What have the authors contributed in "Topological vulnerability of the european power grid under errors and attacks" ?

The authors present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity ( UCTE ) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. The authors conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order facilitate and improve the power grid design and functioning. The authors have found that the node removal behaviour can be logarithmically related to the power grid size. This logarithmic behaviour would suggest that, though size favours fragility, growth can reduce it.

What is the way to prevent disturbances?

One possible way to prevent propagation of disturbances is to design the system to allow for intentional separation into stable islands or interrupt small amounts of load [Madani & Novosel, 2005].

What is the topology of the UCTE?

At the top UCTE level, the need to exchange energy between countries forces long distance connections to cross borders and to connect sites never connected before.

What is the function that fits the results of Fig. 6(b)?

In the case of random failure, the results are very well correlated (r2 = 0.99) by the logarithmic function rrr N ln , with 95.0r and 34.0r , where is the exponent of the exponential function that fits the results of Fig. 6(b) and N is the size of the network.

What is the definition of a geographical stable island?

From a spatial point of view, the definition of a geographical stable island would facilitate and improve the treatment of several different aspects related to power grid design and functioning, ranging from deregulation to spatial load forecasting and maintenance.

What is the formalism of graph theory?

Using the formalism of graph theory, any of these networks can be described in terms of an graph , defined as a pair, EW , , where iwW , Ni ,...,1 is the set of N nodes and ji wwE , is the set of edges or connections between nodes.

what is the critical fraction for a giant component in a graph?

Specifically for the static tolerance to errors, it has been shown [Molloy & Reed, 1998] that the condition for having a giant component S∞ in a graph is 0222 k kPkkkk (10)For randomly deleted nodes, it has been shown [Cohen et al., 2000] that the critical fraction fc is 1 11 2 kk fc (11)Considering the exponential degree distribution of the European power grid (Eq. 1), the authors havek and 22 2k , and thus12 11 c f (12)For 8.1 , the authors have a predicted value 61.0cf .

What is the clustering coefficient for a node?

On one hand, being 1 iji j the set of nearest neighbors of a node Wwi , the clustering coefficient for this node is defined as the number of connections between the components ijw .

What is the value of the exponent that better fits Eq. (6)?

8. As the authors can see, as the size of the network increases, the value of the exponent that better fits Eq. (6), increases at the same time.

What is the relationship between the size of the network and the number of elements removed?

Quite intuitively, as more and more elements are introduced in the network, more prone is the system to failures, whether they come from selective or random removal, and its fragility increases as well.

What is the relationship between the fS exp function and the size of the network?

In spite of this, these behaviours (Fig. 6, b) seem to correlate well with an exponential function of the general form fS exp (13)where S is the relative size of the connected component and f is the fraction of nodes removed.

What is the definition of a stable island?

If grid’s resilience to attacks and failures is somehow related to its size and dimensions, an accurate power grid reliability analysis would have to take into account its relative increase in vulnerability in order to finally give a minimal definition of this stable island.

(Open Access) Topological vulnerability of the european power grid under errors and attacks (2007) | Martí Rosas-Casals

Topological vulnerability of the European power grid

under errors and attacks

Martí Rosas-Casals

1,2

, Sergi Valverde

and Ricard V. Solé

2,3

Càtedra UNESCO de Sostenibilitat, Universitat Politècnica de Catalunya, EUETIT-Campus Terrassa, 08222

Barcelona, Spain

ICREA-Complex Systems Lab., Universitat Pompeu Fabra, Dr. Aiguader 80, 08003 Barcelona, Spain

Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA

Abstract

We present an analysis of the topological structure and static tolerance to errors and attacks of

the September 2003 actualization of the Union for the Coordination of Transport of Electricity

(UCTE) power grid, involving thirty-three different networks. Though every power grid studied

has exponential degree distribution and most of them lack typical small-world topology, they

display patterns of reaction to node loss similar to those observed in scale-free networks. We

have found that the node removal behaviour can be logarithmically related to the power grid

size. This logarithmic behaviour would suggest that, though size favours fragility, growth can

reduce it. We conclude that, with the ever-growing demand for power and reliability, actual

planning strategies to increase transmission systems would have to take into account this

relative increase in vulnerability with size, in order facilitate and improve the power grid design

and functioning.

1 Introduction

Mostly evolved over the last hundred and fifty years, technical infrastructures, from telegraph

[Standage, 1998] to Internet [Pastor Satorras & Vespignani, 2004], are the canvas where almost

every aspect of our economy and society is portrayed. From a broader historical perspective,

networks of energy, transportation and communication constitute the very foundation of all

prospering societies, as the western culture actually knows them. Being usually managed by

different kinds of actors (often with different objectives), formed by a huge quantity of

heterogeneous components (spatially distributed and connected) characterized by complex

interdependencies and relations, the study of these technological systems deserves attention in

order to assure, essentially, structural integrity, efficiency and reliable supply.

In recent years, one particular kind of network has received much attention: the power grid.

Hailed by the US National Academy of Engineers as the 20

century’s engineering innovation

most beneficial to our civilization, the role of the electric power has grown steadily in both

scope and importance during this time and electricity is recognized as a key to societal progress

throughout the world, driving economy prosperity, security and improving the quality of life

[Willis, 2004]. With similar pace, though, increasing frequency and size of malfunctions have

raised general awareness about our real level of comprehension of these networks. In recent

years, both the North American and the (once almost faultless) European grid systems have

experienced numerous examples of such malfunctions in the form of cascading failures and

blackouts [Venkatasubramanian, 2003; UCTE, September, 2003]. The explanations given by

local, national and international electricity coordinating councils for most of these situations go

from aspects related to low investment and maintenance, to those related to generation and

demand inadequateness and, obviously, bad luck. But more than any, the most repeated

explanation is that of a bad comprehension of the interdependencies present in the network.

[Watts, 2003; UCTE, 2004]

In these sense, advances in statistical physics, modeling and computational methods have

stimulated the interest of the scientific community to study electric power grids as complex

networks. In complex network theory, one type of analysis of such interdependencies already

mentioned is usually done under the robustness (or, in the contrary, vulnerability) epigraph

[Boccaletti et al., 2006]. It refers to the ability of a network to avoid malfunctioning when a

fraction of its constitutive elements is damaged. In technical infrastructures, this turns to be a

field of elementary practical reasons since it affects directly the efficiency of the processes

taking place in the network and it can give hints about the resilience of the grid. The analysis of

the robustness of a complex system has been done, traditionally, from two points of view: static

and dynamic. In a static robustness analysis, nodes are deleted without the need of redistributing

any quantity transported by the network [Albert et al., 2000; Crucitti et al., 2003]. In a dynamic

robustness analysis, nodes are deleted and the flow or load carried by them must be distributed

over the rest of the remaining network [Moreno et al., 2002; Motter & Lai, 2002; Crucitti et al.,

2003; Kinney et al., 2005]. At first glance, the theoretical approach to these two types of

robustness seems quite similar, but while static one can be analytically treated, dynamic one

must be, almost always, numerically solved.

In this letter, the static robustness of the European and most of the European countries and

regions electricity transport power grids are investigated. Their tolerance to random loss

(failures) and selective removal (attacks) of the most connected nodes is analyzed. In order to

simplify its topological representation, a simple model of the power grid data is introduced.

Final results and some features worth to notice are discussed in the last part of the letter.

2 European power grid data

In this paper, the vulnerability of the September 2003 actualization of the Union for the Co-

ordination of Transmission of Electricity (UCTE) map has been analyzed.

UCTE associates

most of the continental Europe national power grid operators in order to coordinate the

production and demand of some annual 2,300 TWh and 450 million customers from 24

countries. The map gives data from the transmission network (voltage levels from 110 kV to

400 kV) and ignores the much more extended distribution one. Nonetheless, it deals with more

than 3,000 nodes (generators and substations) and some 200.000 km of transmission lines.

For more than fifty years UCTE has coordinated the international operation of high-voltage

European countries’ grids to ensure adequate balances between offer and demand through

national frontiers. It operates one of the largest electric synchronous interconnections worldwide

in order to optimize the use of installed capacities and reduce the economic cost of power

outages. But more than this, the UCTE transmission network has been shaped by those national

policies and decisions that, for the last one hundred years, have been seeking economic

prosperity, security and quality of life of its inhabitants. From that point of view, and differently

from previous examples considered in the literature [Watts, 1999; Albert et al., 2004], those

different power grids should be a good example of network evolution directed, at the same time,

by technical, economical, political and, lastly, environmental decisions. Differentiated from

country to country, we then would expect to find somehow different patterns and complex

behaviour for every country or territory considered.

Fig. 1 An extremely simplified model for the transmission power grid: two voltage levels (400 kV and 220 kV for the

European network) with generators G and loads L connected by switching stations S. Transformers T connect both

tension levels in order to provide reliability, efficiency and control capacity.

In order to simplify the analysis of the structure of the European power grid, an idealized view

has been adopted (Fig. 1). On one hand, transmission lines have been assumed bidirectional, as

it should be in the electricity transport network, and identical, ignoring the voltage level

variation between lines and other physical characteristics. Although we have different voltage

levels, the transport network works as a whole, using transformers to increase or decrease

voltage depending on time and space requirements, and it would not be suitable or realistic to

split it into different voltage networks, as it has been done in some literature [Crucitti et al.,

2005]. On the other hand, although it is possible to distinguish four different kinds of elements,

namely generators, transformers, switches (considered as stations or substations of any kind)

and, finally, end line points, all these elements have been treated identically in order to avoid, at

this initial point of the study, those difficulties involved in their differentiation and dynamical

behaviour characterization.

http://www.ucte.org

400 kV

220 kV

Lower voltage levels

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Fig. 2 The European electricity transport network (September 2003 actualization of the Union for the Co-ordination

of Transmission of Electricity, UCTE, map) offers the upper topological image, where more than 3,000 nodes act as

stations, substations, transformers and generators, connected by some 200,000 km of high voltage lines (up to 4,300

edges approximately). A closer look gives a more accurate perspective of some national power grids: United

Kingdom and Ireland, bottom left, and Italy, bottom right. Size, and colour as well, indicate the degree of every node:

1 to 2 links, yellow; 3 to 4 links, purple; 5 to 6 links, green; and 7 to 8 links, red.

Bearing these assumptions in mind, five different data sets have been analyzed:

 UCTE as a whole.

 UCTE, United Kingdom and Ireland as a whole.

 UCTE, country by country, plus United Kingdom and Ireland.

 Geographically related regions (Iberian Peninsula, Ireland as an island and England as

an island).

 Traditionally united or separated regions (formers Yugoslavia, Czechoslovakia and

Federal and Democratic Republics of Germany).

Until this time, and as far as we know, no such analysis has been done for the European power

grid and with such depth of detail. A thorough analysis of these data sets will surely give hints

of historical and geographical constraints that might have shaped the structure of the power grid

from country to country, and from time to time. For example, from a geographical point of view,

although neither United Kingdom nor Ireland belong to the UCTE, their isolated geography

might have strongly configured and constrained their national power grids. Similarly, although

Germany is actually united, the former frontier between Federal and Democratic Republics is

still “visible” in form of a very few transmission lines connecting the east and the west of

Germany.

3 Small-world feature of the power grid

The different data sets have been obtained after introducing their topological values, i.e.

geographical positions of stations, substations and longitudes of lines, in a geographical

information system (GIS) (Fig. 2). The national power grid for every country has been obtained

from a typical GIS query: the selection of the part of the UCTE’s network constrained by every

country’s frontier. So far, data analyses of 33 different networks have been performed.

Using the formalism of graph theory, any of these networks can be described in terms of an

graph

 , defined as a pair,



EW ,

, where











Ni ,...,1



is the set of N nodes and





wwE , is the set of edges or connections between nodes. Here,





jiji

ww ,





indicates

that there is an edge (and thus a link) between nodes w

and w

. Two connected nodes are called

adjacent, and the degree k of a given node is the number of edges connecting it with other nodes.

In this case, the UCTE graph,

UCTE

 , is defined as



iUCTE

1



(1)

where



,...,1

are the set of national power grids analyzed.

As well as k, an additional property to be considered is the degree distribution



kP . This is

defined as the (normalized) probability that a node chosen uniformly at random has a degree

or, similarly, as the fraction of nodes in the graph having

k edges. In this sense, it has been

suggested that degree distributions can be classified in three types, namely exponential

(gaussian or random), potential (scale-free) or some mixture of both, exhibiting each one

different dynamic characteristics and adaptive behaviours [Amaral

et al., 2000]. Most of the real

networks degree distributions follow a power law of the form









 kkP with the exponent 

being, mostly, between 2 and 3.

For the five different data sets presented in Section 2, the graph model used considers

undirected and unweighted edges. Though every single network contains hundreds of stations,

Topological vulnerability of the european power grid under errors and attacks

Figures

Citations

Reliability Engineering and System Safety

The Power Grid as a Complex Network: a Survey

The Power Grid as a Complex Network: a Survey

Cyber-physical attacks and defences in the smart grid: a survey

Power grid vulnerability: a complex network approach

References

Collective dynamics of small-world networks

Emergence of Scaling in Random Networks

Complex networks: Structure and dynamics

Error and attack tolerance of complex networks

Classes of small-world networks

Related Papers (5)

Structural vulnerability of the North American power grid

Collective dynamics of small-world networks

Emergence of Scaling in Random Networks

Error and attack tolerance of complex networks

Cascade-based attacks on complex networks.

Frequently Asked Questions (17)

Q1. What have the authors contributed in "Topological vulnerability of the european power grid under errors and attacks" ?

Q2. What is the way to prevent disturbances?

Q3. What is the topology of the UCTE?

Q4. What is the function that fits the results of Fig. 6(b)?

Q5. What is the definition of a geographical stable island?

Q6. What is the formalism of graph theory?

Q7. What is the definition of the term complex network theory?

Q8. what is the critical fraction for a giant component in a graph?

Q9. What are the main reasons for the failures of the power grid?

Q10. What is the power law of the real networks?

Q11. What is the clustering coefficient for a node?

Q12. What is the common explanation for the failures of the grid?

Q13. What is the value of the exponent that better fits Eq. (6)?

Q14. What is the relationship between the size of the network and the number of elements removed?

Q15. What is the meaning of complex network theory?

Q16. What is the relationship between the fS exp function and the size of the network?

Q17. What is the definition of a stable island?