Robustness of the European power grids under intentional attack
Ricard V. Solé,
1,2
Martí Rosas-Casals,
1,3
Bernat Corominas-Murtra,
1
and Sergi Valverde
1
1
ICREA-Complex Systems Lab, Universitat Pompeu Fabra, Dr. Aiguader 80, 08003 Barcelona, Spain
2
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
3
Catedra UNESCO de Sostenibilitat, Universitat Politecnica de Catalunya, EUETIT-Campus Terrassa, 08222 Barcelona, Spain
共Received 28 October 2007; published 7 February 2008
兲
The power grid defines one of the most important technological networks of our times and sustains our
complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well
understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn
into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore
the fragility of the European power grid under the effect of selective node removal. A mean field analysis of
fragility against attacks is presented together with the observed patterns. Deviations from the theoretical
conditions for network percolation 共and fragmentation兲 under attacks are analysed and correlated with non
topological reliability measures.
DOI: 10.1103/PhysRevE.77.026102 PACS number共s兲: 89.75.Fb, 02.50.⫺r, 84.70.⫹p
I. INTRODUCTION
The power grid defines, together with transportation net-
works and the Internet, the most important class of human-
based web. It allows the success of advanced economies
based on electrical power but it also illustrates the limitations
imposed by environmental concerns, together with economic
and demographic growth: The power grid reaches its limits
with an ever growing demand 关1兴. A direct consequence of
this situation is the fragility of this energy infrastructure, as
manifested in terms of sudden blackouts and large scale cas-
cading failures, mostly caused by localized, small scale fail-
ures, ocurring at an increasing frequency 关2,3兴.
The fragility of the power grid is an example of a gener-
alized feature of most complex networks, from the Internet
to the genome 关4–8兴. Specifically, real networks are often
characterized by a considerable resilience against random re-
moval or failure of individual units but experience important
shortcomings when the highly connected elements are the
target of the removal. Such directed attacks have dramatic
structural effects, typically leading to network fragmentation
关9–12兴. This behavior has been studied for skewed power-
law distributions of links, which are found in many small-
world networks 关13,14兴. But recent studies have shown that
similar responses are not unique to small-world, scale-free
networks: Power grids, having less skewed exponential de-
gree distributions and often without small-world topology,
display similar patterns of response to node loss 关15兴.
An additional feature of the power grid is its spatial struc-
ture. The geographic character of this network implies that a
number of constraints are expected to be at work. Other well
known spatially extended nets include the Internet 关16兴,
street networks 关17兴, railroad and subway networks 关18兴, ant
galleries 关19兴, electric circuits 关20兴, or cortical graphs 关21兴.
One fundamental aspect concerning the analysis of com-
plex networks is the increasing evidence of mutual influence
between dynamical behavior and topological structure. The
topology of human contact networks, for example, deter-
mines the emergence of epidemics 关22兴; similarly, the correct
dynamics in cellular networks are rooted in the topology of
the regulatory networks 关23,24兴. Here we present evidence of
a plausible relation between topological and nontopological
reliability measures for the power grid, suggesting that topol-
ogy might be capturing the robustness 共or fragility兲 of the
real system, when dynamics are at work. This evidence has
been obtained analyzing the resilience of 33 different power
grids: 共a兲 The 23 different EU countries, 共b兲 four geographi-
cally related zones 共Iberian Peninsula, Ireland as island, En-
gland as island, and United Kingdom and Ireland as a
whole兲, 共c兲 four traditionally united or separated regions
共former Yugoslavia, Czechoslovaquia and Federal and
Democratic Republics of Germany兲, 共d兲 continental Europe,
and 共e兲 continental Europe plus United Kingdom and Ireland.
The paper is organized as follows. In Sec. II the data set
on European power grids is presented and their basic topo-
logical features summarized. In Sec. III we present both ana-
lytical and numerical estimations of the boundaries for net-
work collapse under attack, using a mean field theoretical
approach. Two classes of networks are shown to be present.
In Sec. IV, evidence for correlation between these two
classes and nontopological reliability indexes is shown to
exist. In Sec. V we summarize our findings and outline their
implications.
II. POWER GRID DATA SETS
Europe’s electricity transport network is nowadays the en-
semble of more than twenty different national power grids
coordinated, at its higher level, by the Union for the Co-
ordination of Transmission Electricity, UCTE 共http://
www.ucte.org兲. The distribution and location of transmission
lines, plants, stations, etc., can be found in the last version
共July 2007兲 of the UCTE Map. The different data sets ana-
lyzed here have been obtained after introducing the topologi-
cal values 共i.e. geographical positions and longitudes兲 of
more than 3000 generators and substations 共nodes兲 and
200 000 km of transmission lines 共edges兲 in a geographical
information system 共GIS兲. The national power grid for every
country or region has been obtained from a typical GIS
query: the selection of the part of the UCTE’s network con-
PHYSICAL REVIEW E 77, 026102 共2008兲
1539-3755/2008/77共2兲/026102共7兲 ©2008 The American Physical Society026102-1
strained by every country’s frontier. The power grid can then
be formally described in terms of a graph ⍀ =共V ,E兲. Here
V=兵
v
i
其 indicates the set of N nodes 共transformers, substa-
tions or generators in our context兲. Figure 1 shows an ex-
ample of such graphs with its geographical 共a兲 and topologi-
cal 共b兲 structures, respectively. These nodes can be
connected, and E =兵e
ij
其 indicates the set of actual links be-
tween pairs of nodes. Specifically, e
ij
=兵
v
i
,
v
j
其 indicates that
energy is being transported between the nodes in the pair
兵
v
i
,
v
j
其. Our system can be analyzed at two main levels: The
whole power grid ⍀
EU
including all countries within the EU
and at the country level. If ⍀
k
indicates the kth power grid of
one of the n =33 countries and regions involved, we have
⍀
EU
=艛
k=1
n
⍀
k
.
The global organization of these webs has been previ-
ously analyzed 关15兴, revealing a very interesting set of com-
mon regularities: 共a兲 Most of them are small worlds 共i.e.,
very short path lengths are typically present兲 and the larger
webs display clustering coefficients much larger than ex-
pected from a random version of the network analysed; 共b兲
they are very sparse, with an average of 具k典=2.8 over all the
webs available 共see Table I兲; 共c兲 the link distribution is ex-
ponential: The probability of having a node linked to k other
nodes is P共k兲=exp共−k /
␥
兲/
␥
关Fig. 1共c兲兴; and 共d兲 these net-
works are weakly or not correlated. This exponential distri-
bution is thus characterized by the constant
␥
which actually
corresponds to the average degree 共i.e., 具k典=
␥
兲.
Correlations were measured using the average nearest
neighbor connectivity of a node with the degree k, i.e., the
average 具k
nn
典=兺
k
⬘
k
⬘
P共k
⬘
兩k兲 where P共k
⬘
兩k兲 is the conditional
probability that a link belonging to a node with connectivity
k points to a node with connectivity k
⬘
关25兴. For these webs,
it was found that 具k
nn
典⬇const, as expected if no correlations
were present. This is a very useful property in our analysis,
since makes mean field predictions valid in spite that we
ignore the planar character of these networks, thus replacing
the geographical pattern by a topological one. Nonetheless,
as these webs are geographically embedded, some care needs
to be taken 共see 关27兴 in connection with epidemic spreading兲.
III. ATTACKS IN EXPONENTIAL NETWORKS:
MEAN FIELD THEORY
In our previous paper, we analyzed the effects of both
random and selective removal of nodes on the EU grids 关15兴.
Nonetheless, in that paper we were mostly interested in the
average behavior of the networks analyzed 共see Fig. 2兲. Here
we want to extend these results to the analysis of the differ-
ences observed in EU power grids with the goal of interpret-
ing the different patterns exhibited compared to the predic-
tions from mean field theory on intentional attacks.
In order to compute the effect of random removal of
nodes, we compute the percolation condition for the graph
assuming it is sparse and uncorrelated. Let f be the fraction
of removed nodes and P共k兲 the link degree distribution of
our graph. The damaged graph will be characterized by the
following degree distribution P共k兲关28兴:
P共k兲 =
兺
i艌k
⬁
冉
i
k
冊
f
i−k
共1−f兲
k
P共k兲. 共1兲
Note that such an equation corresponds to the case when a
fraction f of nodes are removed but it also holds when a
fraction f of links are removed 共or lead to unoccupied sites兲.
In order to study percolation properties, we use the stan-
dard generating function methodology. The two first gener-
ating functions of the damaged graph are
F
0
共x兲 =
兺
k
⬁
P共k兲共1−f兲x
k
, 共2兲
F
1
共x兲 =
1
具k典
兺
k
⬁
kP共k兲共1−f兲x
k−1
. 共3兲
The averages 共i.e., the values at x=1兲 are F
0
共1兲=F
1
共1兲=1
− f, respectively. Here F
0
共1兲 is the fraction of nodes from the
original graph belonging to the damaged graph. Similarly,
F
1
共1兲 is the relation among 具k典 and the average number of
nodes from V that can be reached after deleting a fraction f
of nodes. The generating function for the size of the compo-
0
5
10
15
k
10
-3
10
-2
10
-1
10
0
Cumulative distribution
UCTE
UK and Ireland
Italy
bca
FIG. 1. Power grids define a spatial, typically planar graph with nodes including generators, transformers, and substations. Here we show
共a兲 the geographical and 共b兲 the topological organization of the Italian power grid. These webs are homogeneous, having an exponential
degree distribution, P共k兲=exp共−k /
␥
兲/
␥
, as shown in 共c兲.
SOLÉ et al. PHYSICAL REVIEW E 77, 026102 共2008兲
026102-2
nents, other than the giant one, which can be reached from a
randomly choosen node is
H
1
共x兲 = f + xF
1
关H
1
共x兲兴 共4兲
and the generating function for the size of the component to
which a randomly choosen node belongs to is 关26兴
H
0
共x兲 = f + xF
0
关H
1
共x兲兴. 共5兲
Thus the average component size, other than the giant com-
ponent, will be
具s典 = H
0
⬘
共1兲 =1−f + F
0
⬘
共1兲 ⫻ H
1
⬘
共1兲. 共6兲
After some algebra, we see that this leads to a singularity
when F
1
⬘
共1兲=1. To ensure the percolation of the damaged
graph, the following inequality has to hold:
兺
k
k共k −2兲P共k兲 ⬎
兺
k
k共k −1兲fP共k兲. 共7兲
The above expression can be expressed as
具k
2
典 −2具k典 ⬎ f共具k
2
典 − 具k典兲 共8兲
which leads to a critical probability of node removal f
c
given
by
TABLE I. A summary of the basic features exhibited by some of the European power grids analyzed, ordered by increasing
␥
, the
exponential degree distribution exponent. The critical probability of node removal f
c
is shown for both cases, theoretical and real, and
random 共errors兲 and selective 共attacks兲 removal of nodes. The absolute difference 兩⌬f
c
兩 between theoretical and observed critical probability
diminishes as
␥
increases in general terms. Number of nodes N, number of links L, and mean degree 具k典 are also shown as reference.
Countries in italics have been used to evaluate reliability indexes. EU results 共i.e., results for the ⍀
EU
graph兲 are shown for comparative
purposes.
Country
␥
Errors Attacks
NL具k典f
c
theor
f
c
real
兩⌬f
c
兩 f
c
theor
f
c
real
兩⌬f
c
兩
Belgium 1,005 0,011 0,395 0,384 0,010 0,131 0,121 53 58 2,18
Holland 1,086 0,147 0,387 0,240 0,034 0,126 0,092 36 38 2,11
Germany 1,237 0,322 0,565 0,243 0,097 0,229 0,132 445 560 2,51
Italy 1,238 0,322 0,583 0,261 0,097 0,241 0,144 272 368 2,70
Austria 1,409 0,450 0,506 0,056 0,159 0,191 0,032 70 77 2,20
Rumania 1,418 0,455 0,579 0,124 0,162 0,238 0,076 106 132 2,49
Greece 1,457 0,477 0,492 0,015 0,174 0,183 0,009 27 33 2,44
Croatia 1,594 0,543 0,525 0,018 0,214 0,202 0,012 34 38 2,23
Portugal 1,606 0,548 0,595 0,047 0,217 0,250 0,033 56 72 2,57
EU 1,630 0,557 0,629 0,072 0,223 0,275 0,052 2783 3762 2,70
Poland 1,641 0,562 0,594 0,033 0,226 0,249 0,023 163 212 2,60
Slovakia 1,660 0,569 0,563 0,006 0,231 0,227 0,004 43 52 2,41
Bulgaria 1,763 0,604 0,570 0,034 0,256 0,232 0,024 56 67 2,39
Switzerland 1,850 0,629 0,610 0,020 0,275 0,260 0,015 147 186 2,53
Czech Republic 1,883 0,638 0,634 0,004 0,281 0,279 0,003 70 88 2,51
France 1,895 0,641 0,647 0,006 0,285 0,289 0,004 667 899 2,69
Hungary 1,946 0,654 0,617 0,036 0,295 0,266 0,029 40 47 2,35
Bosnia 1,952 0,655 0,588 0,067 0,295 0,244 0,052 36 42 2,33
Spain 2,008 0,668 0,689 0,020 0,307 0,324 0,017 474 669 2,82
Serbia 2,199 0,705 0,655 0,051 0,339 0,296 0,054 65 81 2,49
0 0,2 0,4
0,6
0,8
1
f
c
0
0,2
0,4
0,6
0,8
1
S
inf
0 0,2 0,4
0,6
0,8 1
f
c
0
0,2
0,4
0,6
0,8
1
S
inf
FIG. 2. Effects of attacks and failures on the topology of the EU
power grids. Static tolerance to random 共white circles兲 and selective
共black circles兲 removal of a fraction f of nodes, measured by the
relative size S
inf
of the largest connected component. Whiskers
stand for the standard deviation. Inset: Evolution of the static toler-
ance to random and selective node removal for Italy 共dashed lines兲
and France 共continous lines兲. Though in the case of random removal
共failures兲 both networks exhibit a similar response, for the selective
one 共attacks兲, Italy behaves in a slightly stronger manner 共i.e., for a
fixed fraction of eliminated nodes, the relative size of the largest
connected component in Italy always remains higher than that of
France兲.
ROBUSTNESS OF THE EUROPEAN POWER GRIDS UNDER… PHYSICAL REVIEW E 77, 026102 共2008兲
026102-3
f
c
=1−
1
0
−1
, 共9兲
where
0
=具k
2
典/ 具k典, and in agreement with 关28兴. In our case,
we have an analytic estimate
0
=2
␥
. Using the average
value 具
␥
典=1.9, we obtain a predicted critical probability f
c
=0.61.
Although random removal is an interesting scenario, it
considers chance events that are not correlated to network
structure. Intentional attacks strongly deviate from random
failures: Even a small fraction of removed nodes having
large degrees has dramatic consequences. In order to predict
the effects of such directed attacks on network structure, the
critical probability associated to network breakdown can be
computed. Here we follow the formalism developed by Co-
hen et al. 关29兴. Roughly speaking, this formalism enables us
to translate an intentional attack into an equivalent random
failure and study the problem in terms of standard percola-
tion using Eq. 共9兲. When the selective removal of the most
connected nodes is considered, a fraction of order O共1/ N兲 is
removed by eliminating elements with a degree larger than a
given k=K. This upper cutoff is then easily computed from
the continuous approximation:
兺
K
⬁
P共k兲⬇
冕
K
⬁
1
␥
e
−k/
␥
dk =
1
N
共10兲
and the new cutoff K
˜
can be obtained 共again under a con-
tinuous approximation兲 from
冕
K
K
˜
1
␥
e
−k/
␥
dk =
冕
K
⬁
1
␥
e
−k/
␥
dk −
1
N
= p, 共11兲
which gives 共assuming K large enough兲 a new cutoff
K
˜
=−
␥
ln p. 共12兲
Following 关29兴, we translate the problem of intentional
attack to an equivalent random failure problem. The removal
of a fraction f of nodes with the highest degree is then
equivalent to the random removal of those links connecting
the remaining nodes to those already removed. Thus, the
probability that a specific link leads to a deleted node will be
given by
p
˜
=
冕
K
K
˜
kP共k兲
具k典
dk, 共13兲
具k典 being the average degree of the undamaged graph. It is
not difficult to show that this gives
p
˜
=
冉
K
˜
␥
+1
冊
e
−K
˜
/
␥
. 共14兲
Using Eq. 共12兲 it is straightforward to see that
p
˜
= 共ln p
c
−1兲p
c
, 共15兲
where we assume that K is large enough to ignore the term
exp共−K/
␥
兲. Thus an equivalent network with maximal de-
gree K
˜
has been built after a random removal of p
˜
nodes due
to the fact that the absence of correlations implies a random
failure of links. In order to obtain the degree distribution of
the damaged graph, such a failure can be introduced into Eq.
共3兲. But this will be formally equivalent to the removal of p
˜
nodes. Thus, to study stability properties, we only need the
resulting probability p
˜
to be introduced in the critical condi-
tion for percolation 共9兲. Replacing p
c
=p
˜
, we obtain
1+共ln p
c
−1兲p
c
=
1
2
␥
−1
, 共16兲
whose solutions 共for each fixed
␥
兲 provide the conditions for
network percolation under attacks. In Fig. 3 and Table I,we
show the result of our calculations. As expected, a much
lower value of f
c
is required to break a power grid network
through intentional attack.
Now we can compare this mean field prediction, evalu-
ated as f
c
theor
, with available data. Using the whole dataset of
EU grids, we can estimate f
c
real
for all EU countries. The
result are shown, for both f
c
’s, in Fig. 3共b兲. As we can see,
there is a very good agreement 共given their small size兲 be-
tween observed 共real兲 and predicted 共theoretical兲 f
c
values,
but some nontrivial deviations are also obvious. We can see
that aproximately for
␥
⬎1.5 the expected f
c
values are very
similar to those predicted by theory. However, the power
grids having lower exponents 共when
␥
⬍1.5兲 strongly devi-
1,0 2,0 3,0 4,
0
γ
0,0
0,2
0,4
0,6
0
,
8
f
c
1 1,5
2
γ
0,0
0,1
0,2
0,3
0,4
0,5
f
c
A
B
Breakdown
Connected
random
attack
a
b
FIG. 3. 共a兲 Phase space for exponential uncorrelated networks
under random removal of nodes and directed attack towards highly
connected vertices. Here
␥
is the average degree of the exponential
network and f
c
indicates the fraction of removed nodes required in
order to break the network into many pieces. The upper curve is the
critical boundary for network percolation under random removal of
nodes. Below it, a network experiencing such random failures
would remain connected 共i.e., with a giant component兲. The lower
curve corresponds to the critical boundary for attacks. In 共b兲 we
display the estimated values of f
c
共
␥
兲 for attacks from the thirty-
three EU power grids 共circles兲 to be compared with the mean field
prediction 共continuous line兲.
SOLÉ et al. PHYSICAL REVIEW E 77, 026102 共2008兲
026102-4
ate from the predicted values. These agreements and devia-
tions are not due to some simple statistical trait, such as
network size. As indicated in Table I, very large power grids
are in both sides 共i.e., the German and Italian grids are in the
first group, whereas the Spanish and French ones belong to
the second兲 and mixed with smaller ones. Although the effect
of geography on the properties of some networks is impor-
tant 共see 关27,30兴 for example兲, this last observation would
suggest that the geographical embedding of these networks
might have a small effect.
IV. CORRELATIONS WITH NONTOPOLOGICAL
RELIABILITY MEASURES
The reliability of a power grid evaluates its ability to con-
tinuously meet demand under major events like overloads,
general failures, external impacts and alike. At the engineer-
ing level, and due to the different dimensions of service qual-
ity involved in a power grid 共i.e., consumers, companies, and
regulators兲, reliability has been traditionally measured by
different indexes as 共a兲 the amount of energy not supplied,
共b兲 the total loss of power, or 共c兲 the equivalent time of
interruption, which measures the number and duration of in-
terruptions experienced by customers 关31兴. In this sense we
would expect a correlation between the critical percolation
fraction f
c
, the exponent that characterizes the grids’ cumu-
lative degree distribution
␥
, and some of 共if not all兲 these
reliability indexes presented.
In order to explore the problem, three reliability indexes
have been obtained from the UCTE monthly reliability mea-
sures 关32兴. They are related to four major events. Namely,
overloads, general failures, external impacts and exceptional
conditions, and finally other reasons 共including unknown
reasons兲. For every major event and transmission grid, the
following indexes have been considered and normalized: 共1兲
Energy not supplied, normalized by the gross UCTE electric-
ity consumption; 共2兲 total loss of power, normalized by the
UCTE peak load on the third Wednesday of December; and
共3兲 equivalent time of interruption 共also known as average
interruption time or AIT兲, which is the ratio between the total
energy not supplied and the average power demand per year,
measured in minutes per year 共normalized by definition兲.
In order to avoid statistical deviations due to the limited
historical data available 共UCTE monthly statistics have been
published only from January 2002 onwards兲, we have dev-
ided UCTE networks in two groups. Group 1 includes those
countries whose critical breakdown probability f
c
real
agrees
with that predicted f
c
theor
共i.e., countries with
␥
⬎1.5兲. Group
2 includes those countries whose f
c
real
deviates positively
from f
c
theor
共i.e., countries with
␥
⬍1.5兲, with an expected
more robust topology than that predicted.
Figure 4 gives the acummulated percentage values for the
formerly presented reliability indexes and for each group of
networks. As we can see, networks in group 1 共i.e., networks
with f
c
real
⬵ f
c
theor
兲 represent 63% of the whole UCTE nodes,
they manage 48 and 51% of the UCTE energy and power,
respectively, but acummulate 85, 68, and 79% of the UCTE
average interruption time, power loss and energy not deliv-
ered, respectively. On the contrary, though networks in group
2 共i.e., networks with f
c
real
⬎ f
c
theor
兲 represent a mere 33% of
the whole UCTE nodes, they manage 46 and 44% of the
UCTE energy and power respectively 共similar to those of
group 1兲 but, even so, they acummulate only 15, 32, and
21% of the UCTE average interruption time, power loss and
energy not delivered, respectively. This fact would suggest a
positive correlation between static topological robustness
and nontopological reliability measures and, as a conse-
quence, a clear diferentiation between two classes of net-
works in terms of their level of robustness.
V. DISCUSSION
In this paper, we have extended our previous work on the
robustness of the European power grid under random failures
with the intentional attacks scenario. A mean field theory
approach has been used in order to analytically predict the
fragility of the networks against selective removal of nodes
and a significant deviation from predicted values has been
found for power grids with an exponent
␥
⬍1.5. For these
networks, the real critical fraction f
c
real
is higher than the
theoretical one f
c
theor
for the same
␥
. This suggests an in-
creased robustness for these networks compared to those
with
␥
⬎1.5.
In order to evaluate the real existence of this two classes
of networks, namely robust and fragile, real reliability mea-
sures from the Union for the Co-Ordination of Transport of
Electricity 共UCTE兲 have been used. It has been found that
there seems to exist indeed a positive correlation between
static topological robustness measures and real nontopologi-
cal reliability measures. This correlation shows that networks
in the robust class 共i.e., networks with f
c
real
⬎ f
c
theor
兲, though
representing only 33% of the UCTE nodes under study and
0
20
40
60
80
%
Reliability indexes
0
20
40
60
80
%
Power grid indexes
AIT
Power
Loss
Energy not
delivered
Energy
share
Power
share
Size
Group 1 (γ
i
> 1.5)
Group 2 (γ
i
< 1.5)
a b
FIG. 4. Power grid indexes vs reliability indexes. 共a兲 Networks
in group 1 共i.e.,
␥
⬎1.5 and f
c
⬵ f
c,p
兲, though representing two-
thirds of the UCTE size, share almost as much power and energy as
networks in group 2 共i.e.,
␥
⬍1.5 and f
c
⬎ f
c,p
兲. 共b兲 Nonetheless,
these same networks of group 1 acummulate more than five times
the average interruption time 共AIT兲 of the latter, more than two
times their power losses, and almost four times their undelivered
energy.
ROBUSTNESS OF THE EUROPEAN POWER GRIDS UNDER… PHYSICAL REVIEW E 77, 026102 共2008兲
026102-5
