robust and systemwide fault location in …1431/...the department of electrical and computer...

ROBUST AND SYSTEMWIDE FAULT LOCATION

IN LARGE-SCALE POWER NETWORKS

VIA OPTIMAL DEPLOYMENT OF

SYNCHRONIZED MEASUREMENTS

A Dissertation Presented

by

Mert Korkalı

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in

Electrical Engineering

Northeastern University

Boston, Massachusetts

December 2013

http://www.korkali.com

http://www.ece.neu.edu/

http://www.northeastern.edu/

c© copyright by Mert Korkalı 2013

All Rights Reserved

Northeastern University

Abstract

Department of Electrical and Computer Engineering

Doctor of Philosophy in Electrical Engineering

by Mert Korkalı

http://www.northeastern.edu/

http://www.korkali.com

iii

This dissertation addresses a novel method for fault location in power systems, while

providing a new vision for the deployment of wide-area measurement systems and

the application of robust estimation techniques in an effort to achieve systemwide,

cost-effective, and resilient fault-location capability in large-scale power systems.

The first part of this dissertation introduces a novel methodology for synchro-

nized-measurement-based fault location in large-scale power grids. The method is

built on the notion of traveling waves that propagate throughout the power network.

The approach is based upon capturing the arrival times of the fault-initiated traveling

waves using a few synchronized sensors and triangulating the location of the fault

with the aid of the recorded arrival times of these waves. In order to pinpoint (locate)

the faults, these sparsely distributed sensors are exploited to capture point-on-wave

samples of transient voltages after the occurrence of a fault.

The second stage of this dissertation complements the fault-location system

developed in the first part of the study. The optimal deployment strategy for syn-

chronized measurements is devised in such a way that the power grid is rendered

observable from the viewpoint of fault location. Accordingly, the concept of fault-

location observability is described and the restrictive cases, which may exist due to

system topology as well as transmission-line lengths and lead to the occurrence of

fault-unobservable segments (blind spots) on transmission lines, are illustrated with

examples.

The final part of this dissertation harnesses the results of the previous parts

of the dissertation so as to make the fault-location capability of the power grid

robust against unwanted changes in synchronized measurements, which may occur

as a result of sensor failures and measurement tampering due to cyberintrusions,

thus adversely affecting reliable fault-location estimation. Two bad-data processing

techniques that enable the fault-location scheme to remain insensitive to corruption

of data in a certain number of redundant measurements are introduced.

Acknowledgments

I am deeply indebted to my mentor and research adviser, Professor Ali Abur, for an

incomparably rewarding educational and personal experience. I have been indescrib-

ably enlightened and inspired by his patient teaching and vast technical expertise.

His constant support, gentle guidance, and warm encouragement gave a positive

impetus to the successful completion of my dissertation. His inspiring ability to

treat problems from a new perspective integrated with many hours of constructive

discussions were the main factors of the progressive improvements in this disserta-

tion. Indeed, being a research assistant to him will definitely fortify my competence

to stay in the forefront of my research area. I have always been greatly impressed

by his fatherly attitude and gracious personality, which turned my long and often

arduous doctoral journey into an unexpectedly memorable and pleasant one. I will

always strive to incorporate his professional and personal qualities into my academic

personality.

In the meantime, this is an opportunity to thank some of the people who have

shaped my academic personality prior to my arrival to Northeastern. Special thanks

go to my undergraduate advisor, Professor Bulent Bilir, for his invaluable support

and incessant encouragement throughout my studies at Bahcesehir University and

Professor H. Fatih Ugurdag for his irreplaceable endeavor that undoubtedly paved

the way for my being a graduate student in the United States. I am more than grate-

ful for experiencing a mentor–younger friend relationship as well as an instructor–

student relationship with them. As a matter of fact, I am really fortunate to have

been in close contact with Professor Bilir during his sabbatical at Northeastern. He

always helped me greatly, not just in shaping my professional career path in the

United States, but in every aspect of life as a caring and an encouraging mentor.

I would like to express my heartfelt gratitude to Professor Hanoch Lev-Ari and

Professor Aleksandar M. Stankovic not only for serving on my dissertation committee

and for their insightful comments on my work, but also for giving me an inspiration

with their immense knowledge in their areas of expertise. I am very grateful to

Professor Lev-Ari for his exceptionally elegant contribution to the development of

the fault-location methodology that shed light on most of the advancements in this

dissertation.

iv

v

I feel very fortunate to meet with great friends during my graduate studies.

I would also like to thank my officemates at Northeastern—Liuxi (Calvin) Zhang,

Murat Gol, Alireza Rouhani, and Cem Bila—for their great friendship, help, and

support, and for all the fun times we spent together. Special thanks go to my friends

Cihan Tunc, Umut Orhan, Surasak (Fa) Chunsrivirot, Seyhmus Guler, Kıvanc Kerse,

Adnan Korkmaz, and Ye Zhao for taking part in the enjoyable moments of my

doctoral years in Boston.

I would also like to dedicate my dissertation work to Professor Yaman Yener,

Senior Associate Dean of Engineering for Faculty Affairs at Northeastern, who passed

away on Friday, June 14, 2013. He was a true inspiration for and a father figure

of Turkish students not only at Northeastern, but also in the Greater Boston area.

He will always remain a role model in the many lives he touched (like mine). His

priceless effort that allowed me to pursue a worthwhile academic career will always

be remembered as one of the cornerstones of my lifetime. I feel very fortunate to

have known such a great scholar during my years at Northeastern, prior to his early

departure from this world. He will be remembered by me and many others with

bottomless affection.

Finally, my deepest appreciation and love is reserved for my parents, Hasan and

Selma Korkalı, for their endless support and love, and for making me who I am.

Their love embraces me everywhere despite the long geographic distance between

us.

The research documented in this dissertation was supported in part by the Na-

tional Science Foundation (NSF) Grant ECCS-08-24005, and by Grant #2574520-

47177-A from the Global Climate and Energy Project (GCEP) at Stanford Univer-

sity. The work made use of Engineering Research Center Shared Facilities supported

by the Engineering Research Center Program of the NSF and the Department of

Energy under NSF Award #EEC-1041877 and the CURENT Industry Partnership

Program.

Contents

Abstract ii

Acknowledgments iv

List of Figures x

List of Tables xii

1 Introduction 1

1.1 Motivations for the Study . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Contributions of the Dissertation . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Wide-Area Synchronized-Measurement-Based Fault Location . 5

1.2.2 Optimal Sensor Deployment for Fault-Location Observability . 6

1.2.3 Robustification of Fault-Location Technique . . . . . . . . . . 6

1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Fault Location in Power Networks 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 General Classification of Fault-Location Methods . . . . . . . . . . . 10

2.3 Accuracy of Fault-Location Algorithms [55] . . . . . . . . . . . . . . . 12

2.4 Use of Traveling Waves for Fault Location . . . . . . . . . . . . . . . 13

2.5 Emerging Use of Synchronized Measurements for Fault Location . . . 16

2.6 Key Features of the Proposed Fault-Location Strategy . . . . . . . . 18

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Analysis of Electromagnetic Transients 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Traveling-Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Traveling-Wave Velocity and Characteristic Impedance . . . . 23

3.2.2 The Telegrapher’s Equations . . . . . . . . . . . . . . . . . . . 25

vi

Contents vii

3.2.3 The Lossless Line . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Transmission-Line Models for Transient Analysis . . . . . . . . . . . . 30

3.3.1 Bergeron’s Tranmission-Line Model . . . . . . . . . . . . . . . 32

3.3.2 Frequency-Dependent Transmission-Line Model . . . . . . . . 34

3.4 Numerical Transient Analysis . . . . . . . . . . . . . . . . . . . . . . 38

3.4.1 Companion Equivalents of Circuit Elements Based on Trape-zoidal Integration . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1.1 Resistance . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1.2 Inductance . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1.3 Capacitance . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.2 Computation of Transients in Linear Networks . . . . . . . . . 41

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Wavelet Analysis—Fundamentals and Its Applications in PowerSystems 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 A Motivation for Wavelets . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Wavelet Transforms versus Fourier Transforms . . . . . . . . . 47

4.2.2 The Short-Time Fourier Transform (STFT) . . . . . . . . . . 48

4.2.3 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . . 50

4.3 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.1 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 The Continuous Wavelet Transform (CWT) . . . . . . . . . . 55

4.3.3 The Wavelet Series . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.4 The Discrete Wavelet Transform (DWT) . . . . . . . . . . . . 58

4.3.4.1 Multiresolution Analysis . . . . . . . . . . . . . . . . 58

4.3.4.2 Wavelet Analysis by Multirate Filtering . . . . . . . 60

4.3.4.3 Wavelet Synthesis by Multirate Filtering . . . . . . . 61

4.3.4.4 The Relationship between Wavelets and Filters . . . 63

4.4 Applications of Wavelet Analysis in Power Systems . . . . . . . . . . 66

4.4.1 Applications in Power Quality . . . . . . . . . . . . . . . . . . 66

4.4.2 Applications in Analysis of Power-System Transients . . . . . 70

4.4.3 Applications in Power System Protection . . . . . . . . . . . . 71

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Traveling-Wave-Based Fault Location in Power Networks 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Proposed Fault-Location Methodology . . . . . . . . . . . . . . . . . 76

5.2.1 The Functions “ζk,`(α(`))” . . . . . . . . . . . . . . . . . . . . 78

5.2.2 A Nonlinear Optimization Problem . . . . . . . . . . . . . . . 81

5.2.2.1 A Two-Step Optimization Approach . . . . . . . . . 82

5.2.2.2 A Sensor-Guided Line-Splitting Approach . . . . . . 83

Contents viii

5.3 Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.1 Fundamentals and Stages of the Implementation . . . . . . . . 87

5.3.2 Computation of the Shortest Propagation Delays . . . . . . . 88

5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Optimal Deployment of Synchronized Sensors Based onFault-Location Observability 99

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 Proposed Formulation for Optimal SensorDeployment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.1 Fault on an “Observable” Line Segment . . . . . . . . . . . . 105

6.3.2 Fault on an “Unobservable” Line Segment . . . . . . . . . . . 108

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Enhanced Robustness via Least-Absolute-Value Estimation andLargest-Normalized-Residual Test 115

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2 Measurement Imprecision due to Low Sampling Rate . . . . . . . . . 118

7.3 Treatment of Sensor Measurements Containing Gross Errors . . . . . 118

7.3.1 Bad Data Identification via Least-Absolute-Value (LAV) Es-timation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3.2 Bad Data Identification via Largest-Normalized-Residual(rNmax

)

Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3.2.1 Detection and Identification of Bad Sensor Measure-ments . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3.2.2 Elimination/Correction of Identified Bad Measure-ments . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.4.1 Measurement Imprecision due to Low Sampling Rate . . . . . 126

7.4.2 Measurements Containing Gross Errors . . . . . . . . . . . . . 128

7.4.2.1 Identifying Erroneous Measurements via LAV Esti-mation . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.4.2.2 Identifying Erroneous Measurements via rNmax Test . 132

7.4.3 Limiting Cases of the Proposed Method Under CoordinatedCyberattacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8 Concluding Remarks and Further Study 143

8.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.1.1 Wide-Area Synchronized-Measurement-Based Fault Location . 144

Contents ix

8.1.2 Optimal Deployment of Synchronized Sensors for Wide-AreaFault Location . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.1.3 Robust Estimation of Fault Location . . . . . . . . . . . . . . 145

8.2 Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.2.1 Investigation of Novel Time-Frequency Methods for TransientAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.2.2 Simultaneous Occurrence of Multiple Line Faults . . . . . . . 146

8.2.3 Inclusion of Additional Network Components . . . . . . . . . 146

8.2.4 Simulation of Various Fault Types . . . . . . . . . . . . . . . . 147

8.2.5 Methods to Mitigate the Effect of Attenuated Traveling Waves 147

8.2.6 Line Modeling and Transient Simulations via Wavelet-LikeTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

A Modal Analysis of Multiphase Transmission Lines 148

A.1 Transmission-Line Equations . . . . . . . . . . . . . . . . . . . . . . . 148

A.2 Modal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.3 Balanced Transformations for Transposed Lines . . . . . . . . . . . . 152

A.3.1 Karrenbauer’s Transformation . . . . . . . . . . . . . . . . . . 152

A.3.2 Clarke’s Transformation . . . . . . . . . . . . . . . . . . . . . 153

B Derivation of A1 and A2 Used in Chapter 3 154

C Proof of Unique Localizability in a Fault-Unobservable Branch 156

Bibliography 158

Vita 175

List of Publications 176

List of Figures

2.1 Lattice diagram for a fault located at a distance x from Bus A. . . . . 14

2.2 Depiction of synchronized-measurement-based fault location utilizingthe theory of traveling waves. . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Classification of power-system transients [56]. . . . . . . . . . . . . . 22

3.2 Heaviside’s model of the differential-length transmission line. . . . . . 25

3.3 Forward- and backward-traveling waves along with their polarity. . . 30

3.4 Decision tree for transmission-line model selection [60]. . . . . . . . . 31

3.5 Equivalent two-port model for a lossless transmission line betweenTerminals s and r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Equivalent circuit representations of (a) resistor, (b) inductor, and(c) capacitor placed between Terminals s and r. . . . . . . . . . . . . 40

3.7 Generic node of a linear network. . . . . . . . . . . . . . . . . . . . . 43

4.1 Fixed-resolution time-frequency planes: (a) narrow window enablesa better time resolution, and (b) wide window enables a better fre-quency resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Multiresolution time-frequency plane. . . . . . . . . . . . . . . . . . . 51

4.3 Examples of common wavelet functions: (a) Haar, (b) Mexican hat,(c) Morlet, (d) Daubechies-8, (e) Meyer, and (f) Gaussian wavelet. . 54

4.4 Analysis filter bank for computing the DWT. . . . . . . . . . . . . . . 61

4.5 Synthesis filter bank for the DWT. . . . . . . . . . . . . . . . . . . . 63

5.1 Illustration of the terms, “origin” and “terminus”, as well as the re-spective propagation delays, D(o)

k,` and D(t)k,`, along the shortest path

from Sensor “k” to faulty Line “`”. . . . . . . . . . . . . . . . . . . . 79

5.2 The intersection of the lines “D(o)k,` + α(`)D`” and “D(t)

k,` + (1− α(`))D`”. 80

5.3 The function ζk,`(α(`)) when (a) βk,` = 0 and (b) βk,` = 1. . . . . . . 82

5.4 The virtual nodes generated at the points “βki,`D`”. . . . . . . . . . . 83

5.5 Computational stages of the devised fault-location algorithm. . . . . . 89

5.6 Single-line diagram of the modified IEEE 30-bus test system. . . . . . 92

5.7 Faulted phase voltages at Buses 1, 17, 21, and 29 after the occurrenceof a short-circuit fault on Line 10-20. . . . . . . . . . . . . . . . . . . 93

x

List of Figures xi

5.8 WTC2s of the aerial-mode voltages at Buses 1, 17, 21, and 29 afterthe occurrence of a short-circuit fault on Line 10-20. . . . . . . . . . . 94

5.9 The value of %(`) for the short-circuit fault occurring on Line 10-20. . 96

5.10 The value of %(`) for the short-circuit fault occurring on Line 12-15. . 97

6.1 Single-line diagram of the modified IEEE 57-bus test system (lengthsof branches are not scaled in proportion to actual line lengths). . . . . 105

6.2 Three-phase voltages and WTC2 of the aerial-mode voltage at Bus 33after the occurrence of a short-circuit fault on Line 24-26. . . . . . . . 107

6.3 Value of %(`) for the short-circuit fault occurring on Line 24-26. . . . . 108

6.4 Unobservable segments (lengths and travel times being designated) ofthe three transmission lines. . . . . . . . . . . . . . . . . . . . . . . . 110

6.5 Values of %(`) for the two short-circuit faults occurring on Line 7-8. . . 111

6.6 WTC2s of the aerial-mode voltages at Bus 7 for the fault events on(a) the near-half and (b) the remote-half unobservable segments ofLine 7-8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1 Single-line diagram of the modified IEEE 118-bus test system (arclengths and actual line lengths are not proportionally scaled). . . . . 125

7.2 Location of a fault occurring at 99 miles away from Bus 63 whichis formed via optimally deployed sensors in presence of (a) roundingerrors (with 20-kHz sensors) and (b) gross errors (with 1-MHz sen-sors). (c) Reestimated location after the elimination of sensors whichare contaminated by gross errors. . . . . . . . . . . . . . . . . . . . . 127

7.3 Convergence of minimum value J min` of the objective function J`. . . 134

7.4 Value of %(`) for the short-circuit fault occurring on Line 47-69. . . . . 135

7.5 Flowchart of the rNmax-Test-based bad-measurement identification al-gorithm for fault location. . . . . . . . . . . . . . . . . . . . . . . . . 136

7.6 Actual location of an “unidentifiable” fault occurring on Line 70-71due to a cyberattack. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.7 Flowchart of the “robustified” fault-location algorithm against com-promised sensor measurements. . . . . . . . . . . . . . . . . . . . . . 141

A.1 Phases a, b, and c of a three-phase transmission line between Termi-nals s and r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

List of Tables

5.1 Transmission-Line Lengths and Wave-Propagation Times for the Mod-ified IEEE 30-Bus Test System . . . . . . . . . . . . . . . . . . . . . 91

5.2 Synchronized Meter Locations versus Wave-Arrival Times for theShort-Circuit Fault Occurring on Line 10-20 . . . . . . . . . . . . . . 91


6.2 Synchronized Measurement Locations versus Wave-Arrival Times forthe Short-Circuit Fault Occurring on Line 24-26 . . . . . . . . . . . . 107

6.3 Lengths and Propagation Times of Modified Transmission Lines . . . 109

6.4 Locations of Synchronized Measurements versus Wave-Arrival Timesfor the Short-Circuit Fault Occurring on the Near-Half UnobservableSegment of Line 7-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.5 Locations of Synchronized Measurements versus Wave-Arrival Timesfor the Short-Circuit Fault Occurring on the Remote-Half Unobserv-able Segment of Line 7-8 . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.6 Values of J min` , `, %(`) (in ms), T

(`)0 (in ms), xfaultest (in mi), and

xfaultcorr (in mi) for the Faults Occurring on Near- and Remote-HalfUnobservable Segments of Line 7-8 . . . . . . . . . . . . . . . . . . . 114


7.2 Transmission-Line Lengths and Wave-Propagation Times for the Mod-ified IEEE 118-Bus Test System (continued from Table 7.1) . . . . . . 124

7.3 Wave-Arrival Times for the Fault Occurring at 99 Miles Away fromBus 63 When 20-kHz Fault-Recording Sensors are Utilized . . . . . . 126

7.4 Synchronized Sensor Locations versus Wave-Arrival Times for theFault Occurring at 99 Miles Away from Bus 63 on Line 63-64 . . . . . 128

7.5 Values of `,∥∥r(`)

LAV

∥∥1, %(`) (in ms), T

(`)0 (in ms), and D` (in ms) for the

Fault Occurring at 99 Miles Away from Bus 63 on Line 63-64 . . . . . 130

7.6 Synchronized Sensor Locations versus Wave-Arrival Times (after cor-recting bad measurements) for the Fault Occurring at 99 Miles Awayfrom Bus 63 on Line 63-64 . . . . . . . . . . . . . . . . . . . . . . . . 131

xii

List of Tables xiii

7.7 Synchronized Sensor Locations versus Wave-Arrival Times for theFault Occurring at 62 Miles Away from Bus 47 on Line 47-69 . . . . . 132

7.8 Changing values in each iteration of rNmax Test . . . . . . . . . . . . . 133

7.9 Synchronized Sensor Locations versus Wave-Arrival Times (after cor-recting bad measurements) for the Fault Occurring at 62 Miles Awayfrom Bus 47 on Line 47-69 . . . . . . . . . . . . . . . . . . . . . . . . 134

7.10 Synchronized Sensor Locations versus Wave-Arrival Times for theFault Occurring at 100 Miles Away from Bus 70 on Line 70-71 . . . . 138

7.11 Values of %(334), T(334)0 , and d with respect to Contaminated Measure-

ments at Sensor 20 for the Fault Case on Line 70-71 . . . . . . . . . . 139

7.12 Miscellaneous Results for the Studied Test Grid . . . . . . . . . . . . 140

7.13 Values of `,∥∥r(`)

LAV

∥∥1, %(`), T

(`)0 , and D` for the Fault Occurring at 100

Miles Away from Bus 70 on Line 70-71 . . . . . . . . . . . . . . . . . 140

To my beloved parents,

Selma and Hasan Korkalı,

with love and gratitude

xiv

“There is a Turkish proverb to the effect that the world

belongs to the dissatisfied.

... the one great underlying principle of all human

progress is that ‘divine discontent’ which makes men

strive for better conditions and improved methods.”

—Charles P. Steinmetz

xv

Chapter 1

Introduction

Operation of the existing power grids is rapidly going through major changes due to

the widespread deployment of synchronized measurement systems. These systems

provide unprecedented advantages in wide-area monitoring of power grids due to

the availability of synchronization among measurements at geographically remote

parts of the system. So far, most of the investigations have focused on the use

of synchronized measurements to improve applications, which require monitoring

and control actions at relatively slow rates, i.e., slow enough to make treatment of

slowly changing system conditions via phasors, which implicitly assume steady-state

operation.

While effectiveness and benefits of synchronized measurements have been well-

documented for such applications, other applications requiring monitoring of the

system conditions at a much shorter time span have not yet been fully explored.

One of the challenges in secure and reliable operation of power grids is to rapidly de-

tect, identify, and isolate faults, which occur due to unexpected equipment failures,

lightning storms, accidental short circuits, etc. Such faults can cause significant dam-

age if not cleared in a matter of fractions of a second. Hence, power-grid protection

systems have so far been designed as control systems that used local measurements

1

Chapter 1. Introduction 2

as decision variables. In this regard, power systems would fully exploit the potential

of synchronized nature of the sampled voltage and current signals and the capability

to access these synchronized values via systemwide communication infrastructure,

enabling accurate and fast methods of fault location and removal.

The newly coined “smart grid” technology embodies several key characteris-

tics, including the capability to self-heal from disturbances, being resilient against

physical and cyberattacks, and having an advanced sensing and intelligent commu-

nications infrastructure. Self-healing feature of a smart grid enables faulty elements

to be detected, isolated, and restored so as to minimize service interruptions as well

as to improve system security in response to malicious attacks. Fault (disturbance)

location and dynamic system instabilities can be identified and immediate corrective

actions can be initiated.

Presently, power grids depend a great deal on the use of synchronized mea-

surement technology that collect synchronized data from multiple nodes to provide

wide-area visibility of system disturbances. In other words, simultaneous data cap-

turing at various locations on the network provides a reliable analysis of system

performance during power-grid faults. In addition to the recent advances in syn-

chronized wide-area measurement systems, improved data-capturing and -processing

capabilities in smart grids facilitate the use of recently introduced signal-processing

techniques. These techniques allow the analysis of sampled waveforms with localized

transients, specifically allowing precise time localization of transient components of

a signal. In this way, needed time information can be collected via the processing

of high-resolution samples of fault-originated signals screened at strategic points on

the power system.


1.1 Motivations for the Study

A power system fault, if not detected or cleared in a sufficiently short amount of time

after its inception, is likely to spread progressively into a systemwide disturbance,

eventually leading to further widespread system outages. Though protection schemes

for transmission systems have been well designed for identifying the location of faults

as well as for isolating the faulty section, reliable event detection and rapid system

restoration even under the threat of compromised security remain a key challenge

for systemwide protection.

Recently, the irrepressible growth in complex power-grid topologies has re-

vealed not only the undeniable importance of corrective measures needed for coun-

teracting the propagation of impulsive changes in the system, but the underlying

challenges of the reliable analysis of fast-growing disturbance events comprising a

large area of the network. On that account, the development of resilient and reliable

protection schemes and efficient emergency actions constitutes the integral part of a

reliable wide-area protection system in evolving power grids.

Wide-area power system protection has been investigated since the first in-

troduction of synchronized measurements. New methods typically aim to optimize

systemwide protection actions in order to avoid cascading outages. These methods

have so far been mainly focused on the utilization of phasor measurements which are

increasingly becoming available in particular in high voltage transmission systems.

In this work, synchronized raw samples (point-on-wave) are assumed to be available

for processing by the protection applications. Traveling-wave-based methods can

then be employed in order to exploit these samples and estimate the location of the

fault in an accurate manner.

Widespread and cost-efficient deployment of the phasor measurement units

(PMUs) allowing for enough redundancy and wide penetration of synchronized mea-

surements is a critical enabler for increasing the wide-area protection capability of the


power grid. Such an enhancement in wide-area protection systems can evidently be

realized by integrating PMU functions with protection and control platforms. Specif-

ically, by collecting raw data from samples of the analog transient signals, synchro-

nized measurement devices can be exploited as high-data-rate disturbance recorders

by virtue of intelligent electronic devices (IEDs) (e.g., microprocessor-based fault

recorders) integrated with synchronized measurement capabilities. Furthermore, a

reliable wide-area disturbance-recording system necessitates the capturing of simul-

taneous recordings of fast-changing dynamic events from PMU-capable IEDs (e.g.,

synchronized sensors) at multiple substations across the transmission grid, thereby

enabling a highly accurate postdisturbance analysis.

With the integration of PMUs, the disturbance recorders can be exploited for

their ability to synchronously measure the quantities such as transient voltages and

currents with high-resolution samples allowing for the highest measurement accu-

racy, thus enabling fast and dependable analysis and localization of power-system

disturbances. Particularly, multiple synchrophasor-enabled IEDs dispersed at se-

lected substations across the electrical grid can provide for the acquisition of key

information forming the systemwide picture of dynamic events through continuous

monitoring of wide-area recordings of transient disturbances. Possessing a high level

of computational capability and standing upon high-speed communications infras-

tructure, emerging modern protection devices can execute a myriad of advanced pro-

cessing algorithms based on collation of data from widely separated meters, where

event records are captured.

Wide-area protection involves the use of systemwide information collected over

a wide geographic area to perform fast decision-making and switching actions in

order to counteract the propagation of large disturbances. The advent of PMUs has

transformed protection from a local concept into a system-level wide-area concept

to handle disturbances. The inherent wide-area nature of the protection schemes

presents several vulnerabilities in terms of possible cyberintrusions to hinder or alter

the normal functioning of these schemes. Even though wide-area protection schemes


like special protection schemes (SPS) are designed to cause minimal or no impact

to the power system under failures, they are not designed to handle failures due

to malicious events like cyberattacks. Also, as more and more SPS are integrated

into the power system, it introduces unexpected dependencies in the operation of

the various schemes and this increases the risk of increased impacts like systemwide

collapse due to a cyberattack. Therefore, it becomes critical to reexamine the design

of the wide-area protection schemes.

1.2 Contributions of the Dissertation

The main contribution of this dissertation is briefly outlined below under three cat-

egories. Detailed description of each part will be given within the body of the

dissertation.

1.2.1 Wide-Area Synchronized-Measurement-Based Fault Lo-

cation

A novel analytical and computational approach to fault location in large-scale power

grids is developed. The proposed methodology involves an online and an offline

stage. The online stage is based solely on the utilization of the time-of-arrival (ToA)

measurements of traveling waves propagating from the fault-occurrence point to

synchronized wide-area monitoring devices installed at strategically selected substa-

tions. The procedure is realized through the processing of captured transient-voltage

waveforms by discrete wavelet transform in order to extract the arrival times of fault-

initiated waves, and to subsequently identify the location of the fault under study.

The main advantage of this method is that it requires a few and strategically de-

ployed synchronized measuring devices to locate faults in a large power system.


1.2.2 Optimal Sensor Deployment for Fault-Location Ob-

servability

This part of the dissertation considers the problem of strategically placing a mini-

mum number of synchronized measurements in a transmission system so that faults

can be detected and uniquely pinpointed irrespective of their locations in the power

network. Hence, it is intended as an extension of the devised fault-location method

to facilitate its efficient implementation via optimal sensor placement. It should be

noted that optimal deployment of synchronized sensors and functionality of fault-

location methodology constitute the two interdependent tasks in this study.

1.2.3 Robustification of Fault-Location Technique

In this part, we propose a robust alternative to the developed fault-location method

to reliably locate power-system faults using simultaneously recorded data from multi-

ple locations. Automatic removal of corrupted measurements resulting from various

factors (e.g., sensor breakdowns and cyberattacks) is accomplished via the use of a

least-absolute-value (LAV) estimator for fault location. Furthermore, inherent lim-

itations of the approach imposed by sensor configurations as well as the effect of

quantization errors incurred by low-precision sensors on the accuracy of estimated

fault locations are described. Additionally, based on largest-normalized-residuals

test, we propose an alternative correction method to handle bad sensor measure-

ments.

1.3 Dissertation Outline

This dissertation comprises eight chapters. It is organized as follows. In the current

chapter, we explain the motivations for conducting this research and outline our


contributions.

In Chapter 2, we first present the general background information about fault-

location problem in power grids and review the relevant literature to the existing

fault-location strategies. Properties of the proposed fault-location method are briefly

provided.

Chapter 3 provides an overview of typical transmission-line models used in elec-

tromagnetic transient analysis along with the theory of traveling waves. Numerical

techniques used in the analysis of electromagnetic transients are also discussed.

In Chapter 4, fundamentals of wavelet analysis are given, followed by its ap-

plications in power systems.

Chapter 5 presents a novel methodology to locate power-system faults which

exploits synchronized measurements that are few in number and disbursed in a sparse

manner across the transmission grid. An analytical derivation of a method by which

location of a disturbance can be accurately determined solely based on sparsely

located synchronized voltage sensors is reviewed first. This derivation illustrates

some of the limitations and practical constraints imposed by the system topology as

well as transmission-line characteristics, which will be overcome by the methods to

be introduced in the succeeding chapter (Chapter 6).

Chapter 6 builds on the results of the fault-location approach described in

the preceding chapter (Chapter 5) in order to devise an optimal scheme for deploy-

ing synchronized voltage sensors in the transmission system based on fault-location

observability analysis.

Chapter 7 expands the scope of the studies explained in the previous two

chapters (Chapters 5 and 6) by making the fault-location capability of the power

grid robust to unwanted changes in synchronized measurements, thereby adversely

affecting reliable fault-location estimation.


Finally, Chapter 8 summarizes the major contributions of this dissertation and

outlines the directions for further study.

Chapter 2

Fault Location in Power Networks

2.1 Introduction

Faults occurring on transmission and distribution lines can be caused by lightning

strikes, short circuits, storms, overloading, equipment failure, insulation breakdown,

tree/animal contact with the line, etc. If the location of a fault can be estimated

with high accuracy, the line can be restored expeditiously. Faults may be temporary

or permanent. Temporary faults, being the most common faults on power lines,

are self-cleared. In the case of temporary faults, accurate fault location can assist

in pinpointing the weak areas on the line, so that these spots can be repaired in

advance to prevent consequent major damages. On the other hand, to restore power

supply can be achieved following a permanent fault, an accurate location of the fault

is a highly valuable information which enables the maintenance crew to immediately

identify the faulty section of a line and to repair the damage caused by the fault.

This not only alleviates all the costs associated with the inspection and repair, but

enables better quality and reliability of supply and faster supply restoration, thereby

avoiding the occurrence of possible blackouts. Distance relays are used as a fast and

9

Chapter 2. Fault Location in Power Networks 10

reliable means to locate a faulted line section; however, they are unable to meet the

need for accurate fault location in all circumstances.

Among the devices that are most commonly used for their fault-location func-

tions are microprocessor-based protective relays and digital fault recorders (DFRs).

Microprocessor-based relays have characteristics of high computational capability

and high-speed communications with remote sites. Also, DFRs offer easy integra-

tion of the fault-location function at little or no additional cost. Depending upon the

needs of utilities, standalone fault-locating devices can equally be specified if sophis-

ticated fault-location algorithms are to be applied, albeit at a higher implementation

cost.

2.2 General Classification of Fault-Location Meth-

ods

Techniques used in locating faults on power lines can be broadly classified into three

main categories:

• techniques based on power-frequency components of voltage and current sig-

nals;

• techniques based on high-frequency components of transient voltage and cur-

rent signals generated by the fault; and

• artificial-intelligence (AI)-based techniques.

The first category is also known as the “impedance-based methods”; however,

the second category is commonly referred to as the “traveling-wave-based” fault-

location methods, primarily due to the use of traveling-wave phenomena which tend

to contain high-frequency signals (varying from several kHz to MHz, depending on


the location of the fault in the network), thus necessitating high-bandwidth data

acquisition. The third category is based on the use of computational-intelligence

methods, and mostly utilizes data-driven models.

Impedance-based methods belong to the following categories depending on

the type of available measurements1: (i) single-end (one-terminal) algorithms [1–

7] where data from only single terminal of the transmission line are available; (ii)

double-end (two-terminal) algorithms [8–21] in which measurements at both ends of

the transmission line can be utilized; and (iii) multi-end algorithms [22–31] that em-

ploy measurements from the (multiple) ends of the multiterminal transmission line.

Typically, double- and multiple-end techniques can be further subdivided into two

classes, i.e., those using synchronized [8–15, 26–31] or unsynchronized measurements

[8, 16–25].

Traveling-wave-based approaches to fault location [32–43] exploit transient sig-

nals generated by the fault. They are based on the correlation between forward- and

backward-traveling waves which propagate along the transmission lines with a ve-

locity close to the speed of light, and yield an explicit determination of the arrival

times of the waves at transmission-line terminals. This information can then be

used to determine the distance to fault-occurrence point. This class of fault-location

techniques is known to be insensitive to fault type; immune to fault resistance,

fault-inception angle, and source parameters of the system; and independent of the

network configuration and the devices installed in the network.

AI-based fault-location methods consist of pattern-recognition and machine-

learning algorithms (e.g., genetic algorithm, artificial neural networks) [44–52] along

with decision-making mechanisms (e.g., fuzzy-set theory) [53, 54], which are shown

to be beneficial tools in coping with the uncertainty in the fault-location problem.

1These can be in the form of “phasor” measurements or digital signals, including synchronizedand/or unsynchronized raw “samples”.


2.3 Accuracy of Fault-Location Algorithms [55]

Several factors may affect the accuracy of abovementioned fault-location algorithms.

These factors can be listed as follows:

• Uncertainty about transmission-line or cable characteristics,

• Accuracy of the line model,

• Sampling frequency and resolution in data acquisition,

• Presence of shunt reactors or series compensation devices equipped with metal

oxide varistors,

• Effect of mutual coupling in zero-sequence components,

• Effect of fault-inception angle,

• Effect of fault resistance,

• Line imbalance,

• Strength of sources behind the line terminals,

• Transient and steady-state errors of voltage and current instrument transform-

ers,

• Frequency response of voltage measurement chains,

• Position of a fault,

• Type of a fault,

• Loss of synchronism among the synchronized recorders, etc.

It should be noted that factors affecting the fault-location accuracy are strongly

linked with the method used in the fault analysis. It is essential to reduce the effects

of errors in order to obtain a sufficient level estimation accuracy in fault location.


2.4 Use of Traveling Waves for Fault Location

Traveling waves have long been applied to identification and location of faults in

transmission lines. As mentioned earlier, the location of a fault is estimated based

on the observation of the arrival time of traveling wavefronts at different locations in

the network where fault-recording devices are installed. In particular, the first or few

subsequent wavefronts may be used to determine the fault location. The propagation

time along the used medium (i.e., line or cable) is also utilized. In addition, these

methods rely on the tower configuration, from which the velocity of propagation for

the line is computed.

The essential idea behind the traveling-wave fault-location methods is that

when a fault occurs on a transmission line, the voltage and current transients travel

toward the line terminals. These transients continue to bounce back and forth be-

tween the fault point and line terminals, further propagating throughout the power

network until a postfault steady state is reached. Evolution of transients at line

terminals can be best understood through the use of the well-known lattice-diagram

method. A lattice diagram showing multiple reflections and refractions of a fault-

initiated traveling wave along a lossless transmission line is illustrated in Figure 2.1.

As illustrated in this figure, a fault occurring at a distance x from Bus A results in

an abrupt injection at the fault point. This injection travels like a surge along the

line in both directions between the fault point and line terminals until a postfault

steady state is reached.

As displayed in Figure 2.1, forward- and backward-traveling waves leave the

fault point traveling in both directions. Transmission-line ends represent a disconti-

nuity where some of the wave energy reflect back to the fault point. The remaining

energy travels to other transmission lines or power-system elements. The lattice

diagram illustrates multiple waves generated at line ends. The amplitudes of waves

are represented by reflection coefficients, which can be determined by the ratios of

characteristic impedances at the discontinuities.


Fault

Bus A

Bus B t1 3t1 5t1

t2 3t2

+2t2

+4t1+2t1 t2t2

t1 3t1+2t2

2t1+3t2

Time (t)

x point

Figure 2.1: Lattice diagram for a fault located at a distance x from Bus A.

The arrival time of the forward-traveling wave at Bus B is t1 = d−xν

; and the

arrival time of the backward-traveling wave at Bus A is t2 = xν, where d is the total

line length and ν is the traveling-wave velocity. This information can then be used

to locate the fault. Detailed discussion on the theory of traveling waves can be found

in the next chapter.

It should be noted that construction of the lattice diagram is computationally

difficult if the attenuation and distortion of the transient signals are taken into ac-

count as they travel along the line. However, time-frequency analysis of the transient

signals can be utilized to extract the travel times of the transients between the fault

point and the line terminals. In lossy multiphase transmission lines, there are three

modes of propagation of waves. Therefore, transient signals must be converted from

the phase domain into the modal domain in order to determine the arrival times.

Traveling-wave-based methods for fault location have gained much popularity

in recent years mainly due to the desire of utilities for faster and more accurate


fault location, coupled with latest improvements in data acquisition thanks to the

advances in communication systems and synchronization achieved via the Global Po-

sitioning System (GPS). Recent developments in transducer technology enable high-

sampling-rate recording of transient signals at the time of faults. Such availability of

broad-bandwidth sampling capability facilitates the efficient use of traveling-wave-

based methods for fault analysis.

The traveling-wave-based fault-location system has proved to be an effective

and viable method of providing accurate and automated distance-to-fault results that

allow operators to rapidly deploy repair teams to the fault site, undertake remedial

action to reduce the instance of transient and intermittent faults, and restore faulted

networks to original operating conditions as soon as possible.

The need for high sampling frequency is the most stated limitation of the

traveling-wave-based fault-location methods. Presence of different line section or

laterals that reflect traveling waves, which may be confused with those initiated

by the fault, can be another difficulty related to application of these methods. In

complex power-grid topologies, detection of abrupt changes may also be complicated

whenever the signal observed at each sensor is a mixture of waves arriving at different

times, following different paths.

Fault-location techniques based on traveling waves are not dependent upon

the network configuration and devices installed in the network. These methods are

highly accurate, but may require high sampling rates, which makes its implementa-

tion more costly than that of impedance-based approaches.


2.5 Emerging Use of Synchronized Measurements

for Fault Location

The fault-location task is usually undertaken with the help of saved transient records

of currents and voltages during the fault. These records may be analog oscillographic

records, digital fault records, or records available from many of the digital computer-

based relays. To enable a comparison of recordings from fault recorders and protec-

tion equipment at different locations, precise synchronization of all fault-recording

and data-collecting devices is necessary. This is ensured by the use of additional

components, such as GPS receiver and sync-transceiver.

A digital fault recorder with a GPS satellite receiver can collect the real-time

fault data, and this can be realized just by adding a GPS satellite receiver to a stan-

dard commercial digital fault recorder. With these synchronously sampled data,

a unique time-domain approach for fault analysis can be established, which is ex-

tremely fast and accurate and provides robust results even under certain very difficult

circumstance such as a time-varying fault resistance. Other fault analysis functions

including fault detection and classification can also be implemented by utilizing syn-

chronously sampled data.

Widespread deployment of the phasor measurement units (PMUs) providing

for both appropriate penetration and redundancy of synchronized measurements is a

key factor. Such widespread deployment can be achieved when integrating the PMU

function within modern microprocessor-based relays in order to enhance the fault-

recording capabilities of the power system. With a large volume of synchronized,

high-resolution, and raw transient data collected through disbursed sensors deployed

over a wide area, power-system operators can obtain a coherent picture of the whole

transient process and extract useful information that allows for pinpointing incipient

problems, and thus taking appropriate corrective actions. The synchronized raw

measurements can be used for observing system dynamic process since power swings


can be recorded via synchronized measurements at a number of buses of the system.

With the aid of GPS and PMUs, fast transient processes can be tracked with a high

rate of sampling.

As an added benefit of the oversampling technique, storage of raw data from

samples of the analog signals enables PMUs to be utilized as high-bandwidth “digital

fault recorders”. Sampled data obtained from around the entire network simultane-

ously can be used in forming a consistent picture of faults and other transient events

as they occur on a power system. Since all the signals recorded by digital relays

or fault recorders are sampled at the same instant, one can obtain a simultaneous

snapshot of the recorded event across the whole set of voltages and currents. Thus,

digital relay data obtained from any substation could be correlated precisely, and

one would have an outstanding tool for postmortem analysis on a systemwide basis.

Furthermore, by making a series of snapshot pictures of unfolding events, one could

trace cause-and-effect phenomena accompanying complex system events.

Intelligent electronic devices (IEDs) such as microprocessor-based relays, me-

ters, DFRs, etc., with synchronized measurement capability will become more preva-

lent in the years to come. These PMU-capable IEDs will be installed in large quan-

tities in power systems mainly for its primary functionalities (protective relaying,

fault-event recordings, etc.). The result is that a substantial number of IEDs in a

power system will become capable of performing the synchronized measurement in

the years to come.

Communications systems are a vital component of a wide-area fault-recording

system. These systems distribute and manage the information needed for operation

of the wide-area relay and control system. However, due to potential loss of commu-

nication, the fault-location system must be designed to detect and tolerate failures

in the communication system.

Data with precise synchronization are extremely valuable in determining the

sequence of events and contributing causes to a catastrophic power system failure.


Thus, widespread use of the PMU data is expected to be a common feature of power

system monitoring, control and protection systems of the future.

2.6 Key Features of the Proposed Fault-Location

Strategy

At present, in parallel with the availability of wide-area synchronized measurements,

computational capabilities in substations allow the utilization of unconventional

techniques, especially those based on traveling waves for fault location. More-

over, recently developed signal-processing tools—notably discrete wavelet transform

(DWT)—allow the analysis of sampled waveforms with localized transients. Wavelet

transform offers the property of multiresolution both in time and frequency domain;

hence, it is appropriate for analyzing a signal containing transient components. In

fact, fast electromagnetic transients produce waveforms that are nonperiodic signals,

involving high-frequency oscillations as well as localized impulses superimposed on

the fundamental frequency and its low-order harmonics.

The idea of using DWT of modal components of the traveling waves initiated

by faults, in order to estimate the fault location is first proposed and associated

algorithm is presented in [37]. Later, this fault-location technique is applied to

the three-terminal (teed) transmission configurations in [39]. The approach to be

described in this dissertation also uses DWT for capturing the time of arrival of trav-

eling waves, yet proposes a wide-area measurement-based solution to fault location

for large-scale power networks.

In this dissertation, we devise a fault-location system in such a way that even

with a relatively sparse penetration of measurements in an interconnection, the faults

can be identified and observed at a wide-area system level. In particular, we propose

a novel analytical and computational approach to fault location for large-scale power


systems. The proposed methodology involves an online and an offline stage. The

online stage is based solely on the utilization of the time-of-arrival (ToA) measure-

ments of traveling waves propagating from the fault-occurrence point to synchronized

wide-area fault-recording devices installed at strategically selected substations. The

captured waveforms are processed together at the time of fault in order to identify

the location of the fault under study. The applicability of the algorithm is indepen-

dent of the fault type and can readily be extended to power grids of any size. The

pictorial representation of the devised fault-location approach is given in Figure 2.2.

x d− x

Fault-occurrence point

Sensor locations

GPS

Traveling waves

Figure 2.2: Depiction of synchronized-measurement-based fault location utiliz-ing the theory of traveling waves.


2.7 Summary

In this chapter, we provide background information regarding fault-location problem

in power systems. We roughly categorize the methods used for fault location, with

particular emphasis on techniques based on traveling waves. The potential benefits

of using synchronized data for fault location are explained. Brief overview of the

proposed fault-location strategy, which will be described in more detail in Chapter

5, is also provided.

Chapter 3

Analysis of Electromagnetic

Transients

3.1 Introduction

Electric power systems are exposed to many types of disturbances, which lead to

transients. For instance, physical phenomena such as lightning may produce tran-

sient overvoltages. Similarly, normal operating procedures (e.g., switching of equip-

ment and breaker reclosing) may initiate electrical transients. Abnormal conditions

such as electrical faults cause transients as well. The physical phenomena associated

with power-system transients can be categorized into two main types:

(i) Interchange between electrical energy stored in capacitors and magnetic energy

stored in inductors; and

(ii) Interchange between electrical energy stored in circuits and mechanical energy

stored in rotating machines.

The first category consists of electromagnetic transients; whereas, the second

category comprises electromechanical transients. Intrinsically, most power-system

21

Chapter 3. Analysis of Electromagnetic Transients 22

transients are oscillatory; hence, they are described with the oscillation frequencies.

Based on the frequency of oscillations, power-system transients can be classified as

shown in Figure 3.1. In this dissertation, our center of interest will be the electro-

magnetic transients.

Electromagnetic transients culminate in irregular voltages (overvoltages) or

irregular currents (overcurrents). Overcurrents may impair power system equip-

ment due to dissipation of excessive heat. Overvoltages may eventuate in flashovers,

insulation breakdown, device outages, and ultimate deterioration of power-system

reliability.

In this chapter, we first provide the overview of the theory of traveling waves

and introduce models of transmission lines used in transient analysis. We also review

the trapezoidal integration method, which is the most popular numerical integration

method used in the analysis of power-system transients.

PowerFrequency

Frequency (Hz)

10−3 10−2 1010−1 1 102 103 104 105 106 107

Electromechanical Electromagnetic

Phenomena Phenomena

Load-FrequencyControl

TransientStability

Short Circuits

StabilizersSubsynchronousResonance

Power-Conversion

Harmonics

SwitchingTransients

Traveling-WavePhenomena

TransientVoltageRecovery

Phenomena

Figure 3.1: Classification of power-system transients [56].


3.2 Traveling-Wave Theory

When we represent an overhead transmission line by means of a number of nominal

π-circuit models, we take the properties of the electric field in a capacitance and

the properties of the magnetic field in an inductance into account and connect these

elements with lossless wires. For steady-state analysis, a lumped-element represen-

tation is sufficient in many cases. For transient analysis, this is no longer the case

and the travel time of the electromagnetic waves has to be taken into consideration

[57].

A representation of overhead lines by means of lumped elements is not helpful

in making us understand the wave phenomena because electromagnetic waves have

a travel time. Only when the physical dimensions of a certain part of the power

system are small in comparison to the wavelength of the transients, the travel time

of the electromagnetic waves can be neglected and a lumped-element representation

of that part of the system can be used.

If the travel time of the voltage and current waves is taken into account, and

we represent the properties of the electric field by a capacitance and the properties

of the magnetic field by an inductance, we call the capacitance and the inductance

distributed. For example, an overhead transmission line has certain physical dimen-

sions; thus, their overall resistance, inductance, and capacitance is considered to be

equally distributed over their size.

3.2.1 Traveling-Wave Velocity and Characteristic Impedance

If a voltage source ϑ is switched on in a two-wire transmission line at t = 0, the

line will be charged by the voltage source. After a small time span δt, only a small

segment δx of the line will be charged instantaneously with a charge δQ = C ′δxϑ.


This charge causes an electric field E around the line segment and the current, or

the flow of charge creates a magnetic field H around the line segment δx.

For an infinitesimal δx, the expression for the current is

i = limδx→0

δQδt

= limδx→0

C ′ϑδx

δt= C ′ϑ

dx

dt= C ′ϑν, (3.1)

where δx/δt is the velocity at which the charge travels along the line. The magnetic

flux present around the line segment is δΦB = L′δxi. If this is substituted in (3.1),

the expression for the induced electromotive force emf in the loop enclosed by the

two wires over the distance δx is

emf = limδx→0

δΦB

δt= L′C ′ϑ

(dx

dt

)2

= L′C ′ϑν2. (3.2)

Since there cannot be a discontinuity in voltage, the emf equals the voltage source

ϑ. Hence, an expression for the wave velocity is given by

ν =1√L′C ′

. (3.3)

The wave velocity depends only on the geometry of the line and the permittivity

and permeability of the surrounding medium. Typically, on an overhead transmission

line, the electromagnetic waves propagate close to the speed of the light; however,

in an underground cable the velocity is considerably lower. When the wave velocity

is substituted in (3.1), we obtain

i = C ′ϑν =C ′ϑ√L′C ′

. (3.4)

Notice that the ratio between the voltage and current wave has a constant value

Zcdef=ϑ

i=

√L′

C ′, (3.5)


where Zc is called the characteristic impedance of a transmission line. The charac-

teristic impedance depends merely on the geometry of the transmission line and its

surrounding medium.

3.2.2 The Telegrapher’s Equations

The transmission-line equations that govern general two-conductor uniform trans-

mission lines are called the telegrapher’s equations. The general transmission-line

equations are named the telegraph equations because they were formulated by Oliver

Heaviside while he was investigating the disturbances on telephone wires.

i(x, t)

δG′ δC ′ϑ(x, t) ϑ(x+ δx, t)

i(x+ δx, t)δR′ δL′

δx

++

−−

Figure 3.2: Heaviside’s model of the differential-length transmission line.

Assume that a series resistance, R′, and a parallel conductance, G′ as well as

a series inductance, L′, and a shunt capacitance, C ′, are evenly distributed along

the wires. When we consider a differential-length transmission line segment δx in

Figure 3.2 with parameters δR′, δG′, δL′, and δC ′ (all in per-unit length), the line

constants for segment δx are R′δx, G′δx, L′δx, and C ′δx. The electric flux ΦE and

the magnetic flux ΦB generated by the electromagnetic wave, which produces the

instantaneous voltage ϑ(x, t) and the current i(x, t), are

dΦE(t) = ϑ(x, t)C ′δx (3.6)


and

dΦB(t) = i(x, t)L′δx. (3.7)

Applying Kirchhoff’s Voltage Law on the loop enclosed by the two wires over the

distance δx, we get

ϑ(x, t)− ϑ(x+ δx, t) = −δϑ = i(x, t)R′δx+∂

∂tdΦB(t)

=

(R′ + L′

∂

∂t

)i(x, t)δx. (3.8)

In the limit, as δx→ 0, this voltage equation becomes

∂ϑ(x, t)

∂x= −L′∂i(x, t)

∂t−R′i(x, t). (3.9)

Similarly, for the current flowing through δG′ and the current charging C ′δx, Kirch-

hoff’s Current Law can be applied as follows:

i(x, t)− i(x+ ∆x, t) = −δi = ϑ(x, t)G′δx+∂

∂tdΦE(t)

=

(G′ + C ′

∂

∂t

)ϑ(x, t)δx. (3.10)

In the limit, as δx→ 0, this current equation becomes

∂i(x, t)

∂x= −C ′∂ϑ(x, t)

∂t−G′ϑ(x, t). (3.11)

The negative sign in these equations is caused by the fact that when the cur-

rent and voltage waves propagate in the positive x-direction, i(x, t) and ϑ(x, t) will

decrease in amplitude for increasing x. In order to solve these equations, they

are transformed into the Laplace domain by substituting the Heaviside operator


h = ∂/∂t; hence we obtain the following partial differential equations:

−∂ϑ(x, h)

∂x= (R′ + hL′)i(x, h), (3.12)

−∂i(x, h)

∂x= (G′ + hC ′)ϑ(x, h). (3.13)

Substituting Z ′ = R′ + hL′ and Y ′ = G′ + hC ′ and differentiating once again with

respect to x, we get the following second-order partial differential equations:

∂2ϑ(x, h)

∂x2= −Z ′∂i(x, h)

∂x= Z ′Y ′ϑ(x, h) = γ2ϑ(x, h), (3.14)

∂2i(x, h)

∂x2= −Y ′∂ϑ(x, h)

∂x= Y ′Z ′i(x, h) = γ2i(x, h), (3.15)

γ =

√(R′G′ +

(R′C ′ +G′L′

)h+ L′C ′h2

)=

1

ν

√(h+$)2 − ϕ2, (3.16)

where

ν =1√L′C ′

is the wave velocity ; (3.17)

$ =1

2

(R′

L′+G′

C ′

)is the attenuation constant (of influence on the

amplitude of the traveling waves); (3.18)

ϕ =1

2

(R′

L′− G′

C ′

)is the phase constant (of influence on the phase shift

of the traveling waves); and (3.19)

Zc =

√Z ′

Y ′=

√L′

C ′

√h+$ + ϕ

h+$ − ϕ is the characteristic impedance. (3.20)

The solutions of (3.14) and (3.15) in the time domain are

ϑ(x, t) = eγxf1(t) + e−γxf2(t) (3.21)


and

i(x, t) = − 1

Zc

[eγxf1(t)− e−γxf2(t)

]. (3.22)

In these expressions, f1(t) and f2(t) are arbitrary functions and independent

of x.

3.2.3 The Lossless Line

For the lossless line, the series impedance R′ and the parallel conductance G′ are

zero. By differentiating (3.9) with respect to x and (3.11) and with respect to t, the

following pair of equations will be obtained:

∂2ϑ(x, t)

∂x2= −L′∂

2i(x, t)

∂x∂t, (3.23)

∂2i(x, t)

∂x∂t= −C ′∂

2ϑ(x, t)

∂t2. (3.24)

Eliminating∂2i(x, t)

∂x∂tand rearranging the terms, we get

∂2ϑ(x, t)

∂x2= L′C ′

∂2ϑ(x, t)

∂t2. (3.25)

Solving (3.9) and (3.11) for i(x, t) instead of ϑ(x, t) yields

∂2i(x, t)

∂x2= L′C ′

∂2i(x, t)

∂t2. (3.26)

Note that transmission-line wave equations are comprised of (3.25) and (3.26).

The propagation constant, γ, and the characteristic impedance, Zc, for the

lossless line become

γ = h√L′C ′ =

h

ν(3.27)


and

Zc = Z0 =

√L′

C ′. (3.28)

Thus, the solutions for the voltage and current waves reduce to

ϑ(x, t) = ehx/νf1(t) + e−hx/νf2(t), (3.29)

i(x, t) = − 1

Z0

[ehx/νf1(t)− e−hx/νf2(t)

]. (3.30)

Writing Taylor series expansion of a function

f(t+ p) = f(t) + pdf(t)

dt+

(p2

2!

)d2f(t)

dt2+ · · · (3.31)

and introducing the Heaviside operator h = d/dt yields

f(t+ p) =

(1 + ph+

p2

2!h2 + · · ·

)f(t) = ephf(t). (3.32)

Applying (3.32) to (3.29) and (3.30), the following solutions for the voltage

and current waves are obtained:

ϑ(x, t) = f1

(t+

x

ν

)

︸︷︷︸ϑ−(t+x

ν )

+ f2

(t− x

ν

)

︸︷︷︸ϑ+(t−xν )

, (3.33)

i(x, t) = − 1

Z0

f1

(t+

x

ν

)

︸︷︷︸i−(t+x

ν )

+1

Z0

f2

(t− x

ν

)

︸︷︷︸i+(t−xν )

. (3.34)

In the above expressions, f1

(t+ x

ν

)is a function describing a wave propagating

in the (-x)-direction, which is mostly referred to as the backward-traveling wave, and

f2

(t− x

ν

)is a function describing a wave propagating in the (+x)-direction, called

the forward-traveling wave (see Figure 3.3).


Forward-traveling waves Backward-traveling waves

ϑ(x, t)i(x, t) i(x, t) ϑ(x, t)

i(x, t)

i(x, t)

+x+x

ν

ϑ(x, t) = Z0i(x, t)

+

−

+

−ϑ(x, t) = −Z0i(x, t)

Terminal s Terminal r(x = 0) (x = d)

Figure 3.3: Forward- and backward-traveling waves along with their polarity.

3.3 Transmission-Line Models for Transient Anal-

ysis

One of the key tasks in power system transient analysis is selection of the model by

which the physical system will be represented. The chosen model must represent

the physical phenomena under study with high accuracy. In this section, we provide

some insight into the models of power system elements for transient analysis.

Approximate nominal π-circuit models are commonly used for short transmis-

sion lines, where the travel time is smaller than the solution time step, but such

models are not practical for transmission distances. Instead, traveling-wave theory

is utilized to obtain more realistic line models.

A traveling-wave model of the lossless transmission-line model is equally ap-

plicable to overhead lines and cables; the main differences arise from the procedures

used in the calculation of the electrical parameters from their respective physical

geometries. Carson’s equations [58] form the basis for the calculation of overhead

line parameters, either as a numerical integration of Carson’s equation, via the use

of a series approximation or through the method of depth of penetration.


Multiconductor lines have been accommodated in the ElectroMagnetic Tran-

sients Program (EMTP) and Alternative Transients Program (ATP) by a transfor-

mation to natural modes to diagonalize the matrices involved. Original stability

problems were assumed to be caused by inaccuracies in the modal-domain repre-

sentation, and thus much effort was devoted to the development of more accurate

fitting techniques. In [59], Gustavsen and Semlyen demonstrate that although the

phase domain is inherently stable, its associated modal domain can be inherently

unstable irrespective of the fitting technique used. This finding has stimulated the

the phase-domain modeling of lines [60].

A decision tree for the selection of the suitable transmission-line model is shown

in Figure 3.4. The lower limit for wave-travel time is d/c where c is the speed of

light, and this can be compared to the solution time step to determine whether a

nominal π-circuit or traveling-wave model is appropriate.

START

Is traveling time greaterthan time step?

No

Yes

Is physicalgeometry of line available(i.e., conductor radius

and positions)?

YesNo

Bergeron’s ModelFrequency-Dependent

Model

Use R, X, and Binformation

Nominal π-Circuit Model

Figure 3.4: Decision tree for transmission-line model selection [60].


3.3.1 Bergeron’s Tranmission-Line Model

Bergeron’s line model is a discrete-time, constant-frequency representation based on

traveling-wave theory, which makes it conducive for computer implementation. This

line model was adopted by Hermann W. Dommel’s widely used ElectroMagnetic

Transients Program (EMTP) [61] in the 1960s.

Now, let us recall the equations we have derived in Section 3.2.3 for the

traveling-wave theory. Multiplying (3.34) by Z0 and adding it to and subtracting it

from (3.33) yields

ϑ(x, t) + Z0i(x, t) = 2f2

(t− x

ν

), (3.35)

ϑ(x, t)− Z0i(x, t) = 2f1

(t+

x

ν

). (3.36)

It should be noted that ϑ(x, t) +Z0i(x, t) is constant whenever(t− x

ν

)is con-

stant. To observe a constant ϑ(x, t) +Z0i(x, t), the traveling time from sending-end

terminal (Terminal “s”) of the line to the receiving-end terminal (Terminal “r”) is

τ =d

ν= d√L′C ′, (3.37)

where d is the transmission-line length. Notice that the expression ϑ(x, t) +Z0i(x, t)

seen by the observer when leaving Terminal r at time (t−τ) must be the same when

he arrives at Terminal s at time t, i.e.,

ϑr(t− τ) = Z0ir,s(t− τ) = ϑs(t) + Z0(−is,r(t)). (3.38)

From this equation, the following set of equations will be obtained:

is,r(t) =1

Z0

ϑs(t) + Is(t− τ), (3.39)

ir,s(t) =1

Z0

ϑr(t) + Ir(t− τ), (3.40)


where the current sources

Is(t− τ) = − 1

Z0

ϑr(t− τ)− ir,s(t− τ) (3.41)

Ir(t− τ) = − 1

Z0

ϑs(t− τ)− is,r(t− τ) (3.42)

from the previously computed values lead to the following update equations:

Is(t) = − 2

Z0

ϑr(t)− Ir(t− τ), (3.43)

Ir(t) =2

Z0

ϑs(t)− Is(t− τ). (3.44)

Z0ϑs(t) ϑr(t)Is(t− τ)

Ir(t− τ)

is,r(t) ir,s(t)

Z0

Figure 3.5: Equivalent two-port model for a lossless transmission line betweenTerminals s and r.

The corresponding equivalent two-port network is depicted in Figure 3.5. The

two line terminals are not directly linked with each other and the conditions at one

terminal are seen with time delays (i.e., traveling times) at the other terminal via

current sources.

Note that (3.39) and (3.40) form the basis for Bergeron’s transmission-line

model and provide an exact solution for the lossless line at its terminals.


3.3.2 Frequency-Dependent Transmission-Line Model

Now, the line parameters are expected to be functions of frequency; hence, the

relevant line equations should first be derived in the frequency domain. Converting

(3.8) and (3.10) into the frequency domain, we obtain

−∂V(x, ω)

∂x= (R′(ω) + jωL′(ω))I(x, ω), (3.45)

−∂I(x, ω)

∂x= (G′(ω) + jωC ′(ω))V(x, ω). (3.46)

Note that line is considered lossy and we only consider this model in the steady-

state (i.e., frequency domain) so that time will be omitted in the derived equations.

Hence, the derivation of (3.45) and (3.46) leads to

∂2V(x, ω)

∂x2= (R′(ω) + jωL′(ω))(G′(ω) + jωC ′(ω))V(x, ω), (3.47)

∂2I(x, ω)

∂x2= (G′(ω) + jωC ′(ω))(R′(ω) + jωL′(ω))I(x, ω). (3.48)

Two important frequency-dependent parameters affecting wave propagation

are the characteristic impedance, Zc(ω), and propagation constant, γ(ω), which are

given by

Zc(ω) =

√Z ′(ω)

Y ′(ω)=

√R′(ω) + jωL′(ω)

G′(ω) + jωC ′(ω), (3.49)

γ(ω) =√Z ′(ω)Y ′(ω) =

√(R′(ω) + jωL′(ω)

)(G′(ω) + jωC ′(ω)

). (3.50)


The propagation constant relation (3.50) can be plugged into (3.47) and (3.48),

and we obtain

∂2V(x, ω)

∂x2= γ2(ω)V(x, ω), (3.51)

∂2I(x, ω)

∂x2= γ2(ω)I(x, ω). (3.52)

(3.51) and (3.52) are ordinary differential equations whose typical solution are given

by

V(x, ω) = A1e−γ(ω)x + A2e

γ(ω)x, (3.53)

I(x, ω) = B1e−γ(ω)x +B2e

γ(ω)x. (3.54)

The constants in the above equations can be computed using the system’s initial

conditions, typically for x = 0 (see Appendix B for full derivation of these constants).

Referring to the relation between voltage and current in (3.22), we observe that the

current relation (3.54) can be rewritten as

I(x, ω) =A1

Zce−γ(ω)x − A2

Zceγ(ω)x. (3.55)

The equations can be further developed using hyperbolic functions. Thus, the

voltage and current relations can be written as

V(x, ω) = Vs(ω) cosh(γ(ω)x)− Zc(ω)Is,r(ω) sinh(γ(ω)x), (3.56)

I(x, ω) = −Vs(ω)

Zc(ω)sinh(γ(ω)x) + Is,r(ω) cosh(γ(ω)x). (3.57)


At the receiving end (i.e., at Terminal “r”) of the transmission line, we know

that x = d, and the respective equations for the receiving end become

Vr(ω) = Vs(ω) cosh(γ(ω)d)− Zc(ω)Is,r(ω) sinh(γ(ω)d), (3.58)

Ir,s(ω) = −Vs(ω)

Zc(ω)sinh(γ(ω)d) + Is,r(ω) cosh(γ(ω)d). (3.59)

Therefore, at any frequency, the steady-state input-output relationship of the trans-

mission line is given by

Vs(ω)

Is,r(ω)

=

cosh(γ(ω)d) Zc(ω) sinh(γ(ω)d)1

Zc(ω)sinh(γ(ω)d) cosh(γ(ω)d)

Vr(ω)

−Ir,s(ω)

. (3.60)

(3.60) provides a basis to develop an equivalent circuit in the frequency domain.

There are various alternative representations for equivalent circuits corresponding

to (3.60). Here, we discuss an equivalent circuit proposed by Snelson [62], which is

suitable for transient analysis. In order to derive the equivalent circuit, four new

functions are defined as follows:

fs(t) = ϑs(t) +R1is,r(t), (3.61)

fr(t) = ϑr(t) +R1ir,s(t), (3.62)

bs(t) = ϑs(t)−R1is,r(t), (3.63)

br(t) = ϑr(t)−R1ir,s(t). (3.64)

Here, fs(t) and fr(t) are called forward-traveling functions ; whereas, bs(t) and br(t)

are named backward-traveling functions. Snelson defines R1 as R1 = limω→∞

Zc(ω).


To relate the time-domain functions to the exact line solution in the frequency

domain, (3.61)–(3.64) are transformed into the frequency domain as follows:

Fs(ω) = Vs(ω) +R1Is,r(ω), (3.65)

Fr(ω) = Vr(ω) +R1Ir,s(ω), (3.66)

Bs(ω) = Vs(ω)−R1Is,r(ω), (3.67)

Br(ω) = Vr(ω)−R1Ir,s(ω). (3.68)

Comparing (3.65)–(3.68) with (3.60), it follows that

Bs(ω) = H1(ω)Fr(ω) +H2(ω)Fs(ω) (3.69)

and

Br(ω) = H1(ω)Fs(ω) +H2(ω)Fr(ω), (3.70)

where

H1(ω) =1

cosh(γ(ω)d) + 12

(Zc(ω)R1

+ R1

Zc(ω)

)sinh(γ(ω)d)

(3.71)

and

H2(ω) =

12

(Zc(ω)R1− R1

Zc(ω)

)sinh(γ(ω)d)

cosh(γ(ω)d) + 12

(Zc(ω)R1

+ R1

Zc(ω)

)sinh(γ(ω)d)

. (3.72)


(3.69) and (3.70) can be transformed into time domain as follows:

bs(t) = h1(t) ∗ fr(t) + h2(t) ∗ fs(t) =

∫ ∞

0

h1(u)fr(t− u) + h2(u)fs(t− u)du,

(3.73)

br(t) = h1(t) ∗ fs(t) + h2(t) ∗ fr(t) =

∫ ∞

0

h1(u)fs(t− u) + h2(u)fr(t− u)du,

(3.74)

where ∗ denotes convolution operation; and h1(t) and h2(t) are the weighting func-

tions for the time-domain convolutions, which are computed by the inverse Fourier

transform. After evaluating the convolutions in (3.73) and (3.74), at each time step

of the solution, (3.63) and (3.64) yield the following equivalent representations for

the line as seen from Terminals “s” and “r”:

is,r(t) =1

R1

ϑs(t) + is,rhist(t), (3.75)

ir,s(t) =1

R1

ϑr(t) + ir,shist(t), (3.76)

where is,rhist(t) and ir,shist(t) are defined from the past values of the variables.

3.4 Numerical Transient Analysis

The transient analysis of large power networks is usually carried out by using numer-

ical techniques. Numerical integration is used to transform the differential equations

of a circuit element into algebraic equations that contain voltages, currents, and

history (past) values. These algebraic equations represent a resistive-companion

equivalent of the circuit element. The equations of the entire resistive-companion

network are assembled and solved as a function of time at discrete instants.


In this section, we present the principles of the numerical technique proposed

by Dommel [63], which is the most popular algorithm currently implemented in sim-

ulation tools designed to analyze electromagnetic transients in power systems. This

algorithm is based on a combination of the trapezoidal integration, which is used to

obtain the resistive-companion equivalent of lumped-parameter circuit elements; and

the Bergeron’s line model, which is used to obtain the resistive-companion equivalent

of distributed-parameter circuit elements.

The following subsections are devoted to the principles of the trapezoidal inte-

gration to obtain resistive-companion equivalents, and to computation of the time-

domain solution of linear networks.

3.4.1 Companion Equivalents of Circuit Elements Based on

Trapezoidal Integration

During fast-changing transient events, we do not need to represent the power sources

in great detail. Indeed, the lumped components representing generators, transform-

ers, and loads can be simulated by equivalent circuits consisting of voltage and

current sources, resistances, inductances, and capacitances. Given predetermined

time step, these components can be discretized, and substituted by a current source

in parallel with a resistance. In the following, derivation of such components will

be provided using a numerical integration method (i.e., trapezoidal integration), as

used in the electromagnetic-transient analysis programs.

3.4.1.1 Resistance

The simplest case is a resistor connected between the terminals, as shown in Figure

3.6(a), and it is represented by the following equation:

ϑs(t)− ϑr(t) = R′is,r(t) (3.77a)


or

is,r(t) =1

R′(ϑs(t)− ϑr(t)) . (3.77b)

3.4.1.2 Inductance

The differential equation for the inductor shown in Figure 3.6(b) is

ϑs(t)− ϑr(t) = L′dis,r(t)

dt(3.78)

which must be integrated from a known state at (t−∆t) to the unknown state at t,

i.e.,

is,r(t) = is,r(t−∆t) +1

L′

∫ t

t−∆t

(ϑs(t)− ϑr(t))dt. (3.79)

Applying the trapezoidal rule yields

is,r(t) = is,r(t−∆t) +1

L′∆t

2[(ϑs(t)− ϑr(t)) + (ϑs(t−∆t)− ϑr(t−∆t))]

= Is,r(t−∆t) +∆t

2L′(ϑs(t)− ϑr(t)) , (3.80)

is,r(t)

ϑs(t) ϑr(t)

R′

(a)

(b) (c)

is,r(t)

ϑs(t) ϑr(t)

L′ C ′

is,r(t)

ϑs(t) ϑr(t)R′ =2L′

∆t

Is,r(t−∆t)

⇔

is,r(t)

ϑs(t) ϑr(t)

is,r(t)

ϑs(t) ϑr(t)

Is,r(t−∆t)

⇔

R′ =∆t

2C ′

Figure 3.6: Equivalent circuit representations of (a) resistor, (b) inductor, and(c) capacitor placed between Terminals s and r.


where

Is,r(t−∆t) = is,r(t−∆t) +∆t

2L′(ϑs(t−∆t)− ϑr(t−∆t)). (3.81)

3.4.1.3 Capacitance

The capacitor is represented by the equation

is,r(t) = C ′d(ϑs(t)− ϑr(t))

dt. (3.82)

Integrating and rearranging gives

ϑs(t)− ϑr(t) =1

C ′

∫ t

t−∆t

is,r(t)dt+ ϑs(t−∆t)− ϑr(t−∆t). (3.83)

Integration by the trapezoidal rule yields

ϑs(t)− ϑr(t) =∆t

2C ′[is,r(t) + is,r(t−∆t)] + ϑs(t−∆t)− ϑr(t−∆t). (3.84)

Hence, the current in the capacitor is given by

is,r(t) =2C ′

∆t(ϑs(t)− ϑr(t))− is,r(t−∆t))− 2C ′

∆t(ϑs(t−∆t)− ϑr(t−∆t))

=2C ′

∆t[ϑs(t)− ϑr(t)] + Is,r(t−∆t), (3.85)

where

Is,r(t−∆t) = −is,r(t−∆t)− 2C ′

∆t[ϑs(t−∆t)− ϑr(t−∆t)] . (3.86)

3.4.2 Computation of Transients in Linear Networks

Companion equivalents derived above consists of resistances, whose values remain

constant if ∆t is constant, and current sources, whose values must be updated at any


integration time step. Therefore, the solution of a linear network during a transient is

the solution of the equations of a purely resistive network whose parameters remain

constant during the transient and for which only current source values must be

updated at any time step.

The derivation of the equations of a linear network is illustrated with the case

depicted in Figure 3.7, which shows several circuit elements connected to a given

node. The application of the Kirchhoff’s current law to Node 1 yields

i12(t) + i13(t) + i14(t) + i15(t) = i1(t). (3.87)

The following equations are derived from the companion equivalent of each

branch connected to this node:

i12(t) =1

R′(ϑ1(t)− ϑ2(t)), (3.88)

i13(t) =∆t

2L′(ϑ1(t)− ϑ3(t)) + I13(t−∆t), (3.89)

i14(t) =2C ′

∆t(ϑ1(t)− ϑ4(t)) + I14(t−∆t), (3.90)

i15(t) =1

Zcϑ1(t) + I15(t− τ), (3.91)

where

I13(t−∆t) =∆t

2L′(ϑ1(t−∆t)− ϑ3(t−∆t)) + i13(t−∆t), (3.92)

I14(t−∆t) = −2C ′

∆t(ϑ1(t−∆t)− ϑ4(t−∆t))− i14(t−∆t), (3.93)

I15(t− τ) = − 1

Zcϑ5(t− τ)− i51(t− τ). (3.94)


1

2

3

45

i1

i12i13

i14i15

Line

R′ L′

C ′

i51

i21

i31

i41

Figure 3.7: Generic node of a linear network.

Upon substitution of (3.88)–(3.91) to (3.87), the following equation is obtained:

(1

R′+

∆t

2L′+

2C ′

∆t+

1

Zc

)ϑ1(t)− 1

R′ϑ2(t)− ∆t

2L′ϑ3(t)− 2C ′

∆tϑ4(t)

= i1(t)− I13(t−∆t)− I14(t−∆t)− I15(t− τ).

(3.95)

Following the procedure with all nodes, the equations of a network of any size

can be assembled and written as follows:

G · ϑ(t) = i(t) + IHist, (3.96)

where G is the symmetric nodal conductance matrix; ϑ(t) is the vector of nodal

voltages; i(t) is the vector of current sources; and IHist is the vector of current

sources representing history terms.


In general, some nodes of a power system have known voltages because voltage

sources are connected to them, and the order of the vector of unknown nodal voltages

can be reduced. Assume that U and K denote the set of unknown and known nodal

voltages, respectively. Nodal admittance equations can be rewritten as follows:

GUU GUK

GKU GKK

ϑU(t)

ϑK(t)

=

iU(t)

iK(t)

+

IU−Hist

IK−Hist

=

IUIK

. (3.97)

Using Kron’s reduction, the unknown voltage vector can be obtained from

GUUϑU(t) = iU(t) + IU−Hist −GUKϑK(t). (3.98)

Also, the current in voltage sources can be computed via

GKUϑU(t) + GKKϑK(t)− IK−Hist = iK(t). (3.99)

3.5 Summary

In this chapter, we provide an overview of the traveling-wave theory and discuss

transmission-line models used in transient analysis. Moreover, we present the deriva-

tion of trapezoidal integration method, which is considered numerically stable and

simple to apply. Computation of transients in linear networks based on nodal ad-

mittance equations is also explained.

Chapter 4

Wavelet Analysis—Fundamentals

and Its Applications in Power

Systems

Notations and Conventions

In the following, L2(R) signifies the set of square-integrable (quadratically inte-

grable) functions, i.e., the set of functions defined on the real line (−∞,+∞), sat-

isfying the condition that∫∞−∞ |f(t)|2dt <∞. The square-integrable functions form

an inner product space whose inner product is given by

〈f, g〉 def=

∫ ∞

−∞f(t)g∗(t)dt, (4.1)

where f and g are square-integrable functions and the superscript ∗ indicates the

complex conjugation. The associated norm is ‖f‖ def=√〈f, f〉.

Below, we use the convention f(t) ←→ F (ω) to represent the Fourier pair,

where F (ω) designates the Fourier transform of f(t), which is defined as F (ω)def=

45

Chapter 4. Wavelet Analysis—Fundamentals and Its Applications in PowerSystems 46

∫∞−∞ f(t)e−jωtdt. Since ‖f(t)‖2 =

∫∞−∞ |f(t)|2dt designates the energy of the signal

f(t), the space L2(R) is also called the space of finite-energy signals.

4.1 Introduction

Over the past three decades, wavelet analysis has drawn much attention in various

application domains including, but not limited to, signal processing (particularly

for nonstationary signal analysis), data compression, image processing and compres-

sion, seismic geology, quantum mechanics, acoustics, radar, optics, and turbulence.

Theoretical advancements in the wavelet theory have strong connections with de-

velopment of new wavelet bases for suitable function spaces and construction of

compactly supported orthonormal wavelets. Wavelets may be understood as func-

tions that chop up a signal into different frequency components, and analyze each

of these components based on a resolution associated with its scale. In analyzing

signals containing sharp spikes and discontinuities, wavelet analysis has superiorities

over Fourier analysis.

In this chapter, we provide an overview of wavelet theory and its applications

to analysis of transient signals. Our discussion begins with a comparison of wavelet

transforms and Fourier transforms. Then, we state the definition of a wavelet along

with its properties and introduce the wavelet transform. We conclude with some

examples of applications of wavelet transforms to electromagnetic transient analysis.

4.2 A Motivation for Wavelets

Historically, the short-time Fourier transform was being used extensively for ana-

lyzing nonstationary signals (i.e., signals whose frequency characteristics vary over

time). Nevertheless, it is not an effective tool for analyzing time-localized events.


Even though the wavelet transform bears a resemblance to the short-time Fourier

transform, its time-localization feature makes it unparalleled choice in dealing with

nonstationary data. In the following, we review the Fourier and short-time Fourier

transforms, touch upon some features that the short-time Fourier transform remains

incapable of, and describe the wavelet transform.

4.2.1 Wavelet Transforms versus Fourier Transforms

In this subsection, we discuss some similarities and differences between wavelet trans-

forms and the Fourier transforms.

Both the Fourier transform and wavelet transform are given by integral equa-

tions in a correlation form. In the Fourier transform, the correlation is with dilations

of the exponential function e−jt. In the wavelet transform, the correlation is with

dilations and translations of the analyzing wavelet of any type. Thus, the wavelet

transform is said to be a bivariate function, with dilation and translation variables.

Both the Fourier transform and wavelet transform may take real- or complex-

valued input signals. The output of the Fourier transform is always complex.

Nonetheless, there are both real- and complex-valued wavelets. If a complex-valued

analyzing wavelet is chosen, the wavelet transform becomes complex-valued. If a

real-valued wavelet is chosen as the analyzing wavelet, the wavelet transform be-

comes real-valued if the input signal is real-valued and complex-valued if the input

signal is complex-valued.

The Fourier transform maps time into frequency and phase; whereas, the

wavelet transform maps time into time and scale. For each frequency, the Fourier

transform produces a phase and an amplitude. Therefore, a signal may be rep-

resented as the sum of sine waves whose amplitude and phase are yielded by the

Fourier transform. Likewise, the wavelet transform produces an amplitude for each


time and scale. A signal can be represented as the sum across scales of wavelets that

are centered on time and whose amplitudes are yielded by the wavelet transform.

Both the Fourier transform and wavelet transform allow for localization in

frequency. However, the wavelet transform has a capability of localizing in time.

Short-time (windowed) Fourier transform, which will be discussed later, can be used

to partial localization in time while achieving its localization property in time by

trading off both bandwidth and frequency resolution. The wavelet transform does

not need to make any such tradeoff.

Unlike the Fourier transform, which uses only sine and cosine basis functions,

the wavelet transform does not have a single set of basis functions. Rather, the

wavelet transform has an infinitely possible basis functions. Therefore, any infor-

mation that could be concealed by the Fourier transform can be provided by the

wavelet transform.

A major advantage of the wavelet transform is that the analyzing windows

change. Indeed, one can utilize short high-frequency basis functions to separate

signal discontinuities; whereas, to obtain a detailed frequency information, one can

choose long low-frequency basis functions. However, the short-time Fourier trans-

form uses a single analyzing window for all frequencies, which truncates the signal to

fit a window with a certain width. Hence, the resolution remains the same through-

out the analysis.

4.2.2 The Short-Time Fourier Transform (STFT)

The Fourier transform is very suitable for processing stationary periodic signals

(e.g., sinusoidal signals). The limitation of the Fourier transform is that it provides

information only in the frequency domain. However, the short time Fourier transform

(STFT) is able to retrieve both time and frequency information from a signal; thus,

it is suitable for the analysis of nonstationary signals. The STFT takes the Fourier


transform of a windowed part of a signal, where the window slides along the time

axis. That is, the signal is windowed by a time-limited window centered at a specific

time. The STFT employs a windowed function to obtain the approximate frequency

contents of a signal in the vicinity of a desired time location. Mathematically, the

STFT of a signal f ∈ L2(R) at time b is given by

Fgf (b, ω) =

∫ ∞

−∞f(t)g∗(t− b)e−jωtdt = 〈f, g(t− b)ejωt〉, (4.2)

where the function g(t) is called the windowing function to be selected by the user.

That is why, the STFT is also called the windowed Fourier transform. It can be

understood that the STFT corresponds to the Fourier transform of the signal f(t)

multiplied by a window g(t).

The original signal f(t) can be uniquely recovered from its STFT. The recovery

formula for the STFT is given by

f(t) =1

2π

∫ ∞

−∞

∫ ∞

−∞Fgf (b, ω)g(t− b)ejωtdωdb. (4.3)

The performance of the STFT depends highly on the selection of a proper win-

dow width. A narrower window provides better time resolution; whereas, a wider

window gives better frequency resolution (see Figures 4.1(a) and 4.1(b)). These are

commonly referred to as “wideband” and “narrowband” transforms, respectively.

For better localization, the time-frequency window is desired to be of small area;

however, it is not feasible to obtain a good time and frequency resolution simultane-

ously. Equivalently, we cannot obtain precise localization both in time and frequency

domain. This is known as Heisenberg uncertainty principle, which states that the

product of time resolution ∆t and the frequency resolution ∆ω is constant and the

area of the time-frequency window is limited by

∆t∆ω ≥ 1

2. (4.4)


Time (t)

Frequen

cy(ω

)

Time (t)

Frequen

cy(ω

)(a) (b)

Figure 4.1: Fixed-resolution time-frequency planes: (a) narrow window enablesa better time resolution, and (b) wide window enables a better frequency resolu-

tion.

Note that the width of the windowing function of the STFT is fixed; therefore,

∆t and ∆ω are constant. With constant ∆t and ∆ω, the time-frequency plane is

divided into equal-area blocks as shown in Figure 4.1. This resolution does not

effectively represent signals with varying frequencies. A higher time resolution is

required for analyzing high-frequency components, which usually show up as series

of short-lived spikes. On the other hand, a higher frequency resolution is required for

processing low-frequency components, which often span a longer time period. For

this reason, the windowing function should be narrow for analyzing high frequencies

and wide for low frequencies. Indeed, to attain the desired level of accuracy, the

STFT has to be applied to the signal repeatedly with a varying window width each

time.

4.2.3 Multiresolution Analysis

The time-frequency resolution problem can be attributed to the Heisenberg uncer-

tainty principle and holds for any analysis technique. Using an approach called the


Time (t)

Frequen

cy(ω

)

Figure 4.2: Multiresolution time-frequency plane.

multiresolution analysis (MRA), a signal can be analyzed at various frequencies with

changing resolutions. MRA gives good frequency resolution but poor time resolu-

tion at low frequencies while providing good time resolution with poor frequency

resolution at high frequencies. The pertinent time-frequency plane is displayed in

Figure 4.2. Notice that the product ∆t∆ω remains the same for each segment in

the plane although the heights and widths of the boxes change, as in the STFT

case. For any window function in STFT and mother wavelet in WT, the area of

each partitioned segment is fixed; whereas, different values for areas can be selected

depending on the choice of windowing or wavelet functions. Still, the areas of the

boxes are lower-bounded by (4.4). In other words, the areas of the segments cannot

be arbitrarily decreased owing to the Heisenberg’s uncertainty principle.

In Figure 4.2, each segment has the equal portion of the time-frequency plane,


but providing different proportions of time and frequency. Notice that at lower fre-

quencies, the height of the segments is shorter while their widths become wider.

This translates to less uncertainty regarding the value of the exact frequency; thus

allowing for better frequency resolution; and more uncertainty regarding the value

of the exact time, thus leading to lower time resolution. On the other hand, at

higher frequencies, the width of the segments decreases, meaning that time resolu-

tion increases; whereas, the height of the segments increases, which indicates that

frequency resolution decreases.

4.3 Wavelet Analysis

This section provides an overview of the main properties of wavelet analysis. The

definition of a wavelet is given first, which is followed by the formulations regarding

the wavelet theory.

4.3.1 Wavelets

Wavelets are finite-energy functions that can be used to efficiently represent transient

signals, thanks to their localization properties. We mean by efficiency that a small

number of coefficients are needed to represent any complicated signal. As opposed

to the sinusoidal functions that have infinite energy, a wavelet is “a small wave”,

which has its energy concentrated in time. If a wavelet is more localized (i.e., its

energy is concentrated in a smaller region), it provides a better representation of the

signal in the time-frequency plane. In our case, better representation implies higher

resolution while requiring fewer number of coefficients.


A signal or function ψ(t) ∈ L2(R) is a wavelet if it satisfies the following

admissibility condition

Cψdef= 2π

∫ ∞

−∞

|Ψ(ω)|2|ω| dω < +∞, (4.5)

where Ψ(ω) is the Fourier transform of the wavelet ψ(t) and Cψ is called the admis-

sibility constant. The above condition requires that

Ψ(0) =

∫ ∞

−∞ψ(t)dt = 0, (4.6)

meaning that the signal ψ(t) behaves like a wave, oscillating up and down the t-axis.

This property implies that the mean of wavelets must be equal to zero. Furthermore,

|Ψ(ω)| must decay to zero rapidly for |ω| → 0 and |ω| → ∞. In other words, ψ(t)

must be a bandpass impulse response that resembles a small wave.

Instead of representing a signal by sinusoidal functions of different frequencies

in Fourier analysis, wavelet analysis seeks to represent a transient signal as a linear

combination of a scaled and translated version of the mother wavelet.

As explained above, the term wavelet comes from the requirement that a func-

tion should integrate to zero, waving above and below the t-axis. Unlike Fourier

analysis, which is an ideal tool for the efficient representation of stationary and very

smooth signals (whose frequency content do not change over time); wavelet anal-

ysis is well suited for representing signals containing discontinuities or for signals

that are intermittent, noisy, and nonstationary, owing to the fact that wavelet basis

functions are localized both in time (via translations) and frequency/scale (via dila-

tions); whereas, Fourier basis functions are localized only in frequency. In addition,

the number of wavelet basis functions is usually much less than the number of sine-

cosine basis functions required to achieve a good approximation. On the other hand,

small deviations of frequency in the Fourier transform result in changes everywhere

in the time domain.


0

−0.2

0.2

0.4

−0.4

−0.6

−0.8

0.8

0.6

0 1−1

−2 2 3−3−4 4−1

1

86420−8 −6 −4 −2

0

−0.2

−0.4

−0.6

−0.8

−1

0.2

0.4

0.6

0.8

1

1.2

0.5

−0.5

−1

0

1

0 0.5 1

0 2 4 61 3 5 7

0.5

1

−0.5

−1

0

1.5

0

0.2

−0.2

−0.40 1−2 2 3−3−4 4−1 5−5

0.4

0.8

0.6

1

0 1−2 2 3−3−4 4−1 5−5

0

−0.2

0.2

0.4

−0.4

−0.6

−0.8

0.8

0.6

−1

1

1.2

(a) (b)

(c) (d)

(e) (f)

t

t t

t

t t

ψ(t)

ψ(t)

ψ(t)

ψ(t)

ψ(t)

ψ(t)

Figure 4.3: Examples of common wavelet functions: (a) Haar, (b) Mexican hat,(c) Morlet, (d) Daubechies-8, (e) Meyer, and (f) Gaussian wavelet.


Wavelet analysis can be carried out in three ways, i.e., in the form of a con-

tinuous wavelet transform, a discretized continuous wavelet transform, and a purely

discrete wavelet transform.

4.3.2 The Continuous Wavelet Transform (CWT)

Given a signal f(t) ∈ L2(R), its continuous wavelet transform with respect to wavelet

ψ(t) is defined by

CWT ψf (a, b)def=

1√|a|

∫ ∞

−∞f(t)ψ∗

(t− ba

)dt; a, b ∈ R, a 6= 0, (4.7)

where a and b denote the scaling (dilation) and time shift (translation) variables,

respectively. More compactly, by defining ψa,b(t) as

ψa,b(t)def=

1√|a|ψ

(t− ba

), (4.8)

an alternative expression can be written as

CWT ψf (a, b) =

∫ ∞

−∞f(t)ψ∗a,b(t)dt = 〈f, ψa,b〉, (4.9)

where ψa,b(t) are commonly known as the daughter wavelets of the mother wavelet

ψ(t). Mother wavelets, which are also known as wavelet functions, are chosen as basis

functions in representing the signals just as sines and cosine waves are utilized for

the case of Fourier analysis. Various wavelet functions have already been developed,

some of which are shown in Figure 4.3.

The normalizing factor of 1√|a|

ensures that the energy of the daughter wavelets

is the same as that of the mother wavelet, i.e.,

∫ ∞

−∞|ψa,b(t)|2dt =

∫ ∞

−∞|ψ(t)|2dt. (4.10)


The scaling factor a shifts along ω-axis and controls the width of the wavelet;

whereas, the translation factor b shifts along t-axis and controls the location of

the wavelet. Scaling a wavelet means expanding it (if |a| > 1) or contracting it

(if |a| < 1), while translating it means shifting its position in time. Therefore,

the wavelet transform provides us with information both on time and frequency

simultaneously.

The inverse of the continuous wavelet transform is given by

f(t) =1

Cψ

∫ ∞

−∞

∫ ∞

−∞

1

a2CWT ψf (a, b)ψa,b(t)dadb. (4.11)

This transformation allows the original signal to be recovered from its continuous

wavelet transform by integrating over all scales a and locations b.

Large wavelet scales correspond to low frequencies and provide detailed infor-

mation hidden in the signal by dilating it. On the other hand, small scales correspond

to high frequencies and provide global information about the signal by compressing

it. For this reason, the wavelet analysis is very convenient for most practical applica-

tions, which involve signals composed of high frequencies that do not last very long

(appearing as bursts) as well as low frequencies which may last a very long duration.

4.3.3 The Wavelet Series

If we use the discrete values of the dilation and translation parameters, a and b, i.e.,

a = am0 and b = nb0am0 (a0 > 1 and b0 6= 0), then we obtain the wavelet series

WSψf (am0 , nb0am0 )

def=

1√am0

∫ ∞

−∞f(t)ψ∗

(a−m0 t− nb0

)dt = 〈f, ψm,n〉, (4.12a)

with

ψm,n(t) = a−m/20 ψ(a−m0 t− nb0), m, n ∈ Z. (4.12b)


Here, a0 is the fixed dilation parameter, and b0 is the location parameter.

The translation parameter b is dependent on the dilation am0 . Different values of

m correspond to wavelets with different widths. Indeed, broad (lower-frequency)

wavelets are translated by larger steps, while narrow (high-frequency) wavelets are

translated by small steps. For some particular choices of a0 and b0, the ψm,n(t) form

an orthonormal basis for L2(R). Specifically, if we select a0 = 2 and b0 = 1, then

the family of dilated mother wavelets

ψm,n(t) = 2−m/2ψ(2−mt− n) (4.13)

constitute an orthonormal wavelet basis and called dyadic-grid wavelets since the

continuous-time wavelet coefficients CWT ψf (a, b) are sampled in a dyadic grid. This is

done to minimize the heavily redundant representation of the signals (hence alleviate

the computational burden) resulting from the CWT, which is due the fact that a

and b are continuous over R.

Discrete dyadic-grid wavelets are selected to be orthonormal, i.e., they are

orthogonal to each other1 and normalized to have unit energy2 [64, 65]. This can be

expressed as

〈ψm,n, ψm,n〉 =

∫ ∞

−∞ψm,n(t)ψ∗m,n(t)dt = δmmδnn =

1 if m = m and n = n,

0 otherwise

(4.14)

By choosing an orthonormal wavelet basis, ψm,n(t), the reconstruction of the original

signal is realized by

f(t) =∞∑

m=−∞

∞∑

n=−∞

WSψf (am0 , nb0am0 )ψm,n(t), (4.15a)

1Two different functions are orthogonal to each other if the inner product of these functions iszero., i.e., 〈f, g〉 =

∫∞−∞ f(t)g∗(t)dt = 0.

2If we take g(t) = f(t), and 〈f, f〉 = ‖f‖2 =∫∞−∞ f(t)f∗(t)dt = 1, we say that function f(t) is

normalized.


which can be alternatively written as

f(t) =∞∑

m=−∞

∞∑

n=−∞

dm(n)ψm,n(t), (4.15b)

with

dm(n) = 〈f, ψm,n〉, m, n ∈ Z. (4.15c)

Types of orthonormal wavelet basis include the Haar wavelets, the Daubechies

orthonormal wavelet bases of all orders, and the Meyer wavelets of all orders.

4.3.4 The Discrete Wavelet Transform (DWT)

The sampled version of the CWT (wavelet series) shall not be regarded as a purely

discrete transform. In effect, the provided information may still be much redundant

for the signal reconstruction, and this redundancy requires an immense computa-

tional time based on the size of the signal and the desired resolution. Nonetheless,

the discrete wavelet transform (DWT) enables significantly reduced computing costs

for wavelet analysis and synthesis of the signal. Efficient realization of the DWT

is based on multirate filter banks [66–68], which will be addressed in the following

subsections.

4.3.4.1 Multiresolution Analysis

Since a basis consists of linearly independent functions, L2(R) can be written as the

direct sum of subspaces

L2(R) = · · · ⊕W−1 ⊕W0 ⊕W1 ⊕ · · · (4.16)

with

Wm = spanψ(2−mt− n), n ∈ Z, m ∈ Z. (4.17)


Each subspace Wm covers a particular frequency band. For the subband sig-

nals, we obtain from (4.15b) that

gm(t) =∞∑

n=−∞

dm(n)ψm,n(t), gm(t) ∈ Wm. (4.18)

Any signal f(t) can be represented as

f(t) =∞∑

m=−∞

gm(t). (4.19)

Now, we can define the subspaces Xm as the direct sum of subspaces Xm+1 and

Wm+1:

Xm = Xm+1 ⊕Wm+1. (4.20)

Here, we assume that the subspaces Xm contain lowpass signals and the bandwidth

of the corresponding signals decreases with increasing m.

Using the fact that scaling of f(t) by the factor two (f(t)→ f(2t)) makes the

scaled signal an element of a larger subspace

f(t) ∈ Xm ⇔ f(2t) ∈ Xm−1, (4.21)

the subspaces Xm are assumed to be spanned by scaled and shifted versions of a

function φ(t):

Xm = spanφ(2−mt− n), n ∈ Z. (4.22)

Hence, the subband signals fm(t) ∈ Xm can be expressed as

fm(t) =∞∑

n=−∞

cm(n)φm,n(t), (4.23a)


with

φm,n(t) = 2−m/2φ(2−mt− n), (4.23b)

where the function φ(t) is called a scaling function.

4.3.4.2 Wavelet Analysis by Multirate Filtering

Since X0 = X1 ⊕ W1, the functions φ0,n(t) = φ(t − n) ∈ X0 can be written as

linear combinations of the basis functions for the subspaces X1 and W1. Using the

coefficients p0(2l − n) and p1(2l − n), l, n ∈ Z, the approach is

φ0,n(t) =∞∑

l=−∞

p0(2l − n)φ1,l(t) + p1(2l − n)ψ1,l(t), (4.24)

or equivalently,

√2φ(2t− n) =

∞∑

l=−∞

p0(2l − n)φ(t− l) + p1(2l − n)ψ(t− l). (4.25)

The above equations are known as the decomposition relations [69].

Considering a known sequence c0(n) and substituting (4.24) into (4.23a) for

m = 0, we obtain

f0(t) =∞∑

n=−∞

c0(n)φ0,n(t)

=∞∑

n=−∞

c0(n)∞∑

l=−∞

p0(2l − n)φ1,l(t) + p1(2l − n)ψ1,l(t)

=∞∑

l=−∞

∞∑

n=−∞

c0(n)p0(2l − n)

︸︷︷︸c1(l)

φ1,l(t) +∞∑

l=−∞

∞∑

n=−∞

c0(n)p1(2l − n)

︸︷︷︸d1(l)

φ1,l(t)

= f1(t) + g1(t),

(4.26)


P0(z)

P1(z)

P1(z)

P0(z)

2 ↓

2 ↓

2 ↓

2 ↓

c0(n)

c1(m)

d1(m)

d2(l)

c2(l)

Figure 4.4: Analysis filter bank for computing the DWT.

where f0(t) ∈ X0, f1(t) ∈ X1, and g1(t) ∈ W1. Using this approach, we can compute

cm+1(l) and dm+1(l) from cm(n) as follows:

cm+1(l) =∞∑

n=−∞

cm(n)p0(2l − n),

dm+1(l) =∞∑

n=−∞

cm(n)p1(2l − n).

(4.27)

Notice that the sequences cm+1(l) and dm+1(l) occur with half the sampling rate

of cm(n). The decomposition in (4.27) is equivalent to a two-channel filter bank

analysis with the analysis filters h0(n) and h1(n).

Assuming that f0(t) is a good approximation of f(t) and knowing the coef-

ficients c0(n), we can compute the coefficients cm+1(n) and dm+1(n), m > 0, and

hence the values of the DWT using the discrete-time filter bank displayed in Figure

4.4. This is the most efficient way of computing the DWT [69].

4.3.4.3 Wavelet Synthesis by Multirate Filtering

Consider two sequences q0(n) and q1(n) that allow us to express the functions

φ1,0(t) = 2−1/2φ(t/2) ∈ X1 and ψ1,0(t) = 2−1/2ψ(t/2) ∈ W1 as linear combinations of


φ0,n(t) = φ(t− n) ∈ X0, n ∈ Z in the form [69]

φ1,0(t) =∞∑

n=−∞

q0(n)φ0,n(t),

ψ1,0(t) =∞∑

n=−∞

q1(n)φ0,n(t),

(4.28)

or equivalently as

φ(t) =∞∑

n=−∞

q0(n)√

2φ(t− 2n),

ψ(t) =∞∑

n=−∞

q1(n)√

2φ(t− 2n).

(4.29)

(4.28) and (4.29) are referred to as the two-scale relation. For time-shifted functions

the two-scale relation becomes

φ1,l(t) =∞∑

n=−∞

q0(n− 2l)φ0,n(t),

ψ1,l(t) =∞∑

n=−∞

q1(n− 2l)φ0,n(t).

(4.30)

From (4.30), (4.18), (4.20), and (4.23a), one can derive

f0(t) = f1(t) + g1(t)

=∞∑

l=−∞

c1(l)φ1,l(t) +∞∑

l=−∞

d1(l)ψ1,l(t)

=∞∑

l=−∞

c1(l)∞∑

n=−∞

q0(n− 2l)φ0,n(t) +∞∑

l=−∞

d1(l)∞∑

n=−∞

q1(n− 2l)φ0,n(t)

=∞∑

n=−∞

( ∞∑

l=−∞

c1(l)q0(n− 2l)φ1,l(t) + d1(l)q1(n− 2l)

)φ0,n(t)

=∞∑

n=−∞

c0(n)φ0,n(t).

(4.31)


Q1(z)↑ 2

↑ 2

c1(m)

c2(l)

d2(l)

Q0(z)

+

+

Q1(z)↑ 2

↑ 2 Q0(z)

+

d1(m)

c0(n)

Figure 4.5: Synthesis filter bank for the DWT.

The generalization of (4.31) yields

cm(n) =∞∑

l=−∞

cm+1(l)q0(n− 2l) + dm+1(l)q1(n− 2l). (4.32)

The sequences q0(n) and q1(n) can be regarded as the impulse responses of the

discrete-time filters, and (4.32) describes a discrete-time two-channel synthesis filter

bank, which is depicted pictorially in Figure 4.5.

4.3.4.4 The Relationship between Wavelets and Filters

Let us assume that the sets φm,n(t) and ψm,n(t) are orthonormal bases for Xm and

Wm,m ∈ Z, respectively. Taking the inner product of (4.24) with φ1,l(t) and ψ1,l(t)

yields [69]

〈φ0,n, φ1,l〉 =∞∑

m=−∞

p0(2m− n)〈φ1,m, φ1,l〉+ p1(2m− n)〈ψ1,m, φ1,l〉,

〈φ0,n, ψ1,l〉 =∞∑

m=−∞

p0(2m− n)〈φ1,m, ψ1,l〉+ p1(2m− n)〈ψ1,m, ψ1,l〉.(4.33)


Observing that

〈φ1,m, φ1,l〉 = δlm,

〈ψ1,m, ψ1,l〉 = δlm,

〈φ1,m, ψ1,l〉 = 0,

〈ψ1,m, φ1,l〉 = 0,

(4.34)

we obtain

p0(2l − n) = 〈φ0,n, φ1,l〉,

p1(2l − n) = 〈φ0,n, ψ1,l〉.(4.35)

In a similar way, the two-scale relation (4.28) yields

q0(n) = 〈φ1,0, φ0,n〉,

q1(n) = 〈ψ1,0, φ0,n〉.(4.36)

Substituting (4.28) into (4.34), we obtain

δl0 =∞∑

n=−∞

q0(n)〈φ0,n, φ1,l〉,

δl0 =∞∑

n=−∞

q1(n)〈φ0,n, ψ1,l〉,

0 =∞∑

n=−∞

q0(n)〈φ0,n, ψ1,l〉,

0 =∞∑

n=−∞

q1(n)〈φ0,n, φ1,l〉,

(4.37)


and by comparing (4.37) with (4.35), we obtain

δl0 =∞∑

n=−∞

q0(n)p0(2l − n),

δl0 =∞∑

n=−∞

q1(n)p1(2l − n),

0 =∞∑

n=−∞

q0(n)p1(2l − n),

0 =∞∑

n=−∞

q1(n)p0(2l − n).

(4.38)

Now, substituting the two-scale relation (4.30) into (4.35), we get

p0(2l − n) =∞∑

k=−∞

q∗0(k − 2l)〈φ0,n, φ0,k〉,

p1(2l − n) =∞∑

k=−∞

q∗1(k − 2l)〈φ0,n, φ0,k〉.(4.39)

and observing that 〈φ0,n, φ0,k〉 = δnk, we derive

p0(n) = q∗0(−n) ←→ P0(z) = Q0(z),

p1(n) = q∗1(−n) ←→ P1(z) = Q1(z).(4.40)

Therefore, (4.38) becomes

δl0 =∞∑

n=−∞

q0(n)q∗0(n− 2l),

δl0 =∞∑

n=−∞

q1(n)q∗1(n− 2l),

0 =∞∑

n=−∞

q0(n)q∗1(n− 2l),

0 =∞∑

n=−∞

q1(n)q∗0(n− 2l).

(4.41)


and by z-transform of (4.41), we obtain

2 = Q0(z)Q0(z) +Q0(−z)Q0(−z),

2 = Q1(z)Q1(z) +Q1(−z)Q1(−z),

0 = Q0(z)Q1(z) +Q0(−z)Q1(−z),

0 = Q1(z)Q0(z) +Q1(−z)Q0(−z).

(4.42)

4.4 Applications of Wavelet Analysis in Power Sys-

tems

The potential benefits of using wavelet methods for analyzing power-system phe-

nomena have recently been explored. The most popular applications of wavelets in

power systems can be grouped as follows:

• Power quality (including analysis of power-system harmonics, voltage flickers,

and power-system disturbances) [70–91],

• Power-system transient analysis (including analysis of switching events and

transformer inrush, and modeling of line transients) [92–103],

• Power-system protection (including power-system fault detection, identifica-

tion, classification, and location as well as transformer protection) [37, 39–

41, 43, 104–116]

4.4.1 Applications in Power Quality

In the literature, wavelets were first applied to power system transients in the early

1990s. The introductory paper by Ribeiro [70] suggests the possible uses of wavelets

in power systems. It is proposed in the paper that wavelets can be used for various


power-system applications, such as waveform signature and analysis, nonstationary

distortion analysis, distortion propagation, and general power-system analysis.

In [71], the authors use wavelets to accurately reconstruct nonstationary power

system disturbances. Wavelets are applied to several short-term events, namely, a

capacitor switching transient, an autoreclosure, and a voltage dip.

In [72], the authors exploit a dyadic-orthonormal wavelet transform to de-

tect, localize, and examine the classification of numerous power-quality (PQ) dis-

turbances. The main purpose of this paper is to decompose a disturbance into its

wavelet coefficients using MRA techniques. The authors utilize the squared versions

of the wavelet coefficients of the analyzed waveform so that noise is taken out from

the wavelet representation of the disturbance. Using the squared wavelet transform

coefficients at each scale of the disturbance, the authors suggest the use of a classi-

fication tool (e.g., neural networks) for automated classification of PQ disturbances.

In [73], Heydt and Galli propose the use of the Morlet wavelet for the analysis

of voltages and currents that propagate throughout the system due to a transient

disturbance.

Reference [74] makes use of the wavelet transform for the compression of power-

system data. Santoso et al. have achieved to compress PQ disturbance signals.

The motivation behind this study is that due to a vast number of available PQ

disturbance-recording devices, which are being used to monitor the quality of the

power being delivered, there is a massive amount of stored data. From the com-

pressed data, the disturbances are shown to be accurately reconstructed.

In [75], the authors propose the use of the wavelet transform for the analysis

of voltage flicker phenomena. In [76], Angrisani et al. attempt to choose an optimal

wavelet for power quality analysis. The optimal selection of a mother wavelet is a

crucial topic and deserves much attention. The authors choose three PQ disturbances

and analyze them using nine different wavelets in a filter-bank implementation. In


[77], Huang et al. utilize the Morlet wavelet to in the identification of disturbances

such as voltage sag, oscillatory transients, and momentary interruptions as well as

for the harmonic analysis of arc furnace current. In [78], Chen and Meliopoulos

propose a Gaussian-wavelet-based algorithm, which is able to extract the voltage

flicker components by the direct modulation of the voltage signal.

In [79], Pham and Wong present a WT-based approach for the identification

of harmonic contents of power-system waveforms. The proposed procedure involves

the decomposition of waveforms using discrete wavelet packet transform (DWPT)

filter banks and the analysis of nonzero decomposed components using CWT. In

[80], the authors investigate the analysis and subsequent compression properties of

the discrete wavelet and wavelet packet transforms, and evaluate both transforms

utilizing a power-system disturbance from a digital fault recorder. The authors sug-

gest an application of wavelet compression in power monitoring to mitigate against

data communications overheads.

In [81], the authors examine the application of wavelet transform in power-

system transient and time-varying harmonic analyses. Based on the discrete-time-

domain approximation, the power-system components are modeled in discrete wavelet

domain for transient and steady-state analyses. Numerical results from an arc fur-

nace system are also presented in the paper.

In [82], the authors introduce the use of the multiresolution signal decomposi-

tion and the wavelet transform as powerful tools in analyzing transient events. The

ability of multiresolution signal decomposition to detect and localize transient events

as well as to classify different PQ disturbances is shown in the paper with examples.

In [83], the authors present an approach for systemwide subharmonic estima-

tion using state-estimation algorithm together with the WT-based algorithm. The

measured waveforms are analyzed at a few selected nodes in the system using WT-

based algorithm to obtain the subharmonic contents. The number of measured nodes


and the locations of measurements are identified by the symbolic observability anal-

ysis method. The measured results are then used in the harmonic state-estimation

algorithm to estimate the subharmonic levels for the whole system.

In [84] and [85], Santoso et al. provide a thorough theoretical background for

and the implementation of the wavelet-based neural classifier. The point of interest

of this work is the neural networks’ capability to classify a disturbance based on the

wavelet spectrum of the disturbance.

The authors of [86] propose a wavelet transform approach for the analysis of

time-varying power system harmonics using a Morlet wavelet basis function. In order

to show the performance of the approach, the inrush current of a transformer and

the current signals of an arc furnace are utilized as test waveforms.

In [87] and [88], the authors propose the use of wavelet packet transform to

reformulate the power quality indices (e.g., voltage and current total harmonic dis-

tortion, transmission efficiency power factor, and displacement power factor) in the

time-frequency domain. Morsi et al. [89] apply the wavelet analysis for assessing and

monitoring reactive power and energy in the presence of PQ disturbances. A com-

parison of the effectiveness of using different wavelets (i.e., orthogonal, bi-orthogonal,

and reverse bi-orthogonal wavelets) is also made in the paper.

In [90], Ren and Kezunovic propose a method for estimating phasor param-

eters (i.e., frequency, magnitude, and angle) in real time using a recursive wavelet

transform. In addition, an approach for eliminating a decaying DC component is

also proposed by using the recursive wavelet transform. Furthermore, in [91], the

authors propose an adaptive approach for estimating phasors while eliminating the

impact of power transient disturbances on voltages and currents. A wavelet-based-

method is used to identify and locate the disturbance as well as to discriminate it

from noise within a given data window.


4.4.2 Applications in Analysis of Power-System Transients

In [92], Robertson et al. apply a nonorthogonal spline wavelet for the analysis of

capacitor-switching transients. The authors provide a discrete-time implementation

of the wavelet transform via filter bank analysis. The authors make use of the

wavelet decomposition of a transient to analyze the transient features at certain

scales, thereby classifying the disturbance.

Wilkinson and Cox [93] resort to wavelet analysis to decompose an arc furnace

current waveform into its Daubechies-20 wavelet coefficients. In addition, analysis

of machine vibrations is done using the same wavelets.

In [94], Meliopoulos and Lee propose an alternative method for transient anal-

ysis based on wavelet series expansion. The authors solve the algebraic equations for

the given network in terms of the wavelet expansion coefficients of the node voltages,

which can then be reconstructed via the wavelet series reconstruction.

In [95], Styvaktakis et al. present a method for identifying and classifying

the transients, which occur due to synchronized capacitor switching events in three-

phase systems. Switching events are detected in each individual phase by utilizing

the discrete wavelet transform.

In [96], the authors show the application of a dyadic wavelet to detect the signa-

ture of the simulated transformer inrush current signals. In [97], the authors present

an algorithm for transformer inrush identification based on the wavelet packet trans-

form. It is shown in their study that the scheme is capable of distinguishing different

types of transformer inrushes from various transformer internal faults while also dif-

ferentiating external transformer faults from the internal ones.

In [98] and [99], the authors present a method for analyzing transients in

nonuniform transmission lines with nonlinear loads by using the combination of

wavelet expansion and finite elements in time and space domain. The line equations


are transformed into algebraic equations where the differential operator is repre-

sented by a matrix and the unknown variables are the coefficients of the wavelet

expansion of voltages and currents. In [100], the authors present a method for es-

timation of mean curve of impulse voltage waveforms by taking an approach based

on multiresolution signal decomposition. This method is shown to be applicable to

both full and chopped impulses, while enabling removal of high-frequency noise from

the data.

In [101], Magnago and Abur propose an approach to model frequency-dependent

transmission lines based on wavelet transform. In [102], Abur et al. build on the

results of [101] by taking into account the effect of the strong frequency dependence

of modal transformation matrices on the line transients in the time-domain simula-

tions via the use of the wavelet transform of the signals. In [103], the authors further

extend the results of their work by incorporating the previously developed wavelet-

based, frequency-dependent modeling of transmission lines into the simulation of

overall network transients.

4.4.3 Applications in Power System Protection

In [37], Magnago and Abur propose the use of the wavelet transform for the identifi-

cation of fault location in transmission lines. Utilizing the theory of traveling waves

on three-phase transmission lines, and the phase-domain transient signals are first

decomposed into their modal components by means of the modal transformation

matrices. Modal signals are then transformed via the wavelet transform to extract

the travel times between the fault point and relay locations. In [104], the authors

develop a method for the identification of the faulty lateral, and subsequently for lo-

cating a fault in a radial distribution system. Use of time delays between the modal

components of a transient signal is suggested in [105]. The proposed method makes

use of single-ended recordings of the fault signals and processes them via the DWT

in order to calculate the fault distance.


In [39], Evrenosoglu and Abur extend the results presented in [37] to the case

of a three-terminal configuration to determine the location of a fault in a teed circuit.

Again, the procedure is based on wavelet-transform-based processing of fault-induced

traveling waves in order to reveal the arrival times to the measurement points.

In [40], Spoor and Zhu suggest a method for fault location based on continuous

wavelet transform by taking the traveling-wave data in a transmission systems into

account. Using the polarities of wavelet coefficients, both the nature of the fault

and the location of the faults are identified. In [41], Gilany et al. propose a wavelet-

based fault-location scheme for the multiend system of aged power cable lines. The

proposed scheme utilizes the fault-originated voltage traveling waves, while requiring

the knowledge of cable lengths and the sample of voltage waveforms at both ends of

each cable.

In [43], the authors propose a traveling-wave-based protection method that

makes use of principal component analysis (PCA) to identify the dominant pattern

of the signals preprocessed by wavelet transform. The needed information from

voltage and current signals are extracted by wavelet transform at different scales. In

the proposed method, the PCA method enables better detection of traveling waves

in the case of close-in and faint faults.

In [111], Livani and Evrenosoglu make use of the discrete wavelet transform to

obtain the transient information from the recorded voltages in three-terminal lines.

The authors make use of support vector machines to classify fault types and faulty

section in the transmission systems, and wavelet transform coefficients are then used

to locate the fault. In [112], the authors take the similar approach to identify the

fault point in a hybrid transmission line, where an overhead line is combined with

an underground cable.

Reference [106] presents a method for real-time classification of transmission-

system faults using fuzzy-logic-based multicriteria technique. The line currents are

processed using online wavelet transform algorithm.


In [107], Michalik et al. propose a wavelet-based algorithm for detection of

high-impedance ground faults in distribution networks. The ground faults are de-

tected by the estimation of phase displacement between wavelet coefficients of zero-

sequence voltage and current waveforms.

In [108], Borghetti et al. present a fault-location algorithm for distribution net-

works based on the analysis of fault-generated traveling waves by means of the CWT.

In addition, the authors propose an algorithm to build specific mother wavelets in-

ferred from the recorded fault-originated voltage transients. In [109], the authors

propose a slightly different procedure for fault location for distribution networks by

utilizing integrated time-frequency wavelet decompositions of the voltage transients

associated with the fault-originated traveling waves. Contrary to [108], their new

approach integrates both time and frequency information obtained from the wavelet

decompositions of the fault-transient signals.

In [110], Nanayakkara et al. propose an algorithm to locate faults on HVDC

transmission lines consisting of both cable and overhead-line segments. In their

method, continuous wavelet transform coefficients of the input signal are utilized

to identify the arrival time of wavefronts at dc line terminals. Specifically, the

DC voltage measured at the converter terminal and the current through the surge

capacitors connected at the DC line ends are examined.

In [113], Mao and Aggarwal present a technique that discriminate between an

internal fault and a magnetizing inrush current in the power transformer by combin-

ing wavelet transforms with neural networks. In particular, the wavelet transform is

first applied to decompose the differential current signals of the power transformer

into a series of wavelet components, signals energies of which are computed and

exploited to train a neural network to distinguish between the faults.

In [114], the authors suggest the use of decision trees and wavelet analysis for

the protection of power transformers. In [115], Saleh and Rahman propose an al-

gorithm for differential protection of three-phase power transformers based on the


wavelet packet transform (WPT). In the presented method, the selection of both

the optimal mother wavelet and the optimum number of resolution levels is per-

formed using the minimum description length (MDL) data criteria. In [116], the

authors develop a WPT-based differential relay for protecting power transformers

using Butterworth passive (BP) filters. In order to detect fault currents, the BP fil-

ters are designed to extract second-level high-frequency components of three-phase

differential currents.

4.5 Summary

In this chapter, we highlight the superiorities of the wavelet analysis over the Fourier

analysis and give an overview of the theory of wavelets. We also list the applications

of wavelet analysis to various domains in power systems.

Chapter 5

Traveling-Wave-Based Fault

Location in Power Networks

5.1 Introduction

Operation of the existing power grids is rapidly going through major changes due to

the widespread deployment of synchronized measurement systems. These systems

provide unprecedented advantages in wide-area monitoring of power grids due to

the availability of synchronization among measurements at geographically remote

parts of the system. So far, most of the investigations have focused on the use

of synchronized measurements to improve applications, which require monitoring

and control actions at relatively slow rates, i.e., slow enough to make treatment of

slowly changing system conditions via phasors, which implicitly assume steady-state

operation.

While effectiveness and benefits of synchronized measurements have been well-

documented for such applications, other applications requiring monitoring of the

system conditions at a much shorter time span have not yet been fully explored.

75

Chapter 5. Traveling-Wave-Based Fault Location in Power Networks 76

One of the challenges in secure and reliable operation of power grids is to rapidly de-

tect, identify, and isolate faults, which occur due to unexpected equipment failures,

lightning storms, accidental short circuits, etc. Such faults can cause significant dam-

age if not cleared in a matter of fractions of a second. Hence, power-grid protection

systems have so far been designed as control systems that used local measurements

as decision variables. In this regard, power systems would fully exploit the potential

of synchronized nature of the sampled voltage and current signals and the capability

to access these synchronized values via systemwide communication infrastructure,

enabling accurate and fast methods of fault location and removal.

The work to be introduced in this chapter is intended to facilitate the uti-

lization of synchronized measurements, which are rapidly populating today’s power

transmission grids. Unlike the existing applications which focus on synchronized

phasors, this study will illustrate the use of “raw” synchronized measurements for

a practical application, namely, for fault location. Many transmission grids do not

have synchronized measurements at every bus, but at only a few selected buses. The

results of this work will enable accurate and reliable fault location by using these

few strategically located synchronized measurements. The broader impact will be a

reduction in the duration of service interruptions, and consequently reduced loss of

revenues for the industry and higher reliability for operation of transmission grids.

5.2 Proposed Fault-Location Methodology

Transmission grid can be modeled as a weighted graph G = (V,E) consisting of a

set of |V| = N vertices (buses) and |E| = L edges (transmission lines), together with

a distance measurement (transmission-line length), d`L`=1, associated with each

edge. The waveform generated by a fault occurring on a transmission line can be

decomposed into several electromagnetic-transient waveforms, each representing a

mode and propagating throughout the network at its own propagation speed ν. For


each mode, by recording the instants at which the wavefronts of the fault-generated

waveform arrive at key buses in the system, the location of a fault can be identified.

We shall assume that K sensors are deployed in the network, each one measur-

ing the time of arrival (ToA) of the fault-originated traveling wave. The data set,

Tk; 1 ≤ k ≤ K, of these ToAs should allow us to determine:

(a) which transmission line (arc) has experienced a fault;

(b) at what point on this transmission line did the fault occur; and

(c) the time instant at which the fault has occurred.

We discuss here only a single-fault event. In addition to the topology (graph struc-

ture) of the transmission grid, we also know the propagation time for each trans-

mission line, which depends both on the length of the line and the speed of the

wave propagation along the line. For a given system with L transmission lines, the

propagation times, D` = d`/ν`; 1 ≤ ` ≤ L, are known in advance.

The time of propagation (i.e., delay) from the point of fault occurrence to

Sensor “k” depends on the network topology, the propagation times, D`, and

three unknown quantities:

(i) the identity of the faulty line (say, “`”);

(ii) the location of the fault on the line (say, α(`)D`, from a designated end of

the line, so that 0 ≤ α(`) ≤ 1); and

(iii) the instant, T(`)0 , of the fault occurrence.

Thus, Tk − T (`)0 = ζk,`(α

(`)), which gives us an overdetermined system of equations,

assuming that K > 3. Our challenge is to solve this system of equations for T(`)0 ,

`, and α(`), taking into account limited accuracy of the measurements, Tk, and of

the propagation times, D`.


5.2.1 The Functions “ζk,`(α(`))”

Assuming that the fault occurred on Line “`”, the shortest propagation time from

the point of fault occurrence to Sensor “k”, which we denote as ζk,`(α(`)), belongs

to a path in the graph that must include one of the two endpoints (nodes) of the

faulty line. We shall designate a priori one of these endpoints as the line origin and

measure the distance to the point of fault occurrence from this end. We shall call

the opposite endpoint the terminus of the line. Since we do not know in advance

which endpoint of the line lies on the shortest path from the point of fault occurrence

to Sensor “k”, we conclude that

ζk,`(α(`)) = min

D(o)k,` + α(`)D`, D(t)

k,` + (1− α(`))D`

. (5.1)

where D(o)k,` is the delay along the shortest path from the origin of Line “`” to Sensor

“k”, and similarly, D(t)k,` is the delay along the shortest path from the terminus of

the same line to the same sensor. The delays, D(o)k,` and D(t)

k,`, can be determined in

advance for every “k” and every “`”. The pictorial representation of the described

approach, along with the related terms, is highlighted in Figure 5.1.

The expression “D(o)k,` + α(`)D`” describes a straight line with a positive slope,

while “D(t)k,` + (1− α(`))D`” describes a straight line with a negative slope as shown

in Figure 5.2. The plot in this figure assumes that D(o)k,` < D

(t)k,` +D`. A similar plot

can be drawn under the alternative assumption (that D(o)k,` > D

(t)k,` +D`). Notice that

we must always have∣∣D(o)

k,` −D(t)k,`

∣∣ ≤ D`, (5.2)

so that we have either

D(o)k,` ≤ D

(t)k,` +D` (5.3)


Sensor“k”

origin

terminus

D(o)

k,ℓ

D(t)

k,ℓ

︷︸︸︷

︷︸︸︷α

(ℓ)Dℓ

Dℓ

point of faultoccurrence

Buses

Line“ℓ”

Figure 5.1: Illustration of the terms, “origin” and “terminus”, as well as the

respective propagation delays, D(o)k,` and D(t)

k,`, along the shortest path from Sensor“k” to faulty Line “`”.

or

D(t)k,` ≤ D

(o)k,` +D`. (5.4)

A more compact expression for ζk,`(α(`)) can be expressed in terms of (βk,`, ξk,`)—

the point where the two straight lines intersect. From

D(o)k,` + α(`)D` = D(t)

k,` + (1− α(`))D`, (5.5)

we conclude that

2α(`)D` = D(t)k,` −D

(o)k,` +D` ≥ 0, (5.6)

so that

βk,` =D(t)k,` −D

(o)k,` +D`

2D`

(5.7a)


D(o)k,ℓ + α(ℓ)Dℓ

D(t)k,ℓ + (1− α(ℓ))Dℓ

D(t)k,ℓ +Dℓ

D(o)k,ℓ

βk,ℓ

ξk,ℓ

ζk,ℓ(α(ℓ))

1 α(ℓ)0

Figure 5.2: The intersection of the lines “D(o)k,`+α

(`)D`” and “D(t)k,`+(1−α(`))D`”.

and βk,` ≥ 0. Also, in view of (5.2), D(t)k,` −D

(o)k,` ≤ D`, so βk,` ≤ 1, viz.,

0 ≤ βk,` ≤ 1. (5.7b)

Thus, the alternative expression is

ζk,`(α(`)) = ξk,` − |α(`) − βk,`|D`, (5.8)

where

ξk,` , ζk,`(βk,`) =D(t)k,` +D(o)

k,` +D`

2. (5.9)


The required network information is completely captured by the row vector

D =[D1 D2 · · · DL

](5.10a)

and the two K × L matrices

B =[βk,`]

(5.10b)

and

Ξ =[ξk,`], (5.10c)

where k ∈ 1, · · · , K and ` ∈ 1, · · · , L. In addition, the set of sensor measure-

ments defines a column vector

T =[T1 T2 · · · TK

]>. (5.10d)

The function ζk,`(α(`)) becomes linear when either βk,` = 0 or βk,` = 1. Both

cases correspond to∣∣D(o)

k,` −D(t)k,`

∣∣ = D`. Indeed:

(a) When D(o)k,` = D(t)

k,` +D`, we get βk,` = 0 and ξk,` = D(o)k,` [see Figure 5.3(a)].

(b) When D(t)k,` = D(o)

k,` +D`, we get βk,` = 1 and ξk,` = D(t)k,` [see Figure 5.3(b)].

5.2.2 A Nonlinear Optimization Problem

The system of equations we need to solve is

T1

T2

...

TK

︸︷︷︸T

−T (`)0

1

1...

1

︸︷︷︸η

=

ζ1,`(α(`))

ζ2,`(α(`))

...

ζK,`(α(`))

︸︷︷︸ζ`(α

(`))

. (5.11)


1 α(ℓ)

D(o)k,ℓ

ζk,ℓ(α(ℓ))

D(t)k,ℓ

0

1 α(ℓ)

ζk,ℓ(α(ℓ))

D(o)k,ℓ

D(t)k,ℓ

0(a) (b)

Figure 5.3: The function ζk,`(α(`)) when (a) βk,` = 0 and (b) βk,` = 1.

It is linear in T0, piecewise-linear in α(`), and highly nonlinear in the integer

index “`”. To address issues of limited accuracy, we redefine our problem as the

constrained optimization problem (with any norm of choice) as follows:

minimize`, α(`), T

(`)0

∥∥T− T (`)0 η − ζ`(α(`))

∥∥ (5.12a)

subject to 0 ≤ α(`) ≤ 1; ` ∈ 1, 2, · · · , L. (5.12b)

5.2.2.1 A Two-Step Optimization Approach

One way to solve the optimization problem (5.12) is to split our optimization effort

into two subtasks:

(I) Fix “`” and determine the optimizing T0 and α values for the given “`”, say,

T(`)0 and α(`).

(II) Search over “`” values to minimize the modified cost function

∥∥T− T (`)0 η − ζ`(α(`))

∥∥. (5.13)


If we choose ‖ ·‖ as the standard (Euclidean) vector norm, then the derivatives

∂

∂T(`)0

∥∥T− T (`)0 η − ζ`(α(`))

∥∥2

2(5.14a)

and∂

∂α(`)

∥∥T− T (`)0 η − ζ`(α(`))

∥∥2

2(5.14b)

can be determined in closed form, allowing a closed-form expression for T(`)0 and α(`).

Also, in searching for “`”, we can exclude transmission lines that are too far from

the set of sensors. For instance, we can restrict our search to those arcs that are

closest to the sensor with the earliest Tk.

5.2.2.2 A Sensor-Guided Line-Splitting Approach

One way to facilitate obtaining closed-form expressions for T(`)0 and α(`) is by “lin-

earizing” the dependence of ζk,`(α(`)) on the variable α(`). This can be achieved by

splitting the `th transmission line at the points defined by βk,`. We first sort the

set βk,`; 1 ≤ k ≤ K in ascending order, say,

0 ≤ βk1,` ≤ βk2,` ≤ · · · ≤ βkK ,` ≤ 1, (5.15)

and then introduce a virtual node at each one of the points “βki,`D`” as depicted in

Figure 5.4.

origin terminus

βk2,ℓDℓβk1,ℓDℓ βkK ,ℓDℓ· · ·

Figure 5.4: The virtual nodes generated at the points “βki,`D`”.

The number of line segments (“virtual arcs”) created in this way does not

exceed K + 1. If some of the βk,` are equal to zero or unity, this number will


be smaller. In any case, the number of arcs in the new graph defined by the split

transmission lines does not exceed (K+1)L. This is the upper bound on the number

of “`” values we will need to consider.

In this new graph, βk,` ∈ 0, 1 are the only possible values for every line

segment. Notice that we have now redefined “`” as an index of a line segment, so

that 1 ≤ ` ≤ Lmax and Lmax ≤ (K + 1)L. Now, ζk,`(α(`)) is linear in α(`), viz.,

ζk,`(α(`)) = ξk,` − α(`)D`

= D(o)k,` − α(`)D` (5.16)

when βk,` = 0 (recall discussion in Section 5.2.1), and

ζk,`(α(`)) = ξk,` −D` + α(`)D`

= (D(t)k,` −D`) + α(`)D`

= D(o)k,` + α(`)D` (5.17)

when βk,` = 1.

We can write this compactly as

ζk,`(α(`)) = D(o)

k,` + Sk,`α(`)D`, (5.18a)

where Sk,` = 2βk,` − 1, or, simply,

Sk,` =

1 if βk,` = 1,

−1 if βk,` = 0(5.18b)


Now, by letting %(`) = α(`)D`, our cost function becomes

J` =∥∥D` + %(`)S` −T + T

(`)0 η

∥∥2

2, (5.19a)

where

D` =

D(o)1,`

D(o)2,`

...

D(o)K,`

and S` =

S1,`

S2,`

...

SK,`

. (5.19b)

By setting up the equations

∂J`∂%(`)

= 2ST` (D` + %(`)S` −T + T

(`)0 η) = 0 (5.20)

and∂J`∂T

(`)0

= 2ηT(D` + %(`)S` −T + T(`)0 η) = 0, (5.21)

we obtain closed-form expressions for %(`) and T(`)0 . The partial-derivative expressions

give us K S>

` η

η>S` K

%

(`)

T(`)0

=

S>

` (T−D`)

η>(T−D`)

, (5.22)

which is a set of two linear equations. Rewriting the inner products as

S>` η = η>S` =

K∑

k=1

Sk,`, (5.23)

a “correlation coefficient” can be defined such that

ρ` ,1

K

K∑

k=1

Sk,`, (5.24a)


so that we obtain

K

1 ρ`

ρ` 1

%

(`)

T(`)0

=

[S` η

]> (T−D`

). (5.24b)

Notice that most of the quantities in this set of equations can be determined a

priori and stored. Only the “products” S>` T and η>T must be evaluated after the

fault has occurred and the ToAs, Tk, have been measured. In fact, these products

are sums, namely,

η>T =K∑

k=1

Tk

S>` T =

K∑

k=1

Sk,`Tk (Sk,` = ±1). (5.25)

Returning to the original cost function

J` =

∥∥∥∥∥∥(D` −T

)+[S` η

]%

(`)

T(`)0

∥∥∥∥∥∥, (5.26)

we obtain, via (5.24),

J` =

∥∥∥∥∥∥

IK − 1

K(1−ρ2` )

[S` η

] 1 −ρ`−ρ` 1

[S` η

]>(T−D`

)∥∥∥∥∥∥

(5.27)

or

J` =∥∥∥M`

(T−D`

)∥∥∥ , (5.28a)


whereM` is the easily computable matrix

M` = IK −1

K(1− ρ2`)

[S` η

] 1 −ρ`−ρ` 1

[S` η

]>, (5.28b)

which represents, in fact, an orthogonal projection. This matrix depends only on

the column vector S`, and can be easily reconstructed whenever S` is available.

Since the expression inside the norm in (5.26) is linear in both %(`) and T(`)0 ,

the two-step optimization approach proposed in Section 5.2.2.1 can be implemented.

The resulting modified cost function is optimized by an integer search over a subset

of “`” values, determined by the proximity of the corresponding transmission lines

to the sensor with the smallest Tk.

5.3 Practical Implementation

Our discussion in Section 5.2 portrays the analytical fragments of the proposed fault-

location technique. Different from the preceding section, this section shows the stages

related to the computational constituents of the overall fault-location procedure as

well as the validation of its performance on a sample transmission grid.

5.3.1 Fundamentals and Stages of the Implementation

In this subsection, the principal steps establishing the fault-location technique are

briefly discussed. In order to estimate the fault point precisely, the measured voltage

waveforms are initially converted to their modal components using Clarke’s real


transformation matrix introduced in Appendix A.3.2, which is given by

V0

V1

V2

=

1√3

1 1 1√

2 − 1√2− 1√

2

0√

3√2−√

3√2

Va

Vb

Vc

(5.29)

since all transmission-line models are assumed to be fully transposed. Note that, in

(5.29), Va, Vb, and Vc denote the phase voltages; V0 is the ground-mode voltage; and

V1 and V2 are the aerial-mode voltages. Then, the modal components are processed

through the DWT and the squares of the wavelet-transform coefficients (WTC2s)

are retrieved and employed to detect the ToA instant of the fault-initiated traveling

wave at which signal energy reaches its first local maximum. During the course of

simulations, Daubechies-8 mother wavelet [117] with the level-4 approximation co-

efficients is chosen for the wavelet transformation. At the same time, aerial-mode

voltage (e.g., V1) WTC2s in scale-1 have formed a basis for the fault-location com-

putations. The stages related to the computational part of the devised methodology

is shown in Figure 5.5.

5.3.2 Computation of the Shortest Propagation Delays

Recall the discussion on calculation of shortest propagation time in Section 5.2.1.

Further, assume that the transmission grid consists of N buses and L transmission

lines. For a particular bus on which a sensor is deployed (i.e., at which the transient-

voltage signal is monitored), the calculation of the arrival time of the fault-initiated

traveling wave to that bus is performed via (5.1). In the computation of the short-

est propagation delay for each pair of buses, Dijkstra’s well-known algorithm for

shortest-path computation [118] has been employed, recognizing the fact that trans-

mission grids can be thought of as real-weighted undirected graphs. It should be

noted that these computations need to be carried out only once for a given network

topology. If the topology changes due to any line switching, these calculations will


ModalTransformation

Discrete WaveletTransform

WTC2

V0

V1

V2

=

1√3

1 1 1√2 − 1√

2− 1√

2

0√3√2

−√3√2

Va

Vb

Vc

Extract ToA oftraveling waves

Obtain

FaultedPhaseVoltages(Va, Vb, Vc)

⇒

(of V1)

Aerial-ModeVoltages

Figure 5.5: Computational stages of the devised fault-location algorithm.

have to be repeated. Identically, after the line-splitting process described in Section

5.2.2.2 is implemented, the shortest propagation times have to be recalculated for

the newly formed network consisting of fictitious transmission lines.

It is worth pointing out that the best-case computational complexity of Dijk-

stra’s shortest-path algorithm is proportional to O(L log(N)). As we have mentioned

earlier, this cost is considered admissible since the computational effort can definitely

be shifted to the offline phase.


5.3.3 Simulation Results

All simulations are carried out in Alternative Transients Program (ATP) and MAT-

LAB with a sampling frequency of 1 MHz corresponding to a sampling time interval

of 1 µs. The fault-occurrence time is chosen to be 20 ms. In addition, the tower con-

figuration of transmission lines is retrieved from [39]. Transmission lines represented

by frequency-dependent models are utilized throughout the simulations.

Preceding the simulations of various fault scenarios, all transmission lines are

modeled as balanced, lossless, and fully transposed lines. The aerial-mode prop-

agation speed in scale-1 is calculated as 1.85885 × 105 mi/s. Also, for the sake

of simplicity, all of the transmission-line configurations are assumed to be identi-

cal in order to avoid the differences in traveling-wave speeds for each transmission

line. However, the proposed method is evidently applicable to transmission grids

with varying line configurations since wave-propagation time for each transmission

line is defined by (3.37). The traveling-wave speeds can be extracted based on the

knowledge of the electrical characteristics of all transmission lines in the power grid.

The studied system has been simulated under short-circuit faults along vari-

ous line segments to assess the overall performance of the proposed fault-location

technique. Line lengths, along with wave-propagation times, are provided in Table

5.1.

Now, consider the modified IEEE 30-bus system whose single-line diagram

is depicted in Figure 5.6. At first, we simulate a short-circuit fault occurring 28

miles away from Bus 10 on a 65-mile-long transmission line connecting Buses 10 and

20. Meanwhile, the three-phase synchronized measurements of voltages are assumed

to be available. Then, aerial-mode WTC2s for each modal voltage are obtained

following the decoupling of the phase quantities into the modal voltages. Four of

these faulted voltage measurements and the pertinent WTC2s of the aerial-mode

voltages are illustrated in Figures 5.7 and 5.8, respectively.


Table 5.1: Transmission-Line Lengths and Wave-Propagation Times for theModified IEEE 30-Bus Test System

Length Time Length TimeLine

(mi) (µs)Line

(mi) (µs)

1− 2 42 225.95 12− 13 19 102.221− 3 274 1, 474.05 12− 14 228 1, 226.592− 4 278 1, 495.57 12− 15 271 1, 457.912− 5 51 274.37 12− 16 99 532.602− 6 241 1, 296.52 14− 15 149 801.583− 4 80 430.38 15− 18 37 199.054− 6 112 602.53 15− 23 135 726.274− 12 27 145.25 16− 17 155 833.865− 7 173 930.70 18− 19 230 1, 237.346− 7 31 166.77 19− 20 187 1, 006.016− 8 216 1, 116.20 21− 22 67 360.446− 9 194 1, 043.67 22− 24 155 833.866− 10 165 887.66 23− 24 138 742.416− 28 139 747.79 24− 25 240 1, 291.148− 28 193 1, 038.29 25− 26 119 640.199− 10 96 516.46 25− 27 234 1, 258.869− 11 61 328.17 27− 28 192 1, 032.9110− 17 116 624.05 27− 29 238 1, 280.3810− 20 65 349.68 27− 30 162 871.5210− 21 220 1, 183.55 29− 30 111 597.1510− 22 204 1, 094.70

Optimal placement of synchronized recording instruments is a problem inter-

related with fault-location algorithm. The relevant problem formulation and its

associated solution will be covered in Chapter 6. The locations of synchronized

recorders are strategically specified in such a way that a short-circuit fault occur-

ring anywhere in the transmission grid can be located uniquely and using a minimal

Table 5.2: Synchronized Meter Locations versus Wave-Arrival Times for theShort-Circuit Fault Occurring on Line 10-20

Buses 1 3 5 8 11 13 14ToAs (ms) 22.560 22.068 22.132 22.198 20.990 21.886 23.010

Buses 17 18 21 26 29 30ToAs (ms) 20.770 22.438 21.328 24.006 24.094 23.686


1 2

3 4

5

6

7

89

10

11

12

13

14

15

16

17

18 1920

21

2223 24

25

26

27

29 30

28

synchronized voltage sensors

Figure 5.6: Single-line diagram of the modified IEEE 30-bus test system.

number of such recorders.1 The chosen locations for the synchronized recorders in

the studied network are listed in Table 5.2. The table also illustrates the instants

when the first local peaks of WTC2s are detected via the synchronized recorders on

the respective buses. Hence, the captured times in milliseconds are stored in the

(13 × 1)-column-vector T right after the occurrence of the fault. As explained in

Section 5.3.1, these instants refer to the ToA measurements of the fault-generated

traveling waves at the corresponding substations. With these ToA measurements,

it is easy to verify that the sensors at Buses 11 and 17 are anchored within close

1Reckoning with the concept of fault-location observability, a different optimal deploymentstrategy for synchronized recorders was devised in [119].


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'

Time (ms)

Voltage(kV)

Time (ms)Voltag

e(kV)

Time (ms)

Voltage

(kV)

Time (ms)

Voltage

(kV)

Three-Phase Voltages at Bus 1 Three-Phase Voltages at Bus 17

Three-Phase Voltages at Bus 21 Three-Phase Voltages at Bus 29

Figure 5.7: Faulted phase voltages at Buses 1, 17, 21, and 29 after the occurrenceof a short-circuit fault on Line 10-20.

proximity to the fault location; however, Buses 26 and 29 are the two most distant

substations with respect to the location of the fault.

Succeeding the optimal deployment of the synchronized recorders, the line-

splitting technique is employed so that the regenerated network consists of 110 buses

and 121 transmission-line segments after the inclusion of virtual nodes and arcs.

Obviously, the total number of buses and line segments constructed in this way is

dependent upon the grid topology, the locations of the synchronized measurements,

and the lengths of the transmission lines. Notice that for the transmission system

under investigation, the number of transmission-line segments is considerably less


Time (ms)W

TC

2

WTC

2

Time (ms)22.6622.48 22.52 22.56 22.6

0.26

0

0.05

0.1

0.15

0.2

20.8220.58 20.62 20.66 20.7 20.74 20.78

9000

0

1000

2000

3000

4000

5000

6000

7000

8000

WTC2 of the Aerial-Mode Voltage at Bus 1 WTC2 of the Aerial-Mode Voltage at Bus 17

Time (ms)

WTC

2

WTC2 of the Aerial-Mode Voltage at Bus 29

21.721.25 21.3 21.35 21.4 21.45 21.5 21.55 21.6 21.65

1100

0

100

200

300

400

500

600

700

800

900

1000


WTC

2

Time (ms)24.2223.98 24.02 24.06 24.1 24.14 24.18

0.0024

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0.0022

Figure 5.8: WTC2s of the aerial-mode voltages at Buses 1, 17, 21, and 29 afterthe occurrence of a short-circuit fault on Line 10-20.

than their maximum possible number, i.e., (K + 1)L = (13 + 1)× 41 = 574.

For the fault-scenario example above, the minimizing value of the cost function,

i.e., J` = 0.0057 ≈ 0, is attained on Line 59 in the resulting split network, following

labeling the line segments. In the meantime, the aforementioned line-splitting tech-

nique is found not applicable to Line 10-20. As a result, the corresponding values of

%(`) and T(`)0 are found to be

%

(59)

T(59)0

=

0.1506

19.9959

ms.


with the associated 13× 1 matrices being calculated as

D59 =

2.4106

1.9206

1.9852

2.0497

0.8447

1.7377

2.8625

0.6241

2.5927

1.1840

3.8621

3.9485

3.5400

and S59 =

1

1

1

1

1

1

1

1

−1

1

1

1

1

.

Here, the value “−1” in the column vector S59 implies that the shortest distance

from the sensor at Bus 18 to Line 10-20 is from the terminus side; whereas, the

values “1” mean that the sensors at the other buses are closer to the same line from

the origin than from the terminus side. Furthermore, the value of T(59)0 is noticeably

close to the actual instant of fault occurrence. Nonetheless, identifying the location

of the fault should be the focal point of our investigation. Accordingly, Figure 5.9

displays the location of the fault on Line 10-20 in terms of the propagation delay,

%(`), associated with the fault. The distance to fault from the origin (i.e., Bus 10) of

Line 10-20 is then computed to be

%(59)ν = (1.506× 10−4 s)× (1.85885× 105 mi/s) = 27.9943 mi.


10 20︸︷︷︸(59) = 0.1506

D10−20 = 0.3497 ms

Figure 5.9: The value of %(`) for the short-circuit fault occurring on Line 10-20.

To serve as a metric for the fault-location accuracy, one can specify the absolute

percentage error (APE) in fault-location estimate based on the total transmission-

line length, according to [120] such that

APE =

∣∣xfault − xfault

∣∣d`

× 100, (5.30)

where xfault is the estimated fault location found by the proposed algorithm and

xfault is the actual distance to fault from the sending end of the faulty transmission

line.

Utilizing the result above, one obtains the APE as follows:

APE =

∣∣27.9943− 28∣∣

65× 100 = 0.0088%.

This certainly indicates a negligible percentage value for an error. Indeed, the

distance-measurement error, ε, will be on the order of only a few meters, viz.,

ε = (28− 27.9943)× 1, 609.34 = 9.1732 m.

In yet another scenario, a short-circuit fault is simulated at the point 260 miles

away from Bus 12 on the 271-mile-long transmission line connecting Buses 12 and

15. According to results of this scenario, the location of the fault is detected on the

virtual line segment (i.e., Line 79) connecting Terminals 15 and 82, as demonstrated


12 1580 81 82

0.2152 0.3013 0.1345

D12−15 = 1.4580 ms

︸︷︷︸(79) = 0.7477

0.8070

Figure 5.10: The value of %(`) for the short-circuit fault occurring on Line 12-15.

in Figure 5.10. The numbers shown right below the virtual line segments represent

the calculated propagation times (in milliseconds) for each of these lines. Similar to

the previous case, the distance to fault from Bus 12 is found to be

d =((2.152 + 3.013 + 1.345 + 7.477)× 10−4 s

)× (1.85885× 105 mi/s) = 259.9974 mi,

with

APE =

∣∣259.9974− 260∣∣

271× 100 = 0.001%

and

ε = (260− 259.9974)× 1, 609.34 = 4.1843 m.

5.4 Summary

This chapter presents a fault-location procedure for large-scale power grids based on

wide-area synchronized voltage measurements. The method relies on synchronized

measurements of transient voltage samples during faults by sparsely distributed

fault-recording devices which utilize GPS receivers. The introduced procedure is

realized through the processing of traveling waves by DWT in order to extract the

arrival times of fault-initiated waves. The main advantage of this method is that it


requires a few and strategically deployed synchronized measuring devices to locate

faults for a large transmission system.

Based on fault-location observability, a problem formulation for synchronized

meter placement to ensure accurate fault location for the entire system will be given

and illustrated with examples in the following chapter.

Chapter 6

Optimal Deployment of

Synchronized Sensors Based on

Fault-Location Observability

6.1 Introduction

With the ever-increasing deployment of synchronized wide-area measurements, novel

control and protection functions are being investigated to improve system stability,

protection, and reliability [121]. Postdisturbance analysis requires accurate infor-

mation from multiple transmission substations, where synchronized measurements

are made available by highly accurate GPS-synchronized phasor measurement units.

While such measurements are very useful for capturing events that occur in pseudo-

steady state, fast transients such as those caused by short-circuit faults will require

capturing samples at much higher resolution, i.e., in the order of microseconds. In

this chapter, it is assumed that such high-resolution raw samples of voltages are

available from synchronized intelligent electronic devices.

99

Chapter 6. Optimal Deployment of Synchronized Sensors Based on Fault-LocationObservability 100

When analyzing wide-area disturbances in a utility system, the value of wide-

area disturbance-recording devices becomes more apparent, especially when discrete

records are simultaneously captured at key substations to provide high-resolution

snapshots from a disturbance event. While existing transmission grids have not yet

extensively adopted the wide-area disturbance-monitoring sensors (such as intelli-

gent electronic devices) for capturing transient signals to postmortem-analyze power

system disturbances, their deployment appears imminent based on various studies

indicating potential benefits to be derived from such devices for enhanced wide-area

disturbance monitoring.

The main motivation of this chapter is to extend the study of fault location

based on wide-area synchronized sensors and develop a method to optimally place

these time-synchronized measurements in order to achieve fault-location observabil-

ity over the entire transmission grid. The concept of fault-location observability is

not entirely new and was used earlier by Lien et al. [119]. It refers to the unique

localizability of any fault-occurrence point in a power grid using a set of readily

deployed fault-recording devices. Likewise, recent work by Liao [122] proposed an

optimal meter placement methodology to reliably and uniquely estimate the loca-

tion of faults in transmission networks based on fault-location observability analysis.

Using an approach based on integer linear programming, Avendano-Mora and Mi-

lanovic [123] also formulated an optimal placement algorithm for fault-monitoring

devices to achieve a full fault-location observability. Unlike these studies, which

exploit impedance-based techniques for locating faults in a grid, the observability

investigation in this study is based upon a traveling-wave-based fault-location pro-

cedure. Taking into account the topology of the transmission grid and using the

previously derived technique for fault location in Chapter 5, an optimal strategy

for placing synchronized measurements is developed. This chapter will describe this

optimal scheme for strategically deploying a minimum number of synchronized volt-

age sensors in a transmission system so that faults can be detected and identified

irrespective of their locations.


6.2 Proposed Formulation for Optimal Sensor

Deployment Problem

It is evident from (5.22) that if the off-diagonal entries,K∑

k=1

Sk,`, are equal to ±K, the

coefficient matrix will become singular; that is, a fault occurring on corresponding

Line Segment “`” cannot be detected. Thus, it is essential we take into account

those infeasible cases before selecting the locations for sensors. In the sequel, we

will introduce the formulation for the optimal placement of synchronized sensors,

through which we address these unsolvable cases and provide the needed solution.

In general, a system is regarded as “fault-observable” if any fault occurring in

the system can be uniquely located using the available set of measurements. Other-

wise, the system is said to be “fault-unobservable”. Those line segments which are

constituted by points whose faults cannot be uniquely identified will be referred to as

“fault-unobservable” line segments. Given the substation locations of the synchro-

nized sensors, the analysis of determining fault-observable and fault-unobservable

segments of the network lines will be referred to as fault-location observability anal-

ysis. It should be noted that fault-location observability is directly related to the

number and location of the installed sensors and is independent of the captured

ToAs. More precisely, optimal sensor deployment is an offline process and place-

ment strategy depends only on the knowledge of grid topology and transmission-line

lengths.

The values “−1” in the column vectors S` in (5.19b) imply that the shortest

distances from the sensors at the corresponding buses to Line “`” are from the

terminus side; whereas, the values “1” indicate that the sensors placed at those buses

are closer to the same line from the origin than from the terminus side. Therefore,

the line segments, for which the elements Sk,` are all equal to “−1” or “1” in (5.23),

will be regarded as unobservable (or “nonlocalizable”) segments.


In order to design a sensor system that ensures full network fault observability,

one simple solution will be to place a synchronized voltage sensor at every bus. While

feasible, this will be a costly solution. A better and more systematic solution can be

obtained by formulating the problem as a binary integer programming problem. The

solver of this binary integer programming problem will then search for a reduced set

of sensors which will accomplish the same task. Furthermore, as discussed in Section

5.2.2.2, every line will be split into several segments by introducing virtual buses in

order to simplify the optimization formulation. As a result, a large number [i.e., L

(a number much larger than the number of lines in the original topology)] of virtual

branches, will be created.

A matrix Υ, which contains 2L rows and N columns, where N is the total

number buses including the virtual ones created due to the splitting of lines, will

then be built. Assuming that binary variables, zj, represent existence of sensors at

Bus j, where a nonzero value indicates existence, the following constraint will be

imposed for each Branch `:

−K < S`,1z1 + S`,2z2 + · · ·+ S`,NzN < K. (6.1)

The above two-sided inequality constraint can be expressed as two single-sided

inequality constraints in two separate rows inside the matrix Υ as shown below:

Υ =

S1,1 S1,2 · · · · · · S1,N

−S1,1 −S1,2 · · · · · · −S1,N

S2,1 S2,2 · · · · · · S2,N

−S2,1 −S2,2 · · · · · · −S2,N

· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·SL,1 SL,2 · · · · · · SL,N

−SL,1 −SL,2 · · · · · · −SL,N

Branch 1

Branch 2

Branch `

Branch L

(6.2)


This will ensure that the sum of S`,j corresponding to those sensors placed at

buses where zj = 1 will not add up to K or −K, i.e., the corresponding virtual-

branch faults will be observable (detectable).

In light of these justifications, the optimization problem for sensor deployment

can be explicitly formulated as the following integer programming problem:

minimize Wz (6.3a)

subject to Υz < K (6.3b)

z = [z1 z2 · · · zN ]> (6.3c)

K = K · 12L×1; K ≥ 0 (6.3d)

zj ∈ 0, 1 (6.3e)

W =[w1 · 11×Na

∣∣ w2 · 11×(N−Na)

](6.3f)

K =N∑

j=1

zj (6.3g)

where 1 is the vector of ones; Na is the number of (actual) buses in the system;

W and z are the weight and the (binary) sensor placement vectors, respectively.

In vector W , w1 and w2 are the weights assigned for actual and fictitious buses,

respectively. Then, the sum of nonzero elements in z gives the total sensor count

required for full fault-location observability.

While the virtual buses are introduced in order to simplify the problem formu-

lation, since they cannot really be used to place actual sensors, it is not desirable

for the optimization algorithm to place sensors at such buses. Hence, the solution

is forced to use such buses only as a last resort. This is accomplished by assigning

w1 w2 (e.g., w1 = 10−2 and w2 = 106), hence making the optimization algorithm

strongly favor placement of sensors at the “actual” buses over the fictitious ones.

As an example, consider the transmission system shown in Figure 6.1. The

above integer programming problem is solved for this system. Following the inclusion


of virtual buses and line segments and optimal placement of the synchronized sensors,

the modified network consists of 220 buses (including the virtual ones) and 241

transmission-line segments. It is worth pointing out that in this example, the solution

set for sensor deployment problem does not include virtual buses. Obviously, the

total number of buses and line segments generated in this way is dependent upon

the transmission-grid topology, the transmission-line lengths, and the locations of

synchronized measurements. In Section 6.3.2, we will introduce the case where the

virtual buses are selected in the solution set of the proposed sensor deployment

algorithm.

6.3 Simulation Results

All simulations are carried out in ATP and MATLAB with a sampling frequency

of 500 kHz. The fault-occurrence time is chosen to be 20 ms with respect to the

simulation start time. Frequency-dependent transmission-line models are used in

the simulations. To solve the binary integer programming problem formulated for

the optimal placement strategy in Section 6, IBM ILOG R© CPLEX R© Optimization

Studio [124] is utilized.

All transmission lines used in the simulations are modeled as balanced, lossless,

and fully transposed lines, and bear the same configuration as those used in Chapter

5. Transmission-line lengths, along with wave-propagation times, are provided in

Table 6.1.

In the following subsections, we will illustrate how power-system faults can be

localized on observable sections of transmission lines (in a fault-observable system)

and on unobservable line segments (in a fault-unobservable system) which will come

into existence after modification of particular line lengths.


6.3.1 Fault on an “Observable” Line Segment

The fault scenarios are realized on the modified IEEE 57-bus system whose single-line

diagram is illustrated in Figure 6.1. As an illustration, we simulate a short-circuit

fault at a point 95 miles away from Bus 24 on the 229-mile-long transmission line

between Buses 24 and 26. Aerial-mode WTC2s for each modal voltage are obtained

following the transformation of the three-phase synchronized voltage measurements

into the modal voltages. The voltage measurements at Bus 33 and the pertinent

aerial-mode voltage WTC2 are displayed in Figure 6.2.

12345

6

7

8 9

10

11

1213

1415

16

17

18 1920

21

22

23

24

25

26

27

29

30

28 31 32

33

34

35

3637

38 39

40

41

42 43

44

45

464748

49 50

51

525354

55

56

57

synchronized voltage sensors

Figure 6.1: Single-line diagram of the modified IEEE 57-bus test system (lengthsof branches are not scaled in proportion to actual line lengths).


Table6.1:

Tra

nsm

issi

on-L

ine

Len

gth

san

dW

ave-

Pro

pag

atio

nT

imes

for

the

Mod

ified

IEE

E57-B

us

Tes

tS

yst

em

Length

Tim

eL

ength

Tim

eL

ength

Tim

eL

en

gth

Tim

eL

ine

(mi)

(µs)

Lin

e(m

i)(µ

s)L

ine

(mi)

(µs)

Lin

e(m

i)(µ

s)

1−

230

61,

646.

219−

5590

484.

1822−

3822

11,

188.

9338−

4413

472

0.89

1−

1514

477

4.69

10−

1231

21,

678.

4823−

2425

01,

344.

9438−

4810

858

1.01

1−

1621

41,

151.

2710−

5185

457.

2824−

2522

11,

188.

9338−

4912

667

7.85

1−

1730

71,

651.

5911−

1321

81,

172.

7924−

2622

91,

231.

9639−

5737

199.

052−

347

252.

8511−

4113

170

4.75

25−

3059

317.

4140−

5616

86.0

83−

430

41,

635.

4511−

4316

789

8.42

26−

2796

516.

4641−

4226

71,

436.

403−

1532

11,

726.

9012−

1317

91.4

627−

2816

990

9.18

41−

4367

360.

444−

527

41,

474.

0512−

1632

21,

732.

2828−

2917

292

5.32

41−

5615

080

6.96

4−

618

599

5.26

12−

1726

61,

431.

0229−

5223

61,

269.

6242−

5625

81,

387.

984−

1837

199.

0513−

1422

11,

188.

9330−

3144

236.

7144−

4552

279.

755−

697

521.

8413−

1588

472.

4231−

3232

21,

732.

2846−

4731

31,

683.

866−

751

274.

3713−

4915

281

7.72

32−

3311

762

9.43

47−

4822

91,

231.

966−

870

376.

5814−

1516

588

7.66

32−

3479

425.

0048−

4961

328.

177−

811

360

7.91

14−

4628

51,

533.

2334−

3519

81,

065.

1949−

5023

91,

285.

767−

2989

478.

8015−

4526

91,

447.

1535−

3623

41,

258.

8650−

5125

41,

366.

468−

954

290.

5118−

1932

11,

726.

9036−

3727

71,

490.

1952−

5329

91.

608.

559−

1027

41,

474.

0519−

2032

31,

737.

6636−

4011

059

1.77

53−

5432

21,

732.

289−

1128

21,

517.

0920−

2157

306.

6537−

3896

516.

4654−

5518

61,

000.

639−

1280

430.

3821−

2232

51,

748.

4237−

3920

107.

6056−

5731

91,

716.

149−

1389

478.

8022−

2325

51,

371.

84


24.6824.56 24.58 24.6 24.62 24.64 24.66

1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Time (ms)W

TC

2

Phase aPhase b

Time (ms)

Voltag

e(V

)

Three-Phase Voltages at Bus 33

Phase c

Figure 6.2: Three-phase voltages and WTC2 of the aerial-mode voltage at Bus33 after the occurrence of a short-circuit fault on Line 24-26.

The strategically selected locations (substations) for the synchronized sensors

in the studied network are shown and listed in Figure 6.1 and Table 6.2, respec-

tively. Table 6.2 also illustrates the instants when the first local peaks of WTC2s

are detected via the synchronized sensors at the corresponding substations. Hence,

the captured times in milliseconds are stored in the (13× 1)-column-vector T right

after the occurrence of the fault.

For the fault-scenario example above, the minimizing value of the cost function,

i.e., J174 = 0.0048 ≈ 0, is attained on Line 174 in the resulting split network, after

labeling the line segments. As a result, the corresponding values of %(`) and T(`)0 are

found to be %

(174)

T(174)0

=

0.1184

19.9991

ms.

Table 6.2: Synchronized Measurement Locations versus Wave-Arrival Times forthe Short-Circuit Fault Occurring on Line 24-26

Buses 3 5 16 17 19 30 33ToAs (ms) 26.456 24.344 26.612 26.310 26.744 22.016 24.616

Buses 42 43 45 46 51 53ToAs (ms) 28.108 26.864 25.416 27.648 26.380 25.948


24 26

0.3739

D24−26 = 1.2320 ms

︸︷︷︸

(174) = 0.1184177 178 179 180

0.0054 0.0134 0.8343 0.0050

Line 171 Line 174

Figure 6.3: Value of %(`) for the short-circuit fault occurring on Line 24-26.

In Figure 6.3, the location of the fault on Line 24-26 is displayed in terms of

the propagation delay, %(`), associated with the fault. As illustrated in the figure, the

location of the fault is detected on the virtual line segment (i.e., Line 174) connecting

Terminals 179 and 180. The numbers shown right below these line segments represent

the calculated propagation times (in ms). The distance to fault from Bus 24 is thus

computed to be

xfault =((3.739 + 0.054 + 0.134 + 1.184)× 10−4 s

)× (1.85885× 105 mi/s)

= 95.006 ≈ 95 mi.

6.3.2 Fault on an “Unobservable” Line Segment

Despite being not prevalent, for some cases unobservable segments (or “blind spots”)

may appear if the solution set for the optimal sensor deployment algorithm contains

one or more virtual buses. This happens essentially when length of a certain trans-

mission line is longer than the total length of the shortest path traveled from one

endpoint of the line to another via neighboring paths. In this case, these sensors will

be moved from the virtual buses to the closest actual buses. This move may lead

to creation of unobservable transmission-line segments which can be made fault-

observable by introducing a second-stage procedure which will be described here.

Thus, a robust deployment strategy that ensures unique identification of fault loca-

tion for any fault will be developed and validated by simulation examples.


To illustrate the existence of any possible unobservable segments on such

special-case systems, we modify the transmission system by varying the lengths

of four transmission lines as shown in Table 6.3. It is worth pointing out that in this

example the solution set for sensor deployment problem includes virtual buses that

physically do not exist. In the studied system, out of 16 buses on which sensors are

deployed, three of them correspond to virtual ones, which are Buses 177, 280, and 323

generated on Lines 7-8, 9-12, and 38-48, respectively. Therefore, these nonexistent

buses will have to be disregarded and replaced by actual buses before determining

the optimal set of sensor locations required for fault-location observability. In this

system, these three buses are replaced by the actual buses closest to them, which

are Buses 7, 9, and 48. Employing the so-called “line-splitting approach” after this

refinement process, it is observed that the newly split transmission grid becomes a

network with 216 buses and 237 transmission-line segments.

Table 6.3: Lengths and Propagation Times of Modified Transmission Lines

Length Time Length TimeLine

(mi) (µs)Line

(mi) (µs)

7− 8 253 1, 361.08 38− 48 238 1, 280.389− 12 120 645.57 38− 49 16 86.08

According to the discussion in Section 6.2, redeployment stage of synchronized

sensors on three (actual) buses leads to creation of six unobservable line segments

as highlighted in Figure 6.4. In order to verify the true applicability of the proposed

fault-location technique, we simulate two short-circuit faults on both “unobservable”

segments of Line 7-8.

When a fault is detected on a transmission line with unobservable segments, a

second-stage verification procedure will be initiated in order to guarantee accurate

fault location. The procedure is described in the following:


38 48

d38−48 = 238 mi

0.4331 ms 0.4327 ms

80.5055 mi 80.4311 mi

7 8

d7−8 = 253 mi

66.0067 mi 65.9881 mi

0.3551 ms 0.3550 ms

9 12

d9−12 = 120 mi

7.0078 mi 7.0078 mi

0.0377 ms 0.0377 ms

Figure 6.4: Unobservable segments (lengths and travel times being designated)of the three transmission lines.

• Second-Stage, Single-Ended Fault-Location Procedure:

To identify the location of the fault on transmission lines with unobservable

segments, the proposed fault-location technique is first applied. This is followed by

a second-stage fault-location procedure, details of which will be described using two

examples.

A short-circuit fault, occurring at 35 miles away from Bus 7 (within the near-

half unobservable segment) on the 253-mile-long transmission line between Buses 7

and 8, is simulated. The first local-peak instants detected on the respective synchro-

nized sensors are listed in Table 6.4.

Table 6.4: Locations of Synchronized Measurements versus Wave-Arrival Timesfor the Short-Circuit Fault Occurring on the Near-Half Unobservable Segment of

Line 7-8

Buses 3 5 7 9 16 17 19 23ToAs (ms) 23.092 20.984 20.188 21.128 23.432 23.128 23.384 25.072

Buses 33 42 43 44 46 48 51 53ToAs (ms) 27.896 24.788 23.544 23.232 24.328 22.752 23.060 23.544


7 8185 186187

0.3551 0.3577

D7−8 = 1.3611 ms

(189) = 2.1636× 10−4

0.0243

188 189

0.25010.0188

0.3550

(192) = 0.2498

︸︷︷︸Line 188 Line 193

Figure 6.5: Values of %(`) for the two short-circuit faults occurring on Line 7-8.

It is again noted that faulty transmission line is not known in advance. Nev-

ertheless, if a faulted line segment is suspected to be on a transmission line having

unobservable line segments, fault-location computations have to be updated via a

procedure introduced in [37] that utilize single-ended recording. Also, due to the

way sensors are deployed by the proposed method, at least one sensor will already

be placed on each transmission line with unobservable segments.

As pictured in Figure 6.5 and indicated in the first row of Table 6.4, the fault

is detected very close to the “terminus” of Line 188 created between Terminals 7

and 185. Since this line segment belongs to Line 7-8, a single-ended fault-location

procedure using the recording provided at Bus 7 will be employed. Particularly,

by measuring the arrival-time difference between the first two successive peaks [as

shown in Figure 6.6(a)] of the WTC2 of aerial-mode voltage signal (in scale-1) at

Bus 7, and taking the product of the traveling-wave speed and half of that arrival-

time difference, the distance to the fault can be calculated. Hence, the corrected

fault-distance estimation will be given by

dcorr =t2 − t1

2ν (6.4)

=(20.564− 20.188)× 10−3 s

2× (1.85885× 105 mi/s)

= 34.9463 ≈ 35 mi.

This result vindicates that fault has occurred on the near-half unobservable segment

of Line 7-8. Indeed, the fact that the fault was previously detected near the terminus


of Line 188 (i.e., dest = 66.0469 ≈ 66.0067 mi) also indicates that the fault is

suspected to have occurred on the near-half unobservable segment.

For the second fault-scenario example, a short-circuit fault is simulated at a

point 42 miles away from Bus 8 (within the remote-half unobservable segment) on the

same transmission line. The first local-peak instants captured on the synchronized

sensors are provided in Table 6.5.

Table 6.5: Locations of Synchronized Measurements versus Wave-Arrival Timesfor the Short-Circuit Fault Occurring on the Remote-Half Unobservable Segment

of Line 7-8

Buses 3 5 7 9 16 17 19 23ToAs (ms) 23.196 21.124 20.876 20.516 22.820 22.516 23.524 24.460

Buses 33 42 43 44 46 48 51 53ToAs (ms) 27.284 24.174 22.932 22.620 23.716 22.140 22.448 23.732

As displayed again in Figure 6.5 and pointed out in the second row of Table 6.6,

the fault is detected very close to the “origin” of Line 193 created between Terminals

8 and 189. It is evident that the line segment also belongs to Line 7-8; therefore, the

single-ended recording values at Bus 7 will be used. Now, subtracting the arrival-

time difference between the first and the third peaks [as illustrated in Figure 6.6(b)]

of the WTC2 of aerial-mode voltage signal from twice the wave-propagation time on

that line, and multiplying them by half of the traveling-wave speed will yield the

distance to the fault. For this case, the corrected distance to fault is computed as

follows:

xfaultcorr =2D7−8 − (t3 − t1)

2ν (6.5)

=

(2× 1.3611− (21.328− 20.876)

)× 10−3 s

2× (1.85885× 105 mi/s)

= 210.9981 ≈ 211 mi (from Bus 7).

Similarly, the above fault-location estimate implies that the fault has occurred on

the remote-half unobservable segment of Line 7-8. Looking at the results of Table


WTC2s of the Aerial-Mode Voltages at Bus 7

5× 106

1.4× 107

107

020.18 20.3 20.4 20.5 20.6

4× 105

6× 105

20.5 20.52 20.56 20.6 20.65

2× 105

0

Time (ms)

WTC

2

(a)

105

2× 105

3× 105

4× 105

5× 105

20.872 21 21.1 21.2 21.340

21.301 21.32 21.34 21.36 21.377

500

1000

1500

2000

2500

Time (ms)

WTC

2

0

(b)

Figure 6.6: WTC2s of the aerial-mode voltages at Bus 7 for the fault events on(a) the near-half and (b) the remote-half unobservable segments of Line 7-8.


6.6 reveals that location of the fault was firstly estimated to be close to the origin

of Line 193 (i.e., xfaultest = 186.9369 ≈ 187.0119 mi), indicating that the fault is

suspected to have occurred on the remote-half unobservable segment.

Table 6.6: Values of Jmin` , `, %(`) (in ms), T

(`)0 (in ms), xfaultest (in mi), and

xfaultcorr (in mi) for the Faults Occurring on Near- and Remote-Half UnobservableSegments of Line 7-8

Jmin` ` %(`) T

(`)0 xfaultest xfaultcorr

0.0037 189 2.1636× 10−4 19.8322 66.0469 34.94630.0031 192 0.2498 19.8703 186.9369 210.9981

6.4 Summary

In this chapter, we develop a practical and an effective strategy for rendering the

transmission grid “fault-observable” by optimal deployment of sensors that record

GPS-synchronized voltage measurements. As a result, previously proposed wide-

area measurement-based fault-location procedure can uniquely identify the location

of a fault irrespective of its location in the power grid. The method can be used to

design synchronized-measurement-based fault-location schemes from scratch or to

upgrade existing measurement designs in order to minimize fault-unobservable seg-

ments. Simulation results are included to illustrate the effectiveness of the proposed

design.

Chapter 7

Enhanced Robustness via

Least-Absolute-Value Estimation

and

Largest-Normalized-Residual Test

7.1 Introduction

A power system fault is typically detected and isolated by fast protective relaying

systems for customer safety and to prevent damage to power system equipment.

Though protection schemes for transmission systems have been well designed for

identifying the location of faults as well as for isolating the faulty section, reliable

event detection and rapid system restoration even under the threat of compromised

security remain a key challenge for systemwide protection.

Transmission-grid topologies have grown more complex; as a result, disturbance

events become more frequent, so do the challenges of reliable analysis of these events

to accurately identify the location of a disturbance or a fault comprising a large

115

Chapter 7. Enhanced Robustness via Least-Absolute-Value Estimation andLargest-Normalized-Residual Test 116

area of the network [125]. On that account, the development of resilient and reliable

protection schemes and efficient emergency actions constitutes the integral part of a

reliable wide-area protection system in evolving power grids.

With the availability of high-bandwidth optical instrument transformers, we

expect the new-generation phasor measurement units (PMUs) to provide synchro-

nized point-on-wave measurements at high sampling frequencies. Having access to

these measurements will facilitate the acquisition of key information forming the

systemwide picture of dynamic events through continuous monitoring of wide-area

recordings of transient disturbances [126, 127]. Possessing a high level of compu-

tational capability and standing upon high-speed communications infrastructure,

emerging modern protection devices can execute a myriad of advanced processing

algorithms based on collation of data from widely separated meters, where event

records are captured. Furthermore, attaining enough level of redundancy can help

minimize the effect of device failure on the sensitivity of protection functions.

Through the remote access to smart meters from external locations, substa-

tion IEDs have been attractive targets for cybervulnerabilities since measurement

readings can be easily manipulated by intruders into the electrical grid if appropriate

security margins are not achieved. In particular, IED data can be compromised with

the intent of jeopardizing the desired protection functions of the grid. Coordinated

cyberattacks on the monitoring and protection schemes can inject false data regard-

ing the existing operating conditions for emergent wide-area disturbance-monitoring

applications. For instance, an adversary could create attack vectors by introducing

time delays on synchronized measurements, which, in turn, results in desynchroniza-

tion of IEDs installed at multiple line terminals [128]. Therefore, countermeasures

(e.g., bad-data identification algorithms) need to be developed in order to prevent

such attacks or mitigate the effect from attacks on the power grid; otherwise, relia-

bility and security of the grid could be threatened to a great extent.


In Chapter 5, utilizing the synchronized measurements of transient voltages at

different locations in the grid, arrival times of fault-induced traveling wavefronts are

extracted with high accuracy to multilaterate (locate) the occurrence point of faults.

To achieve a thorough wide-area disturbance monitoring capability, we have devised

a “multi-sensor network”, where synchronized sensors that are strategically deployed

over a wide area are utilized for the retrieval of the systemwide recordings of transient

signals following the disturbance event. As an extension of this work, a cost-effective

deployment scheme for installing the fault-recording sensors across a bulk power

transmission system is set forth in Chapter 6. The formulated placement strategy is

shown to be suitable for any grid structure (including meshed and radial networks)

and allows for unique localization of any possible fault-occurrence point within the

grid. In this chapter, post-estimation bad-data processing techniques are developed

and implemented in order to detect, identify, and correct bad measurements. Bad-

data rejection properties of the least-absolute-value (LAV) estimator together with

the largest-normalized-residuals test are exploited to reliably estimate the location

of faults even in the presence of faulty sensors and imperfect measurements.

This section features the impact of possible measurement errors on the reliable

estimation of fault location. Investigated error types are (i) errors incurred by

sensor imprecision or (ii) a gross error. The following subsections will provide the

necessary tools for understanding the effect of small measurement deviations on the

fault-location accuracy as well as a feasible approach for eliminating and mitigating

the effect from the measurement deviations.


7.2 Measurement Imprecision due to Low Sam-

pling Rate

Accurate timing synchronization with the precision requirement of a few microsec-

onds is a key to achieving a reliable grid protection. Time stamping of the sam-

pled analog transient waveforms can be utilized to align the values retrieved from

PMU-enabled intelligent electronic devices (IEDs) that perform the critical protec-

tion functions, thereby allowing for obtaining a simultaneous snapshot of the fault

event across the entire set of recorded voltages. A difference in timing at the data-

recording IEDs by several microseconds will result in malfunction of a fault-location

algorithm.

Fast-changing transient events necessitate very high sampling rates (i.e., higher

precision) for analog-to-digital conversion of a disturbance waveform; simply put,

the resolution of disturbance-recording devices directly impact the accuracy of the

captured data essential for postfault analysis. Measurement errors incurred by the

sampling process will lead to an unreliable distance estimate; hence, incorrect fault

location can prolong the needed duration for the tasks of fault diagnosis and system

restoration.

7.3 Treatment of Sensor Measurements Contain-

ing Gross Errors

Conventionally, gross measurement errors can be caused by sensor failures and inac-

curate measurement scans. From the viewpoint of cybersecurity, an attacker can also

introduce bad measurements to intentionally manipulate the protection functions of

the grid. This section shows how grossly erroneous measurements can be identified

using the least-absolute-value estimator and the largest-normalized-residual test.


7.3.1 Bad Data Identification via Least-Absolute-Value (LAV)

Estimation

For the system of equations in (5.11), the state variables are related to the sensor

measurements through the following model

∆T(`) = h(`)(θ(`)) + e, (7.1)

where ∆T(`) = T−D` are the (modified) sensor measurements; h(`)(θ(`)) = T(`)0 η+

%(`)S` are the measurement functions relating the state vectors θ(`) =[%(`) T

(`)0

]>

to the measurements ∆T(`); e =[e1, e2, · · · , eK

]>, and ek 6= 0 only if Sensor “k” is

compromised.

The first-order derivatives of functions, h(`)(θ(`)), with respect to the state

variables, %(`) and T(`)0 , i.e.,

∂h(`)(θ(`))

∂%(`)= S` and

∂h(`)(θ(`))

∂T(`)0

= η, (7.2)

form the matricesH(`) =[S` η

]. The optimization problem for the LAV estimator,

which is a systematic way to identify the outliers, can be modeled as follows [129]:

minimize C>Θ(`) (7.3a)

subject to A(`)Θ(`) = ∆T(`) (7.3b)

A(`) =[H(`) −H(`) IK − IK

](7.3c)

C> =[01×4 11×2K

](7.3d)

[Θ(`)

]>=

[[θ(`)u

]> [θ(`)v

]> [u(`)]> [

v(`)]>]

(7.3e)

Θ(`) 0(2K+4)×1 (7.3f)

where 0 and 1 represent the vectors of zeros and ones, respectively.


7.3.2 Bad Data Identification via Largest-Normalized-Resid-

ual(rNmax

)Test

When sensor-measurement error is assumed to be normally distributed with zero

mean, the weighted least-squares (WLS) estimator of the state vector is given by

θ(`)

WLS =

((H(`))>

WH(`)

)−1(H(`))>

W∆T(`), (7.4)

where W is a diagonal matrix whose elements are reciprocals of the measurement-

error variances of the sensors (provided that errors are independent), i.e.,

W = diag(σ−2

1 , σ−22 , · · · , σ−2

K

). (7.5)

Then, assuming unit-variance errors, the estimated value of the (modified) sensor

measurements is obtained as

∆T(`) =H(`)

((H(`))>H(`)

)−1(H(`))>

∆T(`). (7.6)

Now, the sensor measurement-residual vector can be expressed as follows:

r(`) = ∆T(`) −∆T(`). (7.7)

7.3.2.1 Detection and Identification of Bad Sensor Measurements

In order to identify and subsequently eliminate bad sensor measurements, we utilize

the test known as the largest normalized residual(rNmax

)test, steps of which are

presented in the following:

Step 1. Solve the WLS estimation and retrieve the elements of the measurement-

residual vector:


r(`)k = ∆T

(`)k −∆T

(`)k ; k = 1, · · · , K. (7.8)

Step 2. Compute the normalized residuals:

(r

(`)k

)N=

∣∣r(`)k

∣∣√

Ω(`)kk

; k = 1, · · · , K, (7.9)

where Ω(`)kk is the diagonal element of the matrix

Ω(`) = IK −H(`)

((H(`))>H(`)

)−1(H(`))>. (7.10)

Step 3. Identify j such that(r

(`)j

)Nis the largest among all

(r

(`)k

)N; k =

1, · · · , K.

Step 4. If(r

(`)j

)N> c, then the jth measurement will be suspected as bad mea-

surement; otherwise, no bad measurements will be suspected. Here, c is the

predetermined identification threshold.

Step 5. Eliminate the jth measurement from the measurement set and go to Step

1.

7.3.2.2 Elimination/Correction of Identified Bad Measurements

Assuming that all measurements are free of errors except the jth measurement, the

below approximation [130] can be used to correct the bad sensor measurement

∆T(`)j ≈ ∆T

(`),badj −

r(`),badj

Ω(`)jj

, (7.11a)


from which we obtain

T(`)j ≈ ∆T

(`),badj +D(o)

j,` −r

(`),badj

Ω(`)jj

, (7.11b)

where ∆T(`)j , ∆T

(`),badj , and r

(`),badj are the true value, the measured value, and the

bad measurement residual associated with the jth measurement, respectively; and

D(o)j,` is the propagation delay along the shortest path from the origin of Line “`” to

Sensor “j”.

After the correction of the bad measurement using (7.11), state estimation can

be repeated. The results of this estimation will result in approximately the same

state estimate that would have been found if the (bad) measurement were actually

eliminated from the measurement set.

7.4 Simulation Results

Simulations are carried out in ATP and MATLAB using a sampling frequency of

1 MHz. In all simulations, frequency-dependent line models are used. The fault-

occurrence time is chosen to be 20 ms with respect to the simulation start time.

In addition, the tower configuration and models of transmission lines used in the

simulations are the same as those used in Chapters 5 and 6. Again, identical tower

configurations are assumed for all transmission lines in order to simplify the simula-

tions without loss of generality. The traveling-wave speed of the transients is approx-

imated by evaluating it at the frequency corresponding to the midpoint of scale-1 of

the utilized wavelet transform. This corresponds to the interval [fsamp/4− fsamp/2]

(midpoint of which is 3fsamp/8 = 375 kHz), where fsamp is the sampling frequency

used in the transient simulations. A lookup table (see Tables 7.1 and 7.2) for the

transmission-line lengths and the corresponding wave-propagation times is created

and used in the calculations.


Table7.1:

Tra

nsm

issi

on

-Lin

eL

engt

hs

and

Wav

e-P

rop

agat

ion

Tim

esfo

rth

eM

od

ified

IEE

E118-B

us

Tes

tS

yst

em

Length

Tim

eLength

Tim

eLength

Tim

eLength

Tim

eLine

(mi)

(µs)

Line

(mi)

(µs)

Line

(mi)

(µs)

Line

(mi)

(µs)

1−

2338

1,81

8.36

27−

32

40

215.1

955−

5664

344.

3082−

83

73

392.7

2

1−

3165

887.6

627−

115

283

1,5

22.4

755−

5994

505.

7082−

96

177

952.2

2

2−

12

273

1,46

8.67

28−

29

88

473.4

256−

5724

21,3

01.9

083−

84

134

720.8

9

3−

5122

656.3

329−

31

293

1,5

76.2

756−

5828

11,5

11.7

183−

85

249

1,3

39.5

6

3−

12

332

1,78

6.08

30−

38

88

473.4

256−

5933

177.

5384−

85

192

1,0

32.9

1

4−

5272

1,46

3.29

31−

32

128

688.6

159−

6015

281

7.72

85−

86

251

1,3

50.3

2

4−

11

241

1,29

6.52

32−

113

191

1,0

27.5

359−

6115

683

9.24

85−

88

109

586.3

9

5−

6327

1,75

9.18

32−

114

151

812.3

459−

6337

199.

0585−

89

61

328.1

7

5−

8319

1,71

6.14

33−

37

282

1,5

17.0

960−

6120

71,1

13.6

186−

87

340

1,8

29.1

2

5−

11

216

1,16

2.03

34−

36

81

435.7

660−

6276

408.

8688−

89

60

322.7

9

6−

7282

1,51

7.09

34−

37

308

1,6

56.9

761−

6221

31,1

45.8

989−

90

275

1,4

79.4

3

7−

12

179

962.9

834−

43

212

1,1

40.5

161−

6495

511.

0889−

92

208

1,1

18.9

9

8−

9321

1,72

6.90

35−

36

243

1,3

07.2

862−

6680

430.

3890−

91

128

688.6

1

8−

30

221

1,18

8.93

35−

37

282

1,5

17.0

962−

6719

11,0

27.5

391−

92

116

624.0

5

9−

10

234

1,25

8.86

37−

38

328

1,7

64.5

663−

6411

159

7.15

92−

93

125

672.4

7

11−

1232

172.1

537−

39

205

1,1

02.8

564−

6524

21,3

01.9

092−

94

43

231.3

3

11−

13221

1,18

8.93

37−

40

203

1,0

92.0

965−

6621

51,1

56.6

592−

100

232

1,2

48.1

0

12−

14276

1,48

4.81

38−

65

44

236.7

165−

6833

51,8

02.2

292−

102

97

521.8

4

12−

16267

1,43

6.40

39−

40

113

607.9

166−

6721

21,1

40.5

193−

94

120

645.5

7

12−

117

292

1,57

0.89

40−

41

284

1,5

27.8

568−

6933

31,7

91.4

694−

95

190

1,0

22.1

5

13−

1530

161.3

940−

42

249

1,3

39.5

668−

8132

81,7

64.5

694−

96

271

1,4

57.9

1

14−

1557

306.6

541−

42

142

763.9

368−

116

339

1,8

23.7

494−

100

208

1,1

18.9

9


Table7.2:

Tra

nsm

issi

on

-Lin

eL

ength

san

dW

ave-

Pro

pag

atio

nT

imes

for

the

Mod

ified

IEE

E118

-Bu

sT

est

Syst

em(c

onti

nu

edfr

omT

able

7.1)

Length

Tim

eLength

Tim

eLength

Tim

eLength

Tim

eLine

(mi)

(µs)

Line

(mi)

(µs)

Line

(mi)

(µs)

Line

(mi)

(µs)

15−

17101

543.3

642−

49

135

726.2

769−

7023

41,

258.

86

95−

96

291

1,5

65.5

1

15−

19325

1,74

8.42

43−

44

141

758.5

569−

7523

21,

248.

10

96−

97

169

909.1

8

15−

33340

1,82

9.12

44−

45

208

1,11

8.9

969−

7719

81,

065.

19

98−

100

172

925.3

2

16−

1754

290.5

145−

46

126

677.8

570−

7128

61,

538.

61

99−

100

284

1,5

27.8

5

17−

18264

1,42

0.26

45−

49

140

753.1

770−

7473

392.7

210

0−

101

156

839.2

4

17−

30301

1,61

9.31

46−

47

3317

7.5

370−

7536

193.6

710

0−

103

254

1,3

66.4

6

17−

31114

613.2

946−

48

111

597.1

571−

7268

365.8

210

0−

104

272

1,4

63.2

9

17−

113

264

1,42

0.26

47−

49

110

591.7

771−

7319

61,

054.

43

100−

106

310

1,6

67.7

2

18−

1982

441.1

447−

69

9048

4.1

874−

7573

392.7

210

1−

102

177

952.2

2

19−

2064

344.3

048−

49

3217

2.1

575−

7733

21,

786.

08

103−

104

144

774.6

9

19−

34344

1,85

0.64

49−

50

5227

9.7

575−

118

215

1,15

6.65

103−

105

65

349.6

8

20−

21281

1,51

1.71

49−

51

3217

2.1

576−

7752

279.7

510

3−

110

198

1,0

65.1

9

21−

22132

710.1

349−

54

103

554.1

276−

118

346

1,86

1.40

104−

105

205

1,1

02.8

5

22−

23233

1,25

3.48

49−

66

157

844.6

277−

7810

355

4.1

210

5−

106

260

1,3

98.7

4

23−

24342

1,83

9.88

49−

69

9048

4.1

877−

8013

371

5.5

110

5−

107

209

1,1

24.3

7

23−

2576

408.8

650−

57

308

1,65

6.9

777−

8217

393

0.7

010

5−

108

83

446.5

2

23−

3232

172.1

551−

52

163

876.9

078−

7914

980

1.5

810

6−

107

201

1,0

81.3

3

24−

70293

1,57

6.27

51−

58

100

537.9

879−

8016

186

6.1

410

8−

109

120

645.5

7

24−

72320

1,72

1.52

52−

53

245

1,31

8.0

480−

8117

091

4.5

610

9−

110

264

1,4

20.2

6

25−

26224

1,20

5.07

53−

54

320

1,72

1.5

280−

9611

561

8.6

711

0−

111

46

247.4

7

25−

2754

290.5

154−

55

4323

1.3

380−

9711

360

7.9

111

0−

112

113

607.9

1

26−

30275

1,47

9.43

54−

56

9852

7.2

280−

9816

488

2.2

811

4−

115

211

1,1

35.1

3

27−

2859

317.4

154−

59

9048

4.1

880−

9916

990

9.1

8


45

6

78

9 10

14

16

17

18

19

20

21

22

23

26

29

30

31

32

35

39

43

44

45

46

47

48

49

50

51

52

5354

55

56

57

11

21

312

13

15

28

27

25

24

34

33

38

37

40

42

41

36

114

115

117

113

69

70

71

72

74

73

75

118

58

59

76

77

66

67 6

0

6162

63

64

65

116

68

81

78

79

80

82

105

108

111

112

109

110

106

107

103

104

101

102

83

84 8

5

86

87

88

89 90

91

92

93

94

95

96

97

9899

100

Figure

7.1:

Sin

gle-

lin

ed

iagra

mof

the

mod

ified

IEE

E11

8-b

us

test

syst

em(a

rcle

ngt

hs

and

actu

al

lin

ele

ngth

sare

not

pro

por

tion

ally

scal

ed).


To motivate the discussion, consider the modified IEEE 118-bus system whose

single-line diagram is displayed in Figure 7.1. Also, the optimally chosen locations

for the 43 synchronized sensors in the studied grid are listed in Table 7.3.

It is worth pointing out that the shortest propagation delay between each

pair of buses is obtained by employing shortest-path algorithm since power grid is

modeled as an undirected graph. Given the locations of optimally deployed sensors,

the (newly partitioned) “pseudogrid” formed in this way involves 698 buses and 759

line segments.

In the following, two different types of simulations are performed in the cases

of sensors being subjected to different types of error.

7.4.1 Measurement Imprecision due to Low Sampling Rate

Conventional digital fault recorders’s (DFRs’s) sampling capabilities are usually lim-

ited up to 20 kHz, which corresponds to a sampling time interval of 50 µs. Hence,

the sampling rate of synchronized fault-recording sensors is decreased to emulate

samples taken by such DFRs. These samples are used to illustrate the impact of

Table 7.3: Wave-Arrival Times for the Fault Occurring at 99 Miles Away fromBus 63 When 20-kHz Fault-Recording Sensors are Utilized

Buses 1 2 4 6 10 14 20 29

ToAs (ms) 26.550 26.900 26.450 26.750 26.250 24.550 25.900 25.850

Buses 35 39 41 46 53 55 57ToAs (ms) 24.900 24.450 23.250 22.550 22.950 21.250 22.200

Buses 58 60 61 67 73 74 79ToAs (ms) 22.400 21.550 20.600 22.750 25.950 23.750 24.500

Buses 84 87 88 90 93 95 97ToAs (ms) 25.200 29.000 26.400 27.500 26.600 26.050 24.500

Buses 99 101 104 106 107 109 111ToAs (ms) 24.800 26.500 27.150 27.350 28.450 28.500 28.350

Buses 112 113 114 115 116 117 118ToAs (ms) 28.700 25.100 25.800 26.550 25.000 27.000 24.500


63 64229

0.1775 0.3685

230

0.0512

(178) = 0.3551 ︸︷︷︸

63 64363 364 365

0.0350 0.1425 0.2367

366

0.1318 0.0512

︸︷︷︸

(b)

(c)

Line 315Line 312

Line 179Line 177

63 64363 364 365

0.0350 0.1425 0.2367

366

0.1318 0.0512

(314) = 0.1189 ︸︷︷︸ Line 315Line 312

(a)

(314) = 0.1182

Figure 7.2: Location of a fault occurring at 99 miles away from Bus 63 which isformed via optimally deployed sensors in presence of (a) rounding errors (with 20-kHz sensors) and (b) gross errors (with 1-MHz sensors). (c) Reestimated location

after the elimination of sensors which are contaminated by gross errors.

errors in determining the local peaks of WTC2s, which are used in extracting the

ToAs of traveling waves on selected line terminals [131].

Table 7.3 illustrates the ToA measurements captured at 20-kHz fault sensors

after the fault occurrence taking place on Line 63-64. In Figure 7.2(a), the location

of the fault on Line 63-64 is displayed in terms of the propagation delay %(314). The

identified location of the fault is shown within the borders of the virtual line segment

(i.e., Line 314) linking Terminals 365 and 366. The values shown beneath the line

segments represent the computed propagation times (i.e., D`). Thus, the distance

to fault from Bus 63 is computed as follows:

d =

((5.972−

((1.318− 1.189) + 0.512

))× 10−4 s

)× (1.85885× 105 mi/s)

= 99.1 ≈ 99 mi.

The error in fault-location estimation will then be (99.1−99)×1, 609.34 = 160.93 m.

Clearly, this indicates that the decreased sampling rates of fault sensors will lower

the accuracy of fault-location estimate, albeit lessening the sampling burden.


7.4.2 Measurements Containing Gross Errors

To illustrate the robustness of the fault-location algorithm against incorrect mea-

surements, random gross errors are introduced to the true ToA measurements.

7.4.2.1 Identifying Erroneous Measurements via LAV Estimation

A fault which occurs at 99 miles away from Bus 63 on Line 63-64, will be used to

illustrate the robust fault-location approach using LAV. Notice from Table 7.4 that

out of 43 sensors where ToAs are captured following the fault occurrence, 26 of them

(designated by the color red) are contaminated with huge errors, and will later be

suspected as corrupted measurements. For this case, the proposed LAV estimator

can estimate the correct fault location using the remaining “uncompromised” mea-

surements. In fact, LAV-based state estimator yields an optimal estimate of state

vector, θ(`), for the suspected Line “`”, at which the objective function is minimized.

As a result, one obtains

θ(314) = θ(314)u − θ(314)

v =

%

(314)

T(314)0

=

0.1182

20.0001

ms

Table 7.4: Synchronized Sensor Locations versus Wave-Arrival Times for theFault Occurring at 99 Miles Away from Bus 63 on Line 63-64

Buses 1 2 4 6 10 14 20 29

ToAs (ms) 55.390 37.603 −8.240 47.452 47.112 24.546 16.477 41.190

Buses 35 39 41 46 53 55 57ToAs (ms) 24.885 24.444 44.255 22.539 51.150 9.242 13.801

Buses 58 60 61 67 73 74 79ToAs (ms) 48.054 11.877 −0.761 11.410 25.939 23.728 24.508

Buses 84 87 88 90 93 95 97ToAs (ms) 16.155 14.094 44.236 63.309 60.573 60.568 24.476

Buses 99 101 104 106 107 109 111ToAs (ms) 24.777 26.515 27.139 27.343 28.425 43.264 28.355

Buses 112 113 114 115 116 117 118ToAs (ms) 8.242 25.116 65.510 26.574 24.992 −4.207 12.455


with θ(314)u =

[0.1182, 20.0001

]>and θ(314)

v =[0, 0]>

. Accordingly, the resulting

residue vector is given by

r(314)LAV = u(314) − v(314) =

28.8641

10.7113

−34.6851

20.7107

20.8595

0...

0.0005

−0.0009

−31.2000

−12.0377

,

where

u(314) =

28.8641

10.7113

0

20.7107

20.8595

0...

0.0005

0

0

0

and v(314) =

0

0

34.6851

0

0

0...

0

0.0009

31.2000

12.0377

.

Note that among the elements of r(314)LAV , the residuals belonging to the corrupted

measurements (e.g., Sensors 1-5, 42, and 43) have excessively huge absolute values


as shown in bold in the vector above.

Table 7.5 shows the varying values of∥∥r(`)

LAV

∥∥1

(in ascending order) correspond-

ing to selected transmission-line segments. Among the listed line segments, %(314) is

chosen to be the optimum solution for the sought fault-associated propagation de-

lay. Indeed, this line has the minimum `1-norm value for the residual vector, r(`)LAV,

satisfying the inequality, 0 ≤ %(`) = α(`)D` ≤ D`.

Table 7.5: Values of `,∥∥r(`)

LAV

∥∥1, %(`) (in ms), T

(`)0 (in ms), and D` (in ms) for

the Fault Occurring at 99 Miles Away from Bus 63 on Line 63-64

`∥∥r(`)

LAV

∥∥1

%(`) T(`)0 D` 0 ≤ %(`)≤ D`?

680 496.9527 −5.1157 15.4384 0.1049 No317 514.8469 5.6004 14.3306 0.2717 No700 538.5804 0.7611 20.0001 0.0296 No30 538.7489 −23.4815 41.2165 0.1049 No302 539.8386 0.7799 19.8899 0.0484 No314 542.4916 0.1182 20.0001 0.1318 Yes157 547.0346 0.2042 19.8413 0.1345 No330 552.2862 0.1720 21.7698 0.1722 Yes188 553.2218 0.0352 20.9898 0.0890 Yes618 566.6676 0.0021 17.0978 0.0027 Yes

......

......

......

In Figure 7.2(b), the location of the fault on Line 63-64 is displayed in terms

of the propagation delay %(314). Similar to computation of fault distance in Section

7.4.1, the distance to fault from Bus 63 is computed as follows:

xfault =

((5.972−

((1.318− 1.182) + 0.512

))× 10−4 s

)× (1.85885× 105 mi/s)

= 98.97 ≈ 99 mi.

To further enhance the fault-location accuracy, we first remove the sensors

which are suspected to have carried erroneous measurements and then recompute the


fault distance using the previously devised fault-location method in Chapter 5. This

requires the elimination of faulted sensors, which, in turn, gives rise to a newly formed

virtual network consisting of 410 buses and 471 transmission-line segments. Figure

7.2(c) displays the newly created Line 63-64 and the propagation delay associated

with the fault on Line 178. After the corrupted sensors are discarded, the remaining

(true) measurements are used to determine the corrected distance to fault as follows:

xfaultcorr =((

1.775 + 3.551)× 10−4 s

)× (1.85885× 105 mi/s) = 99 mi.

In other words, the accuracy of the fault-location estimate is improved by (99 −98.97)× 1, 609.34 = 48.28 m.

The corrected vector of ToA measurements, Tcorr, will then be computed by

the following relation:

Tcorr = T− r(`)LAV = T− r

(314)LAV . (7.12)

The obtained values of corrected measurements are displayed in Table 7.6.

Table 7.6: Synchronized Sensor Locations versus Wave-Arrival Times (aftercorrecting bad measurements) for the Fault Occurring at 99 Miles Away from Bus

63 on Line 63-64Buses 1 2 4 6 10 14 20 29

ToAs (ms) 26.526 26.892 26.445 26.741 26.252 24.546 25.901 25.842

Buses 35 39 41 46 53 55 57ToAs (ms) 24.886 24.444 23.260 22.539 22.938 21.237 22.211

Buses 58 60 61 67 73 74 79ToAs (ms) 22.421 21.549 20.576 22.750 25.939 23.728 24.508

Buses 84 87 88 90 93 95 97ToAs (ms) 25.197 28.995 26.402 27.489 26.590 26.053 24.476

Buses 99 101 104 106 107 109 111ToAs (ms) 24.777 26.515 27.138 27.343 28.424 28.483 28.354

Buses 112 113 114 115 116 117 118ToAs (ms) 28.714 25.116 625.810 26.574 24.993 26.994 24.492


7.4.2.2 Identifying Erroneous Measurements via rNmax Test

A fault which occurs at 62 miles away from Bus 47 on Line 47-69, will be used to

illustrate the robust fault-location approach using rNmax test. Notice from Table 7.7

that out of 43 sensors where ToAs are captured following the fault occurrence, 11

of them (designated by the color red) are contaminated will later be suspected as

corrupted measurements.

Applying the steps of rNmax-test introduced in Section 7.3.2.1, the normalized

residual vectors(r(`))N

are produced after successive state estimation processes until

no bad measurements are suspected, i.e., until all measurements stay below certain

threshold c. After the first iteration of state estimation is completed, the measure-

ment at Sensor 25, which is placed on Bus 88, has been suspected as bad measure-

ment; therefore, we have replaced its value by T25 = 23.418. Based on (7.11b),

similar updates are done after each run of state estimation to obtain new values of

Tj for each suspected measurement j as shown in Table 7.8. Displayed in Figure

7.3 is the convergence plot of the minimizing value of the objective function, J`, for

Line “`” selected in each state-estimation run that is shown in Table 7.8. Table 7.9

shows the rectified values of sensor measurements which were formerly suspected to

have been contaminated with huge errors.


Buses 1 2 4 6 10 14 20 29

ToAs (ms) 34.369 18.236 27.548 27.844 27.354 25.649 26.402 26.004

Buses 35 39 41 46 53 55 57ToAs (ms) 8.612 23.142 21.958 20.511 22.744 21.253 22.405

Buses 58 60 61 67 73 74 79ToAs (ms) 15.111 22.152 22.346 22.453 19.608 21.791 22.572

Buses 84 87 88 90 93 95 97ToAs (ms) 12.962 27.058 38.167 25.552 24.653 24.116 16.452

Buses 99 101 104 106 107 109 111ToAs (ms) 22.841 24.578 25.202 25.407 26.488 19.433 26.418

Buses 112 113 114 115 116 117 118ToAs (ms) 26.778 21.598 25.810 26.735 23.766 34.919 22.555


Table 7.8: Changing values in each iteration of rNmax Test

Iteration # ` of J min`

(r

(`)j

)Nmax

j of(r

(`)j

)Nmax

T(`)jcorr

1 149 14.5090 25 23.4182 149 9.4627 2 28.1193 149 8.5942 9 17.3484 379 6.8076 9 24.2555 379 6.3161 35 25.8416 347 6.0408 42 28.5837 337 5.9870 16 21.1978 340 5.8913 29 22.4389 340 4.9117 1 29.21810 354 2.9777 38 24.92711 354 3.6851 20 23.72812 333 1.5611 1 27.61213 333 0.9467 38 25.89314 405 0.9147 25 24.36315 333 0.8966 9 25.17816 333 0.6581 23 23.19317 333 0.6515 35 26.50618 661 0.4944 42 28.08319 661 0.2641 20 23.99520 661 0.1337 2 27.98421 661 0.1237 38 26.01822 661 0.0961 25 24.46023 661 0.0971 29 22.53624 661 0.0650 23 23.25925 661 0.0399 35 26.54626 661 0.0353 9 25.14227 661 0.0202 16 21.17728 661 0.0153 1 27.62829 661 0.0131 42 28.09530 661 0.0095 2 27.99431 661 0.0072 38 26.02532 661 0.0073 20 24.00233 661 0.0052 25 24.46534 661 0.0029 29 22.53935 661 0.0013 9 25.14336 661 0.0012 42 28.096


Minim

um

valueof

J ℓ

Iterations

0

5

10

15 20 25 30 355

15

20

25

30

100

Figure 7.3: Convergence of minimum value Jmin` of the objective function J`.

Table 7.9: Synchronized Sensor Locations versus Wave-Arrival Times (aftercorrecting bad measurements) for the Fault Occurring at 62 Miles Away from Bus

47 on Line 47-69Buses 1 2 4 6 10 14 20 29

ToAs (ms) 27.628 27.994 27.548 27.844 27.354 25.649 26.402 26.004

Buses 35 39 41 46 53 55 57ToAs (ms) 25.143 23.142 21.958 20.511 22.744 21.253 22.405

Buses 58 60 61 67 73 74 79ToAs (ms) 21.177 22.152 22.346 22.453 24.002 21.791 22.572

Buses 84 87 88 90 93 95 97ToAs (ms) 23.259 27.058 24.465 25.552 24.653 24.116 22.539

Buses 99 101 104 106 107 109 111ToAs (ms) 22.841 24.578 25.202 25.407 26.488 26.546 26.418

Buses 112 113 114 115 116 117 118ToAs (ms) 26.778 26.025 25.810 26.735 23.766 28.096 22.555


47 69

0.2448

D47−69 = 0.4842 ms

︸︷︷︸

(661) = 0.2287619 620 621

0.0081 0.13450.0968

Line 662

Figure 7.4: Value of %(`) for the short-circuit fault occurring on Line 47-69.

Succeeding the correction of suspected measurements, a fault-location proce-

dure can now be implemented. For the studied fault scenario, the (converged) min-

imizing value of the cost function is attained on Line 661 (with J661 = 0.0033 ≈ 0)

in the resulting split network, for which we obtain

%

(661)

T(661)0

=

0.2287

20.0001

ms.

In Figure 7.4, the location of the fault on Line 47-69 is displayed in terms of

the propagation delay, %(`)). The location of the fault is detected on the virtual line

segment (i.e., Line 661) connecting Terminals 620 and 621. Thus, the distance to

fault from Bus 47 is computed as

xfault =((0.081 + 0.968 + 2.287)× 10−4 s

)× (1.85885× 105 mi/s)

= 62.02 ≈ 62 mi.

The flowchart depicted in Figure 7.5 visualizes the connections between pro-

posed fault-location algorithm and rNmax-Test-based bad-measurement identification

algorithm.


Assume “ℓ” to bethe faulted branch

Find the fault location

in terms of

((ℓ)

T(ℓ)0

)

Pick “ℓ” that yieldsminimum Jℓ

Compute(r(ℓ)

)Nfor the chosen branch

Identify & correctbad measurement

Stop!

Yes

NoIs ∃

(r(ℓ)j

)N> c ?

Figure 7.5: Flowchart of the rNmax-Test-based bad-measurement identificationalgorithm for fault location.

7.4.3 Limiting Cases of the Proposed Method Under Coor-

dinated Cyberattacks

The objective of this section is to present the exceptional cases when the simulta-

neous attacks on synchronized measurements can evade the bad-data identification

algorithm, thus rendering the location of the faults unidentifiable. Nonetheless, such


attacks will be shown to be detectable in a sense that the proposed algorithm will en-

sure the proper detection of merely faulty line segment in the presence of deliberately

attacked measurements.

The line segments which are declared “defenseless” against coordinated cyber-

attacks will be determined by referring to the shortest-path mapping vectors, S`,

introduced in (5.19b). Note that these vulnerable line segments can be identified

in advance since shortest electrical distance between every pair of buses (including

the virtual ones created at the points defined in Figure 5.4, based on the deployed

sensor locations) is known with the readily available knowledge of the grid topology

and the line lengths. In our case, the matrix SK×L, which is of dimension 43× 759,

is created. Keeping in mind that Sk,` can take either “−1” or “1”, the condition

Sk,` = 1 implies that the shortest propagation path from Sensor “k” to Line “`” is

from the origin of that line; whereas, the shortest electrical pathway from the same

sensor would be from the terminus of that line whenever Sk,` = −1. In particular,

the sensor, for which the value Sk,` has a different sign from the remaining elements

of the column vector S`, will be regarded as a fault-critical sensor for the vulnerable

Line Segment “`”, which can be logically expressed as

∃! Sensor “m” such that Sm,` = −Sk,`; k ∈ 1, · · · , K − m. (7.13)

Figure 7.6 designates the location of a fault on one of the line segments (i.e.,

Line 334) that is vulnerable to a cyberattack on a specific sensor measurement.

Indeed, the elements of the column vector S334 are all “1” except the one corre-

sponding to Sensor 20 (i.e., S20,334 = −1). Therefore, the sensor placed at Bus 73

(i.e., Sensor 20) is detected as a “fault-critical” sensor for Line 70-71. If the cor-

responding measurement at that sensor is corrupted with some degree of error, the

remaining measurements (even assuming all of them are error-free) will not be able

to accurately pinpoint the location of any fault on the respective vulnerable line

segment.



Buses 1 2 4 6 10 14 20 29

ToAs (ms) 38.144 28.624 28.618 29.490 19.346 27.028 27.429 25.132

Buses 35 39 41 46 53 55 57 58

ToAs (ms) 32.494 24.788 23.604 7.322 24.097 22.900 24.051 22.101

Buses 60 61 67 71† 72† 73 74 79

ToAs (ms) 23.799 19.224 24.099 21.001 21.366 38.480 −7.179 30.296

Buses 84 87 88 90 93 95 97 99

ToAs (ms) 37.997 28.360 25.767 26.854 25.955 25.417 23.841 24.142

Buses 101 104 106 107 109 111 112 113

ToAs (ms) 36.844 35.172 27.314 37.946 36.383 23.885 28.080 25.154

Buses 114 115 116 117 118ToAs (ms) 24.939 25.864 25.412 28.726 21.888

† designates the sensors added with the intent to render the fault points on the cor-responding line segment identifiable.

70 71380 381 382

0.0720 0.2050

D70−71 = 1.5386 ms

1.0250 0.2370

Line 334100 mi

(d70−71 = 286 mi)

Figure 7.6: Actual location of an “unidentifiable” fault occurring on Line 70-71due to a cyberattack.

If a fault is detected on a vulnerable segment, the value of the residual corre-

sponding to the “fault-critical” sensor will be equal to zero, and the measurement

might be suspected to have been contaminated. We should note that the residual

belonging to the fault-critical sensor will always appear as “zero” even when its

associated sensor measurement contains gross errors.

As can be seen from Table 7.10, multiple sensors (including the fault-critical

one) have been attacked by intruders. The sensors that are placed at Buses 71 and 72

are also highlighted with a blue color. This is done to increase the redundancy, hence

transforming the sensor located at Bus 73 (formerly Sensor 20, but currently Sensor

22) and making it no longer fault-critical. Indeed, it is observed that Sk,` = −1 for

these newly deployed sensors (i.e., Sensors 20 and 21). With the presence of extra

sensors, Line 334 now becomes Line 338.


Table 7.11 illustrates the changing values of %(334), T(334)0 , and x for different

measurements at Sensor 20 (i.e., T20).

Table 7.11: Values of %(334), T(334)0 , and d with respect to Contaminated Mea-

surements at Sensor 20 for the Fault Case on Line 70-71

T20 (ms) %(334)(ms) T(334)0 (ms) x (mi)

15.812 3.6593 16.8786 680.198023.455 −0.1620 20.7000 −30.112923.131 0 20.5380 022.631 0.2500 20.2880 46.470522.000 0.5655 19.9725 105.1163

It should be noted that the deployed sensors ensures full fault-location ob-

servability for the entire grid; however, a coordinated cyberattack on a fault-critical

sensor renders one segment on a particular line fault-unobservable. Moreover, it is

probable that a sensor that is fault-critical for a certain line segment can bear the

same identity for another line segments in the grid.

Table 7.12 shows some typical results obtained with regard to the vulnera-

bility of the studied grid to measurement tampering in fault-critical sensors such

as the number of optimally deployed sensors (K); the ratio of deployed sensors to

the number of (actual) buses in the transmission grid (K/N); the number of de-

fenseless line segments against cyberattacks (LA) along with its ratio to the total

number of transmission-line segments (LA/L); and the proportion of total length of

attack-vulnerable line segments [dA (in kilomiles (k-mi))] to total length of the entire

grid [dtotal (in k-mi)]. It has been observed from the simulations that sensor redun-

dancy has to be improved in order to minimize the number of attack-vulnerable line

segments.


Table 7.12: Miscellaneous Results for the Studied Test Grid

K K/N LA LA/L dA dtotal dA/dtotal

43 0.3644 111 0.1462 10.534 32.415 0.3250

Table 7.13 lists various values of∥∥r(`)

LAV

∥∥1

for different Line Segments “`”. Based

on the discussion in Section 7.4.2, %(338) is now selected as the optimum solution

for the fault-associated propagation delay. Increased redundancy enables precise

calculation of the distance to fault as given below:

d = (0.538 ms)× (1.85885× 105 mi/s) = 100 mi.

Table 7.13: Values of `,∥∥r(`)

LAV

∥∥1, %(`), T

(`)0 , and D` for the Fault Occurring at

100 Miles Away from Bus 70 on Line 70-71

`∥∥r(`)

LAV

∥∥1

%(`) T(`)0 D` 0 ≤ %(`)≤ D`?

399 126.0293 14.0548 6.4831 0.2959 No403 130.3979 −13.7645 6.4887 0.2959 No352 151.2493 −9.2133 28.2126 1.0540 No349 151.5438 2.7896 20.5382 0.7823 No338 153.0630 0.5380 20 1.0250 Yes400 153.5468 −0.0002 20.2422 0.0914 No356 153.7290 0.6967 20.5382 0.4194 No746 154.1382 −0.0002 20.5381 0.0915 No358 154.1383 0.0723 20.5382 0.0720 No365 154.1385 0.1719 20.5380 0.1720 Yes357 154.2842 0.2773 20.5382 0.2050 No364 154.8246 0.2878 20.5377 0.1160 No354 155.1218 1.1110 20.5382 0.0485 No355 155.1224 1.0625 20.5382 0.3658 No

......

......

......

The flowchart depicted in Figure 7.7 visualizes the connections between each

module of the robustified fault-location algorithm.


Run the LAV-basedcountermeasure algorithm


in terms of

((ℓ)

T(ℓ)0

)

Pick “ℓ” that yields

minimum∥∥r(ℓ)LAV

∥∥1

Does Line “ℓ” havea “critical” sensor?

Yes

Fault isUNIDENTIFIABLE!

No

Fault is IDENTIFIABLE!

Check r(ℓ)LAV to identify

the bad measurement(s)

Remove sensor(s) havingbad measurement(s)

Run the initialfault-location (FL) algorithm

Is minJℓ < c?(c is a threshold,

e.g., c = 1)

Yes

No Errors are minor!Fault is

LOCALIZABLE!


in terms of

((ℓ)

T(ℓ)0

)

Utilize extra sensorsto make the faults onLine “ℓ” localizable

Re-run the initialFL algorithm

...


in terms of

((ℓ)

T(ℓ)0

)...Re-run the initialFL algorithm to

enhance estimation accuracy

Figure 7.7: Flowchart of the “robustified” fault-location algorithm against com-promised sensor measurements.


7.5 Summary

In this chapter, fault-location approach, which is resilient against intentional or

random measurement tampering, is presented. The approach is based on a least-

absolute-value estimation of the fault location and exploits the automatic bad-data

rejection property of such estimators. Another bad-data processing technique based

on largest-normalized-residuals test is also introduced as a less robust alternative

to the LAV-based fault-location estimation. The effect of measurement errors due

to low-resolution sensors on the estimation accuracy of the fault location is also

investigated.

As part of the study, a novel concept of “fault-critical sensor” is also developed,

where it is shown that under specific sensor configurations, certain sensors will be-

come critical in that their errors cannot be detected, thus making them vulnerable to

tampering. Detailed simulation results on a typical transmission grid are provided

to illustrate the effectiveness of developed methods.

Chapter 8

Concluding Remarks and Further

Study

The research presented in this dissertation has dealt with the development of wide-

area synchronized-measurement-based solution to fault-location problem in power

systems; optimal sensor deployment for fault-location observability; and making the

fault-location algorithm resilient against defective measurements. An account of the

main contributions of this dissertation follows, along with the directions for future

work.

8.1 Concluding Remarks

Holistically speaking, we consider the problem of robust identification of electrical

faults in a systemwide level via the use of synchronized measurement sensors that

are optimally deployed across a large-scale electric power network. Indeed, we aim

to present an optimal sensor deployment procedure that ensures unique localization

of line faults appearing in power grids. In analogy to synchronized sensor networks

in wireless communications, we initially model the grid as a wired mesh network,

143

Chapter 8. Concluding Remarks and Further Study 144

in which nodes are connected through power lines. Based on the topology of the

network and the distance measurements among electrical nodes, a fault-location tech-

nique is then established. Specifically, we develop a method that relies on recorded

time-of-arrival (ToA) measurements at nodes which are equipped with synchronized

sensors. Identification of the location for a power-grid fault is achieved via sensors,

which are deployed in an optimal manner. Furthermore, adverse impacts of gross

measurement errors in synchronized measurements on the accuracy of the devel-

oped fault-location technique are eliminated with the successful application of two

bad-data processing algorithms.

The overall network design has the advantages of being both “location-aware”

of any power-system fault and “resilient” against cyberintrusions while concurrently

achieving “cost-effectiveness” via optimal sensor deployment. Consequently, cyber-

security elements can be upgraded to protect the sensor-measurement integrity and

cyberinfrastructure can tolerate any potential attack vectors allowing the grid to op-

erate resiliently under disturbances and malicious attacks. The power grid can thus

be engineered with redundancies to withstand physical failures as well as with error-

detection capabilities, which gracefully handle faulty scenarios due to both electrical

and informational abnormalities in the system. These attributes are expected to

provide enhanced resiliency that can be used synergistically with cyberprotection

mechanisms within the supporting infrastructure.

8.1.1 Wide-Area Synchronized-Measurement-Based Fault Lo-

cation

We develop a novel analytical and computational approach to wide-area fault loca-

tion in large-scale power systems. In order to pinpoint (locate) the faults, optimally

distributed (synchronized) sensors in a power network are exploited to capture dis-

crete samples of transient voltages after the occurrence of a fault. Indeed, signatures

of fault are observed at different substations where these synchronized sensors are


located. These synchronized wide-area measurements are placed in an optimal fash-

ion, and this way any fault anywhere in the network can be uniquely pinpointed.

After the fault occurrence, the fault point is triangulated by the aid of the recorded

times of arrival of fault-induced waves at these strategic buses.

8.1.2 Optimal Deployment of Synchronized Sensors for Wide-

Area Fault Location

The salient contribution of this study is to provide one viable strategy for rendering

the power network observable from fault location point of view, by means of opti-

mally deployed sensors that record GPS-synchronized voltage measurements. As a

result, the proposed fault-location procedure can uniquely identify the location of a

fault regardless of where it is originated from on the power grid.

8.1.3 Robust Estimation of Fault Location

We aim to robustify the developed algorithm for fault location in order to reliably

locate power-system faults using simultaneously recorded data from multiple loca-

tions. In effect, self-correction of corrupted measurements resulting from various

factors (e.g., sensor breakdowns and cyberattacks) is accomplished by the efficient

integration of the least-absolute-value (LAV) state estimator and largest-normalized-

residuals test within the fault-location approach. In addition to revealing the inher-

ent limits of applying the robust state estimator, the effect of quantization errors

incurred by low-precision sensors is taken into consideration.


8.2 Further Study

The following recommendations for future work are by no means exhaustive. How-

ever, they are considered worthy to study with the intent to expand the scope of the

establishment of the field state of the art.

8.2.1 Investigation of Novel Time-Frequency Methods for

Transient Analysis

Though wavelets have been well-applicable in various application domains in power

systems, one can choose to investigate emerging time-frequency methods such as

fractional Fourier transform [132] and fractional wavelet transform [133] to analyze

power-system transients as well as to locate power-system faults. Moreover, cus-

tomization of wavelets for fault location can also be further investigated [134, 135].

8.2.2 Simultaneous Occurrence of Multiple Line Faults

Multiple line faults that occur simultaneously at many different network locations are

very rare. However, new fault-location strategies can be formulated to handle such

events. One possible solution would be to use data-clustering techniques in order to

determine which measurement at a particular network node belongs to which fault.

8.2.3 Inclusion of Additional Network Components

The fault-location strategy developed in this dissertation can be enhanced by inves-

tigating the effects of untransposed lines, double-circuit (parallel) lines, underground

cables, transformers, and FACTS devices such as static VAR compensators (SVCs)

and thyristor-switched capacitors (TSCs).


8.2.4 Simulation of Various Fault Types

We focus our attention merely on single-line-to-ground faults, which are the most

frequently occurring faults in power systems. The methodologies developed in this

dissertation can be applied to other types of power-system faults, which can be either

unsymmetrical (i.e., line-to-line and double-line-to-ground faults) or symmetrical

(i.e., three-phase faults).

8.2.5 Methods to Mitigate the Effect of Attenuated Travel-

ing Waves

As traveling waves propagate throughout the network, they become attenuated or

damped out at distant network nodes due to the multiple reflections and refractions

initiated by fault. Hence, this may result in ambiguities in identifying the precise

arrival times of traveling waves, especially on fault recorders deployed at remote

substations with respect to a fault location. This is expected to be a more influencing

factor in very-large-scale power networks.

8.2.6 Line Modeling and Transient Simulations via Wavelet-

Like Transform

Simulation of transients involving circuits with frequency-dependent line and cable

parameters is another problem to be investigated. A new wavelet-like transform may

assist in improving the accuracy of the transient-simulation results by minimizing

the approximation errors due to the choice of modal transformation matrices for

each subband of frequencies.

Appendix A

Modal Analysis of Multiphase

Transmission Lines

A.1 Transmission-Line Equations

At a given frequency ω, the series voltage and shunt current drops are defined as

−dVphase

dx= Z ′

phaseIphase (A.1)

−dIphase

dx= Y ′

phaseVphase (A.2)

where the vectors

Vphase =

ϑa(x)

ϑb(x)

ϑc(x)

and Iphase =

ia(x)

ib(x)

ic(x)

(A.3)

indicate voltages and currents for Phases a, b, and c at some point x along the line

as shown in Figure A.1. Notice that the sending end of the line is Node “s” at x = 0,

and the receiving end is Node “r” at x = d.

148

Appendix A. Modal Analysis of Multiphase Transmission Lines 149

a

b

c

s r

x

ϑa(x)

ϑb(x)

ϑc(x)

ia(x)

ib(x)

ic(x)

Figure A.1: Phases a, b, and c of a three-phase transmission line between Ter-minals s and r.

Taking the second derivative of (A.1) and (A.2) with respect to x, and relating

the two equations, yields

−d2Vphase

dx2= Z ′

phase

dIphase

dx= −Z ′

phaseY ′phaseVphase (A.4)

−d2Iphase

dx2= Y ′

phase

dVphase

dx= −Y ′

phaseZ ′phaseIphase (A.5)

(A.4) and (A.5) govern the propagation of the voltages and currents, respectively.

It should be noted that Y ′phaseZ ′

phase 6= Z ′phaseY ′

phase since matrix products are non-

commutative.

The solutions of linear second-order differential equations, (A.4) and (A.5), can

be given by

Vphase = Vphase–se−γϑx + Vphase–re

γϑx (A.6)

Iphase = Iphase–se−γix + Iphase–re

γix (A.7)

where γϑ = (Z ′phaseY ′

phase)1/2 and γi = (Y ′

phaseZ ′phase)

1/2. Matrix Vphase–s designates

the phase voltages at the sending end “s” of the line assuming that a wave travels

from Terminal “s” toward the receiving end “r”, and there is no wave reflection at

Terminal “r”. By the same token, matrix Vphase–r designates the phase voltages at


the receiving end “r” of the line assuming that a wave travels from Terminal “r”

toward Terminal “s”, and there is no wave reflection at Terminal “s”. The same

approach applies to the currents.

In general, (A.4) and (A.5) are difficult to solve due to the coupling between

the phases. However, it is possible to transform them into decoupled equations,

which will be described in the next section.

A.2 Modal Transformations

By transforming phase voltages/currents into modal voltages/currents,

Vphase = Tϑ Vmode ⇒ Vmode = Tϑ−1Vphase (A.8)

Iphase = Ti Imode ⇒ Imode = Ti−1Iphase, (A.9)

where Tϑ and Ti are the voltage and current transformation matrices, respectively;

and substituting (A.8) and (A.9) into (A.1) and (A.2); we obtain

−TϑdVmode

dx= Z ′

phaseTi Imode (A.10)

−TidImode

dx= Y ′

phaseTϑ Vmode. (A.11)

It follows from (A.10) and (A.11) that

−dVmode

dx= Tϑ−1Z ′

phaseTi Imode = Z ′modeImode (A.12)

−dImode

dx= Ti−1Y ′

phaseTϑ Vmode = Y ′modeVmode, (A.13)

where Z ′mode and Y ′

mode are the modal series impedance and modal shunt admittance

matrices, respectively.


Now, substituting (A.8) and (A.9) into (A.4) and (A.5), we get

d2Vmode

dx2= Tϑ−1Z ′

phaseY ′phaseTϑ Vmode = Z ′

modeY ′modeVmode = ΛVmode (A.14)

d2Imode

dx2= Ti−1Y ′

phaseZ ′phaseTi Imode = Y ′

modeZ ′modeImode = ΛImode. (A.15)

Notice that the eigenvalues of the matrix products Z ′modeY ′

mode and Y ′modeZ ′

mode are

identical. Matrix Λ is a diagonal matrix of eigenvalues; and the transformation ma-

trices Tϑ and Ti are the matrices of eigenvectors of the matrix products Z ′phaseY ′

phase

and Y ′phaseZ ′

phase, respectively.

Since Tϑ and Ti lead to the same eigenvalues; Λ is diagonal; and Z ′phase and

Y ′phase are symmetric; we can write

Λ = Λ>

Tϑ−1Z ′phaseY ′

phaseTϑ = (Tϑ−1Z ′phaseY ′

phaseTϑ)>

= Tϑ>Y ′phaseZ ′

phase

(Tϑ>)−1

. (A.16)

Hence,(Tϑ>

)−1becomes a modal matrix for the matrix product Y ′

phaseZ ′phase, whose

modal matrix is Ti; thus, we can conclude that Ti =(Tϑ>

)−1.

We can now diagonalize (A.12) and (A.13) as well. Indeed, we can show that

Z ′mode and Y ′

mode are diagonal as shown in the following:

Λ = Tϑ−1Z ′phaseY ′

phaseTϑ =(Tϑ−1Z ′

phaseTi)(Ti−1Y ′

phaseTϑ)

= Z ′modeY ′

mode. (A.17)

Similarly, we can write

Λ = Ti−1Y ′phaseZ ′

phaseTi =(Ti−1Y ′

phaseTϑ)(Tϑ−1Z ′

phaseTi)

= Y ′modeZ ′

mode. (A.18)

Hence, we can infer that Z ′mode and Y ′

mode are diagonal since Λ = Z ′modeY ′

mode =


Y ′modeZ ′

mode. This holds for both the transposed and untransposed line configura-

tions.

A.3 Balanced Transformations for Transposed Lines

The modal transformation discussed above is used to decouple any geometrical con-

ductor arrangement in a transmission line. However, for transposed lines, we can

simplify the eigendecoupling problem, assuming that matrices Z ′phase and Y ′

phase are

balanced1, i.e.,

Z ′phase =

Z ′s′ Z ′m′ Z′m′

Z ′m′ Z ′s′ Z ′m′

Z ′m′ Z′m′ Z ′s′

and Y ′

phase =

Y ′s′ Y ′m′ Y′m′

Y ′m′ Y ′s′ Y ′m′

Y ′m′ Y′m′ Y ′s′

, (A.19)

where Z ′s′ , Z′m′ , Y

′s′ , and Y ′m′ are the self- and mutual impedances and admittances,

respectively. Note that the generalized modal analysis approach is equally applicable

to the specific case above. When solving the eigendecoupling problem for balanced

matrices in (A.19), the conditions for the solution are more relaxed thanks to the

symmetry of the problem. The following are the two well-known transformation

matrices for balanced systems.

A.3.1 Karrenbauer’s Transformation

In Karrenbauer’s transformation, Tϑ = Ti and

Ti =

1 1 1

1 −2 1

1 1 −2

⇒ Ti−1 =

1

3

1 1 1

1 −1 0

1 0 −1

. (A.20)

1A matrix is balanced when all elements in the main diagonal are identical, and all of theoff-diagonal elements are also equal to each other.


A.3.2 Clarke’s Transformation

In Clarke’s transformation, Tϑ = Ti and

Ti =1√3

1√

2 0

1 − 1√2

√3√2

1 − 1√2−√

3√2

⇒ Ti−1 =

1√3

1 1 1√

2 − 1√2− 1√

2

0√

3√2−√

3√2

. (A.21)

In both transformations, it can be seen that the elements of the first column

in Ti are identical and that the elements of the second and third columns add up to

zero. However, the advantage of Clarke’s transformation is that Ti−1 = Ti>, which

makes the computation of Tϑ or modal quantities easier.

For both transformation matrices, the modal impedances and admittances are

identical, viz.,

Z ′mode =

Z ′s + 2Z ′m 0 0

0 Z ′s − Z ′m 0

0 0 Z ′s − Z ′m

(A.22a)

and

Y ′mode =

Y ′s + 2Y ′m 0 0

0 Y ′s − Y ′m 0

0 0 Y ′s − Y ′m

. (A.22b)

Appendix B

Derivation of A1 and A2 Used in

Chapter 3

Based on the boundary conditions at the sending end of the line (x = 0), i.e.,

V(0, ω) = Vs (B.1)

I(0, ω) = Is,r, (B.2)

the voltage and current equations become

V(0, ω) = Vs(ω) = A1e−γ(ω)(0) + A2e

γ(ω)(0) (B.3)

I(0, ω) = Is,r(ω) =A1e

−γ(ω)(0) − A2eγ(ω)(0)

Zc, (B.4)

or equivalently,

Vs(ω) = A1 + A2, (B.5)

Is,r(ω) =A1 − A2

Zc. (B.6)

154

Appendix B. Derivation of A1 and A2 Used in Chapter 3 155

Substituting Vs(ω) = Is,r(ω)Zc + A2 + A2, we get

A2 =Vs(ω)− Is,r(ω)Zc

2, (B.7)

and solving for A1, we obtain

A1 =Vs(ω) + Is,r(ω)Zc

2. (B.8)

Appendix C

Proof of Unique Localizability in a

Fault-Unobservable Branch

Claim. A fault occurring on a so-called “fault-unobservable” branch of the power

grid can still be “uniquely” located via the synchronized measurements from the avail-

able set of sensors, i.e., the solution of the sensor deployment algorithm is declared

to be “optimal”.

Proof. Intuitively, the claim can be proven by a simple induction.

Let B = BO ∪ BU denote the complete set of branches (line segments) in the

transmission grid, where BO and BU represent the sets of fault-observable and fault-

unobservable branches, respectively; and ` denotes the faulty line segment. Then, it

follows that:

• If ` ∈ BO, then fault-location algorithm can uniquely identify the location of

the fault point.

• If ` ∈ BU , then fault-location algorithm cannot identify the location of the

fault point. Thus, second-stage, single-ended fault-location procedure pro-

posed in Section 6.3.2 needs to be carried out to accurately locate the fault.

156

Appendix C. Proof of Fault Localizability in a Fault-Unobservable Branch 157

• The conditions that ` ∈ BU and that proposed algorithm indicates ` ∈ BOcannot occur simultaneously. Indeed:

– If this occurs, it implicates multiple solutions.

– However, the two conditions cannot exist simultaneously owing to the

uniqueness of fault localizability provided by the sensor deployment algo-

rithm.

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Vita

Mert Korkalı received the B.S. degrees both in electrical and electronics engineering

and in industrial engineering from Bahcesehir University, Istanbul, Turkey, in 2008;

and the M.S. and Ph.D. degrees in electrical engineering from Northeastern Univer-

sity, Boston, MA, in 2010 and 2013, respectively. His current research interests lie at

the broad interface of fault location in large-scale power networks; state estimation

in electric power systems; cybersecurity of smart grids; and cascading failures in

interdependent power-communication networks. Dr. Korkalı is a member of IEEE

Power and Energy Society.

175

List of Publications

The following list includes all the papers published by the author during his graduate

studies. Papers designated by “?” are directly linked with research presented in this

dissertation.

Journal Papers

[J4] ?M. Korkalı and A. Abur, “Robust Fault Location Using Least-Absolute-

Value Estimator,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4384–

4392, Nov. 2013.

[J3] ? M. Korkalı and A. Abur, “Optimal Deployment of Wide-Area Syn-

chronized Measurements for Fault-Location Observability,” IEEE Transactions on

Power Systems, vol. 28, no. 1, pp. 482–489, Feb. 2013.

[J2] ? M. Korkalı, H. Lev-Ari, and A. Abur, “Traveling-Wave-Based Fault-

Location Technique for Transmission Grids via Wide-Area Synchronized Voltage

Measurements,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 1003–

1011, May 2012.

[J1] A. Abur, J. Chen, J. Zhu, and M. Korkalı, “Topology, Parameter, and

Measurement Error Processing Using Synchronized Phasors,” European Transac-

tions on Electrical Power (ETEP)—Special Issue: Power System Measurement Data

and their Applications, vol. 21, no. 4, pp. 1600–1609, May 2011.

176


Conference Papers

[C9] ? M. Korkalı and A. Abur, “An Attack-Resilient Fault-Location Al-

gorithm for Transmission Grids Based on LAV Estimation,” 44th North American

Power Symposium (NAPS), Sep. 9–11, 2012, Urbana, IL.

[C8] ? M. Korkalı and A. Abur, “Detection, Identification, and Correction

of Bad Sensor Measurements for Fault Location,” Proceedings of 2012 IEEE PES

General Meeting, Jul. 22–26, 2012, San Diego, CA.

[C7] ? M. Korkalı and A. Abur, “Optimal Sensor Deployment for Fault-

Tolerant Smart Grids,” 2012 IEEE International Workshop on Signal Processing

Advances in Wireless Communications (SPAWC), Jun. 17–20, 2012, Cesme, Turkey.

[C6] ? M. Korkalı and A. Abur, “Use of Sparsely Distributed Synchronized

Recorders for Locating Faults in Power Grids,” Proceedings of 7th International

Conference on Electrical and Electronics Engineering (ELECO), Dec. 1–4, 2011,

Bursa, Turkey.

[C5] ? M. Korkalı and A. Abur, “Transmission System Fault Location Using

Limited Number of Synchronized Recorders,” Proceedings of International Confer-

ence on Power Systems Transients (IPST), Jun. 14–17, 2011, Delft, The Nether-

lands.

[C4] ? M. Korkalı and A. Abur, “Fault Location in Meshed Power Net-

works Using Synchronized Measurements,” 42nd North American Power Symposium

(NAPS), Sep. 26–28, 2010, Arlington, TX.

[C3] M. Korkalı and A. Abur, “Impact of Network Sparsity on Strategic

Placement of Phasor Measurement Units with Fixed Channel Capacity,” Proceedings

of 2010 IEEE International Symposium on Circuits and Systems (ISCAS), May 30–

Jun. 2, 2010, Paris, France.


[C2] M. Korkalı and A. Abur, “Reliable Measurement Design Against Loss of

PMUs with Variable Number of Channels,” 41st North American Power Symposium

(NAPS), Oct. 4–6, 2009, Starkville, MS.

[C1] M. Korkalı and A. Abur, “Placement of PMUs with Channel Limits,”

Proceedings of 2009 IEEE PES General Meeting, Jul. 26–30, 2009, Calgary, AB,

Canada.

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