My title - UNSWorks

Novel Approach for Analyzing

Interconnected Power Systems using

Complex Network Theory

A. B. M. Nasiruzzaman

A thesis submitted in fulfilment

of the requirements of the degree of

Doctor of Philosophy

SCIENTIA

MANU E T MENTE

School of Engineering and Information Technology

The University of New South Wales Canberra

November 2013

Copyright Statement

‘I hereby grant the University of New South Wales or its agents the right to archive

and to make available my thesis or dissertation in whole or part in the University

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retain the right to use in future works (such as articles or books) all or part of this

thesis or dissertation.

I also authorise University Microfilms to use the 350 word abstract of my thesis

in Dissertation Abstract International (this is applicable to doctoral theses only).

I have either used no substantial portions of copyright material in my thesis or I

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granted I have applied/will apply for a partial restriction of the digital copy of my

thesis or dissertation.’


28 Oct. 2013

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Authenticity Statement

‘I certify that the Library deposit digital copy is a direct equivalent of the final

officially approved version of my thesis. No emendation of content has occurred and

if there are any minor variations in formatting, they are the result of the conversion

to digital format.’


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Originality Statement

‘I hereby declare that this submission is my own work and to the best of my knowl-

edge it contains no materials previously published or written by another person,

or substantial proportions of material which have been accepted for the award of

any other degree or diploma at UNSW or any other educational institution, except

where due acknowledgement is made in the thesis. Any contribution made to the

research by others, with whom I have worked at UNSW or elsewhere, is explicitly

acknowledged in the thesis. I also declare that the intellectual content of this thesis

is the product of my own work, except to the extent that assistance from others in

the project’s design and conception or in style, presentation and linguistic expression

is acknowledged.’


28 Oct. 2013

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Abstract

The first contribution of this dissertation is the use of complex network theory for the

vulnerability analysis of power systems, taking into considerations various electrical

parameters of the power grid into forming a graph corresponding to the power grid.

An algorithm is presented to find critical elements from any power grid once the load

flow analysis data is available on the system. A relation between the vulnerability

and stability is found to exist, i.e., removal of important components found from

proposed methodology shows transient behavior in the power angle oscillations.

Various standard IEEE test systems have been used to demonstrate the utility of the

proposed method. The proposed betweenness centrality based method is novel than

previously existing ones employing complex network framework in better capturing

several electrical characteristics of the electricity grid as well in finding stable and

unstable regions within the grid.

Secondly, this thesis develops a modified flow based betweenness centrality mea-

sure to identify critical elements of a power system. The proposed method overcomes

the limitation of concentrating on shortest paths in calculating centrality indices;

instead the method considers all possible paths through which the power can flow

from source nodes to load nodes, giving a more realistic modeling choice of the power

grid. Standard IEEE test systems have been used to exhibit the utilization of the

method in finding critical components of the grid.

iv

Abstract v

Finally, various centrality measures like degree centrality, closeness centrality,

and betweenness centrality used in complex network framework based analysis are

applied for the power grid application. New definitions of these measures are pro-

posed to capture the realistic power flow scenario of the grid. A new matrix, which

contains the information of dependency of bus pairs in a power system, is also

presented. An algorithm of finding the bus dependency matrix from the system

is demonstrated with real power grid examples. Several characteristics, e.g., the

correlation between bus dependency matrix with the electrical closeness centrality

and the electrical betweenness centrality; non-symmetric property, are analyzed in

detail.

Acknowledgements

All praise is due to Allah, the Entirely Merciful, and the Especially Merciful. Thanks

to the Almighty for giving me a chance to fulfil my parent’s dream of completing

the Ph.D. degree.

I would like to express my deepest appreciation and regards to the person without

whom I wouldn’t be able to reach at this stage of life, Associate Professor Hemanshu

Roy Pota, for his trust, dedication and guidance all through my life after starting

the Ph.D. He has been a great mentor both for my academic problems as well as

family concerns. Indeed the journey has been a research training opportunity for

me with his grand professionalism, demanding nature, and analytical insights.

I wish to acknowledge the continuing contribution of Dr. Md. Jahangir Hossain

towards the power system research group in UNSW Canberra, Australia, which gave

me a great opportunity to work closely with future geniuses of the power system

research. I missed a nice opportunity to collaborate with him, but his reputation

and expertise were always there for me to control my trajectory back on track.

Our champ, Dr. Md. Apel Mahmud, has been a great counselor of my daily life

in Australia. His attentiveness to little details, keen professional eye on every aspect

of research, immense drive and huge success, I wish I could strive to emulate!

Special thanks go to Dr. F. M. Rabiul Islam and Adnan Anwar for their daily

commitments to sit and collaborate over a cup of tea, special thanks to the school

vi

Acknowledgements vii

for continuing to provide milk and tea for refreshments in spite of huge budget

cuts. Insights that I achieved during our regular jaunts were vital for completing

the work. Your insightful stories and discussions have spurred me on throughout

the entire candidature.

Thanks to all group mates for all the red ink. All co-authors please accept my

sincere gratitude for your support in preparing, submitting, and publishing papers.

I would like to extend my appreciation towards table tennis partners whose active

participation in the game, keeping works aside, gave me a relief from mundane

routine work of searching and bug-fixing.

Presence of Md. Abdul Barik, Dr. Md. Shamim Anower, Dr. S. M. Shahidul

Islam, Dr. Abu Barkat Ullah, Dr. Naruttam Kumar Roy, Md. Sohel Rana, Habibul-

lah, Md. Asikuzaman, Md. Sawkat Ali, and Dr. Md. Jahangir Alam made my life

much easier in Canberra with their steadfast support, suggestion and guidance. My

deepest regards to UNSW Canberra Bangladeshi Community, whose unparalleled

socialism helped me turning each of my many spectacular failures into the butt of

a joke.

Furthermore, I would like to thank all staffs of UNSW Canberra for contributing

to such an inspiring and pleasant atmosphere. I would like to thank Elvira Berra,

Christa Cordes, Elizabeth Carey for the all the supports they provided to me for

continuing my research. I would like to thank ICTS, ETS for their support during

the candidature.

Acknowledgements viii

Specially, I would like to thank Australian government and UNSW Canberra for

providing me with the scholarship to support my research. I would also like to be

grateful for the contribution of colleagues of Rajshahi University of Engineering &

Technology for giving me a chance to continue my higher education abroad. My

sincere thanks go to Dr. M. G. Rabbani, and Dr. M. R. I. Sheikh, for introducing

me to the research of the power system. Thanks to the reviewers and editors of all

my papers and this thesis, for their insightful comments and suggestions.

Eunice has been very kind in proving me housing so close to work in a reasonable

price. I would like to thank McDonalds Queanbeyan and McDonalds Dickson for

providing casual jobs, which was essential to maintain recreational activities. Special

thanks to sisters and brothers for preparing enticing recipes on various occasions,

the journey would not be as enjoyable without them.

Even an epic cannot capture the depth of my love and gratitude for my parents

A. K. M. Nuruzzaman and Nasrin Begum, who have set the standards early and

always inspired me to go beyond. I really appreciate my gregarious brother A. B.

M. Shakeruzzaman, and my ever intrepid A. B. M. Karimuzzaman for taking care

of parents during my absence. Thanks to my in-laws, grandparents, uncles, aunts,

cousins, neighbors, and friends for providing a nice environment for me to grow up

and mature.

I just have written the manuscript, but the contributions behind it are even more

complex than the power grid itself, as I reckon. It’s time to zip now, but not without

Acknowledgements ix

acknowledging the most important factor of this thesis, Most. Nahida Akter, the

love of my life. Her dedication and sacrifice for this thesis is beyond description, but

it is her achievement that she provided an atmosphere where I could get out of bed

in the middle of the night and drive towards university with her in the passenger sit

to investigate an idea, which has evolved to be this dissertation, that stroke in my

dream.

Dedicated to my family, my wife, Most. Nahida Akter,

&

ADFA Banladeshi Community

List of Publications

Refereed Book Chapters

1. A. B. M. Nasiruzzaman and H. R. Pota, “ Resiliency analysis of large-scale

renewable enriched power grid – a network percolation based approach,” in

Large Scale Renewable Power Generation: Advances in Technologies for Gen-

eration, Transmission and Storage, M. J. Hossain and M. A. Mahmud, Eds.

Springer-Verlag: Berlin Heidelberg, November 2013, In Press.

2. A. B. M. Nasiruzzaman, M. N. Akter and H. R. Pota, “ Impediments and

model for network centrality analysis of a renewable integrated electricity

grid,” in Renewable Energy Integration: Challenges and Solutions, M. J. Hos-

sain and M. A. Mahmud, Eds. Springer-Verlag: Berlin Heidelberg, November

2013, In Press.

Refereed Journal Papers

3. A. B. M. Nasiruzzaman and H. R. Pota, “ Bus dependency matrix of electrical

power systems,” International Journal of Electrical Power & Energy Systems,

Volume 56, March 2014, Pages 33-41.

Refereed Conference Papers

4. A. B. M. Nasiruzzaman and H. R. Pota, “Transient stability assessment of

xi

xii

smart power system using complex networks framework,” Power and Energy

Society General Meeting (PESGM 2011), pp.1–7, Detroit, MI, USA, 24–29

July 2011.

5. A. B. M. Nasiruzzaman and H. R. Pota, “Critical node identification of smart

power system using complex network framework based centrality approach,”

North American Power Symposium (NAPS 2011), pp.1–6, Boston, MA, USA,

4–6 August 2011.

6. A. B. M. Nasiruzzaman, H. R. Pota and F. R. Islam, “Complex network frame-

work based dependency matrix of electric power grid,” 2011 21st Australasian

Universities Power Engineering Conference (AUPEC 2011), pp.1–6, Brisbane,

QLD, Australia, 25–28 September 2011.

7. A. B. M. Nasiruzzaman, H. R. Pota and M. A. Mahmud, “Application of

centrality measures of complex network framework in power grid,” 37th Annual

Conference on IEEE Industrial Electronics Society (IECON 2011), pp.4660–

4665, Melbourne, VIC, Australia, 7–10 November 2011.

8. A. B. M. Nasiruzzaman and H. R. Pota, “A new model of centrality mea-

sure based on bidirectional power flow for smart and bulk power transmission

grid,” 11th International Conference on Environment and Electrical Engineer-

ing (EEEIC 2012), pp.155–160, Venice, Italy, 18–25 May 2012.

9. A. B. M. Nasiruzzaman, H. R. Pota and F. R. Islam, “Method, impact and

rank similarity of modified centrality measures of power grid to identify critical

xiii

components,” 11th International Conference on Environment and Electrical

Engineering (EEEIC 2012), pp.223–228, Venice, Italy, 18–25 May 2012.

10. A. B. M. Nasiruzzaman, H. R. Pota and M. A. Barik, “Implementation of

bidirectional power flow based centrality measure in bulk and smart power

transmission systems,” IEEE PES Innovative Smart Grid Technology Asia

(ISGT Asia 2012), pp.1–6, Tianjin, China, 21–24 May 2012.

11. A. B. M. Nasiruzzaman, H. R. Pota and A. Anwar, “Comparative study of

power grid centrality measures using complex network framework,” IEEE In-

ternational Power Engineering and Optimization Conference (PEOCO 2012),

pp.176–181, Melaka, Malaysia, 6–7 June 2012.

12. A. B. M. Nasiruzzaman, H. R. Pota, A. Anwar and F. R. Islam, “Modified

centrality measure based on bidirectional power flow for smart and bulk power

transmission grid,” IEEE International Power Engineering and Optimization

Conference (PEOCO 2012), pp.159–164, Melaka, Malaysia, 6–7 June 2012.

13. A. B. M. Nasiruzzaman and H. R. Pota, “Modified Centrality Measures of

Power Grid to Identify Critical Components: Method, Impact, and Rank Sim-

ilarity, ” Power and Energy Society General Meeting (PESGM 2012), pp.1–8,

San Diego, CA, USA, 22–26 July 2012.

Others Group Works

14. F. R. Isman, H. R. Pota and A. B. M. Nasiruzzaman, “PHEV′s park as a

virtual active filter for HVDC networks,” 11th International Conference on

xiv

Environment and Electrical Engineering (EEEIC 2012), pp.885–890, Venice,

Italy, 18–25 May 2012.

15. F. R. Islam, H. R. Pota, A. Anwar and A. B. M. Nasiruzzaman, “Design a Uni-

fied Power Quality Conditioner using V2G technology,” IEEE International

Power Engineering and Optimization Conference (PEOCO 2012), pp.521–526,

Melaka, Malaysia, 6–7 June 2012.

Contents

Copyright Statement i

Authenticity Statement ii

Originality Statement iii

Abstract iv

Acknowledgements vi

List of Publications xi

Chapter 1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation of Current Research . . . . . . . . . . . . . . . . . . . . . 7

1.3 Contribution of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2 Centrality Analysis and Transient Stability Assessment 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Power System as a Complex Network . . . . . . . . . . . . . . . . . . 21

2.3 Topological Statistics Parameter in the Power Grid . . . . . . . . . . 27

2.3.1 Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

xv

Contents xvi

2.3.2 Clustering Coefficient . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Characteristic Path Length . . . . . . . . . . . . . . . . . . . 32

2.4 Stability Assessment of the Micro Grid . . . . . . . . . . . . . . . . . 33

2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 3 Maximal-Flow Based Critical Node Identification Approach 42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Modeling of a Power System for Critical Node Identification . . . . . 45

3.3 Critical Node Identification of the Power Grid . . . . . . . . . . . . . 49

3.3.1 Shortest Electrical Path . . . . . . . . . . . . . . . . . . . . . 49

3.3.2 Node Removal and Network Efficiency of Power Grid . . . . . 50

3.3.3 Maximum Flow Based Critical Node Analysis . . . . . . . . . 52

3.3.4 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.6 Simulation of Various Standard Test System . . . . . . . . . . 58

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 4 Electrical Centrality Measures and Bus Dependency Matrix 60

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

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Measure of Connectivity-Degree Centrality . . . . . . . . . . . . . . 67

4.4 Measure of Independence-Closeness Centrality . . . . . . . . . . . . . 69

4.5 Measure of Control of Communication-Betweenness Centrality . . . . 72

Contents xvii

4.6 Simulation of Various Standard IEEE Test Systems . . . . . . . . . . 74

4.7 Measure of Pair Dependence of Various Buses . . . . . . . . . . . . . 78

4.7.1 Shortest Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7.2 Bus Dependency Matrix . . . . . . . . . . . . . . . . . . . . . 81

4.7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7.4 Steps to Find Bus Dependency Matrix from System Data . . . 87

4.8 Characteristics of Bus Dependency Matrix . . . . . . . . . . . . . . . 87

4.8.1 Relationship with Other Centrality Measures . . . . . . . . . . 87

4.8.2 Several Observations . . . . . . . . . . . . . . . . . . . . . . . 89

4.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 5 Bidirectional Power Flow Based Criticality Assessment 91

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 System Model and Methodology . . . . . . . . . . . . . . . . . . . . . 96

5.3 Measure of Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.1 Path Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.2 Connectivity Loss . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.3 Load Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Rank Similarity of Critical Nodes . . . . . . . . . . . . . . . . . . . . 107

5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 6 Conclusions 116

6.1 Directions for Future Research . . . . . . . . . . . . . . . . . . . . . . 119

List of Tables

2.1 Elements of Weight Matrix for IEEE 30 Bus System . . . . . . . . . . 29

2.2 Degree of Various Nodes of IEEE 30 Bus System . . . . . . . . . . . . 30

2.3 In-Degree and Out-Degree of Various Nodes of IEEE 30 Bus System . 30

2.4 Statistical Parameters of Standard IEEE Test Systems . . . . . . . . 32

2.5 Comparison of Betweenness Index . . . . . . . . . . . . . . . . . . . . 37

2.6 Sensitivity of Betweenness Index for IEEE 30 Bus System . . . . . . . 38

2.7 Top Ten Critical Lines of Various Standard Test Systems . . . . . . . 41

3.1 System Data for the Network in Fig. 3.1 . . . . . . . . . . . . . . . . 47

3.2 Various Power in Maximum Flow Network of Fig. 3.1 . . . . . . . . . 55

3.3 Betweenness of Simple 5 Bus System . . . . . . . . . . . . . . . . . . 58

3.4 Critical Nodes of IEEE 30 Bus System . . . . . . . . . . . . . . . . . 58

4.1 System Data for Network in Fig. 4.1 . . . . . . . . . . . . . . . . . . 67

4.2 Degree Centrality for Network in Fig. 4.1 . . . . . . . . . . . . . . . . 68

4.3 Closeness Centrality for Network in Fig. 4.1 . . . . . . . . . . . . . . 71

4.4 Betweenness Centrality for Network in Fig. 4.1 . . . . . . . . . . . . . 76

4.5 Top Ten Critical Nodes According to Degree Centrality of Various

Standard IEEE Test Systems. . . . . . . . . . . . . . . . . . . . . . . 76

4.6 Top Ten Critical Nodes According to Closeness Centrality of Various

Standard IEEE Test Systems. . . . . . . . . . . . . . . . . . . . . . . 76

xviii

List of Tables xix

4.7 Top Ten Critical Nodes According to Betweenness Centrality of Var-

ious Standard IEEE Test Systems. . . . . . . . . . . . . . . . . . . . 77

4.8 System Data for Network in Fig. 4.6 . . . . . . . . . . . . . . . . . . 79

4.9 Various Possible Connection Between Buses 1 and 4 of the System of

Fig. 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10 Maximum Power Flowing in Various Electrical Shortest Path Sets of

the Network in Fig. 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.11 Maximum of In and Out Flow at Various Buses within Various Elec-

trical Shortest Path Sets of the Network in Fig. 4.6 . . . . . . . . . . 86

5.1 Top Ten Nodes in Nondirectional & Bidirectional Power Flow Models 99

5.2 Top Ten Critical Nodes in the Bidirectional Power Flow Model for

IEEE 30 Bus System Under Various Changed Topological Conditions 109

5.3 Six Normal Steady-State Operating Conditions of the Australian Power

Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Ranks of Various Buses of Australian Test System Based on Closeness

Centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5 Variation of Ranks of Several Buses of Australian Test System Based

on Betweenness Centrality . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Figures

2.1 The IEEE-30 bus system. . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Physical topology graph of IEEE 30 bus system. . . . . . . . . . . . . 24

2.3 Power flow diagram of IEEE 30 bus system. . . . . . . . . . . . . . . 28

2.4 Degree sequence distribution of IEEE 30 bus system. . . . . . . . . . 31

2.5 Transient stability analysis of the IEEE 30 bus system with fault in

line 1-2 cleared at 1 sec. Unstable. . . . . . . . . . . . . . . . . . . . 38


line 1-3 cleared at 1 sec. Unstable. . . . . . . . . . . . . . . . . . . . 39


line 6-7 cleared at 1 sec. Stable. . . . . . . . . . . . . . . . . . . . . . 40

2.8 Normalized betweenness for IEEE 57 bus system. . . . . . . . . . . . 41

3.1 Simple 5 bus test system. . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Physical topology graph of simple 5 bus system. . . . . . . . . . . . . 48

3.3 Several possible paths between nodes 2 and 3 of the simple 5 bus

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Network efficiency deterioration of IEEE 30 bus system with targeted

node and line removal. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5 Maximal flow in the simple 5 bus test system. . . . . . . . . . . . . . 54

4.1 Simple 5 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

xx

List of Figures xxi

4.2 Classical closeness of various nodes of the simple 5 bus system in

Fig. 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Electrical closeness based on line impedance of various nodes of simple

5 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Illustration of betweenness in 10 possible shortest path set of the test

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Ten possible shortest path set in terms of electrical distance in simple

5 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6 Modified simple 5 bus system. . . . . . . . . . . . . . . . . . . . . . . 78

4.7 Power flow diagram of modified simple 5 bus system. . . . . . . . . . 79

4.8 Shortest path set for the network of Fig. 4.6. . . . . . . . . . . . . . . 84

5.1 Nominal unidirectional flow in IEEE 30 bus test system. . . . . . . . 97

5.2 Reverse unidirectional flow in IEEE 30 bus test system. . . . . . . . . 98

5.3 Change in path length in IEEE 57 bus test system for removal of

critical nodes based on two different measures. . . . . . . . . . . . . . 102

5.4 Connectivity loss of IEEE 118 bus test system as a function of removal

of critical nodes from two different point of views. . . . . . . . . . . . 103

5.5 Two different effects on load loss due to loss of functionality of im-

portant nodes in IEEE 300 bus test system. . . . . . . . . . . . . . . 106

5.6 Simple cascading failure model. . . . . . . . . . . . . . . . . . . . . . 108

List of Figures xxii

5.7 Variation of ranks of nodes in unidirectional model of IEEE 30 bus

test system when the network is modified slightly. . . . . . . . . . . . 110

5.8 Rank similarity of nodes in the bidirectional power flow model is

better than that of unidirectional one. . . . . . . . . . . . . . . . . . 111





Chapter 1

Introduction

1.1 Background

Power grid is one of the most complex networked systems that the human race has

ever made. The individual components of the grid are interconnected, operated

and controlled in such a way that they behave collectively in an orderly way, but

sometimes small initial failures lead to very complicated chain of events and the

grid behaves destructively and finally when situations go out of control large scale

blackouts occur. This chaotic behavior of the grid often costs up to billions of

dollars without considering social implications and effects on other infrastructural

systems (telephone, internet, computer, traffic, water, gas etc.) where the cascade

may propagate. The power system is intertwined with modern society in a way that

the issue of cascading failure leading to infrequent but large-scale blackouts requires

serious attention of researchers, system operators and policy makers to maintain

grid reliability and to develop new methods to manage the risks of blackouts.

Cascading failures may be considered as sequences of dependent failures, which

generally initiates from a single event of random failure within the grid and weakens

the grid progressively as the cascade propagate through the grid. The definition of

the power system might have a wide area. Power system components may include

1

Section 1.1 Background 2

software, method, group, and organizations involved with the power grid planning,

operating, and regulating the grid. Generally, the cascade initializes from a random

failure of the power grid components, but there exists a connecting link with suc-

ceeding failures. The failure may also cause due to inappropriate response of human

operators to control the event due to lack of necessary global information, or poor

training or experience to handle transient situations. These reasons do not manifest

themselves until it is very late to take action to avoid the cascading; hence these are

considered hidden.

Blackout in a power system can be triggered anywhere in the system initiating

various cascading damages of several components within the grid, which can propa-

gate to any place in the system and costs up to billions of dollars. It is initiated as a

sequence of dependent failures of various components that successively deteriorates

the ability of the power grid to continue its intended functionality [1]. Technology is

progressing day by day and there have been huge investments in system reliability

and security. But blackout is still occurring all over the world. The latest reported

large-scale blackout is found to be California blackout in the early September of

2011 [2].

Several researchers have come forward for the risk assessment of cascading out-

ages. Various attempts have been made by the researchers in improving the under-

standing of cascading outages, which can be broadly categorized [3] as Monte-Carlo

simulation methods and analytical techniques. Examples of these methods and


techniques include several probabilistic, deterministic method as well as approxi-

mate and heuristic techniques. Pros and cons of these methods along with their

limitations are addressed in [4].

ASSESS [5], CAT (Cascade Analysis Tool) [6], POM-PCM (Physical and Oper-

ational Margins - Potential Cascading Models) [7] are deterministic tools used by

industries to analyze and simulate cascading events. There is a huge number of rare,

unforeseeable phenomena that could trigger cascading which could lead to blackout.

Some events are so complicated that cannot be analyzed deterministically [8]. So,

several researchers have taken the probabilistic approach of determining vulnerabil-

ity.

Some methods are starting to emerge based on statistical analysis of cascading

failure. Hidden relay failures are modeled probabilistically and some countermea-

sures are proposed to prevent cascading effect [9]. Short-circuit analysis together

with reactive reserve calculations are used to identify vulnerable regions in a power

grid [10]. Since cascading is a complicated phenomenon and a complete enumeration

of all possibilities is computationally prohibitive, there are limitations in modeling

techniques or methods while assessing the cascading risk. Since it is not feasible to

include all possibilities in a model, there are significant limitations in the methods

based on probabilistic approach of cascading outages [8].

In recent years, there have been significant involvement of researchers in ana-

lyzing the power grid from the perspective of complex network theory. Power grid


topology is shown to possess the characteristic of small-world network in the seminal

paper [11–13]. The power grid is also shown to inherit the ability to cope with ran-

dom attacks but it is very vulnerable to targeted attacks since the abstract network

model of the grid shows scale-free distribution [14–19].

These preliminary results intrigued the interest of the scientific community to

analyze the power grid from a holistic point of view. Debate is going on whether

the purely topological approach of analysis is sufficient or does it provide useful

information about the vulnerability of the power grid [20]. Several researchers have

considered both topological and electrical characteristics of the network when using

complex network based analysis to model the cascading effect in power system [21,

22].

It is well established that the power grid functionality can be significantly re-

duced by removing a small number of components. So, it is necessary to identify

those critical components that can cause severe cascading effect in power system

which could lead to blackout and cost billions of dollars. Identification of critical

components is one of several directions of research in power system based on com-

plex network theory. The identification process takes a system level approach rather

than a component based method to find a set of critical nodes or lines for cascad-

ing failure. This set of nodes or lines have been named critical components, attack

vectors, vulnerable components etc. in various literature. To show the effectiveness

of the proposed methods, several measures of impact are adopted. These measures


show the degradation of network functionality as cascading progresses.

Centrality indices are used in social network literature to quantify an intuitive

feeling that in most networks some vertices or edges are more central than oth-

ers [23]. Several centrality indices namely degree centrality, betweenness centrality

and closeness centrality are proposed to find out influential person within a social

network [24]. In degree centrality, the most central node is the one which has highest

links. Betweenness centrality measures how often a node comes in the shortest path

connecting two different nodes in the system. Closeness centrality quantifies how

close a node is to all other nodes within the system.

Removing node with the highest degree damages the connectivity of the system.

Betweenness central node has the ability to control flow among all other nodes

since it comes in the transmission path many more times than other nodes. The

node which has the shortest distance with the other remaining nodes than all the

other nodes is the closeness central. Sometimes these set of centrality measures

are adopted directly in power system literature and sometime they are modified to

include electrical parameters.

Maximum-flow, minimum-flow, degree and betweenness based attack vectors are

used in [20] to judge the effectiveness of topology based critical component analysis

methods. Results are quite unsatisfactory in the sense that they produce different

types of impact on the grid in terms of connectivity loss, characteristic path length

and blackout size from a simple model of cascading failure.


Critical transmission line analysis is carried out to find which lines show the most

impact when removed from the system [25]. Complex network theory based shortest

path algorithm [26] is used find influential lines in terms of triggering cascading

events in power grid. It is argued that power does not necessarily flow over the

shortest path and utilizing maximal possible flow that a network can sustain under

different conditions the model of cascading failure is revised and new model based

on maximal flow approach [23] is proposed in [27]. This approach takes huge time

to execute and the method is used to find out critical lines in standard IEEE test

systems [21]. A more realistic approach based of Power Transfer Distribution Factor

(PTDF) is used to simulate cascading event in an attempt to identify correlated

lines [28].

Network efficiency loss and amount of load shedding due to removal of critical

components are used by some researchers as a measure of impact. Bus admittance

matrix is used to model the power system as a graph [29] and DC load flow is used

to find flows in different lines which comes with its inherent limitation of finding

real power only. A hybrid approach combining DC load flow with hidden failures in

relays is considered for improving the previous method [30].

Several new measures like net-ability, path redundancy and survivability is de-

fined and used to assess the vulnerability of the system [31]. Various lines in the

system are removed and change in these parameters are observed which gives a set of

critical lines whose removal would cause maximum impact. An extended topological

Section 1.2 Motivation of Current Research 7

method is proposed which incorporate electrical distance, power transfer distribution

factors and line flow limits to find out critical lines in the system [32].

There is no accepted standards on which set of critical components can result in

maximal efficiency loss in the system and research is ongoing on this issue. The main

reason why we can not be certain of the results is that the blackout model which

is used to quantify the impact is an approximate assumption. It is not possible to

capture all the real-world dynamics into mathematical or simulation methods. The

dissertation focuses on the complex network theory to develop tools and methods in

order to analyze the vulnerability of the power grid to prevent cascading outages.

1.2 Motivation of Current Research

We can summarize the issues relating to analyses of power system vulnerability using

complex network framework as follows:

• In case of the power system, the number of contingencies to analyze is very

huge, and the computational burden is more than ultra-modern computers

can handle. Complex network theory may be helpful, in such cases, to quickly

assess various contingency scenarios during emergency situations where small

preventive measures could stop spreading of large-scale cascade.

• Complex network framework based analysis of the power grid provides an

elegant, alternative approach to identify vulnerability of the power grid which

requires increased attention as the system is being stressed regularly with

augmented loads and generations to match the inflated demands.


• It is necessary to model the power grid properly considering both the topologi-

cal and several electrical characteristics under the complex network framework.

• Alternative modeling approaches have a considerable effect on simulation out-

comes. So a comprehensive analysis of effects of different modeling choices

upon the results should be investigated properly. Under complex network

framework, the power grid has been modeled mainly as abstract network. An-

alytical strategies should be developed considering the electrical structure of

the grid.

• In order to make the best out of the limited resources, critical nodes and

lines should be identified properly and monitored regularly to prevent large-

scale outages. The effects of removing critical elements from the system on

the structure and functionality of the grid should also be analyzed. Results

obtained from such analysis provide useful information to rank large-scale

critical infrastructures.

• Degree distribution, frequency distribution of node degrees, is one of the most

fundamental properties of networks. Degree distributions of various electricity

transmission networks need to be investigated deeply since it is a defining

characteristic of network structure and provides valuable information about

local properties of a network.

• Network scientists have categorized various networks into three groups accord-

ing to their distinctive natures and features e.g., random network, scale-free


network and small-world network. The topological, as well as functional char-

acterization of the power grid within the subclasses of networks, provides a

better understanding of system dynamics like cascading failures and blackouts.

• Analysis of network percolation leads to the realization of cascading phe-

nomenon in the power grid making connections between network structures

and functionality. Proper investigation in percolation behavior of the power

grid leads to an elegant theory of robustness of the interconnected systems to

either random or targeted failure of their constituents e.g., buses or transmis-

sion lines.

• Power grid has shown substantial robustness against random failures, but the

same grid becomes very much vulnerable when critical components are made

dysfunctional. Further exploration of this robust yet fragile nature of the

power grid is needed in terms of topological structure and functionality of the

power grid, which provides valuable insights into cascading failure mechanism.

• Effects of different attack strategies on the power grid should be simulated

in order to find out the consequence of various intentional and unintentional

faults occurring throughout the system on a daily basis which helps better

understanding of the power grid resilience as well as vulnerability to random

or targeted node or link removal.

• Identification of critical elements (typically nodes and sometimes links) of the

power grid has gained considerable attention in the literature. In case of the


power system nodes are typically transmission or distribution buses that are

well protected within closed enclosure and with continual supervision. On the

other hand transmission lines, represented by links in graphical models of the

power grid, runs thousands of if not hundreds of thousands of kilometers in

open air under the influence of wind, snow, vegetation etc. Moreover, history

of large scale cascading failure shows us that most of the events initiated from

small disturbance caused by removal of a single transmission system. So,

critical link analysis needs to be performed in great details.

• Identification of closeness central nodes spans a considerable portion of liter-

ature from social science since it identifies cohesiveness of components within

the network, but this property has lacked the interest of the power system

researchers which could have important implication in defining system’s ro-

bustness. So closeness centrality in terms of the electrical distance needs to

be defined and analyzed within the complex network framework.

• Over the past few years, betweenness centrality has become very popular strat-

egy to identify key nodes within a network. Of course, there is another link

version of this centrality measures. Both of these quantities need to be explored

further in order to achieve comprehension about central factors initiating cas-

cading failures in a large extent.

• Australia, being populated mainly in the coastal regions and due to the lack

of interconnection between the western and eastern part of the grid shows

Section 1.3 Contribution of this Thesis 11

an unique radial topology. Difference in radial and meshed grid structure

should be properly modeled and analyzed which has not been addressed in

any previous literature.

• In a grid the power flows from generators through various intermediate trans-

mission nodes towards load nodes. The directionality of the power flow from

source nodes to load nodes should be taken into consideration while modeling

the grid under complex network framework. Effect of choice of bidirectional

flow pattern in the future smart power grid should also be taken into consid-

erations.

1.3 Contribution of this Thesis

This thesis provides a novel complex network framework based investigation into

the structural and functional vulnerability of the power grid to cascading failures.

This research work is aimed at identifying critical components of the dynamically

evolving power grid in a computationally efficient and fast manner using complex

network based approach. This dissertation intends to improve the power system vul-

nerability analysis methodology which implements graph theory based approach by

incorporating various electrical characteristics of the power grid into the traditional

abstraction of connectivity and DC load flow based model. The methodologies, al-

gorithms and simulations provided in this thesis are focused on providing deeper

insights into fragility of the power grid as well as improving the existing dynamic se-

curity analysis and management system. The main contributions of this dissertation


in this direction are as follows:

• modeling the power grid as a graph to analyze the topological and functional

vulnerability of the system using complex network framework and investigating

the effect of various modeling choices on the robustness or fragility of the

electrical power transmission network;

• establishing an elegant complex network theory based vulnerability analysis

framework for the power system network which provides alternative contin-

gency analysis mechanism for planning purpose or helps fast and reliable de-

cision taking, during emergency situations, to prevent large-scale cascades;

• identifying important elements (transmission lines or buses) of the power grid,

which cause significant damages in system performance when removed ei-

ther intentionally or by accident and can initiate devastating cascading failure

mechanism;

• studying relationships between transient stability and vulnerability of the

power grid upon intentional removal of important transmission lines and an-

alyzing the consequence of random removal of links between buses due to

unintentional causes on the power-angle oscillations of the generators of the

system;

• analyzing the effect of dynamic behavior of the power flow on centrality mea-

sures of the power system and implementing a maximal flow based criticality

analysis approach to take into considerations continual changing nature of


generations and loads;

• proposing an electrical power grid counterpart of degree centrality based on

the power flow through the grid and empirically explaining the scale-free char-

acteristics of the power grid and the effect of this characteristic on the power

grid vulnerability;

• conducting studies on the electrical closeness centrality measure of the power

grid by quantifying this metric with the electrical impedance of the transmis-

sion lines, which limits the flow of the electricity throughout the grid and an

inherent property of transmission lines;

• establishing a betweenness index for the electrical power grid based on the

power flow characteristics as well as exploring its strengths and limitations to

analyze the vulnerability of the system;

• summarizing the importance of various buses of a power transmission network

in a matrix form which can be used to find information regarding vulnerability

of various nodes in a given operating condition, in determining the dependence

of various buses for transmitting electricity and providing closeness and be-

tweenness centrality of various buses in a condensed way;

• providing a bidirectional power flow based model, which is used to capture the

changed pattern of the power flow in the future smart power grid in order to

analyze the robustness and fragility of the electricity grid when various large

scale renewable sources will be integrated in distribution levels and the grid

Section 1.4 Thesis Outline 14

will encounter a whole paradigm shift in the power flow pattern;

• proposing and analyzing various topological and functional measures of im-

pacts of removal of components from the functioning electricity grid and de-

veloping fast and efficient algorithms to calculate these impact matrices;

• understanding and thoroughly investigating percolation behavior of various

test case power grids under random and intentional removal of nodes and

edges;

• conducting case studies on various standard test cases to develop and analyze

algorithms and then applying on a practical power grid dataset to validate the

results obtained from simulations; and

• focusing results under various operating conditions of the power grid to show

the robustness of proposed algorithms under continual loads and generations

growth, as well as system parameter fluctuations and to find super stable

nodes.

Various proposed algorithms and methods are simulated in various flagship com-

mercial packages like MATLAB, PSS/E and Power World Simulator etc.

1.4 Thesis Outline

Motivated by the limitations of current works several achievements have been made

in the field of the power system vulnerability assessment employing complex net-

work based framework for vulnerability assessment, which is demonstrated in this

dissertation as outlined below:


Chapter 1 provides a brief introduction to the field of complex network based

analysis of the power grid, motivations behind the current research as well as a sum-

mary of contributions made. A brief overview of all the chapters in the dissertation

is also presented at the end of this chapter.

Chapter 2 demonstrates the use of complex network theory for vulnerability

analysis of power systems after taking into considerations various electrical param-

eters of the power grid into modeling a graph corresponding to the power grid. A

betweenness centrality based approach of finding critical elements from a social net-

work have been adopted and extended to capture the true power flow scenario within

the grid. An algorithm is presented to find critical elements from any power grid

once the load flow analysis data is available on the system. A relation between the

vulnerability and stability is found to exist i.e., removal of important components

found from proposed methodology shows transient behavior in the power angle os-

cillations. Various standard IEEE test systems have been used to demonstrate the

utility of the proposed method. The proposed betweenness based method is novel

than previously existing ones employing complex network framework in better cap-

turing several electrical characteristics of the electricity grid as well as it finds stable

and unstable regions within the grid. Although, the proposed approach is a simple

extension of previous abstract network based model, however, it provides a plausible

new direction for complex system network research in the power system.

Chapter 3 deals with the vulnerability analysis of the power grid concentrating


on maximal-flow of the system. This approach also uses a modified betweenness

centrality based measure to identify critical elements of a power system. The Floyd-

Warshall algorithm has been used to calculate maximal-flow of a network in a given

operating condition. The proposed method overcomes the limitation of concentrat-

ing on shortest paths in calculating centrality indices; instead the method considers

all possible paths through which the power can flow from source nodes to load

nodes, giving a more realistic modeling choice of the power grid. Several standard

IEEE test systems have been used to exhibit the utilization of the method in finding

critical components of the grid.

Chapter 4 explores various centrality measures like degree centrality, closeness

centrality, and betweenness centrality used in complex network framework based

analysis; and adopts them for various power system applications. New definitions

of these measures have been proposed to capture the realistic power flow scenario of

the grid. A new matrix, which contains the information of dependency of bus pairs

in a power system is also presented. The correlation between bus dependency matrix

with the electrical closeness centrality and the electrical betweenness centrality has

been explored in detail. A step-by-step procedure is demonstrated, with an example,

to evaluate the matrix from the power flow data of the grid. Several characteristics

of the bus dependency matrix have been explored.

Chapter 5 addresses a critical node analysis procedure based on complex net-

work theory. Credibility of the proposed modified centrality index, i.e., the electrical


closeness and the betweenness centrality measures has also been investigated. It is

found that, impact of removing critical nodes found from proposed analysis is seri-

ous and hampers system’s ability to maintain intended function since connectivity is

lost and the amount of load need to be shedded increases. Rank similarity analysis

of critical nodes has also been carried out to demonstrate that the proposed method

is fairly stable, although numerical stability is not achieved. Various measures of

impact have been proposed.

Chapter 6 concludes the dissertation focusing on the current research and pro-

viding future research direction.

Chapter 2

Centrality Analysis and Transient

Stability Assessment

2.1 Introduction

Many of the public infrastructures like the electric power network are subject to two

types of threats: intentional and accidental [33]. Intentional attack can be subdi-

vided into physical or cyber attacks. According to US government accountability

office, in 2002, 70 percent of power companies experienced some kind of severe cyber

attack to their computing or energy management systems [34]. Whether it is going

to be a physical or cyber attack, the modern smart grid must resist. The designers

of the modern grid should plan for a dedicated, well planned attack prevention strat-

egy. For the modern grid to resist attack it must reduce the vulnerability of the grid

to attack by protecting key assets from cyber, physical, or accidental attacks. The

complex networks approach to the electric power network security would identify

key vulnerabilities, assess the likelihood of threats, and determine the consequences

of attacks. One of the particular goals of the security program is to identify critical

sites and systems.

18

Section 2.1 Introduction 19

Complex networks, which had been the main research area of graph theory,

have drawn interest of researchers from various disciplines as graph theory began

to focus on statistics and analytics [35]. A complex random network model was

proposed in [36]. There are other networks whose behaviour falls in between regular

and random, and these are classified as small-world networks [11, 37–39]. A power

system usually falls into the small-world network category [40].

Complex network theory has been used to model the power system and analyze

its several aspects [25, 40–42]. The structural vulnerability of the North Ameri-

can power grid was studied after the August 2003 blackout affecting the United

States [43]. Similarly, the large scale blackouts and cascading failures motivated

analysis of the Italian power grid based on the model for cascading failures [1,44–92].

The effect of the redistribution of loads on nodes due to failure of certain important

nodes on the cascading failure was also demonstrated in [93–96].

A model of cascading failure is introduced in [97], which is different from be-

tweenness based approach in that the cascading failure is considered as the process

of organization, infusion, and relaxation of congestion effect in the network. A com-

plex network based qualitative analysis about the Indian blackout [98] is carried out

in [99], where it is assumed that a node fails, if either of the real (P ) or reactive

(Q) power capacity of the node becomes lower than the actual load, distributing the

power to adjacent nodes while initiating cascade through the network. Cascading

failure in Watts-Strogatz small-world network is analyzed in [100]. Results from


the cascading failure model suggests that, small-world networks have homogeneous

degree distribution, but the betweenness distribution is heterogeneous. Cascade in

the power grid is modeled as load redistribution of broken nodes, where the dispatch

of load follows local preferential rule, in [101].

A simple probabilistic model of line outage is integrated with the hidden failure

model of the power grid in order to model the cascading phenomenon in a power

system in [102]. A simple probabilistic model considering variation and uncertainty

of the motor load is considered to model the cascading failure in [103–105], where an

estimate of parameters of cascading failure model is obtained from system data. An

statistical estimator, based on a series of simulated blackouts, is provided in [106].

The propagation of load shed, as estimated by the estimator, is consistent with the

estimate of line outage.

Vulnerability analysis models [11, 25, 37–42] were initially proposed for com-

plex abstract networks and were then used in power systems [14, 33, 43–46, 48, 107,

108]. However, those physicists’ work neglected some concrete engineering features.

Therefore, there are good prospects for researchers, further, to investigate the com-

plex problems by considering various power system characteristics and complex net-

work theory together. Electric power networks are different from these abstract

networks. Electric networks are governed by Ohm’s and Kirchhoff’s Laws which are

used to form the bus admittance matrix. These special characteristics result in a

unique pattern of interaction between nodes in power grids. Therefore, for better

Section 2.2 Power System as a Complex Network 21

explaining complex blackouts of power systems, an improved model which is based

on the system bus admittance matrix is proposed, representing the special electrical

topological structure [29, 30].

Till now the power system research based on complex network theory has been

mainly on fault study or vulnerability studies [104, 109–148, 148, 149, 149]. Since

the nodes and edges of the power grid increase as the human race develops, and the

interaction of the components in the power system becomes more and more complex,

attention must be paid in new research approaches to solve load flow, fault analysis,

and stability analysis problems [35]. Since smart grids add a new dimension and

complexity in the power system, a method for addressing transient stability issue

in the smart grids based on topographical information of the power grid has been

proposed in this chapter.

The rest of the chapter is organized as follows. Section 2.2 describes a model for

analyzing the power system within the context of complex networks. Section 2.3 de-

scribes some statistic parameter for complex network. Section 2.4 gives an index for

assessing the stability of the complex power system network using complex network

framework. Some concluding remarks are given in Section 2.5.

2.2 Power System as a Complex Network

To analyze the power system within the context of complex network theory, the first

step is to model the system as a graph [30]. From the perspective of network theory, a

graph is an abstract representation of a set of objects, called nodes or vertices, where


some pairs of objects are connected via links or edges. The power system of today

is a complex interconnected network which can be subdivided into four major parts:

generation, transmission, distribution and loads [150]. To portray the assemblage of

various components of power system, engineers use single-line or one-line diagrams

provide significant information about the system in a concise form [151]. Power is

supplied form generator nodes to load nodes via transmission and/or distribution

lines. Since, for a given operating condition, power flows only in one direction, a

directed graph can be easily constructed from the single-line representation of the

power system considering various generators, bus bars, substations, or loads of the

system as nodes or vertices and transmission lines and transformers as edges or

links between various nodes of the system. The principle of mapping is described as

follows:

a) all impedances between a bus and neutral are neglected,

b) all transmission and/or distribution lines are modeled except for the local

lines in plants and substations,

c) all transmission lines and transformers are modeled as weighted lines, the

weight is equal to the admittance between the buses, and

d) parallel lines between buses are modeled as an equivalent single line.

To illustrate mapping of a single-line diagram to a directed graph, a simple ex-

ample of the IEEE 30 bus system [152] is used here. Fig. 2.1 depicts the IEEE 30


Figure 2.1. The IEEE-30 bus system.


Figure 2.2. Physical topology graph of IEEE 30 bus system.


bus system with 30 bus bars, and 41 links connecting them. Fig. 2.2 is the cor-

responding mapped graph from the original IEEE 30 bus system. It contains 30

nodes/vertices, which correspond to the slack, voltage-controlled, and load bus bars

of the original system. The transmission lines are represented by the 41 links/edges

which connects various nodes. Now, the weight matrix from the graph has to be

formulated. The traditional modeling approach only considers the physical connec-

tion [30, 35, 45, 48, 96, 153], the weight matrix, W, (also called adjacency matrix or

Boolean matrix) is calculated by considering only the physical topology of the graph.

If there is a connection between node i and node j then the corresponding element

of the weight matrix wij = 1, otherwise wij = 0. The weight matrix found in tradi-

tional models has no sense of directionality, i.e., when nodes i and j are connected

wij = wji = 1. This model does not capture the electrical power system’s most

important trait like impedance which plays a significant role in the flow of power,

losses, stability of the system. Several researchers have considered the reactance of

the line [25, 154], neglecting the line resistance which is very small for transmission

systems, but, in order to generalize the model for both the transmission and the dis-

tribution system, the impedance, (i.e., both the reactance and resistance) needs to

be taken into consideration. This approach based on bus admittance matrix is well

adopted by various research in the power system [27, 29]. In this case, the weight

matrix can be found from the off-diagonal elements of the bus admittance matrix.

For an n bus system the node-voltage equation is written in the matrix form as:


I1

I2

.

Ii

.

In

=

Y11 Y12 . Y1i . Y1n

Y21 Y22 . Y2i . Y2n

. . . . . .

Yi1 Yi2 . Yii . Yin

. . . . . .

Yn1 Yn2 . Yni . Ynn

V1

V2

.

Vi

.

Vn

(2.1)

or

Ibus = YbusVbus (2.2)

where, Ybus is the bus admittance matrix. The diagonal elements of the bus

admittance matrix correspond to the sum of the impedances of the lines connected

to each bus of the system. Since diagonal elements are not included in weight

matrix, in effect, the role of various impedances connected from the bus to the

neutral is not considered here. The off-diagonal elements are equal to the negative

of the equivalent admittance between the nodes. They are known as the mutual or

transfer admittances. So, in this case the ij-th element of the weight matrix can

be found from wij = Yij. Here, it is obvious that Ybus is a symmetric matrix, i.e.,

Yij = Yji, except when there are phase-shifting or tap-changing transformers in the

system. So, the directionality of the power flow is not considered in this model. The

information of the direction of power flow within a network can be found from load

Section 2.3 Topological Statistics Parameter in the Power Grid 27

flow analysis. By conducting power flow, we can find the voltage magnitudes and

angles of all the buses within the system. If there is a link between bus i and bus j,

if voltage angle of bus i is higher than that of bus j, then power flows from bus i to

bus j, otherwise power flows in the reverse direction, i.e., from bus j to bus i. The

weight matrix is constructed using the following rule:

wij =

Yij if Pij > 0

∞ if Pij ≤ 0

(2.3)

where, Pij indicates the flow of power from node i to node j. Fig. 2.3 shows the

directionality of the IEEE 30 bus system in steady-state. Table 2.1 summarizes the

elements of the weight matrix for IEEE 30 bus system.

2.3 Topological Statistics Parameter in the Power Grid

This section describes some basic statistic parameter of the power grid within a

complex network framework. All of these parameters come from graph theory, the

branch of mathematics that deals with networks [155].

2.3.1 Degree

The number of links, directed or undirected, connected with a node i in a graph is

called the degree of the node, di. For the IEEE 30 bus system, in Fig. 2.1, the degree

of various nodes is given in Table 2.2. When the graph is directed, the out-degree of

a node is equal to the number of outward-directed links, and the in-degree is equal

to the number of inward-directed links. The in-degree and out-degree of the IEEE


Figure 2.3. Power flow diagram of IEEE 30 bus system.


Table 2.1. Elements of Weight Matrix for IEEE 30 Bus System

Element Weight Element Weight

w1−2 0.0192 + 0.0575i w12−15 0.0662 + 0.1304iw1−3 0.0452 + 0.1852i w12−16 0.0945 + 0.1987iw2−4 0.0570 + 0.1737i w13−12 0.0000 + 0.1400iw2−5 0.0472 + 0.1983i w14−15 0.2210 + 0.1997iw2−6 0.0581 + 0.1763i w15−18 0.1073 + 0.2185iw3−4 0.0132 + 0.0379i w15−23 0.1000 + 0.2020iw4−6 0.0119 + 0.0414i w16−17 0.0824 + 0.1923iw4−12 0.0000 + 0.2560i w18−19 0.0639 + 0.1292iw6−7 0.0267 + 0.0820i w20−19 0.0340 + 0.0680iw6−8 0.0120 + 0.0420i w22−21 0.0116 + 0.0236iw6−9 0.0000 + 0.2080i w22−24 0.1150 + 0.1790iw6−10 0.0000 + 0.5560i w23−24 0.1320 + 0.2700iw6−28 0.0169 + 0.0599i w25−24 0.1885 + 0.3292iw7−5 0.0460 + 0.1160i w25−26 0.2544 + 0.3800iw9−11 0.0000 + 0.2080i w27−25 0.1093 + 0.2087iw9−10 0.0000 + 0.1100i w27−29 0.2198 + 0.4153iw10−20 0.0936 + 0.2090i w27−30 0.3202 + 0.6027iw10−17 0.0324 + 0.0845i w28−27 0.0000 + 0.3960iw10−21 0.0348 + 0.0749i w27−8 0.0636 + 0.2000iw10−22 0.0727 + 0.1499i w29−30 0.2399 + 0.4533iw12−14 0.1231 + 0.2559i other ∞

30 bus system is given in Table 2.3. The hub of a graph is the node with the largest

degree. So node 6 with degree 7 is the hub of IEEE 30 bus system. The degree

sequence distribution of nodes of the IEEE 30 bus system is shown in Fig. 2.4.

2.3.2 Clustering Coefficient

Every node directly connected with a given node is called the neighbor of that node.

If there are di such neighbors of a node i, it means that there may be [di(di − 1)]/2

potential links among the neighbors of the node i. Suppose that the neighbors share

c links; then the clustering coefficient of node i, Cc(i), is the ratio between the actual


Table 2.2. Degree of Various Nodes of IEEE 30 Bus System

Node Degree Node Degree Node Degree

1 2 11 1 21 22 4 12 5 22 33 2 13 1 23 24 4 14 2 24 35 2 15 4 25 36 7 16 2 26 17 2 17 2 27 48 2 18 2 28 39 3 19 2 29 210 6 20 2 30 2

Table 2.3. In-Degree and Out-Degree of Various Nodes of IEEE 30 Bus System

NodeDegree

NodeDegree

NodeDegree

In Out In Out In Out

1 0 2 11 1 0 21 2 02 1 3 12 2 3 22 1 23 1 1 13 0 1 23 1 14 2 2 14 1 1 24 3 05 2 0 15 2 2 25 1 26 2 5 16 1 1 26 1 07 1 1 17 2 0 27 1 38 2 0 18 1 1 28 1 29 1 2 19 2 0 29 1 110 2 4 20 1 1 30 2 0


1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

50

Figure 2.4. Degree sequence distribution of IEEE 30 bus system.

number of links and maximum possible links.

Cc(i) =2c

di(di − 1)(2.4)

The clustering coefficient of an entire graph is the average over all node clustering

coefficients. If there are n nodes in the whole system the clustering coefficient of the

whole system or graph G, CC(G), is

CC(G) =1

N

N∑

i=1

Cc(i) (2.5)

The clustering coefficient of IEEE 30 bus system is 0.2348.


Table 2.4. Statistical Parameters of Standard IEEE Test Systems

Parameters 30 Bus 57 Bus 118 Bus 300 Bus

Node 30 57 118 300Edge 41 80 186 411Average Degree of Node 2.73 2.81 3.15 2.74Clustering Coefficient 0.2348 0.1211 0.1592 0.0851Characteristic Path Length 3.43 5.12 2.95 5.95Diameter 7 13 9 17Number of Maximum Shortest Path 3 4 3 1

2.3.3 Characteristic Path Length

The length of a path is equal to the number of links between starting and ending

nodes of the path. Path length is measured in hops-the number of links along the

path. The distance between two nodes along a path is equal to the number of hops

that separate them. It is possible for a graph to contain multiple paths connecting

nodes. Generally, the shortest path is used to calculate the distance between nodes

i and j. This is also known as the direct path between two nodes. The average

path length of a graph is equal to the average over all direct paths. This metric

is also known as the characteristic path length of the graph. The diameter of the

graph is the maximum distance between any pairs of nodes [37, 39, 155–158]. The

characteristic path length and diameter of the IEEE 30 bus system is 3.43 and 7

hops respectively.

Table 2.4 summarizes various statistic parameters of several IEEE test systems.

Section 2.4 Stability Assessment of the Micro Grid 33

2.4 Stability Assessment of the Micro Grid

A small power system that includes self-contained generation, transmission, distri-

bution, sensors, energy storage, and energy management software is called a micro

smart grid [159]. This micro grid has a seamless and synchronized connection to a

utility power system but can operate independently as an island from that system.

Interconnections are required within several micro smart grids or between today’s

regional grid layouts and planned renewable energy generators to form future mega

grids [159] to transmit the electricity to any region where needed. The vision of the

grid is, also, to eliminate congestion problems and balance loads from intermittent

energy sources across regions. It is also known as super grid or national grid.

When multiple micro smart grids will be interconnected, they could have a sub-

stantial influence on grid stability. Undesirable dynamic interactions could cause key,

heavily loaded transmission lines to trip, interrupting power exports and imports

between areas. However, if micro grids are designed with their dynamic impact on

the transmission system taken into account, i.e., analyzing transient stability before-

hand, they can enhance the stability of the transmission lines, which could permit

the transmission power limits to increase. The transient stability of the system de-

pends on the transfer reactance which is heavily reliant on the topological structure

of the power network. Hence some of the complex network concepts and techniques

may be applicable to help analyze the stability of smart grid systems. Ideas from

the complex network theory have been used, in this chapter, to find whether a smart


power system will be stable or not when subjected to transmission line removal from

the system due to fault or overloading.

Research is ongoing on the power system vulnerability analysis using complex

network theory. There are some critical links in every network which can make

the system very vulnerable to attacks. Complex network theory has been used to

explain some phenomenon like cascading effects in a power system and identification

of vulnerable line. In this chapter, we address the stability or synchronization issue

which is an immediate consequence of random or intentional attack on a network

by introducing a new vulnerability index called line betweenness, which relates to

the system stability. Betweenness measures the extent to which a line or edge lies

in the shortest paths between various sets of nodes [44]. In order to calculate the

betweenness we follow the following steps:

(a) Model the power system as a directed graph from the power flow solution

according the mapping procedure described earlier.

(b) Calculate the weight matrix from the mapped-directed graph according

to (2.3).

(c) Form a shortest path set including all possible shortest paths from all (source)

nodes containing generators to all other nodes using Floyd-Warshall algorithm [160].

(d) Find the betweenness of every line of the directed graph from the shortest

path set. If any line is included in the shortest path between generator node i and

other node j, then the real power flowing in the line is called the betweenness of that


line. For the lines that are in multiple shortest paths, add up all the betweenness

indices.

(e) Sort and rank the lines according to the betweenness in descending order.

IEEE 30 bus system is analyzed, in this manner, to find the vulnerability of

the system. Table 2.5 gives critical lines of the system. To test our hypothesis we

performed the multimachine stability analysis of the system. The system is faulted

initially and to clear the fault a line is removed from the system at 1 second, and the

relative swing of the generators with respect to the slack bus is observed to check

whether the machines are swinging back to the equilibrium position or going out of

sync. It is found that if the lines with high vulnerability are removed the machines

cannot maintain synchronism. The lower the vulnerability, the higher is the chance

for the post-fault system to be stable.

Table 2.5 also compares two different approach of calculating betweenness. In

past approach researchers ignored the load of the system [154]. It can be seen

from the Fig. 2.1 of the IEEE 30 bus system that this system consists of only two

generators one at bus 1 and other at bus 2. So the impact of removing line 1-3

should be higher than removing lines 6-7, or 6-8. The past approach gives priority

to lines 6-7 or 6-8 than the line 1-3 in terms of betweenness index. This is clearly a

shortcoming of the past approach since removing line 1-3 would leave only one path

to flow the power from source to the rest of the system via bus 1 making the system

more susceptible to collapse. The proposed approach improves the betweenness of


line 1-3 and gives it priority that line 6-7.

To verify our assumption, we simulated the swing equations for this multimachine

system and the results are depicted in Figs. 2.5–2.7. The simulation results are

also tabulated in the third column of the present and past approach of Table 2.5.

Transient stability analysis of the network was performed [150, 151, 161–163]. We

can remove any line of the interconnections and see the effect on the relative swings

of the machines. The swing equation is the very basic form that we used as given

in (2.6), (2.7)

dδ

dt= ∆ω (2.6)

d∆ω

dt=

πf0H

(Pm − Pmax sin δ) (2.7)

Next, in order to find the sensitivity of the proposed betweenness index with

topology the generator of bus 1 of IEEE 30 bus system is shifted to other buses.

This causes change in network topology since changing the generator bus causes a

redistribution of the power flow. Hence, critical lines of the system change, as well.

Table 2.6 lists top ten critical lines of the IEEE 30 bus system with generation of

bus 1 shifted to buses 3, and 23 respectively.

The simulation is repeated for IEEE 57, 118, and 300 bus test system. Table 2.7

summarizes top fifteen critical lines in IEEE 57, 118, and 300 bus systems. From


Table 2.5. Comparison of Betweenness Index

Proposed Approach Past Approach

LineNormalized

BetweennessStability Line

Normalized

BetweennessStability

L1−2 1.0000 Unstable L1−2 1.0000 UnstableL1−3 1.0000 Unstable L2−4 1.0000 UnstableL2−4 1.0000 Unstable L2−5 1.0000 UnstableL2−5 1.0000 Unstable L2−6 1.0000 UnstableL2−6 1.0000 Unstable L6−7 1.0000 StableL6−7 0.9621 Stable L6−8 1.0000 StableL6−8 0.9621 Stable L6−9 1.0000 StableL6−9 0.9621 Stable L6−28 1.0000 StableL6−28 0.9621 Stable L1−3 0.9635 UnstableL9−10 0.4810 Stable L9−10 0.5000 StableL9−11 0.4810 Stable L9−11 0.5000 StableL3−4 0.4000 Stable L10−17 0.3889 StableL10−17 0.3741 Stable L10−20 0.3889 StableL10−20 0.3741 Stable L10−21 0.3889 StableL10−21 0.3741 Stable L10−22 0.3889 StableL10−22 0.3741 Stable L3−4 0.3854 StableL4−12 0.3500 Stable L4−12 0.3372 StableL12−14 0.3207 Stable L12−14 0.3333 StableL12−15 0.3207 Stable L12−15 0.3333 StableL12−16 0.3207 Stable L12−16 0.3333 StableL28−27 0.3207 Stable L28−27 0.3333 StableL27−25 0.2672 Stable L27−25 0.2778 StableL27−29 0.2672 Stable L27−29 0.2778 StableL27−30 0.2672 Stable L27−30 0.2778 StableL15−18 0.1603 Stable L15−18 0.1667 StableL15−23 0.1603 Stable L15−23 0.1667 StableL20−19 0.1069 Stable L20−19 0.1111 StableL22−24 0.1069 Stable L22−24 0.1111 StableL25−26 0.1069 Stable L25−26 0.1111 Stable


0 2 4 6 8 10−2.5

−2

−1.5

−1

−0.5

0x 10

5 Phase angle difference (fault cleared at 1s)

t, sec

Del

ta, d

egre

e

Figure 2.5. Transient stability analysis of the IEEE 30 bus system with fault in line 1-2

cleared at 1 sec. Unstable.

Table 2.6. Sensitivity of Betweenness Index for IEEE 30 Bus System

Generator Bus 3 Generator Bus 3

LineNormalized

BetweennessStability Line

Normalized

BetweennessStability

L3−1 1.0000 Unstable L23−15 1.0000 UnstableL3−4 1.0000 Unstable L23−4 1.0000 UnstableL4−2 1.0000 Unstable L15−12 1.0000 UnstableL4−6 1.0000 Unstable L15−14 1.0000 UnstableL4−12 1.0000 Unstable L15−18 1.0000 UnstableL2−6 0.6786 Stable L12−4 0.7059 StableL6−7 0.6786 Stable L12−13 0.7059 StableL6−8 0.6786 Stable L12−16 0.7059 StableL6−9 0.6786 Stable L24−22 0.7059 StableL6−28 0.6786 Stable L24−25 0.7059 Stable


0 2 4 6 8 10−18

−16

−14

−12

−10

−8

−6

−4

−2

0x 10

4 Phase angle difference (fault cleared at 1s)

t, sec

Del

ta, d

egre

e


cleared at 1 sec. Unstable.

simulation, it can be concluded that there exists a margin in the proposed normalized

betweenness index. It can be said that lines with normalized betweenness higher

than 0.5 is most critical and can cause stability problem when subject to fault.

So, special attention must be given to these critical lines. Again, a line whose

normalized betweenness index falls below 0.5 requires less care and maintenance.

The normalized betweenness of various IEEE 57 bus test systems is shown in Fig. 2.8

which also a margin of stability.

Section 2.5 Chapter Summary 40

0 2 4 6 8 10−55

−50

−45

−40

−35

−30

−25

−20Phase angle difference (fault cleared at 1s)

t, sec

Del

ta, d

egre

e


cleared at 1 sec. Stable.

2.5 Chapter Summary

In this chapter, we demonstrated the use of complex network theory for vulnerability

analysis of power systems after taking actual electrical parameters into considera-

tions. Various IEEE test systems were used to find critical transmission lines using

the proposed method. Other methods used by earlier researchers show clearly that

the proposed method is more realistic and draws a margin between stable and unsta-

ble region. Although, the proposed approach is simple, it provides a new direction

for complex system network research.


Table 2.7. Top Ten Critical Lines of Various Standard Test Systems

IEEE 57 Bus System IEEE 118 Bus System IEEE 300 Bus System

LineNormalized

BetweennessLine

Normalized

BetweennessLine

Normalized

Betweenness

L1−2 1.0000 L9−8 1.0000 L2−3 1.0000L1−15 1.0000 L10−9 1.0000 L3−1 1.0000L1−16 1.0000 L8−5 0.9697 L3−4 1.0000L1−17 1.0000 L8−30 0.9697 L3−7 1.0000L3−15 1.0000 L89−85 0.8175 L3−129 1.0000L15−13 1.0000 L89−88 0.8175 L249−3 1.0000L15−14 1.0000 L89−90 0.8175 L4−16 0.9877L15−45 1.0000 L89−92 0.8175 L16−15 0.9768L14−46 0.5820 L92−91 0.8175 L16−36 0.9768L46−47 0.5542 L92−93 0.8175 L33−36 0.9441L47−48 0.5265 L92−94 0.8175 L36−28 0.9441L48−38 0.4988 L92−102 0.8175 L36−35 0.9441L38−22 0.4711 L49−42 0.7676 L36−40 0.9441L38−37 0.4711 L49−45 0.7676 L15−31 0.6840L9−13 0.3325 L49−47 0.7676 L31−32 0.6840

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Line →

Nor

mal

ized

bet

wee

nnes

s →

Figure 2.8. Normalized betweenness for IEEE 57 bus system.

Chapter 3

Maximal-Flow Based Critical Node

Identification Approach

3.1 Introduction

There are some critical components in networks which when removed either acci-

dentally or deliberately, make the system vulnerable to failures or hazards. In the

case of a power system, either a link or a node removal can have a serious impact on

the normal system operation. Removal of a transmission line from the system shifts

the load of that line to other intact lines in order to match the load demand of the

network, but this may overload other portions of the network which can trigger relay

operation to disconnect some more links from the system. As a result, a cascading

effect can occur, and eventually a large portion or the whole network may suffer

from total power loss. In the case of a node removal much more serious and fast

network loss may occur. Generally, nodes, i.e., generating stations or substations

are connected to other nodes via several links. Malfunction of one node can cause

removal of some lines resulting in a cascading failure.

Complex network theory has been used to model and analyze several aspects of

power system networks. The structural vulnerability of the North American power

42


grid was studied after the August 2003 blackout affecting the United States [43].

Similarly, the large scale blackouts and cascading failures motivated the analysis of

Italian power grid based on the model for cascading failures in [48]. Vulnerability

analysis models [11, 37, 38, 44] were initially proposed for complex abstract network

analysis and were then used in power systems [40, 42, 43, 48, 107]. The application

of the ideas in [11, 37, 44] did not capture essential power system characteristics

since the ideas were based on abstract networks. There is a good motivation to

investigate further the complex problems of the grid by considering power system

characteristics and complex network theory together. Particularly, electric power

networks have additional features which are not captured by the prevalent abstract

networks. Power networks are governed by Ohm’s and Kirchhoff’s laws, as well as

flow limits imposed on companies. These special characteristics result in a unique

pattern of interaction between nodes in power grids.

In order to measure the performance of the power grid globally and locally,

the concept of network efficiency was introduced in [37] while discussing the model

of cascading failure in the power grid. In it the power system was modeled as a

Boolean network. A weighted line betweenness,which considered only generations

and neglected loads of the system, was used to find out critical links in [154]. A

betweenness index based on the position and power flowing in the line was proposed

to identify critical lines in a power system [25], where the reactance was considered

as the weight of the network and the resistance was neglected. It was observed


that the system is quite robust to random attacks, and there is hardly any effect

on the efficiency if lines are randomly selected and removed, but it was fragile to

targeted attacks. A hybrid approach for structural vulnerability analysis of power

transmission networks, in which a DC power flow model considering overloading

of lines is embedded into the traditional methodology, was proposed in [30]. The

admittance of the transmission line was considered as the weight matrix.

The redistribution of load on nodes due to cascading failure of certain important

nodes was demonstrated in [94]. A simple model was introduced to explain why

large but rare cascades triggered by small initial shocks are present in most of the

complex communication or transportation networks [44]. It was shown that it is

only the breakdown of a selected minority of nodes that can trigger the collapse

of the system. A vulnerability index was proposed based on power flowing in the

connecting lines of a node [107]. These models are initial attempts to explain and

model the cascading event in a power system. Power flow may vary from time to

time depending on load demand and generations available. So instead of using a

fixed power flow model we can use a dynamic one. Also, power does not always flow

via the shortest path from source node to sink node which was the main assumptions

of previous researchers. This limitation was identified, and a new maximum-flow

based centrality approach was proposed to identify critical lines in a power system

in [27]. In this chapter, we apply this concept to find critical nodes of the power

system [164].

Section 3.2 Modeling of a Power System for Critical Node Identification 45

3.2 Modeling of a Power System for Critical Node

Identification

To analyze the power system within the context of complex network theory, the

first step is to model the system as a graph [29]. From the perspective of network

theory, a graph is an abstract representation of a set of objects, called nodes or

vertices. Some pairs of the objects are connected via links or edges in the graph.

The power system is a complex interconnected network which can be subdivided

into four major parts of generation, transmission, distribution and loads [150]. To

portray the assemblage of various components of power system, engineers use single-

line or one-line diagrams, which provide significant information about the system

in a concise form [163]. Power is supplied from the generator nodes to the load

nodes via transmission and/or distribution lines. For a given operating condition

power flows only in one direction, a directed graph can be easily constructed from

the single-line representation of the power system considering various generators,

bus bars, substations, or loads of the system as nodes or vertices and transmission

lines and transformers as edges or links between various nodes of the system. The

principle of mapping is described as follows:

• all impedances between a bus and neutral are neglected,

• all transmission and/or distribution lines are modeled except for the local lines

in plants and substations,


0.0

2+

j0.0

6

0.08+j0.24 0.01+j0.03

0.04+j0.12

0.0

8+

j0.2

4

20 MW

15 Mvar 50 MW

30 Mvar

60 MW

40 Mvar

20 MW

10 Mvar

30 MW, V=1.03 pu

40 MW, V=1.045 pu

V=1.06 pu

Figure 3.1. Simple 5 bus test system.

• all transmission lines and transformers are modeled as weighted lines, the

weight is equal to the optimum power flowing in the network, and

• parallel lines between buses are modeled as equivalent single lines.

A single-line diagram to a graph, a simple 5 bus test system [150] is used to

illustrate the mapping of a single-line diagram to a graph. Fig. 3.1 depicts the the

test system with 5 bus bars and 7 links connecting them. Columns 1–5 of Table 3.1

gives the system data for the network of Fig. 3.1. Fig. 3.2 is the corresponding

mapped graph for the original 5 bus system. It contains 5 nodes/vertices, which

correspond to the slack, voltage-controlled, and load bus bars of the original system.

The transmission lines are represented by the 7 links/edges which connects various

nodes.

The weight of the lines in Fig. 3.2 is the optimum power flowing in the lines.

The fuel cost functions for three thermal power plants in buses 1, 2, and 3 in $/h

required for optimal power flow solution are given by:


Table 3.1. System Data for the Network in Fig. 3.1

From To R X 12B Optimum

Bus Bus in pu in pu in pu Power

1 2 0.02 0.06 0.030 16.131 3 0.08 0.24 0.025 7.522 3 0.06 0.18 0.020 4.612 4 0.06 0.18 0.020 13.212 5 0.04 0.12 0.015 47.703 4 0.01 0.03 0.010 50.954 5 0.08 0.24 0.025 13.71

C1 = 200 + 7.0P1 + 0.008P 21

C2 = 180 + 6.3P2 + 0.009P 22 (3.1)

C3 = 140 + 6.8P3 + 0.007P 23

The real power limits of these generators are:

10MW ≤ P1 ≤ 85MW

10MW ≤ P2 ≤ 80MW (3.2)

10MW ≤ P3 ≤ 70MW

The optimum power flow calculation for this network is continued until the ab-

solute value of difference between the scheduled slack generation, determined from


16.1

3 M

W

7.52 MW 50.95 MW

47.70 MW

13.7

1 M

W

Figure 3.2. Physical topology graph of simple 5 bus system.

the coordination equation, and the slack generation, obtained from the power flow

solution, is within a pre-specified limit. In this case, we have taken this limit as

0.001 MW. The optimum power flowing in different lines is taken as the weight of

the directed transmission lines, which, in this case, represents the maximum power

flow limits of various lines. The maximum power flow limits for various lines in this

network is given in column 6 of Table 3.1.

Any power network can be represented by a graph G = (V,E,W ) comprising

of a set V , whose elements are called vertices or nodes, a set E of ordered pairs of

vertices, called edges or lines, and a set W , whose elements are weights of the edge

set elements. From the node set V , we can find two subsets S and L; where, s ∈ S

represents source nodes in the power network, and l ∈ L is the set of load nodes in

the system. An element e = (x, y) of the edge set E, is considered to be directed

from x to y, where, y is called the head, and x is called the tail of the edge. A

one-to-one correspondence exists between set E and set W .

Networks can have weights on their edges, which indicate that some edges are

stronger or more prominent than others. In some cases, these weights can represent

Section 3.3 Critical Node Identification of the Power Grid 49

capacities of the edges to conduct a flow of some kind [155]. In the case of a power

network, the edge weights can represent the strength of the lines to aid in power

flow through the network. Alternatively, every line has some maximum capacity of

power flow, which can act as a weight of edges if needed. In our current modeling

in this chapter, the maximum power transfer capacity of a line is considered as the

weight of the lines.

For the network in Fig. 3.1, V = {1, 2, 3, 4, 5}, E = {(1, 2), (1, 3), (2, 3), (2, 4),

(2, 5), (3, 4), (4, 5)}, and W = {16.13, 7.52, 4.61, 13.21, 47.70, 50.95, 13.71}. Also,

S = {1, 2, 3} and L = {2, 3, 4, 5}.

3.3 Critical Node Identification of the Power Grid

This section explains maximum flow based critical node identification procedure of

the electrical power grid.

3.3.1 Shortest Electrical Path

In a power grid with n buses represented as a graph G = (V,E,W ), the shortest

electrical path between any two buses is the path which has minimum electrical

distance between them. The distance can be measured in various ways. Power

World Simulator [165] provides several important distance measure options like per

unit series reactance, magnitude of series impedance, length of the transmission

lines, number of nodes in the path, etc. In this chapter, we have used absolute

measure of impedance, |Z|, as the weight of the line. For example, if we want to


find shortest electrical path between buses 2 and 3, several paths are possible as

given in Fig. 3.3. We can clearly see that the shortest path between buses 2 and 3

is 2− 3 whose weight is 0.19 pu. Several efficient algorithms are available to find all

possible shortest path in a network. In this chapter, we have used bioinformatics

toolbox of MATLAB to find shortest electrical path between various buses which

uses Johnson’s algorithm that has a time complexity of O(Nlog(N) +NE), where

N and E are the number of nodes and edges respectively.

3.3.2 Node Removal and Network Efficiency of Power Grid

Removal of nodes with a high degree, i.e., nodes which have more connections than

nodes with low degree causes more damages in the network. In addition, targeted

links removal also can cause significant deterioration of performance of a network.

A targeted node and link removal from IEEE 30 bus system and the calculated

network efficiency, E, given in is shown in Fig. 3.4, which suggests in our case node

removal have much serious consequence.

E =1

n(n− 1)

∑

i 6=j∈V

1

|Zij|(3.3)

where, |Zij| is the absolute value of the series impedance of the shortest electrical

path between buses i and j.

An alternative formulation of the network efficiency, considering line reactance

and voltages at the two ends of the transmission lines, is given in [166], as follows:

Sectio

n3.3

Critica

lNodeIdentifi

catio

nofthePower

Grid

51

1

2

3

5

0.0

6

0.25 0.03

0.13

0.2

50.190.19

4 1

2

3

5

0.0

6

0.25 0.03

0.13

0.190.19

4

1

2

3

5

0.0

6

0.25 0.03

0.13

0.2

50.190.19

4 1

2

3

50

.06

0.25 0.03

0.13

0.190.19

4

0.2

50

.25

Figure 3.3. Several possible paths between nodes 2 and 3 of the simple 5 bus system.


Figure 3.4. Network efficiency deterioration of IEEE 30 bus system with targeted node

and line removal.

E =1

NLNG

∑

i∈NL

∑

j∈NG

ViVj

Xij

(3.4)

where, NL and NG are the number of load and generator nodes, respectively. Vi

and Vj are voltages of nodes i and j, respectively, and Xij is the reactance of the

transmission line between nodes i and j.

3.3.3 Maximum Flow Based Critical Node Analysis

Given a power grid, we can find how much maximum power can be transferred

through the network from a source node s ∈ S to a load node l ∈ L, where s 6=

l ∈ V from the solution of the maximum-flow problem [167]. There are many

algorithms to solve this problem, and in this chapter we have used MATLAB’s

bioinformatics toolbox which uses Goldberg’s algorithm [168] to solve maximum

flow problem. The idea behind this problem is to push power as much possible

to transfer from the source node s to load node l within the network modeled as


a graph G = (V,E,W ). Various extremities in the network are explored with all

source and load combinations.

Solutions of maximum flow problem for various source-load combinations are

given in Fig. 3.5. For example, in the network of Fig. 3.5(b) to transfer power from

source bus 1 to load bus 3 two possible paths are 1 − 3 and 1 − 2 − 3. The path

1− 3 has a capacity of 7.52 MW. In the path 1− 2− 3, path 1− 3 has a capacity of

16.13 MW, but the maximum limit of path 2− 3 is 4.61 MW. So, only 4.61 MW of

power is transferred via path 1 − 2 − 3. So, the maximum possible power transfer

in the network of Fig. 3.5 (b) is 7.52 + 4.61 = 12.13 MW. In this maximum flow of

12.13, node 2 carries power 4.51 MW. There is no possible flow from node 3 to 2.

This case is illustrated in Fig. 3.5(h). Similar data for the simple 5 bus system is

given in Table 3.2.

Now, the importance of a node has within a network can be quantified by how

much power is transferred via the node within various possible maximum flow net-

works. This gives us a measure of centrality, called betweenness in social science,

which can be used to identify critical lines in the network.

3.3.4 Definition

Let, Fk be the net maximum power flowing through intermediate node k in net-

works with source node s ∈ S and load node l ∈ L. Then Fk could be defined

mathematically as:

Sectio

n3.3

Critica

lNodeIdentifi

catio

nofthePower

Grid

54

1

2

3

5

4

16

.13

MW

7.52 MW 50.95 MW

47.70 MW

13

.71 M

W

4.61 MW

13.21 MW

1

2

3

5

4

4.6

1 M

W

7.52 MW 50.95 MW

47.70 MW

13

.71 M

W

4.61 MW

13.21 MW

1

2

3

5

4

16

.13

MW

7.52 MW 10.44 MW

47.70 MW

13

.71

MW

2.92 MW

13.21 MW

1

2

3

5

4

16

.13

MW

7.52 MW 7.52 MW

16.13 MW

7.5

2 M

W

4.61 MW

13.21 MW

1

2

3

5

4

16

.13

MW

7.52 MW 50.95 MW

47.70 MW

13

.71

MW

4.61 MW

13.21 MW

1

2

3

5

4

16

.13

MW

7.52 MW 4.61 MW

47.70 MW

13

.71

MW

4.61 MW

13.21 MW

1

2

3

5

41

6.1

3 M

W

7.52 MW 0.50 MW

47.70 MW

13

.71

MW

0.50 MW

13.21 MW

1

2

3

5

4

16

.13

MW

7.52 MW 50.95 MW

47.70 MW

13

.71

MW

4.61 MW

13.21 MW

1

2

3

5

4

16

.13

MW

7.52 MW 50.95 MW

47.70 MW

13

.71 M

W

4.61 MW

13.21 MW

1

2

3

5

41

6.1

3 M

W7.52 MW 13.71 MW

47.70 MW

13

.71 M

W

4.61 MW

13.21 MW

a) 1-2 b) 1-3

c) 1-4 d) 1-5 e) 2-3

f) 2-4 g) 2-5 h) 3-2

i) 3-4 j) 3-5

Figure 3.5. Maximal flow in the simple 5 bus test system.


Table 3.2. Various Power in Maximum Flow Network of Fig. 3.1

From To Maximum Intermediate Bus PowersBus Bus Power Bus 1 Bus 2 Bus 3 Bus 4 Bus 5

1 2 16.13 0.00 0.00 0.00 0.00 0.001 3 12.13 0.00 4.61 0.00 0.00 0.001 4 23.65 0.00 16.13 10.44 0.00 0.001 5 23.65 0.00 16.13 7.52 7.52 0.002 3 4.61 0.00 0.00 0.00 0.00 0.002 4 17.83 0.00 0.00 4.61 0.00 0.002 5 61.41 0.00 0.00 0.50 13.71 0.003 2 0.00 0.00 0.00 0.00 0.00 0.003 4 50.95 0.00 0.00 0.00 0.00 0.003 5 13.71 0.00 0.00 0.00 13.71 0.00

Fk =∑

s∈S

∑

l∈L

F slk (3.5)

where, s 6= l 6= k.

Also, let, FG be the net maximum power flowing in networks with source node

s ∈ S and load node l ∈ L, which is defined mathematically as:

FG =∑

s∈S

∑

l∈L

F slG (3.6)

The ratio of these two power could be used as a measure of importance, called

betweenness, of various nodes of the system. The betweenness of node k is defined

as:

CB(k) =Fk

FG

(3.7)


F slk is the maximum possible flow through node k in the network G = (V,E,W )

with source node s and load node l. F slG is the maximum flow in the network

G = (V,E,W ) with source node s and load node l.

The numerator of the centrality equation is the sum of the maximum possible

flow through vertex k in various source-node combinations of the network where k

is neither source nor load The denominator is the maximum flow in the network for

various source-node combinations where k is neither source nor load. This gives us

a normalized measure of betweenness for vertex k, which was originally proposed

in [21] for unidirectional networks, but now since bi-directional power flow is a vision

of modern smart power system we have modified the equations for our use.

3.3.5 Example

For the network in Fig. 3.1 the betweenness of node 2 is:

CB(2) =F2

FG

(3.8)

where, F2, the summation of maximum powers flowing through intermediate

node 2 in networks with source node s ∈ {1, 2, 3} and load node l ∈ {2, 3, 4, 5} can

be found as:


F2 =∑

s∈{1,2,3}

∑

l∈{2,3,4,5}

F sl2

= F 132 + F 14

2 + F 152 + F 34

2 + F 352

= (4.61 + 16.13 + 16.13 + 0.00 + 0.00)MW

= 36.87MW

(3.9)

and FG for the overall network in Fig. 3.1 can be found as follows:

FG =∑

s∈{1,2,3}

∑

l∈{2,3,4,5}

F slG

=F 12G + F 13

G + F 14G + F 15

G

+ F 23G + F 24

G + F 25G

+ F 32G + F 34

G + F 35G

=16.13 + 12.13 + 23.65 + 23.65 + 4.61

+ 17.83 + 61.41 + 0.00 + 50.95 + 13.71

=224.07

(3.10)

So, the betweenness of node 2, CB(2), is 36.87/224.07 = 0.16. Similarly, be-

tweennesses for other buses could be found which is given in Table 3.3.


Table 3.3. Betweenness of Simple 5 Bus System

Bus Betweennessk CB(k)

1 0.002 0.163 0.104 0.165 0.00

Table 3.4. Critical Nodes of IEEE 30 Bus System

Node Betweenness Node Betweenness

4 0.6851 15 0.16996 0.4177 10 0.14822 0.3587 28 0.14093 0.3381 27 0.097412 0.3276 9 0.0948

3.3.6 Simulation of Various Standard Test System

Various standard test systems [152] are simulated to identify critical lines in the

system. The top ten critical lines for IEEE 30 bus system are tabulated in Table 3.4.

3.4 Chapter Summary

Complex network theory is utilized for analyzing vulnerability of the power grid

in this chapter. Critical nodes are identified using the maximum-flow algorithm.

Resource can be allocated, regularly, to monitor and service the critical nodes of the

system and large scale blackouts can be prevented. The method is applicable for

bi-directional power flow in modern smart grid system, and used to identify critical

nodes rather than links which have prominent influence in network vulnerability.


The vulnerability analysis framework presented in this chapter identifies critical

nodes in a power grid when subjected to maximum possible flow of power. Since

the actual power flow scenario might be quite different depending on the system

operating characteristics, and capturing all the uncertainties is beyond the capacity

of available computing packages or instruments, the work is a step of identifying

worst case scenario. It is a matter of continuing research to develop and modify the

metrics and tools presented in this chapter in order to capture the true vulnerability

of the interconnected power system.

Chapter 4

Electrical Centrality Measures and Bus

Dependency Matrix

4.1 Introduction

The power grid is one of the most complex man-made infrastructures; for example,

the Australian power grid, commonly known as the National Electricity Market

(NEM), operated under the Australian Energy Market Operator (AEMO) is the

worlds longest interconnected power system that runs for more than 5, 000 kilometers

from Port Douglas in Queensland to Port Lincoln in South Australia and supplies

more than $10 billion worth of electricity annually to meet the demand of more

than 8 million end users [169]. NEM interconnects five regional market jurisdictions

including Queensland, New South Wales, Victoria, South Australia and Tasmania.

Power systems play an indispensable role in modern society. However, there have

been several large-scale blackouts in recent years, in spite of technological progress

and huge investments in system reliability and security. For instance, in August

1996, more than 4 million people in several western states of the USA were out of

the power service [170].

60


In August 2003, a historic blackout was triggered in the power grid of the United

States and Canada, which disconnected 61, 800 MW of power to an area spanning

most of the north-eastern states of the USA and two provinces of Canada, totally

containing more than 50 million people [171]. Besides, in the summer and autumn

of the year 2003, several large-scale blackouts happened, such as London blackout

in the UK, Sweden-Denmark blackout and Italy blackout, etc. [171].

A severe blackout occurred in India on two consecutive days [99]. On 30 July

2012, overloading of one of the Northern regional grid initiated a cascading failure

event affecting more than 370 million people in India. Although the system was

brought to normal condition around 7 PM, the next day, the cascade was propagated

to Eastern and Northeastern grids due to another grid station failure. Twenty

states out of 28 were affected by this event; more than 700 million were left without

electricity

Prevention of large scale outages is attributed to the security assessment and

monitoring system. Recent series of blackouts occurring all over the world shows

that the system designated for prevention of blackouts is not working well, which

stimulates researchers to seek solutions from alternative means. Recently advances

of research in complex network field have attracted the interest of researchers of

the power grid to model and analyze the century old power grid under the complex

network framework.

In case of a power system, the number of possibilities to be analyzed is huge.


Suppose we want to analyze the consequence of every line getting tripped with

faults in several locations in the Australian power grid. It is just too complicated,

time consuming and does not make any sense [172]. So, first of all, from some the

topological characteristics of the network we have to find few cases which we should

study in depth. The number of contingency is too large, somehow we have to decide

which contingencies are important and which are not. Complex network framework

can be used for this purpose.

If the network structure is known, several measures or matrices could be devel-

oped, which can identify particular features of the network. Social scientists have

used several centrality measures [24,173–175] to explain a person’s influence within

a network. Among these centralities most widely used measures are degree central-

ity, betweenness centrality, and closeness centrality. To analyze the vulnerability

of the power grid or to measure which nodes are more important within a power

network these centrality approaches were used by researchers [176–179]. Some of

these research considered the power grid as an abstract network and neglected con-

crete engineering features, whereas some literatures considered various features like

impedance or admittance of various lines.

Based only on the topological information of the power grid, [180] proposed an

evolutionary algorithm based approach to find critical lines in groups by formulating

a variant of betweenness centrality, the group betweenness centrality, as defined


in (4.1):

CB(g) =

∑

i,j∈G,i<j

si,j(g)

si,j

(N − dim(g)− 1)(N − dim(g))(4.1)

where, g is the subset of edges of the graph,and si,j represents the number of the

shortest paths connecting nodes i to j that pass through g. Implementation of the

proposed algorithm on the Italian high-voltage electrical transmission network shows

the utility of the genetic algorithm based approach in identifying critical links, but

the vulnerability due to operational parameters is not reflected by this simple net-

work structure dependent algorithm. Also, the huge computational burden required

for the optimum convergence of the method limits its applicability for the dynamic

network vulnerability assessment and monitoring system.

Traditional betweenness centrality approaches [24, 173, 175] rely on the assump-

tion that flow occurs using the shortest possible paths between source and load

nodes, which is not the case always, specially for the power system. This inherent

shortcoming is the definition of the centrality measure is taken into consideration,

and current flow betweenness centrality, somewhat overcoming this limitation, is

proposed in [174] for the electrical system where the flow quantity is current, as

in (4.2):

CB(CF ) =

∑N

i=1,s<t Isti

12N(N − 1)

(4.2)

where, Isti is the current flowing from source node, s, to load node, t, within an

electrical circuit. This measure is based on random walks between source and load


nodes, and essentially includes contributions from all paths between nodes, not just

the shortest one, but giving the shortest path more weights than the other paths.

This centrality measure is utilized in [181], as random-walk centrality, to show its

implementation in case of power system, with a typical example of IEEE 14 bus test

system.

Essentially, the quantity of interest for a power grid is real power. To include this

quantity as a measure of importance, [182] modified the current flow betweenness

centrality as in (4.3):

CB(PF ) =

∑N

i=1,s<t Psti

12N(N − 1)

(4.3)

where, P sti is the power flowing from source node, s, to load node, t, within a power

system. This centrality, known as power flow centrality, takes a global approach of

computation rather than the random sampling proposed in [174], and computation-

ally more efficient, but there is an inherent problem of convergence of the load flow

analysis.

In this chapter, various electrical centrality measures based on the power flow in

the transmission system is proposed [183]. A new matrix which captures the infor-

mation of pair dependency of various buses is also developed [184,185]. This matrix,

the bus dependency matrix, is used to find two centrality measures (betweenness

and closeness) of buses of the grid. A generalized methodology is developed to find

out bus dependency matrix for an n-bus system.

The rest of the chapter is organized as follows. Section 4.2 describes a model

Section 4.2 System Model 65

for analyzing the power system within the context of complex networks. Section 4.3

– 4.5 gives various centrality measures as applied to the power system. Section 4.6

deals with simulation of standard IEEE test systems, which finds out various cen-

trality measures in those systems. Section 4.7 introduces the bus dependency matrix

and gives an example to construct it from the system data. Section 4.8 compares

and relates two previously defined centrality measures with bus dependency matrix.

Some concluding remarks are given in Section 4.9.

4.2 System Model

In order to develop a dependency matrix of the electric power grid based on the

complex network framework, the first thing to do is to construct a graph from the

system model [29, 186–195]. From the perspective of network theory, a graph is an

abstract representation of a set of objects, called nodes or vertices, where some pairs

of the objects are connected via links or edges.

To portray the assemblage of various components of the power system, engineers

use single-line or one-line diagrams, which provide significant information about the

system in a concise form [162]. Power is supplied form the generator nodes to the

load nodes via transmission and/or distribution lines. The principle of constructing

a graph from the single-line diagram of the power grid is described as follows [196]:

• all impedances between a bus and neutral are neglected,

• all transmission and/or distribution lines are modeled except for the local lines

in plants and substations,

Section 4.2 System Model 66

Figure 4.1. Simple 5 bus system.

• all transmission lines and transformers are modeled as weighted lines, the

weight is equal to the optimum power flowing in the network, and

• parallel lines between buses are modeled as equivalent single lines.

A power system network is represented by a graph G = (V,E,W ) comprising

of a set V , whose elements are called vertices or nodes, a set E of ordered pairs of

vertices, called edges or lines. An element e = (x, y) of the edge set E, is considered

to be directed from x to y; where y is called the head, and x is called the tail of the

edge. A set W , whose elements are weights of the edge set elements. There exists

a one-to-one correspondence between set E and set W . In this model, we consider

the transmission line impedances in pu as weights of the edges between nodes.

A simple example of 5 bus system [150] is used in this chapter to illustrate various

concepts of complex network in the power system. Fig. 4.1 depicts the system with

5 bus bars, and 7 links connecting them. We can model the system as a graph

Section 4.3 Measure of Connectivity-Degree Centrality 67

Table 4.1. System Data for Network in Fig. 4.1

From To R X 12B

Bus Bus in pu in pu in pu

1 2 0.20 0.6110 0.0301 3 0.08 0.1123 0.0252 3 0.60 0.5139 0.0202 4 0.06 0.5663 0.0202 5 0.04 0.1155 0.0153 4 0.10 0.5727 0.0104 5 0.08 0.2725 0.025

which contains 5 nodes/vertices, which correspond to the slack, voltage-controlled,

and load bus bars of the original system. The transmission lines can be represented

by the 7 links/edges which connects various nodes. The system data is given in

Table 4.1.

For the network in Fig. 4.1, V = {1, 2, 3, 4, 5} , E = {(1, 2), (1, 3), (2, 3), (2, 4), (2, 5),

(3, 4), (4, 5)} , and W = {0.20+ j0.61, 0.08+ j0.11, 0.60+ j0.51, 0.06+ j0.57, 0.04+

j0.12, 0.10 + j0.57, 0.08 + j0.27}.

4.3 Measure of Connectivity-Degree Centrality

Degree centrality is the simplest form of centrality measures for networks. Although

it is very simple, it has a great significance. It represents the connectivity of a

node to the network [156]. Individuals who have more links with other persons are

more connected to the network in the sense that they have more resource, access of

information than others. A non-social network example is the use of citation counts

in the evaluation of scientific papers. The number of citations of a paper can be

regarded as its impact on research [155].

Section 4.3 Measure of Connectivity-Degree Centrality 68

Table 4.2. Degree Centrality for Network in Fig. 4.1

Bus CD(k) CED(k)

1 0.50 21.582 1.00 42.333 0.75 16.134 0.75 40.135 0.50 15.80

For example, node 2 in Fig. 4.1 is adjacent to four other nodes, it’s degree is

four. In a 5 node graph, any node can be adjacent to only remaining four nodes. So,

this node has the highest connectivity. In the literature degree centrality is defined

as:

CD(k) =deg(k)

n− 1(4.4)

where, deg(k) is the degree of node k.

In case of electrical network, the power flowing in the adjacent links of the node

in concern can be regarded as a degree of the node and the definition of the electrical

degree centrality can be given as:

CED(k) =

∑

k∼t

Pkt

n− 1(4.5)

where, k ∼ t indicates that node k are t are connected. Pkt indicates the power

flowing in line connected in between nodes k and t.

Table 4.2 shows the degree centrality of simple 5 bus system in Fig. 4.1 using

classical and proposed approach.

Section 4.4 Measure of Independence-Closeness Centrality 69

Figure 4.2. Classical closeness of various nodes of the simple 5 bus system in Fig. 4.1.

4.4 Measure of Independence-Closeness Centrality

This approach of centrality measure is based upon the degree to which a node is

close to all other nodes in the network [24]. Fig. 4.2 shows closeness in a classical

sense and to illustrate the idea of electrical closeness centrality Fig. 4.3 is drawn to

show the closeness of various nodes of the simple 5 bus system in Fig. 4.1 in terms of

electrical distance found in Table 4.1. It is clear from Fig. 4.3 that node 2 is adjacent

to three other nodes (nodes 1, 3, and 4) in terms of electrical distance, while nodes

1, 3, and 4 being adjacent to two nodes. Node 5 is adjacent to one node only. So

node 2 is the closest to other nodes than the rest of the nodes in the network.

In social network theory, closeness is a sophisticated measure of centrality. It is

defined as the mean geodesic distance (i.e., the shortest path) between a vertex k

and all other vertices reachable from it [197]. In mathematical form, the closeness

centrality of a vertex k, CC(k) in a network of n vertices is given by:


CC(k) =

∑

t∈V \k

d(k, t)

n− 1(4.6)

where, d(k, t) being the shortest path length between vertices k and t. This definition

of closeness centrality gives a measure of distance of particular vertex from other

vertices. So, some researchers have used the reciprocal of the shortest path to

quantify closeness centrality as follows:

CC(k) =1

∑

t∈V \k

d(k, t)(4.7)

The electrical closeness centrality was defined in [179] as:

CCz(k) =n− 1

∑

t∈V \k

dz(k, t)(4.8)

where, dz(k, t) is taken as the shortest electrical distance between nodes k and t.

Resistance was neglected since they considered only transmission systems; but in

order to generalize the concept to both transmission and distribution systems we

cannot neglect resistance of the network lines, which is a significant portion of the

line impedance in case of distribution lines. The numerator was taken as n−1. This

was adopted in (4.6) to average the distance, but when it comes in the numerator

it just scales the parameter. So, in this chapter we propose our electrical closeness

centrality as:


Figure 4.3. Electrical closeness based on line impedance of various nodes of simple 5 bus

system.

Table 4.3. Closeness Centrality for Network in Fig. 4.1

Bus CC(k) CEC (k)

1 0.17 1.282 0.25 2.133 0.20 1.234 0.20 1.925 0.17 1.14

CEC (k) =

1∑

t∈V \k

d(k, t)(4.9)

where, d(k, t) is the weight of the shortest electrical path from node k to all other

nodes t reachable from k.

Table 4.3 shows the closeness centrality of simple 5 bus system in Fig. 4.1 in

classical as in (4.7) and proposed approach.

The independence of a node is determined by the closeness centrality of the

node [24]. In Fig. 4.1, node 2 is in direct contact with nodes 1, 3, and 4. It must

Section 4.5 Measure of Control of Communication-Betweenness Centrality 72

depend upon node 4 to communicate with node 5. So, node 5 needs only one relayer

to communicate with all other nodes of the network. On the other hand, node 1

needs node 2 to communicate with node 4 and both need 2 and 4 to communicate

with node 5. So we can say that node 2 is more independent than node 1. So

closeness centrality can be used to quantify independence of various nodes within

an electrical power grid.

4.5 Measure of Control of Communication-Betweenness

Centrality

This type of centrality is based upon the frequency with which a node falls between

pairs of other nodes on the shortest or geodesic paths connecting them [24]. This

idea is illustrated by ten possible shortest paths in the network of Fig. 4.1 as shown

in Fig. 4.4. Node 2 comes four times between other points in the six geodesics. Node

4 comes three times. So node 2 is more central in terms of betweenness.

The betweenness centrality CB(k) for vertex k is computed as follows [197]:

1. Find the shortest path set of the network.

2. Find out the fraction of the shortest path containing node k for each pair of

vertices.

3. Sum this fraction over all pairs.

Mathematically,

Section 4.5 Measure of Control of Communication-Betweenness Centrality 73

Figure 4.4. Illustration of betweenness in 10 possible shortest path set of the test system.

Section 4.6 Simulation of Various Standard IEEE Test Systems 74

CB(k) =

n∑

s=1

n∑

t=1

σst(k)

σst

, s 6= t 6= k ∈ V (4.10)

where, σst is the number of shortest paths from s to t, and σst(k) is the number of

shortest paths from s to t that pass through a vertex k.

As in closeness centrality the shortest paths for an electrical network can be

calculated from the line impedance, and the power flowing in the line is taken as a

measure of betweenness [196]. The electrical betweenness centrality of a node k in

a network of n nodes is defined as:

CEB (k) =

n∑

s=1

n∑

t=1

Pst(k)

Pst

, s 6= t 6= k ∈ V (4.11)

where, Pst is the maximum power flowing in the shortest electrical path between

buses s and t , and Pst(k) is the maximum of inflow and outflow at bus k within

the shortest electrical path between buses s and t. Fig. 4.5 illustrates the concept

of electrical shortest path and shows ten possible geodesics in the simple 5 bus test

system.

Table 4.4 shows the betweenness centrality of the simple 5 bus system in Fig. 4.1

using classical and proposed approach.

4.6 Simulation of Various Standard IEEE Test Systems

There are some critical nodes in every network which when removed from the system

can make the system very vulnerable to attack. Previously researchers used complex


Figure 4.5. Ten possible shortest path set in terms of electrical distance in simple 5 bus

system.


Table 4.4. Betweenness Centrality for Network in Fig. 4.1

Bus CB(k) CEB (k)

1 010

0670

2 110

192670

3 110

0670

4 110

93.3670

5 010

0670

Table 4.5. Top Ten Critical Nodes According to Degree Centrality of Various Standard

IEEE Test Systems.

30 Bus CEB (k) 57 Bus CE

B (k) 118 Bus CEB (k)

2 12.5841 1 7.9668 12 12.21146 9.2330 4 5.3512 69 5.63721 9.0528 2 4.9898 70 5.19164 7.0180 3 4.4910 80 4.82333 5.7375 15 3.9604 7 4.72275 3.7382 6 3.5725 11 4.720010 2.4665 17 3.4561 32 4.41569 2.3013 24 2.9787 46 4.15327 2.2122 23 2.0769 75 3.816012 2.1811 13 2.0628 34 3.2639

Table 4.6. Top Ten Critical Nodes According to Closeness Centrality of Various Standard

IEEE Test Systems.

30 Bus CEC (k) 57 Bus CE

C (k) 118 Bus CEC (k)

6 2.2366 14 1.7785 65 3.05534 2.1676 13 1.7596 68 3.024928 2.0587 46 1.7399 116 2.98848 2.0438 47 1.7120 81 2.93663 2.0108 48 1.7042 38 2.87739 2.0029 15 1.6993 64 2.835310 1.9662 38 1.6775 69 2.82987 1.9069 11 1.6663 80 2.815812 1.8110 3 1.6167 66 2.811921 1.7877 12 1.6149 30 2.7189


Table 4.7. Top Ten Critical Nodes According to Betweenness Centrality of Various

Standard IEEE Test Systems.

30 Bus CEB (k) 57 Bus CE

B (k) 118 Bus CEB (k)

2 0.6117 1 0.6117 12 0.70301 0.5546 2 0.4862 7 0.43316 0.3114 17 0.4142 11 0.43104 0.3103 3 0.3380 2 0.34313 0.2972 15 0.2271 3 0.07805 0.1668 16 0.1427 6 0.06297 0.0547 4 0.1398 14 0.03508 0.0490 6 0.0566 117 0.03409 0.0420 14 0.0544 13 0.028610 0.0406 5 0.0498 4 0.0219

network theory to explain blackouts or cascading effects in the power system. Very

few works were done to identify critical nodes of the system. System reliability can

be improved a lot if these critical nodes can be identified beforehand by monitoring

them regularly and servicing them when subjected to deterioration. Critical nodes

can be found from calculating various centrality measures as outlined in previous

sections.

IEEE 30, 57, and 118 bus systems were used to simulate various centrality mea-

sures and results are given in Tables 4.2 – 4.4. Results in Tables 4.2 – 4.4 show

that different approaches give different nodes as critical in order of priority. This

is expected because these three centrality measures are based on three different ap-

proaches. So if we are dealing with connectivity, degree based centrality is the one

to consider. If some cases required the criticality measure based on independence

on the node, closeness centrality would be the best option to consider. However, the

Section 4.7 Measure of Pair Dependence of Various Buses 78

Figure 4.6. Modified simple 5 bus system.

last option – control of communication can be measured using betweenness central-

ity. Since the future smart grid will rely on control of communication along with

the power transfer, the third measure of centrality could be very useful for future

control room engineers and planners to take necessary action in critical events.

4.7 Measure of Pair Dependence of Various Buses

To illustrate the idea of pair dependency, we slightly modified the system parameters

in order to accommodate various flows within the system as given in Fig. 4.6. Various

data for the system in given in Table 4.8. The simultaneous nonlinear algebraic

equations for the power flow problem of this network are solved using standard

Gauss-Seidel method [150]. Power flowing in various lines, line losses, generations

and loads in various buses in a steady-state of the network are given in Fig. 4.7.


Figure 4.7. Power flow diagram of modified simple 5 bus system.

Table 4.8. System Data for Network in Fig. 4.6

From To R X 12B

Bus Bus in pu in pu in pu

1 2 0.20 0.6110 0.0301 3 0.08 0.1123 0.0252 3 0.60 0.5139 0.0202 4 0.06 0.5663 0.0202 5 0.04 0.1155 0.0153 4 0.10 0.5727 0.0104 5 0.08 0.2725 0.025


4.7.1 Shortest Path

The concept of the shortest path is used by the researchers of the power system who

use complex network framework for network vulnerability analysis [25,30,153,154].

In order to assess the vulnerability of a power grid, the researchers used a dynamic

power system model where the concept of network flow is introduced [30]. The flow

between two nodes, s and t, takes on the shortest path between them. If there are

two or more paths between two buses then the path that has less weight is regarded

as the shortest path between those two buses.

Only the physical connection is considered in traditional modeling approach by

the complex network researchers. The weight of the line between nodes simply

reflects the topology of the network. If there is a connection between node s and

node t then the weight of the corresponding line is taken as 1, otherwise it is 0 in

the traditional approach [30, 153]. In case of a power system, the main parameter

of a transmission line, which has a significant effect in the power flow in the line

between buses, is its impedance which is not considered in this model.

Several researchers have considered the reactance of the line [154], neglecting the

line resistance which is very small for transmission systems. In order to generalize

the model for both the transmission and the distribution system, the impedance,

(i.e., both the reactance and resistance) needs to be taken into consideration [25,196].

In this chapter, we have used absolute measure of impedance, |Z|, as the weight

of the line. If we want to find the shortest electrical path between buses 1 and 4,


Table 4.9. Various Possible Connection Between Buses 1 and 4 of the System of Fig. 4.6

Connection Weight (pu)

1–2–4 1.211–2–3–4 2.011–2–5–4 1.041–3–4 0.721–3–2–4 1.501–3–2–5–4 1.33

several paths are possible as given in Table 4.9. We can clearly see that the shortest

path between buses 1 and 4 is 1− 3− 4 whose weight is 0.72 pu. This approach of

measuring shortest path by combining absolute values of complex numbers |Z| may

appear theoretically troublesome, but the rationale here is to find shortest possible

paths for the power to flow from various source nodes to target nodes in an electric

circuit. Using both resistance and reactance in the calculation of line weights provide

a better way of modelling electricity grid under complex network framework than

to use resistance or reactance alone.

Finding the shortest path set for a network is a problem of graph theory and

several efficient algorithms are available.

4.7.2 Bus Dependency Matrix

In the context of complex network theory, when a pair of buses in the power system

are connected via a transmission line without any other buses in between (interme-

diaries), they are said to be adjacent [156]. A bus s adjacent to bus k, another bus

t adjacent to bus k, creates a transmission path between buses s and t via bus k.

The shortest electrical path linking a pair of buses is called a geodesic [155].


Let, Pst be the maximum power flowing in the shortest electrical path between

buses s and t , and Pst(k) be the maximum of inflow and outflow at bus k within the

shortest electrical path between buses s and t. Then, let their fraction is represented

by rst(k) as in:

rst(k) =Pst(k)

Pst

(4.12)

where, the ratio rst(k) is an index of the degree to which buses s and t needs bus k

to transmit the power between them along the shortest electrical path.

The pair dependency of nodes in a network is defined in [198]. The concept of pair

dependency in [198] is used here in case of an electrical power grid. The dependency

of bus pairs can be regarded as the degree to which a bus s must depend upon

another bus k to transmit its power along the shortest electrical path or geodesic

to and from all other reachable buses t’s in the network. For a power grid with n

buses, the dependency of bus s upon bus k to transmit power on any other buses in

the network can be represented as follows:

dsk =

n∑

t=1s 6=t6=k∈V

rst(k) =

n∑

t=1s 6=t6=k∈V

Pst(k)

Pst

(4.13)

The dependency of bus pairs for the whole system can be calculated and the result

can be summarized in a matrix D as follows:


D =

d11 d12 · · · · · · d1n

d21 d22 · · · · · · d2n

......

. . ....

...

dn1 dn2 · · · · · · dnn

(4.14)

Each element of D is an index of the degree to which a bus designated by row

number must depend upon another bus designated by column number to transmit its

power along the shortest electrical path or geodesic to and from all other reachable

buses in the network. Thus, this matrix captures the information of importance of

a bus as an intermediary with respect to other buses in the network. So we can call

the matrix D as bus dependency matrix.

4.7.3 Example

The shortest electrical path set is found here using Johnson’s algorithm for sparse

network [199]. Fig. 4.8 portrays the power flowing in various shortest electrical

paths within the network of Fig. 4.6. For example, the shortest 2 − 5 − 4 is the

shortest electrical path between buses 2 and 4. 81.02 MW of power is injected into

line 2 and 76.97 MW of power reaches at bus 5 via line 2 − 5. Similarly, for line

5 − 4, 16.97 MW of power is injected at bus 5 and 16.59 MW reaches at bus 4.

This situation is illustrated in Fig. 4.8(f). So, the maximum power in this shortest

electrical path between buses 2 − 4 is 81.02 MW. Table 4.10 lists maximum power

in various shortest paths within the simple 5 bus test system.


Figure4.8.Shortestpath

setforthenetwork

ofFig.4.6.


Table 4.10. Maximum Power Flowing in Various Electrical Shortest Path Sets of the

Network in Fig. 4.6

Bus Bus Shortest Pst

s t Path Set MW

1 2 1–2 69.821 3 1–3 94.491 4 1–3–4 94.491 5 1–2–5 81.022 1 2–1 69.822 3 2–1–3 94.492 4 2–5–4 81.022 5 2–5 81.023 1 3–1 94.493 2 3–1–2 94.493 4 3–4 67.693 5 3–4–5 67.694 1 4–3–1 94.494 2 4–5–2 81.024 3 4–3 67.694 5 4–5 16.975 1 5–2–1 81.025 2 5–2 81.025 3 5–4–3 67.695 4 5–4 16.97

Now, P24(5) is 76.97 MW since in the shortest electrical path between buses

2 and 4 inflow at bus 5 is 76.97 and outflow is 16.97. So, maximum of inflow

and outflow is 76.97. Since no other buses fall within this shortest electrical path

P24(1) = P24(3) = 0 MW. Most of these values are zero for this specific network.

Only non-zero elements of Pst(k) are given in Table 4.11.

An example of evaluating a component of the dependency matrixD, for example,

the element d13 is given in (4.15):


Table 4.11. Maximum of In and Out Flow at Various Buses within Various Electrical

Shortest Path Sets of the Network in Fig. 4.6

Pst(k) Power (MW)

P23(1) 94.49P32(1) 94.49P15(2) 81.02P51(2) 81.02P14(3) 87.99P41(3) 87.99P35(4) 62.33P53(4) 62.33P24(5) 76.97P42(5) 76.97

d13 =∑

t∈{2,4,5}

r1t(3)

=∑

t∈{2,4,5}

P1t(3)

P1t

=P12(3)

P12+

P14(3)

P14+

P15(3)

P15

=0

69.82+

87.99

94.49+

0

81.02

= 0 + 0.93 + 0

= 0.93

(4.15)

Similarly, other elements of the matrix could be found, and the resulting bus

dependency matrix for the simple 5 bus system is given in (4.16):

Section 4.8 Characteristics of Bus Dependency Matrix 87

D =

0.00 1.00 0.93 0.00 0.00

1.00 0.00 0.00 0.00 0.95

1.00 0.00 0.00 0.92 0.00

0.00 0.00 0.93 0.00 0.95

0.00 1.00 0.00 0.92 0.00

(4.16)

4.7.4 Steps to Find Bus Dependency Matrix from System Data

The procedural steps to find bus dependency matrix from the system data is as

follows:

1. Model the system as a graph as described in Section 4.2.

2. Find a shortest path set for the graph using Johnson’s algorithm.

3. Find flow in various lines of the system solving load flow problem.

4. Find the maximum power flowing in the shortest electrical path between buses

s and t, Pst, for the shortest path set.

5. Find Pst(k), the maximum of inflow and outflow at bus k within the shortest

electrical path between buses s and t.

6. Evaluate bus dependency matrix D from Pst and Pst(k).

4.8 Characteristics of Bus Dependency Matrix

4.8.1 Relationship with Other Centrality Measures

If we take a column sum of the k-th column of the bus dependency matrix:

Section 4.8 Characteristics of Bus Dependency Matrix 88

n∑

s=1

dsk =

n∑

s=1

n∑

t=1

rst(k), s 6= t 6= k ∈ V

=n

∑

s=1

n∑

t=1

Pst(k)

Pst

, s 6= t 6= k ∈ V

= CEB (k)

(4.17)

So, it is clear that the sum of the elements of the k-th column of bus dependency

matrix is the electrical betweenness of the k-th bus of the system. So, the column

sum of (4.16)

[

2.00 2.00 1.86 1.84 1.90

]

represents electrical betweenness centrality

of 1, 2, ..., 5 bus respectively of the simple 5 bus system.

Similarly, if we take a row sum of the s-st row of the bus dependency matrix:

n∑

k=1

dsk =n

∑

k=1

n∑

t=1

rst(k), s 6= t 6= k ∈ V

=

n∑

k=1

n∑

t=1

Pst(k)

Pst

, s 6= t 6= k ∈ V

(4.18)

This summation represents the power ratio of all available paths starting from

bus s.

So, we can say that, the s-th row sum of the bus dependency matrix represents

the closeness centrality of s-th bus of the network. where the weight of the path

is taken as the power ratio instead of electrical impedance or admittance. The row

sum of (4.16)

[

1.93 1.95 1.92 1.88 1.92

]T

represents electrical closeness centrality


of 1, 2, ..., 5 bus respectively of the simple 5 bus system.

4.8.2 Several Observations

Several observations about the bus dependency matrix is enumerated as follows:

• The (s, t)-th element of the matrix represents the dependency of bus s on bus

t.

• Diagonal elements of the bus dependency matrix is zero.

• This matrix is non-symmetric.

• The row sum of the matrix could be used as a electrical closeness centrality

measure.

• The column sum of the matrix is electrical betweenness centrality measure.

4.9 Chapter Summary

This chapter presents a new matrix which contains the information of dependency

of bus pairs in a power system. The correlation between bus dependency matrix

and electrical closeness centrality and electrical betweenness centrality is explored

in this chapter. A step by step procedure for evaluating the matrix is also given.

The example of simple 5 bus system clarifies the concept of bus dependency matrix

and its relation with other centrality measures is also investigated.

Several characteristics of the bus dependency matrices are explored. The matrix

described in this chapter could have various implementation in the future smart

power grid, where both information and power are transmitted via the transmission


lines. Since there will be communication, there will be control of communication.

Bus dependency matrix, in fact, represents how much control a bus can exert on

another bus since it is a measure of one bus’s dependency on another in case of

transmitting the information or the power through transmission lines. Bus depen-

dency matrix could be used as a measure of how independently a bus can transmit

the power or the information through the lines in the grid.

Chapter 5

Bidirectional Power Flow Based

Criticality Assessment

5.1 Introduction

Existing power transmission grids around the world are being made much smarter

by integrating smart and new technologies by utilities [200]. The smart grid can

manage various generation sources efficiently, primarily in the distribution side –

near consumers. Engagement of customers with the energy management systems is

the most lucrative part of smart grid from the point of view of regulating energy

usage. Excess of generation after local use can be transmitted long distances to

meet the energy shortage of the destination area, which introduces a new concept of

the power flowing from customer end towards the grid. The bidirectional power flow

changes the whole power flow pattern of the existing grid [161]. Analytical methods,

technical strategies, control system and protecting devices need to be changed along

with, to mention a few. Metering and protecting equipments will experience flows

coming from the reverse side. Proper operation of the equipments used earlier can

be ensured either by changing the instruments themselves or by incorporating new

measurement techniques.

91


From the frequent events of large scale-blackouts [98, 99, 170, 171], it is clear

that the existing dynamics security assessment and monitoring system has not been

working well [29]. The motivation of complex network framework based analysis

approach comes from the necessity of new, alternative and improved methodologies

to assess the risk associated with cascading events in the power system. Degree

centrality, betweenness centrality and closeness centrality measures are commonly

used in the social network research to find a person with the most influence. [24].

The research on the power grid from a system point of view has been triggered

after the publications of the preliminary topology based analytical results. Since

results from pure topological approach is quite misleading [20], several researchers

have a mix of both topological and electrical characteristics based complex network

analysis of the power system to find reasonably improved results [21, 22].

Motivated by the topology based analytical result [14], that found the power grid

robust against random failure but vulnerable to targeted attacks [20], critical node

and link analysis of the power grid have been carried out to explore the criticality

of the power grid. If critical components, which can initiate cascading effect, can

be spotted out, special preventive actions could be exercised to prevent large scale

blackouts from happening.

Network efficiency, a topological measure of performance change after the inclu-

sion or removal of nodes or lines from a grid, is analyzed in [35]. A weighted line


betweenness based approach is utilized to find critical lines responsible for spread-

ing of large scale blackouts from small initial shock [154]. Vulnerable regions of the

power system are identified employing complex network theory based qualitative

simulation in [41]. Transmission line reactance is incorporated to compute a new

vulnerability index to identify critical lines [25].

A link is explored between the power system reliability and small world ef-

fect [153]. Maximum flow based centrality approach is used to find out critical lines,

which removes the shortcoming of the assumption of the power flowing through the

shortest path between source and load nodes [27]. The flow based method has slow

convergence, but can be useful when used in conjunction of planning issues. A DC

power flow model is used, and hidden failure of protective equipment is considered to

model the structural vulnerability of the power grid [30]. Electrical parameters are

incorporated extensively to improve the centrality indices for the power system [179].

An extended topological approach proposed in [31] takes into consideration tra-

ditional topological metrics as well as operational behavior of the power grid like

the real power flow allocation and the line flow limits. The power transfer distri-

bution factor (PTDF) is used to simulate cascading event in an attempt to identify

correlated lines [32].

Purely topological analysis cannot capture many important features of the power

grid, so a weighted directed model is considered in [201]. The power grid is modeled

into two directional graphs one containing real power flow (P ) information, and


another signifying reactive power flow (Q) within the network. The vulnerability

of a node is defined to be dependent on the betweenness value and an operational

state parameter, ε, of the node. The vulnerability of a node i, M , is given by:

Mi =√

εiCB(i) (5.1)

where, εi is the operational state vulnerability parameter, and CB(i) is the between-

ness centrality of node i, respectively. εi is given as:

εi =1

|xi(t)− xcri |

(5.2)

where, xi(t) is the state value at time t, and xcri is the critical value of the state.

Either voltage or phase angle of nodes can be considered as the state value. Due

to natural cohesion, phase angle is considered as state value in case of real power

(P ) graph, while voltage is taken as state when reactive power (Q) is taken into

consideration.

All these analyses are carried out mainly on nondirectional models where the

direction of the power flow has not been considered, but since with the inclusion of

distributed generations the power flow pattern is going to change, new methodolo-

gies have to be proposed, which take into account bidirectional power flow. Since

communication is an important factor in smart grid; identifying those nodes,which

are important for communication in the system, would be very much useful.


In this chapter, a comparison of the bidirectional flow based method has been

made with a nondirectional flow based method. This method is a modification of

closeness centrality, which takes into account the power flow distribution among

various transmission lines during steady state [202–204]. This work is a reason-

able extension of previous work carried out by the researchers since it captures the

power flow in smart grid environment. The power flow pattern in traditional power

grid and the envisioned smart electricity grid are different. Generally the power

flows from generation nodes throughout the network towards the destination nodes

or distribution systems. Recently, more and more renewable generations has been

included in the distribution systems and the excess of the generation in the distri-

bution system will flow back to the transmission grid which will be utilized in a

different area of need, or will be stored in a battery for future use. Hence, the power

flow pattern is different in traditional power systems and future smart power grid.

The impact of removing critical components is identified using well known impact

metrics like path length, connectivity loss, and load loss.

The organization of the rest of the chapter is as follows. Section 5.2 provides a

model for the analysis of the smart power grid under complex network framework.

A new model based on the bidirectional power flow is considered, and a method is

discussed to find critical nodes in the power grid. Effects of removal of critical nodes

on various topological and electrical measures are addressed in Section 5.3. Rank

similarity analysis is carried out is Section 5.4. Conclusion is drawn, and future

Section 5.2 System Model and Methodology 96

research direction is provided in Section 5.5.

5.2 System Model and Methodology

The power flow analysis is conducted for the given test system during nominal

condition. Newton-Raphson method is used to solve the simultaneous nonlinear

algebraic power flow equations [150]. The direction of the real power flowing through

the lines is taken as the direction of edges in the modeled graph as shown in Fig. 5.1.

From this point, this graph will be known as nominal unidirectional flow graph. In

order to consider the bidirectional flow in smart grid, a backward unidirectional

flow graph is also modeled. The direction of edges in the reverse unidirectional

flow graph is exactly opposite to the nominal unidirectional flow graph as shown

in Fig. 5.2. Superposition of the two models gives bidirectional model. Here, we

show the difference of two modeling approaches: (a) nondirectional model and (b)

bidirectional model.

Assume that, k represents the intermediate bus within the shortest path origi-

nating from bus s and ends at bus t. Let Pst represents the maximum power flowing

in the shortest electrical path between buses s and t , and Pst(k) is the maximum

of inflow and outflow at bus k within the shortest electrical path between buses s

and t. Then, let their fraction is represented by rst(k) as in:

rst(k) =Pst(k)

Pst

(5.3)


Bus 1

Bus 2 Bus 3

Bus 4Bus 5

Bus 6

Bus 7

Bus 8

Bus 9

Bus 10Bus 11

Bus 12

Bus 13 Bus 14

Bus 15Bus 16

Bus 17 Bus 18

Bus 19

Bus 20

Bus 21

Bus 22 Bus 23

Bus 24

Bus 25

Bus 26 Bus 27

Bus 28 Bus 29

Bus 30

Figure 5.1. Nominal unidirectional flow in IEEE 30 bus test system.


Bus 1

Bus 2Bus 3

Bus 4 Bus 5

Bus 6

Bus 7

Bus 8

Bus 9

Bus 10Bus 11

Bus 12

Bus 13Bus 14

Bus 15

Bus 16

Bus 17

Bus 18

Bus 19

Bus 20

Bus 21

Bus 22Bus 23

Bus 24

Bus 25

Bus 26Bus 27

Bus 28Bus 29

Bus 30

Figure 5.2. Reverse unidirectional flow in IEEE 30 bus test system.


Table 5.1. Top Ten Nodes in Nondirectional & Bidirectional Power Flow Models

Nondirectional Bidirectional

1 53 82 74 66 413 2412 199 1314 1228 14

where the ratio rst(k) is an index of the degree to which buses s and t need bus k

to transmit the power between them along the shortest electrical path. If a double

sum is taken of (5.3) over all intermediate buses k and all destination buses t for

the source buses s,

CEC (s) =

n∑

k=1

n∑

t=1

Pst(k)

Pst

, s 6= t 6= k ∈ V (5.4)

a centrality measure for bus s within the grid is obtained. This measure (5.4) adds

up the real power of the lines originating at bus s and terminating at all other buses.

This quantity takes high values if the difference between numerator and denominator

term is low. This fact represents that a very small amount of the power is lost in

the shortest path. Such buses might have direct influence on other buses since a

small amount of the power is lost. Table 5.1 lists top ten critical nodes in IEEE 30

bus test system [150] found from nominal and backward unidirectional as well as

bidirectional model.

Section 5.3 Measure of Impact 100

In summary, the method to identify can be summarized as:

1. Model a power system as a directed graph.

2. Calculate power flowing through various lines.

3. Construct a reverse directional graph.

4. Find the shortest path set of the graph from source nodes to load nodes.

5. Find rst(k) and calculate CEC (k).

6. Sort and rank in the descending value of CEC (k).

5.3 Measure of Impact

At first, the nominal network is solved, and nodes are removed from the system one

by one in the descending order of centrality measure. In order to measure the impact

of removing critical nodes from the system various measures are being used [205].

In this chapter, four measures are considered. The first two of them, path length

and connectivity loss, are purely topological. The last measure is the percentage of

load lost due to the removal of critical nodes.

5.3.1 Path Length

The path length is used by researchers as a measure of network connectedness. It is

the average length of the shortest paths between any two nodes in the network [108].

It is found that if a node is removed from a system, it generally increases the

distance between other nodes. So, the increase in network characteristic path length


is considered as a measure of impact analysis of removing critical nodes from the

system.

Distance between two vertices can be computed as:

d(u, v) = min|P | (5.5)

where P is a path from u to v. Path length can be defined as:

d =1

k

∑

u 6=v∈V

d(u, v) (5.6)

where 0 ≤ d(u, v) ≤ ∞. k is the number of connected pairs.

This is topological path length. Another electrical path length is also measured

where the distance is computed in terms of the impedance of transmission lines. A

simple IEEE 57 bus test system is used to simulate the consequence of node removal

on path length and the result is depicted in Fig. 5.3. It is clear that the impact

of removing critical nodes based on the bidirectional flow rather than bidirectional

flow model is comparatively higher. Initially the impact is higher in nondirectional

measure but after four nodes removal the bidirectional model shows the impact in

a large scale. Electrical path length based measure shows similar characteristic. In

the later measure the impact is always higher than the topological path length in

bidirectional flow model.


0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5

Pat

h L

eng

th

Number of Removed Node

Nondirectional (Topological) Bidirectional (Topological)

Nondirectional (Electrical) Bidirectional (Electrical)

Figure 5.3. Change in path length in IEEE 57 bus test system for removal of critical

nodes based on two different measures.

5.3.2 Connectivity Loss

This is a purely topological measure of impact a power grid encounters when some

nodes are removed from the system. In this measure, we calculate how much con-

nectivity is lost in terms of how many generators a transmission or distribution node

can access due to effect of removing a node from the system. The less is the number

of generators a node is connected with; the less is the redundancy and the more is

the vulnerability of the node. It is given as (5.7) originally proposed in [43].

C = 1−

⟨

N ig

Ng

⟩

i

(5.7)

where the averaging is done over each intermediate nodes, i.e., substations. Ng is

the total number of generators and N ig is the number of generators that a node i can


0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

Co

nn

ecti

vit

y L

oss


Nondirectional Bidirectional

Figure 5.4. Connectivity loss of IEEE 118 bus test system as a function of removal of

critical nodes from two different point of views.

reach. Impact on connectivity loss for two different models is presented in Fig. 5.4

for IEEE 157 bus test system.

It is found that connectivity is lost to a great extent in both cases, although the

effect is higher in case of bidirectional flow model than the unidirectional counter-

part.

In case of bidirectional flow model, almost 50% connectivity is lost after six nodes

removal while if we remove nodes according to nondirectional model even after 10

nodes removal the connectivity is very high. It takes 17 nodes removal according to

nondirectional model to decrease the connectivity loss to 50%.

5.3.3 Load Loss

Last measure of impact is found from a simple model of cascading failure that

is presented here. Since it is not possible to exactly model the blackout, various


approximate measures have been taken, by several researchers, to mimic the situa-

tion [20, 206–208].

The power system is a very much complex interconnected system whose exact

modeling would require consideration of dynamics of rotating machines and devices

within the system, discrete dynamics of switchgear elements, non-linear algebraic

equations that govern line flows and social dynamics of governing and operating

bodies.

In this chapter, a fairly simple model of cascading failure of the power grid is

proposed by incorporating important electrical features ignoring those which are too

complicated but have little effects. The detail of the model is described here.

At first, the AC power flow is used to calculate the steady state condition of the

network. The real and reactive power of transmission lines are found from numerical

solution of line flow equations given in (5.8) and (5.9)

Pi =

n∑

j=1

|Vi||Vj||Yij|cos(θij − δi + δj) (5.8)

Qi = −n

∑

j=1

|Vi||Vj||Yij|sin(θij − δi + δj) (5.9)

where the symbols have their usual meanings as found in the power system literature.

During the analysis, generator and load dynamics are not included. Although

the limitation of not using dynamics of generators and loads are well understood, but


it is at least useful for modeling one mechanism of cascading failure that is cascad-

ing overload. Also, Generation Shift Factors (GSF) and Line Outage Distribution

Factors (LODF) [163] are used to recalculate flows in lines after disturbance. This

helps achieving fast results without using actual load flow after each disturbance.

The transmission lines are removed if overloaded. Also, time delayed over current

relays are used in every line so if there is a lot of overload it trips fast and if there

is a little bit of overload it trips slowly. Another thing that is added to the model

is ramping up of generators. As the system separates into sub grids, generators are

allowed to ramp up or ramp down to rebalance a little bit.

So, if a component failure disturbs the supply-demand balance, through genera-

tor set-point adjustment this balance is achieved, but if there is not enough ramping

ability, then the ultimate choice is to trip lowest possible system load. The total

amount of load lost during the successive removal of nodes is used as a measure of

impact.

Fig. 5.5 shows load loss as a percentage of total system load. Up to six node

removal the load loss is nearly equal and does not increase much for both unidirec-

tional models. After five node removal, more than 50% loads of the system need to

be shedded to ensure secure and reliable operation of the remaining system.

The overall steps are summarized below while the flowchart in Fig. 5.6 represents

the same.


0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10

Load L

oss


Unidirectional

Bidirectional

Figure 5.5. Two different effects on load loss due to loss of functionality of important

nodes in IEEE 300 bus test system.

Step 1 The AC power flow is solved to find out the steady-state condition of the

network.

Step 2 Find out if the power flow is converged or not.

Step 3 If the power flow is not converged, reduce some load from the system and

go to Step 1.

Step 4 If the power flow is converged, initialize counter i.

Step 5 Increment counter i by 1.

Step 6 Record total load shedded during the process.

Step 7 Remove i -th critical node from the critical list.

Step 8 Line Outage Distribution Factor (LODF) [163] is used to calculate redis-

tribution of the power flow without actually solving the load flow problem

again.

Step 9 Check for system load-generation balance. If the balance is not achieved,

ramp-up or ramp-down generators accordingly to adjust generator set-point

Section 5.4 Rank Similarity of Critical Nodes 107

in order to achieve the balance. Generation Shift Factor (GSF) is used to

calculate redistribution of the power flowing among lines.

Step 10 Check if there is any overloaded line.

Step 11 If any overloaded line is found trip the line and go to step 8. Time-

delayed overcurrent relays are used in every line so if there is a lot of overload

they trip faster and if there is a little bit of overload they trip slowly.

Step 12 If there is no overloaded line, go to Step 5.

5.4 Rank Similarity of Critical Nodes

From the results of Section 5.3, it is clear that, the nodes found from bidirectional

flow model has much more impact than nominal and backward unidirectional mod-

els. In order to analyze the effect of system change on ranks of critical nodes a

rank similarity analysis is performed [209–212]. A structural change like change in

the direction of the power flow is incorporated in the model, and critical nodes are

found out for the modified system. This change in network corresponds to a situa-

tion when there is a pushback of power from low voltage network via transmission

system to meet energy needs in other area.

Table 5.2 compares the changes in the top ten critical nodes in IEEE 30 bus

test system. This analysis is carried out for the bidirectional power flow model.

Top row of Table 5.2 corresponds to the topological state of the system. The first

column gives the top ten critical nodes from the bidirectional model. The rest of the

columns list changes in critical nodes for changed topology. For example, the third


Solve AC Power Flow

Problem for the

Nominal Case

Shed Load

Power Flow

Converges?

Calculate

Total Load

Shedded

Remove i-th

Critical Node

from Critical List

Update Flows of

Lines using

LODF

Overloaded

Line?

Match Supply

Demand Balance

using GSF

Trip Line

i = 0

i = i+1

No

Yes

NoYes

Figure 5.6. Simple cascading failure model.


Table 5.2. Top Ten Critical Nodes in the Bidirectional Power Flow Model for IEEE 30

Bus System Under Various Changed Topological Conditions

Nominal Line Line Line Line Line Line Line Line LineCase 24–25 29–27 6–2 17–10 4–3 10–6 18–15 30–29 15–141 1 1 1 1 1 1 1 1 13 3 3 2 3 2 2 3 3 32 2 2 3 2 4 4 2 2 24 4 4 6 4 6 6 4 4 46 24 24 4 13 24 24 6 6 624 13 6 24 12 19 19 24 24 2419 6 19 19 24 13 13 19 19 1913 12 29 13 6 12 12 18 13 912 19 13 12 16 14 14 9 12 2614 14 12 14 19 9 9 26 14 139 9 14 9 17 26 26 23 9 18

column represents the top ten critical nodes when the nominal direction of flow is

changed through line 29–27. It is clear that, changed topology does not affect much

the node criticality.

On the other hand, slightly more change is observed in criticality for the unidi-

rectional model as shown in Fig. 5.7. Each color represents different nodes of the

system in Figs. 5.7 - 5.10. When the power flow pattern through the grid is uni-

directional, nominal unidirectional method is effective, but, in order to model the

situation in the future smart grid, bidirectional model gives better result in terms

of rank similarity as given in Fig. 5.8.

In a typical power system, load varies from time to time and generation have to

match the load and line loss. For this reason, various power flow profiles are found

in the system during various seasons of the year. Even the scenario is different at


1 2 3 4 5Node 1

Node 5

Node 10

Node 15

Node 20

Node 25

Node 30

No. of observations →

Ran

k c

han

ge

of

var

iou

s n

od

es →

Figure 5.7. Variation of ranks of nodes in unidirectional model of IEEE 30 bus test

system when the network is modified slightly.

different times in a day. To demonstrate that the proposed power flow based cen-

trality method gives critical nodes which is insensitive to system load and generation

change, an Australian test system is considered which provides six test cases from

heavy to light load conditions [213].

Table 5.3 gives the six normal steady-state operating conditions for the system.

Betweenness and closeness centrality of the test system is measured for various test

cases.


1 2 3 4 5 6 7 8 9 10

Node 1

Node 5

Node 10

Node 15

Node 20

Node 25

Node 30

No. of observations →

Ran

k c

han

ge

of

var

iou

s n

od

es →

Figure 5.8. Rank similarity of nodes in the bidirectional power flow model is better than

that of unidirectional one.

Table 5.3. Six Normal Steady-State Operating Conditions of the Australian Power Grid

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Load Condition Heavy Medium Heavy Peak Light Medium LightestGeneration (MW) 23030 21590 25430 15050 19060 14840

Load (MW) 22300 21000 24800 14810 18600 14630


1 2 3 4 5 6rank 5

rank 4

rank 3

rank 2

rank 1

Test Case →

Ran

k→

Figure 5.9. Rank similarity of nodes in the bidirectional power flow model is better than

that of unidirectional one.

Table 5.4. Ranks of Various Buses of Australian Test System Based on Closeness Cen-

trality


Bus 13 44 44 44 44 44 44Bus 8 5 4 3 7 3 8Bus 14 45 45 45 45 45 45Bus 44 46 46 46 46 46 46Bus 2 2 3 1 2 1 4Bus 24 24 23 23 26 26 27Bus 57 17 18 19 19 18 18Bus 47 37 37 38 39 39 38Bus 28 56 56 53 56 55 56Bus 48 8 9 9 7 8 10


1 2 3 4 5 6rank 5

rank 4

rank 3

rank 2

rank 1

Test Case →

Ran

k→

Figure 5.10. Rank similarity of nodes in the bidirectional power flow model is better

than that of unidirectional one.

Variation of ranks of the Australian test system in six test cases are shown

in Fig. 5.9 and presented in Table 5.4 for closeness centrality measures and for

betweenness centrality measures in Fig. 5.10 as well as in Table 5.5. From the figures,

it is clear that betweenness based measure is more rank stable than the closeness

one. In case of betweenness based measure, first three rank positions do not change

in six different operating conditions, whereas in case of closeness centrality there are

small variations in ranks two to four positions. In case of rank five there is a large


Table 5.5. Variation of Ranks of Several Buses of Australian Test System Based on

Betweenness Centrality


Bus 41 58 59 58 59 59 57Bus 43 56 57 57 55 57 53Bus 46 1 4 2 5 1 4Bus 45 59 58 59 58 58 59Bus 42 49 51 46 46 47 46Bus 44 51 46 49 50 50 47Bus 15 16 17 18 18 15 18Bus 32 32 30 31 29 30 32Bus 59 45 44 45 43 46 44Bus 34 11 11 11 10 11 10

variation of two ranks.

5.5 Chapter Summary

The changing power flow pattern demands for improved analytical techniques to

solve the global problem of cascading failure. This issue has been addressed under

complex network framework. An improved model of closeness centrality measure

has been proposed. The bidirectional power flow pattern of the future smart power

system have been taken into consideration while modeling the system. Three dif-

ferent measures of impact have been evaluated to observe the effect of removal of

critical nodes found from centrality measures.

Closeness centrality based critical node analysis has been carried out and re-

sults from nondirectional and bidirectional flow based methods have been compared.

Large changes in path length, connectivity and load loss implies the efficacy of the

proposed method. IEEE 30 bus, 57 bus, 118 bus and 300 bus test systems has been


used to demonstrate the applicability of proposed modified centrality measures in

critical node analysis of the power system. The analysis of a real Polish transmission

system based on the proposed methodology is our future work.

Chapter 6

Conclusions

This chapter summarizes key facts and findings of this research work, presents con-

clusions drawn and discusses future areas of exploration. In this work the emphasis

was on to identify the vulnerability of the power system, specially the topological

weakness of the transmission system. Various criticality assessment methods have

been proposed to identify important elements of the power transmission grid. Effect

of removal of the elements (nodes or lines) on the performance of the system has

also been analyzed.

The results presented in this thesis are based on the formulation of various cen-

trality measures for the power system to overcome the limitations of current topology

based criticality measures to assess the vulnerability of the system. However, before

designing new measures, an overview of the shortcomings of the existing measures

has been provided.

Complex network theory has been used for the vulnerability analysis of power

systems, taking into consideration critical electrical parameters which affect stability

of the power grid. Various simplified mapping techniques have been used without

neglecting key parameters. In order to capture the true power flow scenario within

the grid, a novel centrality analysis approach has been provided. The result of

116

Section Conclusions 117

the centrality analysis provides critical transmission lines of the grid, which when

removed from the system have found to affect the transient stability.

Another criticality assessment procedure has been provided, considering the max-

imum possible flow within the grid. Power flow solution technique has been com-

bined with the maximum flow finding algorithms to demonstrate the applicability

of complex network based analysis techniques into power systems in order to as-

sess vulnerability of the system. This criticality analysis procedure provides critical

nodes of the system. Network efficiency is greatly affected when top critical nodes

are removed from the system. This explores the vulnerability of the power system

on targeted node removal.

Motivated from social network based centrality analysis procedures, various cen-

trality based analysis procedures have been modified to use in case of the power

system. Electrical degree centrality, closeness centrality, and betweenness centrality

measures have been analyzed. Comparisons are carried out between pure topology

based centrality measures and the proposed electrical centrality measures. Bus de-

pendency matrix have been developed, which provides a succinct representation of

the closeness and betweenness centrality of the power grid. An algorithm is provided

to illustrate the step-by-step procedure of formulating centrality measures from the

system data.

Considering the bidirectional flow of the future smart power system, a new frame-

work has been established within the context of the complex network to identify

Section Conclusions 118

vulnerability of the power system. The bidirectional power flow model is a superpo-

sition of the generic power flow situation of the grid with the backward flow model.

The superiority of the bidirectional flow based model in finding the critical elements

of the smart power system has been demonstrated with examples.

From this work, the following conclusions can be drawn:

• Critical element analysis of the power system is crucial for system planning as

well as dynamic security assessment and monitoring system.

• Both nodes and links play important roles in criticality analysis.

• Although rare, but the node removal causes much more significant loss of

system’s performance than the link removal.

• Targeted removal of components cause more damage to the system than the

random removal.

• System reliability and security could be enhanced by regularly monitoring and

servicing the most critical components of the grid.

• Large scale blackouts could be prevented if actions are taken immediately in

case of a fault or overloading of high betweenness lines.

• The betweenness index could also be used as a measure of transient stability

of the system.

• The criticality analysis provides a margin between unstable and stable oper-

ating region.

• Removal of flow central nodes affects network efficiency.

Section 6.1 Directions for Future Research 119

• Power doesn’t flow only through the shortest path between two nodes, it flows

through all possible combination of originating at source node and terminating

in load nodes.

• The degree centrality is a measure of connectivity.

• The independence of a node is determined by the closeness centrality of the

node.

• The betweenness centrality provides a measure of control of communication.

• The bus dependency matrix contains the information of dependency of bus

pairs in a power system.

• The bidirectional power flow based model captures the changed topological

and pattern of power flow in smart grid.

• The rank similarity analysis shows the robustness of the proposed methods

under system parameter changes.

• The proposed methods of vulnerability analysis are proven to be computation-

ally efficient, improve system reliability and security, help dispatchers to take

prompt corrective action, and reduce the probability of large-scale cascading

failure leading to blackouts.

6.1 Directions for Future Research

Although this research achieved promising results in analyzing the vulnerability of

the power grid, and identifying critical components of the system, the work does not

Section 6.1 Directions for Future Research 120

end here. The proposed power system mapping, and criticality analysis procedure

may be further improved and consolidated by the following processes,

1. Considering both real and reactive power flow when measuring centrality of

the system.

2. Identifying how system robustness can be improved by simply adding links

while cascading is in progress.

3. Extending the vulnerability analysis into practical power systems.

4. Analyzing the effect of loop flow on vulnerability of power grid.

5. Conducting a detailed study to ascertain the impacts and possible benefits of

the inherent variability of renewable sources, particularly wind and solar, and

their correlations with load profiles.

6. Investigating into the network functional vulnerability.

7. Exploring the concept of network percolation in power system analysis.

8. Conducting case studies on electrical distribution networks.

This dissertation focuses on developing complex network based system metrics to

identify electrical power grid vulnerability. However, there are many other metrics

available which focus on contingency ranking. Although, this thesis does not address

contingency ranking [134], but it will be interesting to compare the performance of

complex network based metrics with conventional approaches to critical component

detection.”

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