage

of using these indicators is shown using extensive application examples.

2. The introduction of a non-repairable aging state into the Markov approach,

which is explicitly modelled as part of the normal life mode of power system

equipment. This model uses frequency and duration method to evaluate

network reliability. The advantage of this model with respect to earlier

developments is to include non-repairable aging as part of maintenance, repair

and switching states, and to be able to include these as part of the frequency

and duration evaluations.

3. A revised contingency enumeration method for the frequency and duration

technique was developed to include the aging components. This model was

developed to fill the gap in addressing the non-repairable aging condition in

the frequency and duration approach. In capturing non-repairable aging

effects, selected triple contingencies are added to the evaluation technique.

The effect and the contribution of non-repairable aging equipment to load

indices are evaluated using the algorithms developed in the first contribution

to this research thesis. Extensive applications examples are illustrated to show

the advantages of this model.

4. The use of statistical methods to determine load types. A customer load type

and its average load is essential information in power system reliability

evaluations, specifically in a distribution network. A distribution load is

normally measured at a main substation supply point with the type of

customers being served, and the average load of each customer sector

contributing to the measured value, usually unknown. This research has

5

recognised the gap and statistical methods were used to estimate the ingredient

of the measured load. Using statistical clustering, correlation and validation

approaches, an unknown distribution customer feeder load is decomposed into

the summation of known load types.

5. The development of an application to make a link between the non-repairable

aging facilities in higher voltage level serving a distribution network. The non-

repairable aging failure effects of facilities in substation and sub-transmission

systems supplying distribution network are critical for adequate planning of

corrective actions. The effect of substation/sub-transmission non-repairable

aging to distribution customer load points is estimated.

6. A systematic cost benefit method was developed to determine the optimal

replacement time of non-repairable equipment in high voltage substation and

sub-transmission systems. Customer outage costs and equipment investment

for a replacement are used in this method to find a year in the future planning

when the estimated economic benefit or risk criteria indicate the replacement.

1.4 Literature Review

The basic function of an electric power system is to supply electrical energy to its

customers as economically as possible with a reasonable degree of continuity, safety

and quality [1]. Modern society, because of life styles and work requirements, has

come to expect the supply to be continuously available on demand. This creates a

difficult problem of balancing the need for continuity of power supply and the cost

involved in providing that continuity. The relationship between the investment cost

and reliability is not normally a linear function, as demonstrated in Figure 1.1.

During initial stages of improving reliability of a power system a small level of

expenditure may result in an appreciable improvement in reliability. However, over

time expenditure can increase but the incremental improvement in reliability

decreases as a result of aging equipment [1, 2].

6

Reliability Worth

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23

Investment cost (k multiple of $)

Rel

iab

ility

Reliability achievement

Incremental Reliability w orth

Fig 1.1 Reliability worth, with investment cost.

There needs to exist a trade off between the investment cost and the improvement in

reliability. The optimal decision is not easily determined and may not be the same for

all systems. The ideal planning target in this regard depends on various factors such

as the network configuration, cost of outages to customers, equipment reliability and

investment cost for reliability improvement.

Power utility engineers have always tried to provide a continuous and high quality

electricity supply to their customers and to make the supply available with minimum

cost. To achieve this, a balance has to be reached between the amount of investment

needed to improve reliability and the actual level of reliability achieved or achievable.

1.4.1 Power System Reliability

In power systems, the term reliability refers to the ability of a power system to

provide customers with an adequate supply of electrical energy [3]. The concept of

Reliability improvement versus investment cost

Investment cost

7

power system reliability is very broad and not associated with an exact definition but

does include all aspects of the ability of the power system to satisfy the customers’

load requirements. However, power system reliability evaluation can classified into

the two domains of adequacy and security assessments [1, 4-6], as shown in Figure

1.2.

Fig 1.2 Subdivision of Power System Reliability.

‘Adequacy’ relates to the existence of sufficient facilities within the power system to

satisfy the customer load demand. This includes the necessary facilities to generate

sufficient electrical energy and the associated transmission and distribution physical

assets required to transfer the energy to the actual load points. Therefore, adequacy is

related with a system static condition. ‘Security’ however, relates to the ability of the

system to respond to disturbances. As a result, security is concerned with the

response of the system to whatever perturbations it is subjected to. These include the

conditions associated with both local and widespread disturbances and the loss of

major generation and transmission facilities. Consequently, adequacy involves the

steady state post outage analysis of power systems whereas security involves the

analysis of both static and dynamic conditions.

1.4.2 Functional Zones and Hierarchical Levels

In the modern world electric power networks are extremely integrated and normally

extensively vast and complex. The electrical energy is generated by various

Power System Reliability

System Adequacy System Security

8

generating units and supplied to load customers through proper transmission and

distribution facilities. It is not practical, nor is it deemed worthy, to analyse the

complete power system as a one entity. It is therefore feasible to divide a power

system into segments that can be closely examined but separately. These segments

are called “functional zones” [1]. The three basic functional zones for the purpose of

planning, organization and analysis are: Generation, Transmission and Distribution,

as shown in Figure 1.3. These functional zones can be combined to form a series of

hierarchical levels for the purpose of performing system reliability analysis. For

example the evaluation of the generating facilities is designated as hierarchical level I

(HL I) assessment while the assessment of the composite generating station and

transmission system is designated as a hierarchical level II (HL II) study. The

assessment of the complete power system including the distribution functional zone is

designated as hierarchical level III (HL III) appraisal.

Fig 1.3 Hierarchical levels for power system facilities.

Generation Facilities

Transmission Facilities

Distribution Facilities

Hierarchical Level I

Hierarchical Level II

Hierarchical Level III

9

Classically there are two main approaches in the design, planning and operation of a

power system to maintain high degree of supply continuity, namely; deterministic and

probabilistic. A deterministic approach typically includes the following information:

• The installed capacity which is equal to the expected maximum demand,

• Or, the spinning capacity which is equal to the expected load demand

• Plus a reserve margin equal to one or more of the largest units,

• And commonly an n-1 criterion which means that the system must meet the

demand even with the loss of one piece of equipment.

These and other similar methods are deterministic and allow for chance of failures.

The major downside to these approaches is that they do not and cannot account for the

probabilistic nature of system behaviour, customer demands or of component failures.

In other words, the use of a deterministic approach can lend itself to an over or under

estimation of reliability during design, planning and operational phases which can

introduce substantial monetary mismanagement.

Stochastic techniques however capture and incorporate the inherent probabilistic

nature of the network components in power system reliability evaluations. This

approach is explained and applied to substation and sub-transmission systems in this

thesis with detailed illustrations and examples.

1.5 An Overview of Probabilistic Reliability Evaluation Developments of

High Voltage Network

Different reliability theories are developed and applied in different areas of research

such as the fields of aerospace and avionics, communications, computer hardware and

software, the manufacturing of components and systems etc., and also power system

reliability analysis. Among these areas, although the core principals and the very

10

basic concept theories may be the same or similar, the extended research development

models and methods and the application techniques can be quite different. This is

understandable, as diverse fields and practices may have their own set of required

functions, process considerations and operational requirements. Due to overwhelming

diversity of reliability concepts, theories and applications that have been developed,

used and applied in various fields of study techniques will differ. Therefore, this

section will focus on the background developments of reliability theory in the field of

power system reliability. In particular, an emphasis is placed on electrical power

substation and sub-transmission systems, as part of composite high voltage generation

and transmission facilities.

Similar to the hierarchical structure of power systems shown in Figure 1.3, the

techniques developed for reliability evaluation can be classified in terms of their

application to the same structure and with the corresponding functional zones. Over

the last six decades reliability appraisals for different hierarchical levels and

functional zones has undergone continuous development and application. Research

papers [7-14] published in IEEE Transactions present extensive research

developments on the reliability assessment of power systems, using probabilistic

approaches. This thesis uses the references to develop new concepts which focus on

reliability evaluation at the HL II level and the effects on the distribution functional

zone.

Reliability assessment of power systems in North America and Europe was largely

concentrated on the generation area prior to 1960’s. It was however, from this date

that composite generation and transmission system reliability evaluation started

developing. This approach used adequacy evaluation of a composite generation and

transmission system and involved the simulation and computation of the system

conditions for each possible outage condition in order to determine system operation

violations.

11

Two different methods, ‘contingency enumeration’ and ‘Monte Carlo simulation’,

have been applied to the reliability assessment of power systems and are two

fundamentally different approaches to reliability assessment. Considerable research

in both the areas of simulation and contingency enumeration has been published in the

literature [15-19]. The comparison between these to method indicates the conceptual

differences in modelling and problem perception and allows a better understanding of

the merits and demerits of the two approaches. For example, the enumeration method

provides an exact result whereas the Monte Carlo technique gives converging values

around a mean. The Monte Carlo method can incorporate operational constrains and

conditions much easier that the enumeration method but the enumeration method is

normally faster in computational time. However, the distribution of results and more

information can be extracted using the Monte Carlo approach.

The reliability evaluations at HL II level is a complex process. Determination of the

effect of unexpected contingencies may require successive load flow and

corresponding operational simulations, such as switching actions. The purpose of the

simulations is to determine the contingencies that ultimately contribute to load

outages. Part of the simulation is to determine if the system is able to withstand some

of the contingencies, without violation of the constraints. The composite power

system reliability assessment can include a security evaluation, whereby steady state

and transient effects can be incorporated.

An overwhelming majority of the research work in power system reliability does not

consider equipment degradation as part of system evaluations. In some applications

simple sensitivity studies are performed by varying failure, repair time and

maintenance parameters to give some indication for aging. This approach is very

crude and does not correspond to any failure characteristics and is not modelled

properly in the calculation processes. There is a developing research area that has

started to address the aging modelling and considerations in power system reliability.

This emerging interest and development is in a direct response to the current state of

12

most electricity utilities which are confronted with increasingly non-repairable aging

equipment.

1.6 Development of Scope of the Thesis

Most electricity networks have been built and established in last decades. Many of

these facilities are still operating but are aging and getting close to their mean

functional life where failures are normally due to fatigue and degradation and are

considered non-repairable. Consequently, there is an increasing interest in using

aging considerations in power system reliability assessments. This interest is

naturally focused on the network facilities where failure can have the most adverse

effect and cause system outages. These facilities are normally costly and their

replacement is very time consuming particularly if advance ordering, shipping and

delivery is required.

The research gap in the consideration of aging of power system facilities has been

recognised. Currently, there are research publications that have developed techniques

to address certain aspect and applications of aging in power systems [20-40].

However, there is very limited research literature available on the aging of power

networks [24-27, 30, 31, 37]. A few papers [24-26] use non-homogeneous Poisson

Process characterization of the aging components and have applied a sequential

Monte Carlo method for the evaluations. In these works, the model used for the aging

characteristic did not include evidence of a good fit for the actual power system

equipments aging attribute. An additional drawback can be the excessive

computation time for the Monte Carlo technique.

The Bayesian Network method is also cited as having addressed aging in power

system reliability [22, 23]. This method developed an evaluation technique using

conditional probability, with cut set, tie set or fault tree methods employed as part of

13

modeling and evaluations. Additionally, this modeling of a power system network

can be complicated and it is not readily achievable. As noted before, the use of cut

set, tie set and fault trees in power system Bayesian Network analysis can also

introduce an extensive burden on computations while reliability modeling is only part

of the considerations. This method may be suitable for certain applications, but it can

impede operational requirements and considerations that are part of the power system

reliability evaluations.

Another simple probabilistic method is developed and used by WenYuan Li [30, 31].

In this method, probability distribution suited for aging components are used to obtain

aging probabilities for each year of study. Using the combination of each component

for yearly probabilities will yield a set of system states for each year of study. Each

system state effect in a year is evaluated and combined to assess the system. Using a

union concept, Li has combined the useful life and aging periods of a component with

repair considerations. However modeling maintenance and switching considerations

in this way may not be readily applicable. In addition, the method does not include

frequency as part of the model. In this thesis, an aging model is developed as part of

the frequency and duration approach using the state space Markov model.

The basic objective of the research work described in this thesis is to model and

examine the significant contributions and the effects of non-repairable aging

components in substation or sub-transmission power networks. Specifically, studies

on bulk load supply points and customer load points in distribution systems, including

the risk of outages and the financial appraisals associated with replacements, are

targeted. The cost worth analysis is included in number of published papers [41-47]

related to power system reliability, where largely the worth of incremental reliability

were considered. In this thesis, a new systematic approach in cost worth model is

developed for the replacement decision on the non-repairable aging components. In

this model, the reliability attributes of facilities such as supply generating units,

transmission line facilities and their protection, control devices like circuit breakers,

14

bus sections and transformers are included in the assessments. Amongst these

facilities, the emphasis is placed on substation equipment where their non-repairable

aging can have large financial and risk consequences. The research developments

carried out in this thesis are in the area of adequacy with non-repairable aging

reliability evaluation and using a frequency and duration approach. More closely

related work in this area of research is described in References [30, 31].

Significant techniques, approaches and tools that have been developed as a result of

this research work are presented in Chapters 2-6. These techniques are:

• Percentage contribution of component failures to substation and sub-

transmission systems risk indices,

• Inclusion of equipment random and non-repairable aging failures in reliability

evaluations,

• The impact of non-repairable aging failures on bulk load supply and customer

load points,

• Customer load classification and unknown feeder load decomposition,

• Replacement decision using risk and financial appraisals.

These studies are illustrated by application to published test cases and IEEE

Reliability Test Systems.

1.7 Review of Outline of the Thesis

The motivation and overview of the thesis research topic and the aims and the

objectives of the research work are discussed in the earlier part of this chapter. This is

followed by a brief presentation and specification of the scope of the thesis. Original

15

contributions the research work is then summarized. Finally, a thesis outline is

presented.

Previously published literature on power system reliability evaluation methods is

reviewed in this chapter. The publications show that the methods associated with

power system reliability evaluation are classified by either being deterministic or

probabilistic. Background development in the research literature on composite power

systems is then discussed. This is because, the topic of this research contribution

relates to substation and sub-transmission facilities which are a part of composite

power systems. Further, the research literature relating to degradation and non-

repairable aging in power systems is discussed and a gap in the research associated

with this area is introduced as the basis for the research in this thesis.

In order to show the newly developed methodology in this thesis, the standard

frequency and duration technique used in reliability studies is explored and reviewed

in Chapter 2. A detailed application of the system state enumeration method for

power systems, using Markov Space State, is presented. Here the major operational

states of components with transitional rates moving from one state to the next are

illustrated. The individual states considered are; operating, switching, repair and

maintenance and these can be combined to form system states. The combinations for

a moderate size system will run into an unreasonable number of system states which

are not only impractical to solve but are unnecessary, as a large number of these

system states have insignificant probability value. The applications in Chapter 2 use

single and the double contingencies because the probability for the higher order

outages are insignificant in a power system’s evaluation.

Given the network configuration and the reliability data for the individual components

and the load values, a system state failure mode is developed for the network and the

resulting effects are evaluated and the final indices in Chapter 2 consolidated. Two

flow methods are incorporated in this chapter, connectivity and maximum flow

16

methods. The connectivity method only examines if there are links established

between the sources and the load. Maximum flow takes the capacity or the rating of

the network components into consideration in examining if the loads can be supplied.

Maximum flow is shown to be useful for networks that may have capacity limitations.

Numerous load and source points and system indices are evaluated and illustrated in

this chapter.

As part of the research work in Chapter 2, it is recognized that the network topology

and the components reliability data together with other constraints in the network such

as limitation of capacity can influence the outcome indices. From the design,

planning and asset management point of view it is very valuable to identify the

importance and the effects of individual components in system reliability

measurement. This information can assist system designers, operators and those who

maintain and manage it to allocate resources accordingly and to balance the reliability

with costs in an effective way. Recognizing the gap in the frequency and duration

approach, exclusive algorithms related to the enumeration technique are developed

and systematically incorporated as part of the contribution to this research work.

Extensive examples are illustrated in Chapter 2 to show the advantage of this method.

The indicators developed include; the percentage effect and the contribution of each

component to the individual load and source outages, as well as the overall system

loads and the weighted load indices. The values are systematically collected during

the system state enumerations using the approaches provided in this research work.

A new technique for non-repairable aging, as an exclusive state in Markov model is

developed and illustrated in Chapter 3. In this model an extra state is introduced to

exclusively designate non-repairable aging. This state is described as a state where,

once a component transits from its operating state to this non-repairable aging state

(defined by aging fatigue and degradation or any other irreversible fault), the

component must either be refurbished or replaced. Therefore, returning from this

non-repairable aging state to an operating state is entails an average time to refurbish

17

or replace factor. The failure from the operating state to a non-repairable aged state is

associated with an increasing failure rate with time, which is usually associated with a

normal or a Weibull distribution function. The Markov state space model is designed

to include a non-repairable aging state, as a constant for each year of computation,

while the failure rate increases year by year. The new model developed here is shown

to include both modes of equipment life cycle that is, random failure and the

increasing non-repairable aging failure modes. The increasing failure rate of the non-

repairable state will introduce yearly changes to state probabilities, that is, while a

component is in its early life period aging effects are almost absent but will

continuously indicate an increasing effect due to non-repairable failures. The

advantages of this method are discussed later in this chapter.

A set of third order contingencies, including aging system states, are included to

provide an adequate level of accuracy in the evaluated results. This addition has a

minimal influence on computing time but does increase the accuracy of the calculated

indices. The component failure contribution to probability and frequency indices are

upgraded to include the new non-repairable aging state and the triple contingencies.

The network calculations are designed to consider and execute the assessment over a

number of planning years but displaying the complete output results for each year of

evaluation may be impractical. In addition calculations to determine selected yearly

indices and yearly component failure contributions to some of the critical indices are

demonstrated in chapter 3. This method has proved to be extremely useful, where a

wide range of information is efficiently captured and utilized.

Chapter 4 is devoted to customer load analysis and evaluations. The customer loads

are normally measured at the 11KV distribution feeders nearest to the electricity

supply customers. However, at this point there is no information about the customer

load types and their profiles. This information can be very important in reliability

studies as well as cost/benefit evaluations of electricity networks that provide power

to consumers. This information together with increasing effects of substation and

18

sub-transmission network outages on distribution system due to non-reparable aging

can provide very useful information for planning and reinforcing decisions.

Information about the loads can greatly assist in a range of cost/benefit assessments

that, together with risk indices, can justify managerial decisions.

In chapter 4, an approach is developed and used to facilitate the decomposition of

unknown load profiles that are measured at the 11KV feeders. The decomposing

method is used to extract information about the type of (sub)-sector/customer loads

that contribute to a feeder load. This method is tested and compared with 11 KV load

data provided by a local electricity supply company.

Additionally, the method developed as part of this research should prove to provide

further benefits for electricity suppliers in areas such as; economic load management,

load diversification relating to electricity block purchasing, load forecasting, retailing

and strategic network planning.

A sound decision plan for electricity networks is normally the one that can be directly

related and measured in some ways with customer load reliability attributes. The

focus of this thesis is on the non-repairable aging considerations at substation/sub-

transmission system, where the network is complex and highly interconnected.

However, reliability assessment of combined composite and distribution systems are

extremely difficult and is not practical. In Chapter 5, the effects of outages in

substation or sub-transmission and how they impose on the customer loads in a

distribution system are studied.

The impact of aging for higher voltage facilities is normally measured at the bulk load

buses but these points are still a distance away from distribution customer loads.

Reliability evaluations at the distribution level, where the network is commonly

arranged in a radial form, is different and has a different set of customer related

indices. Ultimately, the outages experienced due to aging in higher voltage networks

19

are seen at the customer load points. It is therefore very useful to include the aging

effects of the facilities in substation/sub-transmission networks which supply power to

distribution customers, as part of the distribution reliability assessments.

Initially detailed analytical reliability evaluations for distribution systems are

reviewed in Chapter 5. Two flow approaches were examined as part of the

evaluations and two methods of load shedding are developed for distribution systems

as part of this research work and applied to achieve the results show in this chapter.

The provision of capacity limitation in distribution reliability assessments and the

outages caused due to aging facilities at higher voltage network can prove to be useful

in long term planning, where feeders and transformers may enhance the nominal

capacity as the system as expands to accommodate load growth. Detailed results for

both RBTS distribution systems at Bus 2 and 4 are provided for all methods in this

chapter together with related discussions on the findings.

The inclusion of substation/sub-transmission aging as part of distribution reliability

evaluations is included in Chapter 5, and the results for both RBTS distribution

systems at Bus 2 and 4 are provided. The effect of aging is shown as part of the

distribution system indices; SAIFI, SAIDI, CAIDI, ASI and ENS. A comparison of

the results provide useful indicators and measures that assist in making corrective

planning decisions, where cost/benefit consideration can systematically be included.

Chapter 6 shows the development of a risk based economic model used to evaluate

the replacement time for non-repairable components in a power system. The trade-off

between the customer outage costs due to non-repairable failure of aging components

and the investment cost for replacement are the bases in the evaluations. Failure at

substation or sub-transmission systems can cause outage to the customers in

distribution system. There are different types of electricity customers and the

financial losses due to outage duration are different for the customer sectors. On the

other hand, the equipment in high voltage power systems are extensive and requires

20

long times of manufacturing and shipping to be available to install. All these factors

are considered in replacement time evaluation, based on risk based economic

modeling of non-repairable aging components. Various techniques and developments

provided in this thesis are included and used in the economic model developed and

applied in Chapter 6. Considerable examples are used to illustrate the method, with

various applications.

A brief presentation and illustration of the software developed as part of this research

is provided in Chapter 7. Two software programs were designed and coded for

substation/sub-transmission and distribution network reliability evaluations. The

software programs have graphical user interfaces and software codes written using

Microsoft C sharp Object language. The software is efficient, user friendly, includes

numerous useful and valuable applications and functionalities and has many flexible

network applications and evaluation options.

Random failure and non-repairable aging modes for components is successfully

incorporated in the substation software. The program output indices have been

successfully tested against practical field results illustrating that this application

software is very useful and instrumental in the research work presented here.

Chapter 8 provides conclusions and recommended future research on this topic.

21

Chapter 2

Contribution of Equipment to Power System Reliability Performance

using the Enumeration Method

2.1 Introduction

There are a considerable number of publications that use Markov State Space models

to evaluate power system network reliability [7-14]. The main focus of these research

works are on the load points and overall system reliability performances, where

normally a series of sensitivity studies are conducted to examine the effect of

individual component attributes, such as; failure and maintenance rates, repair,

switching and maintenance times, etc., on the system reliability indices.

There are two major contributing factors that affect the reliability of an electricity

network. These are; the reliability data attributes of individual components and the

way the network is configured. Thereby, the location of equipment in a network can

also have different levels of criticality in contributing to the load outages [48-60].

The methods developed and used in these research works are attributed to cut set

method theories. However, the cut set techniques and the Bayesian methods [22, 23,

61-68] that use cut set in network reliability have limited applications in power

systems. This is due to the fact that consideration of cut set in a network can lead to

huge unnecessary calculations that make the process impractical. In addition, the

components in a power system may have multiple state representations while other

aspects of power system operations also need to be considered. In this regard, the

reliability evaluation becomes a part of power system assessment, and that can pose

many limitations on the method. The enumeration method [15, 17-19, 69-74]

however is designed to relax the calculations by selecting in advance the

22

contingencies which together with the operational and switching actions necessary,

are combined before using the flow algorithm. In this respect, there is a need for

calculating the importance or the contribution of components in the power system

reliability algorithms, in conjunction with the enumeration technique. In this chapter,

a few unique and new algorithms are developed and applied using the Markov state

space enumeration method [1, 28] and these provide the evaluation of percentage

contribution of each component to load outages. This technique can be extremely

useful in power network reliability design and planning, reinforcement and asset

management. Generally most decisions which need to be made at the component

level, take into account the equipment location and its effect on network reliability.

The reliability software applications in this thesis are tested with substation and sub-

transmission configurations. These network facilities are important connections

between a generation and bulk transmission supply system, and between transmission

and distribution networks. Substations and sub-transmission configurations are used

to link supplies and loads in a way as to enhance service reliability and security,

operation flexibility, simplicity, easily support future extensions and modifications,

etc. In general, the substations have a variety of interconnections and as such, they

are very good examples to examine the complexity of an interconnection. However,

this approach can equally be applied to any other power system network as well, as

will be shown in later chapters.

2.2 State Space Model for Substation sub-transmission

Components

The substation and sub-transmission components may include; generators,

transmission feeder connections, bus sections, circuit breakers (both normally closed

and normally open) and isolators and transformers.

23

The computations in this thesis are, as discussed earlier, based on a frequency and

duration method using state space Markov models, with constant transitions from one

state to the other. Each component can have four states, namely; an operating state,

switching state, repair state and maintenance state. This is shown in Figure 2.1.

Fig. 2.1 General four sates model representing substation and sub-transmission

equipment modes.

Where, λa, λp, λm, µsw, µr, µm are active failure, passive failure, maintenance transition,

switching transition, repair transition and maintenance transition rates, respectively.

An active failure, λa refers to a permanent outage and will activate automatic

protection switches when, for instance, a short circuit occurs. Upon an active failure

event, the state of the component moves from a fully operational (up) state to a

switching state before going to repair state. The switching state is defined by all

manual/automatic protection actions that take place which isolate the faulty

component(s). The time spent in this state is the average isolation switching time

(Tsw) and the transition from the switching state to the repair state is designated as µsw.

A passive failure, λp is referred to as a failure event that will not activate protection

devices, that is, failures that cause an open circuit. In this case, the state of the

component moves from fully operating (up) directly to the repair state. The time a

µsw

λa

µr µm

λp λm

Maintenance outage state

Repair state

Up state

Switching state

24

component spends in the repair state is equal to the average repair time (Tr) and the

transition from repair state back to up state is designated as µr. Component states can

also transit (λm) from fully operational to a maintenance state, which is referred to as

random maintenance in long term planning. The time component spent in a

maintenance state, Tm is the average scheduled maintenance time and the transition

from maintenance state back to up state is designated as µm.

2.2.1 Markov State Probabilities

There are number of techniques [75] available to obtain the probabilities associated

with every state in Figure 2.1. A frequency balance approach is used where the

frequency of leaving and entering a state are made equal. Using this concept and

applying it to the state space diagram in Figure 2.1, will give:

mmrrmpaup PPP µµλλλ +=++ )( (2.1)

aupswsw PP µµ = (2.2)

sspuprr PPP µµµ += (2.3)

mupmm PP λµ = (2.4)

1=+++ mrswup PPPP (2.5)

Manipulating equations 2.1-2.5, results in the state probabilities, shown in Figure 2.1:

25

mswrpamswamrmswr

mswrupP

λµµλλµµλµµµµµµµµ

++++=

)( (2.6)

mswrpamswamrmswr

amrswP

λµµλλµµλµµµµµλµµ

++++=

)( (2.7)

mswrpamswamrmswr

pamswrP

λµµλλµµλµµµµµλλµµ

+++++

=)(

)( (2.8)

mswrpamswamrmswr

mswrmP


++++=

)( (2.9)

Where, Pup, Psw, Pr and Pm are the state probabilities associated with up, switching,

repair and maintenance states, respectively. To be able to make use of or simplify the

general formulas above, certain modifications can be made. For example, if there is:

▪ No passive failure, pλ = 0

▪ No maintenance, mµ = 1 and mλ = 0

▪ No maintenance and no passive failure, mµ = 1 and mλ = pλ = 0

▪ No switching and no passive failure, swµ = 1, aλ = pλ = 0)

▪ N maintenance and no switching, mµ = swµ = 1 and mλ = swλ = 0

Making these modifications allows the relevant terms to be removed from the

component states.

For example, if the rate of a component entering the maintenance state mλ = 0, then

the probability of maintenance Pm becomes zero, which is what was desired from the

equations.

26

2.2.2 Enumeration of Selected System States

The enumeration method incorporates single and double contingency situations within

the computations. Single contingencies considered in the calculations are:

• a switching (S) state i (active failure) and,

• a repair (r) state i.

Double (overlapping) contingencies considered are:

• a switching (S) state i of a component overlapping with a switching (S) state j

of another component,

• a repair (r) state i of a component overlapping with a repair (r) state j of

another component,

• a switching (S) state i of a component overlapping with a repair (r) state j of

another component,

• a maintenance (m) state i of a component overlapping with a switching (S)

state j of another component,

• a maintenance (m) state of a component overlapping with a repair (r) state j of

another component.

The single and the double system contingencies can be formed using component state

probabilities as per the frequency and duration method. The probability expressions

for single and double contingencies [16, 17, 28, 76-92] are shown in the following

Equations 2.10-2.16:

( ) ∏≠=

=n

ikk

uksi PPsiP1

( )ni ,...,1= (2.10)

( ) ∏≠=

=n

ikk

ukri PPriP1

( )ni ,...,1= (2.11)

27

( ) ∏≠=

=n

jikk

uksjsi PPPsjsiP

,1

, ( )ijnji ≠= ;,...,1, (2.12)

( ) ∏≠=

=n

jikk

ukrjri PPPrjriP

,1

, ( )ijnji ≠= ;,...,1, (2.13)

( ) ∏≠=

=n

jikk

ukrjsi PPPrjsiP

,1

, ( )ijnji ≠= ;,...,1, (2.14)

( ) ∏≠=

=n

jikk

uksjmi PPPsjmiP

,1

, ( )ijnji ≠= ;,...,1, (2.15)

( ) ∏≠=

=n

jikk

ukrjmi PPPrjmiP

,1

, ( )ijnji ≠= ;,...,1, (2.16)

Where, associated switching, repair, maintenance and up state probabilities of i, j and

k components are used in Equations 2.10-2.16. The flow method, that is, the

transportation model [93, 94] links the sources to sinks (in this case the loads), and is

used for every system state contingency. Each system state, single or double

contingency, is carefully examined during the enumeration to determine the required

switching actions before the flow it executed. For example, if a component has

actively failed the circuit breakers adjacent to the failed component are opened before

a flow analysis is conducted. The maintenance and repair events do not open the

automatic protection switches. In these cases only manual isolators adjacent to the

component are opened while the component is undergoing preventive maintenance or

being repaired.

It is important to note that not every system state contingency will result in load

curtailment or the supply point being disconnected. The consequence to each system

28

state is recorded after the flow analysis is conducted, the consequences would be the

disruption to a load and/or the disconnection of a supply point from any load point.

The accumulation of the results at the end will provide load point and system indices.

2.2.3 Flow Methods

Flow methods are required to test any load on outage after the failure of a component

has occurred but the main concern is to examine if the sources are able to supply the

loads. In an event where a load is not supplied, the amount of load or loads being on

outage is recorded and together with the event probability and frequency, other

indices are calculated. There are two main flow methods used for a network,

connectivity otherwise known as minimal cut set method and the maximum flow

method.

Distribution systems beginning at source and finishing at load are inherently radial by

design and the radial nature makes it more simpler approach to traversing the network

looking for paths to connect source and load. Substation and transmission systems

however tend to have multiple parallel branches due to the designed redundancy of

the system, allowing for multiple routes between source and load points.

2.2.3.1 Connectivity

The connectivity method, an exclusive case of maximum flow [93, 94], can be

thought of as a model of an idealised system. This method will simply find if there is

a connection between sources and load, regardless of the capacity of the path that it

must take. The details of flow method are given in Appendix A.

29

Fig 2.2 Example of connectivity flow with an active failure at component 9.

An example is shown in Figure 2.2, where there is an active failure on Transformer 9.

This active failure requires protective devices Circuit Breaker 1 and Circuit Breaker 5

to operate until the transformer has been isolated. As it can be seen, there is no

possible path between any supply points to the load point 4. This would represent a

loss of supply to load point 4.

Another case is shown in Figure 2.3; this time Circuit Breaker 3 is in a maintenance

state. This requires no protective devices to operate. It does however isolate the

supply from Source 3.

Fig 2.3 An example of connectivity flow with passive failure at component 3.

30

As can be seen in Figure 2.3, even though Source 3 has been disconnected from the

system, the circuit still continues to function and there is no loss of supply to any of

the loads due to the fact that alternate paths exist.

2.2.3.2 Maximum Flow

Maximum Flow is closer to a realistic model, whereby circuits are only able to carry

their rated capacity. This limitation is likely to cause more outages than the infinite

circuit capacity of the connectivity method due to the fact that a loss of a supply will

place more demand on the remaining infrastructure. Maximum flow method

algorithms are given in Appendix A.

Values are assigned to each component in the system, and the system control will not

allow any more power to flow through that component, effectively creating a

bottleneck that will starve components downstream of it, unless there is an alternate

connection.

A simple case is considered in Figure 2.4, which is also the same as the first case

considered (Figure 2.2) in the connectivity section. This is to highlight differences

between the methods.

Fig 2.4 An example of maximal flow with active failure at component 9.

31

As can be seen, there is a complete loss of supply to Load Point 4 with the active

failure of Transformer 9. A major difference between the connectivity method and the

maximal flow however, is that partial supply is a possibility. With connectivity, either

there was supply, or there was none. This can be seen most clearly in the above case,

where Load Points 5 and 6 have had to shed some of their loads due to bottlenecks in

the system.

Another case is considered in Figure 2.5, again the same network of Figure 2.3 is

considered to highlight differences between the two flow methods.

Fig 2.5 An example of maximal flow with passive failure at component 3.

This case highlights the way that capacity is spread throughout the system. As can be

noted from Figure 2.5, there is complete supply to Load Points 4 and 5 but, only

partial supply to Load Point 6.

2.3 Substation and Sub-transmission Indices

The main indices obtained for substation and sub-transmission parts of the network

are the probabilities and the frequencies. All other indices are the by-product of these

two indices. The indices are in two main categories:

32

• Load point indices and

• System indices.

System indices are accumulated from the load point indices. In addition, indices

associated with supply points are also calculated:

• Probability of Load Curtailments (PLC)

• Probability of Source Disconnection (PSD)

• Expected Demand Not Supplied (EDNS)

• Expected Frequency of Load Curtailments (EFLC)

• Unavailability (U)

• Average Duration of Load Curtailments (ADLC)

• Expected Energy Not Supplied (EENS)

These indices are calculated and accumulated during every contingency enumeration.

These indices are listed as follows;

Probability of Load Curtailment, (PLC) ;

For a load point k:

(2.17)

where Pload(i,k) is the system state probability corresponding to the i th component(s)

failure state causing an outage at load point k and Nk is the total number of component

failure states that cause an outage at load point k. In some events more than one failed

component is needed to cause the outage.

33

Probability of Supply Disconnected, (PSD);

For a source point k:

(2.18)

where Psource(i,k) is the system state probability corresponding to the i th component(s)

failure state causing a disconnection at supply point s and Ns is the total number of

component failure states that cause a disconnection at supply point s. Sometimes

more than one failed component at a time is needed to cause the outage.

Expected Demand Not Supplied (EDNS)

For a load point k: MW 1

k

N

iikk LPEDNS

k

∑=

= (2.19)

where Lk (MW) is the average load at load point k during the period considered

(usually one year).

Expected Frequency of Load Curtailments (EFLC)

EFLC is the number of expected load point outages in one year. For each

component(s) failure state that causes an outage at a load point, the frequency of this

state occurring can be obtained using a relationship between the state probability and

frequency.

For load point k: ∑=

=M

jjikik Pf

1

λ occ/yr (2.20)

34

where fik is the frequency of the i th component(s) failure state causing an outage at

load point k and λj is the departure rate of the jth failed component(s) in state i and M

is the number of departure rates from state i.

For example, consider a component in the switching state has caused load point k to

go on outage. M would be the number of all the possible transitions away from the

switching state, normally just the single transition from switching to repair and λj

would be each individual transition rate which in this case is the just the one.

However, if the rest of the components in the network were all in the up state, they

would all have the a higher value for M as it could be possible to go from the up state

to either the switching, maintenance or repair states.

EFLC is then the sum of fik for all the component(s) failure states that cause load point

k to go on outage.

∑=

=kN

iikk fEFLC

1 occ/yr (2.21)

This calculation is a slight approximation due to the fact that the sum in Equation 2.21

excludes the deduction of the sum of all transition frequencies between system failure

states [28]. However it is a difficult process to try and deduct the extra failure states

from the sum and because transitions between system failure states rarely occur in the

operation of electrical networks, this approximation is acceptable.

A similar index as Expected Frequency of Supply Disconnected; EFSDs can be obtain

with respect to a supply point s.

35

Unavailability (U)

Unavailability is the average number of hours per year that the load point is expected

to be on outage.

For load point k: Uk = PLCk x 8760 hrs/yr (2.22) Where 8,760 are the number of hours in a year.

Average Duration of Load Curtailments (ADLC)

ADLC is the average duration of each failure at a load point, measured in hours per

failure.

For load point k: ADLCk = PLCk × 8760 hrs/failure/yr (2.23)

EFLCk

Expected Energy Not Supplied (EENS)

For load point k:

(2.24)

Where; Lik is the average load curtailment for load k in the ith system state and Pik is

the probability of the ith system state in which the load k is curtailed.

It can be noted that; EENSk = 8760×EDNSk.

2.4 Percentage Contribution of Components to Probability and

Frequency of Contingencies

The percentage contribution of components to the probability and frequency of

contingency events can be very useful indicators. An important advantage of these

indicators is to identify the degree of criticality of a component in a network with

respect to particular effect of interest, such as, individual loads on outage, individual

36

supply point disconnections, or the overall system loads being on outage, etc. The

importance contribution in network reliability associated with the cut set method has

been cited in number of references [48-60]. Although the cut set method may be

required for certain applications like communication and manufacturing, etc., using

this method for power system reliability not only is not necessary but it can introduce

extreme complexity to the calculations. However, using the enumeration method,

new algorithms are required to extract the pertinent information.

As part of the power system reliability evaluations in this thesis, new measures are

presented here and applied to obtain the percentage contribution of the components.

In addition, the same concept is used for triple contingencies where it is extended to

calculate the component contributions for the aging model which is introduced in the

next chapter.

The new equations presented in the following sections are used to evaluate the

percentage contribution of a component probability and frequency indices using the

enumeration method applied for power system reliability evaluations.

This index can assist utilities in network configuration design, planning,

reinforcement and maintenance etc. The costs associated with these factors can be

much better managed with knowledge of the contribution of components to individual

loads and system outages. The quantitative information achieved by these indices

stem from three main attributes of a system, namely; the reliability data, the network

topology and the load magnitudes.

2.4.1 Percentage Contribution of Components to Reliability Indices in a Power

System

Considering the Probability of Load Curtailment PLC index as an example, suppose

that kth load has the value PLCk, then the percentage contribution of component C to

single and double contingencies to PLCk is as follows:

37

100

][

×+

+

=

∑∑∈∈∈

k

kikj ij

jij

kjj

PLC PLC

PP

PPP

C M

C MC

C

MC

C

C

k (2.25)

Where;

CjP is the j th single contingency, system state probability associated with

component C, that causes load k to go on outage.

MC ijP is the j th and i th double contingency, system state probability associated

with component C and any other component M, that causes load k to go on

outage.

For every enumerated double contingency system state, there are only two

components on independent outage and the rest are in an operating state. It can be

appreciated that each of the failed or out of service components may have different

state probability values which contribute to the system state contingency. Therefore,

the probability weighted value of the double contingency probability terms associated

with the effect of component C can be extracted and accumulated, as shown in

Equation 2.25.

A similar expression to Equation 3.25 can be constructed for the contribution of each

component to the overall system load outages (LPLCC ). In this case, any system state

that causes one or more loads to go on outage is included as part of either the single

contingency probability term or the double contingency probability term. The double

contingency term is also weighted similar to the one in Equation 2.25. Therefore a

similar equation is constructed to express the percentage contribution of component C

to kth load curtailment (or any other outcome) to frequency as follows:

38

100

][

×+

+

=

∑∑∈∈∈

k

kikj ij

jij

kjj

EFLC EFLC

PP

Pff

C M

C MC

C

MC

C

C

k (2.26)

Where;

Cjf is the j th single contingency, system state frequency associated with component

C, that causes load k to go on outage.

MCijf is the j th and ith double contingency, system state frequency associated with

component C and any other component M, that causes load k to go on outage.

For every double contingency system state, there are two components on outage and

the rest are in an operating state. As shown, the probability weighted value of the

double contingency frequency terms associated with the effect of component C are

extracted and accumulated in Equation 2.26.

A similar expression, as in Equation 2.26, can be constructed for the contribution of

each component to the overall system load outage frequencies (LEFLCC ). In this case,

any system state that causes one or more loads to go on outage is included as part of

either the single contingency frequency term or the double contingency frequency

term. The double contingency term is also weighted similar to the one in Equation

2.26.

2.4.2 Load Weighted Attributes to Probability and Frequency Indices

The purpose of electricity networks is to maintain supplying the loads as reliably as

possible, with a reasonable degree of quality. In addition, the utilities revenue is

directly related to the amount of load being supplied. On the other hand, the loads are

spread in different locations throughout the network. As such, it is desired that

39

facilities making up the network have quantitative indicators ascribed to them. In this

respect, further scaling of the percentage probability and frequency contributions of

the equipment responsible for the degree of load magnitude support during an outage,

are provided in this section.

A component’s contribution to the overall system load probabilities and frequencies

does not control the number of loads lost during an outage, that is, during the system

states. Therefore, to include the effect of the loads that are lost during an outage the

following weighting Equations 2.27 and 2.28 for probability and frequency are used.

The percentage contribution of component C to the system state probability weighted

by load curtailment is as follows:

100

][

][

C all

×

++

++

=

∑ ∑∑

∑∑

MC

MC

C

MCCC

MC

MC

C

MCCC

L

ijij

jijjj

ijij

jijjj

PLC

LPP

PPLP

LPP

PPLP

CW (2.27)

Where;

CjL is the j th single contingency system state, associated with component C,

that causes any load L to go on outage.

MC ijL is the j th and i th double contingency system state, associated with

component C and any other component M, that causes any load L to go on

outage.

The denominator of Equation 3.27 is the total PLC weighted loads for all

components.

And, the percentage contribution of component C to the system state frequency

weighted by load curtailment is as follows:

40

100

][

][

C all

×

++

++

=

∑ ∑∑

∑∑

MC

MC

C

MCCC

MC

MC

C

MCCC

L

ijij

jijjj

ijij

jijjj

EFLC

LPP

PfLf

LPP

PfLf

CW(2.28)

2.5 Application Applied to Substation and sub-transmission

Configurations

The main function of a substation, apart from interfacing the different voltage levels

through transmission lines in the electricity network, is to facilitate multiple link

connections for providing power with adequate switching and protection devices, to

isolate faults and reroute the connection if possible.

Substations normally have a range of equipment including; power transformers,

circuit breakers, VAR compensation reactors and capacitor banks, measuring voltage

and current transformers, isolators etc. Substation equipment is normally very costly

and is a substantial part of the assets in the electricity supply industry.

Due to the complexity of interconnection in substations and the great impact they can

have on the reliability of the rest of the system, it was decided to test the methods and

the indices presented in this chapter on a range of substations; small, medium and

large. The following section will demonstrate the methodology and show results

using the applications introduced above.

2.5.1 Application Applied to a Small Sub-transmissions System

The method described is applied to various sub-transmissions test systems [83] and

the results using the indices provided in this chapter are illustrated in the following

41

pages. Figure 2.6 provides different connection configurations for a small sub-

transmission system..

(a)

(b)

(c)

42

(d)

(e)

Fig 2.6 Small Substation Test Cases [74].

The data used for the substations are provided in Table 2.1, assuming that both Load 1

and Load 2 are 10MW each.

43

Component Active Failure Rate

(occ/yr)

Passive Failure Rate

(occ/yr)

Maintenance Rate

(occ/yr)

Repair Time (hrs)

Switching Time (hrs)

Maintenance Time (hrs)

Breaker 0.01 0.01 0.1 3 1 5 Bus Bar 0.024 2

Transformer 0.1 0.2 50 1 10 Line 0.09 7.33 1

Table 2.1 Substation Component Test Data [2].

The results for the small substation test cases of Figure 2.6 are provided in Tables 2.2

and 2.3 and include the percentage contribution of each component towards the

probabilities and frequencies associated with outages or disconnections of;

o Individual loads,

o Individual sources,

o Overall system loads and,

o Weighted load outage magnitude.

There are two sets of indices; localized load and sources, and system wide overall

load and weighted load contributions. Comparing the results for the system

configurations of Figure 2.6, the contribution of each component changes from one

system to the next to some degree. The variation in the results is only due to

differences in the system layouts, as the data and load values are identical for all

cases.

For example, in case “c” (Figure 2.6c and Table 2.2c), the transformers TX1 and TX2

have the most impact as shown by the percentage weighted values of 43.2%. This is

due to two reasons; case “c” is virtually two disjoint systems and as such the benefit

of supplying power from an alternative source while a component is out on one side

is impossible. Secondly, the transformer amongst the other components in this system

has the highest failure and maintenance rates, maintenance and repair times.

44

On the other hand, the transformers in case “a” have, with percentage weighted value

of 4.17%, the least effect on load outage, as the system is interconnected and the

circuit breakers will initiate instant isolation of the fault from the upstream source, in

many failure situations, allowing the load to be supplied through the other root or

source. The transformers in all case, however, contribute highly to their source side

disconnection as illustrated by the high factors of 85.2% in system (a) and 86.4% in

system (b) for example.

The effect of each component on all loads and weighted load magnitudes are not

greatly pronounced in these cases, as the examples are for small networks and the

loads have the same values. However, the overall system load related indices, for

example; system wide PLC is not equal to the summation of the individual load point

PLCs. In cases like “a” and “b”, where the network has only one load, the system

wide (all loads) and individual loads will give the same component percentage

contribution values. In later examples, with larger systems the diversity of the results

will be shown.

(a)

45

(b)

(c)

(d)

(e)

Table 2.2 Cases “a-e”, Percentage contribution of components to contingency

probabilities.

46

(a)

(b)

(c)

(d)

47

(e)

Table 2.3 Cases “a-e”, percentage contribution of components to contingency

frequencies.

Overall and system wide PLC, unavailability, EFLC and ADLC are provided in Table

2.4. The network of case “a” is clearly the best performing system with a Probability

of Load Curtailment of 9 x 10-6. Although, the PLC and EFLC of each case obtained

through these examples increase or decrease together, this is not a general rule. In

some cases, the PLC can increase where EFLC decreases, or vice versa. Complexity

of a network and the range of data incorporated in the studies can provide very

different result to that anticipated.

(a)

(b)

(c)

(d)

48

(e)

Table 2.4 Overall, system load indices, for cases “a-e”.

The load point indices in Table 2.5 also indicate that, the network of case “a” is the

best performing system, again shown by PLC = 1 x 10-6. There are two additional

subsets of results that are provided with load point indices in Table 2.5. These are for

the single and the double contingency enumeration results. The addition of these two

makes up the overall results for each load point. This breakdown can be very useful,

giving more in depth information as to the sensitivity of the system to different modes

of failure. In some systems, where heavy redundancy is heavily applied, the weight of

the results will tend to be determined by the double contingency events.

(a)

(b)

(c)

49

(d) and (e)

Table 2.5 Load point indices, for cases “a-b”, and each load in “c-e”.

2.5.2 Application Applied to a Medium Substation System

In addition to variations in component reliability data and system load values, the

configuration complexity of a network can lead to large differences in the results.

Therefore, some of the techniques developed earlier with regard to component

contribution to indices can be very useful in providing in-depth information about the

networks viability.

In this section, the application applied to a medium size substation [2] shown in

Figure 2.8 is examined and the data used for this network is provided in Table 2.6.

Each load in this substation is 10MW.

The overall system and the individual load point indices are given in Tables 2.7 and

2.8, respectively. In addition, Table 2.8 provides the results associated with single

and double contingency events, separately. It can be seen from the results that the

single contingency events are the major contributor to the outages in this system.

50

Fig 2.7 Medium substation test network [2].

Component Active Failure Rate

(occ/ yr)


(occ/yr)

Maintenance Rate

(occ/yr)

Repair Time (hrs)

Switching Time (hrs)

Maintenance Time (hrs)

Breaker 0.002 0.0001 0 126 1 0

Bus Bar 0 0.025 0 13 0 0

Transformer 0.026 0 43.1 1 0

Table 2.6 Medium Substation Component Test Data [2].

Table 2.7 Overall, system load indices, for the medium substation.

51

(Load 1)

(Load 2)

(Load 3)

Table 2.8 Load point indices, for a medium substation.

The results in Table 2.8 indicate that load points 1 and 3 have identical indices. This

is because the network is symmetrical with respect to loads 1 and 3, and the data for

the components of the same type in this substation are identical.

More in-depth information can be obtained from the results available in Tables 2.9

and 2.10, where the percentage contribution of each component to the probability and

frequency of individual loads and sources, the overall system loads and the percentage

weighted load contribution is given.

52

Table 2.9 Percentage contribution of components to contingency probabilities. Power transformers TX1, TX2 and TX3 have the highest contribution to the

probability of system and individual loads with approximately 77.7% and 25.9%,

respectively. As described earlier, probability and frequency of the overall system

loads are indiscriminately associated with the system states that cause any one or

more loads to be disconnected. Thus, the effect of each component can spread at a

particular level, as seen in this application. For instance; transformers TX2 and TX3

have insignificant effect on load 1 outages, whereas all transformers have almost the

same effect on overall system loads outages (25.9%). This application is a unique

example where the system is symmetrical and the data are identical for all similar

component types, therefore the effect of each component, e.g. a power transformer, on

the overall system loads is approximately one third of the individual load outage

cases. This relationship can change when using a network that no longer has

homogenous topology and data.

The disconnection of sources to any load points, due to substation originated outages,

are also included as part of Tables 2.9 and 2.10. In this system example (refer Figure

2.7), the effect of Buses 1, 2 and 3 and CBs 5, 4 and 3, on the disconnection

probability of Sources 1, 2 and 3 are respectively considerable. However, with regard

to the frequency of sources being disconnected, the power transformers and the bus

sections have almost similar contributions, 45.5% and 43.9% respectively.

53

Table 2.10 Percentage contribution of components to contingency frequencies.

The weighted load, as a percentage contribution for the probability and frequency

cases, are almost identical to the values in the “all loads” columns. This is because all

of the loads in this example have the same value. In a practical systems, however,

normally the loads throughout a network are not of the same value and also the yearly

load growth rate may not be the same for all loads. Therefore, in practice, the value

of this indicator can be very different from the overall system loads index.

In order to illustrate the sensitivity of the weighted loads indicator, the value of the

loads in this example are changed to 10, 50 and 100 MW for loads 1, 2 and 3

respectively. No other changes are made to the system. The results associated with

the percentage contributions of each component to probability of individual loads and

sources, and the overall system load and weighted load are presented in Table 2.11.

As can be seen, when comparing these values to those in Table 2.9, all values

remained the same except for the weighted load column. With 48.5%, transformer

TX3 has now the highest contribution as it is directly responsible for the highest load

of 100 MW. Similarly, weighted components TX2, Bus 3, Bus 2, and TX1 are ranked

in the order from highest to lowest importance.

54

Table 2.11 Percentage contribution of components to contingency probabilities.

The breakdown of system PLC, Unavailability U, EFLC and ADLC indices in

association with the events

• “no load on outage”,

• “one load on outage”,

• “two loads on outage” and

• “three or more loads on outage”

And in combination with:

• “no source on outage”,

• “one source on outage”,

• “two sources on outage” and

• “three or more sources on outage”

Are provided in Table 2.12.

This table provides additional information as to how the substation oriented outages

can effect loads and sources in different ways. In estimating the reliability of a

network it is desirable to design a system such that the probabilities and the

frequencies of outages that are causing wide spread effects are insignificant. The

55

wide spread effects are failure events that may simultaneously cause the loss of links

from the sources to a number of loads.

The results in Table 2.12 indicate that most outages effecting the loads are localized,

and close to the loads. It is also seen from the results in this table that the chance of

having multiple loads on outage is very small. Part of the design for a reliable system

is to increase the redundancy, only to the extent that would be worth the investment.

Overdesign not only can be costly but it can also have an adverse effect on the system

reliability. In other words, the reliability and the cost/benefit should always

complement each other in design and planning. This will be addressed in a later

chapter in this thesis.

Table 2.12 Breakdown of load point indices for a medium sized substation.

56

The number of load and source contingency events in an incremental range of

probability and frequency, similar to a histogram, can provide extra information when

measuring the reliability performance of a system. The information provided in

Tables 2.13 and 2.14 is systematically extracted through the enumeration method.

The incremental range of probabilities and frequencies are given in the first two

columns and the number of load and source contingency events encountered for the

given range of probability and frequency are listed in these tables. Normally in a

reliable system the contingency events leading to load and/or source disconnections

should occur in the upper part of the table where the range of probability and

frequency are very low. In addition, the lowest number of outage events when

comparing two or more compatible systems with different circuit designs, is usually

desirable.

Table 2.13 The number of system probability contingency events for loads and

sources in the medium substation.

The values given in Tables 2.13 and 2.14 do not necessarily follow the same trends.

That is, a system may encounter outages with very low values of probabilities but can

have higher values of frequencies, or vice versa. The influencing factors, as noted

earlier, are the way a system is designed and configured (leading to better

57

redundancy), and the associated reliability data of the system components. If capacity

constraint is considered as part of the system design, it can also be treated as an

additional contributor and influencing factor in the system reliability performance.

Table 2.14 The number of system frequency contingency events for loads and sources

in the medium substation.

2.5.3 Application of Limited Capacity in System Reliability

Evaluations

The maximum flow method, as discussed earlier, is used to take into account the

limitations in capacity of a network. The capacity limitation is a real issue in practical

applications especially as industry, metropolitan areas, housing, farming and

commercial developments grow. The electricity supply network including generation,

transmission and distribution must plan to adequately fulfill the demand with an

acceptable level of risk. This network upgrading, as the demand increases, is

necessary and the ‘maximum flow’ technique is able to capture the capacity

limitations as part of the system reliability evaluations.

58

The medium sized substation network used in the previous example is employed to

highlight the application of ‘capacity limitation’ and the effect that can have on

reliability indices. This aspect is discussed next.

Each load is kept at 10MW, as in last example. However, to examine the effect of

capacity limitation, every source is set to have a limit of 10MW. As can be seen from

the overall system load indices in Table 2.15, PLC, Unavailability U, EFLC and

ADLC have all increased compared to the results in Table 2.7 and indicates that the

system is more prone to capacity limitations.

Table 2.15 Overall, system load indices, for the medium substation including capacity

limitation from the sources.

Similar trends of increasing individual load indices are shown in Table 2.16.

However, the increasing effect on the single contingency outages is more than the

double contingencies. This means that the double contingency outages are very low

in probability and in many cases they would have caused load outages even without

the capacity limitation. Some of the single contingency events occurring near the

supply points that would have been compensated by the other two sources are now

causing some of the loads to go into outage, due to an insufficient supply.

Table 2.16 Each load point indices, for the medium substation with capacity limited

sources included.

59

The effect of individual components on the probability and the frequency of

individual loads, sources, overall system loads and the weighted load indices are

given in Tables 2.17 and 2.18.

Comparing these two tables with Tables 2.9 and 2.10, it can be seen that as the

capacity limitations from the sources are introduced the percentage effect of

components on the listed contingency probabilities and frequencies are changed. This

table confirms the increasing effect of components near the sources. For example, the

effect of CB5, CB4 and CB3 on loads 1, 2 and 3 are now increased by about 112

times. On the other hand, the effect of transformers and bus sections are reduced. A

similar trend is observed with regard to the effect on overall system loads. However,

the effect on source disconnections are remained the same, as expected. Once again,

the effects on overall system loads and the weighted loads have not changed, because

the loads have the same values. The numerical values obtained for the component

effects are valuable measure for asset management of power systems.

Table 2.17 Percentage contribution of components to contingency probabilities,

including capacity limitation of the sources.

60

A similar comparison can be made with regard to the frequency effect, in Table 2.18.

The effect on the frequency of source disconnections remains the same, as expected.

The effect of CB5, CB4 and CB3 on loads 1, 2 and 3 has increased by about twice. A

similar level of increase is seen on the overall system loads. The effect of

transformers on load outage frequency is decreased slightly for example TX1-Load 1

changes from 64.2 to 62.6 percent. Generally, the frequency effect is not significant

indicating that the number of incidents associated with each component resulting in

the load outages has not varied much. However, the probability of some of those

incidents occurring has increased significantly.

Table 2.18 Percentage contribution of components to contingency frequencies,


The breakdown of load point indices is shown in Table 2.19, for the application of

capacity limitation. The changes compared to no capacity limitation shown in Figure

2.12 are mainly associated with the “no loads down” and “one load down” cases.

This finding is consistent with the conclusions made through the earlier results. The

added information in this table is with regard to the “no loads down” cases. Out of

the contingencies enumerated through this application, the results indicate that the

probability and frequency of the “no loads down” cases have reduced by 40% and 4%,

61

respectively. The details in this category are very similar to the results with no

capacity limitation, with the probabilities and frequencies associated with one and two

generators being disconnected equal to zero. Therefore, in this case, it can be

interpreted that when capacity limitation is imposed the load outages are directly

linked to any one, two or more sources that are disconnected from the network. This

is a logical outcome of these condition and can be expected. It is seen that when

capacity limitation is imposed the system vulnerability, with respect to loads, are

increased.

Table 2.19 Breakdown of load point indices, for the medium sized substation


62

2.6 Summary

This chapter has provided detailed examples of the system state enumeration method

and the Markov Space State model to analyze reliability in power systems. In this

method, the major operational states of components with the transitional rates moving

from one state to the next are incorporated. The states considered are; operating,

switching, repair and maintenance. But it is not required that every component is

modeled with the four states. Some components need only be modeled with two,

operating and repair states, where as some others with three states; operating,

switching and repair. In addition, the maintenance state can also be included for those

deemed appropriate.

Each component can reside in one state, at a time. The combination of the component

states forms a system state. These combinations in a moderate size system will form

an unreasonable number of system states which not only are impractical to solve but

also unnecessary and have, in fact, an insignificant probability. Given the outage

statistics on power systems, the use of single and the double contingencies are

adequate for reliability evaluations.

Knowing the network configuration and the reliability data for the components and

the load values, the system state failure mode analysis can be conducted on the

network and the resulting effects evaluated to accumulate the indices. During the

system failure mode the effects of components that are not in service or undergoing

any switching action can be removed before a network flow analysis is conducted.

Two flow methods are incorporated in this chapter, connectivity and maximum flow

methods. The connectivity method only examines if there are links established

between the sources and the loads. The maximum flow takes the capacity or the

rating of the network components into consideration when examining if the loads can

be supported by a source or sources. The outcome of these two methods will indicate

63

which loads are not being supplied and this information is recorded as ‘loads on

outage’. There are numerous load and source points, and system indices that are

evaluated during the system state calculations. These are accumulated and tabulated.

As part of the research work in this chapter it was recognized that the network

topology and the components reliability data together with other constraints in the

network such as limitation of capacity, can influence the outcome indices. From the

design, planning and the asset management point of view it can be very valuable to

identify the roles and effects of individual components in the makeup of a system

reliability measurement. This information can assist system designers, operators and

those who maintain and manage the system in allocating resources accordingly and in

an effective way to balance reliability with costs. Recognizing this need, exclusive

enumeration technique algorithms were incorporated in a novel way to extract useful

information about the effect components have on a network and its reliability indices.

The indicators include; the percentage effect and the contribution of each component

to individual load and source outages, overall system loads and the weighted load

indices. These values are systematically collected during the system state

enumerations using a method developed as part of this research work.

64

Chapter 3

A Markov Model for Non-Repairable Aging Equipment and its

Contribution to Substation/Sub-transmission Reliability Evaluations

3.1 Introduction

There are many publications in the literature that use Markov State Space models to

evaluate power system network reliability. Most of these publications have only

considered the random failure mode (useful life period) of the components, in their

calculations. Lack of aging considerations in many of the research publications has

lent itself to optimistic results in power system reliability evaluations. However,

much of the equipments in the electric utilities are becoming aged and close to their

end of life, which are normally non-repairable. Therefore, it is essential to include the

aging characteristics of the equipment as part of power system reliability evaluations.

This should assist utilities in making more reliable decisions with regards to an aging

network’s operation, planning, reinforcement and generally asset management.

In recent years there has been growing body of research literature on the reliability

and availability of aging equipment and extensive literature does address aging at the

component level. However, there is very limited research literature available on the

aging of power networks [20-40]. A few conference papers [24-26] use a non-

homogeneous Poisson Process to characterize aging components and have applied the

sequential Monte Carlo method for these evaluations. The model used did not give

any evidence indicating to be a good fit for the actual power system equipment aging

characteristics. In addition, the Monte Carlo technique can introduce a very long

calculation time in order to obtain reduced variance and thus a more accurate result.

65

A limited set of papers [22, 23] present a Bayesian Network method but the power

system model becomes complicated and a solution is not readily achievable. This

latter method develops an evaluation using conditional probability, and cut set, tie set

or fault trees are employed as part of the modeling and evaluations. The use of cut

set, tie set and fault trees in power system Bayesian Network analysis can introduce

an extensive computational burden. Although this method may be suitable for certain

applications, it can impede operational requirements and considerations that are part

of the power system reliability evaluations.

Another simple probabilistic method is developed and used by WenYuan Li [30, 31].

In this method, probability distribution suited for aging components is used to obtain

aging probabilities for each year of study. Using a combination of each component’s

yearly probabilities yields a set of system states for each year of study and these states

are evaluated and combined to assess the system. Using a union concept, Li has

integrated the useful life period and aging period of a component with its repair

considerations. Modeling maintenance and switching considerations in his model

may not be readily applicable. In addition, this method does not include frequency as

part of the model.

In this chapter, an aging model is developed as part of the frequency and duration

approach using a state space Mark model as part of power system reliability

evaluation. This model is introduced and developed in the following section and

examined extensively throughout this chapter.

3.2 Non-Repairable Aging Failure Model

The increasing hazard rate, or failure rate characteristic of equipment is directly

related to its aging degradation in reliability evaluations. For example, the failure

66

characteristics of different components can be similar to the hazard rates illustrated in

Figure 3.1. The flat portion of a hazard rate is normally referred to as random failure

or the useful life zone or mode of a component’s life cycle. The probability

distribution associated with this region is an exponential distribution [75].

-0.01

0.01

0.03

0.05

0.07

0.09

0.11

0.13

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Year

Haz

ard

Rat

e

Fig. 3.1 Aging failure or, hazard rate functions.

The increasing hazard rates in Figure 3.1 correspond to component degradation, and

are normally associated with end of life failures. However, less likely a degradation

effect and non-repairable situation, due to poor design, can also happen at an earlier

point in the life of a component. The failure types and the characteristics of the two

regions, namely; the constant and the increasing parts of the hazard rate function, are

different and normally independent of each other. The independency of these two

regions is a fair assumption as the normal usage of equipment in operational situations

should not produce any adverse or accelerated aging effects. It is worth noting that

the aging characteristic is associated with the functional or operational age of a

component. Although expensive, life testing can be used to estimate the functional

age [21]. However, for long term planning purposes, the natural age of a component

can be used instead of the functional or operational age [29]. The probability

distribution associated with an increasing hazard rate, described as the degradation

67

failure rate for power equipment, is usually described by a normal or a Weibull

distribution function.

Posterior probability associated with a Weibull or normal distribution fits well for a

increasing failure rate of the non-repairable aging state [28], and can be used for the

model presented in this thesis. The failure rate is a conditional function of the failure

density function, the conditional relationship being the survivor function [75]. The

general expression for the failure probability otherwise known as hazard or failure

rate of a component, on the condition that it has survived up to time t years, is

provided in Equation 3.1.

(3.1)

Where t is in years.

Failure rates λ(t) calculated using the Weibull and normal distributions (Figures 3.2 &

3.3) are plotted in Figures 3.4 and 3.5 respectively.

Aging Failure Distribution

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

Time (years)

f(t)

Fig. 3.2 Aging Failure, using Weibull distribution with α = 45 and β = 7.

68

As noted, the failure mode relating to aging is independent and different to that of the

useful life period which is also known as the random failure mode. The characteristic

of the failure rate is dependent on the failure distribution function.

Aging Failure Distribution

0

0.01

0.02

0.03

0.04

0.05

0.06

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Time (years)

f(t)

Fig. 3.3 Aging Failure, using Normal distribution with µ = 45 and σ = 8.

The aging failure rates obtained using the preceding approach can directly be used as

part of component attributes in the network reliability evaluations, as shown in the

subsequent section.

Hazard Rate

0

5

10

15

20

25

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

Time (years)

Failu

re R

ate

(occ

/yr)

Fig. 3.4 Aging, Failure Rate, using Weibull distribution with α = 45 and β = 7.

69

Hazard Rate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

Time (years)

Fa

ilure

Ra

te (o

cc/y

r)

Fig. 3.5 Aging, Failure Rate, using Normal distribution with µ = 45 and σ = 8.

3.3 Including Non-Repairable Aging in Markov State Model

The Markov state space models discussed in Chapter 2 are used to describe the state

of power system components, as part of an enumeration technique in electric network

reliability evaluations. One of the conditions in using this model is that the transition

rates between the states must be constant. This condition suits the constant failure

rate region of the hazard rate curve. However, a changing failure rate, related to non-

repairable aging degradation, cannot directly be used in this model. As a research

contribution in this thesis, the non-repairable aging and the normal life of a

component are combined to form as part of Markov state space model, as shown in

the following sections. This approach provides a number of advantages;

70

• It presents a Markov model for the whole life cycle of a component with

constant failure and aging failure modes, which can be used simultaneously in

network reliability evaluations.

• The model enables the calculation of all load, source and system indices at all

levels including; probability, frequency and unavailability, etc.

• Third order aging contingencies can efficiently be included to provide more

accurate results.

• The model is simple, flexible, effective and efficient and includes most

operational situations deemed necessary.

The model includes an additional state associated with aging failures. This aging state

has different attribute compare to the other states. The probability of entering into this

state changes with time and follows a distribution similar to the ones shown in Figure

3.2 and 3.3. Consequently, the aging failure rate is responsible for moving the

component state from ‘operating’ to the state described as ‘failed due to end of life

aging’. Obviously, this failure rate must increase with aging in years. It is assumed

that routine maintenance and repair due to the random failures in a component’s

useful life period have no effect on the functional aging of a component. However, it

is rightly assumed that once a component fails due to aging it is not repairable. The

component in this state either has to refurbished or replaced. The choice of

refurbishing or replacing equipment is a combined technical and economic issue that

needs to be considered. However, in either case, there is a substantial time required to

resume operation. The timing for major refurbishment or replacement of a component

is a decision which is based on cost and/or risk criteria and will be addressed in the

later chapters.

As noted, in order to include aging in the Markov models presented in Chapter 2, an

aging state is added which is shown in Figures 3.6-3.8. The reliability study of a

power system with aging components may sometimes require simulations for

planning considerations in the range of 10-20 years. In a practical situation,

71

equipment in a power network may be of different ages, in a given year of study. The

model proposed in this thesis can handle and capture many of the practical operating

variations. The aging failure rates and the state probabilities are updated during every

year of planning before a new set of evaluations are performed,.

3.3.1 Non-Repairable Aging using State Space Model

The physics of aging failure is attributed to extensive fatigue in the fundamental

structures of equipments. Thus, normal corrective actions are no longer viable to

make it operational. Both refurbishment and replacement are expensive and time

consuming, and in some cases the refurbishment cost may exceed replacement cost

and therefore the decision may favor replacement. The state space model can equally

be used for either of the two cases. For practical purposes, it is assumed that the time

to refurbish or replace an aging component follows an exponential distribution.

Equipment in power systems consists generally of slow aging components, normally

with mean life of over 40 years. At the same time failure rates associated with the

aging components in a power system are commonly linearly increasing functions [29].

This characteristic can best represented by a normal or Weibull distribution. Because

of the fairly long mean life, with a standard deviation of about 15 years [20], the

failure rates of components will turn out to be a gradually increasing function similar

to Figure 3.5. The time scale in distribution and failure rate functions are normally in

years and given the long life of power network facilities and the slow rate of their

aging failure, it is a reasonable assumption to consider their aging failure rates as

constant over a given year. Using this assumption the Markov state space model can

then be used independently for each planning year as some of the components

continue to age or a new component is added to a network. As provided in later

sections of this chapter, this method proves to be suitable for application to power

systems.

72

In the following sections, the inclusion of a failure state in the models presented in

Chapter 2 is discussed and the formulas to calculate state probabilities are derived.

3.3.1.1 Aging State with Two States

The equations for the two states; ‘up’ and ‘repair’ are provided in Chapter 2 with

these two state models sometimes being best suited to describe the reliability

attributes of a component in a power system. For example, the failure modes of bus-

bars sometimes are limited to operating and repair modes. This can be due to the

nature of the faults, such as a sudden opening of conductors on transformer bushes,

because this action will not activate any protection switches. In other cases like

insufficiency of data, or the location of component in the network, the two states can

be an adequate model to represent the operational situation.

The aging state is added to the two states model and is shown in Figure 3.6. The term

µag in this model is the rate of return to an operating state from an aging state. This

rate is inversely proportional to mean time to refurbish or replace a component. The

aging failure, λag is associated with a failure value in an aging year, corresponding

similar to the graph shown in Figures 3.4 and 3.5.

Fig. 3.6 A three state Markov model, including an aging state.

µag λag

µr

λp

Repair state

Up state

Aging failure state

73

There are several methods [75] that can be applied to deduct the mathematical

relationships between the states and the transiting rates in the model shown in Figure

3.6. One, a frequency balance approach to probability states P is demonstrated in

Equations 3.2 – 3.4.

agagrragpup PPP µµλλ ×+×=+× )( (3.2)

ag

agupagagagup PPP

µλ

µλ =⇒×=× agP (3.3)

r

puprrrpup PPPP

µλ

µλ =⇒×=× (3.4)

The summation of all state probabilities is unity, as given in Equation 3.5.

1=++ agrup PPP (3.5)

Replacing Pag and Pr in Equation 3.5 by their corresponding expressions in Equations

3.3 and 3.4 will enable the up state probability Pup to be calculated in term of all

transiting rates. This is calculated and given in Equation 3.6.

agrpagagr

agrupP

λµλµµµµµ

++= (3.6)

Now, substituting Pup from Equation 3.6 into Equations 3.3 and 3.4 will give the

repair and aging state probabilities in terms of transisting rates only, as shown in

Equations 3.7 and 3.8.

agrpagagr

pagrP

λµλµµµλµ

++= (3.7)

agrpagagr

agragP

λµλµµµλµ

++= (3.8)

74

All transiting rates in Equations 3.6 – 3.8 are constant, except for the aging failure rate

λag. For every aging year, new set o values are therefore calculated. The relationship

between the aging state probability and the other two states are deducted and shown in

Equations 3.9 and 3.10. Carefull consideration of the expressions indicates that the

propabilities of up and repair states have the same form as when there is no aging

state, but weighted by (1 – Pag). The value of Pag is the unavailability of the

component due to an aging failure while the complement of Pag corresponds to the

availability of the component. Therefore, adding the aging state into the existing two

state Markov model will change the other state probabilities proportionally to the

availability of the aging state (1-Pag).

)1( agpr

rup PP −×

+=

λµµ

(3.9)

)1( agpr

pr PP −×

+=

λµλ

(3.10)

3.3.1.2 Aging State with Three States

The mode of equipment failure in a power system and the type of actions that are

required as a consequence, normally can be described by an appropriate Markov

model. For example, a certain mode of failure in equipment may cause a short circuit

and adversely affect the continuity of supply in an electricity network. In such

situations, switching devices like circuit breakers are used to protect the network from

wide spread disruptions. Circuit breakers are designed to disconnect the faulty section

from the rest of the network and this action is done automatically once and

immediately after a fault is detected in the breaker zone. Certain technical and safety

standards and procedures need to be observed before the faulty equipment can be

isolated. Some of these events such as switching actions can be systematically

modeled using Markov state space method. The faulty equipment is normally isolated

75

and sent for repair following the protection switching action. For such events, the

model described in Figure 3.6 can be extended by introducing a switching state as

shown in Figure 3.7.

Fig. 3.7 A four state Markov model, including an aging state.

A similar protective switching action, with a relatively short repair time, will take

place when an aging failure occurs , however, on average the replacement or

refurbishment times are much longer [28]. Therefore, similar to the maintenance state

in the Markov model, switching actions which are at most in the range of few hours,

are not modeled for the aging state.

The state probabilities of the Markov model in Figure 3.7 can be calculated using the

same technique applied for the state space model of Figure 3.6. A basic set of

equations can be derived as shown in the following Equations 3.11 – 3.14.

µag λag

µsw

λa

µr

λp

Repair state

Up state

Switching state

Aging failure state

76

agagrragpaup PPP µµλλλ ×+×=++× )( (3.11)

swswpuprr PPP µλµ ×+×=× (3.12)

sw

aupswaupswsw PPPP

µλλµ =⇒×=× (3.13)

ag

agupagagagagup PPPP

µλ

µλ =⇒×=× (3.14)

Substituting Equation 3.13 in Equation 3.12, will give;

r

paupr PP

µλλ )( +

= (3.15)

The summation of all state probabilities are unity, as stated in the following Equation

3.16.

1=+++ agswrup PPPP (3.16)

Substituting Equations 3.13 – 3.15 into Equation 3.16 will give the probability of an

up state as per the following Equation 3.17;

agswraagrpaagswagswr

agswrupP

λµµλµµλλµµµµµµµµ

++++=

)( (3.17)

Similarly, the other state probabilities are calculated and given in the following



aagrswP

λµµλµµλλµµµµµλµµ

++++=

)( (3.18)

77


paagswrP

λµµλµµλλµµµµµλλµµ

+++++

=)(

)( (3.19)


agswragP

λµµλµµλλµµµµµλµµ

++++=

)( (3.20)

By rearranging the frequency balance of Equations 3.11 – 3.14, every state probability

can be obtained as a function of aging probability. The calculated expressions shown

in Equations 3.21 – 3.23 demonstrate that; as the aging probability of a component

increases other state probabilities will proportionally decrease. This is true due to the

fact that all the transitional rates in these equations are constant and only Pag is

increasing with aging year.

)1()( ag

paswarswr

swrup PP −×

+++=

λλµλµµµµµ

(3.21)

)1()( ag

paswarswr

arsw PP −×

+++=

λλµλµµµλµ

(3.22)

)1()(

)(ag

paswarswr

paswr PP −×

++++

=λλµλµµµ

λλµ (3.23)

Therefore, as a component ages, the probability of an aging failure occurring

dominates the chance of encountering to any other states.

3.3.1.3 Non-Repairable Aging State with Four States

In addition to the switching state, the maintenance state can also be added to the

Markov model. Some of the equipment in a power system such as power

transformers and circuit breakers may require regular maintenance. The maintenance

is normally random or scheduled. Short term planning is associated with scheduled

maintenance, and long term planning is associated with random planning. In general,

78

routine maintenance is planned and should not cause any disruption, unless some

other component fails during a maintenance period. The Markov model of a

component shown in Figure 3.8 includes the maintenance state.

Fig. 3.8 A five state Markov model, including an aging state.

The same approach as in the last sections is used to generate the state probabilities

agagmmrragmpaup PPPP µµµλλλλ ×+×+×=+++× )( (3.24)

swswpuprr PPP µλµ ×+×=× (3.25)

sw

aupswaupswsw PPPP

µλλµ =⇒×=× (3.26)

m

mupmmupmm PPPP

µλλµ =⇒×=× (3.27)

ag

agupagagagagup PPPP

µλ

µλ =⇒×=× (3.28)

µag λag

µsw

λa

µr µm

λp λm

Maintenance

outage state

Repair state

Up state

Switching state

Aging failure

state

79

Substituting Equation 3.26 in Equation 3.25, will give;

r

paupr PP

µλλ )( +

= (3.29)

The sum of all state probabilities is adding to unity, as expressed in Equation 3.30.

1=++++ agswmrup PPPPP (3.30)

Substituting Equations 3.26 – 3.29 into Equation 3.30 will give the probability of up

state as shown in the following Equation 3.17;

agswmraagmrmagswrpaagswmagswmr

agswmrupP

λµµµλµµµλµµµλλµµµµµµµµµµµ

+++++=

)(

(3.31)

Similarly, the other state probabilities are calculated and given in the following



aagmrswP

λµµµλµµµλµµµλλµµµµµµµλµµµ

+++++=

)(

(3.32)


paagswmrP

λµµµλµµµλµµµλλµµµµµµµλλµµµ

++++++

=)(

)(

(3.33)


magswrmP


+++++=

)(

(3.34)


agswmragP


+++++=

)(

80

(3.35) As per the previous sections, Equations 3.36 – 3.39 are derived to show that in this

case too, the probabilities of the up, switching, repair and maintenance states are equal

to the probabilities without having an aging state that is weighted by the addition of

the aging state. These equations are shown merely to deduce some meaningful

relationships between the usual operating states residing in the random mode of the

useful life period of a component with the aging state mode of its life. Throughout the

last few sections which develop the expressions for state probabilities, this

relationship has consistently been true in all cases.

)1()( ag

mswrpamswamrmswr

mswrup PP −×

++++=

λµµλλµµλµµµµµµµµ

(3.36)

)1()( ag

mswrpamswamrmswr

amrsw PP −×

++++=


(3.37)

)1()(

)(ag

mswrpamswamrmswr

pamswr PP −×

+++++

=λµµλλµµλµµµµµ

λλµµ (3.38)

)1()( ag

mswrpamswamrmswr

mswrm PP −×

++++=


(3.39)

The advantage of representing a non-repairable aging state in a Markov model, for

power system equipment is discussed next. Comparisons are made at the component

level between including and excluding non-repairable aging states in the conventional

four states Markov model.

The basic states in a conventional model are; operating, switching, repair and

maintenance states, as shown in Figure 3.9. The transitional rates in the conventional

model are associated with the useful life, otherwise known as random failure mode of

81

the equipment. Each state signifies its unique function or attribute that makes it

different from the rest of the states. A common attribute between the four states in the

conventional model is that the transitional rates between the states are constant owing

to useful life period consideration. A brief description of each state is provided in the

following;

Fig 3.9 Four states, conventional Markov model for power system equipment.

• The up or operating state indicates that the equipment is available and

operational.

• The switching state indicates that the equipment has failed and caused the

protection switches to activate until the failed component is isolated. During

this time, a wider network disconnection due to the protection switches

operating can occur.

• The repair state indicates that the faulty equipment is on repair after the

isolation has occurred. The repair state is reached either through the switching

state or directly through the operating state, if the equipment failure did not

activate protection switches, like the sudden opening of conducting points.

• The maintenance state indicates that the equipment is on planned maintenance

and during this time provisions are made for continuity of supply such that no

outages are experienced based on this event alone.

µsw

λa

µr µm

λp λm

Maintenance state

Repair state Up state

Switching

state

µag λag

Aging

failure state

82

In addition, the non-repairable aging state has three distinct differences with other

states, as described in the followings;

• The failure rate for this non-repairable aging state is not constant and

increases nonlinearly with the aging year of the equipment, which can be

obtained from the aging characteristic of the equipment.

• Equipment failing due to this event must be replaced or refurbished, and

the time for either of the actions is significantly long duration compared

with repair or maintenance times.

• The non-repairable aging event of equipment is NOT a re-occurring event

to be represented by switching, repair or maintenance states. The aging

failure rate and the replacement time for this state, together with the other

states for calculating power system reliability, are merely facilitating an

estimate of the risk whereby a calculated and timely decision can be made.

The preceding discussions and the descriptions logically justify the need,

appropriateness and the advantages of the non-repairable aging model.

For comparison, the following component level examples are undertaken to illustrate

the unique attribute and advantage of the non-repairable aging model. An argument

can be made against modelling the aging state and although conceptually disputable, it

can be said that instead of introducing a non-repairable aging state the failure rate in

the conventional model could be increased. This approach violates the basic

assumptions and descriptions made earlier. However, in the following examples the

application for both cases are examined, compared and discussed. Table 3.1 shows

the data used in these examples. The aging failure rate is obtained using a Weibull

distribution with α = 30 and β = 15.

83

Active Failure Rate


Maintenance Rate

Switching Time

Repair Time

Maintenance Time

Replacement Time

(occ/yr) (occ/yr) (occ/yr) (hours) (hours) (hours) (hours) 0.1 0.01 0.2 1 50 10 1176

Table 3.1 Reliability data used for Markov models.

In order to make the conventional four states model compatible with the aging

included model, the aging failure rate is added to the active failure rate. The results

for all state probabilities and frequencies are provided in Figures 3.10 – 3.19.

As aging increases, the operating state probability and frequency of the two models

shown in Figures 3.10 and 3.14 clearly indicate the difference between them. The

model with the non-repairable aging state can be validated as the decrease in

operating state probability occurs during the same period as the mean life of the

equipment. In addition, Figures 3.11 and 3.15 illustrate the sharp increase in both

switching state probability and frequency of the conventional model without an aging

state, but with an identical total failure rate. The non-repairable aging event is non re-

occurring in operation and therefore should not cause the switching state probability

and frequency to increase, as compared with the conventional model with increasing

failure rate.

With & without Aging State

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Year

Op

erat

ing

Sat

ate

Pro

bab

ilit

y

Incl Aging State

Excl Aging Sate

Fig 3.10 Operating state probability in Markov models; with and without an aging

state.

84

With & Without Aging State

0

0.005

0.01

0.015

0.02

0.025

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58

Year

Sw

itch

ing

/Act

ive

Fai

lure

Sta

te

Pro

bab

ilit

y

Incl Aging State

Excl Aging State

Fig 3.11 Active/switching state probability in Markov models; with and without an

aging state.

As the aging increases and dominates, it is anticipated that the other states probability

and frequency decrease in favour of the aging. Consequently, the probability and the

frequency of repair and maintenance states shown in Figures 3.12, 3.13, 3.16 and 3.17

for the two models, also indicate that the increasing failure rate used in the

conventional model is not compatible with the aging equipment failure characteristics,

and therefore fails to include non-repairable aging mode correctly.


0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Year

Rep

air

Sta

te P

rob

abil

ity

Incl Aging State

Excl Aging State

Fig 3.12 Repair state probability in Markov models; with and without an aging state.

85


0

0.00005

0.0001

0.00015

0.0002

0.00025

1 6 11 16 21 26 31 36 41 46 51 56

Year

Mai

nte

nan

ce S

tate

P

rob

abil

ity

Incl Aging State

Excl Aging State

Fig 3.13 Maintenance state probability in Markov models; with and without an aging

state.

As stated before, the random and the non-repairable failure events of equipment are

separate modes and are considered to be “independent” of each other. Ignoring these

considerations can provide misleading data when modelling the two modes of events,

and therefore, care needs to be taken to treat them appropriately.


0

20

40

60

80

100

120

140

160

180

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58

Year

Op

erat

ing

Sta

te F

req

uen

cy

Incl Aging State

Excl Aging State

Fig 3.14 Operating state frequency in Markov models; with and without an aging

state.

86


020

4060

80100

120140

160180

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Year

Sw

itch

ing

/Fai

lure

Sta

te

Fre

qu

ency

Incl Aging State

Excl Aging State

Fig 3.15 Active/switching state frequency in Markov models; with and without an

aging state.

The non-repairable state probability and frequency are shown in Figures 3.18 and 3.19

and illustrate the close correlation with the equipment aging failure characteristic. It

can be seen that the values increase near the mean life time of the equipment and

converge to a limit thereafter, where the non-repairing state dominates.


02040

6080

100120

140160180

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Year

Rep

air

Sta

te F

req

uen

cy

Incl Aging State

Excl Aging State

Fig 3.16 Repair state frequency in Markov models; with and without an aging state.

87


0

0.05

0.1

0.15

0.2

0.25

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Year

Mai

nte

nan

ce S

tate

F

req

uen

cy

Incl Aging State

Excl Aging State

Fig 3.17 Maintenance state frequency in Markov models; with and without an aging

state.

In addition to the previous discussions, the repair state is associated with an average

repair time. Aging considerations in the conventional model, without the non-

repairable state, lacks the inclusion of replacement time, where the actual replacement

can be implemented during a planning year as it will be discussed in the following

section. Furthermore, frequency indices will also give a gross error if the increasing

failure rate is applied to represent the non-repairable aging in the conventional model.

Aging State Probability

0

0.2

0.4

0.6

0.8

1

1.2

1 6 11 16 21 26 31 36 41 46 51 56

Year

Ag

ing

Sta

te P

rob

abil

ity

Aging State Probability

Fig 3.18 Non-repairable aging state probability.

As the risk reaches a certain level which is beyond an acceptable point, the component

should actually be replaced. For example, Figures 3.20 and 3.21 show that when a

88

component has reached a non-acceptable level of risk at age 31 years, it is replaced

with a new one. It can be seen that once the component is replaced the probability

associated with this state reduces to zero,

Aging State Frequency

0

1

2

3

4

5

6

7

8

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57Year

Exp

ecte

d F

req

uen

cy (

occ

/yr)

Aging State Frequency

Fig 3.19 Non-repairable aging state frequency.

Non-repairable State Probability

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Year

Pro

bab

ilit

y

Non-repairable StateProbability

Fig 3.20 Non-repairable aging state probability, with replacement.

89

Non-repairable State Frequency

0

0.1

0.2

0.3

0.4

0.5

0.6

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71

Year

Fre

qu

ency

(o

cc/y

r)

Non-repairable StateFrequency

Fig 3.21 Non-repairable aging state frequency, with replacement.

As part of this section, two models using the conventional Markov state space with

and without non-repairable state are considered. The examples are used to verify that

the inclusion of an aging state is an appropriate and useful model for representing

non-repairable aging. The treatment of non-repairable aging as a separate state in

combination with the conventional model was shown to work well.

3.4 Procedures in Evaluating Load Point and System Indices,

Including Non-Repairable Aging States

This section explains, using the enumeration method discussed in Chapter 2, the steps

and procedures involved in evaluating the load points and the system indices while

components are aging..

The procedures are very similar to the ones discussed in the last chapter, with a few

exceptions. As the aging components are included in the power system reliability

evaluations, the number of planning years will normally be required for the

90

calculations. These calculations for each year of simulation are performed

independent of any other year.

The contingency enumeration method used in the last chapter included single and

double contingencies. The aging state added to the Markov model in this chapter

introduces the increasing aging probability of the aged states as the planning year

increases. At first, this problem may seem to put enormous demand on the number of

enumeration combinations to obtain more precise results. The advantage of using the

enumeration method over the cut set or tie set and Bayesian models is that, the system

states are depicted by initially defining and selecting the contributing system states of

interest to avoid excessive calculations. This method suits the application in power

systems analysis but may not be useful in other applications. More in-depth

investigation into the calculations suggested that, as the components are aging,

normally corrective actions such as replacement or refurbishment are required at a

time near the mean life of a component. This is where the aging state probabilities are

still far less than when they may dictate the inclusion of more combinations in the

calculations. However, a selected set of triple combinations of contingencies was

found to be sufficient to add in as a trade-off between the accuracy of the results and

the computation intensity. It should be noted that aging contingency is included

automatically as part of the single and the double contingencies for this method. The

selected third order contingencies included are as follow:

• If there is only one aging component;

No changes are required.

• If there are only two aging components in a network, the following third order

contingencies are added;

Aging, Aging, Switching

Aging, Aging, Repair

Aging, Aging, Maintenance

91

• If there are three or more aging components in a network, the following third

order contingencies are added;

Aging, Aging, Aging

Aging, Aging, Switching

Aging, Aging, Repair

Aging, Aging, Maintenance

The percentage contribution of the components in the reliability indices are shown in

Equations 2.25, 2.26, 2.27 and 2.28 in Chapter 2 and are adjusted in accordance with

the inclusion of the third order contingencies in this method.

The detailed programming procedures and calculations of the indices are very

complex and are outside of the scope of this thesis. However, the basic steps in the

system reliability evaluation using this method and starting from year one of the

planning can be explained as follows;

• Calculate the failure rates of the components that are aging, using normal or

Weibull distributions.

• Calculate all state probabilities, for each component.

• Use the enumeration technique to calculate all system state contingency

probabilities.

• For every system state contingency perform a flow to check if there are any

loads or sources disconnected then record the indices and the relevant

calculated information as part of the output results.

• Increment the planning year and follow the same steps again until all planning

years are completed.

92

3.5 Application to Substation and Sub-transmission Networks

In this section extensive example of two small and one medium size substation

networks are used to examine the methodology. Two sets of data are used

simultaneously for these cases. The component reliability data associated with the

random failure of the useful life mode of the networks are the same as in Chapter 2,

and given in Tables 2.1 and 2.6. The loads in all systems have also retained their

values for the applications in this chapter.

The data associated with the aging components are as follows;

• For consistency a normal distribution is used to represent all aging

components, with a mean of 45 years and a standard deviation of 15 years.

• A Weibull distribution for aging can also be used in these calculations.

• The age of all aging components examined in this chapter are the same as the

planning years in simulations.

• In most of the applications in this chapter, the planning years of simulation are

taken from year 1 to 60.

• Only the effect of transformers and circuit breakers aging are presented.

However, the method and the code developed are able to handle the aging of

every other component in the network as well. Transformer and circuit

breaker aging models are included most easily, among the facilities in a power

network substation .

• The average replacement or refurbishment time used for a transformer is 7

weeks (equivalent to 1176 hours), and for a circuit breaker is 4 weeks

(equivalent to 672 hours).

93

3.5.1 Application to Small Networks

In this section the reliability modeling of small section networks is considered. In

order to place a limit on the number of results presented and at the same time be able

to cover adequate applications to demonstrating the method, only selection of small

networks is examined from Chapter 2. Cases “a” and “d” are selected from the small

networks as they represent more practical applications than the rest of the cases. At

the same time these two cases have a greater integration of components included in

them.

As there are sixty years of planning simulations for every application and many cases

to examine, it is appreciated that a detailed presentation and illustration of all results is

impossible within the scope of this thesis. In order to capture adequate information

from the results, only some of the fundamental indices such as the probability and

frequency are considered and is presented in this Chapter. Other indices such as

unavailability, expected energy not supplied, expected demand not applied and

average duration of load curtailment, are the direct or indirect byproduct of the two

probability and frequency indices. The percentage contribution of components at load

points and system indices, can considerably to be useful in achieving the results,

particularly when aging and planning years are incorporated in the calculations. The

use of the percentage contribution index will greatly reduce the number of results that

are needed to convey reliability information.

3.5.1.1 Application to Case “a”

Case “a”, has two routes of supplying the same load, as seen in Figure 2.6. Each route

has a transformer and a circuit breaker that are examined for aging. Table 3.2

provides the results when “TX1”, “TX1 and TX2” and “TXs and CB1” are aging. As

the network is symmetrical, these cases for aging considerations seem to be adequate.

The results shown in this table with reference to the network configuration make

94

logical sense. In year 1, all cases have the same unavailability value. This is because

the components that are chosen to age have not aged yet. As the aging is gradual, the

effect takes a number of years before it makes a significant impact on the system load

unavailability. As expected, the impact of both transformers and the circuit breaker

aging is more than the two transformers, and far less effect can be seen when only one

transformer is aged. In the event of only one transform aging, the second route, where

the second transformer is not aging but has random failure, gives a much better

chance of supplying the load.

Case a, Effect of Components Aging on System Load Unavailability (hrs/yr) Planning Years TX1 TX1,TX2 TXs,CB1

1 0.063443 0.063446 0.063448

5 0.063456 0.063471 0.06348

10 0.063538 0.063638 0.063694

15 0.063926 0.064447 0.064745

20 0.065331 0.067756 0.06914

25 0.069304 0.08032 0.086599

30 0.078456 0.127249 0.15498

35 0.096996 0.299214 0.413388

40 0.133551 0.939928 1.389097

45 0.209997 3.573145 5.392866

50 0.386313 16.314374 24.350005

55 0.828523 89.128248 125.734125

60 1.925044 520.72862 627.548858

Table 3.2 System load unavailability of network case “a”, versus components age.

95

Case a, Effect of Components Aging on System Load Unavailability

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35 40 45 50

Time (year)

Load

Out

age

(hou

rs/y

ear)

TX1

TX1, TX2

TXs, CB1

Fig. 3.22 System load unavailability due to component aging for network case “a”.

As discussed before, there are two major contributors to, for example load outages.

One is the reliability attributes of the components and the other is the level of

redundancy that the configuration provides.

Figure 3.22 shows a graphical presentation of the results of expected outage in hours

per year. The effect of the same aging components on Expected Frequency of Load

Curtailment (EFLC) is provided in Table 3.3 and graphically is shown in Figure 3.23.

A similar effect, but to a lesser degree, is seen for the aging of these facilities and the

effect aging has on the system load EFLC. The range of variation of the system load

EFLC from year 1 to year 60 from the planning simulations are; multiples of 5, 30 and

45 for each aging case.

96

Case a, Effect of Components Aging on System Load EFLC (occ/yr) Planning Years TX1 TX1,TX2 TXs,CB1

1 0.038049 0.038049 0.038049

5 0.03805 0.038051 0.038052

10 0.038056 0.038064 0.038068

15 0.038087 0.038126 0.038148

20 0.038199 0.038349 0.038435

25 0.038514 0.038988 0.039258

30 0.039241 0.040485 0.0412

35 0.040714 0.043634 0.045339

40 0.043619 0.050302 0.0543

45 0.049701 0.066216 0.076452

50 0.063758 0.113117 0.144972

55 0.099197 0.293826 0.421339

60 0.188199 1.138736 1.725509

Table 3.3 System load EFLC of network case “a”, as components age.

Case a, Effect of Component Aging on System EFLC

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 10 20 30 40 50 60

Time (year)

Fre

quen

cy (

occ/

yr)

TX1

TX1, TX2

TXs, CB1

Fig. 3.23 System load EFLC due to component aging for network case “a”.

The components in a network normally have different levels of response to an event

of interest. This response, or effect, and its contribution may also change as aging is

taking place. The percentage contribution of the individual components in “case a”,

while TX 1 is aging, is illustrated in Figure 3.24. It should be noted that the sum of

97

the contribution values shown for each year adds up to 100%. In the early years of

planning the aging shows no effect however, as aging progresses, the effect becomes

more evident. Many factors may affect the rate of change of the contributions. The

results for this example indicate that initially bus section outages are the major factor

causing load unavailability.

Around year 30, it is seen that TX1’s contribution increases followed by TX2. All

other components have a reduced effect on the load PLC. Although TX1 is the only

aging component, TX2 has the next largest contribution. This is due to the fact that

the combined outage probabilities of TX2, and any component in the other branch,

has a much higher value in effecting the load outages than any other components in

the second route. This in fact is a good example for showing where in the network

more attention and resources should be allocated. There can be seen similar effects on

the load EFLC in Figure 3.25 with different degree. The increase in EFLC usually

lags the unavailability index, as it is the product of system state probabilities and the

rate of transitions.

% Contribution of Componenets to System Load PLC, Case a, TX 1 aging.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Time (year)

Per

cen

tag

e Line1

TX1

CB1

Line2

TX2

CB2

Bus

Fig. 3.24 Percentage contribution of components to System load unavailability, as

TX 1 is aging for network case “a”.

The detailed relationship between unavailability and EFLC are very complex.

However, it can be explained that in some systems particular component failure

98

occurrences contribute to shorter outage durations compared the components in other

systems.

% Contribution of Components to System Load EFLC, Case a, TX 1 aging.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (year)

Per

cent

age Line1

TX1

CB1

Line2

TX2

CB2

Bus

Fig. 3.25 Percentage contribution of components to System load EFLC, as TX 1 is

aging for network case “a”.

The case where TX1 and TX2 are aging is shown in Figure 3.26. This is an

interesting case where it shows that both transformers have exactly the same

contributions to a load outage as expected. It can be seen that after to the mean life

age of the transformers, the failure probabilities of the two completely dominate the

system load unavailability events. This change in influence of the two transformers

happens between the ages 15 – 45 years. Note that the horizontal axis planning years

coincide with the aging years of the components for this application.

99

It should also be noted that the yearly percentage contribution of a component to an

index, in this case the system load, corresponds to the year the index is calculated.

% Contribution of Components to System Load PLC, Case a, TX1, TX2 aging.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Time (year)

Per

cen

tag

e Line1

TX1

CB1

Line2

TX2

CB2

Bus

T X1, TX2

Bus

CB1, CB2


TX 1 and TX 2 are aging for network case “a”.

Similar effects are observed with regard to the percentage contribution of the

components to the expected frequency of the system load curtailment. The range of

changes in this case is more widespread and exists between 15 – 65 years. As the

aging failure frequency of transformers increases, the expected frequency of system

load curtailment also increases.

During the early life of components, bus bars and circuit breakers have the highest

contribution to system load outages of about 64% and 14% respectively (Figure 3.26)

and an unavailability and EFLC of 53% and 22% (Figure 3.27). As transformers are

aging identically the system is symmetrical, therefore the patterns of contributions of

100

any pair of similar components on either side of the circuit are the same as in Figures

3.26 and 3.27.

% Contribution of Components to System Load EFLC, Case a, TX1, TX2 aging.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age Line 1

TX 1

CB 1

Line 2

TX 2

CB 2

Bus

TX1, TX2

Bus

CB1, CB2

Fig. 3.27 Percentage contribution of components to System load EFLC, as TX 1 and

TX 2 are aging for network case “a”.

The last application for case “a” is when both transformers and circuit breaker CB1

are aging. As seen in Figures 3.28 and 3.29, the contributions to PLC and EFLC are

the same as in the previous cases for the earlier part of the planning years. However,

in the later part the contribution of transformer TX1 and circuit breaker CB1 are

shared differently and the summation of the TX1 and CB1 contributions adds to 50%,

equal to the TX2, at the mean age of the components (60 years). As there are two

sources on each side of the circuit linked to the load, the outages are equally

contributed between each side.

101

% Contribution of Components to System Load PLC, Case a, TXs, CB1 aging.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX 1

CB 1

Line 2

TX 2

CB 2

Bus


TXs and CB1 are aging for network case “a”.

% Contribution of Components to System Load EFLC, Case a, TXs & CB1 aging.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age Line 1

TX 1

CB 1

Line 2

TX 2

CB 2

Bus

Fig. 3.29 Percentage contribution of components to System load EFLC, as TXs and

CB1 are aging for network case “a”.

102

3.5.1.2 Application to Case “d”

In case “d” of Figure 2.6, each load is connected to an individual bus. Each bus has

similar circuit which is linked to two different sources. The buses in this system are

interconnected through a circuit breaker “CB” thus this network is symmetrical and

each load can have access to two sources. The networks in cases “a” and “d” are both

symmetrical and look similar but have differences. The network in case “d” has two

loads, two buses and one circuit breaker.

The unavailability and expected frequency of system load curtailment in case “d” is

shown in Figure 2.6, while 60 year planning for aging components “TX1”, “TX1 and

TX2”, “CB”, “TX1 and CB” or “TXs and CB” is given in Tables 3.4 and 3.5. The

facilities within this circuit providing a separate supply per load point are more in line

with case “a” than “d”, and thus make “d” it more reliable.

Case d, Effect of Components Aging on Overall System Load Unavailability (hrs/yr) Planning Years TX1 TX1,TX2 CB TX1,CB TXs,CB

1 0.414792 0.414795 0.414792 0.414795 0.414798

5 0.414803 0.414817 0.414805 0.414819 0.414833

10 0.414878 0.414968 0.414889 0.414978 0.415069

15 0.415229 0.415703 0.415283 0.415743 0.416238

20 0.416499 0.418743 0.416711 0.418726 0.421275

25 0.420092 0.430547 0.420751 0.428996 0.442381

30 0.428369 0.475729 0.430063 0.462946 0.529402

35 0.445139 0.644172 0.448961 0.575811 0.869031

40 0.478221 1.278006 0.486352 0.971982 2.172874

45 0.547466 3.897111 0.565102 2.550978 7.561428

50 0.707497 16.607155 0.749666 10.132003 33.070332

55 1.110852 89.350469 1.231377 54.359474 167.962723

60 2.123232 520.812981 2.555978 337.26723 808.377202

Table 3.4 System load unavailability of network case “d”, as components age.

As pointed out in Chapter 2, the overall system load indices relate to any one or more

loads being on outage due to any system state contingency. The indices evaluated

using this consideration provide system wide measurements. The advantage of the

103

measurement is that it will allow a comparison of the networks with different circuit

arrangements. For instant, networks in case “a” and “d” are similar, and their

differences include the circuit arrangement and the number of load points. As

expected, and seen in Tables 3.2 – 3.5, comparing the results of similar aging failures

suggest that case “a” exhibits better reliability than case “d”. But, given the number

of components serving per load in each case, case “d” can be more efficient and cost

effective from the reliability point of view.

Case d, Effect of Component Aging on Overall System Load Unavailability

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45 50

Time (years)

hour

s/ye

ar TX1

TX1, TX2

CB

TX1, CB

TXs, CB

Fig. 3.30 System load unavailability due to component aging for network case “d”.

104

Similarly to case “a”, the first year indices are the same as when there are no aging

failures but as the aging failures increase the load unavailability is seen to increase

rapidly.

The mean and the variance of the failure probability distribution function due to aging

components will influence the magnitude and the spread of the increasing effect of

system reliability indices, as the aging years are increased.

The unavailability and expected frequency of the overall system load are illustrated in

Figures 3.30 and 3.31, for all component aging groups.

Case d, Effect of Components Aging on Overall System Load EFLC (occ/yr) Planning Years TX1 TX1,TX2 CB TX1,CB TXs,CB

1 0.369232 0.369232 0.369232 0.369232 0.369232

5 0.369232 0.369232 0.369232 0.369232 0.369232

10 0.369231 0.369229 0.369234 0.369233 0.369232

15 0.369225 0.369219 0.369244 0.369237 0.369231

20 0.369207 0.369182 0.369278 0.369253 0.369229

25 0.369153 0.369083 0.369375 0.369303 0.369239

30 0.369032 0.368887 0.369598 0.369441 0.36934

35 0.368789 0.368625 0.370053 0.369827 0.369877

40 0.368325 0.36864 0.370957 0.371 0.372237

45 0.367421 0.370945 0.372883 0.375242 0.382684

50 0.365671 0.387984 0.377508 0.394406 0.434492

55 0.363338 0.502459 0.390319 0.502539 0.724308

60 0.370179 1.226699 0.430991 1.184042 2.291083

Table 3.5 System load EFLC of network case “d”, as components age.

As more components are allowed to age for these trail cases, the reliability effect also

increases. In case “d”, the circuit breaker is in the link of alternative supply for both

loads but its effect on the reliability indices can however vary depending on the other

component reliabilities.

Since case “d” has two load points, the reliability effects on each load point is

examined with respect to variation of aging components. The results are provided in

105

Tables 3.6 and 3.7 for unavailability and expected frequency of load curtailment,

respectively.

Case d, Effect of Component Aging on Overall System Load EFLC

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60

Time (years)

Fre

quen

cy (

occ/

year

)

TX1

TX1, TX2

CB

TX1, CB

TXs, CB

Fig. 3.31 System load EFLC due to component aging for network case “d”.

The load point indices are specific to a load point and as such demonstrate the effect

of contingencies and aging on individual loads.

Case d, Effect of Components Aging on Load 1/2, Unavailability (hrs/yr) Years TX1_Ld1 TX1_Ld2 TX1,TX2 CB TX1,CB_Ld1 TX1,CB_Ld2 TXs,CB

1 0.214407 0.214407 0.21441 0.214406 0.214409 0.214408 0.214411

5 0.214419 0.214418 0.214433 0.214412 0.214427 0.214426 0.214441

10 0.214498 0.214491 0.214587 0.214454 0.214549 0.214541 0.214638

15 0.214871 0.214836 0.215339 0.214651 0.215138 0.215082 0.215605

20 0.216222 0.216084 0.218437 0.215363 0.217486 0.217041 0.2197

25 0.220042 0.219614 0.230405 0.217378 0.225964 0.222581 0.236311

30 0.228842 0.227746 0.275965 0.222024 0.255798 0.235315 0.302747

35 0.24667 0.24422 0.445174 0.231452 0.360381 0.261017 0.557246

40 0.281828 0.276709 1.080509 0.250105 0.740442 0.311318 1.525373

45 0.355379 0.34468 3.702723 0.289393 2.286686 0.414901 5.513811

50 0.525157 0.501591 16.419792 0.381471 9.796965 0.645584 24.43819

55 0.951823 0.896014 89.179971 0.62181 53.875714 1.174865 125.7546

60 2.015041 1.879437 520.68135 1.282806 336.579822 2.2058 627.4719

Table 3.6 Load unavailability of network case “d”, as components age.

106

A comparison between the overall system load and individual load indices can

provide the degree of the joint cause of outages for the individual load points in the

network as shown in Figure 3.32.

Case d, Effect of Component Aging on Load 1/2 Unavailability

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35 40 45 50

Time (years)

Out

age

time

(hou

rs/y

ear)

TX1_Ld1

TX1_Ld2

TX1, TX2

CB

TX1,CB_Ld1

TX1,CB_Ld2

TXs, CB

Fig. 3.32 Load unavailability due to component aging for network case “d”.

For example, if one system state contingencies cause a load to go on outage and

another system state causing a second load to go on outage, then the overall system

load outage index would include the summation of the two system state probabilities.

However, if a system state contingency cause the two loads to go on outage at the

same time, then for the overall system load outage index we do not add up twice, for

each load.

107

Case d, Effect of Components Aging on Load 1/2, EFLC (occ/yr) Years TX1_Ld1 TX1_Ld2 TX1,TX2 CB TX1,CB_Ld1 TX1,CB_Ld2 TXs,CB

1 0.189246 0.189246 0.189246 0.189246 0.189246 0.189246 0.189246

5 0.189246 0.189246 0.189247 0.189246 0.189247 0.189246 0.189247

10 0.18925 0.189248 0.189252 0.189247 0.189251 0.189249 0.189253

15 0.189264 0.189256 0.189274 0.189251 0.18927 0.189261 0.18928

20 0.189318 0.189285 0.189357 0.189267 0.18934 0.189306 0.18938

25 0.189468 0.189367 0.189597 0.189313 0.189542 0.189433 0.189671

30 0.189816 0.189556 0.19018 0.189418 0.190032 0.189725 0.190393

35 0.190523 0.18994 0.191486 0.189631 0.191129 0.190315 0.19208

40 0.191923 0.190706 0.194558 0.190055 0.193701 0.191471 0.196279

45 0.194885 0.192341 0.203113 0.19096 0.200848 0.193859 0.208775

50 0.201886 0.196283 0.233842 0.193139 0.226626 0.199215 0.256524

55 0.220482 0.207213 0.378538 0.199203 0.350817 0.211737 0.489123

60 0.272968 0.240728 1.156766 0.218677 1.069199 0.237587 1.724515

Table 3.7 Load EFLC of network “case d”, as components age.

Figures 3.32 and 3.33 illustrate the unavailability and EFLC for individual load points

of case “d”, as some of the components age. The aging of components may have

different effects on each load. If a system is symmetrical, the similar component

aging failures will have the same effect on each load. For example, the failure of each

of the events; “TX1 and TX2”, “CB” and “TXs and CB”, will separately have the

same reliability effect on both loads. The effect of TX1 aging failure on load “1” is

fractionally more so than on load “2”, shown by both unavailability and EFLC

indices. The difference in the effect will depend on the combined reliability of each

branch, while the transformer is aging. A circuit breaker of 60 years of age has a

similar EFLC effect as TX1 has on load “2”. The effect of TX1 on load “1” at 60

years is somewhat higher. The two transformers failure effect is very similar to the

effect of the combined aging failure of “TX1 and CB” on load “1”. However, the

difference in unavailability is substantially larger with the two transformers reaching

to about twice as much as “TX1 and CB” together. This example shows that the

effect on frequency indices will not necessarily be the same as the PLC and

unavailability indices.

108

Case d, Effect of Component Aging on Load 1/2 EFLC

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60

Time (years)

Fre

quen

cy (

occ/

Yea

r)

TX1_Ld1

TX1_Ld2

TX1, TX2

CB

TX1,CB_Ld1

TX1,CB_Ld2

TXs, CB

Fig. 3.33 Load EFLC due to component aging for network case “d”.

Observation of the results for variation of aging events will provide invaluable

information about the reliability performance of a network. The information can be

used for numerous objectives, including network design, reinforcement, and

replacement, etc.

In this section, extensive illustrations are provided in Figures 3.34 – 3.49 to show the

effect of different component aging failures on the unavailability and EFLC indices

for the network of case “d”. The general trends in the early life portion of these

graphs can be classified into two groups. The first group is associated with the

component contributions to load “1” and the second group is related to the component

contributions to load “2”. The common trends in the first group is that; TX1, Line1,

Bus1 and CB have the highest to the lowest contributions respectively to load “1” and

the other component’s contributions are insignificant. Similarly, the common trends

in the second group is that; TX2, Line2, Bus2 and CB have the highest to the lowest

109

contributions respectively to load “2”, and the other component’s contribution is

insignificant.

% Contribution of Componenets to Load 1 PLC/Unavailibility, Case d, TX1 aging

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Pro

babi

lity

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.34 Percentage contribution of components to load 1 unavailability, as TX1

ages for network case “d”.

110

% Contribution of Componenets to Load 1 EFLC, Case d, TX1 aging

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.35 Percentage contribution of components to load 1 EFLC, as TX1 is aging for

network case “d”.

% Contribution of Components to Load 2 PLC/Unavailability, Case d, TX1 aging

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.36 Percentage contribution of components to load 2 unavailability, as TX1

ages for network case “d”.

111

The percentage contribution of components to the load “1” and “2” unavailability and

EFLC, while TX1 is aging are shown in Figures 3.34 – 3.37.

It can be seen that TX1 aging failure has a dominant contribution to load “1” PLC and

EFLC, with a slow increase while it is aging (Figure 3.36). As TX1 aging takes place,

TX2 contribution to load “1” increases fairly rapidly. This shows that load “1”

dependency on the alternative route of supply, where TX2 has a large contribution to

it, increases as TX1 encounters aging failures. The contribution of TX1 on load “2”

however, can be seen in a different way. For the random failure events of TX2 the

availability of the alternative supply route to load “2”, which depends on TX1 aging,

becomes more important. This situation will affect the share of contribution between

TX1 and TX2 in supplying load “2”. As the TX1 probability of aging failure

increases, the importance of this component to load “2” continuity of supply will also

increase. This explains the sharp rise in percentage contribution of TX1 to load “2”

unavailability, shown in Figure 3.36.

% Contribution of Componenets to Load 2 EFLC, Case d, TX1 aging

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.37 Percentage contribution of components to load 2 EFLC, as TX1 ages for


The decrease in TX2 contribution to load “2” EFLC shown in Figure 3.37, is due to

the sharp increase in the probability of TX1 failures.

112

% Contribution of Componenets to Load 1 PLC/Unavailability, Case d, TX1 & CB aging

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cen

tage

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.38 Percentage contribution of components to for load 1 unavailability, as TX1

and CB are aging for network case “d”.

The percentage contribution of components for load “1” and “2” unavailability and

EFLC, while TX1 and CB are aging is shown in Figures 3.38 – 3.41.

The contribution of TX1 and CB aging failure events are greater for load “1” than

load “2” unavailability. As seen in the case “d” network, these two components are

located one on each supply route. This is why CB contribution to load “1”

unavailability and EFLC increases fairly quickly, as shown in Figures 3.38 and 3.39.

113

% Contribution of Componenets to Load 1 EFLC, Case d, TX1 & CB aging

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.39 Percentage contribution of components to load 1 EFLC, as TX1 and CB are

aging for network case “d”.

% Contribution of Components to Load 2 PLC/Unavailability, Case d, TX1 & CB aging.

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Pe

rcen

tag

e

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.40 Percentage contribution of components to load 2 unavailability, as TX1 and

CB are aging for network case “d”.

114

% Contribution of Components to Load 2 EFLC, Case d, TX1 & CB aging.

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Pe

rce

nta

ge

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.41 Percentage contribution of components to load 2 EFLC, as TX1 and CB are


% Contribution of Componenets to Load 1 PLC/Unavailability, Case d, CB aging

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Per

cent

ag

e

Line 1

TX1

Line 2

TX 2

CB

Bus 1

Bus 2

Fig. 3.42 Percentage contribution of components to load 1 unavailability, as CB is


115

The contribution of the aging event of load “2” outages is far less than to load “1”, as

it enjoys a route to the supply with no aging components. The effects of this event are

shown in Figures 3.40 and 3.41. As given in Table 3.6, the unreliability of this event

on load “1”, at age 60 is 150 times more than that of load “2”. However, EFLC of

load 1, given in Table 3.7, is 4.5 times more than that of load 2 at age 60 years.

The effects of aging circuit breaker event are shown in Figures 3.42 and 3.43. The

circuit breaker effect is the same for loads “1” and “2”.

The difference between this case and the case where TX1 and CB are aging is that the

rate of the contributions here are slower to rise. This is because only one component

CB is aging and it blocks the alternative routes from both supplies.

% Contribution of Componenets to Load 1 EFLC, Case d, CB aging

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.43 Percentage contribution of components to load 1 EFLC, as CB is aging for


In the next study the aging effects of both transformers and the circuit breaker on load

“1” is considered. As the TXs and CB are aging the contribution of components to

load “1” PLC and EFLC are shown in Figures 3.44 and 3.45.

116

% Contribution of Componenets to Load 1 PLC/Unavailability, Case d, TXs & CB aging

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.44 Percentage contribution of components to load 1 unavailability, as the TXs

and CB are aging for network case “d”.

The effect to the outage index is shared by the contributing aging components which

is based on the degree of the chance in their occurrence. Figure 3.44 shows that TX1

has the most contribution, followed by TX2 and then CB as aging passes the mean life

of the components.

% Contribution of Componenets to Load 1 EFLC, Case d, TXs & CB aging

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.45 Percentage contribution of components to load 1 EFLC, as TXs and CB are


117

In the next study, the effect of circuit breaker aging on all system load PLC and EFLC

are considered. The contribution of components to PLC and EFLC associated with

any one or more load outage events are shown in Figures 3.46 and 3.47.

% Contribution of Componenets to All Loads PLC/Unavailability, Case d, CB aging

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70

Time (years)

Per

cen

tage

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.46 Percentage contribution of components to all loads unavailability, as CB are


In early planning years the contributing components to both unavailability and EFLC

of all load outage events, from higher to lower contribution, are transformers, lines,

buses and the circuit breaker.

% Contribution of Componenets to All Loads EFLC, Case d, CB aging

0

5

10

15

20

25

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

Line 1

TX1

Line 2

TX2

CB

Bus 1

Bus 2

Fig. 3.47 Percentage contribution of components to all loads EFLC, as CB are aging

for network case “d”.

118

As can be seen in Figures 3.46 and 3.47, the circuit breaker provides the largest

contribution of all components as it ages. Transformers stay almost steady while the

contribution of both the lines and the buses reduce. It is interesting to note that the

initial lowest contributor to load outage events becomes the largest contributor as

aging progresses. As discussed before the aim of these applications is to understand

the relationship between the aging components in a network and their contributions to

outage indices. This particular study provides a special insight into the effects of

aging of all components in a power system.

3.5.1.3 Application to Medium Network

In previous applications or examples associated with small networks, the circuits and

connections were simple and straight forward. But, even with this simplicity the

effect of aging components on the system performance in small networks, is not

necessarily trivial or easy to understand. In a more complex interconnected system it

is inconceivable to draw any conclusions without methodologically establishing some

sort of logical and systematical relationships to evaluate the effects and the

performances.

In this section the medium sized system of Figure 2.7 is used to apply the methods

developed in this thesis to include component aging as part of power system reliability

evaluations. This medium sized system is somewhat more complex and includes

more components as part of its functional requirements and the same data as in Table

2.6 is used as the random mode, or the useful life period, part of the application.

Each Load is taken as 10MW and the aging data is same as used in the last two

sections. The aging data and the planning years for calculation are the same as in the

previous applications. In order to limit the results for evaluation and discussion, a

119

single selected case was examined to give a representative snapshot of the system and

its interacting components.

In the following example, the effects of aging components TX2, TXs, Source CBs,

CBs 1 and 2, CBs 1,2, 4 on the overall medium network load unavailability and EFLC

are studied, and results given in Tables 3.8 and 3.9.

Med Sub, Effect of Components Aging on Overall System Load Unavailability (hrs/yr)

Planning Years TX2 TXs Source CBs CBs_1,2 CBs_1,2,4 1 4.058358 4.065097 4.054986 4.054987 4.054986

5 4.072682 4.108067 4.054974 4.054979 4.054974

10 4.167361 4.392093 4.054895 4.054928 4.054895

15 4.612007 5.725708 4.054527 4.054687 4.054527

20 6.221839 10.55045 4.053192 4.053815 4.053192

25 10.77698 24.172025 4.049419 4.05135 4.049418

30 21.267963 55.373614 4.040754 4.045676 4.040752

35 42.521628 117.861275 4.023378 4.034204 4.023373

40 84.434666 238.286665 3.990389 4.01167 3.990378

45 172.106453 478.478476 3.932069 3.964915 3.932048

50 374.432766 975.825369 3.914992 3.859023 3.914947

55 882.607758 1922.766437 5.641549 3.60602 5.641457

60 2147.2184 3020.018139 42.849583 3.067179 42.849435

Table 3.8 System load unavailability of the medium network, as components age.

120

Med Sub, Effect of Components Aging on Overall System Load EFLC (occ/yr) Planning Years TX2 TXs

Source CBs CBs_1,2 CBs_1,2,4

1 0.220975 0.220981 0.220972 0.220972 0.220972

5 0.220987 0.221017 0.220972 0.220972 0.220972

10 0.221066 0.221254 0.220967 0.220969 0.220967

15 0.221438 0.22237 0.220948 0.220956 0.220948

20 0.222785 0.226409 0.220876 0.220908 0.220876

25 0.226594 0.237839 0.220673 0.220772 0.220672

30 0.235369 0.264168 0.220207 0.220459 0.220205

35 0.253147 0.317529 0.219266 0.219827 0.219261

40 0.288214 0.422824 0.217425 0.218584 0.217414

45 0.361604 0.643218 0.213664 0.216005 0.213643

50 0.531155 1.151174 0.205727 0.210156 0.205683

55 0.958155 2.405364 0.195017 0.196141 0.194925

60 2.027871 5.236386 0.323297 0.165973 0.323149

Table 3.9 System load EFLC of medium network, as components age.

The effect of the same components aging is also examined for the individual load

points PLC/unavailability and EFLC and the results provided in Tables 3.10 and 3.11.

Med Sub, Effect of Components Aging on Load, Unavailability (hrs/yr) Years TX2_Ld2 S_CBs_Ld1 S_CBs_Ld2 CBs_1,2_Ld1 CBs_1,2_Ld2 CBs_1,2,4_Ld2

1 1.357701 1.352513 1.354328 1.352513 1.354328 1.354328

5 1.372034 1.352509 1.354324 1.352511 1.354326 1.354324

10 1.466776 1.352486 1.354297 1.352494 1.354307 1.354297

15 1.911713 1.352376 1.354168 1.352416 1.354222 1.354168

20 3.522601 1.351977 1.353704 1.352134 1.353912 1.353703

25 8.08073 1.350851 1.352392 1.351338 1.353034 1.352389

30 18.578587 1.348273 1.349388 1.349504 1.351016 1.349381

35 39.846152 1.343194 1.343449 1.345797 1.346936 1.343434

40 81.786496 1.334315 1.332892 1.338514 1.338921 1.332861

45 169.514946 1.325829 1.320971 1.323401 1.322295 1.320907

50 371.969683 1.420682 1.408284 1.289162 1.284664 1.408147

55 880.452829 3.409831 3.380846 1.207278 1.194902 3.380543

60 2145.7409 41.16413 41.107919 1.032303 1.004853 41.107326

Table 3.10 Load unavailability of medium network, as components age.

121

The results in Table 3.10 illustrate that after 20 years the aging of TX2 and its effect

on Load 2 starts to dominate. Table 3.11 shows that the dominating effect starts to

occur around 25 years.

Med Sub, Effect of Components Aging on Load, EFLC (occ/yr) Years TX2_Ld2 S_CBs_Ld1 S_CBs_Ld2 CBs_1,2_Ld1 CBs_1,2_Ld2 CBs_1,2,4_Ld2

1 0.076129 0.074295 0.076126 0.074295 0.076126 0.076126

5 0.076141 0.074295 0.076126 0.074295 0.076126 0.076126

10 0.076224 0.074294 0.076124 0.074294 0.076125 0.076124

15 0.076611 0.074289 0.076117 0.074289 0.07612 0.076117

20 0.078014 0.07427 0.076092 0.074273 0.076103 0.076091

25 0.081982 0.074219 0.076022 0.074227 0.076054 0.076019

30 0.091122 0.074101 0.07586 0.074122 0.075943 0.075853

35 0.10964 0.073862 0.075534 0.073908 0.075718 0.075519

40 0.14616 0.073401 0.0749 0.073489 0.075275 0.074869

45 0.222564 0.072494 0.073642 0.072618 0.074357 0.073577

50 0.398949 0.070983 0.071355 0.070643 0.072274 0.071217

55 0.842354 0.075437 0.074089 0.065911 0.067281 0.073786

60 1.948173 0.235696 0.231428 0.055719 0.056523 0.230834

Table 3.11 Load EFLC of medium network, as components age.

Graphical illustrations of these effects on the overall and individual load’s

unavailability and EFLC are shown in Figures 3.48 – 3.51.

As expected, the effect of TXs aging are more than TX2 aging on load outages given

in Figures 3.48 and 3.51.

122

Effect of Load Transformers Aging on Overall/Single Load PLC/Unavailability.

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35 40 45

Time (years)

Out

age

Tim

e (h

ours

/yea

r)

TX2

TXs

TX2_Ld2

Fig. 3.48 Overall and single load unavailability due to transformers aging for medium

network.

As the aging failures are treated like maintenance or repair states, their failure effect

does not causing widespread switching and therefore causes more system outages.

As it is shown in Figures 3.50 and 3.51, the effect of circuit breakers is steady over

time and raise sharply following the mean life of these devices. The aging failure of

CB 1 and 2 do not directly block the sources links to loads. The routes through CB1

and CB2 are used in the events that a direct link from source to load is disrupted

123

Effect of Load Transformers Aging on Overall/Single Load EFLC.

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70

Time (years)

Fre

quen

cy (

occ/

year

)

TX2

TXs

TX2_Ld2

Fig. 3.49 Overall and single load EFLC due to transformers aging for medium

network.

Effect of Circuit Breakers Aging on Overall/Single Load PLC/Unavailability.

0

1

2

3

4

5

6

0 10 20 30 40 50 60

Time (years)

Out

age

Tim

e (h

ours

/yea

r)

Source CBs

CBs_1,2

CBs_1,2,4

S_CBs_Ld1

S_CBs_Ld2

CBs_1,2_Ld1

CBs_1,2_Ld2

CBs_1,2,4_Ld2

Source CBs, CBs 1,2,4 CBs 1,2

CBs 1,2_Ld 1,2

S_CBs Ld 1,2CBs 1,2,4 Ld2

CBs 1,2

Fig. 4.50 Overall and single load unavailability due to circuit breakers aging for the

medium sized network.

124

Effect of Circuit Breakers Aging on Overall/Single Load EFLC.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50 60 70

Time (years)

Fre

quen

cy (

occ/

year

)

Source CBs

CBs_1,2

CBs_1,2,4

S_CBs_Ld1

S_CBs_Ld2

CBs_1,2_Ld1

CBs_1,2_Ld2

CBs_1,2,4_Ld2

Source CBs, CBs 1,2,4

S_CBs Ld 1,2CBs 1,2,4 Ld2

CBs 1,2_Ld 1,2

Fig. 3.51 Overall and single load EFLC due to circuit breakers aging for the medium

sized network.

As the system is almost symmetrical, the contribution of some of the component’s

aging is measured against load “2” which is in the middle of the system connections.

The contribution of source circuit breakers on load “2” is shown in Figure 3.52 and

3.53. As expected, during the earlier life of the components, TX2 followed by Bus2

have the most effect on load “2”. However, the aging of source circuit breakers will

start to dominate following the mean life of the circuit breakers.

The aging failure effects of circuit breakers “1” and “2” on load “2” are similar but to

a lesser extend as shown in Figures 3.54 and 3.55.

125

%Contribution of Components to Load 2, PLC/Unavailability, Med Sub, Source Circuit Breakers Aging.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

TX2

Bus2

Bus 1,3TX 1,3CBs 1,2

CBs 3,4,5

Fig. 3.52 Percentage contribution of components to load 2 unavailability, as source

circuit breakers are aging for the medium sized network.

126

% Contribution of Components to Load 2, EFLC, Med Sub, Source Circuit Breakers Aging.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

TX2

Bus2

CBs 3,4,5

Fig. 3.53 Percentage contribution of components to load 2 EFLC, as source circuit

breakers are aging for the medium sized network.

% Contribution of Components to Load 2, PLC/Unavailability, Med Sub, Circuit Breakers 1,2 Aging.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

TX2

Bus2

CBs 1,2

Fig. 3.54 Percentage contribution of components to load 2 unavailability, as circuit

breakers 1 and 2 are aging for the medium sized network.

127

% Contribution of Components to Load 2, EFLC, Med Sub, Circuit Breakers 1 & 2 Aging.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

TX2

Bus2

CBs 1,2

Fig. 3.55 Percentage contribution of components to load 2 EFLC, as circuit breakers

1 and 2 are aging for the medium sized network.

The aging failure effects of circuit breakers 1, 2 and 4 on load “2” is very similar to

the three source breaker aging effect on load “2”. In fact, by inspecting the medium

network configuration, it can be seen that these two aging events will limit the supply

to load “2” in a similar manner.

The overall system load indices for unavailability and EFLC, as all transformers and

circuit breakers in the system age, are given in Table 3.12. This aging affects all roots

and connections. Therefore, the severity of this event is more than any other events.

The unavailability for this case starts rising from year 5 and increases fairly rapidly.

This is not so with the increase in EFLC in Figure 3.57.

128

% Contribution of Componenets to Load 2, PLC/Unavailability, Med Sub, Circuit Breakers 1,2,4 Aging.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

CBs 1,2,4

TX2

Bus2

Fig. 3.56 Percentage contribution of components to load 2 unavailability, as circuit

breakers 1, 2 and 4 are aging for the medium sized network.

% Contribution of Components to Load 2, EFLC, Med Sub, Circuit Breakers 1,2,4 Aging.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

TX2

Bus2 CBs 1,2,4

Fig. 3.57 Percentage contribution of components to load 2 EFLC, as circuit breakers

1, 2 and 4 are aging for the medium sized network.

The outage times for the overall system load events are more than those for the

individual loads as shown in Tables 3.13 and 3.14. This is due to the fact that not

129

every load will share an outage event with every system state. If this was the case

then the overall system load outage events would have resulted in the same individual

load values. The individual load point indices are also provided with an average

Energy Not Supplied, ENS index.

Table 3.12 Overall System Load Outage Indices, as all transformers and breakers are

aging in the medium substation.

130

Table 3.13 Load 1/3 indices as all transformers and breakers age in the medium

substation.

The effect of all transformers and circuit breakers aging for individual loads “1” or

“3”, and load “2” are provided in Tables 3.13 and 3.14. As loads “1” and “3” are

symmetrical their load point indices are exactly the same. But, load point “2” enjoys

more access to the alternative supply points as it is located in the middle of the

network. As such, the outage indices for this load are shown to be slightly less than

that the other load points. The gap between load “2” and the other load points is

shown to widen gradually as years increase. Finally, Figures 3.58, 3.59 and Figures

3.60, 3.61 compare the individual versus the overall load effects with respect to

component contributions while all circuit breakers and transformers age.

131

Table 3.14 Load 2 indices as all transformers and breakers are aging in the medium

substation.

% Contribution of Components to Load 3, PLC/Unavailability, Med Sub, All Transformers and Breakers Aging.

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

Fig. 3.58 Percentage contribution of components to load 3 unavailability as all

transformers and circuit breakers age for the medium sized network.

132

% Contribution of Components to Load 3, EFLC, Med Sub, All Transformers and Breakers Aging.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

Fig. 3.59 Percentage contribution of components to load 3 EFLC as all transformers

and circuit breakers age for the medium sized network.

% Contribution of Components to All Loads, PLC/Unavailability, Med Sub, All Transformers and Breakers Aging.

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

Fig. 3.60 Percentage contribution of components to all loads unavailability as all


The general trend in these graphs is very similar which indicates that although overall

load indices may cover different ranges of incident, as opposed to the individual load

indices, but the trend and effect may follow similar patterns as they both are subject to

all transformers and circuit breakers aging.

133

% Contribution of Components to All Loads, EFLC, Med Sub, All Transformers and Breakers Aging.

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70

Time (years)

Per

cent

age

CB5

CB4

CB3

CB1

CB2

TX1

TX2

TX3

Bus1

Bus2

Bus3

Fig. 3.61 Percentage contribution of components to all loads EFLC, as all


134

3.6 Summary

This chapter developed a technique for modelling non-repairable/aging as part of an

existing Markov model for power system equipment. In this model an extra state has

been introduced to designate non-repairable/aging or end of life failure. This state is

described as a state where once a component transits from its operating state to this

state, due to aging fatigue and degradation or due to any irreversible fault, the

component must either be refurbished or replaced. Therefore, returning from this

state to operating state is associated with an average time to refurbish or replace. The

failure from operating state to this non-repairable/aged state is associated with an

increasing failure rate with time which is usually associated with a normal or a

Weibull distribution function. The Markov state space model is designed to include

this state as constant in each year of computation while the failure rate increases year

by year. This model exclusively includes both modes of life cycle which are a

random failure mode and an increasing non-repairable/aging failure mode. The

increasing failure rate of the non-repairable state will introduce yearly changing state

probabilities. For a component is in its early life period the aging effect is almost

absent but as it ages the aging probability increases and other state probabilities

decrease. One important attribute of this method is that; in addition to the evaluation

of the probability indices, the frequency related indices can also be calculated while

aging is taking place.

Sets of equations have been developed in this chapter to describe the relationships

between the transitional rates and the state probabilities. The equations can describe

many forms of Markov state space diagram which have options to include or exclude

the following equipment states; aging state, switching state and maintenance state. A

set of third order aging system states are also included. These additions minimally

influence the computing time however, accuracy of the calculated indices is increased.

The component failure contribution to the probability and frequency indices were

135

upgraded to include the new non-repairable/aging state and the triple contingencies.

The network calculations are designed to consider and execute the assessment over a

number of planning years. However, displaying the complete output results for each

year of evaluation can be impractical. In addition to the selected yearly indices, the

yearly component failure contribution to some of the important indices can be

selected for calculation. These sets of results are extremely useful when a wide range

of information needs to be efficiently captured and presented.

136

Chapter 4

Classification and Decomposition of Distribution Feeder Loads

4.1 Introduction

The main function of electricity industries is to provide electrical energy to meet

customer load demands with an acceptable level of reliability and quality. However,

the complement to reliability appraisal is normally the cost and the benefit

considerations for an acceptable level of risk. In practice a sound decision is usually a

compromise between the two.

Never-the-less, the cost and the benefit evaluations in power systems reliability

require some information about the loads in the network. There are many attributes

associated with loads that are routinely considered in many studies, such as load

forecasting, demand side management, etc. The load attributes relating to reliability

and cost/benefit assessments can include; magnitude, variation in time and seasons,

customer type and the number of loads, etc. At different levels in power systems,

namely; generation, transmission, sub-transmission and distribution, the cost and

benefit analysis may require special information about the loads. At lower voltage

levels, such as in distribution and sub-transmission systems closer to the customer

load points, the type and number of customers can be very important in evaluating

certain reliability indices such as interruption costs to different type of customers.

However, loads are usually measured at the 11kV supply point nearest the customer

loads. Normally at this point little or no information is retained about the type of

customers and their contributions to the loads. The need for an approach to extract

this information is recognized in this thesis and paves the way for practical

applications in systematic economic assessments of power system reliability.

137

Initially a technique is applied in this thesis to identify different customer load profiles

for distribution network feeders that have load type information. The load

information obtained is then used as part of a decomposing technique on an unknown

feeder to extract moderately accurate information about its customer load types.

Apart from the particular interest in the application of this method in this thesis, the

load attributes and the information on different customer load patterns has always

been of major interest to the electricity retail market. With accurate predictions and

classifications of customer load types, the electricity suppliers can better manage

unpredicted situations such as load forecasting and transformer overloads. The

network planners and operators will also benefit from the customer load determination

in areas such as load diversification, reinforcement, upgrading and load ratings of

transformers.

Using direct measurement devices at voltage levels lower than 11kV, together with

the facilities required to acquire and manage the measured information, can be an

expensive practice which may not be economically viable. While there are many

benefits associated with having information on customer load types, very few

publications have addressed this topic using statistical evaluations and predictions.

Basu [95] has provided some relevant insights on classifying loads, based on data

obtained, into different customer profiles such as residential, commercial and

industrial. Since the classification process was not cited in the paper, a comparison

between Basu’s method and the methods proposed in this thesis is not possible. Also,

an important difference between the classification process promoted in this thesis and

that used by Basu is that Basu focused on load prediction rather than on customer

profiling and feeder decomposition.

138

4.2 Data Collection

Three type of customer loads; residential, commercial and industrial are considered in

this chapter. The data used here was provided by a local distribution authority thus,

due to sensitivity and confidentiality, the detail and the source of this data used here

cannot be provided. The estimated results may be affected by the initial information

collected from known feeder load types but accuracy can be achieved by applying the

following constraints to the half hourly feeder loads under study.

Constrains are as follows:

a) During the customer profile process the selected known feeder loads need to

be composed largely of just one type of customer from one of the three

sectors.

b) Over the data collection period the selected and known feeder loads should

have sufficient data sets to reduce the effect of possible errors; missing load

values and abnormalities in the measurements.

c) Within the data collection period the selected known feeder loads should not

have undergone any major upgrades or changes.

d) To improve the model’s ability to generalize, known feeder loads have been

selected from a range of different area locations.

While all constraint were taken into consideration, 34 feeder loads in total (11 from

each sector and 1 unknown feeder used in section 4.5 for decomposition) were

selected for the study.

139

4.3 Model Formulation

Half an hourly load profile of a feeder describes how one unit of load current is drawn

throughout a week (directly relating to energy). The load profile of a given feeder is

denoted by the sequence:

.7,,1,48,,1,, KK == dtP dt

Where P is a normalized quantity with no unit and t denotes half an hourly discrete

time and d denotes the day in a week. By this definition of a load profile the

following restrictions are imposed on the sequence:

.0,1 ,, ≥=∑∑ dtd t

dt pp (4.1)

It is assumed that the load drawn for each weekday follows the same pattern and so;

.48,1,5,1, KL === tpp tt

The loads drawn on Saturday and Sunday are permitted to differ from each other and

from the load drawn during weekdays. This assumption does not conflict with the

basic load observations and inspecting the load data shown in Figures 4.1 and 4.2

confirms this assumption. There are no further assumptions concerning the structure

or functional form of the load profile.

As noted previously a certain customer type load pattern can well be described by a

normalized load model Pt,d throughout an entire week. So in order to relate the load

pattern to the actual data from the load feeders the following weighting is described:

140

,,2,1,0 K=≥ wmw

Where, m denotes the expected amount of ampere drawn for week w. The observed

amount of ampere y drawn on the feeder load at time t on day d and week w is

modeled in Equation 4.2 as;

,,,,,, wdtdtwwdt epmy += (4.2)

Where, e denotes observational error. There are numerous factors which can have

significant effect on a feeder load such as growth in the area which the feeder supplies

and the effects of seasonal weather. The above model does not account for these

factors explicitly and the expected amount of current drawn for a week will implicitly

act as a surrogate for these factors. This limits the way in which the load pattern can

change from week to week and imposes certain assumptions on the type of growth.

For example, if growth occurs in an area supplied by a feeder then the additional load

is assumed to have a similar pattern to the original load.

The model for load profile and weekly expected load are estimated using the least

squares method. Since the model has nonlinear parameters an iterative method such as

the Gauss-Newton scheme [96, 97] is required to find a solution. Fortunately, the bi-

linear form of the model makes the iterative scheme particularly simple to solve.

Details are as follows:

1. An initial estimate of m is required to start the iterative scheme. In this

analysis the estimate defined by Equation 4.3 gives:

,,2,1,487 K=××= wymw (4.3)

Where, y is used as the sample mean of the feeder load data.

141

2. Update profile: Given an initial estimate of the weekly expected load the

profile can be estimated by solving a linear least squares problem subject to a

linear equality constraint and non-negative constraints. In the data analyzed,

the non-negativity constraint did not need to be explicitly enforced and so the

profile estimate is obtained by solving a linear system via least squares [98,

99] different data the non-negativity constraint may need to be enforced

explicitly in which case an algorithm using monotonic smoothing splines

[100] could be used.

3. Update weekly expected load: Given an estimate of the profile the weekly

expected load can be estimated by solving a linear least squares problem

subject to non-negativity constraints. However, as the parameters in this case

are orthogonal the solution to this problem is simply given in Equation 4.4;

( )wm̂,0max (4.4)

Where, m̂ is the ordinary least squares estimate. As was the case when

updating the profile, the non-negativity constraints do not need to be explicitly

enforced.

The last two steps were iterated until convergence was reached. Convergence is

assessed by changes in residual squared error and by changes in the parameter

estimates.

The results of system measurements are shown in Figures 4.1 and 4.2 and as stated

earlier show the reparative nature of the current drawn during weekdays and the non-

repetitive nature of the current drawn during weekends.

142

TueMon Thur Fri Wed Sun Sat

Mon Tue Wed Thur Fri Sat Sun

0 50 100 150 200 250 300 350

5010

015

0

Industrial feeder

Half hourly

Hal

f Hou

rly A

mps

Usa

ge

Fig. 4.1 Industrial feeder plot for 1 week.

0 50 100 150 200 250 300 350

8010

012

014

016

018

0

Residential feeder

Half hourly

Hal

f Hou

rly A

mps

Usa

ge

Fig. 4.2 Residential feeder plot for 1 week.

Using the iterative process method described above, Figure 4.3 was plotted to

compare the difference between the predicted load values generated from the model

versus the actual measured load values from the feeder load. The predicted values

from the model fit the measured feeder load data with a reasonable degree of accuracy

143

except for two large measured anomalous peaks. Possible explanations for these

peaks are:

1. A sudden changes in the temperature from one day to the next,

2. A breakdown of the feeder supplying the neighboring area causing the feeder

under study to overloaded abnormally, or

3. The possibility that the day was a ceremonial/ public holiday.

Although it is desirable to incorporate a function in the model to predict the

anomalous peaks however, without knowledge of when and why theses peaks occur it

is difficult to predict when it will happen again.

0 500 1000 1500 2000 2500 3000

050

100

150

200

250

300

10 weeks comparison plot

Half hourly

Hal

f Hou

rly A

mps

Usa

ge

real valuespredicted values

Fig. 4.3 Fitted results versus actual results showing two anomalous peaks.

144

By applying the same method to the entire data collection period, estimates for the

load usage pattern of Weekdays, Saturday and Sunday are obtained. These estimates

are shown in Figure 4.4.

0 20 40 60 80 100 120 140

0.00

10.

002

0.00

30.

004

0.00

5

Industrial feeder

Half hourly

Sta

ndar

dise

d Am

ps U

sage

with

sum

=1

Fig. 4.4 Weekday, Saturday and Sunday load plot for Industrial loads.

Repeating the process for each of the 33 load feeders, a general customer profile can

be created for each customer type, as shown in Figure 4.5 which illustrates 11 of the

33 load profiles.

Sunday

Weekday

Saturday

145

0 20 40 60 80 100 120 140

0.00

00.

002

0.00

40.

006

Industrial feeder comparison

Half hourly

Sta

ndar

dize

d Am

ps U

sage

with

sum

=1

Fig. 4.5 Industrial loads comparisons.

4.4 Sub – Sector / Customer Profiles

Up to this point load profiles for given feeders based on the observed load data have

been estimated. Each of the feeder load profiles will be affected to some extent by

local factors which are of little importance to the general sector to which it belongs.

To obtain more accurate sector profiles an average over the feeder load profiles in

each sector is obtained. By doing this the locality effects in the feeder load profiles are

reduced in the sector profile. It is noted that the sector profiles will satisfy the

conditions imposed in Section 4.3 on load profiles.

One possible problem that may arise from the above procedure is that the

classification of the areas which the feeders service namely industrial, residential and

commercial, is too coarse. This would result in the average of the feeder load profiles

146

being a poor representation of the sector profiles. It is thus necessary to perform some

type of clustering of the feeder load profiles within each sector to identify appropriate

sub-sectors. For this purpose we use the K means clustering algorithm [101] where

the distance between two feeder loads is measured by the Euclidean metric. Various

criteria have been proposed to determine the most appropriate number of clusters, for

example by estimation of clusters using gap statistics [102]. However, as the ultimate

aim here is to determine percentages associated with each sector, the number of

clusters in the prediction performance assessed by cross-validation [103] will be used.

Figure 4.6 illustrates three (sub)-sector load profiles obtained from the commercial

sector. This information, along with the (sub)-sector load profiles established from the

industrial and residential sectors will form the bases to decompose any unknown

feeder load.

One assumption made in the process is that the load pattern of an unknown feeder

follows the basic behaviour of a sector. If the unknown feeder load operates in a

completely unique fashion to those sampled for reference then the model will not

accurately define the type of (sub)-sectors contributing to the feeder load. For

example, if a load feeder supplies a unique industrial company that operates 24 hours

a day, 7 days a week, then this load pattern will greatly differ from the basic industrial

load patterns obtained for the samples where operation hours are from 7:30 to around

18:00, 5 days a week.

147

Weekday Saturday Sunday

0 20 40 60 80 100 120 140

0.00

10.

002

0.00

30.

004

0.00

50.

006

Sub-Sectors within commercial

Weekday/Sat/Sun

Clu

ster

cen

ters

Fig. 4.6 (Sub)-Sectors within the commercial sector.

4.5 Decomposing a Load Feeder

In the previous section, load characteristic profiles were identified for each of the

(sub)-sectors observed in the data. These profiles were built on the assumption that

the area serviced by each feeder was homogeneous, that is only one sector was

predominant in the area serviced. In the following a method is formulated to

determine which (sub)-sectors are present and the percentage each one contributes to

an unknown feeder load.

The characteristic profiles for the (sub)-sectors are denoted by;

Sdtdt PP ,

1, ,,K ,

148

Where, S is the number of (sub)-sectors identified in Section 4.4. It is assumed that

the profile of a feeder load for the new (unknown) area can be represented in Equation

4.5.

mdtmdtdt PPp ,

1,1, ωω ++= L (4.5)

Where;

.0,11 ≥=+ im ωωω L

The coefficients ωi determine the proportion of the load that each (sub)-sector draws.

If one of the coefficients ωi is zero then the corresponding (sub)-sector is not present

in the area serviced by the feeder.

The coefficients ωi are estimated here by the least squares method. This linear least

squares problem is subject to a linear equality constraint and non-negative constraints

that can be solved using the algorithm described by Wood [100]. As an example, this

method is applied to a feeder servicing an area whose composition in terms of the

(sub)-sectors is unknown. Once the profile of the feeder is determined using the

method of Section 4.3, it is decomposed as described above. The result of this are

given in Table 4.1.

Sub-Sectors/Profiles Correlation Values

Ind1 0.1433486

Com1 0.1489686

Com2 0.02814233

Com3 0

Res1 0.6795404

Table 4.1 Contribution of (Sub)-Sector Profiles for Industrial (Ind), Commercial

(Com) and Residential (Res) loads types.

149

Saturday Sunday

From the results given in Table 4.1 it can be seem that this feeder supplies mostly

support the residential sector or customers with similar load usage patterns as a

residential sector. The (sub)-sector profile of Com3 has a zero correlation value

implying that this (sub)-sector’s profile is not present in the area supplied by the

feeder (does not contribute to the feeder load). Having the highest correlation value,

Res1 is considered as the dominant (sub)-sector/customer load type on this feeder.

Having determined the weights of each (sub)-sector/customer profile, it is possible to

plot the average weekly load current contribution in amps for each (sub)-sector by

multiplying the respective profile values by the correlation values and the average of

the weekly multiplier for the feeder load under study. Figure 4.7 shows the (sub)-

sector contributions of the load feeder under study.

0 20 40 60 80 100 120 140

050

100

150

200

250

Contribution values of each (Sub)-Sector for a Feeder

Weekday/Sat/Sun (half hourly)

Am

ps

real valuesSum of Contribution valuesInd1 ContributionCom1 ContributionCom2 ContributionRes1 Contribution

Fig. 4.7 Contribution values of Sub-Sectors for an unknown feeder.

It is worth noting that the sum of load contributions closely follows the real load

values with an overall difference of less than 1%.

150

4.6 Cross Validation and Testing

Cross validation is a technique by which we an approximately unbiased estimate of

the predictive accuracy of some statistical method/model [103, 104] can be obtained.

The technique involves splitting the data set into two groups, a training set and a test

set. The parameters of the statistical method/model are estimated using the data in the

training set. The resulting model is then used to predict the test set and the accuracy of

these predictions measured. This can be repeated over different divisions of the data

in training and test sets with the results averaged to provide an approximately

unbiased estimate of accuracy. In this thesis the ‘leave-out-one’ cross validation

technique is used. In the leave-out-one cross-validation the test set is a single data

point and the procedure is repeated for the n = 33 possible divisions of the data into

training and test sets.

First considered is the case where the sector load profiles are formed by averages of

the feeder load profiles. Using the method of Section 4.4, a feeder is classified into

one of the three sectors according to which sector received the largest weight. The

number of misclassifications made under cross-validation is: 1 for Industrial, 4 for

Commercial and 0 for Residential.

As the commercial sector appears to be most problematic, the possibility of (sub)-

sectors within the commercial sector was considered. Two and three possible (sub)-

sectors and their accuracy assessed by cross-validation and was studied. A feeder is

now classified using the method of Section 4.4 according to the sector which received

the largest weight, summing over its (sub)-sectors. With two (sub)-sectors the

number of misclassifications increases to: 3 for Industrial, 6 for Commercial and 0 for

Residential. However, for three (sub)-sectors for the commercial sector gives the

151

least number of misclassifications: 3 for Industrial, 0 for Commercial and 0 for

Residential.

The results from the classifications with three (sub)-sectors for the commercial sector

are summarized in the following plots, Figures 4.8 – 4.10. Ideally, all weights would

be concentrated near one or zero as appropriate.

Figure 4.8 illustrates that one result shows that a commercial feeder 4 has been

recognized as a industrial feeder (circled). This indicates that the two feeders probably

have similar load patterns

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Industrial Feeder

wei

ght

Fig. 4.8 Cross Validation Results with an Industrial feeder: weights o = industrial

sector, ∆ = commercial sector, + = residential sector.

152

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Commercial Feeder

wei

ght

Fig. 4.9 Cross Validation Results with a Commercial feeder: weights o = industrial


Further investigation into this issue discovered that the reason for the commercial

misclassifications may be caused by the similar load pattern of the commercial and

residential sectors. To cater for the shopping needs of the residential customers many

commercial shops and businesses have altered their opening hours to suit those of the

residential sector. One example is the grocery chains that have changed their opening

hours to include Saturday and Sunday for the convenience of customers.

153

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Residential Feeder

wei

ght

Fig. 4.10 Cross Validation Results for Residential feeder: weights o = industrial


The other misclassifications are where an industrial feeder was mistaken as a

commercial feeder and a commercial feeder as an industrial feeder can be explained

after the feeder loads were compared with its respective sector feeder loads. After

comparison it was found that those feeder loads demonstrated completely different

load patterns to the other feeder loads in its respective sector. Since the feeder loads

are unique and different from all the other feeder loads in its sector the model can not

accurately identify the type of sector the feeder load belonged to. With this discovery

it is possible to classify these feeder loads as unique sector feeders for future study

references.

154

4.7 Load Information for Reliability and Cost/Benefit Assessments

The decomposition technique applied to feeder loads in this chapter can obtain

different type of load profiles contained in a feeder load. A load type may also be

divided into its sectors. For example, affluent residential load profiles may exhibit a

different profile to deprived residential areas. The decomposed load profiles provide

data as half-hourly per-unit loads and the summation of the different types of load

profile is equal to the total feeder load value. In this regard, given that the actual

feeder load values are known as well as the proportions of the load types, the actual

values for all load types can easily be obtained.

The reliability and cost/benefit evaluations of an electricity network serving different

loads are commonly performed on per year basis with indices given in per year terms.

However, the indices can also be calculated on per day, per season, or generally per

any period of interest.

The value of load used in assessing the load point reliability indices, such as Expected

Energy Not Supplied (EENS), is either presented as a maximum or an average value.

Therefore, for a given period the maximum or average load can be extracted for each

of the load types. The calculation requires that the area under each load type is

obtained separately and then divided by the period under consideration. It is very

common, with utility companies, to directly relate the maximum load to the average

load using a load factor. In this case, normally Load Duration Curve (LDC),

representing the descending form of a yearly load is used. Using load factor in

obtaining the average load, given that the maximum value of the load is provided in a

year, is based on the assumption that the LDC in per unit is the same in any year.

155

4.8 Summary

Electrical loads are normally measured at an 11KV feeder nearest to the electricity

customers. However, presently there is no information available on the makeup of

load types or their profiles. This information can be very important in reliability and

cost/benefit evaluations of electricity networks providing power to consumers. The

information can assist utilities in network design, planning, operation and

maintenance to ensure compliance with an acceptable level of reliability where

cost/benefits are envisaged.

In this chapter an approach has been used to facilitate the decomposition of unknown

load profiles, measured at 11KV feeders, to known load types of (sub)-

sector/customer loads. Although this model has some imperfections when classifying

(sub)-sector/customer load profiles, further improvements and accuracy can be gained

by incorporating seasonal, weather and temperature parameters.

In addition, this application can provide other benefits for electricity suppliers in areas

such as; economic load management, load diversification relating to electricity block

purchasing, load forecasting, retailing and strategic network planning.

156

Chapter 5

Impact of Non-Repairable Failures of Aging Transmission/Sub-

transmission Network on Distribution Load Point Indices

5.1 Introduction

For many logistic and technical reasons including; organization, control, planning,

operation and monitoring etc., an electric power system is usually divided into

generation, transmission and distribution functional zones. However, the facilities in

these zones or networks have all one common purpose and that is to provide electrical

energy to customers, as economically as possible and with an acceptable degree of

quality and reliability. The main theme of this thesis is the modeling of aging applied

to transmission/sub-transmission networks and the effect of component aging on the

power network at this level was extensively examined in Chapter 3. However, aging

facilities in higher voltage networks and their effects on loads in distribution systems

is examined in this chapter.

As discussed in the earlier chapters, the enumeration technique basically describes the

combinations of first, second and third order contingencies leading to a loss of any

one or more bus loads. It was shown that subsequent to the completion of all

enumerations, the indices are obtained. Among the collected indices, there are three

indicators, namely; failure frequency, annual outage duration and average outage time

at each bus load that can be used to represent the reliability performance of the supply

network upstream of the distribution system [105, 106]. However, this approach can

equally be used to describe the effect of substation/sub-transmission facilities aging to

customers of the distribution system.

157

This chapter initially provides an analytical evaluation technique for solving

reliability performance at the distribution system [105, 107-111]. The reliability

evaluation techniques used for distribution systems are unique, both for the indices

and the applications. Normally distribution systems are radial consisting of main and

lateral feeders and an upstream main feeder is usually connected to a substation bus

where the supply voltage is stepped down to distribution level. The lateral feeders are

‘teed’ off the main feeders to provide electric power to customers at different

geographic locations. At the same time there can be numerous automatic and manual

switches joining the main feeders and/or lateral feeders and it is also a common

practice to use fuses before the location of the load transformers. There can also exist

switches between main feeder branches, that are normally open, which facilitate

alternative supply points in the event of an outage that disrupts supply to downstream

loads. The functionalities and operations in the distribution system are logically and

systematically implemented in a program to simulate every outage that causes

contingencies.

5.2 Distribution System Reliability

Power outages cost utilities lost revenue and can cause financial damage to customers

and certainly leads to customer dissatisfaction. Therefore, ensuring reliability of

power distribution is of considerable importance and it is very important to design and

plan a distribution network with a high level of reliability. Many utilities use their

past performance to measure the level of their system reliability performance. This

approach however does not provide any information for estimating the future

performance of a distribution system in an ever changing and expanding network.

The past performance of an existing distribution network, based on historical data,

may give some insight into certain design aspects, but it is more important and useful

158

to make an estimation in advance when designing a network or examining the cost

effectiveness of network upgrades.

5.2.1 Electric Power Distribution Reliability Indices

The primary output of distribution reliability calculation, as described before, is a

collection of indices which provide information on outage durations, frequency of

outages and the power loss typically in terms of an average customer. These indices

are used widely within the industry and are standardized by the IEEE [107]. These

indices can be categorized as either load point indices, which measure the reliability

of a single load point, or system indices which measure the reliability of an entire

distribution system of interest.

5.2.1.1 Load Point Indices

Failure Rate (λ)

The load point failure rate, described by Equation 5.1, is a measure of the number of

times during a year that a load point is disconnected from its power source due to

component failures.

∑=

=n

ii

1

λλ occ/year (5.1)

Where n is the total number of failures causing a load point to go on outage

and iλ is the i th failure rate, belonging to a failed component.

Annual Unavailability (U)

Unavailability is the average number of hours in a year that a load point is

disconnected from its power source as a result of network failure. This is described

by Equation 5.2.

159

∑=

=n

iii rU

1

λ hours/year (5.2)

Where; n is the total number of failures causing a load point to go on outage

and iλ is the i th failure rate belonging to a failed component. The average

restoration or repair timeir is the time taken for the load to be restored.

Average Outage Duration (r)

The average outage duration is the average length of time in hours per failure per year

that a load point is without power during an outage and is described by Equation 5.3.

λλ

λU

rr

n

iii

==∑

=1 hours/failure (5.3)


The load point expected energy not supplied is the amount of energy per year that

would have been supplied to a load point in the period it was disconnected due to

network failure.

∑=

=n

iiEEENS

1

MWhr/year (5.4)

Where; n is the total number of failures causing a load point to go on outage

and iE is the energy not supplied to a load point due to the failurei . The term

iE is the product of the average load and the time period.

160

5.2.1.2 System Indices

System Average Interruption Frequency Index (SAIFI)

SAIFI measures the average frequency of interruptions incurred per customer in the

system being assessed and is described by Equation 5.5.

SAIFI = total number of customer interruptions

total number of customers served (5.5)

System Average Interruption Duration Index (SAIDI)

SAIDI is the average total time a customer is interrupted annually which is described

by Equation 5.6.

SAIDI = sum of customer interruption durations

total number of customers interruptions (5.6)

Customer Average Interruption Duration Index (CAIDI )

CAIDI measures the length of time in hours an average customer is without power

during an outage, as in Equation 5.7.

CAIDI =

sum of customer interruption durations

total number of customer interruptions interruptions

SAIFI

SAIDI =

(5.7)

Average Service Availability Index (ASAI)

ASAI is the percentage of time an average customer is connected to the network in a

year, as in Equation 5.8.

hours/customer

hours/customer interruption

161

ASAI = customer hours of available service

customer hours demanded (5.8)


The system energy not supplied is equal to the total load point energy not supplied

due to disconnections resulting from outages and is described by Equation 5.9.

∑= EENSPoint LoadEENS System MWhr/year (5.9)

5.3 Analytical Simulation

When an outage occurs in a network it triggers a sequence of switching operations as

efforts are made to isolate the cause of the outage so it can be repaired, all while

minimizing the loss of power to customers. The order and timing of these switching

operations determines the load points that are disconnected during an outage and the

duration during which they are disconnected. The purpose of the analytical

simulation is for every possible individual failure or scheduled outage within a

network to determine the likely sequence of switching operations and the consequent

effect on customers. This information can then be used, with the probability of each

outage, to determine the average number of failures and total amount of disconnection

time that each load point can expect to experience in a year.

For the purposes of this analytical simulation networks are represented as bi-

directional connected graphs, each with a single start and end node. All components,

with the exception of bus bars which are a special case of node, are represented by

edges within the network. Nodes can represent either simple points which join

components within the network or bus bars explicitly included by the user. Two

nodes common to all networks are the source and sink nodes; source components are

162

connected to a source node at one end and a network at the other, likewise load points

connect the network to the sink node. Load point connectivity is determined by

identifying paths and power flow between the source and sink nodes [94, 112], as

shown in Figure 5.1.

Source

Sink

Figure 5.1 User and internal representations of an analytical simulation network.

5.3.1 Switching Operations

The switching sequence that occurs due to an outage varies depending on the cause of

the outage. The switching sequences supported are all simulated using a limited

number of switching operations which, for timing purposes, are all started at the same

time which is the moment the fault occurs and finish in a specific order. Each

operation is timed according to the switching time of the last switch to change state as

part of that operation and no switching operation can complete before the preceding

one. The switching operations used in the analytical simulation are the tripping of

automatic protection devices, manual isolation of a failed component, reclosing and

isolation of insufficiently supplied load points.

163

5.3.1.1 Tripping of Automatic Protection Devices

When a failure such as a short circuit occurs, it is necessary to immediately cut power

to the component that has failed. In such a case the fault condition causes any

automatic protection equipment, such as circuit breakers and fuses with a path of

continuity to the failed component, to trigger thus isolating the fault. The circuit

breakers and fuses that need to be activated to isolate the fault are identified by

searching the network outward from the failed component. As the search continues

outward the closed circuit breakers and fuses that are found are tripped and the search

along the path interrupted by the tripped component, terminated. When the fault is

fully isolated the search region is completely contained by closed switches and

consequently the search is terminated. This is illustrated in Figure 5.2. Any sources

or load points found during this search are treated in a similar manner to the circuit

breakers and fuses to prevent any power being supplied to, or drawn from, the faulted

portion of the network.

3. Trip the protection devices surrounding the failed feeder.

1. A short circuit occurs in a feeder.2. Find the protection devices surrounding the failed feeder

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

Figure 5.2 Tripping automatic protection devices.

*Note that circled switches in network diagrams are in an open state, and crossed out

components in a failed state.

Tripping circuit breakers and fuses in this manner will result in the tripping of

downstream components that would not be tripped in most real situations. This is

done because in situations where this is incorrect the wrongly tripped components will

not impact on the results of a simulation thus making a special case for downstream

1. A short circuit occurs in a feeder

2. Find the protection devices surrounding the failed feeder

3. Trip the protection devices surrounding the failed feeder

164

components unnecessary. Restoration times are only considered in temporary failures

which are typically caused by some form of power surge such as a lightning strike and

in these circumstances downstream tripping would occur to prevent damage at the

load points. In all other cases tripped components are restored with no regard for their

restoration time if they can be isolated from the fault, meaning tripping them

unnecessarily will not result in incorrect behavior.

5.3.1.2 Manual Isolation of Components

Manual isolation is performed to isolate faults that do not trigger automatic protection

devices and to reduce the isolated region surrounding the failed component. Isolation

is performed by opening both isolators and closed circuit breakers that have a direct

path of continuity to the fault, similar to how automatic protection devices are tripped.

Where circuit breakers and fuses have already been tripped within the network the

search for switches to open will be contained to the already isolated region to prevent

that region from growing. After manual isolation has been completed all circuit

breakers and fuses that were tripped beforehand are restored and the tripping

procedure is rerun; this restores the circuit breakers and fuses which are no longer

needed to isolate the fault. This process is shown in Figure 5.3.

Figure 5.3 Manual isolation of components.

. . .

LP 1 LP2

LP 3 LP4

LP1 LP 2

LP3 LP 4

LP1 LP2

LP3 LP4

1. Failure in feeder isolated by circuit breaker

2. Find switches surrounding fault within isolated area

3. Restore circuit breaker and fuse to restore LP1

165

5.3.1.3 Reclosing Alternative Supply

After a fault has been isolated a number of load points can often be identified that

have been isolated from both the fault and a power source. Normally open switches

are included in networks to provide an alternate path to a source of power in these

situations. Re-closing is the process of identifying these switches in the network and

closing them where appropriate. Re-closers that need closing are found by first

identifying the load points that have been disconnected, and then for each of these

load points performing a path of continuity search for re-closers. If a re-closer that

has both a path of continuity to an isolated load point and a power source is found

then that switch is closed and the process moves onto the next isolated load point.

This is shown in Figure 5.4.

1. Failure in feeder isolated by a circuit breaker and an isolator.

2. Search for reclosers that can be closed to restore load points.

3. Normally open switch closed to restore LP3

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

Figure 5.4 Closing of normally open switches.

5.3.1.4 Isolation of Insufficiently Supplied Load Points

The isolation of insufficiently supplied load points isn’t strictly an independent

switching operation, instead it’s an addendum to the manual isolation and re-closing

operations which handles occurrences of line overloading. In cases where an isolated

fault does not cause a complete disconnection of supply but rather a reduction in

supply capacity it may not be possible to adequately supply all load points with a path

of continuity to a source. When this occurs it is necessary to disconnect a number of

loads points to prevent more power being drawn than the network can supply without

1. Failure in feeder isolated by a circuit breaker and an isolator

2. Search for re-closers that can be closed to restore load points

3. Normally open switch closed to restore LP3

166

line overloading. The analytical simulation process provides a number of load

connectivity procedures for determining which load points can and cannot be supplied

without causing overloading. Once the load points which cannot be fully supplied

have been identified it is necessary to isolate them in a manner that both minimizes

the number of isolated healthy load points and retains as many possible paths of

alternate supply. This is done by counting the number of flow deprived load points

which are supplied directly by each switch in a radial branch of a network (a switch

directly supplies a load point if there are no subsequent switches between the switch

and the load point). Then working away from the root of each major radial branch

and only traversing closed switches the switches found that are directly supplying one

or more insufficiently supplied load points are opened until all insufficiently supplied

load points are isolated. This is shown in Figure 5.5.

0/100 100/100

50/100 100/100

1

1

0

0

0/100 100/100

50/100 100/100

0/100 100/100

0/100 100/100

1. The network state has changed and the suppliable flow at each load

point determined.

2. Count the number of insufficiently supplied load point supplied directly

by each switch.

3. Close the switch nearest the supply with non zero load point

count.

Figure 5.5 Isolation of insufficiently supplied load points.

This process of isolating load points that cannot be fully supplied is a departure from

previous maximum flow simulations that used a concept of partial supply whereby it

was assumed there was some switching capacity within load points allowing them to

accept a reduced supply. This would seemingly allow the maximum possible amount

of flow to be delivered to load points under a failure condition until the effects of load

shifting that are resultant from the combination of load point isolation and re-closing

are considered. When a load point is isolated the simulation code will attempt to

3. Close the switch nearest the supply with non-zero load point count to restore LP1

1.The network state has changed and the suppliable flow at each load point determined

2. Count the number of insufficiently supplied load point supplied directly by each switched area

167

close a normally open switch to restore that load point and potentially others, which if

successful can result in the complete restoration of load points that would have

received reduced or no flow in a partial supply simulation.

5.3.2 Outage Causing Modes

The analytical simulation simulates the switching response of networks to four

distinct outage triggers. The switching sequence for each trigger differs in both the

switching operations performed and the timing of the operations. The four causes

simulated are active failure, passive failure, temporary failure and maintenance

outage.

5.3.2.1 Active Failure

As shown in Figure 5.6, an active failure occurs when a component in the network is

damaged requiring repair or replacement and a short circuit or power surge resulting

from the failure causes circuit breakers and fuses in the network to be tripped to

protect the rest of the network. Following an active failure the appropriate isolators

are manually opened to further isolate the failed component while repairs take place

and where possible fuses and circuit breakers are restored and normally open switches

are closed to restore power to any disconnected loads.

1. Failure occurs in feeder tripping circuit breakers and fuses.

2. Normally closed switch opened to isolate fault and fuse restored.

3. Normally open switch closed to restore power tp LP3

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

Figure 5.6 Active failure in a network.

3. Normally open switch closed to restore power to LP3

1. Failure occurs in feeder tripping circuit breakers and fuses

2. Normally closed switch opened to isolate fault and fuse restored

168

5.3.2.2 Passive Failure

Typically occurring in circuit breakers; passive failures occur when a switch

incorrectly opens causing loss of power to sections of the network, as shown in Figure

5.7. Unlike an active failure a passive failure will not cause the automatic opening of

circuit breakers or fuses, however before the fault can be repaired the failed

component must be isolated by manually opening circuit breakers and isolators

following which power may be restored to disconnected loads through re-closers.

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

1. Failure in circuit breaker causes it to incorrectly open.

2. Closed switches are manually opened to isolate failed circuit

breaker.

3. Normally open switch closed to restore power to isolated load point.

LP1 LP2

LP3 LP4

Figure 5.7 Passive failure in a network.

5.3.2.3 Temporary Failure

A temporary failure, as shown in Figure 5.8, is an event such as a power surge that

causes circuit breakers and fuses to be tripped but does not result in any permanent

damage to the component from which the outage originated. In this situation no

switching of isolators or re-closers is employed and the network is only in a failed

state for as long as it takes to restore the tripped circuit breakers and fuses.

3. Normally open switch closed to restore power to isolated load point

1. Failure in circuit breaker causes it to incorrectly open

2. Closed switches are manually opened to isolate failed circuit breaker

169

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

2. No repairs are required so network is restored.

1. Failure in feeder trips circuit breaker and fuses.

Figure 5.8 Temporary failure in a network.

5.3.2.4 Maintenance Outage

Maintenance outages are scheduled outages where a component is disconnected from

the network in order to perform regular maintenance. Because the outages are

scheduled normally open switches can be closed in advance of the outage minimizing

or even eliminating interruptions to customers, as shown in Figure 5.9.

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

LP1 LP2

LP3 LP4

1. Transformer scheduled for maintenance.

2. Transformer isolated and normally open switch closed to restore power

to load points.

3. Maintenance begins on transformer.

Figure 5.9 Maintenance outage in a network.

5.3.3 Load Connectivity / Shedding

After any switching operation has occurred it is necessary to determine what impact

that operation has had on the connectivity of load points within the network. Two

distinct approaches to the determination of load point connectivity are supported;

1. Failure in feeder trips circuit breaker and fuses

2. No repairs are required so network is restored

3. Maintenance begins on transformer

1. Transformer scheduled for maintenance

2. Transformer isolated and normally open switch closed to restore power to load points

170

minimum cut set and maximum flow, the difference between the two being the

consideration of line ratings.

5.3.3.1 Minimum Cut Set

The traditional technique for determining load point connectivity is minimum cut set.

This technique ignores line capacity and assumes that provided a path of continuity

exists between a source and a load point that load point can be fully supplied. In

many cases this is adequate as the capacity available within a network is significantly

greater than what the combined loads draw, or no switching arrangement will result in

a reduced supply capacity. If these conditions are not satisfied however, the results

gained from minimum cut set connectivity are going to be overly optimistic. The

minimum cut set is calculated by performing a search of the network for paths that

start at the source node and end at the sink node not traversing any edges in an open

circuit state. All load points found as part of this search have paths of continuity to a

source and hence form part of the minimum cut set. This is shown in Figure 5.10, and

the algorithms are provided in Appendix A.

1. Need to determine connectivity in a network in which a fault has been

isolated and an alterative path of supply has been introduced.

2. Search for paths from the source to the load points.

3. Load points which were found with paths from the source are allocated

all their required flow.

0/100 0/100

0/100 0/100

0/100 0/100

0/100 0/100

0/100 100/100

100/100 100/100

Figure 5.10 Solution for minimum cut set of a simple network.


Maximum flow introduces line capacity and the possibility of line overloading to the

determination of load point connectivity. Often when an outage occurs it does not

3. Load points which were found with paths from the source are allocated all their required flow

1. Need to determine connectivity in a network in which a fault has been isolated and an alternative path of supply has been introduced

2. Search for paths from the source to the load points

171

completely cut continuity to load points either because the component that failed was

in parallel with another which is still able to supply power, or because a normally

open switch is closed to create an alternate path of supply for the load points

disconnected by the fault. While there is still continuity in these circumstances the

capacities of these alternate supply paths may not be sufficient to supply the

additional loads without being overloaded. Maximum flow provides a method of

determining which of these load points can be supplied without causing any

overloading, as shown in Appendix A. Because there can be any number of possible

solutions for the maximum flow within a network two techniques are provided for

solving the maximum flow; the shortest path technique and the priority technique.

5.3.3.2.1 Shortest Path Maximum Flow

Shortest path maximum flow, shown in Figure 5.11, is a search based technique for

solving for the maximum flow in a network. The name refers to the manner in which

it favors the load points with the shortest path to a source when allocating flow (path

length is measured as the number of edges in a path). Shortest path works by

searching for paths with available capacity between the source and sink nodes, when a

path is found the flow in each edge of that path is augmented by an amount equal to

the available capacity of the edge with the least available capacity. This search

procedure is reset and performed again after each path is found until a search

completes without finding any path of available flow between the source and sink

nodes which indicates the maximum flow of the network has been determined.

172

4. Path with capacity of 50 found to next load point.

5. No further paths with available capacity can be found, so maximum

flow has been determined.

1. Find flow in the network after a transformer has been isolated.

2. Path with capacity of 100 found to nearest load point.

3. Path with capacity of 100 found to next nearest load point.

0/100 0/100

0/100 0/100

0/2500/250

0/2500/250

0/2500/250

0/100 0/100

0/100 0/100

0/2500/250

0/2500/250

0/2500/250

100

0/100 100/100

0/100 0/100

100/2500/250

100/2500/250

0/2500/250

100

100/100 100/100

0/100 0/100

200/2500/250

100/250100/250

0/2500/250 50

100/100 100/100

0/100 50/100

250/2500/250

150/250100/250

50/2500/250

Figure 5.11 Solution for shortest path maximum flow of a simple network.

Shortest path is simple technique and this can be reflected in its results. If a

bottlenecked edge supplies a number of radial feeders all of similar length the loads

that do not receive flow will tend to be spread amongst the far ends of all the feeders.

If one feeder is significantly longer than the others then the deprived loads are more

likely to be confined to that feeder. The optimal situation when considering isolation

of deprived loads is typically going be one where the load points are grouped together

so they can all be isolated by the opening a single switch and then be potentially

reconnected through an alternate path of supply. Given the likelihood of feeder

lengths being sufficiently mismatched this will often be the case, but if this is not the

case then depending on the positioning of switches within the network shortest path

maximum flow may give excessively pessimistic results. The shortest path algorithm

is predictable enough that situations where this will occur can often be easily

identified and accounted for when considering the results.

3. Path with capacity of 100 found to next nearest load point

1. Find flow in the network after a transformer has been isolated

2. Path with capacity of 100 found to nearest load point

4. Path with capacity of 50 found to next load point

5. No further paths with available capacity can be found, so maximum flow has been determined

173

5.3.3.2.2 Priority Maximum Flow

Priority maximum flow uses the concept of edge priority to control the distribution of

flow through a network. This is shown in Figure 5.12. All edges are assigned a

starting priority typically zero which is incremented whenever flow is allocated to an

edge, processing starts at the source node where each connected edge is allocated an

amount of flow equal to its available capacity. When an edge is allocated flow the

node at the opposite end of the edge to the node from which the flow was supplied is

added to a processing queue. As each node is processed the accumulative inbound

flow to that node is redistributed outwards amongst the edges connected to the node.

From the perspective of a node an edge can either have positive or negative flow, if an

edge supplies flow to the node then it is positive and if it sinks flow then it is negative

and consequently the amount of flow that needs to be redirected can be found by

summing the flows of connected edges. The available flow at a node is redistributed

among the connected edges based on the priorities of the edges, the edge with the

lowest priority value will be allocated flow first and the remaining edges are allocated

flow in order from lowest priority value to greatest.

174

Figure 5.12 Solution for priority maximum flow of a simple network.

Each edge is allocated as much of the available flow as its capacity allows and this

amount is deduced from the total available at the node until the total reaches zero and

flow is not allocated to any further edges. All nodes are processed in this manner with

the exception of the source and sink nodes; the source node is able to supply flow

without limit provided the priority of the edge being supplied is below a threshold

value and the sink node will absorb all flow it receives. The algorithm will terminate

3. Enough flow available at the node to supply one of the P0 branches, choose the right

1. Initialise all edge flows and priorities to 0

2. Unlimited flow available from source, so allocate the transformer its maximum 250 and increment its priority

6. Allocate the excess flow to the lowest priority edge with available capacity

4. Allocate the lower capacity load point edge its maximum capacity and the remained to the isolator edge

5. Allocate the load point edge its maximum capacity and return the remainder to isolator edge

9. All available flow has been allocated, maximum flow found

7. Flow is now available to allocate to the left branch

8. Allocate the available flow to the lower capacity load point branch

175

when all the priorities of all edges connected to the source node have reached the

threshold priority and the accumulated flow at all other nodes is zero.

Because the flow allocation decisions are made at a nodal level in priority maximum

flow when there is a bottleneck in the network the load points that are denied supply

are typically contained to a group all fed by a common edge. This means where there

are a limited number of normally closed switches in a network the chances that the

load points that require isolation are confined to a radial branch fed by a single switch

are increased. If the load points are spread among more radial branches multiple

switches may need to be closed resulting in a greater number of healthy load points

being disconnected by the isolation. Edge priority is used when allocating flow to

prevent the algorithm from getting caught in feedback loops caused by parallel paths

in the network. If flow is allocated to a series of edges that form a loop in the network

the priority value of edges in the loop are incremented, while edges that diverge from

the loop that weren’t allocated flow retain a lower priority value. Consequently the

likelihood of the loop edges being allocated flow in series before others connected to

the same nodes is reduced, with every iteration of the flow allocation to that loop.

The other advantage of the priority maximum flow is that non-zero starting priority

values can be assigned to edges to indicate that other options should be explored

before allocating flow to those edges. This is used with closed switches to initially

guide flow away from parts of the network that can easily be isolated and confine

flow deprived load points to groups which are separated from sources by a large

number of switches. Isolators are only assigned small initial values as they should not

obstruct flow too greatly. Re-closers however are assigned large starting values close

to the threshold value so that all alternative possibilities are explored before allocating

flow across a re-closer which prevents a closed re-closer from denying flow to regions

of the network that would otherwise be adequately supplied.

176

5.4 Simulation Tests

Two test systems have been chosen to analyze the results of the analytical

simulations. They are the two RBTS Bus 2 and Bus 4 [107] networks that have been

analyzed in numerous other studies and so provide reference points against which to

compare results when considering active failures in the distribution network only.

The RBTS system and the complete data for the two distribution systems at Bus 2 & 4

are provided in Appendix B.

5.4.1 RBTS Bus 2 Distribution Network

SP1

a b c

F1 F2 F3 F4

12

3

45

6

78

9

10

11

12

13

14

15

16

17

1819

20

2122

23

24

25

26 27

28

29 30

31

32 33

34 35

36

Figure 5.13 Distribution network at RBTS Bus 2.

The RBTS Bus 2 network, shown in Figure 5.13, is a single bus network with four

distribution feeders; each feeder is connected to another by a normally open switch

177

allowing loads disconnected by some failures in the distribution network to be

restored while repairs take place. The complete data for this system is given in Tables

5.1 – 5.5.

Line active failure rate 0.065 failures/km-year

Line repair time 5 hours

Transformer active failure rate 0.015 failures/year

Transformer repair time 200 hours

Isolator switching time 1 hour

Recloser switching time 1 hour

Table 5.1 Component failure rate, repair time and switching time data for distribution

network at RBTS Bus 2.

Lines Length

2, 6, 10, 14, 17, 21, 25, 28, 30, 34 0.6 km

1, 4, 7, 9, 12, 16, 19, 22, 24, 27, 29, 32, 35 0.75 km

3, 5, 8, 11, 13, 15, 18, 20, 23, 26, 31, 33, 36 0.8 km

Table 5.2 Feeder lengths in the distribution network at RBTS Bus 2.

Load Point Number of

Customers

Average

Load

LP1, LP2, LP3, LP10, LP11 210 535 kW

LP4. LP5, LP13, LP14, LP20, LP21 1 566 kW

LP6, LP7, LP15, LP16, LP22 10 454 kW

LP8 1 1000 kW

LP9 1 1150 kW

LP12, LP17, LP18, LP19 200 450 kW

Table 5.3 Load point data of the distribution network at RBTS Bus 2.

178

Lines Capacity

1, 12, 16, 26 4 kW

4, 14, 18, 29 3 kW

7, 10, 21, 24, 32, 34 2 kW

Table 5.4 Distribution line capacities in the distribution network at RBTS Bus 2.

Sub-transmission line Capacity

a 6.5 kW

b 6.5 kW

c 6.5 kW

Table 5.5 RBTS Bus 2 sub-transmission line capacities.


Unsurprisingly the feeders and system indices (Table 5.6), as well as load point

indices (Table 5.7) obtained due to failures in the distribution network using minimum

cut set connectivity match those found in most other published results.

SAIFI

(int/cust-yr)

SAIDI

(h/cust-yr)

CAIDI

(h/cust-int)

ASAI

EENS

(MWh/yr)

Feeder 1 0.2480 3.6184 14.5906 0.999587 13.1721

Feeder 2 0.1398 0.5233 3.7442 0.999940 1.1221

Feeder 3 0.2499 3.6238 14.5014 0.999586 11.2032

Feeder 4 0.2471 3.6051 14.5907 0.999588 12.2484

System 0.2482 3.6126 14.5545 0.999588 37.7457

Table 5.6 Distribution system indices at RBTS Bus 2, using minimum cut set.

179

Load

ID

Failure

Rate

(f/yr)

Unavailability

(h/yr)

Av. Outage

Duration r (h)

Av. Energy Not Supplied

(MWh/yr)

Feeder 1 LP1 0.2393 3.5753 14.9436 1.9128

LP2 0.2523 3.6403 14.4311 1.9475

LP3 0.2523 3.6403 14.4311 1.9475

LP4 0.2393 3.5753 14.9436 2.0236

LP5 0.2523 3.6403 14.4311 2.0604

LP6 0.2490 3.6240 14.5542 1.6453

LP7 0.2523 3.6013 14.2765 1.6350

Feeder 2 LP8 0.1398 0.5428 3.8837 0.5428

LP9 0.1398 0.5038 3.6047 0.5793

Feeder 3 LP10 0.2425 3.5785 14.7567 1.9145

LP11 0.2523 3.6403 14.4311 1.9475

LP12 0.2555 3.6565 14.3112 1.6454

LP13 0.2523 3.5883 14.2250 2.0310

LP14 0.2555 3.6045 14.1076 2.0402

LP15 0.2425 3.5785 14.7567 1.6246

Feeder 4 LP16 0.2523 3.6403 14.4311 1.6527

LP17 0.2425 3.5915 14.8103 1.6162

LP18 0.2425 3.5785 14.7567 1.6103

LP19 0.2555 3.6435 14.2603 1.6396

LP20 0.2555 3.6435 14.2603 2.0622

LP21 0.2523 3.5883 14.2250 2.0310

LP22 0.2555 3.6045 14.1076 1.6364

Table 5.7 Distribution load point indices at RBTS Bus 2, using minimum cut set.


The system indices and load point indices for shortest path and priority maximum

methods are given in Tables 5.8, 5.10 and Tables 5.9, 5.11 respectively. The change

in the load point indices is due to limited capacity in the distribution feeders supplying

normally open switches. The end result of this is that the there is no difference in the

180

results obtained using priority or shortest path maximum flow, and the load point

failure rates are unchanged from those obtained using minimum cut set connectivity.

Both techniques tend to isolate loads at the furthest end of a limited feeder, so when

additional loads are appended to the end of a limited feeder the load points isolated

are usually selected from those appended. These load points have already been

disconnected by the initial outage so being isolated by the maximum flow algorithm

simply extends the duration for which they are disconnected.

There is a difference in the load points that experience outages as those tends favor to

denying flow to load points at the far end of distribution feeders. The energy not

supplied is also greater overall in this application due to the policy of isolating load

points that can only be partially supplied.

SAIFI

(int/cust-yr)

SAIDI

(h/cust-yr)

CAIDI

(h/cust-int)

ASAI

EENS

(MWh/yr)

Feeder 1 0.2480 3.8351 15.4645 0.999562 13.8846

Feeder 2 0.1398 0.6988 5.0000 0.999920 1.5023

Feeder 3 0.2499 3.9523 15.8162 0.999549 12.6224

Feeder 4 0.2471 3.9033 15.7977 0.999554 13.6433

System 0.2482 3.8929 15.6837 0.999556 41.6526

Table 5.8 Distribution system indices at RBTS Bus 2, using shortest path maximum

flow.

181

Load

ID

Failure

Rate (f/yr)

Unavailability

(h/yr)

Av. Outage

Duration r (h)

Av. Energy Not

Supplied (MWh/yr)

Feeder 1 LP1 0.2393 3.7313 15.5956 1.9962

LP2 0.2523 3.7963 15.0496 2.0310

LP3 0.2523 3.9913 15.8226 2.1353

LP4 0.2393 3.9263 16.4107 2.2223

LP5 0.2523 3.7963 15.0496 2.1487

LP6 0.2490 3.7800 15.1807 1.7161

LP7 0.2523 3.6013 14.2765 1.6350

Feeder 2 LP8 0.1398 0.6988 5.0000 0.6988

LP9 0.1398 0.6988 5.0000 0.8036

Feeder 3 LP10 0.2425 3.7735 15.5608 2.0188

LP11 0.2523 4.0303 15.9772 2.1562

LP12 0.2555 4.0465 15.8376 1.8209

LP13 0.2523 4.1863 16.5956 2.3694

LP14 0.2555 4.2025 16.4481 2.3786

LP15 0.2425 4.1375 17.0619 1.8784

Feeder 4 LP16 0.2523 3.7963 15.0496 1.7235

LP17 0.2425 3.7475 15.4536 1.6864

LP18 0.2425 3.9425 16.2577 1.7741

LP19 0.2555 4.0075 15.6849 1.8034

LP20 0.2555 4.2025 16.4481 2.3786

LP21 0.2523 4.1863 16.5956 2.3694

LP22 0.2555 4.2025 16.4481 1.9079

Table 5.9 Distribution load point indices at RBTS Bus 2, using shortest path

maximum flow.

182

SAIFI

(int/cust-yr)

SAIDI

(h/cust-yr)

CAIDI

(h/cust-int)

ASAI

EENS

(MWh/yr)

Feeder 1 0.2480 3.8351 15.4645 0.999562 13.8846

Feeder 2 0.1398 0.6988 5.0000 0.999920 1.5023

Feeder 3 0.2499 3.9523 15.8162 0.999549 12.6224

Feeder 4 0.2471 3.9033 15.7977 0.999554 13.6433

System 0.2482 3.8929 15.6837 0.999556 41.6526

Table 5.10 Distribution system indices at RBTS Bus 2, using priority maximum flow.

Load

ID

Failure Rate

(f/yr)

Unavailability

(h/yr)

Av. Outage

Duration r (h)

Av. Energy Not

Supplied (MWh/yr)

Feeder 1 LP1 0.2393 3.7313 15.5956 1.9962

LP2 0.2523 3.7963 15.0496 2.0310

LP3 0.2523 3.9913 15.8226 2.1353

LP4 0.2393 3.9263 16.4107 2.2223

LP5 0.2523 3.7963 15.0496 2.1487

LP6 0.2490 3.7800 15.1807 1.7161

LP7 0.2523 3.6013 14.2765 1.6350

Feeder 2 LP8 0.1398 0.6988 5.0000 0.6988

LP9 0.1398 0.6988 5.0000 0.8036

Feeder 3 LP10 0.2425 3.7735 15.5608 2.0188

LP11 0.2523 4.0303 15.9772 2.1562

LP12 0.2555 4.0465 15.8376 1.8209

LP13 0.2523 4.1863 16.5956 2.3694

LP14 0.2555 4.2025 16.4481 2.3786

LP15 0.2425 4.1375 17.0619 1.8784

Feeder 4 LP16 0.2523 3.7963 15.0496 1.7235

LP17 0.2425 3.7475 15.4536 1.6864

LP18 0.2425 3.9425 16.2577 1.7741

LP19 0.2555 4.0075 15.6849 1.8034

LP20 0.2555 4.2025 16.4481 2.3786

LP21 0.2523 4.1863 16.5956 2.3694

LP22 0.2555 4.2025 16.4481 1.9079

Table 5.11 Distribution load point indices at RBTS Bus 2, using priority maximum

flow.

183

5.4.2 RBTS Bus 4 Distribution Network

T1 T2

T3

T4

T5

T6

F1 F2 F3

F4

F5

F6

F7

12

34

5

6 7

8

9

10

1112

13

14

15

16

1718

19 20

21

22 23

24

25

26

272829

30

3132

33

34

35

36

37

38

39

40 41 42

43

4445

46

47

48

49

50

51

52

53

54 55

56

575859

60

61

62 63

64

65

66

67

SP1

SP2

SP3

Figure 5.14 Distribution network at RBTS Bus 4.

The RBTS Bus 4 distribution network, shown in Figure 5.14, is a three bus network

with a total of seven distribution feeders. Each distribution feeder is connected by a

normally open switch to another distribution feeder supplied by a different bus. This

allows load points to be restored in both the case of distribution and sub-transmission

failures. The complete data for this system is provided in Tables 5.12 – 5.16.

Line active failure rate 0.065 failures/km-year

Line repair time 5 hours

Transformer active failure rate 0.015 failures/year

Transformer repair time 200 hours

Isolator switching time 1 hour

Re-closer switching time 1 hour

Table 5.12 Component failure rate, repair time and switching time data for

distribution network at RBTS Bus 4.

184

Lines Length

2, 6, 10, 14, 17, 21, 25, 28, 30, 34, 38, 41, 43, 46, 49, 51, 55, 58, 61, 64, 67 0.6 km

1, 4, 7, 9, 12, 16, 19, 22, 24, 27, 29, 32, 35, 37, 40, 42, 45, 48, 50, 53, 56, 60, 63, 65 0.75 km

3, 5, 88, 11, 13, 15, 18, 20, 23, 26, 31, 33, 36, 39, 44, 47, 52, 54, 57, 59, 62, 66 0.8 km

Table 5.13 Feeder lengths in the distribution network at RBTS Bus 4.

Load Point Number of

Customers

Average

Load

LP1, LP2, LP3, LP4, LP11, LP12, LP13, LP18, LP19, LP20,

LP21, LP32, LP33, LP34, LP35

220 545 kW

LP5. LP14, LP15, LP22, L:P23, LP36, LP37 200 500 kW

LP8, LP10, LP26, LP27, LP28, LP29, LP30 1 1000 kW

LP9, LP31 1 1500 kW

LP6, LP7, LP16, LP17, LP24, LP25, LP38 10 415 kW

Table 5.14 Load point data of the distribution network at RBTS Bus 4.

Lines Capacity

31, 33 6 kW

1, 3, 13, 19, 21, 36, 44, 50, 56, 58 5 kW

5, 15, 23, 39, 41, 46, 52, 60 4 kW

7, 10, 17, 26, 28, 48, 54, 63, 65 3 kW

Table 5.15 Distribution line capacities in the distribution network at RBTS Bus 4.

Transformers Capacity

T1, T2 6 kW

T3, T4, T5, T6 6.5 kW

Table 5.16 RBTS Bus 4 sub-transmission transformer capacities.

185


As with Bus 2, Bus 4 reliability indices obtained due to failures in the distribution

network match those obtained in other published results. This is shown in Figures

5.17 and 5.18.

SAIFI

(int/cust-yr)

SAIDI

(h/cust-yr)

CAIDI

(h/cust-int)

ASAI

EENS

(MWh/yr)

Feeder 1 0.3010 3.4682 11.5210 0.999604 12.1922

Feeder 2 0.1896 0.3759 1.9829 0.999957 1.3211

Feeder 3 0.2933 3.4683 11.8266 0.999604 12.0034

Feeder 4 0.3075 3.4746 11.2987 0.999603 13.9267

Feeder 5 0.1863 0.3727 2.0000 0.999957 1.1180

Feeder 6 0.1950 0.3640 1.8667 0.999958 1.2659

Feeder 7 0.2967 3.4723 11.7045 0.999604 12.4660

System 0.2997 3.4653 11.5641 0.999604 54.2933

Table 5.17 Distribution system indices at RBTS Bus 4, using minimum cut set.

Load

ID

Failure

Rate (f/yr)

Unavailability

(h/yr)

Av. Outage

Duration r (h)

Av. Energy Not

Supplied (MWh/yr)

Feeder 1 LP1 0.2945 3.4355 11.6655 1.8724

LP2 0.3043 3.4843 11.4519 1.8989

LP3 0.2945 3.4355 11.6655 1.8724

LP4 0.3075 3.5005 11.3837 1.9078

LP5 0.3043 3.4843 11.4519 1.7421

LP6 0.3075 3.5005 11.3837 1.4527

LP7 0.3043 3.4843 11.4519 1.4460

Feeder 2 LP8 0.1820 0.3380 1.8571 0.3380

LP9 0.1918 0.3868 2.0170 0.5801

LP10 0.1950 0.4030 2.0667 0.4030

Feeder 3 LP11 0.2978 3.4908 11.7238 1.9025

LP12 0.2945 3.4745 11.7980 1.8936

186

LP13 0.2945 3.4745 11.7980 1.8936

LP14 0.2848 3.4258 12.0307 1.7129

LP15 0.2945 3.4745 11.7980 1.7373

LP16 0.2848 3.4258 12.0307 1.4217

LP17 0.2945 3.4745 11.7980 1.4419

Feeder 4 LP18 0.3108 3.4908 11.2333 1.9025

LP19 0.3010 3.4420 11.4352 1.8759

LP20 0.3108 3.4908 11.2333 1.9025

LP21 0.3108 3.4908 11.2333 1.9025

LP22 0.3010 3.4420 11.4352 1.7210

LP23 0.3108 3.4908 11.2333 1.7454

LP24 0.3108 3.4908 11.2333 1.4487

Feeder 5 LP25 0.3010 3.4420 11.4352 1.4284

LP26 0.1885 0.3835 2.0345 0.3835

LP27 0.1918 0.3998 2.0848 0.3998

LP28 0.1788 0.3348 1.8727 0.3348

Feeder 6 LP29 0.1918 0.3478 1.8136 0.3478

LP30 0.2015 0.3965 1.9677 0.3965

LP31 0.1918 0.3478 1.8136 0.5216

Feeder 7 LP32 0.3010 3.4940 11.6080 1.9042

LP33 0.3010 3.4940 11.6080 1.9042

LP34 0.2880 3.4290 11.9063 1.8688

LP35 0.3010 3.4940 11.6080 1.9042

LP36 0.2880 3.4290 11.9063 1.7145

LP37 0.3010 3.4940 11.6080 1.7470

LP38 0.2880 3.4290 11.9063 1.4230

Table 5.18 Distribution load point indices at RBTS Bus 4, using minimum cut set.


Maximum flow connectivity in Bus 4 exhibits similar behavior to what was seen with

Bus 2 when considering failures in the distribution network. There are no changes in

the failure rates of any load point but many have an increase in outage duration due to

187

limited capacity in alternate paths of supply preventing them from being restored.

The system and load point indices for the shortest path and priority maximum flow

are shown in Tables 5.19, 5.21 and 5.20, 5.22 respectively. Unlike what was seen

with Bus 2, failures in the sub-transmission network do cause load point outages; as a

single transformer does not have sufficient capacity to supply all the load points

connected to a supply bus. The normally open switches in Bus 4 connect distribution

feeders supplied by different buses so when a sub-transmission failure occurs some of

the lost supply can be restored through those switches. By spreading the isolated load

points across multiple feeders the shortest path technique minimizes the amount of

flow that needs to be supplied through these switches and allows all load points be

restored. This comes at the expense of disconnecting all the load points on those

feeders for the period between the fault initially occurring and manual switching

taking place. Priority maximum flow groups the isolated load points on the one

distribution feeder which reduces the number of load points experiencing outages but

not all the load points can be restored following manual switching. The energy not

supplied and outage durations due to load points not being restored for the duration of

the outage using priority maximum flow outweigh that due to additional load points

being disconnected by shortest path, so for Bus 4 shortest path gives the more

favorable overall indices.

There is generally an improvement in the system indices, the exception being SAIDI

and CAIDI for priority maximum flow. The improvement in energy not supplied

particularly is a result of the isolation of partially supplied load points and the

subsequent load shifting which restores load points that would have simply been

denied power using partial supply. An increase in failure rate and decrease in outage

durations would be expected to accompany this change in energy not supplied but the

SAIFI and SAIDI do support this, SAIFI and SAIDI however are weighted by

customer numbers making accurate comparisons difficult without load point indices.

188

SAIFI

(int/cust-yr)

SAIDI

(h/cust-yr)

CAIDI

(h/cust-int)

ASAI

EENS

(MWh/yr)

Feeder 1 0.3310 4.0477 12.2276 0.999538 13.7987

Feeder 2 0.1896 0.5839 3.0800 0.999933 2.1531

Feeder 3 0.3233 3.8269 11.8384 0.999563 12.9891

Feeder 4 0.3375 3.8822 11.5020 0.999557 15.2659

Feeder 5 0.1863 0.5633 3.0233 0.999936 1.6900

Feeder 6 0.1950 0.5633 2.8889 0.999936 1.8639

Feeder 7 0.3267 3.8966 11.9283 0.999555 13.8371

System 0.3296 3.9055 11.8491 0.999554 61.5977

Table 5.19 Distribution system indices at RBTS Bus 4, using shortest path maximum

flow.

Load ID

Failure Rate

(f/yr)

Unavailability

(h/yr)

Av. Outage

Duration r (h)

Av. Energy Not

Supplied (MWh/yr)

Feeder 1 LP1 0.2945 3.6305 12.3277 1.9786

LP2 0.3043 3.8873 12.7765 2.1186

LP3 0.2945 4.0465 13.7402 2.2053

LP4 0.3075 4.3065 14.0049 2.3470

LP5 0.3043 4.2903 14.1011 2.1451

LP6 0.3075 3.5005 11.3837 1.4527

LP7 0.3043 3.4843 11.4519 1.4460

Feeder 2 LP8 0.1820 0.5460 3.0000 0.5460

LP9 0.1918 0.8028 4.1864 1.2041

LP10 0.1950 0.4030 2.0667 0.4030

Feeder 3 LP11 0.2978 3.6858 12.3787 2.0087

LP12 0.2945 3.8255 12.9898 2.0849

LP13 0.2945 4.0335 13.6961 2.1983

LP14 0.2848 3.9848 13.9939 1.9924

LP15 0.2945 3.4745 11.7980 1.7373

LP16 0.2848 3.4258 12.0307 1.4217

LP17 0.2945 3.4745 11.7980 1.4419

Feeder 4 LP18 0.3108 3.6988 11.9027 2.0158

189

LP19 0.3010 3.8580 12.8173 2.1026

LP20 0.3108 3.9068 12.5720 2.1292

LP21 0.3108 4.1148 13.2414 2.2425

LP22 0.3010 4.0660 13.5083 2.0330

LP23 0.3108 3.4908 11.2333 1.7454

LP24 0.3108 3.4908 11.2333 1.4487

Feeder 5 LP25 0.3010 3.4420 11.4352 1.4284

LP26 0.1885 0.5915 3.1379 0.5915

LP27 0.1918 0.7638 3.9831 0.7638

LP28 0.1788 0.3348 1.8727 0.3348

Feeder 6 LP29 0.1918 0.5428 2.8305 0.5428

LP30 0.2015 0.7995 3.9677 0.7995

LP31 0.1918 0.3478 1.8136 0.5216

Feeder 7 LP32 0.3010 3.6890 12.2558 2.0105

LP33 0.3010 3.8450 12.7741 2.0955

LP34 0.2880 3.9750 13.8021 2.1664

LP35 0.3010 4.0400 13.4219 2.2018

LP36 0.2880 4.1700 14.4792 2.0850

LP37 0.3010 3.4940 11.6080 1.7470

LP38 0.2880 3.4290 11.9063 1.4230

Table 5.20 Distribution load point indices at RBTS Bus 4, using shortest path

maximum flow.

SAIFI

(int/cust-yr)

SAIDI

(h/cust-yr)

CAIDI

(h/cust-int)

ASAI

EENS

(MWh/yr)

Feeder 1 0.3310 6.3272 19.1135 0.999278 20.0373

Feeder 2 0.1896 0.5839 3.0800 0.999933 2.1531

Feeder 3 0.2933 3.7969 12.9471 0.999567 12.8851

Feeder 4 0.3375 3.8822 11.5020 0.999557 15.2659

Feeder 5 0.1863 0.5633 3.0233 0.999936 1.6900

Feeder 6 0.1950 0.5633 2.8889 0.999936 1.8638

Feeder 7 0.3267 3.8966 11.9283 0.999555 13.8371

System 0.3228 4.4233 13.7022 0.999495 67.7324

Table 5.21 Distribution system indices at RBTS Bus 4, using priority maximum flow.

190

Load

ID

Failure Rate

(f/yr)

Unavailability

(h/yr)

Av. Outage

Duration r (h)

Av. Energy Not

Supplied (MWh/yr)

Feeder 1 LP1 0.2945 3.6305 12.3277 1.9786

LP2 0.3043 3.8873 12.7765 2.1186

LP3 0.2945 4.0465 13.7402 2.2053

LP4 0.3075 4.3065 14.0049 2.3470

LP5 0.3043 4.2903 14.1011 2.1451

LP6 0.3075 3.5005 11.3837 1.4527

LP7 0.3043 3.4843 11.4519 1.4460

Feeder 2 LP8 0.1820 0.5460 3.0000 0.5460

LP9 0.1918 0.8028 4.1864 1.2041

LP10 0.1950 0.4030 2.0667 0.4030

Feeder 3 LP11 0.2978 3.6858 12.3787 2.0087

LP12 0.2945 3.8255 12.9898 2.0849

LP13 0.2945 4.0335 13.6961 2.1983

LP14 0.2848 3.9848 13.9939 1.9924

LP15 0.2945 3.4745 11.7980 1.7373

LP16 0.2848 3.4258 12.0307 1.4217

LP17 0.2945 3.4745 11.7980 1.4419

Feeder 4 LP18 0.3108 3.6988 11.9027 2.0158

LP19 0.3010 3.8580 12.8173 2.1026

LP20 0.3108 3.9068 12.5720 2.1292

LP21 0.3108 4.1148 13.2414 2.2425

LP22 0.3010 4.0660 13.5083 2.0330

LP23 0.3108 3.4908 11.2333 1.7454

LP24 0.3108 3.4908 11.2333 1.4487

Feeder 5 LP25 0.3010 3.4420 11.4352 1.4284

LP26 0.1885 0.5915 3.1379 0.5915

LP27 0.1918 0.7638 3.9831 0.7638

LP28 0.1788 0.3348 1.8727 0.3348

Feeder 6 LP29 0.1918 0.5428 2.8305 0.5428

LP30 0.2015 0.7995 3.9677 0.7995

LP31 0.1918 0.3478 1.8136 0.5216

191

Feeder 7 LP32 0.3010 3.6890 12.2558 2.0105

LP33 0.3010 3.8450 12.7741 2.0955

LP34 0.2880 3.9750 13.8021 2.1664

LP35 0.3010 4.0400 13.4219 2.2018

LP36 0.2880 4.1700 14.4792 2.0850

LP37 0.3010 3.4940 11.6080 1.7470

LP38 0.2880 3.4290 11.9063 1.4230

Table 5.22 Distribution load point indices at RBTS Bus 4, using priority maximum

flow.

5.5 Effect of Substation Sub-transmission Aging on Distribution

Network

The focus of this thesis is on the aging substation reliability. The choice of substation

is because it is the most integrated and commonly used circuit in all levels of electric

supply hierarchies; generation, transmission and distribution. The technique

developed for aging in this thesis, however, can equally be applied to

transmission/sub-transmission facilities. Chapter 3 addressed the aging in substation

facilities, and extensive application was provided to examine the effects. However,

the loads supplied through the substations are normally bulk loads and can include an

entire distribution system. Therefore, it won’t comprise the failures associated with

the distribution system. On the other hand, the reliability evaluation of distribution

systems is normally excludes the failures in the substation or sub-transmission system.

As the facilities in transmission and substations age, the aging failure inevitably will

affect the distribution loads. This is an area where no research work is reported and

can be important in decisions relating to investments in upgrading the facilities to

meet certain level of reliability. In this chapter the effects of substation aging

facilities on distribution system and loads are examined.

192

From the point of supply at the buses in sub-transmission through feeders to the loads

in distribution system, the circuit is normally arranged in series at any given time.

The method of distribution system evaluations therefore are unique as discussed and

evaluated in the previous sections. The set of system indices evaluated are also

unique to distribution system, as they are mainly customer related indices. The

electric energy is however generated at generating units, usually located far distance

away from consumers, and carried through the transmission systems and by

successive substations to distribution networks to supply the loads. Therefore, the

outages anywhere in the network may ultimately affect the loads in distribution

system and the customers. Although the majority of load outages are caused by

distribution system facilities, but as aging in transmission and substation system

facilities are taking place, the effects of outages at this level to the loads at distribution

need to be considered more rigorously. In order to cater the effect of substation

outages in the distribution systems the following approach is applied.

The electric supply to distribution network is normally provided through sub-

transmission facilities. This is tapped at substation linked to distribution system. The

reliability evaluation performed at substation/sub-transmission level, in terms of a set

of average failure rate and average annual outage time at each load bus, are

representing a distribution system supply point. The inclusion of these indices,

together with the distribution system reliability evaluation, will provide us with

overall indices. The comparison of the results between the inclusion and the

exclusion of substation/sub-transmission, with provide us with the effects. In the

following section applications are provided to examine the aging effects to

distribution system.

193

5.5.1 Application of Substation/Sub-transmission Non-repairable Aging on

Distribution System Reliability

In this section numerical examples are used to illustrate the effect aging components

in substation/sub-transmission networks on distribution system. The RBTS Bus 2 and

4 are distribution systems are used to show the results. The aging substation/sub-

transmission systems examined in Chapter 3 of Figure 2.6(d) and 2.7 are used as the

supply links to the distribution systems. In the first instance, the distribution systems

are assumed to have been supplied through load Bus 1 and load Bus 2 of the systems

in Figures 2.6(d) and 2.7, respectively. The average failure frequency and

unavailability for the two load buses, as the transformer and circuit breaker facilities

in both systems are aging, are provided in Tables 5.23 and 5.24. For all practical

purposes, the values are extracted for every five years of aging.

Year Failure probability Failure frequency (f/yr) Unavailability (hr/yr) Average repair time (hr) 1 0.000024 0.189246 0.214411 1.132975 5 0.000024 0.189247 0.214441 1.133128 10 0.000025 0.189253 0.214638 1.134133 15 0.000025 0.189280 0.215605 1.139080 20 0.000025 0.189380 0.219700 1.160101 25 0.000027 0.189671 0.236311 1.245899 30 0.000035 0.190393 0.302747 1.590116 35 0.000064 0.192080 0.557246 2.901114 40 0.000174 0.196279 1.525373 7.771453 45 0.000629 0.208775 5.513811 26.410303 50 0.002790 0.256524 24.438194 95.266696 55 0.014356 0.489123 125.754673 257.102351

60 0.071629 1.724515 627.471903 363.854129

Table 5.23 Bus 1 load indices of Figure 2.6(d).

The impact of the aging substation/sub-transmission system of Figure 2.6(d) is

examined on RBTS Bus 2 distribution system. In order to exclusively examine the

aging effect on distribution system performance, the capacity limitations and load

growth were not included. Therefore, the method of minimum cut set was used to

obtain the results in Table 5.25. The first thirty years of aging in substation facilities

194

have little effect on the distribution system. As expected, the adverse effect of aging

appears at about year 35, close to the mean life of the components which in this case

is 45 year the mean life for transformers and the circuit breakers.

Year Failure probability Failure frequency (f/yr) Unavailability (hr/yr) Average repair time (hr) 1 0.000155 0.076129 1.357698 17.834176 5 0.000157 0.076140 1.372019 18.019687 10 0.000167 0.076219 1.466678 19.242945 15 0.000218 0.076587 1.911083 24.953099 20 0.000402 0.077923 3.518087 45.148249 25 0.000919 0.081731 8.048667 98.477530 30 0.002099 0.090655 18.390507 202.862578 35 0.004446 0.109372 38.951037 356.133535 40 0.008904 0.148687 77.999460 524.588296 45 0.017529 0.240294 153.555785 639.032955 50 0.034174 0.489813 299.360037 611.172094 55 0.059878 1.214282 524.530436 431.967563

60 0.061056 2.237983 534.854596 238.989571

Table 5.24 Bus 2 load indices of Figure 2.7.

The application associated with the maximum flow methods discussed in earlier

sections can be appreciated when the load growth and capacity limitation are

introduced, during the aging years of planning. Therefore, the results presented in

Table 5.25 can be considered as conservative figures.

There are many different aspects in making decision in planning. Reliability

evaluation can be one and important part of the decision. As noted earlier, the aging

components in a system can have different importance in continuity of supply to

customer load points. This aspect lends itself to how a system is configured and to

what degree redundancy is available. The next application to some degree can

facilitate to explore and compare the redundancy aspect of the two configurations.

195

Year SAIFI

intrp/cust yr SAIDI

hrs/cust yr CAIDI

hrs/cust intrp ASAI

EENS MWh/yr

1 0.437 3.83 8.75 0.999563 40.38

30 0.439 3.92 8.93 0.999553 41.47 35 0.440 4.17 9.47 0.999524 44.59 40 0.444 5.14 11.56 0.999413 56.49 45 0.457 9.13 19.97 0.998958 105.52 50 0.505 28.05 55.58 0.996798 338.12 55 0.737 129.37 175.45 0.985232 1583.40

60 1.973 631.08 319.91 0.927958 7750.00

Table 5.25 RBTS Bus 2 distribution system – supplied from aging load Bus 1 of

Figure 2.6(d), using minimum cut set.

In the next example, the effect of load Bus 2 of the substation in Figure 2.7 on RBTS

Bus 2 distribution system is examined. The results are provided in Table 5.26.

Year SAIFI

intrp/cust yr SAIDI

hrs/cust yr CAIDI

hrs/cust intrp ASAI

EENS MWh/yr

1 0.324 4.97 15.32 0.999433 54.43 10 0.324 5.08 15.66 0.999420 55.77 20 0.326 7.13 21.86 0.999186 80.99 25 0.330 11.66 35.34 0.998669 136.67 35 0.358 42.56 119.03 0.995141 516.49 40 0.397 81.61 205.63 0.990684 996.44 45 0.489 157.17 321.73 0.982058 1925.10 50 0.738 302.97 410.52 0.965414 3717.18 55 1.462 528.14 361.13 0.939710 6484.75

60 2.486 538.47 216.58 0.938531 6611.64


Figure 2.7, using minimum cut set.

Starting from the first aging year, both RBTS Bus 2 results in Tables 5.25 and 5.26

compared with Table 5.6 indicate the effect of the substation/sub-transmission

systems. The effects manifest in different way for the two cases, as the aging years

are progressed. In the first aging year, Bus 1 in Figure 2.6(d) experiences far more

failure frequency than that of Bus 2 in Figure 2.7, as seen in Tables 5.23 and 5.24.

However, this trend reverses as the aging years increase. Conversely, the first load

196

bus unavailability is less than that of the second one. Similarly, the trend between the

two reverses as the aging years increase. Consequently, the effect of the two buses on

the RBTS Bus 2 distribution system follows accordingly. The effects on the

frequency and the duration related indices can be seen in Tables 5.25 and 5.26,

accordingly. The reliability data used for the components in the two substation/sub-

transmission systems are the same. Therefore, the variations in which the two

systems exhibit are owing to the way they are configured.

In the followings, the aging effects of substation/sub-transmission systems are

examined on the RBTS Bus 4 distribution system. The results are illustrated in

Tables 5.27 and 5.28.

Year SAIFI intrp/cust yr

SAIDI hrs/cust yr

CAIDI hrs/cust intrp

ASAI EENS MWh/yr

1 0.489 3.68 7.53 0.999580 59.56 35 0.492 4.02 8.18 0.999541 67.99 40 0.496 4.99 10.06 0.999430 91.79 45 0.508 8.98 17.66 0.998975 189.82 50 0.556 27.90 50.17 0.996815 654.98 55 0.789 129.22 163.82 0.985249 3145.34

60 2.024 630.94 311.70 0.927975 15477.60


Figure 2.6(d), using minimum cut set.

Similar impacts, as in the previous cases, are seen on all distribution system indices.

The distribution system of RBTS Bus 4 is larger than that of Bus 2 with more

normally open links and load points. Therefore, the expected energy on outage in this

distribution system is naturally more so. However, a larger distribution system will

not necessarily give higher frequency and duration indices.

197

Year SAIFI

intrp/cust yr SAIDI

hrs/cust yr CAIDI

hrs/cust intrp ASAI

EENS MWh/yr

1 0.376 4.82 12.83 0.999449 87.67 20 0.378 6.98 18.50 0.999203 140.77 30 0.390 21.86 56.00 0.997505 506.33 40 0.448 81.46 181.70 0.990700 1971.52 50 0.789 302.83 383.58 0.965431 7412.56 55 1.514 528.00 348.76 0.939727 12947.30

60 2.538 538.32 212.13 0.938548 13201.00


Figure 2.7, using minimum cut set.

Obviously in all the application cases examined, aging of the substation/sub-

transmission systems have direct effect on the distribution system indices. The effects

in the earlier years are much less than that of closer to mean life of the components.

5.5.2 Effects of Aging Equipment’s Mean Life on Distribution System

Reliability

Different manufacturers provide equipment for power system substations and sub-

transmission networks. The mean life of equipment can be affected by the design, the

materials and the manufacturing specifications. The environment and the operational

conditions can also have effect on the mean life. The non-repairable aging method

presented and applied in this thesis can provide the different aging effects in the

indices. In this method, the effects are included through frequency and duration

approach. Therefore wide range of effected indices that are produced using duration

and/or frequency of outages can be provided to make an appropriate and timely

planning decision for reinforcement or replacement.

A case of non-repairable aging is examined in the followings to illustrate the

discrepancies between the results between two aging cases. Consider the medium

substation in Figure 2.7 of Chapter 2. Transformer TX2 in this sub is taken for a case

198

study. The mean life of the transformer in case (a) is taken as 45 years, and 30 years

in case (b). The failure probability and frequency of the bulk load point 2 supplied by

transformer TX2, are shown in Figures 5.15 and 5.16. The differences between cases

(a) and (b) are clearly observable and indicate that; as the mean life of TX2 passes the

30 years in case (b), the failure probability and frequency increase rapidly compared

to case (a).

Bus 2 of Medium Sub Failure Probability

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 5 10 15 20 25 30 35 40 45 50 55 60

Year

Fai

lure

Pro

bab

ilit

y

Failure Prob (a)

Failure Prob (b)

TX2 is aging

Case (a): mean: 45 yr S.D. 10 yrCase (b): mean: 30 yr S.D. 10 yr

Figure 5.15 Bulk Load 2 of medium sub in Figure 2.7, failure probability as TX2 aging with two different mean lives.

Bus 2 of Medium Sub Failure Frequency

0

1

2

3

4

5

6

7

8

1 5 10 15 20 25 30 35 40 45 50 55 60

Year

Fai

lure

fre

qu

ency

Failure Freq (b)

Failure Freq (a)

TX2 is aging

Case (a): mean: 45 yr S.D. 10 yrCase (b): mean: 30 yr S.D. 10 yr

Figure 5.16 Bulk Load 2 of medium sub in Figure 2.7, failure frequency as TX2 aging with two different mean lives.

199

Once more, RBTS Bus 2 distribution system is considered to be supplied at the bulk

load point 2 of the medium substation. The frequency and duration outage effects of

TX2 of the medium sub on the distribution system indices are provided in Table 5.29,

for both cases (a) and (b). The effects on all the indices; SAIFI, SAIDI, CAIDI, ASAI

and ENS consistently indicate the disadvantages of case (b). The indices provided at

the distribution level can be taken as the risk information to facilitate for planning

engineers to make an informed engineering judgment for appropriate remedial action.

Table 5.29 RBTS Bus 2 distribution system indices – supplied from medium sub

load Bus 2 of Figure 2.7 TX2 aging with mean life a) 45, b) 30 years.

Year SAIFI intrp/cust yr SAIDI hrs/cust

yr CAIDI hrs/cust

intrp ASAI EENS MWh/yr

a b a b a b a b a b

1 0.331257 0.331961 5.09 5.90 15.38 17.78 0.999419 0.999326 55.94 65.88

5 0.331271 0.333310 5.11 7.45 15.42 22.36 0.999417 0.999149 56.14 84.94

10 0.331361 0.337630 5.21 12.42 15.73 36.78 0.999405 0.998582 57.41 145.97

15 0.331783 0.347581 5.70 23.85 17.17 68.63 0.999350 0.997277 63.38 286.52

20 0.333312 0.367740 7.46 47.02 22.37 127.86 0.999149 0.994633 84.97 571.24

25 0.337639 0.407490 12.43 92.69 36.81 227.47 0.998581 0.989419 146.09 1132.61

30 0.347603 0.490617 23.88 188.19 68.69 383.58 0.997274 0.978517 286.83 2306.39

35 0.367789 0.682363 47.07 408.39 127.99 598.49 0.994626 0.953380 571.93 5012.84

40 0.407591 1.163380 92.81 960.26 227.70 825.41 0.989406 0.890381 1134.04 11795.90

45 0.490828 2.356810 188.43 2326.29 383.91 987.05 0.978489 0.734441 2309.36 28585.80

50 0.682815 4.613440 408.91 4897.00 598.85 1061.46 0.953321 0.440981 5019.21 60182.40

200

Figure 5.17 SAIFI index of RBTS Bus 2 distribution system – supplied from medium

sub load Bus 2 of Figure 2.7 as TX2 aging with mean life a) 45, b) 30 years.

Figures 5.17 and 5.18 provide graphical illustrations of SAIFI and CAIDI results.

Figure 5.17 shows the frequency index (SAIFI) difference between cases (a) and (b).

As seen in this figure, SAIFI in case (b) increases much more than case (a) once TX2

of case (b) passes its mean life.

Figure 5.18 CAIDI index of RBTS Bus 2 distribution system – supplied from medium sub load Bus 2 of Figure 2.7 as TX2 aging with mean life a) 45, b) 30 years.

Effect of TX2 in Medium Sub on CAIDI Index of RBTS Bus 2 Distribution System

0

200

400

600

800

1000

1200

1 5 10 15 20 25 30 35 40 45 50 60 Year

CAIDI (hrs/cust intrp)

Case (a)

Case (b)

TX2 is aging

Case (a): mean: 45 yr S.D. 10 yr Case (b): mean: 30 yr S.D. 10 yr

Effect of TX2 in Medium Sub on SAIFI Index of RBTS Bus 2 Distribution System

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1 5 10 15 20 25 30 35 40 45 50 60 Year

SAIFI (intrp/cust yr)

Case (a)

Case (b)

TX2 is aging

Case (a): mean: 45 yr S.D. 10 yr Case (b): mean: 30 yr S.D. 10 yr

201

The CAIDI differences are the widest between cases (a) and (b) starting from year 35

to 45. This is due to the non-repairable effects that are most effecting near the means

in each case.

202

5.6 Summary

The power system as a whole, from generating units located in different site in the

network to the transmission lines transferring the electrical energy to bulk load buses

and to the sub-transmission systems where the high voltages are reduced to medium

range voltage in facilitation to bring power to distribution systems, and to the

substations spreading around the whole network in either stepping up the voltage to

transmission lines from the generating units, or successively stepping down the

voltages from transmission to sub-transmission and further to distribution centers, or

simply providing switching arrangements in the network, they all have a common

goal, and that is to supply power to load customers.

Such interconnected network is very complex where inadequacy in design, operation

and maintenance anywhere in the system can diversely affect other parts of the

network and ultimately the customers. In addition to that, the facilities in the network

are aging and getting close to their mean functional life, while the demand for

electrical energy is increasing. As part of system planning and asset management, in

addressing this issue, it is imperative to device systemic approaches for reinforcement

and replacement policies that are adequate to meet the intended level of availability

for the customers, while economically justifiable.

The model for aging components was introduced in Chapter 3. This model was

shown to evaluate the aging effects of the facilities as part of power system reliability.

The focus of this thesis is the aging at substation/sub-transmission system where the

network is looped and interconnected and the effect of aging facilities such as power

transformers and circuit breakers are not easily obtainable. The impact of aging at

higher voltage facilities are normally measured at bulk load buses. These points are

still distance away from distribution customer loads. The reliability evaluations at

distribution level, where the network is commonly arranged in a radial form, is

203

different and has different set of indices, customer related indices. Ultimately, the

outages experienced due to aging at higher voltage networks are received at customer

load points. It is therefore commendable to include the aging effects of the facilities

in substation/sub-transmission that supply power to distribution customers, as part of

the distribution reliability assessments.

This chapter initially provided detail reliability evaluations for distribution systems.

Several flow approaches were used as part of the evaluations. Minimum cut set and

maximum flow methods are the two main approaches used. The minimum cut set

method is a connectivity method and do not include any capacity limitations in the

network. However, the maximum flow can include capacity limitations and therefore

load shedding needs to be considered. Two methods of load shedding were

introduced and applied in this chapter. The provision of capacity limitation in

distribution reliability assessments can prove to be useful in long term planning,

where feeders and transformers may rich the nominal capacity as the system expands

and load grows. Detail results for both RBTS Bus 2 and Bus 4 were provided for all

methods in this chapter together with related discussions on the findings.

The inclusion of substation/sub-transmission aging as part of distribution reliability

evaluations later was included and the results for both RBTS distribution systems

were provided. The effect of aging were shown as part of the distribution system

indices; SAIFI, SAIDI, CAIDI, ASI and ENS. The comparison of the results can

provide useful indicators and measures, assisting in making corrective planning

decisions, where cost worth consideration can systematically be included.

204

Chapter 6

Economic Consideration for Substation and Sub-transmission

Equipment Replacement

6.1 Introduction

The basic function of an electric power system is to meet the customer load demand at

the lowest possible cost, while maintaining acceptable level of supply continuity. The

‘acceptable’ level of continuity may normally be conventional risk indices set by

regulatory bodies that are governed by legislators, and do not systematically relate to

customer outage costs. However, a rational reliability planning decision for power

systems needs to be associated with cost/worth analysis. A large portion of the cost is

related to the outages inflicted on customers [108, 113-122]. This portion of the cost

plays an important role in cost analysis. The concept of cost/worth estimation is well

developed in generation, transmission and distribution systems reliability. The

extrapolated cost of bulk load losses due to outages at generation and composite

generation and transmission levels may not directly represent the customer financial

losses and therefore would not be the best avenue to evaluate the cost/worth

estimation. Almost all utility customers are directly supplied through distribution

networks. Usually, the type of customers and the amount of load consumed varies

from one distribution network to other. The financial damages to utility customers are

measured as a function of outage duration, and each type of customer has a different

cost function. Consequently, the customer financial losses due to electricity outages,

initiated from anywhere in the power system, are most appropriate to be calculated in

the distribution system zone affected.

As was noted in preceding chapter, high voltage substations and sub-transmission

networks have equipment that is expensive and the non-repairable aging failure of

205

these facilities can have grave consequence such as widespread outages on the

customer loads, including distribution level. In this thesis, methods are developed to

include the failure and replacement time of the non-repairable aging equipment as part

of the reliability assessment of substation and sub-transmission systems. The

approach to measure the impact of non-repairable aging failures within high voltage

networks to distribution customer loads is also presented with application to IEEE

Bus 2 and 4 distribution feeders. The types of customer load and the contribution of

each type to a load measured on 11kV substation are necessary for outage cost

calculations. This information is not readily available and can be difficult to measure

directly. Statistical methods to extract this information were presented in Chapter 4.

Replacement is required for the non-repairable aging failure of equipment in the high

voltage networks which is causing increased risk of customer outage cost. However,

this equipment is often expensive and normally not available on demand.

Manufacturing, shipping and handling can take significant amount of time, in addition

to the installation time, before the equipment is ready to use. A careful financial

assessment is required to find the trade-off between the consumer cost of outages and

the reliability benefit of investment in equipment replacement. The economic trade-

off between the cost and the worth may financially justify a time for replacement.

However, the regulatory requirements based on certain level of risk indices may

necessitate the equipment to be replaced earlier. Early reliability standards focused on

average performance across all customer feeders [123]. Later, regulations made refer

to more specific groups of feeders [124]. This chapter considers the possibility of

minimum performance standard being applicable for individual feeders or feeders’

zone. The basic objective of this chapter is to illustrate how the trade-off between the

customer outage cost and equipment investment cost can be used to make a timely

replacement decision. The factors which influence and are required to evaluate the

outage cost are: interruption duration, frequency and load curtailed. The non-

repairable aging model developed in the earlier chapters of this thesis produces the

required variables for the intended cost evaluations.

206

6.2 Customer Interruption Costs

Reliability worth assessment provides the opportunity to incorporate cost analysis and

quantitative reliability assessment into a common framework [45]. An indirect method

[2] of reliability worth is frequently assessed through interruption costs. In this

method, surveys are usually carried out to evaluate the impact of outage interruptions

to customer service. This includes the effect on different type or sector of customers,

such as; residential, commercial, industrial, etc. A comprehensive statistical analysis

is first performed on the raw data collected from different sectors of customers. The

outcome of the study provides two set of customer damage functions, Sector

Customer Damage Function (SCDF) and Composite Customer Damage Function

(CCDF). The SCDF provides the unit interruption cost as a function of the

interruption duration for an individual customer sector, as shown in Figure 6.1.

Figure 6.1 Sector customer damage function [2].

The SCDF duration and cost values are given in Table 6.1.

207

Interruption cost of customer sectors ($/kW)

Duration Agri. Large user Resid. Gover. Indus. Commer. Office

1 min 0.060 1.005 0.001 0.044 1.625 0.381 4.778

20 min 0.343 1.508 0.093 0.369 3.868 2.969 9.878

1 hr 0.649 2.225 0.482 1.492 9.085 8.552 21.065

4 hr 2.064 3.968 4.914 6.558 25.163 31.317 68.830

8 hr 4.120 8.240 15.690 26.040 55.808 83.008 119.160

Table 6.1 Sector customer damage functions for the seven customer categories [2].

The CCDF is the measure of the interruption cost as a function of the interruption

duration for a customer mix at a bus, in a service area, or in a whole system, as shown

in Figure 6.2.

Figure 6.2 Composite customer damage function [2].

The CCDF duration and cost values are given in Table 6.2.

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Interruption duration Interruption cost ($/kW)

1 min 0.67

20 min 1.56

1 hr 3.85

4 hr 12.14

8 hr 29.41

Table 6.2 Composite customer damage function for the seven customer mixture [2].

6.3 Expected Interruption Cost (EIC)

Expected interruption cost (EIC) is the most appropriate index that can represent the

outage cost due to the failure contingencies discussed in Chapter 3. EIC can be

calculated for every system state failure contingency event that causes any load to go

on outage. This approach can be applied to distribution, substation and sub-

transmission in a similar way. The outage of every load is associated with a duration,

frequency and curtailed load. These parameters are used to calculate EIC for each

load. The SCDF is used to calculate the EIC for distribution systems, where CCDF is

considered to evaluate the EIC for substation and sub-transmission systems.

The EIC for every failing system state contingency k can be calculated using Equation

6.1.

(6.1)

Where; L and F are average load and failure frequency, and I(D) is interruption cost,

which is a function of duration D as found from either SCDF or CCDF functions, and

i is the failing system state event that causes load k go on outage. The overall EIC of

a network having M number of loads is obtained using Equation 6.2.

(6.2)

$/yr )(1∑

==

N

iiiik DIFLEIC

$/yr 1∑

=

=M

jjEICEIC

209

In the non-repairable aging method developed in this thesis, load point and system

indices are calculated for every year of planning. The estimated interruption cost is

calculated for each year of planning separately. The calculated EIC is already

represented in cash flow form as it is calculated yearly.

However, the investment cost of each component in a substation or sub-transmission

system also needs to be computed into a cash flow form. This will allow computation

and comparison of the EIC that resulted from the aging of a component with its

investment cash flow. The time to replace a component can be found when the

expected interruption cost exceeds the investment cash flow.

Equation 6.3 can be used to calculate the investment cost of a component.

(6.3)

Where A, S and CRF are annual capital investment cash flow, actual capital

investment in some year, and capital return factor. The capital return factor is

calculated using Equation 6.4.

(6.4)

Where i is the discount rate and n is the economic life of the investment, in years.

The economic life is technically defined as the functional life of equipment. The

functional life is normally estimated by life testing, which can be very expensive and

not practical. The life testing approach requires induced life acceleration, which

normally is prepared in a controlled environment for a sample of populations. Instead

of life testing, for all practical purposes, the mean life of the natural age of equipment

can be used.

$/yr CRFSA ×=

1)1(

)1(

−++=

n

n

i

iiCRF

210

6.4 Application to Substation and Sub-transmission Systems

The application of economic considerations is illustrated in this section. This

application assists the decision for timely replacement of non-repairable equipment,

when economically viable. The substation and the sub-transmission networks

considered for this application are the same ones (Figures 2.6 and 2.7) used

throughout this thesis, with the same reliability data. Two sets of results are provided

for the application. In the first set, the economic considerations are performed only

on the substation and sub-transmission networks, using composite customer damage

function (CCDF). In this application, the CCDF is representing an aggregated cost

function used for loads at bulk supply networks. In the second set, the economic

evaluations resulting from the effect of non-repairable failures in high voltage

networks are examined at distribution load centre, using sector customer damage

function (SCDF). In this application, the SCDF contains the actual cost functions for

every type of load in the distribution system.

6.4.1 Economic Evaluations at Substation and Sub-transmission Systems

In this section, the direct economic evaluation of substation and sub-transmission

networks is examined. In this method, the decision to replace non-repairable

equipment is based on the financial considerations using CCDF.

The small sub-transmission case “d” of Chapter 2 is shown in Figure 6.3. Case “d”,

among the other cases, is most suited for this example, as there is a higher level of

interconnection. The circuit breaker (CB) linking Bus 1 and 2 can play an important

role in supplying both loads 1 and 2, via the two branches, in the event that the supply

from one of the branches is not available. As shown in Table 3.4 of Chapter 3, the

effect on the system unavailability changes from 0.4 to 2.5 hours over sixty years,

when only the CB is aging. However, the combined aging of CB with one or both

transformers increases the unavailability to 337 and 808 hours, respectively.

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Similarly, single transformer aging that causes maximum 2 hours unavailability per

year does not pose grave consequence as opposed to the case when both transformers

are aging resulting in 520 hours unavailability per year. These results indicate how

redundancy in a network can improve system performance and reduce dependency of

the load outages on some of the equipment failures. More importantly, this

observation can assist us in making a rational strategy for designing the economic

evaluations.

Figure 6.3 Small sub-transmission, Case “d”.

In view of the preceding review of the system shown in Figure 6.3, a case study with

TX1 and CB aging is considered for the economic application. In this case study,

both TX1 and CB have mean life of 45 and standard deviation of 10 years. The

CCDF for this application is the cost functions provided in this chapter. Other

reliability data are the same as the ones used in Chapter 3 for this system, with 12

MW for each load. The period of study is taken as 14 years. The actual capital

investment for a power transformer and a circuit breaker are taken as $1.9M and

$200k respectively. Using a discount rate of 8% in Equation 6.4, an annual equivalent

investment of $173433 is obtained.

Two EIC values are evaluated for this example, one with TX1 and CB non-repairable

aging and the other with no aging. The difference between the two will give us the

EIC due to non-repairable aging of the two components, as shown in Table 6.3. Once

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the difference exceeds the equivalent investment cost, it is economically time to

replace the components. The difference indicates that the damage cost exceeds the

investment cash flow. As it is seen in this case, the redundancy in a network can

prolong the utilization of components.

year Age EIC-with aging EIC-no aging difference

1 36 37333 22895 144382 37 41131 22895 182363 38 45994 22895 230994 39 52257 22895 293625 40 60370 22895 374756 41 70952 22895 480577 42 84852 22895 619578 43 103249 22895 803549 44 127792 22895 104897

10 45 160802 22895 13790711 46 205575 22895 18268012 47 266825 22895 24393013 48 351339 22895 32844414 49 468952 22895 446057

Table 6.3 Case “d” EIC evaluation for replacement target time, TX1 & CB aging.

Owing to redundancy, the economic evaluation suggests the replacement of TX1 and

CB at age 46 years, in this case study. The basic reliability indices for this case study

are provided in Table 6.4. If the reliability indices exceed the regulators’ setting prior

to the year financially suggested, then replacement time needs to be adjusted

accordingly.

year Age Unavailability (hrs/yr) EFLC (f/yr)

1 36 0.74 0.44

2 37 0.80 0.443 38 0.89 0.444 39 1.01 0.445 40 1.15 0.446 41 1.35 0.447 42 1.61 0.44

8 43 1.95 0.449 44 2.41 0.44

10 45 3.03 0.4511 46 3.87 0.4512 47 5.04 0.4513 48 6.64 0.4514 49 8.88 0.46

Table 6.4 Case “d” major reliability indices, TX1 & CB aging.

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In the following section, Case “d” with both transformers aging is taken under

considerations. The same reliability and financial data is used for this example. An

EIC evaluation for replacement target time, similar to the last case, is developed and

shown in Table 6.5.

In this case study, the actual capital investment for the two transformers is $3.8M.

Therefore, the annual equivalent investment turns out to be $313831. The economic

analysis in this case suggests the replacement in the 9th year of planning, when the

transformers are 44 years old.


1 36 59182 22895 362872 37 70179 22895 472843 38 84556 22895 616614 39 103415 22895 805205 40 128254 22895 1053596 41 161126 22895 1382317 42 204867 22895 1819728 43 263413 22895 2405189 44 342282 22895 319387

10 45 449238 22895 42634311 46 595303 22895 57240812 47 796222 22895 77332713 48 1074590 22895 105169514 49 1463024 22895 1440129

Table 6.5 Case “d” EIC evaluation for replacement target time, TX1 & TX2 aging.


1 30 30236 22895 73412 31 35210 24803 104073 32 44313 28619 156944 33 52379 30526 218535 34 66402 34342 320606 35 80351 36250 441017 36 103569 40066 635038 37 134510 43882 906289 38 183205 49605 133599

10 39 241301 53421 18787911 40 331322 59145 27217712 41 456525 64869 39165613 42 648745 72500 57624514 43 899996 78224 821772

Table 6.6 Case “d” EIC and replacement target time, TX1 & TX2 aging with 10%

yearly load increase.

214

The EIC cost is a function of the load on outage. Therefore, increase load can change

its value. The previous case is reapplied by setting the transformers age at 30 years,

and using a 10% increase in yearly load value and the result is provided in Table 6.6.

Table 6.6 indicates that, the yearly load increase has caused the replacement target

time to move from age 44 to 41 years. As there are no linear relationships between

EIC and the changes in load, outage frequency and duration and system configuration,

the replacement time is case specific.

The same analysis is applied to the medium substation case of Chapter 2, shown in

Figure 6.4. Each load is assigned 12 MW, equivalent to the total loads of IEEE Bus 2

distribution system. In this application, initially the effect of TX2 aging on system

EIC is evaluated. The age of TX2 is taken as 20 years and the planning study is taken

as 14 years. All other data reliability and cost data are taken to be the same as before.

As the non-repairable aging of TX2 only can affect load 2, the EIC results and

evaluations are taken for load 2 only. The annual equivalent investment for TX2 is

found to be $156915. For this case the replacement target time is evaluated and

shown in Table 6.7.

Figure 6.4 Medium substation.

215

The difference between the two EICs, as TX2 is aging and without aging increases

and at the age of 22 exceeds the annual equivalent investment cost. Economically,

this is the year when the replacement should take place.


1 20 195640 73222 1224172 21 229427 73222 1562053 22 270772 73222 1975504 23 320902 73222 2476805 24 381155 73222 3079336 25 452980 73222 3797577 26 537943 73222 4647208 27 637742 73222 5645209 28 754229 73222 681006

10 29 889437 73222 81621511 30 1045632 73222 97241012 31 1225372 73222 115214913 32 1431583 73222 135836114 33 1667665 73222 1594443

Table 6.7 Medium Bus EIC and replacement target time, TX2 aging with 12MW

load.

If the load on transformer TX2 increased to 24 MW, equivalent to IEEE Bus 4, then

the result will change as shown in Table 6.8.


1 15 209383 146445 629382 16 230622 146445 841773 17 257946 146445 1115014 18 292737 146445 1462925 19 336580 146445 1901356 20 391280 146445 2448357 21 458855 146445 3124108 22 541544 146445 3950999 23 641805 146445 495360

10 24 762310 146445 61586511 25 905959 146445 75951412 26 1075885 146445 92944113 27 1275485 146445 112904014 28 1508458 146445 1362013

Table 6.8 Medium Bus EIC and replacement target time, TX2 aging with 24 MW

load.

216

The increased load on TX2 causes the replacement time to change from age 22 to 19.

The frequency and duration reliability indices, corresponding to age 22 and 19, can be

seen in Table 6.9. The reliability indices may suggest advancing the replacement

time, if exceeding the regulatory requirement.

year Age Unavailability (hrs/yr) EFLC (f/yr)

1 15 2.09 0.082 16 2.29 0.083 17 2.55 0.084 18 2.89 0.085 19 3.31 0.086 20 3.84 0.097 21 4.50 0.098 22 5.29 0.099 23 6.26 0.09

10 24 7.43 0.0911 25 8.81 0.0912 26 10.46 0.0913 27 12.38 0.0914 28 14.64 0.09

Table 6.9 Medium Bus major reliability indices, TX2 aging.

The previous case for a medium bus considering the effect of TX2, CB1 and CB2

aging on the replacement target time was evaluated and it was found that the

replacement time remained the same. This is due to number of reasons. Firstly, the

reliability of circuit breakers is higher than transformers. Secondly, the circuit

breakers are further away from load 2. Thirdly, the only supply path to load 2 is

through TX2. The fourth reason is that the replacement time for circuit breakers is

much less than that of transformers. Lastly, the contingency enumeration is increased

to third order, which is important while including non-repairable equipment. All

these factors together are making the load to be much more sensitive to the load

transformers.

In the following example, an advanced breaker and a half substation, as shown in

Figure 6.5, is used for an application. Breaker and a half substation configurations are

commonly designed and used by many utilities around the world. The breaker and a

217

half substation is a highly redundant system. There are many alternative routes from

sources to loads in this configuration, and makes it a very reliable system. In this

example, the same set of data as in medium substation is applied. Each load is

assigned 24 MW, equivalent to the total loads of IEEE Bus 4 distribution system.

Figure 6.5 Advance breaker and a half substation.

The non-repairable aging values and parameters of the circuit breakers in this example

are the same used in Chapter 3. The age of the circuit breakers are assumed to be 35

years. Fourteen years of planning is considered in this example. The loads are

assumed to increase 10% per year.

Transformer non-repairable aging consideration will be very similar to what was

examined in the medium substation. In this example, the non-repairable aging effects

of circuit breakers are examined. There are nine circuit breakers in this system. The

218

investment cash flow for these circuit breakers was calculated and the value is

equivalent to $148657. The same procedure as in previous example is applied and the

results are provided in Table 6.10.


1 35 239665 231344 83212 36 263812 250623 131903 37 300603 279540 210624 38 341203 308458 327445 39 387241 337376 498656 40 452625 375934 766917 41 530475 414491 1159848 42 626202 453048 1731549 43 747562 491606 255957

10 44 938629 549442 38918711 45 1173379 597638 57574112 46 1514667 655474 85919213 47 2013583 722950 129063414 48 2747557 800064 1947493

Table 6.10 Breaker and a half substation, EIC and replacement target time, with

circuit breakers aging.

The economic evaluations suggest that the replacement should take place in the 8th

year of the planning window. The results in Table 6.10 confirm the earlier statement

that the breaker and a half configuration is very reliable system. In this example,

there are four sources in different locations that have access to loads through various

routes.

6.4.2 Economic Evaluations at Distribution System

As discussed before, EIC in the distribution network is evaluated by using the sector

customer damage function (SCDF). Using SCDF can reduce the approximation in the

calculation of expected interruption cost. In this section, the expected frequency and

the average down times at a bulk supply load point to a distribution network, which

were calculated for the higher voltage network, are included in the distribution

reliability and cost evaluation. Through this approach, the effects of non-repairable

219

aging failures in high voltage networks to distribution system are extracted and used

as part of EIC calculations. Then a similar approach, as in the previous section, is

applied to evaluate the replacement time.

Two examples using a medium bus network are provided in this section. This is

applied to the IEEE Bus 2 and IEEE Bus 4 networks. Table 6.11 gives the calculated

frequency and average repair time at load point 2 of the medium bus network, which

is required to be applied to the distribution system.

TX2 Age Planning year Failure probability Failure frequency (f/yr) Unavailability (hr/yr) Average repair time (hr)15 1 0.000238 0.083572 2.09 24.9516 2 0.000261 0.083751 2.29 27.3517 3 0.000292 0.083981 2.55 30.4218 4 0.000330 0.084273 2.89 34.3019 5 0.000378 0.084642 3.31 39.1620 6 0.000439 0.085101 3.84 45.1521 7 0.000513 0.085669 4.50 52.4722 8 0.000604 0.086365 5.29 61.3023 9 0.000715 0.087207 6.26 71.8224 10 0.000848 0.088220 7.43 84.1925 11 0.001006 0.089428 8.81 98.5726 12 0.001194 0.090856 10.46 115.0927 13 0.001414 0.092534 12.38 133.8428 14 0.001671 0.094493 14.64 154.88

Table 6.11 Medium Bus major reliability indices at load 2, as TX2 aging.

A set of EIC values can be obtained, once the frequency and average repair time is

incorporated in IEEE Bus 2 calculations, including a corresponding SCDF. Table

6.12 presents the results considering the non-repairable aging of transformer TX2.

Similar analysis can be conducted to find the economical replacement time of other

equipment in the high voltage network where their failure can cause load outage. If

the failure of a component in the high voltage network has insignificant effects, an

average life can be taken as the replacement time.

TX2 Age year EIC-with aging EIC-no aging difference15 1 309119 193319 11580016 2 319630 193319 12631117 3 335260 193319 14194118 4 355496 193319 162177

Table 6.12 Replacement time by application to RBTS Bus 2 distribution.

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This approach and the data in Table 6.11 is applied to IEEE Bus 4 and the results are

illustrated Table 6.13.

TX2 Age year EIC-with aging EIC-no aging difference

15 1 382994 227762 155231.716 2 399317 227762 171555.517 3 420319 227762 192556.918 4 447059 227762 219296.5

Table 6.13 Replacement time by application to RBTS Bus 4 distribution.

The replacement times in Tables 6.12 and 6.13 are found to be few years different to

those given in Tables 6.7 and 6.8. In one set, EIC is obtained directly in high voltage

network using CCDF, and SCDF is used in the second set to find EIC through the

outage effects in distribution system. Obviously, the SCDF approach using the effect

of outages to distribution system provides accurate results. This approach, in

addition, will provide the risk indices like; SAIDI, SAIFI, CAIDI at distribution level,

that can also measure the effect of aging failures of the high voltage network on the

distribution system. The distribution indices like SAIDI, SAIFI and CAIDI can be

used as risk criterion to make a replacement decision if any one precedes the

economic criteria. The approaches provided in this thesis should assist utilities in

design, planning and operational decisions.

221

6.5 Summary

This chapter was the focal point, where all the methods and techniques developed in

this thesis were brought together with cost analysis. The main objective of power

systems is to provide electric energy to their customers as economically as possible

with an acceptable degree of quality and adequacy. The main theme of this chapter

was to bring economies and reliability together in power systems where non-

repairable aging equipment needs to be included.

The failures in service due to non-repairable aging components can produce a

financial burden both on customers as well as the suppliers of electrical energy. The

approach for replacement of the aging components in this chapter intended to create a

balance between the cost of investment and the cost of outages. Because of this

factor, systematic expected interruption cost, using established composite customer

damage cost and sector customer damage cost functions, were included to produce

meaningful cost values. The blend of cost and reliability evaluations can interactively

give appropriate results that are responsive to both these aspects.

The approach provided in this chapter enables an economic decision to be made with

respect to timing of replacement. However, the regulatory bodies may set a risk

criterion where by a power system will require to perform accordingly. The technique

provided in this chapter can be used to address both avenues of economic and risk

criterion, in an effective way.

The techniques provided in this chapter will enable the utility engineer planners and

operators to evaluate the analyze costs both for high voltage and the effect through

distribution application. Various parameters such as; yearly load growth, SCDF or

CCDF cost functions, non-repairable failure characteristics, and random failures can

222

easily incorporated in this application. The information on costumer types and their

average loads that make up a distribution load are critical as part of the application in

this chapter. A statistical method and evaluation technique was provided in Chapter 4

to extract the required load information.

223

Chapter 7

Software for Substation, Sub-transmission and Distribution

Reliability Evaluations

7.1 INTRODUCTION

As part of this research thesis, two software tools were developed. These are

substation/sub-transmission and distribution reliability programs. The programs are

Graphical Users Interface (GUI) using visual design with Object Orientated

programming principles, developed using Microsoft visual C# (C Sharp).

These programs are designed to facilitate variety of reliability calculations for

substation and distribution systems. All the substation none-repairable/aging models

were incorporated in the software. The effect of substation failure on distribution

system also evaluated using the developed software. More capabilities, including cost

analysis is also considered in the program. Some future extension functionalities, like

Monte Carlo method and data base interfaces, are also envisaged in the software.

Most applications coded in this software are associated with the research work done in

this thesis, and not available in any commercial software. There are few educational

and research provoked software reported in the literature [125-131], starting from

1982. However, these software tools are only designed for the conventional

application and the traditional techniques used in either distribution or substation.

Some of the technical details, such as switching and searching, flow method, etc. are

not addressed, in some of the recent applications. References [126] has applied

simple graphical user interface but with old data entry application which suggest that

the code is not fully object oriented. In addition, the switching and searching

algorithms are not automated and the user is required to enter the details of protection

224

switching actions of every component with active failure, and limited substation

switching arrangements are applicable.

In the following sections brief overview of the software are presented. The technical

descriptions and algorithms in reliability evaluations discussed in chapter Chapters 2,

3 and 5 are coded in the program.

7.2 General Program Flow Chart

Figure 7 .1 High level flow chart gives the overview calculating process of reliability

indices.

Start

All component state probabilities are

calculated

All enumerating system states are

formed

Examine a system state mode and

perform all switching required and perform flow

calculations

Record relevant indices and

information, if any one or more loads

are curtailed, or any source(s)

disconnected

All system states done?

no

yes

End

225

Figure 7.1 provides a very high level flow chart of the principle behind the reliability

evaluation process is presented. The same process is repeated in the event when

planning years, using none-repairable aging is considered. Some in-depth process of

calculations is discussed in Chapters 2, 3 and 5.

In addition to the system reliability evaluation, there is complex and sophisticated

coding and functional processes behind the front end of the GUI program, including

building and integrating the input at user display and interactively communicating

with different parts of the program functions. It is outside the scope of the thesis to

cover details on all coding aspect of the software.

7.3 Software application guide

In this section, snapshots of the software front end is shown and explained. In order

to draw, provide data and proceed to perform evaluation for a case study, series of

logical steps needs to be taken before the user execute the program. The design of

graphical user interface and object oriented coding has made the software user

friendly and flexible in application.

Once the program is run, the user can either open an existing saved file from the

software topside menu bar or open a new file from a dialogue box, as shown in Figure

7.2. The dialogue box has two options for creating a new file, either distribution or

substation. For either option, it is preferred that the user give a referenced name to the

file being created. Once this step is completed, an existing or a newly created file will

appear in a window, as shown in Figures 7.3 or 7.4.

Figure 7.2 Dialogue box for creating a new substation or distribution case study.

226

As shown in Figures 7.3 and 7.4, the tool bars on the left hand side of the screen

contain the components for building a substation or a distribution network. Almost all

components are visually common between the two networks, except that, fuse is an

additional component provided for distribution network. However, the data attributes

of the components, in most cases, are different.

Using the select button, a component can be selected for drawing. By default, the

selection button is always invoked unless a component is selected for drawing. Once

a component is selected and drawn on the screen, the cursor is automatically returns to

the default selection mode for a new component selection. However, if the user wants

to draw same type of component several times, the lock tool button will ease the

drawing process by locking the selection on a component.

As the default drawing of a component is a straight coordination between the two

ends of the component, therefore, creating a large network with many interconnecting

components can pose overlapping drawing problems. Bending button is introduced to

avoid this problem. Using bending option for a component will enable to make the

visual drawing with breaking point(s), where it can be drawn with several segments

each with different angle.

All objects on the menu bar have two points of connections for building a network,

except source, load and bus bar. Source and load have only one connection points, as

a source provides at a node and load consumes from a node point in a network.

Transformer drawing in a network should follow the arrow direction, as oriented from

supply to load point(s). Some components in both networks can be given with limited

capacity for flow method.

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Figure 7.3 Screen view of substation software program, with component menu and

tool bar.

In the GUI part of the software, the bus bar is designed to look schematically like a

bus bar, while having attributes of an object. Therefore, every point on a bus bar is

the same point with infinite connecting point to other components. Design of the bus

bar as described, and including it as part of the network flow in a network was a

challenge in the code.

Figure 7.4 Screen view of distribution software program, with component menu and

tool bar.

Select button

Bus bar

Line

Circuit breaker

Isolator

Normally open switch

Load point

Source

Transformer

Bending point

Lock tool

Fuse

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Component data entry dialogue boxes for substation and distribution networks are

shown in Figures 7.5 and 7.6, respectively. All components, except isolator, normally

open switch and load, in substation software have similar data fields. Data include

random transition rate information and aging data. Non-repairable/aging failure data

for a component is optional and can be provided using either a normal or a Weibull

distribution. Once a distribution is chosen for a component, values of the

corresponding distribution parameters are required in their field. In addition, age and

refurbishment/replacement time of the component also need to be provided, as shown

in Figure 7.5. Isolator and Normally open switches do not require any data, in

substation software.

Figure 7.5 Screen-view of substation components and the data input dialogue boxes,

using graphical users interface.

As noted earlier the distribution software with the exception of fuse, has the same

components as in substation. However, the attributes to some of the objects in

distribution software are different to that of substation. There is no aging option

available for distribution components. The data dialogue boxes for the components in

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distribution software are shown in Figure 7.6. Source and bus bar have similar data

field. Feeder has also similar data, except that the failure rate of a feeder is related to

the length of the feeder. Feeder failure rate is therefore provided in occ/yr/length,

where its length is required. Specifying underground option for a feeder is merely for

visual purpose on the screen. Circuit breaker and isolator have no data requirement

and are assumed to be 100% reliable in distribution network software. For the

purpose of capturing only an individual feeder section output results, transformers

high voltage side between transmission or sub-transmission network and distribution

network can be specified, using the option in transformer dialogue boxes. A

switching time is also required for normally open switch, for distribution software.

Figure 7.6 Screen-view of distribution components and the data input dialogue boxes,

using graphical users interface.

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Similar to substation, the main components in distribution network can be assigned

with a capacity limit value.

Figure 7.7 Screen-view of load point for both distribution substation, and the load

data input dialogue box, using graphical users interface.

The load dialogue box is similar between the substation and the distribution software.

Load and customer numbers are common between the two programs. However, the

percentage load and number of customer growth are not included in distribution

system software yet. Figure 7.7 gives a visual display of a load point and the

corresponding dialogue box.

Figure 7.8 shows a display of most pull down menus on the topside tool bar. Most

command instructions are common to any other software such as; save, cut, copy,

paste, select all and delete. The zoom in, the zoom out and extend are very useful in

view and constructing a network.

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All components in the software have an option of name associated with them. The

names are tagged to the object and are appearing in output results where association is

made. This feature is also very useful in drawing and displaying a network. For

convenience purposes, an option is also considered to hide names on the screen.

Output results are provided in tabular forms both in html and excel format files. After

a system wide simulation run selected load points on the network, such as the ones on

a feeder section, can be calculated to obtain a set of system type indices, separately.

This feature is also very useful for different application and the application can be

executed using the command shown on tool bar menu in Figure 7.8.

Figure 7.8 Screen-view of the program pull-down menu bar and functions.

Another very useful feature in the software is the system wide data entry for both

substation and distribution software programs. This feature is particularly very

valuable and efficient in saving time in data entry for large networks consisting of

many similar type components. System wide data can be overridden at individual

component dialogue box. The snapshot of the dialogue boxes for the substation and

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the distribution system wide data entry are shown in Figure 7.9 (a) and (b),

respectively. Load points deliberately were excluded from system wide data entry as

normally this feature does not apply to them.

(a) (b)

Figure 7.9 Screen-view of system wide component data entry for a) substation and b)

distribution, programs.

The simulation settings shown in Figure 7.10 are as part of data options for substation

program.

(a) (b)

Figure 7.10 Screen-view of simulation settings for a) substation and b) substation &

distribution, programs.

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These options include; number of years of simulation planning, contingency

enumeration options and costs associated with load outage situations. However, the

cost plan is also includes as part of distribution software.

The number of simulation planning are currently designed for substation software

were non-repairable aging components are applied. Contingency enumeration options

are too part of substation software, as multiple contingencies are evaluated. This

option will give the user to see the effect of a desired contingency order on the output

indices. A pat of loss of load outages, a flat revenue loss cost is used in substation

reliability. However, the costing plan associated with customer damage cost function

is a varying cost with respect to average down time of each load type. The data for

cost functions for different customer types are provided in the costing plan dialogue

box.

7.4 Software Output Result Files

As noted earlier, all results for the substation and the distribution examples included

in this thesis were generated using the software being discussed in this chapter. In

most cases, the results were directly imported with the same table formats that were

created in the software output files. The outputs are created in html and Excel format

files. The Excel format file was considered and designed as an option at user disposal

to provide flexibility and ease with any further manipulations of output results. There

are many reliability indices and performance measurements that are produced as part

of the output files. The output results are provided in tabular format as they are

included and shown in Chapter 2, 3 and 5.

In the case of planning year application option, for each planning year an output file is

generated. In addition, a summary of important indices are generated in a file named

major output indices. If many years of planning are chosen by the user, the tabulated

234

results in the major output file are given for every five years. Otherwise, for every

year of planning the important results are displayed in the major output file.

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7.5 Summary

As part of the thesis research work, two software programs were developed, a

substation and a distribution software programs. The software programs are graphical

user interface and the codes are written using Microsoft C sharp Object language.

The software is efficient and user friendly. It includes numerous useful and valuable

application and functionalities. The random failure and non-repairable/aging modes

of components are successfully incorporated for the substation software. There are

many flexible network application and evaluation options that are included in the

software. The software program output indices were tested for approved results. This

application was proved to be very useful and instrumental in this research work.

236

Chapter 8

Conclusions and Future Work

8.1 Introduction

The basic objective of the research work described in this thesis was to examine the

contribution and the effect of non-repairable/aging components in substation or sub-

transmission power networks on bulk load supply points, and customer load points in

distribution systems, including risk of outages as an indication for reinforcement or

replacement decision. In this approach, facilities such as; supply generating units,

transmission line facilities and their protection and control devices like circuit

breakers, bus sections and transformers were included. Amongst these facilities, the

emphasis was put on substation equipment that their non-repairable/aging can have

extensive adverse consequences to customer load points. The studies carried out in

this thesis investigate the area of adequacy and risk of outages as the result of non-

repairable/aging reliability evaluation using frequency and duration approach. The

significant techniques and approaches that have been developed in this research work

are presented in Chapters 3-6. These techniques were utilized to conduct substation

and sub-transmission systems risk assessment, contribution of equipment random and

non-repairable/aging failures to the reliability indices, impact of these failures on bulk

load supply and customer load points, customer load classification and unknown

feeder load decomposition, network redundancy level. These studies are illustrated by

application to published test cases and IEEE Reliability Test Systems.

237

8.2 Summary

Chapter 1 provides the motivation and overview of the thesis research topic. The

aims and the objectives of the research work are then discussed. It is followed by

brief presentation and specification of scope of the thesis. Original contribution of

thesis research work is then summarized. Finally, thesis outline is presented.

The previously published literature on power system reliability evaluation methods is

also reviewed and presented in Chapter 1. The publications show that the methods

associated with power system reliability evaluation are classified by either being

deterministic or a probabilistic. This assessment can be conducted in the two domains

of system adequacy and system security. In adequacy domain, a gap associated with

the contribution of non-repairable/aging equipments to frequency and duration of bulk

and customer load points is recognized. The merits and demerits of the available

literatures relating to this topic are discussed in Chapter 1.

Chapter 2 of this thesis discusses the analytical simulation technique used in

switching facility and substation/sub-transmission reliability evaluation and the basic

concepts, indices and procedures are presented. The detail application of frequency

and duration using Markov state enumeration method is also provided. Two flow

methods, connectivity and maximum flow, are used and the results are shown in this

chapter. As part of the frequency and duration approach, algorithms were developed

to measure the contribution of equipment outages to obtained indices. An extensive

application is performed and the results provided in tabular and graphical

presentations.

Chapter 3 of this thesis developed an extended Markov model for non-

repairable/aging equipments. This model is used to represent the useful or random

failure mode and non-repairable/aging mode of equipment, simultaneously. Set of

equations were developed in this chapter to describe the relationships between the

238

transitional rates and the state probabilities. The equations can describe many forms

of Markov state space diagram, suitable in having options to include or exclude the

equipment states including; aging state, switching state and maintenance state. An

approach is taken to include the aging contingencies as part of the enumeration to

avoid excessive calculations while retaining accuracy. An advantage of this model is

to adopt well with the frequency and duration technique and therefore; the frequency

related indices can also be calculated. In combination with the developments in the

third chapter of this thesis, the contribution of non-repairable/aging equipments are

easily extracted and shown on a yearly basis. This approach, in addition of providing

the information on the importance of equipments in relation to a network and weak

links, it also gives an efficient way of providing results.

As the information on loads in distribution systems are very useful in many planning

applications, an approach is introduced and implemented to identify the customer load

types in Chapter 4. The loads are normally measured at 11KV feeders, nearest to the

electricity customers. However, at this point there is no information on the ingredient

of load types and their profiles. This information can be very important in reliability

and cost/benefit evaluations of electricity networks providing power to load

consumers. In this chapter, an approach is successfully used to facilitate the

decomposition of unknown load profiles measured at 11KV feeders, to the type of

(sub)-sector/customer load profiles contributing to the feeder loads.

The facilities in power systems; generation, transmission and distribution have all one

common goal and that is to provide electrical energy to customers as reliable and as

economically possible. Therefore, it makes much better sense to measure the impact

of reliability performance of the facilities used in a power network at the customer

load points. Chapter 5 discusses and presents distribution reliability evaluations and

uses an approach to present the effect of non-repairable/aging equipments in higher

voltage levels on the distribution customer load points. The effect of aging are shown

as part of the distribution system indices; SAIFI, SAIDI, CAIDI, ASI and ENS. The

239

comparison of the results can provide useful indicators and measures, assisting in

making corrective planning decisions, where cost worth consideration can

systematically be included.

Chapters 3-5 contain the contributions associated with component and system

reliability modeling and evaluations, including non-repairable aging equipment, and

load type and profile estimation. The concepts and the outcomes resulted from those

chapter are brought together and used as part of a risk base economic model

developed and presented in Chapter 6. The research developments prior to Chapter 6

are related to reliability and risk assessments. In the real world, the reliability

evaluations are normally accompanied and dressed with cost and worth appraisals. A

systematic economic method was developed and presented in Chapter 6 to include

both reliability and cost for an optimal planning decision in power systems, where

replacement time can be decided for the non-repairable aging equipment.

The model developed in Chapter 6 is based on reliability and cost worth analysis. In

this model, the cost is associated with the outages that are incurred to customers. This

is done with the inclusion of various outage cost characteristics associated with

different type of customers. The worth is measured as the difference between the

accumulated costs to the customers and the cash flow linked with investment cost.

This method is based on a replacement time policy which include many attributes of a

power network, including; customer outage damage cost based on the outage duration

and average, equipment investment cost, equipment non-repairable aging failure and

replacement/refurbishment characteristics, estimated probability, frequency and

duration of outages, and finally the system configuration.

Chapter 7 of this thesis discusses and presents the software programs developed as

part of the research works. As part of the research activities, two software programs

are developed, substation and distribution software tools. The software tools are

designed as graphical user interface and the codes are written using Microsoft C sharp

240

Object language. The software programs are users friendly and efficient. The

software tools include numerous useful and valuable application and functionalities.

The random failure and non-repairable/aging modes of components are successfully

incorporated for the substation software. There are many flexible network application

and evaluation options that are included in the software programs. The software tools

output indices are tested for approved results. This application was proved to be very

useful and instrumental in this research work.

8.3 Future Research

This research has contributed to several areas of substation/sub-transmission

reliability evaluations, including non-repairable/aging equipments. However, there

are several areas of future work that can be addressed. The areas of future work that

can add value to the techniques presented in this thesis and extend the research work

and possibly open a window to a new research area, is listed as in the followings;

• Monte Carlo approach as an additional technique to complement the approach

used in this thesis is proposed. The Monte Carlo approach although provides

results with some degree of deviations and it is very intensive in calculation

requirements and has considerable computational time, but it has some

advantages. Sequential Monte Carlo approach is such that it can include many

operational considerations. Many indices can easily be calculated and the

distribution of the result indices is obtainable. In addition, data with different

distributions can also be use in this technique.

• Dependency between the maintenance and non-repairable/aging equipments

can be investigated and included as part of the present model. As maintenance

can change the characteristic of failure, this provision can introduce an

optimistic results compare to the existing ones. However, a comprehensive

241

approach and give a tradeoff between the maintenance cost and the reliability

worth.

• This research work provides very useful raw information between the non-

repairable aging equipments and their locations in a network configuration

where they operate. An extended research work can utilize this information to

find an optimal network reconfiguration. As the aging changes with time the

reconfiguration can be dynamic with optimal cost objectives.

• Distribution feeder load type identification is very useful for load

diversification and distribution planning. In addition, the information on load

types can be used for effective assessment in investment cost and reliability

worth analysis. Therefore, the load type prediction used in this thesis can be

improved including new techniques and parameters such as; seasonal

variations, temperature and weather effects.

• As oppose to rural areas, the interconnected distribution systems with many

open points of alternative supply are the candidates for aging equipment

analysis. Appropriate model can be designed to extend non-repairable aging

considerations to this form of distribution systems reliability.

242

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251

APPENDIX A

MAXIMUM FLOW, LABELING ALGORITHM

Three steps are required in computing the maximum flow through a graph [93, 94].

The steps are:

1. Using the labeling algorithm, the nodes of the graph must be labeled in such a

way that a source-load path is identified through which flow can be increased.

If it is not possible to label the load node starting from the source node, the

algorithm terminates and the circuits between the labeled and unlabelled nodes

are minimum cut sets.

2. Using the augmentation algorithm and the labeled graph, flow is augmented

along the source-load paths until the flow cannot be augmented further

because of circuit capacity constraint.

3. Return to step 1.

As can be seen, there are two main routines, the labeling algorithm and augmentation

algorithm. The two routines are detailed in the followings:

A.1 Labeling Algorithm

The labeling algorithm is used to assign labels to the nodes of the graph in such a way

that a path can be traced from the source node to the load node in which the flow can

be increased for all lines along the same path. The methods of the labeling network

are explained in the following:

When a node j is labeled, it is assigned with three values (Aj, Bj, Cj), where;

Aj node number of a labeled node i to which the node being labeled is connected.

Bj indicator of the flow change available to the node being labeled:

Bj = F when the flow can only be increased in the forward direction

Bj = R when the flow can only be decreased in the forward direction

Cj the amount of line flow that can be increased,

If Bj = F, then Cj is the minimum of (Ci, Cm – Lij) (a.1)

252

If Bj = R, then Cj is the minimum of (Ci, Cm + Lji) (a.2)

Where Ci is the flow available at node i,

Cm is the line rating,

Lij is the flow from node i to node j,

Lji is the flow from node j to node i.

Labeling starts at node 1 which is assigned the values (S, F, 99999). The source node

S is now labeled and not scanned. Not scanned means flow is not yet extracted from

this node.

For any labeled and not scanned node (starting with node 1), then the following steps

will be followed:

1. For all unlabeled and not scanned nodes j connected to node i in which line

flow is from i to j and its value is below rating, labeled bus j by (i, F, Cj) where

Cj is given in Equation (a.1).

2. For all unlabeled and not scanned nodes j connected to node i in which line

flow is from j to i and its value below rating, labeled bus j by (I, R, Cj) where

Cj is given in Equation (a.2).

3. Mark node i to show that it has been scanned.

4. Repeat 1 to 3 until the load node is labeled or no more labels can be assigned.

The augmentation routine as in the following section can then be followed when a

complete labeled path can be found from the source to the load node, and if the flow

can be increased. Otherwise, the maximum flow routine is terminated.

A.2 Augmentation Routine

All line flows are initially assigned zero values. The augmentation routine is the

procedure which actually increases the flow in the lines based on a complete path of

the labeled nodes. The process starts from the last node (load node) that is labeled

and the flow in the lines are increased as in the following steps:

253

1. If the label on the node is (Aj, F, Cj), then increase the flow through the line

connecting node j to node Aj by Cj. However, if the label on the node is (Aj,

R, Cj), then decrease the flow through the same line by Cj.

2. Advance to node number Aj and repeat step 1 until Aj equals to the source

node.

3. Erase all labels and return to the labeling algorithm.

As noted earlier, if labeling the graph from the source to the load is not possible, then

the maximum flow algorithm terminates and therefore, the flow in the lines

connecting the load point to the load node is the maximum load that can be supplied

at the corresponding load points. For these lines, the difference between the line flow

and the rating equals the amount of load not supplied for the load point connected.

The maximum flow method can be easily applied to produce results similar to that of

the minimal cut set method. This can be achieved by simply setting the circuit rating

to very high value, i.e. no limit on the circuit rating.

254

APPENDIX B

SINGLE LINE DIAGRAMS OF THE RELIABILITY TEST

SYSTEMS

Figure B.1: Single Line Diagram of the RBTS

1×40 MW 4×20 MW 2×5 MW

2×40 MW 1×20 MW 1×10 MW

255

SP1

a b c

F1 F2 F3 F4

12

3

45

6

78

9

10

11

12

13

14

15

16

17

1819

20

2122

23

24

25

26 27

28

29 30

31

32 33

34 35

36

Figure B.2: Single Line Diagram of the Distribution System of the RBTS Bus 2

256

T1 T2

T3

T4

T5

T6

F1 F2 F3

F4

F5

F6

F7

12

34

5

6 7

8

9

10

1112

13

14

15

16

1718

19 20

21

22 23

24

25

26

272829

30

3132

33

34

35

36

37

38

39

40 41 42

43

4445

46

47

48

49

50

51

52

53

54 55

56

575859

60

61

62 63

64

65

66

67

SP1

SP2

SP3

Figure B.3: Single Line Diagram of the Distribution System of the RBTS Bus 4

257

DISTRIBUTION NETWORK OF THE RBTS: SYSTEM DATA

Customer type Peak loads (MW) Bus 2 Bus 4 Bus 6

Residential 7.25 19.00 12.1147

Small Industrial 3.50 16.30 07.8004

Governmental/Institutions 5.55 ------ ----------

Commercial 3.70 04.70 ----------

Farms ---- ------ ----------

Totals 20.00 40.00 20.0000

Table B.1: Peak Loads in the RBTS Length Feeder section number (km)

BUS 2:

0.60 2, 6, 10, 14, 17, 21, 25, 28, 30, 34 0.75 1, 4, 7, 9, 12, 16, 19, 22, 24, 27, 29, 32, 35 0.80 3, 5, 8, 11, 13, 15, 18, 20, 23, 26, 31, 33, 36

BUS 4:

0.60 2, 6, 10, 14, 17, 21, 25, 28, 30, 34, 38, 41, 43, 46, 49, 51, 55, 58, 61, 64, 67

0.75 1, 4, 7, 9, 12, 16, 19, 22, 24, 27, 29, 32, 35, 37, 40, 42, 45, 48, 50, 53, 56, 60, 63, 65

0.80 3, 5, 11, 13, 15, 18, 20, 23, 26, 31, 33, 36, 39, 44, 47, 52, 54, 57, 59, 62, 66

Table B.2: Feeder Lengths

258

No. of load customer average peak No. of load points type load point load point customers

points MW MW

BUS 2:

5 1-3, 10, 11 Res 0.535 0.8668 210 4 12, 17-19 Res 0.450 0.7291 200 1 8 Ind 1.000 1.6279 1 1 9 Ind 1.150 1.8721 1 6 4, 5, 13, 14, 20, 21 Gov & Ind 0.566 0.9167 1 5 6, 7, 15, 16, 22 Com 0.454 0.7500 10

Totals 12.291 20.0000 1908

BUS 4:

15 1-4, 11-13, 18-21, 32-35

Res 0.545 0.8869 220

7 5,14,15,22,23,36,37 Res 0.500 0.8137 200 7 8, 10, 26-30 Ind 1.000 1.6300 1 2 9, 31 Ind 1.500 2.4450 1 7 6, 7, 16, 17, 24, 25,

38 Com 0.415 0.6714 10

Totals 24.580 40.0000 4779

Table B.3: Customer Data

259

feeder load average peak No. of number points load load customers

(MW) (MW)

BUS 2:

F1 1-7 3.645 5.934 652 F2 8-9 2.150 3.500 2 F3 10-15 3.106 5.057 632 F4 16-22 3.390 5.509 622

BUS 2 TOTALS 12.291 20.000 1908

BUS4:

F1 1-7 3.510 5.704 1100 F2 8-10 3.500 5.705 3 F3 11-17 3.465 5.631 1080

SP1 Totals 10.475 17.040 2183

F4 18-25 4.010 6.518 1300 F5 26-28 3.000 4.890 3

SP2 Totals 7.010 11.408 1303

F6 29-31 3.500 5.705 3 F7 32-38 3.595 5.847 1290

SP3 Totals 7.095 11.552 1293

BUS 4 TOTALS 24.580 40.0000 4779

Table B.4: Loading Data

260

Component λp λa λt λm r rp rm Tsw

occ/yr occ/yr occ/yr occ/yr hour hour hour hour

transformers:

138/33 0.010 0.010 0.05 0.5 15 168 1 33/11 0.015 0.015 0.05 1 15 120 1

11/0.415 0.015 0.015 200 10 1(OHL) 3(UGC)

33/0.415 0.015 0.015 200 10 1(OHL) 3(UGC)

breakers:

138 0.0058 0.0035 0.05 0.2 8 108 1 33 0.0020 0.0015 0.02 0.5 4 96 1 11 0.0060 0.0040 0.06 1 4 72 1

busbars:

33 0.001 0.001 0.01 0.5 2 8 1 11 0.001 0.001 0.01 1 2 8 1

lines (single weather state):

33 0.046 0.046 0.06 0.5 8 8 2 11 0.065 0.065 5 1

33 kV lines (two weather states):

normal 0.0139 0.0139 0.018 0.5 8 8 2 adverse 5.86 5.86 7.6 cables:

11 0.04 0.04 30 3

Table B.5: Reliability and System Data

recloser time =0.083 hr average duration of normal weather =724 hr average duration of adverse weather = 4 hr line failures occurring in adverse weather = 70% of total 33kV line lengths: SP1-SP2 and SP2-SP3 = 10 km; SP1-SP3 = 15 km Transformer ratings: SP1 (BUS 4), SP (BUS 2) = 16 MVA each SP2 and SP3 (BUS 4) = 10 MVA each λp = permanent failure rate (occ/yr) [for lines/cables (occ/yr.km)] λa = active failure rate (occ/yr) [for lines/cables (occ/yr.km)] λt = temporary failure rate (occ/yr) [for lines/cables (occ/yr.km)] λm = maintenance outage rate (occ/yr) r = repair time (hr) rp = replacement time by a spare (hr) rm = maintenance outage time (hr) Tsw = switching time (hr)

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