Recent Researches in Automatic Control and Electronics
An approach to model the European Interchange
Energy Network as a Small World Net
LEONARDA CARNIMEO
Dip. di Elettrotecnica ed Elettronica
Politecnico di Bari
Via Orabona, 4 - 70125 Bari
ITALY
e-mail: carnimeo@poliba.it
MICHELE DASSISTI
Dip. di Ingegneria Meccanica e Gestionale
Politecnico di Bari
Via Orabona, 4 - 70125 Bari
ITALY
e-mail: m.dassisti@poliba.it
Abstract: European energy policy pursues the objective of a sustainable, competitive and reliable supply of
energy. In 2007, the European Commission adopted an energy policy for Europe with several documents and
included an action plan to meet the major energy challenges Europe has to face. A farsighted diversified yearly
mix of energies was suggested to countries, aiming at increasing security of supply and efficiency, but a wide
and systemic view of energy interchanges between states are still missing. In a previous work of the same
authors, energy import/export interchanges between European States were used to develop a geographic
overview at one glance. In this paper, the European Interchange Energy Network (EIEN) is approached from a
modelling point of view, as a Small World Net, by supposing that connections between States are characterized
by a probability depending on economic/political relations between countries.
Keywords: Energy Mix, Small World Net, Energy Import/Export, European Countries
low energy consumption from an economic point of
view.
Thus, European energy policy must pursue the
objective of a sustainable, competitive and reliable
supply of energy. In 2007, the European
Commission adopted an energy policy for Europe
[2], supported by several documents on various
aspects of energy and including an action plan to
meet the major energy challenges Europe has to face
[3-10]. For this purpose, the adoption of a farsighted
diversified yearly mix of energies was suggested to
all Member States, aiming at increasing security of
supply and efficiency. Unfortunately, a wide and
systemic view of energy interchanges between states
was not available. On this proposal, the Authors
used a graphical representation of energy
import/export exchanges between European States
in [11] to identify the existence of a European
interchange energy network at one-glance. The
same network is herein investigated from a
modelling point of view, as a Small World Net [12],
by supposing that connections can exist between
States with a probability depending also on
economic/political relations between countries.
For this purpose, the hypothesis that the smallworld phenomenon is not only a feature of social
networks [12], nor an artefact of an idealized model
is assumed in this paper, i.e., the Small World Net is
probably a valid model for the European
Interchange Energy Network (EIEN) .
1 Introduction
Energy is essential to daily life. For this purpose, it
is actually essential to face big energy challenges
posed by climate changes, increasing national
dependence on imports, pressure on energy
resources and secure energy supply at affordable
prices for all consumers. In 2006 the European
Commission invited Member States to properly
choose their Energy Mix, namely, the diversification
of national energy supply, with guidelines set out in
the Green Paper [1]. This goal has to be reached by
taking simultaneously into account the different
nature of energy exchanges between pairs of
countries. In most cases energy trade may not be
optimal. On this proposal, the Green Paper can be
considered as an important step in the development
of EU energy policy, aiming at achieving economic,
social and environmental challenges by facing
Europe in the energy sector: increasing dependence
on imports, volatility in fuel prices, climate change,
increased demand and barriers to domestic energy.
Moreover, in [2] the European Commission states
that the implementation of an ambitious European
energy strategy, covering all available energy
sources, whether fossil fuels (oil, gas, coal),
renewable (solar, wind, biomass, geothermal, hydro,
tidal) or nuclear ones, aims at beginning a new
industrial revolution, with the objective of making
EU more secure, competitive, sustainable and at a
ISBN: 978-1-61804-080-0
159
Recent Researches in Automatic Control and Electronics
purchases and sales made by persons living in that
country, by considering that import/export
exchanges take place when the corresponding
commodity crosses the national boundaries, that is,
when clearance by customs authority has taken
place. In order to keep external trade figures for
fuels and energy consistent with the main economic
indicators, purchases should be considered, at least
partly for domestic uses. As a consequence,
quantities in temporary transit through a country
should not be included as import/export figures.
Thus, import/export physical amounts of exchanged
energy found in officially provided data reveal
sometimes not coincident. Origins and destinations
will therefore be considered in neighbouring
countries. This constitutes a major difference with
the reporting of trade of most of the other fuels.
It is well known that the structure of electricity
grids can be relevant concerning with efficiency and
robustness of power networks [13]. Moreover,
networks emerging from unpredictable exchanges of
energies seem to have their own features: it in this
work, the logic behind the structure of these
networks is explored from a systemic point of view,
to derive sound hints for eventual regulations or
possible improvements of the overall efficiency. By
embedding national economical agreements and
exchanges, the resulting network depends on both
social issues and technical ones. Scientific research
questions can be articulated as follows: what kind of
global pattern can be found in the European energy
network, as a whole? Which model could be
adequate for this network in order to represent the
collective dynamics of all National Energy Mixes in
Europe?
In this paper a first attempt is performed to
evaluate if the mixed nature of structural energy
exchanges in Europe, can be dealt with the so-called
small-world phenomena [12]. In this way, a proper
composition of annual National Energy Mix [3-12]
can be justified by a comparison of energy
import/export values for each country. More in
detail, an analysis of the proposed approach is
provided after focusing on the available statistical
data set. European collected data in the period 1996
– 2008 are considered. Finally, results are reported
and discussed.
2.2 Notes on the electricity data structure
The first step in the performed analysis was an
accurate data collection. Data related to Electric
Energy imports by Country of origin and data
related to Electric Energy exports by Country of
destination have been derived from official Eurostat
Databases. It is well known that Energy import
values are quite diversified in values in EU Area
Countries in the considered time period. In
particular, it results that Italy and Germany are the
principal electric energy importers, and in all
Countries the energy import trend is positive with
exception for UK, Turkey and Norway. Concerning
with energy export values, it can be noticed that
France and Germany are the main European electric
energy exporters, followed by Switzerland. The
condition of Germany reveals of a particular
interest, due to the fact that this country presents
high electric energy flows both in import cases and
in export ones, differently from other countries,
such as Italy.
2 European Energy Exchanges
In order to perform an energetic data analysis all
annual electric energy import/export flows have
been analyzed for each European Country in the
period 1996 – 2008. The structure of data set as well
as nature and availability of data are here
commented.
2.1 Data sources
3 Small World Net Modelling
An official data survey has been firstly refined
concerning Energy import/export values between
pairs of European Countries. These data can be
grouped with respect to country, year and type of
imported sourced (crude oil, gas, electric energy…).
After processing available data to make them
effectively comparable, they were analyzed with the
aim of identify which type of energy sources is the
most frequent in internal European exchanges.
Therefore, electric Energy import/export trends have
been analyzed within the period 1996 - 2008.
Imports and exports of commodities are quantities
entering and leaving a given country as a result of
ISBN: 978-1-61804-080-0
The ultimate goal of the study of European
Interchange Energy Network (EIEN) is to
understand and explain how the unpredictable
structure of exchanges can affect the overall
efficiency of electrical fruition within EU, i.e., to set
different scenarios and derive sound conclusions on
the potential effects of ruling strategies. The
methodology here suggested to analyze this network
is based on graph theory. The main hypothesis is
that the energy flow from/toward a country is
represented by an edge of a network, whose length
is provided by the overall annual energy exchange.
The network then consists of n vertices and k edges,
160
Recent Researches in Automatic Control and Electronics
Another important index necessary for the
analysis is the Reciprocity Index (REC) defined as
the ratio of all possible connected couples (A, B) of
vertices which have a reciprocal behaviour[13]. To
a certain extent, the measure of REC turns to be an
interesting indicator of the degree of interconnection
of the EIEN, since the higher the index value, the
stronger the mutual relationships between countries.
An analogous measure is defined as the
transitivity (the probability that adjacent vertices of
the i-th vertex are connected), also called Clustering
Coefficient CCi:
where vertices represent countries at a high level of
abstraction, whereas edges distinguished by colours
represent a specific type of energy flow. It is critical
to consider the whole set of flows with all their
colour intensities: in this sense, the analysis of
overlapping networks (multi-dimensional networks)
is not an easy process to perform, with particular
reference to the criticalities coming from
interactions between networks. In this case, the
analysis here performed is for a single network.
Adopted variables are here detailed. Let’s say n
the number of vertices and k the edges connected to
each vertex. One can start to quantify the structural
properties of the network by evaluating the
Characteristic Path Length (CPL) evaluated as the
geodesic distance (shortest path) dij from vertex i to
vertex j:
CCi = 3 ⋅
CC =
1
CPL =
⋅ ∑ d ij
1 / 2 ⋅ n ⋅ (n − 1) i≥ j
No.(triangles _ centered )
∑ CC
[vertex _ i ]
i
i
An interesting value that may help us to understand
or predict the behaviour of these systems is the
Small World Coefficient, (SMW) calculated as
The Closeness Centrality (CC) is a variable which
measures how many steps are required to access
every vertex starting from a given one.
Considering each vertex as the geographical centre
of a generic state at an higher level of abstraction, it
can be supposed that the higher the distance with
other states is, the stronger the commercial
exchange. Thus, the higher the distance, to a certain
extent, the stronger the connection between vertices
should be. This appears just as the opposite of the
small world phenomenon. Nevertheless, indicators
still maintain their effectiveness and will be used to
draw accordingly sound conclusions. For this
purpose, the amount of energetic exchange will be
represented by the weight of the corresponding
edges.
The Diameter of the network (DIAM) has been
also considered, by evaluating the weighted length
of the longest geodesic path between any two
vertices weighted by the amount of exchanged
energy. In this analysis, this measure has not a
physical meaning; rather it represents a measure of
the entity of each exchange, in an opposite sense:
the higher is the Net diameter, the stronger the
exchanges amongst countries are. It has to be noted
that in this kind of analysis, energetic exchanges are
typically point-to-point ones. Also temporary
exchanges are possible, but they are not visible in
the available statistical data. Due to the fact that
these edges are connected in a network, the meaning
of the variable distance dij can be interpreted as a
sort of energy “fluidity”, that is, the easiness of
exchanges within the EU Area.
ISBN: 978-1-61804-080-0
1
⋅
n
No.(triangles _ connected )
SMW= CPL/CC
for the real network and taking as a reference the
fully interconnected network (complete oriented
graph with n·(n-1) edges) providing the information
on edges sense: exiting/entering from/in a vertex in
case of export/import, respectively.
When analyzing the distribution of degrees of all
vertices, it is interesting to draw an oriented graph
of this small network. The Degree of vertices (DEG)
can be adopted to analyse the EIEN, being the
degree of the i-th vertex defined as the number of
edges connected to that vertex: the maximum DEG
is an interesting synthesis indicator for a scenario
analysis.
Another interesting analysis to be performed is
based on skewed degree distributions, i.e., the
cumulative distribution of the number of vertices
connected to any given one. This is an intriguing
analysis to distinguish scale properties of the EIEN
and to characterise it with respect to the small-world
features too. The small-world effect could also
result mathematically evident, if the number of
vertices within a distance from a typical vertex
grows exponentially with the distance. Logarithmic
scaling has been be proved for a variety of network
models [12], where a typical distribution follows a
power law, with exponents ranging around values
2÷3.
An important part of the analysis is the study of
Resilience [12] of the EIEN, which provides the
sensibility of the EIEN to different critical
scenarios, analysed by removing target vertices,
161
Recent Researches in Automatic Control and Electronics
until a critical condition is reached. This may
correspond to a hypothetical fault in the
relationships of a country or, for our purposes, to a
scenario where fully unregulated energetic policies
are carried on by countries. The measures adopted
to understand effects of this targeted vertex removal
may give hints on the opportunity of improving
exchanges regulation. The edges removal strategy
here adopted has been developed starting from the
vertex with highest DEG (Germany) and then going
further up, by observing significant changes in
network features, and trying to verify if vertex pairs
become progressively disconnected and when
communication between them through the network
EIEN becomes impossible. By reading the evolution
of values of DIAM, REC and SMW, sound hints
about the stability of the same EIEN can be derived
and, thus, on the need to rule it.
Different network configurations can eventually be
analysed: connectivity is, in fact, the true point to
address in our case: which vertex in this network
would prove to be the most crucial for the whole
connectivity if removed? Such a question has a deep
meaning in a network of a reduced number of
vertices. In this work this kind of analysis is
performed by considering the edge Betweenness
[13] of vertices (BTW), being the vertex and edge
Betweenness defined by the number of geodesic
(shortest paths) going through a vertex or an edge.
Selected measures are the maximum value and the
average one of BTW. By analysing the “census” it is
also possible to understand the change in the
structure of the EIEN: the quantity MUT is defined
as the number of pairs with mutual connections,
whereas ASYM is the number of pairs with nonmutual connections and NULL is intended as the
number of pairs with no connection between them
[13].
(a)
(b)
Fig.1: Radial representation of electrical exchanges:
(a) in 1997; (b) in 2008
In Fig.1 (a)-(b), the network of exchanges does
appear almost stable in time: it is in fact based on
historical relationships between adjacent countries
for political or social reasons. This network, on the
other hand, can be considered as unpredictable,
since no general rule procedures for electrical
exchanges seemed to be present at the time of
analysis. Moreover, by observing the values of CPL
in Fig.5 the idea that the EIEN can show a smallworld effect, because of its spontaneous structure,
reveals justified.
CPL
4 Analysis
of
the
European
Interchange Energy Network EIEN
3,1
3
As above mentioned, the whole EIEN, composed of
28 vertices has been analysed. This network is
almost a virtual one, since its edges represent
European energy exchange flows, even though these
flows are subjected to the presence of physical
connections. Conclusions herein drawn have to be
accordingly considered. The analysis here is
developed over time and not in a static network
configuration, due to the fact that official sources
provide data from 1997 up to 2008.
A comparative radial picture of the network EIEN is
firstly provided in selected years, at the extreme of
the period.
ISBN: 978-1-61804-080-0
2,9
2,8
2,7
2,6
2,5
2,4
2,3
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Fig.5: Values of CPL over years for the whole Net
162
Recent Researches in Automatic Control and Electronics
Concerning the analysis of the resiliency of the
network, a target removal strategy has been here
adopted by suppressing successively Germany D,
then Switzerland CH, then France F, then Austria A,
then Italy I and finally, Hungary HR over the
network and observing the same parameter before
mentioned. It is interesting to note the behaviour of
the cumulative distribution of degrees as follows:
the typical length of each path increases
significantly (i.e., exchanges of electricity decrease
as well), but the system seems still to maintain its
connection capability, with a SMall World
coefficient that increases from 3,44 in 1997 up to
7,83 in 2008. The electricity exchange net can be
said to be resilient enough, even though its features
change, as it results from the cumulative degree
distribution (see tables below), where a completely
different distribution can be observed.
Table 1 – Year 1997
Table 2 – Year 2008
phenomena are present, by supposing that
connections exist between States with a probability
depending also on economic/political relations
between countries.
This modelling technique allows to interpret the
evolution of the EIEN over the years, highlighting
some interesting behavioural features. The proposed
methodology can be performed at a deeper detail to
investigate several scenarios and evaluate different
ruling strategies of energetic exchanges at a
European level.
The basic predictions coming out from the
proposed modelling technique, could reveal useful
for a further analysis of renewable energy flows,
5. Conclusions
A different view of the imports and exports of
electric energy flows between Countries of
European Area is proposed for potential use in
ruling these exchanges. The basic idea of the paper
is to show that such a “spontaneous” network has
features similar to small world networks, thus
presenting complex features than a technological
network (such as the electrical grid network).
In order to show the features and potentialities, a
panel data from 1996 to 2008 as part of a network
of exchanges was considered. From a modelling
point of view, the European import/export energy
network has been dealt with to verify if Small World
ISBN: 978-1-61804-080-0
163
Recent Researches in Automatic Control and Electronics
[6] COM (2008) 781Communication from the
Commission to the European Parliament, the
Council, the European and Social Committee
of the Regions, 2nd Strategic Energy Review,
An EU Energy Security & Solidarity Action
Plan.
[7] European Commission (2009), EUROSTAT Statistical Book : Energy - Yearly statistics
2007, Luxembourg: Office for Official
Publications of the European Communities.
[8] Menna P., Gambi R., Hercsuth A., Gillett W.,
Tondi G., Piontek A. (2009), EUROPEAN
PHOTOVOLTAIC
ACTIONS
AND
PROGRAMMES-2009, Brussels.
[9] Schäfer G., Ivan D., Augier A.J., Piirto J.,
Wieland U. (2009), Eurostat Statistical Books:
Europe in figures - Eurostat yearbook 2009,
Luxembourg: Office for Official Publications
of the European Communities.
[10] Bolla V., Hauschild W., Hofmeister O., Jung
D., Lock G., Pavlovic A., Scheller A., Tronet
V. (2009), Sustainable development in the
European Union - 2009 monitoring report of
the EU sustainable development strategy,
Luxembourg: Office for Official Publications
of the European Communities.
[11] Carnimeo L., Dassisti M., Abrescia A. (2010),
A Graphic Tool for Analyzing Energetic Mix
between European Countries, 4th Int. Conf. on
Sustainable Energy and Environmental
Protection (SEEP2010), June 29th- July 2nd,
Bari, Italy, 2010, ISBN: 978-88-905185-2-2.
[12] M.E.J. Newmann, (2000), Models of the Small
World, J. of Statistical Physics, Vol. 101, Nos.
3/4, 2000.
[13] Gabor Csardi, (2010), Package ‘igraph’,
Repository CRAN, Date/Publication 2010-0819 12:12:22.
which would be interesting to evaluate, as far as
reliable statistical data become available.
Trade of electricity has been growing with the
increasing globalisation and opening of national
economies. Thus, it reveals important to collect
information about trade, disaggregated by countries
of origin and destination. These statistics could also
help in identifying potential transmission congestion
and could provide means for a more efficient
operation of an evolving international transmission
grid.
References:
[1] COM(2006)105 GREEN PAPER, A European
Strategy for Sustainable, Competitive and
Secure Energy, {SEC(2006) 317}.
[2] COM(2007)
Communication
from
the
Commission to the European Council and the
European Parliament, An Energy Policy for
Europe, {SEC(2007) 12}.
[3] COM(2006)545 final Communication from the
Commission, Action Plan for Energy
Efficiency:
Realising
the
Potential,
SEC(2006)1173- 1174-1175.
[4] Commission Staff Working Document:
Summary report on the analysis of the debate
on the green paper, A European Strategy for
Sustainable, Competitive and Secure Energy,
SEC (2006)1500.
[5] Commission staff working document Accompanying
document
to
the
Communication from the Commission to the
Council and the European Parliament
Renewable Energy Road Map Renewable
energies in the 21st century: building a more
sustainable future IMPACT ASSESSMENT,
COM(2006) 848final, SEC(2006) 1720,
SEC(2007) 12.
ISBN: 978-1-61804-080-0
164