Content uploaded by Sergey Makrushin
Author content
All content in this area was uploaded by Sergey Makrushin on Aug 28, 2017
Content may be subject to copyright.
Analysis of Russian Power Transmission
Grid Structure: Small World Phenomena
Detection
Sergey Makrushin
Abstract In this paper, the complex network theory is used to analyze the spatial
and topological structure of the Unified National Electricity Grid (UNEG)—Russia’s
power transmission grid, the major part of which is managed by Federal Grid Com-
pany of the Unified energy system. The research is focused on the applicability of the
small-world model to the UNEG network. Small-world networks are vulnerable to
cascade failure effects what underline importance of the model in power grids analy-
sis. Although much research has been done on the applicability of the small-world
model to national power transmission grids, there is no generally accepted opinion
on the subject. In this paper we, for the first time, used the latticization algorithm
and small-world criterion based on it for transmission grid analysis. Geo-latticization
algorithm has been developed for a more accurate analysis of infrastructure networks
with geographically referenced nodes. As the result of applying the new method, a
reliable conclusion has been made that the small-world model is applicable to the
UNEG. Key nodes and links which determine the small-world structure of the UNEG
network have been revealed. The key power transmission lines are critical for the
reliability of the UNEG network and must be the focal point in preventing large
cascade failures.
1 Introduction
The object of the current research is the Unified National Electricity Grid (UNEG)
of Russia, the major part of which is managed by Federal Grid Company of Unified
Energy System (FGC UES Inc.). The aims of the research are to find the key network
topology properties and to identify the UNEG network model. The main focus is on
the detection of the small-world phenomena in the UNEG network and on finding
appropriate methods for this analysis. These methods are based on the complex
S. Makrushin (B
)
Financial University Under the Government of the Russian Federation,
49 Leningradsky Prospekt, GSP-3, 125993 Moscow, Russia
e-mail: SVMakrushin@fa.ru
© Springer International Publishing AG 2017
V.A. Kalyagin et al. (eds.), Models, Algorithms, and Technologies
for Network Analysis, Springer Proceedings in Mathematics & Statistics 197,
DOI 10.1007/978-3-319-56829-4_9
107
108 S. Makrushin
network theory and use for analysis the UNEG computer model. Complex network
theory methods are widely employed for power transmission grid analysis all over
the world but have not been used for Russian’s UNEG.
In the majority of papers on the analysis of power transmission grids with the
complex network theory methods the small-world model [23] applicability is con-
sidered [13]. This attention can be explained by the fact that the small-world model
is one of the crucial models of the complex network theory. One of the major prop-
erties of small-world networks is their capability of fast diffusion of information (or
any other process) in a network [14]. As concerns electricity transmission networks
it means that small-world networks are vulnerable to cascade failure effects what
underline importance of the model in power grids analysis [13]. Although much
research has been done on the applicability of the small-world model to national
power transmission grids, there is no common opinion on that subject [6,13,22].
One of the reasons for this is the lack of appropriate methods for the identification
of small-world topology in a network. The current paper considers new more accurate
methods for the identification problem; these methods rely on the results which were
gained by Telesford et al. [21] but have not been used for electrical network analysis.
The paper also proposes a modification for that method which is adapted for a network
with geographically referenced nodes.
2 The UNEG Network Computer Model
For the complex network analysis UNEG appears as a network with electric power
stations and substations as nodes and high-voltage power lines as links. For the current
research, data on 514 nodes and 614 links has been gathered. This data includes the
networks topology, geographical coordinates of nodes, region binding, voltage levels,
and other properties. The data for the computer model creation have been taken from
official papers [12], online UNEG map from FGC UES [19], OpenStreetMap GIS
[11]. The UNEG computer model has been created for the main operating regions
of UNEG except from the unified national energy systems of Siberia, East and Urals
(partially). Visualization of the UNEG computer model with nodes in accordance
with their geographical coordinates can be seen in Fig. 1a.
The computer model processing and the algorithm development have been done
with Python programming language and NetworkX library [10]. Using the computer
model of UNEG, the basic properties of the network have been found (see Table 1).
3 The Generally Accepted Small-World Network Criterion
According to the definition, small-world networks have an clustering coefficient close
to that of regular lattices with a similar quantity of nodes and links and an average
path length close to that of random networks [5] with a similar quantity of nodes
Analysis of Russian Power Transmission Grid Structure … 109
Fig. 1 Geo-latticizated UNEG network aInitial UNEG model (nodes shown in accordance with
their geographical coordinates) bGeo-latticizated UNEG network: if possible, long links are
replaced by shorter ones, node’s degrees are preserved from changing
and links [23]. In practice, however, to analyze the applicability of the small-world
model for a certain network the average path length and the clustering coefficient
are compared only between the network and its random analog. There is a wide-
spread small-world model applicability criterion which was presented for the first
timein[7]:
110 S. Makrushin
Tabl e 1 Basic properties of the UNEG network and its analogs
1 2 3 4 5
UNEG network analogsa
UNEG network Erdös–Rényi
random network
Results of
random
relinkage
procedure
Latticizated
UNEG network
Geo-latticizated
UNEG network
N514 514/459.6G514 514 514
L642 645.2/637.5G642 642 642
l11.93 6.51G7.35 67.25M/62.16 16.65M/17.63
Ltot 48107 –––40237M/41201
C0.0942 0.0047/0.0047G0.0037 0.2210M/0.2026 0.1614M/0.1641
Cavg0.0807 0.0034/0.0038G0.0026 0.2328M/0.2027 0.1771M/0.1770
aThe mean values for 100 results of generation process
GThe value for the giant component of the network
MThe value for the network with the minimum total length of links
N—number of nodes in the network, L—number of links in the network, l—the average path
length (in hops) between all node pairs of the network (or of its giant component), Ltot—total
length of links (in kilometers), C—the global clustering coefficient, Cavg—the average clustering
coefficient
σ=C/Crnd
l/lrnd ,(1)
where Cand lare the clustering coefficient and the average path length (in hops)
between all node pairs of the current network, and Crnd and lrndare the clustering
coefficient and the average path length of the randomized analog of the network.
There are more than 400 citations of the paper [7], and currently this criterion of
small-world model applicability is generally accepted among researchers in the field
of complex network theory.
In graph theory, a clustering coefficient is a measure of the degree to which nodes
in a graph tend to cluster together. Currently, there are two definitions of the clustering
coefficient of a network: the global clustering coefficient and the average clustering
coefficient. The global clustering coefficient is defined as:
C=3×number of triangles
number of paths of leng th 2,(2)
where a ‘triangle’ is a set of three nodes in which each node is connected to the
other two. The average clustering coefficient of a network Cavgis the mean of local
clustering coefficients Cavg
iover all nodes, where Cavg
ifor node iis defined as:
Cavg
i=2Li
ki(ki−1),(3)
Analysis of Russian Power Transmission Grid Structure … 111
where Liis the number of links between all neighbors of iand kiis the degree of
node i.
Both definitions capture intuitive notions of clustering but though often in good
agreement, values for Cand Cavgcan differ by an order of magnitude for some
networks. Use of the global clustering coefficient has some advantages for our task.
In case of Erdös–Rényi (ER) random network we can use the probability of link
creation pas the mathematical expectation of the global clustering coefficient C.
Moreover, this definition of the clustering coefficient is used as a primary one in
[7], and if we use it, we will get more comparable results. Further, unless otherwise
specified, we will use the global clustering coefficient as the measure of clustering.
Nevertheless, in Table 1the values of Cavgarealsoshown.
According to the definition of a small-world network, its clustering coefficient Csw
has a property: Csw≈Clat , where Clat is the clustering coefficient for the analog of
the network with properties of a regular lattice. The expression for the small-world
average path length follows from the second part of the definition: lsw≈lrnd.
Consequently, for small-world networks lsw/lrnd ≈1, and from the estimation
Clat Crnd we will get an estimation for clustering coefficients ratio: Clat/Crnd 1.
Inserting the estimations of the clustering coefficient and the average path length into
the expression (1) we get the estimation for the criterion of the small-world model
applicability: σsw1. By the same reasoning, we can conclude the higher the value
of σ, the closer the network to the small-world state.
Values of the global clustering coefficient Cand the average path length lfor
the initial UNEG network are shown in column 1 of Table 1. We can calculate the
values of Crnd and lrndfor the equivalent ER random graph [5]. An ER random
graph is constructed by uniquely assigning each edge to a node pair with the uniform
probability. Parameters of the ER random graph generation process are the quantity
of nodes Nand the probability pof creating an edge. Value of pis calculated as
follows:
p=L
N(N−1),(4)
where Lis the quantity of links and Nis the quantity of nodes in the network. For the
UNEG L=642, N=514 and, consequently, p=0.0049. Due to the randomness
of the ER graph generation process, values of L,C, and lare different in each case.
We repeated the generation process 100 times and calculated the mean values which
are shown in column 2 of Table 1.
The mean value of links quantity in ER random graph creation process for the
UNEG is quite near to L. Mean value of the global clustering coefficient for ER
network corresponding to the UNEG network is 0.0047. Since probability of link
creation p in ER network is the expectation of its global clustering coefficient, the
value of Cis quite near to p=0.0049.
However, using the ER random graph creation process could not help find the
correct value of the average path length lrndbecause the networks, which were
generated by the process with parameters N=514 and p=0.0049, are discon-
nected. In disconnected networks, paths between some pairs of nodes are absent,
112 S. Makrushin
and calculation of the average path length between all node pairs of a network is
incorrect. Since, in our case, each ER random graph has a giant component (a con-
nected subgraph with the largest number of nodes), as a workaround, we can calculate
the average path lengths for the giant components of ER random graphs (see values
with index ‘G’ in column 2 of Table 1). The weak point of this solution is that in
our case the giant components have only near to 90% of nodes and about 99% of
links of the whole network. Hence smaller quantity of nodes and higher density of
links will lead to underestimation of the value of lrndif the average path lengths of
giant components of ER random graphs are used. Since both ‘triangles’ and paths
of length 2 from formula (2) are very rare outside giant components of ER random
graphs, generated with our parameters, the global clustering coefficient of a giant
component is almost the same as of the whole ER random graph (see Cvalues in
column 2 of Table 1).
To avoid this problem we can use an alternative network generation process with
random relinkage procedure. This procedure is quite similar to the relinkage process
in the Watts–Strogatz model for generating small-world networks from lattice net-
works [23]. The main difference is that in the current procedure the connectedness
of the network and the degrees of all nodes are preserved.
A basic step of the random relinkage procedure consists of a random choice of
two pairs of linked nodes, disruption of the old links within each pair and creation of
two new links between two nodes from different pairs. Relinkage step is rolled back
if it leads to violation of network connectedness. When a random relinkage step is
repeated many times it saves the degree of every node and the connectedness of the
network but shuffles links between the nodes. That process leads to a random network
with the same degrees of nodes as in the initial network and to the preservation of
connectedness of the network.
We have created the software implementation of the random relinkage procedure,
and it has been applied to the UNEG network computer model. For neutralization of
the stochastic effects of the random relinkage procedure, it was applied to the UNEG
network 100 times. The average values of network properties for the randomized
UNEG network analogs are shown in column 3 of Table 1.
The value of the average path length for the random analog of the UNEG obtained
by the random relinkage procedure is 7.35. This value is more than 10% greater than
the average path length for the giant component of the ER random analog of the
UNEG. As mentioned above, this is the consequence of smaller quantity of nodes and
higher density of links for giant components along with the effect of different nodes
degree distributions in networks which are generated by different procedures (the
direction of this effect requires a separate study). Moreover, the distinction between
nodes degree distributions is also the cause of significant difference in magnitude of
the global clustering coefficients for the two random network generation procedures.
The values of σcriterion 1 for the UNEG are shown in Table 2. For all calculation
methods of Crnd and lrnd values of σgreatly exceed 1 and may be interpreted as
values of a network with small-world properties. But as shown in Telesford et al.
paper [21], the generally accepted approach for the detection of small-world structure
in networks has significant flaws.
Analysis of Russian Power Transmission Grid Structure … 113
4 Criticism of the Generally Accepted Small-World
Network Criterion
Watts–Strogatz small-world network model is defined as a hybrid between the regular
lattice (in terms of its high clustering coefficient) and the random network (in terms of
its short average path length). Many succeeding small-world models, e.g., Kleinberg
model [9], use the same main idea. Using the generally accepted small-world criterion
σwe compare a network with only one extreme case of the small-world model the
random network, and ignore another extreme case the regular lattice. Comparison
with the regular lattice using the traditional criterion is commonly ignored due to the
absence of a simple and widely known algorithm to generate the regular analog of a
network.
Depending on the method of calculating Crnd the value of ratio C/Crnd for the
UNEG network is in the range [19.85,25.79](see Table 2). In any case, this ratio
has a quite big value what means that global clustering coefficient of the UNEG
network is very different from a random analog of the network. But from CCrnd
and Clat Crnd it does not follow that C≈Clatt. It means that such a big value
of C/Crnd does not ensure that clustering of the UNEG network has a value close
to regular lattices with a similar quantity of nodes and links. Consequently, when
we have a high C/Crnd ratio and consequently high value of σcriterion we cannot
make a correct conclusion about satisfaction of one of the two requirements from
the small-world definition.
Moreover, identification of the small-world structure for a network with the help
of the generally accepted criterion 1is ambiguous because value of Crnd is highly
dependent on the size of the network. As mentioned above, the value of the global
clustering coefficient of an ER random network could be estimated by the probability
of creating an edge p
CER
rnd ≈p=2L
N(N−1)=k
N−1≈
N1
k
N(5)
where kis the average node degree. If we consider several networks which have
the same structure (first of all, the same average node degree kand the same global
clustering coefficient C) but have different quantity of nodes N, we will get the
estimation for the numerator of the fraction 1:
Tabl e 2 σsmall-world criterion calculation
1 2 3
Random analog of the network C/Ca
rnd l/lrnd aσ
Erdös-RényiG19.85 1.83 10.83
Relinkage process 25.79 1.62 15.89
aThe mean values for 100 results of generation process
Gthe value for the giant component of the network
114 S. Makrushin
C/CER
rnd ∼N(6)
In particular, if the considered networks are small-world networks with the value
of the fraction l/lrnd close to 1, then the relation between σcriterion and the size
of the network Ncan be estimated as:
σ=C/Crnd
l/lrnd ∼N(7)
This means that the value of σcriterion for small-world networks with the same
structure is proportional to the quantity of nodes in those networks. Consequently,
the size of a network is crucial when we use sigma criterion to check the small-world
model applicability, and this fact makes using σcriterion not reliable.
For example, the two power transmission networks considered in this paper [6]
are classified as a random network and a small-world network on the basis of their
values for C/Crnd and l/lrnd. For the first network (classified as a random network)
C/Crnd =3.34 and l/lrnd =1.46. For the second network (classified as a small-
world network) C/Crnd =22.00 and l/lrnd=1.46. Average degrees of nodes
for both networks are almost the same and have values close to 2.6 links per node.
Values of σcriterion for the networks are 2.29 and 15.04, respectively. These values
confirm the classification made in [6]. However, if we take into account the cause of
difference in the ratios C/Crnd for the networks, then our conclusions will become
more ambiguous.
For the first network the ratio is calculated on the basis of the following values:
C/Crnd =0.107/0.032 =3.344, and for the second network the ratio is: C/Crnd =
0.088/0.004 =22.000. This shows that the first network has a significantly greater
global clustering coefficient than the second network, and what means that the greater
ratio value for the second network is explained only by different values of the global
clustering coefficients of the random analog networks. Since both networks have the
same average degree of nodes, the difference in the clustering coefficients for the
random networks arises from the difference in the networks size: the first network
has only 84 nodes, while the second one has 769 nodes. The value of Crnd depends
heavily on the network size, but Clat should not have that dependency. Thus, while
considering C/Crnd in the criterion σfor testing C≈Clatt , we implicitly add the
incorrect dependency of σon the size of the network to the criterion. Consequently,
the usage of the criterion based on the ratios C/Crnd and l/lrnd for classifying
networks of different sizes in the case from [6] led to ambiguous conclusions.
In Telesford’s paper [21] an analysis of families of small-world networks gener-
ated by the Watts–Strogatz model was performed. It has revealed that the unilateral
comparison in the traditional small-world criterion has several significant flaws. In
the Watts–Strogatz model from [23] the links in a regular lattice are randomly relinked
with a certain relinkage probability p. Telesford’s paper shows that with the growth
of probability pfrom 0 to 1 in networks generated by the Watts–Strogatz model the
small-world criterion value σsteadily increases to the maximum value and steadily
decreases after reaching the maximum. It means that the dependency of pon σis not
Analysis of Russian Power Transmission Grid Structure … 115
single-valued; therefore, one criterion value has two different interpretations in terms
of the Watts–Strogatz small-world model. Moreover, the traditional criterion σdoes
not have any certain value interval, and for different sizes of networks maximum
values of σcould differ by almost two orders of magnitude. That means that the
same value of the criterion σin different cases could mean a fundamentally different
small-world status of a network.
The analysis presented above demonstrated that the interpretation of the value of
the σcriterion for the UNEG is very ambiguous. Despite the large value of the σ
criterion for the UNEG network we cannot say that the clustering coefficient of the
network is close to a regular lattice with a similar quantity of nodes. We do not know
the maximum value of σcriterion for networks with the same quantity of nodes and
links as in the UNEG network. Consequently, we do not know how close the UNEG
network is to a perfect small-world structure. Moreover, due to the existence of the
two interpretations of σcriterion, we do not know if the UNEG network differs from
the perfect small-world structure in the direction of the random network structure or
in the direction of the regular lattice structure.
In Kim and Obah paper [8] there is another example of difficulties in the inter-
pretation of σcriterion. In that research the generally accepted small-world criterion
σis used to analyze the changing topology of a power transmission grid in different
scenarios of failures of power transmission lines. Kim and Obah found that σvalue
significantly decreased in scenarios which led to major cascade failures. These facts
have been interpreted in [8] as a shift from the small-world network topology to
the random network structure. But the ambiguousness of the σcriterion suggests
that there can be another possible interpretation of its decrease: a shift in the direc-
tion of the regular lattice structure. Moreover, there are some signs that the second
interpretation is more adequate. In particular, the decrease of the σvalue considered
above is caused by a significant decrease in the average path length. It is a typical
consequence of the long links removal from a small-world network, and it leads to
a topology shift in the direction of the regular lattice structure.
5 Latticization Algorithm and New Small-World
Network Criterion
To overcome the problems of the generally accepted small-world criterion in [21]
a new criterion was offered by Telesford et al. for the identification of the small-
world structure in networks. It is based on the latticization algorithm described in
[3,18,20] which is used to generate the regularized analog of a network. The new
criterion uses the comparison of the clustering coefficients of the current network
and its regularized analog along with the comparison of the average path lengths of
the current network and its random analog. The new criterion is as follows:
116 S. Makrushin
ω=lrnd
l−C
Clatt
(8)
where Clatt is the clustering coefficient of the latticizated analog of the current net-
work.
For Watts–Strogatz networks the following conditions for the fractions from Eq. 2
are met: 0 ≤lrnd /l≤1and0≤C/Clatt ≤1. It follows that for Watts–Strogatz
networks and for the broad class of networks in which these conditions are met the
ωcriterion has a certain value interval: from −1 to 1. The criterions values close to 0
conform to the structure of a small-world network, values close to −1 conform to the
structure of a regular lattice, and values close to 1 conform to the random structure
of the network. Dependence of pon ωin the Watts–Strogatz small-world model is
single-valued; consequently, we can uniquely identify the direction of the network
differences from the perfect small-world structure. Also in [21] it was shown that
values of the criterion ωare almost independent from the size of a network. Thus,
the new criterion does not have the fundamental shortcomings of its predecessor. It
has a certain value interval for a network: from −1 to 1, and values of the criterion
are almost independent from the size of a network.
The main idea of the latticization algorithm is to repeatedly execute a relink-
age procedure which is similar to the relinkage procedure in the random relinkage
process. The distinguishing feature of the new variant of the relinkage procedure is
that the relinkage process step is executed only if the total length of the new pair
of links is greater than the total length of the old ones. The testing of this condition
is shown on line 9in the pseudocode realization of the latticization algorithm (see
Fig. 2). The previous steps of the relinkage algorithm are aimed to randomly choose
the two pairs of linked nodes which are suitable for cross relinkage. After choosing
a correct pair of nodes the relinkage procedure starts (see line 14–19 in Fig. 2). This
procedure consists of the disruption of the two old links within each pair and the cre-
ation of two new links between nodes from different pairs. The relinkage procedure
is rolled back if it leads to the violation of the network connectedness.
The evaluation of links lengths (or the distances between the nodes) in the latti-
cization algorithm is described in [3,18,20]; it is based on the definition of a closed
one-dimensional sequence of nodes (‘ring of nodes’) and on the metric of links length
induced by that sequence. The distance between neighboring nodes in this ring (and
the length of links between these nodes) is minimal and has the value of 1, while the
distance between nodes in opposite parts of the ring is maximum and has the value
of [N/2]. This rings structure corresponds to the initial one-dimensional lattice in
the Watts–Strogatz small-world model. As the result of the latticization algorithm,
the network is transformed into a quasiregular network wherein the majority of links
connect nearest neighbors in the one-dimensional sequence of nodes.
Unlike in the case of the Watts–Strogatz model, we do not have any information
about the initial one-dimensional sequence of nodes, and due to this in the latticization
algorithm ring sequences are generated randomly. The latticization procedure is
repeated with different initial sequences (e.g., 100 repetitions were performed for
Analysis of Russian Power Transmission Grid Structure … 117
Fig. 2 Realization of the latticization algorithm in pseudocode
Tabl e 3 ωsmall-world criterion calculation
1 2 3
Algorithm of network regularization lrnd /lbC/Ca
latt ω
Latticization 0.6161 0.4261 0.1900
Geo-latticization 0.6161 0.5833 0.0328
aValues given for networks with minimum total length of links (among the results of 100 repetitions
of each generation process)
bThe mean values for the random relinkage process
the UNEG), and the result with the minimum total length of links is accepted as the
end result of latticization.
The latticization algorithm and the small-world criterion based on it have not
been used for power transmission grids analysis yet. For the current research the
118 S. Makrushin
implementation of the latticization algorithm has been made using Python program-
ming language and has been applied to the UNEG computer model. The basic prop-
erties of the latticizated UNEG network are shown in column 4 of Table 1.Forthe
UNEG network the value of the new small-world criterion ωcalculated taking into
account the latticizated network is equal to 0.19 (see Table 3). It is near to 0 which is
typical for a small-world network; this means that the UNEG network is quite close
to having small-world structure. The criterion value for the UNEG network is greater
than 0 which means that this network has a random rather than a regular structure.
6 Geo-latticization Algorithm
The Latticization algorithm is universal and not targeted for application to power
transmission grid or any other infrastructure network. However, the nature of
infrastructure networks is quite specific. First of all, their nodes have a geographic
binding. Thus building quasi-regular network for random one-dimensional sequence
of nodes is not adequate in this case. Infrastructure networks have well defined two-
dimensional structure of nodes and metric for links length based on geographical
distance.
Widespread usage of two-dimensional modifications of Watts–Strogatz small-
world model [9] makes implementation of two-dimensional modification of the
algorithm particularly justified. Realizing specific of infrastructure networks in the
current research we have developed and programmed a new modification of the lat-
ticization algorithm and call it geo-latticization. In the geo-latticization algorithm,
geographical coordinates of nodes and geodesic distance are used for creation of
metric for calculating length of link in relinkage process. For implementation of the
geo-latticization algorithm only new realization of function Distance is needed (see
line9inFig.2).
For improving geo-latticization algorithm performance random selection of node
pairs in relinkage procedure was changed to selection algorithm based on a function
of distance. The New algorithm chooses the second pair of nodes not accidentally, but
given the additional condition: one node from the second pair must have a distance
to one node from the first pair not greater than a certain value. The additional con-
dition due to the obligatory nearness of nodes from different pairs greatly increases
probability of success relinkage and increases the speed of decreasing the total links
length of a network. In the improved algorithm, a geohash technique [17]isusedfor
fast search of the nearest nodes. For implementation of this feature new realization
of function ChooseLinkedNodes (seeline5inFig.2) has been done.
The result of geo-latticization of the UNEG network is shown in Fig.1b. In the
figure, we can see that some long links have been changed to shorter ones. That
replacement has been done only if it was possible without changing the degree of
nodes and network connectedness violation. Because of the stochastic nature of
geo-latticization process this result is not determined and network configuration in
Fig. 1b is only one from many possible geo-latticization results. But results in the
Analysis of Russian Power Transmission Grid Structure … 119
geo-latticization algorithm are more determined than in the latticization algorithm
because in geo-latticization algorithm there is explicitly defined metric for the links
length. This metric is determined by geographical coordinates of nodes instead of
the randomly defined ring sequence of nodes in latticization algorithm metric.
The basic properties of the geo-latticizated UNEG network are presented in
column 5 of Table1. As for the latticization algorithm 100 repetitions of the geo-
latticization were performed for the UNEG and the value for network with the min-
imum total length of links is accepted as the end result of latticization (in column 5
of Table 1this values have label ‘M’).
The average path length for the geo-latticizated UNEG network is 16.65 and this
value is much less than value 67.25 for the latticizated UNEG network. Predictably,
the two-dimensional quasiregular network shows a sufficiently less average path
length in comparison with quasiregular network for the one-dimensional sequence
of nodes. The global clustering coefficient for geo-latticizated UNEG network is
0.1614 and this value is less than value 0.2210 for the latticizated UNEG network.
That difference is not accidental and has the following explanation. Nodes in two-
dimensional space have more nearest neighbors than in one-dimensional space and
it makes creation of clusters with same quantity of links less probable than in one-
dimensional case.
The value of the small-world criterion ωbased on geo-latticization for the UNEG
network is 0.03. A more accurate method, which takes into account the UNEG net-
work geographic nature shows that this network is sufficiently closer to the perfect
small-world structure than that which was found by using the latticization algorithm
(see Table 3). But qualitative characteristics of the UNEG network stay the same: the
UNEG network has rather a random than a regular structure. Closeness of the UNEG
network to a perfect small-world structure makes it relevant to analyse reliability to
cascade failure effects from a network topology point of view.
7 Long Links Analysis
Analysis of real cases of cascade failures in power transmission grids in [8] and
computer modeling of cascade failures in small-world networks in [15] have revealed
that in power transmission grids with small-world structure a special role in cascade
failure effects belongs to long links (or shortcuts). In small-world networks, shortcuts
are responsible for strong reduction in the average path length. In the Watts–Strogatz
small-world model a major part of new links which emerged in the relinkage process
became long links. Although a large length of a power transmission line in kilometers
does not necessarily mean that this line belongs to network shortcuts, there is a strong
correlation of a large length of line and the possibility of it reducing the average path
length in a network. The geo-latticization process virtually rules out all long links
which could be eliminated from a network. Therefore, comparison of an original
network with its geo-latticizated analog could help identify long links.
120 S. Makrushin
Fig. 3 The complementary cumulative distribution function (CCDF) of the length of links and
fitted distribution laws: afor the UNEG network (distribution law: f(x)∼e−0.0133x)andforthe
geo-latticizated UNEG network (distribution law: f(x)∼e−0.0159x); (b) for long links of the UNEG
network (distribution laws: f(x)∼e−0.0123xand f(x)∼x−4.58)
In the geo-latticizated UNEG network all links except from 4 (0.4% of total
quantity links) have lengths of less than 223km but in the original UNEG network
29 have lengths greater than 223km (see Fig.3a). The distribution in the figure shows
that in the geo-latticizated UNEG network there are many links a little shorter than
223 km. Therefore, links shorter than this limit cannot be definitely identified as
shortcuts. Due to these facts, the length of 223 km has been taken as a threshold for
shortcuts.
It is necessary to notice that the selected long links of the UNEG network are
relatively short compared to geographical size of the UNEG network. For example,
the mean of links lengths in the randomly relinked UNEG network is 914km while
the length of the longest line in the UNEG network is only 607 km. This means that
Analysis of Russian Power Transmission Grid Structure … 121
the UNEG network cannot be described by the Watts–Strogatz model since it does
not have long links comparable to the spatial diameter of the network.
Some other small-world models (i.e., Kleinergs model [9]) admit shorter long
links but have constraints on link length distribution [4,16]. In order to have the
average path length l∼log(N)(i.e., an important property of the small-world
model) in a two-dimensional spatial network with randomized links (i.e., Kleinergs
model), the distribution of the length of lines has to have a fat power law tail with the
value of the exponent α≤4(see[4]). As seen in Fig. 3a, the empirical cumulative
distribution function of the length of links for the UNEG network is approximated
well by exponential distribution. The empirical cumulative distribution function of
the length of shortcuts in the UNEG network can be approximated with the same
quality (measured by a likelihood function) by exponential distribution or by power
law distribution with the high value of the exponent α=4.58 (see Fig. 3b). This
means that the distribution of the length of long lines does not have a sufficiently fat
tail to imply slow growth in the average path length l∼log(N)according to the
requirements described in the Kleinergs model.
But Kleinergs model (and the majority of other spatial small-world models) is
not suited for infrastructure networks such as the UNEG because many underlying
assumptions of this model do not reflect the principles of their construction. Kleinergs
model uses a homogeneous lattice network to construct a small-world network. But
the UNEG has a spatially inhomogeneous nodes structure and a highly uneven dis-
tribution of nodes degree, and this is typical for many infrastructure networks. These
features are very important for the average path length estimation because spatial
areas with high density of nodes and nodes with high degree in a network make it
possible to build small-world structure using relatively short shortcuts. Moreover,
the estimation of dependence of the average path length from growth in number of
nodes in [4,16] assumes that linear size of a network grows as N1/Dwhere D is
number of dimensions in the space where network exists (D=2 for the UNEG and
other infrastructure networks). However, growth of real infrastructure networks is
achieved not only by extending their area but also by increasing spatial density of
nodes. It means that the effect from a relatively short shortcut could increase with
the growth of a network.
In generally accepted spatial small-world models long links are added by randomly
choosing two end points, and this is totally different from infrastructure networks.
In most real cases some special (‘trunk-line’) type of links is used for shortcuts (e.g.,
lines of extra-high voltage (EHV) in power transmission grids). Usually, ‘trunk-line’
links form long paths (or even grids of higher order) into infrastructure networks. In
such a way several relatively short shortcuts could form a path almost equivalent to
a long shortcut.
In Fig. 4diameter of a node is proportional to the total length of two longest links
incident to it. This way a large diameter of a node means that it is on a path of two
shortcuts. There are many large diameter nodes in the figure and almost all of them
have the voltage of 500 kV or 750 kV (i.e., belong to EHV class). Moreover, from
the figure we can see that in the UNEG there are several long paths formed by long
122 S. Makrushin
Fig. 4 Visualization of the UNEG network model. Size of a node is proportional to the total length
of two longest links incident to the node (large size of a node indicates that it is on the path between
two long links); color of a node is defined by voltage level of the electricity transformer in the node
(see the legend, values in kV); colors of links are defined by their betweenness centrality (see color
map in the legend)
links. This example confirms what was said above about ‘trunk-line’ links structures
in infrastructure networks.
The arguments presented above show that due to the inadequacy of Kleinergs
model for the UNEG case, the requirements for the distribution of links length [4,
16] must be refined using a more adequate network model. Moreover, comparison
between the UNEG network and its geo-latticizated analog (see Table 1) shows that
elimination of long links increases the average path length lfrom 11.93 to 16.65
Analysis of Russian Power Transmission Grid Structure … 123
hops. This confirms the high role of long links (representing only 2% of the total
quantity of links) in the UNEG topology which is typical for small-world networks.
Identification of long links in the UNEG network is very important for practical
UNEG operations due to the closeness of this network to a small-world structure.
Identification of long links in the UNEG network will show the power transmission
lines and power substations which must be points of special attention in preventing
large cascade failures.
From small-world model analysis in papers [1,2], we know that nodes incident to
shortcuts have a high value of betweenness centrality. This property could be used as
an alternative method for shortcuts identification. Betweenness centrality is defined
as follows:
g(u)=
s=t=u
σst (u)
σst
(9)
where σst (u)is the number of shortest paths between nodes sand tpassing through
node uand σst is the number of all shortest paths between nodes sand t. A similar
definition exists for betweenness centrality of links. High betweenness centrality of
a node indicates its importance as a transit node in shortest paths between different
pairs of nodes. Particularly, in the case of removing nodes with high betweenness
centrality from a network, shortest paths between some pairs of nodes become longer
and the average path length of the network increases.
Betweenness centrality was calculated for all nodes of the UNEG network com-
puter model; the results are visualized in Fig. 5. From the figure we can see that in the
UNEG network there are a few nodes of very high betweenness centrality. A matter
of special interest is the fact that all those nodes are linked into one chain that runs
through the center of the UNEG. The values of betweenness centrality for the nodes
from the chain are in range from 0.19 to 0.35. The average value of betweenness
centrality for the UNEG nodes is 0.021, the median value is 0.004, the value of 95th
percentile is 0.112 and the value of 98th percentile is 0.188. Thus, the betweenness
centrality values for the nodes from the chain are within 2.SS
Visual analysis of Fig. 5can explain the special role of the nodes with a high
betweenness centrality from the chain: this chain of nodes provides a topologically
short path through the central part of the UNEG network. This result is achieved
by large lengths of power transmission lines in the chain. It is especially important
due to high density of nodes and consequently a relatively small length of power
transmission lines in the central part of the UNEG network. Large lengths of power
transmission lines in the chain are determined by extra-high voltage 500 and 750kV
which is economically viable for electricity transfer over such long distances.
124 S. Makrushin
Fig. 5 Visualization of the UNEG network model. Sizes of nodes are defined by their degree; colors
of nodes are defined by their betweenness centrality (blue nodes have the maximum centrality value)
8 Conclusion
In this paper, new methods for infrastructure networks analysis have been developed.
The latticization algorithm and a new small-world criterion based on it have been used
for power transmission grid analysis for the first time. The geo-latticization algorithm
has been developed for a more accurate analysis of networks with geographical refer-
ence of nodes. This method helps to more accurately identify small-world properties
in power transmission grids and in other infrastructure networks with geographical
reference of nodes.
In this paper, complex network theory has been used to analyze the spatial and
topological structure of the Unified National Electricity Grid for the first time. The
new methods described above have been applied to the UNEG network. Through the
use of these methods a reliable conclusion that the small-world model is applicable
to the UNEG network has been made. Consequently, we have proved the necessity
to conduct an analysis on the UNEG vulnerability to cascade failures with taking
into account the networks topology features.
Key nodes and links which determine the small-world structure of the UNEG
network have been revealed. Identification of the key power transmission lines is
Analysis of Russian Power Transmission Grid Structure … 125
critical to control the reliability of the UNEG network and must be a point of special
attention in preventing cascade failures. This research also discovers that the key
UNEG nodes and links are combined into one chain which passes through the central
part of the UNEG and transforms the whole network into a topologically compact
structure.
References
1. Barrat, A., Barthelemy, M., Vespignani, A.: The effects of spatial constraints on the evolution
of weighted complex networks. J. Stat. Mech. p. 503 (2005)
2. Barthelemy, M.: Spatial networks. Condens. Matter. Stat. Mech. Phys. Rep. 499, 1–101 (2011).
arXiv:1010.0302
3. Brain Connectivity Toolbox. http://www.brain-connectivity-toolbox.net (2016). Accessed 31
Aug 2016
4. Caretta Cartozo, C., De Los Rios, P.: Extended navigability of small world networks: exact
results and new insights. Phys. Rev. Lett. 102, 238703 (2009)
5. Erdös, P., Rényi, A.: On random graphs I. Publ. Math. Debrecen 6, 290–297 (1959)
6. Han, P., Ding, M.: Analysis of cascading failures in small-world power grid. Int. J. Energy Sci.
IJES 1(2), 99104 (2011)
7. Humphries, M.D., Gurney, K.: Network small-world-ness: a quantitative method for determin-
ing canonical network equivalence. PLoS One 3, e0002051 (2008)
8. Kim, C.J., Obah, O.B.: Vulnerability assessment of power grid using graph topological indices.
Int. J. Emerg. Electr. Power Syst. 8(6) (2007) (Article 4)
9. Kleinberg, J.M.: Navigation in a small world. Nature 406, 845 (2000)
10. NetworkX. https://networkx.github.io (2016). Accessed 31 Aug 2016
11. OpenStreetMap. http://www.openstreetmap.org (2016). Accessed 31 Aug 2016
12. Order of the Ministry of Energy of Russia: Shema i programma razvitiya ENES na 2013–2019
godi (Scheme and development program of the UNES on 2013–2019 years). Order of the
Ministry of Energy of Russia from 19.06.2013 309 (2013)
13. Pagani, G.A., Aiello, M.: The power grid as a complex network: a survey. Phys. A Stat. Mecha.
Appl. 392(11) (2011)
14. Pandit, S.A., Amritkar, R.E.: Random spread on the family of small-world networks. Phys.
Rev. E 63 (2001)
15. Pepyne, J.: Topology and cascading line outages in power grids. J. Syst. Sci. Syst. Eng. 16(2)
(2007). doi:10.1007/s11518-007-5044-8 (Systems Engineering Society of China & Springer)
16. Petermann, T., De Los Rios, P.: Physical realizability of small-world networks. Phys. Rev. E
73, 026114 (2006)
17. Python module to decode/encode Geohashes to/from latitude and longitude. Available at:
https://github.com/vinsci/geohash/ (2016). Accessed 31 Aug 2016
18. Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: uses and interpre-
tations. Neuroimage 52, 1059–1069 (2010)
19. Services for technological connection: power distribution centers. http://portaltp.fsk-ees.ru/
sections/Map/map.jsp (2016). Accessed 31 Aug 2016
20. Sporns, O., Zwi, J.: The small world of the cerebral cortex. Neuroinformatics 2, 145–162 (2004)
21. Telesford, Q.K., Joyce, K.E., Hayasaka, S., Burdette, J.H., Laurienti, P.J.: The ubiquity of
small-world networks. Brain Connect 1(5), 367–375 (2011)
22. Watts, D.J.: Small Worlds: The Dynamics of Networks between Order and Randomness. Prince-
ton University Press, Princeton, NJ, USA (2003)
23. Watts, D.J., Strogatz, S.: Collective dynamics of small-world networks. Nature 393, 440–442
(1998)
