Topology Analysis of the Mexican Power Grid using

Complex Network Theory

Jorge Sanchez-Jaime Francisco Rivas-Davalos J. Horacio Tovar Hernandez
DEPI-PGIIE DEPI-PGIIE DEPI-PGIIE
TecNM/Instituto Tecnológico de TecNM/Instituto Tecnológico de TecNM/Instituto Tecnológico de
Morelia Morelia Morelia
Morelia Mich, México Morelia Mich, México Morelia Mich, México
jsancezj@toluca.tecnm.mx frivasd@hotmail.com horacio.tovar@yahoo.com

Abstract— In this paper, the topological properties of the centers will receive the necessary power from the other
Mexican power transmission grid were investigated. For this generators or transmission lines. Nonetheless, even with the
purpose, the study included the voltage levels of 400 kV and 230 presence of redundant paths, cascading failures and large
kV. Both voltage levels were analyzed independently and as a blackouts continue to occur across the globe [2]. In Mexico
single combined grid. Similarly, the vulnerability of the power there have been several serious blackouts, such as the one that
grid to random failures and intentional attacks was investigated. occurred in 2017 in the Southeast area of Mexico and the
Based on the complex network theory, some topological metrics Yucatan Peninsula, causing huge economic losses [3].
were calculated for this investigation. It was found that the
Mexican power grid displays some features of small-world The electric power grids are usually subject to many types of
networks, namely large clustering coefficient and small average hazards and events, which can be categorized in three major
shortest-path length. The degree distribution of the power grid groups [4]: random failures, natural hazards, and intentional
reveals exponential behavior. Additionally, it was found that the attacks. Random failures affect all similar components (nodes
power grid is more vulnerable to targeted attacks on nodes with and lines) of a power system network with the same probability
high degree than random failures. In general terms, the distribution. The position of a specific component in the network
conclusions of this work improve the current understanding of has no impact on the probability of failure. Examples of random
the Mexican power grid, which is essential for its control and failures can be those that are due to component aging,
security.
communication system failures, etc. In Mexico, as in many other
Keywords— Complex network analysis, Intentional attacks, parts of the world, there is already a growing concern about the
Power systems, Random failures, Vulnerability, Topology. aging of some of the components of the power grid [5]. The
impact of random failure events is usually investigated by
I. INTRODUCTION randomly removing several system components [4]. The system
behavior without these elements highlight its vulnerability and
In recent years, a theme that constantly emerges in the robustness.
analysis and evaluation of network-based critical infrastructure
is vulnerability. The analysis of infrastructure vulnerability
consists of assessing the physical, operational and geographical
characteristics of infrastructure components, and their role in
the system with which they interact, in terms of fragility to
disruptive events and the impact of these events on the
condition of the infrastructure. For example, consider the
infrastructure network ilustred in Fig. 1, depicting the power
grid of Mexico. Some questions that may arise in this case are
[1]: What is the vulnerability of this critical infrastructure to
disruptions? What are the probable consequences of a particular
disruption scenario? What are the critical nodes or lines that, if
damaged or destroyed, would cause the most damage to the
system? Fig. 1. The Mexican 400 kV and 230 kV power grids with 462 nodes and 653
The electric power transmission grid is one of the most lines.
complex man-made networks. The principal components of
Natural events, such as earthquakes, hurricanes, cold and
these systems are lines and nodes where the latter can be power
heat waves, and lightning, can directly and indirectly affect
generators, substations that connect high voltage lines, and load
power grid components. Contrary to random failures, the
centers that distribute electrical energy to end users. The power
geographic position affects the probability of disruption of a
grid has a complex structure, where there are several redundant
component. In September 2014, the hurricane Odile roared into
paths in order to ensure that, in the event of failure or shutdown
Mexico's Baja California Peninsula, where most of the area's
of any generator, substation or transmission line, the load
power poles were blown over, leaving 239,000 people in the

XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE

state of Baja California Sur without electricity [6]. In February different networks. In [10], the authors presented the case of
2018, a prolonged 7.2 magnitude earthquake that rocked scale-free networks, whose degree distribution (the number of
Mexico left nearly a million homes and businesses without lines attached to each node) follows a power law. Scale-free
power in the capital and south [7]. networks have been shown to be robust against random failures.
An intentional attack targets the critical components of the However, they are extremely vulnerable in case of deliberated
power grid. Intentional attacks might be cyber or physical. In attacks, because the loss of some prominent nodes or links have
2013, government officials from Mexico, U.S. and Canada, the potential to disrupt the whole network [10, 20]. The work in
staged a massive emergency drill, called GridEx II, that [10, 21] suggests that the degree distribution of the power grid
involved thousands of utility workers, business executives, seems to be following a power law distribution function, but
National Guard officers, FBI anti-terrorism experts. The exponential cumulative degree distribution functions are found
scenario envisioned by GridEx II is where terrorists or an in Californian power grid [22], the whole US power grid [23],
enemy country stages a combination of cyber-attacks and and thirty-three different European transmission power grids
physical attacks that destroy or render inoperable crucial power [24].
facilities and take down large sections of the grid [8]. The In [25], random networks are examined, which are
impact of such an event is usually investigated by removing a characterized by a low global clustering coefficient and short
number of power grid components, considering the centrality distances among nodes. Random networks are vulnerable under
of each node or line, as defined in Section 5. both random failures and deliberated attacks. On the contrary,
In this context where electric power grids are subject to small-world networks have low characteristic path length, but
hazards, threats and surprising events, understandable concerns their global clustering coefficient is higher than random
are arising on their vulnerability and risk of failure. Recently, networks [9]. The nature of small-world networks is found in
several studies analyzed the topological properties of real power grids such as the Shanghai power grid [26], the Italian
power grids with the aim of linking these properties with system 380 kV, the French 400 kV, the Spanish 400 kV power grids
vulnerability. Within these studies, those that are based on the [14], and the Nordic power grid [19].
theory of complex networks stand out. Complex Network One of the main outcomes of all these topological
Analysis (CNA) is a relatively young field of research. The first vulnerability studies applied to different power grids of several
systematic studies appeared in the late 1990s [9, 10] having the countries is that it has been found topological characteristics
goal of studying the properties of large networks that behave as that all power grids share, such as similar values of node degree
complex systems. In particular, we refer interested readers to and average shortest-path length, and other features that are
[11,12] which provided a thorough review of extant literature common in groups of power grids, for instance, scale-free and
on power network topology and vulnerability analyses research, small-world properties. However, there are characteristics that
and where it is shown that, into the category of complex are very particular of each power network, such as vulnerability
network analysis methods, there are too many works related to level, shortest-path distribution, critical nodes, etc. The results
the analysis of vulnerability in power grids based on pure of this type of analysis have been very useful for understanding
topological approaches. Here we discuss some important works and modeling different dynamic processes that occur in power
in detail to give background on the field. Pure topology-based systems, such as cascade failures and resilience of networks to
analysis methods include the following network topological external and internal events. That is why topological
measures: average shortest-path length, network efficiency, vulnerability analysis of power grids is still important and
clustering coefficient, and the size of the largest component. useful. This paper presents the first topological vulnerability
Average shortest-path length (also called characteristic path analysis of the Mexican power grid (MXPG), using measures
length) [13] is the mean value of geodesic path distance and concepts of the Complex Network Theory. The main
between any two nodes and has been used in vulnerability purpose of this study is to verify that the MXPG also has these
analysis of the European power grid [14], the power grid of the topological characteristics common to all power grids, and to
western United States [9], and the IEEE 300 power grid [15]. identify to which group the MXPG belongs to, as well as to find
Network efficiency [16] is the normalized average value of the out the particular topological characteristics. For this purpose,
inverse of geodesic path distance between any two nodes and three networks from the MXPG were analyzed to have a more
has been used in vulnerability analysis of the power grids of complete idea of the topology properties. Their tolerance to
Europe [14], Italy [17], and North America [18], among others. random failures and intentional attacks on the most connected
Clustering coefficient [13] is the probability that any two nodes is also analyzed.
neighboring nodes of a randomly selected node are also
connected and has been included in performance and II. COMPLEX NETWORK THEORY AND REPRESENTATION OF
vulnerability analysis of several power grids such as the POWER GRID
European power grid [14], western United States power grid [9]
and the Nordic power grid [19]. The size of the largest A. Complex network theory
component is a measure of the fraction of nodes in the
In general, the structure of complex systems can be viewed
connected sub-network that contains the largest number of
nodes and any two nodes can be reachable, and has been used as complex networks. One of the most characteristic features of
in the analysis of the Nordic power grid [19]. complex systems is that some of their properties as such are not
deduced directly from the individual properties of their
Additionally, several studies have tried to characterize
network topology by finding common structures or patterns in components. Another very characteristic feature of these
systems is their networked character, which justifies the use of
networks in their representation in a natural way [27]. Modern C.3. Degree distribution
power systems, also referred to as large-scale power grids, Since the nodes in a given electrical power network do not
belong to a typical class of complex systems. all have the same degree, the node degree alone is not enough
The basic purpose of complex network theory is to describe to characterize the network. Degree distribution (that is the
the form and functionality of real-world complex systems by probability that a randomly selected node has exactly 𝑘 links)
modelling them as networks and using different measures. provides a better approach to explain network topology. Several
Complex network theory methods can be applied to the analysis studies have discussed if the degree distribution in power grids
of power systems for i) performing preliminary assessments of follows a power-law or an exponential function. In the power-
vulnerabilities by topological and dynamical analysis and ii) law degree distribution, the probability of finding a high degree
providing elementary information for further detailed analyses node is relatively small in comparison with the high probability
of critical areas [28]. of finding low-degree nodes (that is the probability of a node to
The topological structure of electric power grids is an have 𝑘 links attached to it decays as a negative power of the
important consideration since it can dramatically influence degree: 𝑝(𝑘)~𝑘 −𝛼 ) [27]. Additionally, these networks are also
performance. That is why topological analysis based on referred as ‘scale-free’ networks since the degree distribution is
complex network theory is quite valuable because it offers the always characterized by the same scale 𝛼 , irrespective of
capability of unveiling relevant properties of the structure of a sample size. With respect to vulnerability, scale-free are robust
network system by identifying components of structural against random failure but vulnerable to targeted attacks.
vulnerability, i.e., network lines and nodes whose failures can Exponential-degree distributions are characterized by having a
induce a severe structural damage to the network through the faster decay to zero than power laws, which mean that the
physical disconnection of its parts [28]. probability of having nodes with high degree is slightly larger
B. Network representations of power grids in scale-free networks [20].
The structure of a power grid is a network in which nodes C.4. Average shortest-path length
represent stations (generators, transmission substations, loads), In an undirected network, the shortest path distance 𝑙𝑖𝑗 is the
and links represent the transmission lines between the nodes. number of links in the shortest path between the nodes 𝑖 and 𝑗.
Graph theory provides a natural framework for the
The average shortest-path length (the average of the shortest
mathematical representation of power grids. A network, or
distance 𝑙𝑖𝑗 between all pairs of nodes) and network diameter
graph, is described by 𝐺 = (𝑁, 𝑀), where 𝑁 is the set of nodes,
(the maximum shortest path) characterize the distances among
and 𝑀 is the set of links. Any given network can be uniquely
nodes globally for a network. The average shortest-path length
represented by an 𝑁 × 𝑁 adjacency matrix, 𝐴. If there exists a
is calculated as:
link from a node 𝑖 to a node 𝑗 , then the element 𝑎𝑖𝑗 is 1; 1
〈𝑙〉 = ∑𝑖,𝑗∈𝑁,𝑖≠𝑗 𝑙𝑖𝑗 (1)
otherwise, it is 0. Several structural properties of networks are 𝑁(𝑁−1)
related to adjacency relationships between nodes. By analyzing C.5. Clustering coefficient
the structure of the network or by assessing properties of the
The clustering coefficient quantifies the degree to which
network when it is changed or degraded due to component
nodes are clustered in a graph. Suppose a node 𝑖 is connected to
failures or deliberate attacks on components, relevant
conclusions about the modelled of the power system can be 𝑘𝑖 other nodes, or neighbors. Then the clustering coefficient for
drawn. In this work, six main topological network property a given node 𝑖 is defined as:
2𝑀𝑖
metrics are covered. 𝐶𝑖 = (2)
𝑘𝑖 (𝑘𝑖 −1)

C. Network topological metrics where 𝑀𝑖 and 𝑘𝑖 are the number of links connected to the
node 𝑖 and node degree of the node 𝑖, respectively.
C.1. Size A clustering coefficient equal to 1 indicates that every
Network size is the most basic metric when describing neighbor of node 𝑖 is connected to every other neighbor of node
network structure. Network size is defined by number of nodes, i. Correspondingly, for the whole network the clustering
𝑁, and total number of links, 𝑀 coefficient 𝐶 is defined as the average of the clustering
coefficients of all nodes as:
C.2. Node degree 1
𝐶 = ∑𝑖 𝐶𝑖 (3)
Node degree (k) is defined as the number of links connected 𝑁

to each node. The idea behind the use of node degree as a Also, in this case the clustering coefficient is 1 only when the
network property is that a node is more central or more network is completely connected (all pairs of nodes are directly
influential than another one in a network if the degree of the connected by a link). In general, the clustering coefficient is
first is larger than that of the second. always less than one [26].
C.6. Global network efficiency IV. STATISCAL PROPERTIES OF THE MEXICAN POWER GRIDS
Global network efficiency represents the ease with which a
A. Basic topological properties
graph can transmit information from node i to node j and this
will depend on its shorter path length. This metric is frequently A.1. Size
used to measure the vulnerability of a network when simulating
Comparing the size of the networks MX400-230 with the
cascading failures, and is defined as [29]:
1 1 size of power networks of some other countries (see Table I).
𝐸𝑔𝑙𝑜𝑏𝑎𝑙 = ∑𝑖,𝑗∈𝑁.𝑖≠𝑗 (4)
𝑁(𝑁−1) 𝑙𝑖𝑗 TABLE II
SOME TOPOLOGICAL PROPERTIES OF THE THREE MX NETWORKS
III. THE STUDIED NETWORKS OF THE MEXICAN POWER COMPARED TO THOSE OF SOME EUROPEAN NETWORKS. (𝑁: NUMBER OF
GRID NODES, 𝑀: NUMBER OF LINES, 〈𝑘〉: AVERAGE NODE DEGREE, 〈𝑙〉: AVERAGE
SHORTEST-PATH LENGTH. 𝐶 : CLUSTERING COEFFICIENT, 𝑑 : NETWORK
In this work, three networks from the Mexican power grid DIAMETER).
were analyzed. Two of them correspond to the transmission Met/Country Mexico France Spain Italy Germany
systems of 400 kV (MX400) and 230kV (MX230), and the third 𝑵 429 1659 798 634 782
one is the transmission system with both voltage levels (MX400- 𝑵𝟐𝟑𝟎 367 1273 597 372 302
𝑵𝟒𝟎𝟎 120 386 201 262 480
230). These networks were obtained by extracting information 𝑴 612 2160 1115 812 1090
from the official document National Electrical System 𝑴𝟐𝟑𝟎 468 1479 731 437 341
Development Program 2017-2031 (PRODESEN, for its 𝑴𝟒𝟎𝟎 160 477 284 321 671
acronym in Spanish) [30]. The transmission systems with other 〈𝒌〉 2.85 2.59 2.79 2.53 2.58
voltage levels and the three isolated systems located in the 𝒌𝟐𝟑𝟎 2.55 2.32 2.45 2.35 2.12
peninsula of Baja California were not considered in this work. 〈𝒌𝟒𝟎𝟎 〉 2.66 2.42 2.83 2.40 2.57
〈𝒍〉 10.62 12.17 10.45 11.98 12.19
To analyze these networks, three graphs were generated. The 〈𝒍𝟐𝟑𝟎 〉 19.11 23.69 13.90 10.17 9.58
power plants and substation buses (transformers and switching 〈𝒍𝟒𝟎𝟎 〉 8.23 8.86 7.63 9.62 11.20
stations) were represented as nodes, and the transmission lines 𝑪 0.179 0.071 0.091 0.046 0.127
as links. Since the focus of the analysis is on the grid topology, 𝑪𝟐𝟑𝟎 0.172 0.061 0.103 0.055 0.119
the links and nodes were considered as homogeneous, 𝑪𝟒𝟎𝟎 0.114 0.026 0.097 0.050 0.153
unweighted and undirected (see Fig. 2). 𝒅 32 30 24 32 29
𝒅𝟐𝟑𝟎 50 54 40 30 26
𝒅𝟒𝟎𝟎 25 20 18 27 26

The Mexican network is the smallest one (N = 429 and M =

612), whereas the biggest one is the French network. Although
it has not been clearly identified the precise factors that
determine the number of nodes and lines in the power grid of
each country, apparently the network size is closely related to
electricity consumption.
A.2. Average node degree and degree distribution
With respect to the values of the average node degree
parameter 〈𝑘〉, it is clear that these values are very similar for
all the networks of the different countries in Table I. This is a
characteristic feature that has been found in most of the power
(a) networks worldwide. It can also be appreciated in Table I that
the 400kV networks have bigger average node degree in
comparison with the 230kV networks. This might be explained
by taking into account that the higher the voltage level, the
higher the network connectivity required since the amount of
energy that is transmitted in a power network grows with
voltage level, as it was mentioned above. The degree
distribution of the three MX networks, plotted in Fig. 3, peaks
at about 𝑘 = 2 (most of the nodes have node degree k = 2) but
there is a large number of nodes with node degree 𝑘 > 2.
This implies that a failed substation disconnected from the
network can easily be overtaken through other paths in the
system. Nodes with 𝑘 = 1 are the boundary substations of the
three power transmission networks. Node degree is a measure
(b) that has been widely used in studies of different types of
Fig. 2. Graph representation of the studied Mexican power networks: (a)
230kV, (b) 400kV.
complex networks to assess the importance or level of influence
of nodes on the performance of a given network.
 functions, is that the probability of having high-degree nodes is
less than in a scale-free network.
A.3. Average shortest path length
In the case of network distances among nodes, all the 230kV
networks in Table 1 have higher values of network diameter and
average shortest-path length, except the network of Germany,
compared with the values of the 400 kV networks. Generally
speaking, in almost any power system construction the 230kV
lines are built to connect lower distances than 400 kV lines.
Therefore, this explains that the diameter and average shortest-
path length of 230 kV networks are larger than those of 400 kV
Fig. 3. Node degree distribution of the three MX networks. networks.
In most electric power grids, the distribution of shortest-path
In the particular case of electric power grids, for example, lengths follows a quasi-normal distribution. However, in some
there are studies based on node degree that seek to identify cases distances spread out to larger values, with a positive
critical nodes that are able to cause the propagation of large- skewness [20]. Here, for the case of the MX400-230 network,
scale cascading failures [31]. Other studies are focused on using Fig. 5 shows its shortest-path length distribution P(l), which has
this measure to design strategies to restore the power system a tail up to 32, implying that one has to pass at most through 32
after a blackout problem [32]. That is why this measure is still nodes for the power to be transmitted from one point to another
present in topology analysis of power systems. in the network. This value is the diameter of the network.
Fig. 4 shows the cumulative degree distribution 𝑃(𝑘 ≥ 𝐾) of Also, in Fig. 5, the largest portion of the distribution is
the three MX networks. For each network, the cumulative concentrated around values of 4 and 14, and the distribution
degree distribution follows an exponential function with peaks at 8, implying that the connectivity of this network is
(fitting) coefficient of determination 𝑅2 values acceptably high. Theoretically, for any network the average shortest-path
𝑁+1
large. length is bounded as 1 ≤ 〈𝑙〉 ≤ , where the lower bound is
3
According to [2], the cumulative degree distributions of the obtained for the complete network (fully connected network)
networks shown in Table I also follow the exponential function. and the upper one is reached for the path of N nodes. For the
MX400-230 network, an average shortest-path length 〈𝑙〉 =
10.62 is found. This clearly reflects that the network has good
global connectivity properties. In fact, all networks shown in
Table 1 have good global connectivity properties, where the
French network has the higher connectivity.

Fig. 4. Exponential cumulative degree distribution of the three MX networks.

The β values of these functions are in the range between -0.37

and -0.64. The β value for the MX400-230 network is -0.55, Fig. 5. Shortest-path length distribution of the MX400-230 network.
which is within that range. As Fig. 4 shows, the smaller the β,
the faster the decline and therefore the probability of finding
A.4. Clustering coefficient
nodes with high degree is lower. Additionally, it can also be
seen that, although there is a significant size difference between Another measure which quantifies the connectivity in a
the MX400 and MX230 networks, their cumulative degree network is the clustering coefficient C. Large values of C would
distributions are very close to each other. This implies that there be better for the robustness of the connectivity: The
is no mathematical relation between β and network size. disconnection of two parts of the network by a node removal
Finally, the implication that the cumulative degree distributions would be overcome by simply passing onto adjacent working
of the three MX networks are approximated by exponential nodes through short-range neighboring nodes. In this view,
comparing the networks of Table I, the MX400-230 and conclude that the MXPG demonstrates the small-world
MX230 networks have the higher clustering coefficients. phenomenon, which is present in the Watts and Strogatz model.
Additionally, for the MX400-230 network most of the nodes The high clustering that the MXPG’s structure exhibits
have no links connecting their neighbors (Ci is zero), as it is provides efficient local distribution with paths that are locally
shown in Fig. 6. Nonetheless, the percentage of nodes with all short. Also, at the same time, the small average shortest-path
their neighbors connected (Ci = 1) is above 10%, which can be length gives shortcuts between the local clusters. All these
considered as a high value for an electric power grid. mean that small-worldness of the MXPG’s structure benefits
from a general robustness against attacks: the absence of big
hubs that keep the network together improves reliability.

V. STATIC TOLERANCE TO RANDOM FAILURES AND

INTENTIONAL ATTACKS
A common approach to analyze static tolerance to random
failures and intentional attacks in complex networks consists of
identifying the relationship between node deletion (without
considering any quantity transported by the network links) and
the existence and relative size of the connected component,
after such a deletion (global connectivity). In this work, in order
to compute the effect of random removal of nodes (to simulate
Fig. 6. Distribution of the local clustering coefficient values of the MX400- random failures) on the MXPG, the percolation condition for
230 network. the graphs of the three MX networks is computed.
The complex networks of real-world systems commonly
A.5. Testing the “small-worldness” of the Mexican power grid reveal to be highly robust to the random deletion of their nodes.
In order to fully understand the structure of a power grid, it The simplest robustness indicator in a network is the variation
is required some reference network models with which the (or lack of variation) in the fraction of nodes in the largest
power grid can be compared. Using random models of the three component of the network. For example, in an electrical power
MX networks it is possible to analyze the influence of the network, two nodes can communicate with one another if there
topological parameters reported in Table I on the structure of exist a connecting path between those two nodes; therefore, the
the power grid. Since in the previous section IV.A.2 it was nodes in the largest component can communicate with an
found that the Mexican power grid is not a scale-free network, extensive fraction of the entire network, while those in the small
in this study it was generated an Erdös-Rényi random network components can communicate with only a few others at most.
(ER) and a Watts-Strogatz small-world network for comparison In the numerical studies reported in [33] and [34], it was found
(WS) (see Table II). that the problem of robustness to random failure of nodes in a
network is equivalent to a site percolation process on the
network. Nodes are randomly designated as working or failed
TABLE II nodes, and the number of nodes remaining that can successfully
COMPARISON BETWEEN THE THREE MX NETWORK STRUCTURES AND communicate is precisely the largest component of the
RANDOM (ER) AND SMALL-WORLD (WS) NETWORK MODELS corresponding percolation model [35,36].
Net 𝑵 𝑴 〈𝒌〉 𝒌𝒎𝒂𝒙 〈𝒍〉 𝒅 𝑪
/Metric Let suppose a configuration model with degree distribution
MX 367 468 2.55 9 19.11 50 0.1727 𝑝𝑘 . Now suppose that only a fraction 𝑞 of the nodes are chosen
MX230

ER 344 474 2.75 8 6.3 50 0.0057 randomly as “working” nodes. Since node failure is random and
WS 367 734 4 7 5.09 10 0.1502
MX 120 160 2.66 6 8.23 25 0.1147
uncorrelated, the subset of all nodes that are working forms
MX400

ER 109 158 2.89 7 4.88 25 0.0247 another configuration model with the following degree
WS 120 240 4 7 3.80 7 0.1179 distribution:
MX 429 612 2.85 11 10.62 32 0.179
𝑘 𝑘′
MX400-

𝑘−𝑘 ′
𝑝 𝑘 ′ = ∑∞
230

𝑘=𝑘 ′ 𝑝𝑘 ( ′ ) 𝑞 (1 − 𝑞)
ER 409 617 3.01 10 5.78 32 0.0052 (5)
WS 429 858 4 8 11.70 11 0.1877 𝑘
The critical fraction of nodes 𝑓𝑐 to be eliminated in order to
make a graph unable to percolate (the fraction of nodes at which
The three MX power networks have clustering coefficients the network no longer has a largest component) can be obtained
significantly larger than the clustering coefficients of the ER by the standard generating function methodology. Thus, the
networks, and very close to the ones of the WS networks. With critical fraction of node removal is [37]:
respect to the average shortest-path length, the MX400-230 1
𝑓𝑐 = 1 − (6)
network and its corresponding WS network have this metric 𝑘0 −1
very similar, whereas in the MX400 and MX230 networks their where 𝑘0 is compute with:
average shortest-path lengths are only somewhat larger than
those of the corresponding WS networks. These results lead to 〈𝑘 2 〉
2𝛾 = 𝑘0 ≈ 〈𝑘〉
(7)
and 〈𝑘 2 〉 is the square average node degree determined with:
𝑘𝑖2
〈𝑘 2 〉 = ∑𝑁
𝑖=1 (8)
𝑁

On the other hand, in order to analyze an intentional attack

in terms of standard percolation, the method developed in [38]
gives the conditions for network percolation under attacks as
follows,
1
1 + (ln 𝑝𝑐 − 1)𝑝𝑐 = (9)
2𝛾−1

where 𝑝𝑐 is the critical fraction of attacked nodes to make a

graph unable to percolate.
Table III shows the results of calculating the critical
fraction for random failures and intentional attacks (node- (b)
degree based attacks) of nodes (𝑓𝑐 and 𝑝𝑐 , respectively) for the
three MX networks, and they are compared with those reported
for France and Iran power networks [29]. Additionally, with the
purpose of illustrating this analysis, Figs. 7 and 8 show the
behavior of the MX networks facing intentional attacks and
random failures.

TABLE III
PERFORMANCE OF THE THREE MX NETWORKS AGAINST INTENTIONAL
ATTACKS (𝑝𝑐 )AND RANDOM FAILURES ( 𝑓𝑐 )
Network N M 〈𝒌〉 𝒑𝒄 𝒇𝒄
MX230 367 454 2.44 0.2333 0.5710
MX400 120 160 2.6116 0.2304 0.5687
MX400-230 429 612 2.8531 0.3083 0.6711
Iran (400kV) 105 142 2.7048 0.2598 0.61
France (400kV) 667 899 2.69 0.2841 0.6415
(c)

As expected, a much lower value of 𝑝𝑐 is requiered to break

into isolated tiny unconnected islands a power grid through
intentional attack. For example, the MX400-230 network will
be unable to percolate at about 30% and 67% of nodes removed
intentionally (based on their degree) and randomly,
respectively. This mean that the Mexican power grid is more
robust to random failures compared with intentional targeted
attacks. This is one of its characteristics as a grid with small-
world architecture.

(d)

Fig. 7. Static tolerance of the MX230 network confronting the intentional

attack (by removing high degree nodes) for different values of the critical
fraction pc.

Fig. 7 shows four moments where we can see how the

MX230 power network degrades based on the target attacks at
the higher-grade nodes, passing through a fraction of 0.0163
until reaching the critical fraction of 0.2333.
(a)
 VI. VULNERABILITY ANALYSIS

In this study, the vulnerability of the Mexican power grid

under random failures and intentional attacks is analyzed by
calculating the drop in the global network efficiency. Also, the
identification of critical nodes is obtained by measuring the
global network efficiency degradation due to the disconnection
of one node at a time.
In Table IV, the values of the global network efficiency of
the three MX networks are shown, and they are compared with
(a) those of some European power networks [14].

TABLE IV
COMPARISON BETWEEN THE GLOBAL NETWORK EFFICIENCY OF THE
THREE MX NETWORKS WHIT THOSE OF SOME EUROPEAN NETWORKS

Network 𝑬𝒈𝒍𝒐𝒃𝒂𝒍
Mexico MX230 0.0858
Mexico MX400 0.1793
Mexico MX400-230 0.1293
France 400kV 0.197
Spain 400kV 0.259
Italy 380kV 0.173
Swiss 220-380kV 0.205

(b)

In complex network theory, small-world networks have

high 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 corresponding to low average shortest-path length.
Therefore, since Mexico’s power grid demonstrates the small-
world phenomenon, it can be said that these 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 values of
the three MX networks are high. Fig. 9 presents the impact of
random failures (random removed nodes) and degree-based
intentional attacks (large-degree removed nodes) on the global
network efficiency of the three MX networks. Results indicate
that all the three networks are more robust against random
failures than intentional attacks. Analyzing the MX400-230
network, after 10% of nodes randomly removed, the value of
𝐸𝑔𝑙𝑜𝑏𝑎𝑙 was 0.115, whereas global network efficiency reached
(c) almost zero for the case of intentional attacks with the same
portion of nodes removed. This means that the impact of
intentional attacks is more prominent than random failures.

(d)

Fig. 8. Static tolerance of the MX230 network confronting random

failures (by randomly removing nodes) for different values of the critical
fraction fc. (a)
 (b) (a)

(c) (b)

Fig. 9. Impact of random failures and intentional attacks of nodes on the global
network efficiency of the three MX networks: (a) MX230, (b) MX400 and (c)
MX400-230

With respect to the identification of critical nodes, in this

work the criticality of a node is determined by estimating its
vulnerability in terms of the amount of relative decrease in the
global network efficiency caused by its removal [29]. Thus, the
vulnerability of a node is obtained as:
∆𝐸𝑔𝑙𝑜𝑏𝑎𝑙
𝑉𝑖 = (10)
𝐸𝑔𝑙𝑜𝑏𝑎𝑙

where ∆𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is the amount of change in global network

efficiency when the node i is removed from the network, and
𝐸𝑔𝑙𝑜𝑏𝑎𝑙 corresponds to the original network. Fig. 10 shows the
histogram of node vulnerability values for the three MX (c)
networks. In order to explain these histograms, let’s take the Fig. 10. Histograms of node vulnerability defined as the percentage change
result of the MX230 network, whose histogram indicates that in the global network efficiency when a node is removed from the network:
there are few nodes that cause a substantial change in the global (a) MX230; (b) MX400; (c) MX400-230
network efficiency. For example, there are three nodes that reach
a vulnerability value of 0.26, which means the network has an
efficiency loss of 26% if one of these nodes is removed (see In order to illustrate the results of Table V with one
Fig.10 (a)) [30]. On the other hand, there are 77 nodes that their example, Fig. 11 shows the location of the most critical nodes
removal would not cause any global network efficiency loss, and (nodes 200, 220 and 221) of the MX230 network. These nodes
even there are also 71 nodes that their removal would represent are identified as load type and they are located in a power
an increase in the global network efficiency of up to 0.001. In transmission corridor with a capacity of 1500MW [30].
these terms, Table V shows the most critical nodes for the three
MX networks.
 TABLE V the range of 4 and 14 for the case of the 400-230 kV network,
THE THREE MOST CRITICAL NODES FOR THE THREE MX NETWORKS IN TERMS meaning that power energy is transmitted through a high
OF EGLOBAL LOSS BY REMOVING THEM FROM THE NETWORK (NODE
VULNERABILITY)
percentage of shortest-path lengths.
Network Most vulnerable 𝑬𝒈𝒍𝒐𝒃𝒂𝒍 k Node type With respect to the results of the small-worldness test, the
nodes (node loss in %
number)
Mexican power networks lie in the group of power networks
200 26.0 4 Load that were analyzed and reported in the literature as small-world
MX230 220 26.1 2 Load networks. Additionally, the Mexican power networks shows
221 26.6 3 Load exponential cumulative degree distributions in agreement with
132 5.64 5 Power substation some other power networks already identified with this feature.
MX400 87 5.84 6 Power substation
96 6.09 5 Load In relation to the vulnerability analysis to random failures
239 4.1 8 Power substation and intentional attacks, the Mexican power grid is vulnerable to
MX400-230 231 4.1 9 Load degree-based intentional attacks and robust to random failures,
385 4.4 7 Power substation
according to the criteria of critical fraction of node removal (site
percolation process on networks) and global network
efficiency. This feature can be attributed to its skewed node
degree distribution, where a large number of nodes have small
node degree, and a small number of nodes have very high node
degree.
In summary, the main outcome of this work is that, although
the Mexican power grid has evolved and developed under
specific economic, political, and environmental (geographical)
conditions of Mexico, the power grid shares the same basic
topological properties with many other power grids of different
countries. Also, the MXPG is a small-world network with good
global connectivity properties and topological robust to random
failures.
It is expected that this work can be useful for understanding
Fig. 11. Location of the three most critical nodes in the MX230 network. and modeling different dynamic processes that occur in the
Mexican power grid, such as cascade failures and the resilience
to external and internal events.
VII. CONCLUSIONS

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