On Robustness of Power Grids

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EASURING THE

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Yakup Koç

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Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op 16 November 2015 om 12:30 uur.

door

Yakup K

OÇ

Master of Science in Telecommunications geboren te Istanbul, Turkije.

dr. ir. R. E. Kooij.

Composition of the doctoral committee: Rector Magnificus, chairman

Prof. dr. F. M. T. Brazier, Delft University of Technology, promotor

Prof. dr. ir. R. E. Kooij, Delft University of Technology and TNO, promotor Dr. M. E. Warnier, Delft University of Technology, co-promotor Independent members:

Prof. dr. M. Ili´c Carnegie Mellon University Prof. dr. C. M. Scoglio Kansas State University Prof. dr. ir. M. Aiello University of Groningen Prof. dr. ir. P. Van Mieghem Delft University of Technology

Prof. dr. ir. P. M. Herder Delft University of Technology, reservelid

This research is funded by NWO, and has been carried out under the auspices of SIKS, the Dutch Research School for Information and Knowledge Systems. SIKS Dissertation Series No. 2015-31.

Cover design by Hakan Ergun Typeset with LA_{TEX 2ε}

Printed by CPI – Koninklijke Wöhrmann

Copyright © 2015 by Yakup Koç E-mail: yakupkoc@gmail.com ISBN 978-94-6203-956-8

An electronic version of this dissertation is available at

Time is flying. It was nearly four years ago that I decided to start my Ph.D. journey. A decision that I now clearly see that it was a milestone in my life, a decision that I am utmost happy about. During this process, I have met many people, learnt about many cultures, seen many countries, gained and improved many skills, experienced many feelings, and overall, learnt a lot. Many people contributed to this amazing learning process in one way or another. I would like to heartily thank their contribution to this signature experience.

Frances Brazier and Robert Kooij are my promoters, giving me the opportunity to start my Ph.D. research, and to be a part of their groups Systems Engineering and Network Ar-chitectures and Services. Our discussions with Frances and her feedbacks did not only im-prove the quality of technical aspects of my research, but also shaped my understanding of the academical world, and thought me the value of effective communicating of my research to the outside world. She was always there whenever I needed her insight and feedback. Robert Kooij was one of the main reasons that I started my Ph.D. research. His creativity and helicopter view over different disciplines increased the quality of my research. His at-titude towards the research and, in a broader context, towards life inspired me "to do what I like, and to enjoy my time, that is the most valuable and important towards success". I am grateful for all this. Martijn Warnier was my co-promoter. He was also my daily supervisor, mentor, colleague, and a good friend. He took all these roles occasionally to motivate, to cheer, to criticize, and to inspire me. His encouragement to improve myself in different aspects made this Ph.D. process unique. Thank you very much for all the efforts.

In the last four years, I met amazing people, collected wonderful friends and memories. Zulkuf and Engin were the significant part of the joy at the office. It has always been great pleasure to be in company with them. Jordan brought colours here all the way from Rio; every chat with him charged me full of energy. I very much enjoyed my time with Yilin, Selma, Shallini, Cagri, Sertac, Nina, Vasiliki and Mingxin inside and outside the office. Evangelos Pournaras, I am happy that we had the chance to work together in the same project. I hope you are still enjoying Turkish coffee! I always enjoyed sharing moments and discussions with Amir and Alireza. A very special thank to Michel Oey for being very friendly all the time, and for the help whenever needed. I was always pleased to have interesting discussions with Sander, Caroline, Stefan, Jos, and Tanya. Everdine and Diones were always very helpful with all kind of administrative work.

Throughout my Ph.D. research, I spent one day a week at the Network Architectures and Services group at the Faculty of Electrical Engineering, Mathematics, and Computer Science. I would like to thank Piet Van Mieghem, the head of the section, for making this possible. Our discussions with Piet were exciting learning moments form me, combined with fun. His sceptical view to any point in the research helped me significantly to posi-tion/mature my research. Fernando Kuijper was always very kind and helpful whenever needed. A very special thank for Xiangrong Wang and Hale Cetinay. I very much enjoyed our discussions, coffee breaks and collaboration with Xiangrong Wang and Hale Cetinay.

Seeing them evolving into independent researchers makes me also proud of them! I would like to thank Jill, Norbert, Stojan and all other friends at NAS group for being part of a warm and friendly environment.

During my Ph.D. research, I spent 4 months at IBM T.J. Watson Research Center in USA. This was invaluable to my personal development, and future career decisions. I would like to thank Ron Ambrosio, the CTO of Smarter Energy program, and Tarun Kumar, the manager of Smarter Energy Group for giving me the opportunity to join IBM Watson research center. My visit to IBM opened a new perspective to my research from the industry point of view, inspiring me about the applicability of my research. Abhishek Raman, my mentor at IBM, was amazing. We spent almost the half of each office day together, and I enjoyed every moment of it utmost. Football with Kenny Tran and morning coffee with Harsch were joyful moments that I was looking for. Aanchal, Prasand, Yong-hun, and Mark are just a few of many other great people that I met at IBM.

My friendship with several other people made me feel at "home" in the Netherlands. First and foremost, I would like to thank Ibrahim Eroglu who was my house-mate for 5 years, and will be a friend for lifetime. Kubilay Ozdemir was one of the people who con-tributed significantly to how I perceive world; from politics to art. Yavuz Gokce, Erdem Yaktemur, Muhammed Yaktemur, Gurbuz Yaktemur, and John Veltema have a significant role in whatever I have achieved till this day. I am grateful to them. I owe a special thank to van Meer Family and Frans Corpolijn for all the support they provided me. Spending time with Hande, Sevinc, Azad, Sezen, Mitzi, Berend, Zana, Burak, Mustafa, Ozge, Cigdem, Kostas, Orkun, Sinem, Zeki, Ahmet Koray, Gorkem, Harun and Dogan has always been a pleasure for me. Furthermore, there are people back in Turkey who make me feel that we keep our contact and level of friendship as we never left apart: Ugur Kavza, Onur Bayat, Cagatay Yasar, Onur Yalcintas, Ercan Yorulmaz and Onur Karacorlu. Their continuous remote support and encouragement have been extremely helpful.

A Ph.D. process is full of ups and downs. It is crucial to feel the unconditional support, and pure love. In this regards, I had the luxury and blessing to have my family: my mother, my brother, my sisters, and my wife. Despite nearly 3.000 km between us, I felt the support and love of my mother, my brother, and my sisters at every moment as we were always together. My beloved wife Aylin has been the closest witness of every moment of this Ph.D. process. She has been my best listener, supporter, care taker, criticizer, and motivator. Having experienced her care and support made me always wondered about how my single Ph.D. colleagues survive a Ph.D. process! Because of the warmest feelings of my family, I never felt alone in this process. They have been the source of my resilience, success, ambitions, and the reason for this Ph.D. process. They inspired me to dream for the best, to dare trying it and to do whatever it takes to reach it. I am grateful for all they have done to make this possible. Nearly four years ago we started this Ph.D. journey together with them, and now, I am writing these very last lines of this dissertation. Time is truly flying.

Yakup Koç Delft, June 2015

1 Introduction 1 1.1 Research Overview. . . 3 1.1.1 Research Objectives . . . 4 1.1.2 Research Approach. . . 4 1.1.3 Research Contributions. . . 5 1.2 Thesis Outline . . . 6

2 Related Work & Research Positioning 9 2.1 Power Grid, Complex Systems, and Complex Network Theory . . . 10

2.1.1 Power Grid. . . 10

2.1.2 Complex Systems . . . 12

2.2 Robustness of Power Grids: research positioning . . . 18

2.2.1 Short-term Intervention to power grid. . . 19

2.2.2 Long-term Intervention to power grid. . . 22

3 Robustness against Cascading Failures in Transmission Grids: Quantifying the impact of the operative state 29 3.1 Cascading Failures in Power Grids . . . 30

3.2 Modelling Cascading Failures in Power Grids. . . 31

3.3 Robustness Metric . . . 34

3.3.1 Electrical nodal robustness . . . 34

3.3.2 Electrical node significance. . . 37

3.3.3 Network Robustness Metric . . . 37

3.4 Experimental Verification. . . 38

3.4.1 Experimental set-up . . . 38

3.4.2 Metric experimental verification . . . 39

3.5 Cascading failures robustness analysis of electric power grid: Case studies . 41 3.5.1 Effect of line adding . . . 41

3.5.2 Robustness assessment of different IEEE power systems under tar-geted attacks. . . 43

3.5.3 Evolution of network robustness by new operative states . . . 44

3.6 Conclusion and discussion . . . 46

4 Robustness against Cascading Failures in Transmission Grids: Quantifying the impact of the topology 49 4.1 Dynamic Model of Cascading Failures in Power Grids . . . 50

4.2 Effective Graph Resistance in Electric Power Grids. . . 52

4.2.1 Complex networks preliminaries . . . 52

4.2.2 Effective graph resistance and its computation in electric power grids. . . 52

4.3 Effective Graph Resistance as a Robustness Metric . . . 53

4.4 Effective Graph Resistance and Power Grid Robustness: Experimental Ver-ification . . . 56

4.4.1 Power Grids: IEEE power systems and synthetic grids . . . 56

4.4.2 Robustness levels by effective graph resistance and by simulations . 57 4.4.3 Assessing the effectiveness of the effective graph resistance in an-ticipating power grid robustness . . . 58

4.5 Use Cases . . . 62

4.5.1 Upgrading IEEE 118 power system to improve grid robustness . . . 62

4.5.2 Identifying the critical lines in IEEE 118 Power system. . . 64

4.6 Discussion and Conclusion. . . 67

5 Phase Transitions in Transmission Grids: Quantifying the Impact of the Topology 69 5.1 Spectral Graph Metrics. . . 70

5.2 Experimental Methodology. . . 71

5.2.1 Power grids: IEEE power systems and the synthetic grids. . . 71

5.2.2 Attack Strategies. . . 71

5.2.3 Computation of the critical loading threshold. . . 72

5.3 Numerical Analysis . . . 73

5.3.1 Impact of line removal on phase transition in power grids. . . 74

5.3.2 Relating phase transitions to topology in power grids . . . 74

5.4 Discussion and Conclusions . . . 78

6 Robustness of Power Distribution Grids: Quantifying the Impact of the Topology 83 6.1 Redundancy vs. Robusness. . . 84

6.2 Metric: Upstream Robustness. . . 85

6.2.1 Inter-path independency . . . 85

6.2.2 Intra-path independency . . . 87

6.2.3 Upstream robustness . . . 88

6.3 Use Case: Asset Criticality Assessment. . . 88

6.4 Conclusion and Discussion. . . 94

7 Discussion, Conclusion & Future Work 97 7.1 Research Questions Revisited. . . 99

7.2 Conclusions . . . 102 7.3 Future work . . . 103 References 105 Summary 121 Samenvatting 123 Özet 125

SIKS Dissertation Series 127

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I

NTRODUCTION

On 14 August of 2003, the United States of America suffered the largest blackout event in North American history [1]. Four high-voltage transmission line failures near Ohio, and a virus that disabled the alarm system initiated large-scale cascading failures, leaving a large part of the country, including New York City, without electricity.

When the cascading failure hit New York City, traffic lights and subway trains failed immediately. The out-of-order traffic lights caused chaos in traffic, while after subway failure, thousands of people walked through subway tubes potentially putting themselves in life-threatening situations. In the huge sky-crabbers in the city, hundreds of people were captived in elevators, while thousands of people had to use the stairs to evacuate buildings often in the dark. Air traffic was affected too. Because the airports could not restore power for customer screening, the flights were delayed extensively, if not cancelled.

The blackout affected also the water and sewer systems. As the pumps did not function properly, one New York City pump station leaked millions of gallons of sewage, that with the effect of heavy rains flowed into waterways in Detroit and Cleveland. As a result, four million people in these states needed to boil their water for a period of four days due to a risk of contamination between the sewer and water systems [2].

The effects of the blackout would have been even worse if the telecommunication sys-tem had also been affected. If the blackout had lasted longer than the design time of the energy storage system, communication failures could have propagated to other services that rely on telecommunications, such as stock markets or emergency responders [2].

The August 14, 2003 blackout illustrates the key importance of the electric power grid to modern societies. This is not only because of the vital role of electric power for daily life, but also because of the strong dependencies of other critical infrastructures (such as telecommunications, transportation, and water supply [3,4]) on power grids. Disruption of power delivery systems has severe effects on the public order and substantial economic cost for society [2]; the 2003 blackout left more than 50 million people in the dark, with an estimated economic cost of $ 6 billion and contributed to the death of at least 11 people [1]. Other countries have suffered from catastrophic blackouts too [1,5]. Analysis of in-ternational blackout data reveals that the probability distribution of the blackout sizes

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expo-ity suggesting large-scale blackouts are more likely than expected. More recent examples of large-scale blackouts are the one in Brazil in 2009 that left 40% of the country without power for up to five hours [5], the one in India in 2012 that left 630 million people without electricity 14 hours long (the complete power restoration took 2 days) [7], and blackouts in the Netherlands [8] and in Turkey [9] in 2015, causing similar scenarios as experienced earlier in New York City.

NEAR-FUTURECHALLENGES FORPOWERGRIDROBUSTNESS

The analysis of the U.S. Energy Information Administration (EIA) data on power outages (of 100 MW or more) between 1991 and 2005 reveals that the number of outages in North America has been increasing exponentially [10]. The continuous demand increase, over-ageing of the assets, impact of extreme weather conditions, and the increasing threat of terrorist attacks are the major causes of this exponential increase of number of outages. The projections on these factors [11–13] for the next 20 years suggest that their negative impact on the robustness of the system will increase, necessitating preventive actions.

Demand increase According to International Energy Agency (IEA) and Energy Informa-tion AdministraInforma-tion (EIA) data on electric power demand trends in the last 60 years, global power demand has continuously increased, as it is correlated to the increase in economi-cal prosperity [14]. Projections suggest a further increase in electricity demand in the next 20 years by nearly 25 % [11]. Such an increase in electricity demand results in a heavier loading of the power grids. Accordingly, the ability of power grids to absorb disturbances decreases, making power grids more vulnerable to failure.

Over-ageing of the components in power grids The power grid is one of the oldest man-made technological systems on earth. In many developed countries, it has been built at the beginning of20t h century, and not much has changed in these systems. Many of the assets in these systems are near the end of their life-cycle. For instance, in most of Organisation for Economic Co-operation and Development (OECD) member countries, the network infrastructure, including transmission and distribution grid, was constructed over 40 years ago [15], and more than 70% of transmission lines and transformers in US grids are older than 25 years [12]. This over-ageing of assets in the power grids drastically increases the potential failure rate of these assets, and accordingly, threatens the safety of the power grids.

Extreme weather conditions Extreme weather conditions are one of the primary causes of power outages. Power outages due to extreme weather conditions might cause tens of billions of dollars damage in developed countries: for instance, in the period between 2003 and 2012, such outages cost on average $18 billion to $33 billion (only to the U.S. economy). Depending on the severity of the conditions, the cost goes up to $75 billion (in 2008) or $52 billion (in 2012) [16]. The number of power disturbances due to severe weather is expected to increase as the impact of global warming increases the frequency and intensity of the extreme events such as hurricanes, blizzards, floods and other extreme

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weather events. Accordingly, the vulnerability of the power grid is also expected to increase due to severe weather expected as the climate changes [13].

Distributed generation The 2020 goals of the European Union onCO2emission and on the share of renewable energy in the energy mix of EU countries stimulate these countries to subsidize renewable energy generation [17]. As a result, consumers are rapidly becom-ing producers of electricity installbecom-ing e.g. solar panels and wind mills [18,19]. The in-creasing deployment of renewable energy sources (RES) calls for analysis of their negative impact on grid operation and control, as it threatens grid safety [20]: the large-scale pen-etration of RES negatively affects grids in terms of regional overloading of transmission lines, reduction of available tie-line capacities, frequency performance, grid congestions, increasing need for balance power and reserve capacity, increasing power system losses, and increasing reactive power compensation [21]. These effects of large-scale introduction of renewable energy sources to the current centralized power grid makes it more vulnerable to disturbance.

Terrorist attacks The key importance of power delivery systems for nations across the world makes these systems focus of terrorists. A single attack on a critical component in a power grid might have severe effect on daily life, and causes chaos in a country. For example, in 2015, the terrorist attack on a critical transmission line in Pakistan left 80% of the population (140 million) without electric power [22]. The threat of terrorist attacks increases as the instability around the world increases, calling for attention of power grid operators and researchers, to increase the robustness of power grids with respect to deliber-ate attacks.

This thesis is motivated by the increasing need for enhanced robustness of power grids, imposed by environmental, economical, and human-caused factors. To this end, this dis-sertation (i) further investigates the complex notion of robustness of power grids, and (ii) designs measures and metrics to quantitatively assess the robustness of power grids from various perspectives. The results should assist policy and decision-makers in developing and evaluating strategies for effective resource allocation with the ultimate purpose of en-hancing the robustness of power grids.

Within the context of this research, robustness of a power grid refers to the ability of a system to avoid malfunctioning when a fraction of its elements fail as a result of deliberate attacks or random failures that limit the ability of the system to accomplish its tasks [23]. The opposite of robustness is the vulnerability of a power grid: vulnerability refers to the sensitivity of a power grid with respect to deliberate and random failures.

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This section presents an overview of the research in this dissertation. The research ob-jectives introduce the overarching pursuits, the research questions highlight the specific knowledge required to address the research objective, and the research approach introduces an explanation of the tools and means used in this thesis. This section ends with a highlight of the main contributions of this dissertation.

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outages occur. The key importance of electric power grid to society encourages further research into sustaining power system security and developing new methods and tools to evaluate and mitigate the risk of large-scale power system outages. This dissertation’s ob-jective is to better analyse and understand the subtle behaviour of power grids in terms of robustness, to propose a set of measures that assist grid operators to design and operate power grids in such a way that the robustness of the system enhances.

The main research question this thesis aims to answer is:

RQ Can measures be designed to quantify the robustness of a power grid? Answering this research question requires addressing following sub-questions: SRQ1 What are the factors that govern robustness in power grids?

SRQ2 How to capture the impact of operative state of the grid on the robustness of power transmissiongrids?

SRQ3 How to capture the impact of topology on the robustness of power transmission grids?

SRQ4 What other forms of robustness exist in power grids, and what is the relationship between the topology and these robustness forms in a transmission grid?

SRQ5 How to capture the impact of topology on robustness of power distribution grids? The remainder of this dissertation addresses these research questions. Chapter2 an-swers SRQ1. Chapter 3 addresses SRQ2. Chapter 4 and Chapter 5 answer SRQ3 and SRQ4, while Chapter6focusses onSRQ5.

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A research philosophy [24] is a belief about the way in which data about a phenomenon should be gathered, analysed and used. One of the major research philosophies in the Western tradition of science is positivist (also called scientific) [25,26]. Positivist paradigm entails verifiable observations to build up the scientific knowledge and following transparent methods to produce sound knowledge, that makes possible to replicate the research. This, epistemologically, reflects the belief that there is an external, objective reality and it is stable across contexts and people. Post-positivism, on the other hand, acknowledges that involvement of humans impedes the objectivity of knowledge [24]. Humans understand meaning but cannot objectively measure or quantify it [27].

This dissertation presents research on socio-technical systems that involve humans. Therefore, this research follows the post-positivist approach to combine objective mea-surement with human experience.

The research strategy that this thesis follows is research through design that guides a scientific research to provide a solution for a specific problem with emphasis on artefacts

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as the outcome of the research [28]. Artefacts can be concepts, models, methods, or imple-mented systems. The resulting products of this research are concepts, models, and measures relating to the robustness of power grids.

The research instruments that this research uses are literature review, case study, ex-perimentation, and evaluation. A literature review provides background knowledge and deeper understanding of the research domains [29]. A case study provides qualitative and observatory insight into the real-life context of the research phenomenon [30]. Evalua-tion helps researchers to assess the final product of their research. This research employs the literature review and case study instruments to perform an extensive literature survey regarding the theory of Complex Systems, especially the Complex Networks Theory, fun-damentals of power networks and power grid operation. Artefacts are implemented and experimentally validated. After experimentation, the evaluation instrument is used to apply the implemented system to a number of synthetic and real-life cases.

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The main contributions of this dissertation are additional insight into the complex behaviour of robustness of power grids, and a set of measures to quantitatively assess the robustness of power grids, for robust operation and design of these systems. To be more specific, this dissertation proposes metrics to assess the robustness of power grids with respect to cas-cading failures due to line overloads (for transmission grids), and supply security (for dis-tribution grids). The proposed metrics are intended to assist (i) grid operators for dynamic optimization of flow and topology of a given power grid, and (ii) grid analysts in relevant processes for mid-term and long-term grid planning, such as asset management and net-work expansion planning for power transmission and distribution grids. This dissertation designs, implements, and validates the proposed concepts and metrics for power transmis-sion and distribution grids. The high-level contributions of this dissertation is highlighted as follows.

C1 Bringing structure in different interpretation of the robustness concept in different lay-ers of power grids, and highlighting the relevant aspects of these systems that govern the robustness in these different layers (This dissertation).

C2 Proposing a metric that quantifies the robustness of a power grid by, in addition to the impact of topology, incorporating the impact of operative state on the robustness of the system (See Chapter3).

C3 Proposing a metric that captures the impact of topology on the robustness of a power transmission grid with respect to cascading line overloads. The proposed metric captures important electric power flow characteristics including power flow allocation according to Kirchoff Laws (See Chapter4).

C4 Establishing a connection between the power grid robustness and spectral graph the-ory. This enables applying the well-established knowledge in spectral graph theory on power transmission grids to enhance the robustness of power grids (See Chapter4 and Chapter5).

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transition behaviour of power transmission grids (See Chapter5) .

C6 Proposing a metric that quantifies the impact of the topology on the robustness of power distributiongrids with respect to supply security. The proposed metric provides a tool to assess the impact of actions on the grid from the point of topological robustness, in addition to perspectives of economical and capacity aspects (See Chapter6).

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UTLINE

This dissertation consists of 7 chapters. Figure1.1illustrates the structure of this disserta-tion, and how the chapters inter-relate to each other.

Chapter(1( Introduc0on( Chapter(2( Related(Work(and(Research( Posi0oning( Chapter(3( Opera0ve(state(and( robustness(( Chapter(4( Topology(and( robustness(( Chapter(5( Topology(and(phase( transi0ons(( Chapter(6( Topology(and( robustness(

Transmission(grid(

Distribu0on(grid(

Figure 1.1: The outline of this dissertation.

Chapter2introduces basics of Complex Systems, Complex Networks Theory, and Elec-trical Power Systems Theory, and positions the research of this dissertation.

Chapter3 investigates the impact of operative state on the robustness of power grids with respect to cascading line overloads. Relying on the developed insight, this chapter designs a metric to quantify the robustness of a system incorporating the impact of operative state on the robustness.

Chapter 4 and Chapter5 focus on the impact of topology on the various aspects of robustness in power transmission grids. Chapter4investigates the relationship between the

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topology and robustness of a transmission grid with respect to cascading line overloads, and proposes a metric to quantify and to exploit this relationship. Chapter5assesses the impact of topology on phase transitions in the robustness of the system with respect to cascading line overloads.

Chapter6outlines the impact of topology on the robustness against supply security in a power distribution grid. This chapter designs a metric to quantitatively assess the impact of topology on the robustness.

Finally, Chapter7brings the designed metrics together discussing for which purposes the proposed metrics can be deployed. It also answers the stated research questions, pro-vides a discussion on high level conclusions of this thesis, and proposes future research directions.

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OSITIONING

This dissertation investigates the subtle behaviour of robustness of power grids, and presents measures to quantitatively assess the robustness of power grids from a complex network theory perspective. This research is positioned in the intersection of Complex systems, Complex Networks Theory, and Electrical Power Systems Theory (See Figure 2.1). A power grid is assumed to be a complex system, and modelled as a complex network. This makes it possible to combine, extend, and apply the established knowledge in these fields to assess the robustness of electrical power grids.

Complex( Systems( Complex( Networks( Theory( Power( Systems( Theory(

Figure 2.1: Positioning of this dissertation (in the shaded area) in the related fields.

This chapter provides preliminaries of Complex Systems, Complex Network Theory and Electrical Power Systems Theory, and then positions this dissertation within these re-lated fields.

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This section explains the fundamentals of Electrical Power System Theory (Subsection2.1.1), and basics of Complex Systems and Complex Networks Theory (Subsection.2.1.2).

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According to the National Academy of Engineering, the electric power grid is the greatest engineering achievement of the20t hcentury. Their main motivation is [31]:

"In the 20th century, widespread electrification gave us power for our cities, factories, farms, and homes - and forever changed our lives. Thousands of engineers made it happen, with innovative work in fuel sources, power generating techniques, and transmission grids. From street lights to supercomputers, electric power makes our lives safer, healthier, and more convenient."

The success of electrification resulted in heavy dependency of societies on the electrical power grid. The importance of the electric power for daily life in developed countries is ever increasing. This dependence of societies on electric power raises the need for strict assurance of structural integrity, and safety of electric power grids.

Power transmission systems consist of sub-grids that are interconnected over large geographic areas, and interact with each other. This interconnection of grids provides widespread availability of electricity to users and helps to control the flow of electricity throughout a region based on supply and demand. For example, the US transmission grid consists of three interconnections (Eastern, Western, and Texas interconnections), while in Europe, the grid consists of national transmission grids that are maintained and operated by Transmission System Operator (TSO) of the corresponding country.

Power transmission systems are socio-technical systems that consists of three generic layers: physical, ICT, and social layers. The Physical layer consists of physical assets such as lines, transformers, and substations. This layer accommodates power flow. Supervisory Control and Data Acquisition (SCADA) systems acquire data from physical layer on elec-trical parameters and state of devices. This data is sent to regional control centres for state estimation, and security assessment [32]. This layer is the control layer, or ICT layer. On top of these two layers, the human layer resides, representing the involvement of human factor on the system, e.g. grid operators in control rooms, decision-makers, and end-users of electric power. All of these three layers interact with each other, and comprise a sys-tem of syssys-tems [4]. Securing power grids requires assessment of the robustness of these layers separately, and accounting for the interdependencies of these layers with each other. The scope of this dissertation lies in the robustness of the physical layer. Therefore in the remainder of this dissertation, the power grid refers to the physical layer of power systems. A power grid is composed of three functional parts: generation, transmission, and distri-bution. Power is provided from generation buses to distribution stations through transmis-sion buses that are all interconnected via transmistransmis-sion lines. Figure2.2illustrates a typical power grid layout.

The electric power is generated at various generating plants including hydro-based, coal-based, natural gas-based, nuclear-based, and recently renewable sources such as so-lar, wind, and geothermal sources. Depending on the generating plant, power generation

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ranges from 200 MW in hydroelectric plants up to 1500 MW in nuclear power plants, with a voltage level between 12 to 30 kV (See Figure2.2).

These generating plants are typically built far from densely populated load areas. The generated electric power is shipped to distribution regions through bulk transmission sys-tem. Transmission substations step up the voltage level of generated electric power to the level of extra high-voltage (EHV) transmission systems to enable connection of plants to the transmission system. A typical EHV transmission grid operates between 220-1000 kV: the upper limit depends on the standards and regulations of countries. This voltage level limits power loss due to conductor heating, allowing large amount of bulk power (in the order of hundreds of MW) to be shipped over large distances (in the order of hundreds of km) economically. A high-voltage (HV) transmission grid (typically 110 kV and more) is connected to a EHV transmission grid through transmission substations, and feeds large customers (>5 MW) and the medium voltage (MV) distribution grid. Within distribution regions, distribution substations lower the voltage level from HV transmission level to MV distribution level (typically 50 kV). The MV distribution grid feeds medium sized cus-tomers (between 100 kW and 5 MW) and the low voltage (LV) distribution grid, that in turn feeds small customers (<150 kW). From a distribution substation, power is transmitted over low voltage distribution lines towards service locations, where it is again stepped down to the service level required for end users, e.g. residential customers [33,34].

DYNAMICS ANDOPERATION

The complex electric power has two components: real and reactive power. The real power is directly exploited by customers, while reactive power is used to create a magnetic field that is needed to operate machines and devices. The real power mainly affects the phase angles of complex voltage at bus bars, while the reactive power relates to the magnitude of the complex voltage at bus bars [34].

A power grid must be operated to keep the voltage level at each bus bar within certain levels. During normal operation of power grids, the voltage magnitude at a bus bar resides between 95-105% of the rated values. This voltage magnitude relates to reactive power injections/withdrawals, and related flows over the network. To regulate these voltage mag-nitudes to desired levels, a TSO adjusts various system devices such as generators, static VAR compensator, or shunt reactors [35].

The real power injection must equal the withdrawal and lost power in a power grid at any moment in time in the system. This perfect balance between supply and demand is re-flected by a fixed frequency level in the system; 50 Hz for Europe and 60 Hz in US. When there is a mismatch between supply and demand as a result of e.g. sudden disconnection of load, the frequency of the system deviates from the nominal level. Subsequently, additional components potentially trip off for protection purposes, that in turn results in cascading fail-ures. To cope with frequency deviation, a TSO changes e.g. the set point of the generators in the control area to compensate frequency deviation from the nominal level [34] .

In addition to voltage level and frequency constraints, rated limits of transmission lines have to be respected. Power flow through transmission lines all over the grid must re-spect the rated limits of these lines. Otherwise, for example, when a transmission line is overloaded for a period of time, the temperature of the medium increases, and melts the transmission line, permanently damaging it.

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The constraint regarding the balance between the supply and generation is called equal-ity constraint. The constraints on voltage, frequency, line flow limits, and transient stabilequal-ity are called inequality constraints. The equality constraint is crucial for system operation, while inequality constraints define a feasible operation region for a power grid. TSOs de-ploy various control measures to operate their grids within this feasible region in which all constraints are respected.

From a security point of view, a power grid can be operated in three different states; normal state, alert state, and emergency state.

During normal operation, a power grid operates in a feasible region defined by equality and inequality constraints. This feasible region assures a certain level margin for security that is defined by e.g. generation spinning reserve, and transmission capacity reserve.

A power grid is driven from a normal state to an alert state, if the security margin reduces due to e.g. a disturbance, or heavy utilization. In an alert state, a power grid operates without violating any of the constraints. However, as the system might go into an emergency state e.g. as a result of an additional disturbance, proactive measures are taken in an alert state to assure a certain security margin again, i.e. bringing the system back to the normal state. Generation shifting and reserve increase are typical preventive measures taken by grid operators to take the system back from an alert state to a normal state.

Failure of preventive measures in an alert state, or severe disturbances in a normal state potentially drive the system to an emergency state. In an emergency state, a system is still in balance, and synchronized. However, one or more of a system’s inequality constraints are violated which ultimately initiates a cascade of failure and subsequently results in uncon-trolled disintegration of the system. To avoid disintegration of the grid, emergency control actions must be taken to return a system into a normal or an alert state. These actions in-clude cutting of faults, generation re-routing, generation tripping, intentional islanding, and load shedding.

2.1.2.

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S

Researchers in [37,38] define a complex system as "a large group of relatively simple com-ponents with no central control and where organization and emergent non trivial behaviour are exhibited". Although there is no consensus on an official definition of a complex sys-tem, most of the definitions share the common ground that (i) a complex system has a large number of constituents with relatively simple individual behaviour, (ii) these constituents interact with each other at different level of granularity, and (iii) as a result non-trivial sys-tem behaviour emerges. For instance, in Cognitive Science, neurons are the individuals whose local behaviour can be understood, while the intelligence is the emergent behaviour that arises as a result of interaction between neurons, and it can not be explained by be-haviour of individual neurons. In complex systems, simple interactions between the indi-vidual components give rise to unpredicted complex behaviours. These typical emergent features include cascading failures and phase transitions [39].

Cascading failures are a set of successive failures in a complex system. Cascades are triggered usually by a single failure and spreads to the remainder of the system due to inter-actions between the components in the system. Cascades can be observed in many systems including power grids [2,40], computer networks [41,42], financial systems [43], and hu-man bodily systems [44,45]. Phase transitions in a complex system refer to the strong

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Page 1 of 2 file:///Users/ykoc/Desktop/Grid.svg

Figure 2.2: A typical electric power grid layout [36].

qualitative changes in the macroscopic features of a system, due to tuning of a suitable con-trol parameter [46]. Phase transitions are a common feature of complex systems, and occur for example for network synchronisation [47–49] epidemic processes [50,51], and power grids [6,52,53].

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Modelling complex systems and investigating/analysing these emergent features enable interdisciplinary inspiration to solve problems over different disciplines. Complex systems can be modelled by various methods including differential equations, agent-based mod-elling techniques, and complex networks.

Deploying differential equations is one way to describe the behaviour of complex dy-namical systems. The flexibility offered by differential equations enables detailed anal-ysis of a part of the complexity [33]. With the advancements in computational science, simulation-based modelling approaches became an alternative to model complex systems. Cellular automataand its more sophisticated derivative agent-based models [54] illustrate clearly how unexpected behaviour in a complex system emerges when a certain threshold of a system control parameter is crossed. The underlying reason for these simulation-based techniques is the ability to account for the heterogeneity of the constituents of the sys-tem [33].

Another way of modelling complex systems behaviour is to model the interactions between the components in the system. Differential equations and simulation-based ap-proaches successfully model complex systems that are composed of many identical or non identical elements interacting through mainly local interactions. However, modelling sys-tems with many non-identical elements that have diverse and multi-level interactions (local and non-local) is challenging [33]. On the other hand, the ever increasing availability of large databases and the advances in computational capability for storage and manipulation of data enabled modelling large systems as networks containing thousands, millions, even billions of components, making Complex Networks Theory a promising approach to model complex systems [33,55].

COMPLEXNETWORKSTHEORY

Complex Networks Theory builds on graph theory: a graph is a mathematical structure modelling pairwise relations between components in a system. A graph consists of nodes and edges that connect them. A complex network, in turn, is a graph that has non-trivial topological features: features that do not occur in simple theoretical networks (such as lattices or random graphs [56]), but often occur in the representations of real-world systems, e.g. a heavy tail in the degree distribution [56].

The first application of graph theory on a real-world problem dates back to 1735, when Leonhard Euler solved the famous problem of the seven Konigsberg Bridges. The emer-gence of Complex Networks Theory from graph theory has three milestones: the introduc-tion of random network, small-world network, and scale-free network models.

Random Network model Paul Erdös and Alfred Rényi formulate the theory of random network model [57]. A random network consists of N nodes that are connected to each other with a link probability of l, forming a randomly generated topology. Key characteristics of resulted topologies are small average shortest path length, small clustering coefficient, and binomial degree distribution (that turns into Poisson degree distribution for infinitely large number of nodes) [56]. The most remarkable feature of random networks though is the existence of a threshold value for link probability l after which suddenly meaningful topological motives (e.g. giant component) appear. For example, an l that is larger than a certain threshold,lc∼(lnN)/N, results in a connected random graph [58].

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Random graphs contributed to theoretical progress significantly, yet, they are rarely deployed in large-scale to model real-world systems as they cannot capture structural char-acteristics of real-world systems such as highly skewed degree distribution, and relatively large clustering coefficient [59].

Small-world network model Watts and Strogatz [60] categorize real-world networks based on two main characteristics: average shortest path length and the clustering coef-ficient. They conclude that, despite its low average shortest path length, a random network model has a low clustering coefficient failing to capture structural characteristics of real-world systems. The small-real-world network model [60] proposed in 1998 accounts for the deficiencies of random networks (i.e. low clustering coefficient) to capture real-world sys-tem characteristics.

A small-world network model is developed through a random rewiring process, illus-trated in Fig2.3. Through a single parameter, the small-world network model interpolates between a regular lattice network (that has a large clustering coefficient but also a large av-erage shortest path length) and a random graph (that has a small avav-erage shortest path length and a small clustering coefficient). To find a middle ground between these two models, a randomness parameter p is proposed: for p=1 the network turns into a random graph while for p=0 it is a regular lattice network (See Fig.2.3). When considering a regular lattice, upon increasing p value, a set of long-range links appear that form short-cuts throughout the network. The addition of only a small number of these long-range links transforms a regular graph, in which the diameter is proportional to the size of the network, into a small-worldnetwork in which the average path length is proportional to the logarithm of the size of the network, while the clustering coefficient remains large [60].

Figure 2.3: Random rewiring procedure for interpolating between a regular ring lattice and a random network [60].

The small-world network model accounts for the deficiency of random networks to cap-ture structural characteristics of real-world networks in terms of its clustering coefficient: the small-world network model has (as opposed to the random network model) a relatively large clustering coefficient and a relatively small average shortest path length.

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Nevertheless, the degree distribution in small-world network model follows a binomial degree distribution (or a Poisson distribution for infinitely large number of nodes) [59], just as random networks. In a network, node degree distribution is characterized with a distribution function P (k), that gives the probability that a randomly selected node has exactly the degree ofk. For example, in a regular lattice, as all of the nodes have the same degree, the distribution function contains a single spike atk. Any randomness introduced in regular lattice broadens the shape of the peak. Hence, in a random network case, the spike transforms into a bell shaped Poisson distribution that falls off exponentially from its peak value. Because of this exponential characteristic, networks with a Poisson degree distribution are also called exponential networks. This exponential fall-off from the peak value (i.e. k) excludes the existence of a node with a very large degree compared tok: almost all of the nodes in the network have around the same degree with a certain level of variance.

However, the degree distribution of real-world systems significantly deviates from a Poisson distribution. For many of them, their degree distribution can be fitted to a power law distribution, in which the distribution function falls off gradually compared to an expo-nential one, allowing for a few nodes of very large degree to exist [56].

Scale-free network model As a subclass of small-world networks, the scale-free network model [61,62] differs from the small-world network model in degree distribution. Just as many real-world systems, the degree distribution of a scale-free network model has a heavy tail in its degree distribution, following a power law distribution [58].

A power law degree distribution falls off gradually compared to an exponential one. This allows the existence of a small set of nodes with a degree that is an order of magnitude largerthan the average degree, while a very big portion of the nodes have a degree that is an order of magnitude smaller than the average degree of the network. This causes an infinitely large variance in the typical degree of a node: no characteristic scale to define this degree distribution, therefore the term scale-free [33]. Figure2.4illustrates the difference in the shape of Poisson and Power law degree distributions.

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The heavy tail in the node degree distribution of scale-free networks gives rise to the existence of hub nodes. These are nodes with a much larger degree compared to other nodes. Hub nodes correspond to critical, important, or popular nodes (depending on the type of the system that is modelled). Identifying these nodes is crucial e.g. to control the epidemics in social/computer networks [51], or to increase the robustness of power grids with respect to cascading failures [63].

The scale-free network model explains the existence of the power law degree distribu-tion in real-world systems by the mechanisms of growth and preferential attachment [61]. Differing from the e.g. the random network model, the scale-free network model accounts for network growth over time as it happens in real-world systems. For example, the world wide web had only one page in 1990, now billions of them exist, or internet had only a few routers three decades ago, but now millions of them are connected to the system [62].

When a real-world network grows as a result of new nodes, a new node tends to connect to the more connected nodes (i.e. nodes with a relatively larger degree), and accordingly, these connected nodes in the system acquire more links over time than their less connected neighbours. This preferential attachment of these new nodes to the existing system results in rich-gets-richer process, that in turn, creates the hub nodes in the system [61,62].

Possessing the power law degree distribution, scale-free network models capture the key characteristics of many real-world systems including the World Wide Web, the Inter-net, infrastructural networks, networks of airline city connections, scientific collaboration networks, and cellular networks [64].

COMPLEX NETWORKS AS A TOOL TO MODEL POWER GRIDS

The electric power grid has grown into one of the most complex technological networks. The highly interconnected structure of power grid enables it to deliver power over huge distances. Yet, it also propagates local failures into the global network causing system-wide failures. The robustness behaviour emerges as a consequence of the collective dynamics of the complex power grid. The interdependencies between a large number of components in a power grid govern the dynamics of the system ranging from power delivery to spread of the faults in the system. Therefore, understanding and analysing the robustness of a power grid require assessment from the point of view of the system level and from the perspective of the global network [55,65].

A global analysis of a large-scale power grid is a challenge for traditional approaches that rely purely on power flow based analysis (e.g. N-x contingency analysis [32]) due to the complexity and extremely large amount of computational time [66]. The complex nature of power grids and its networked structure however makes it possible to model it as a Complex System and to analyse and better understand interdependencies between components by adopting a Complex Networks approach [4,60,61,67,68] that has shown its promising po-tential to model and analyse power networks at the system level. Such an approach captures interdependencies between components and the collective emergent behaviour of complex power grids rather than the detailed behaviour of a system at the individual components level.

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2.2.

R

OBUSTNESS OF

P

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RESEARCH POSITION

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Robustness of power grids is a complex notion composed of multiple dimensions. The op-erative state (e.g. loading level and power flow distribution), and structural aspects (e.g. type of buses and their interconnection) of a power grid, together with the design choices of engineering systems (e.g. the protection systems, automatic controls, and towers and in-sulators) govern the robustness in a power grid. Accordingly, enhancing the robustness of a power grid requires careful assessment of engineering system design choices and optimiza-tion of the operative state and the topology of the grid. The operative state of a power grid continuously changes while the topology remains mainly unchanged. Optimization of the operative state of a power grid is a short-term optimization problem and requires a dynamic optimization approach [63,69,70]. As opposed to the operative state, optimization of the topology is a long-term optimization problem and requires investigating the impact of the grid topology on the robustness of power grids.

This dissertation proposes a set of measures to assess the robustness of a power grid. A distinction between distribution and transmission grids is made due to differences between these systems. A power transmission system ships bulk electric power over large distances, while a power distribution grid is designed to deliver power over relatively smaller distances to end customers. This difference in functionalities causes these two systems to be different both with respect to the topology and the operation. For instance, a power transmission grid has a meshed structure, unlikely a power distribution grid that has a radial structure.

This difference between the transmission and distribution grids affect also the type and level of interdependencies between the components in these systems. Accordingly, the ro-bustness of these systems has different meaning for each of these systems. Transmission grids have topological redundancy due to their meshed structure. Therefore, on the trans-mission side, the loss of one single component does not result in topological disconnection. However, a transmission grid is typically loaded more extensively compared to a distribu-tion grid, limiting the ability of components in a transmission grid to afford excess power load. Accordingly, a failure of a component in a transmission grid might trigger successive failures in the form of line overloads based on capacity constraints, voltage and frequency level instabilities, and hidden failures in the protection devices, and the cumulative effect might result in the disintegration and impairment of the grid.

On the other hand, in the case of the distribution grid, capacity constraint is not the main reason for failure spread. However, because of its typical radial-like structure, the loss of one single component might potentially result in topological disconnection of cer-tain geographical region of the distribution grid, disrupting the supply availability for the corresponding part of the system.

Because of the difference in the nature of transmission and distribution grids, this dis-sertation considers the robustness from various aspects for these different layers of power grids. When considering the power transmission grids, the robustness of the system with respect to cascading line overloads is considered. On the other hand, when focussing on power distribution grids, a more topological approach is deployed and the aptitude of the system for supply availability is assessed.

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the major application areas of these metrics lie in power grid operation (e.g. dynamic monitoring), and strategic mid-term (e.g. asset management) and long-term (e.g. network expansion) planning of power grids. Therefore, this section positions proposed metrics in the context of short-term and long-term interventions to a power grid for the purpose of robustness enhancement.

The decompositions of the impact of operative state and topology on the robustness of power grids implies that actions for enhancing the robustness of power grids can be catego-rized into (i) short-term interventions (to operative state) and (ii) long-term intervention (to topology) to the system.

From a robustness point of view, interventions to the grid can be classified based on granularity level and response time. Granularity level refers to the level at which the system is intervened, while the response time gives the time frame in which the intervention is performed. Figure2.5positions short-term and long-term interventions in a perspective based on granularity level and response time. Short-term interventions relate to power grid operation. In the current grid, short-term interventions are performed at the component level during (near) run-time (i.e. dynamically), e.g. power shifting at the interconnections between national transmission grids [71]. This requires adjustments of operative state e.g. to bring the system from an alert state to a normal state. On the other hand, long-term interventions are strategic decisions that are performed at the network level in a broader time horizon (i.e. static analysis).

Loc ally' Syste m ' Gra nu la rit y'Level ' Dynamic' Sta4c' Response'Time' Short&term) interven-on) measures) Long&term) interven-on) measures)

Figure 2.5: Positioning of short-term and long-term intervention measures from reaction time and granularity level point of view.

2.2.1.

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I

The short-term interventions are a set of active and proactive measures that are taken in (near) real-time to enhance the robustness of the system. Proactive measures are preventive, and are intended to keep a system in a normal state or to bring it back from an alert to a normal state. Active measures are taken to deal with a contingency on-the-fly, e.g. emergent control actions such as controlled islanding [72], or load shedding [21,73,74].

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In current practice, grid operators asses the security of a power grid based on flow based simulations, i.e. N-x contingency analysis [32]. Given a new loading and accordingly a new generation profile, grid operators simulate the system using either Alternating Current (AC) or Direct Current (DC) load flow analysis [32] to estimate the state of the network. Subsequently, particular parts of the network are disabled to evaluate the impact of these outages on the system, i.e. whether the outage affects consumers, the grid remains stable, or any other component in the system overloads. This analysis helps a grid operator to prepare a roadmap to define what to do under unexpected contingencies. This is to avoid equipment damage, and mitigate and minimize the customer outages.

Transmission grid operators operate their grids according to the N-1 criterion that is compulsory for TSOs; the grid is able to afford the outage of any single component [75]. Although N-2 contingency analysis is still possible from computational time point of view, evaluating scenarios where more than three components fail at the same time is almost impossible due to the complexity of the simulation models. However, a set of tens, or even hundreds, of outages do occasionally occur and result in very large blackouts [76]. Hence, achieving a higher level of robustness requires additional complementary measures to traditional flow based assessment techniques.

The set of proactive short-term intervention measures includes the dynamic adjustment of the (a) topology and the (b) operative state of a power grid. Dynamic adjustment to the topology can be achieved by means of techniques such as intentional islanding [72]. Such adjustment to the topology splits a power grid into controllable parts (or islands) with its own independent generation. Also load shedding is performed to assure a balance between generation and demand in these sub systems. Intentional islanding helps to isolate a fault preventing its spread to the rest of the system.

The on-line adjustments to the operative state is performed in terms of power shifting/re-routingby using power flow controlling devices, e.g. Phase Shifting Transformers (PTS), Flexible AC Transmission System (FACTS) devices, and High Voltage DC installations [77]. Power re-routing through power controlling devices offer promising opportunities. For ex-ample, the transmission capacity of a grid is increased without installing any additional transmission line. This is achieved because of the ability to balance the power flow through-out the network, making it possible to better utilize the existing assets withthrough-out violating the N-1 criteria. Moreover, this flexibility to control power flow makes it possible to intervene a component that is overloaded, avoiding trip and over- ageing of that component.

The potential of the power re-routing concept attracts many researchers [78–83] for further research to mature, and to exploit the concept. Many researchers investigate the possibility of active management of power flow in power grids: Nguyen et al. [79] propose a method to manage the active power relying on multi-actor systems perspective, while the research in [81] introduces an architecture for a smart router which is envisioned to enable power routing in a faster and more efficient way. Other studies focus on deployment of FACTS devices within the system: Lima et al. and Ayman et al. [82,83] focus on optimal placement of FACTS devices in large-scale, while [71,80,84] investigate techniques for coordination of these devices to profit from them at a maximum level.

Moreover, Bajracharya et al. [78,85] investigate the value of the power re-routing con-cept to avoid over-ageing of assets. For example, transformers age at different rates based on various aspects including loading profile [86]. Researchers in [78,85] deploy a model

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to relate the ageing of distribution transformers to their loading profiles. They propose a framework to determine the optimal loading of these components, and subsequently opti-mise the actual loading of these components to maximize the life-cycle of these expensive assets. They demonstrate that adjusting the actual loading of these assets to the optimal loading level, through power re-routing, increases the life- cycle of the transformers.

The high cost and coordination problems limit deployment of power flow controlling devices on a large-scale [87]. Currently, these devices are placed only at critical compo-nents in the system, e.g. at the interconnections between national transmission grids [71]. Consequently, in the current regime, dynamic adjustments to the operative state can only be performed at a limited number of places in power grids. The Smart Grid, on the other hand, enables more sophisticated network flow management techniques [70]. The large-scale integration of advanced devices such as storage systems, FACTS, and smart sensors makes it possible to collect a large amount of system data, to process this data in an efficient way, and to act automatically enabling network flow management at the system level. This envisioned system requires coordination mechanisms to collect and interpret system data in a meaningful way e.g. to re-route power flow and to charge and de-charge storage systems at multiple locations, at the same time.

Pournaras et al. [70] propose a management system at grid level to assure a desired level of robustness of a power grid. The proposed system requires deployment of active devices such as FACTS, or thyristor controlled devices for active power management, and smart relays for dynamic adjustments to the systems. Researchers in [88,89] propose a wide-area, intelligent, adaptive protection and control system that empowers future grids by providing critical and extensive information in real- time, assessing system vulnerability quickly, and performing timely self-healing and adaptive reconfiguration actions based on system-wide analysis.

Network flow management at the system level enhances the overall robustness of the system by avoiding critical situations with fast control actions before contingency situa-tions. This prevents systems going from a normal state to an alert or an emergent state. Such network flow management also enables a better utilization of all the assets and overall capacity in the system.

Power re-routing at component and system level is significantly beneficial to a grid e.g. to enhance the system robustness, or transmission capacity. However, for the purpose of power re-routing, two questions are of crucial importance to answer: (i) when to intervene in the system?and (ii) how and where to intervene in the system?

The former question translates into: how to detect that the robustness of the system is critically low, so that preventive actions should be taken. Dynamic monitoring of the ro-bustness of power grids is key to answer this question. The continuously changing nature of the operative state of a power grid makes the robustness of a power grid also dynamic. Therefore, monitoring and measuring the robustness of a grid requires capturing the im-pact of operative state on the robustness. Although numerous researchers [90–95] propose metrics for measuring the robustness of power grids, a significant part of them proposes (extended) topological metrics measuring only the impact of the topology on the robust-ness. Some others [96,97] design metrics relying on flow based models, that capture the impact of the operative state on the robustness. However, the required time to compute these metrics makes it impossible to deploy them for monitoring the robustness of the grid

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during run-time.

The latter question relates to measures based on which the network will be intervened: having detected a critically low robustness level of the system, for which component(s) and how to do power routing to enhance system robustness. In current practise, power is re-routed when certain components in the system are overloaded to relax these components. This is a local adjustment to fix a problem in one single component. However, as pinpointed also by other researchers [79,87,98], due to the strong interdependencies between compo-nents in a power grid, the actions to solve a local problem may result in another problem in another part of the system, and evolve to a system-wide problem. For example, in case of overloading, shifting load of a transmission line to the remainder of a system without assess-ing its impact on the rest of the system might overload other lines in the system droppassess-ing the overall robustness of the system. Performing this at multiple components in the system potentially makes this effect worse. Hence, a preventive action to enhance the robustness further decreases the robustness of the system. Therefore, assessing the subsequent impacts of the preventive actions on system robustness is of crucial importance.

This requires metrics that are able to monitor the robustness of the system in real-time. Such a metric makes it possible to assess the impact of power re-routing, and to evalu-ate whether system robustness is increased or decreased. On the one hand, such metrics should assess the impact of the power flow distribution on the robustness of the system. On the other hand, because all of the real-time assessment, such metrics should be compu-tationally simple. These prerequisites impose various requirements to these metrics such as their computation should not involve any system simulation (like flow based methods), and they should be computable in parallel or in a distributed manner for the purpose of fast computation.

Chapter3of this dissertation designs a metric for the purpose of dynamic monitoring of power grid robustness. The proposed metric accounts for the impact of the operative state (system loading level and power flow distribution) and the topology on the robustness of a power grid. The proposed metric can be computed during run-time as it does not require system simulation. Moreover, it can be computed in a distributed fashion making it possible to compute the metric in parallel. Figure 2.6illustrates the positioning of the proposed metric in the examples of existing work in the literature. Metrics are classified based on (i) type of computational analysis that is required to compute the metrics, and (ii) aspects that these metrics account for the robustness of the system. Static analysis implies off-line computation of a metric, while run-time analysis corresponds to the dynamical computation of a metric.

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The long-term intervention is a set of strategic long-term decisions/investments to the grid for the purpose of enhancing its robustness. Long-term interventions to a power grid are critical and usually costly. Therefore, decisions on these plans require careful assessment from various perspectives. The long term intervention to a grid can be in a form of (i) asset managementand (ii) network expansion.

Asset management Over-ageing of an asset results in deterioration of the condition of the asset. The result is reflected in the failure probability rate of the asset that decreases the