University of Groningen
The Power Grid as a complex network
Pagani, Giuliano Andrea; Aiello, Marco
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Physica A: Statistical Mechanics and its Applications
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10.1016/j.physa.2013.01.023
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Pagani, G. A., & Aiello, M. (2013). The Power Grid as a complex network: A survey.
Physica A: Statistical
Mechanics and its Applications
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392
(11), 2688-2700. https://doi.org/10.1016/j.physa.2013.01.023
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Physica A 392 (2013) 2688–2700
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The Power Grid as a complex network: A survey
Giuliano Andrea Pagani
∗
, Marco Aiello
Distributed Systems Group, Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands
a r t i c l e i n f o
Article history:
Received 10 October 2012
Received in revised form 3 January 2013
Available online 11 January 2013
Keywords:
Power Grid
Infrastructure reliability
Complex Network Analysis
Graph theory
a b s t r a c t
The statistical tools of Complex Network Analysis are of useful to understand salient prop-
erties of complex systems, may these be natural or pertaining human engineered infras-
tructures. One of these that is receiving growing attention for its societal relevance is that
of electricity distribution. In this paper, we present a survey of the most relevant scientific
studies investigating the properties of different Power Grids infrastructures using Complex
Network Analysis techniques and methodologies. We categorize and explore the most rel-
evant literature works considering general topological properties, physical properties, and
differences between the various graph-related indicators and reliability aspects. We also
trace the evolution in such field of the approach of study during the years to see the im-
provement achieved in the analysis.
© 2013 Published by Elsevier B.V.
1. Introduction
Complex Network Analysis (CNA) is a relatively young field of research. The first systematic studies appeared in the late
1990s [1–4] having the goal of studying the properties of large networks that behave as complex systems. The research
owes a great deal of its foundations to the seminal work on Random Graphs of Erdős and Rényi [5,6] who studied
asymptotic properties of stochastic graph processes. The research on the topic has embraced the spatial structure of
networks [7], the dynamical aspects [8], and more and more applications in the natural and artificial world benefit from
analyzing systems using the network approach [9]. Complex Network Analysis has been used in many different fields of
knowledge, from biology [10] to chemistry [11], from linguistics to social sciences [12], from telephone call patterns [13]
to computer networks [14] and web [15] to virus spreading [16] to logistics [17] and also inter-banking systems [18].
Men-made infrastructures are especially interesting to study under the Complex Network Analysis lenses. Especially, those
characterized by large scale and growth following a decentralized and independent fashion, thus not the result of a global,
but rather of many local autonomous designs. The Power Grid is a prominent example. But what do we mean by Power
Grid in the context of the present treatment?
We focus on the electricity transmission and distribution Power Grid as it is essential for today’s society as an enabling
infrastructure. Power Grid efficiency and operations have major consequences, among other things, for the environment.
Blackouts seem to have a special role in reminding us of the importance of the Grid and how much we give its availability for
granted. From the technological point of view, the electrical system and Power Grid involve many scientific knowledge areas
that contribute to the design, operations and analysis of power systems: Physics (electromagnetism, classical mechanics),
Electrical engineering (AC circuits and phasors, 3-phase networks, electrical systems control theory) and Mathematics
(linear algebra, differential equations). Traditional studies tend to focus on specific aspects of the Grid, e.g., defining how
to design a transformer and predicting its functioning. Typically, studies tend to consider on the physical and electrical
∗
Corresponding author. Tel.: +31 503633984.
E-mail addresses: g.a.pagani@rug.nl (G.A. Pagani), m.aiello@rug.nl (M. Aiello).
0378-4371/$ – see front matter © 2013 Published by Elsevier B.V.
doi:10.1016/j.physa.2013.01.023
G.A. Pagani, M. Aiello / Physica A 392 (2013) 2688–2700 2689
properties (e.g., Ref. [19]), or the characteristics of the Power Grid as a complex dynamical system [20], or again, the control
theory aspects [21]. The move from a ‘‘local’’ to a ‘‘global’’ view of the Power Grid as a complex system is possible by resorting
to Complex Network Analysis and statistical graph theory.
The goal of the present treatment is to provide a survey and compare the most relevant and well-known scientific
studies conducted using Complex Network Analysis techniques concerning Power Grid systems. We consider the field
mature enough to deserve a survey classifying the approaches, the geographies, the models used and the results of network
reliability analysis. To the best of our knowledge, there are only two other surveys using Complex Network Analysisas its
foundations and they are both older than five years [22,23]. The work of Sun [22] is a basic comparison of the then available
works using Complex Network Analysis for the Power Grid more to show a new method than showing the differences and
peculiarities followed by the studies. On the same tone is the work of Bai et al. [23]: very few works are surveyed (less than
five) belonging to few geographies without stressing the commonalities or differences between the Grids and giving few
space to the reliability aspects. Our survey is more comprehensive, with more than 30 works analyzed, considering several
studies that have emerged recently that take into account not only pure topological analysis, but also physical parameters are
considered in modeling of the network. In addition, our work examines several parameters (e.g., node degree, betweenness,
geography, type of network, small-world, network attacks and improvement strategies) to assess the differences between
the various studies under survey with particular stress on the resilience aspects of the analysis. Thus, our modern study
allows us to have a broader view on the topic, emphasizing the differences between works that analyze different Power
Grids and considering the latest results obtained in the recent years using enriched Complex Network Analysis models that
use also physical properties of electrical lines. In addition, with the growing interest on the Smart Grid topic, we consider
beneficial to have a broad view of the characteristics of the Grids in several geographies around the globe since the Smart
Grid will also require changes and updates in the Power Grid infrastructures worldwide. The paper is organized as follows:
we start by introducing the methods and metrics that are evaluated in this work (Section 2). Section 3 provides the main
characteristics of all the studies surveyed. The actual comparison of these using CNA metrics are reported and discussed in
Section 4, while Section 5 concludes the paper.
2. Background and survey methodology
Before going further analyzing the various studies in detail, we consider the reader familiar with basic notion of graph
theory and Complex Network Analysis, otherwise we refer to established literature for a general introduction [24,25].
As described in Section 1, all the works that are examined in the present manuscript consider the Power Grid networks
as graphs following the mathematical meaning of the term.
The main investigation that is usually performed when analyzing the Power Grid and that is almost always the motivation
that drives Complex Network Analysis studies related to electrical infrastructures is the investigation of reliability. Usually,
the investigation involves evaluating the disruption behavior of the graph when its nodes or edges are removed.
Other terms to compare the various Power Grid studies involve more general characteristics of the network under
analysis. In particular, the geographical location of the analyzed Grid is responsible for topological properties due to the
different morphological characteristics of different countries. Another relevant aspect deals with the layer of the Power
Grid under investigation since differences can emerge from a topological perspective investigating the different ends in
which the Grid is usually partitioned: High, Medium and Low Voltage. It is also important to have information if the type
of Power Grid graph under analysis comes from a real network infrastructure or it is a synthetic sample extracted from
blueprint models for the Power Grid such as the Bus models of IEEE.
The motivations to include the works in this survey are based on the quality of the research performed, the rigor in the
application of Complex Network Analysis methodologies and the geography of the Power Grid analyzed in order to cover a
broad spectrum of the infrastructure realized in the different countries and identify possible differences.
3. The Power Grid as a complex network
Complex Network Analysis studies are becoming more and more popular given the amount of natural and human
complex systems. The Power Grid is clearly amenable to such studies and a number of these have been performed on the
High Voltage Grid. Here we describe the most important aspects of each work under investigation. In particular, the works
that are considered in this review are: Refs. [22,26–57]. These have been chosen based on the following factors: they are
specifically about the Power Grid, they cover US, European, Chinese Grids or synthetic topologies of electrical engineering
literature, they have samples of different sizes and, most importantly, these are the best-known and most representative
works on the topic of CNA and Power Grid.
3.1. Basic Power Grid characteristics
The aspects considered in this first basic assessment of the studies take into account general and non-technical aspects so
to give a global idea of the Grid considered, Table 1. Several aspects of comparison are considered: the number of nodes and
2690 G.A. Pagani, M. Aiello / Physica A 392 (2013) 2688–2700
Table 1
Comparison between studies using CNA for the Power Grid.
Work Number of nodes Number of lines Sample type Network type Geography
[26] ∼14,000 ∼19,600 Real HV North America
[27] ∼300 ∼500 Real HV Italy
[28] ∼314,000 NA Real HV North America
[29] ∼4800 ∼5500 Real HV Scandinavia
[30] ∼2700 ∼3300 Real HV Europe
[31] ∼3000 ∼3800 Real HV Europe
[32] ∼3000 ∼3800 Real HV Europe
[33] ∼370 ∼570 Real HV Italy, France and Spain
[34] ∼370 ∼570 Real HV Italy, France and Spain
[35] ∼4900 ∼6600 Real HV Western US
[58] ∼8500 ∼13,900 Synthetic and real HV Western US and New York State Area
[36] ∼4850 ∼5300 Real MV/LV Netherlands
[37]
a
∼210 ∼320 Synthetic and real HV China
[38] NA NA Real HV Europe
[39] 300 411 Synthetic HV
[40] ∼6400 ∼8700 Synthetic and real HV North America, Scandinavia and Korea
[41] 300 411 Synthetic HV
[42] ∼8500 ∼13,900 Synthetic and real HV Western US and New York State Area
[43] ∼30 ∼13,900 Synthetic and real HV Western US and New York State Area
[44] ∼900 ∼1150 Real HV China
[45] ∼3200 ∼7000 Synthetic and real HV New York State Area
[46] ∼4900 ∼6600 Real HV Western US
[47] ∼1700 ∼1800 Real HV China
[48] ∼39 ∼46 Syntethic HV
[49] ∼39 ∼46 Syntethic HV
[50] ∼2500 ∼2900 Real HV China
[22] ∼15,400 ∼18,400 Real HV North America and China
[51] ∼550 ∼800 Synthetic HV
[52] ∼14,000 ∼19,600 Real HV North America
[53] ∼90 ∼120 Synthetic HV
[54] ∼550 ∼700 Synthetic and real HV Italy
[55] ∼29,500 ∼50,000 Synthetic and real HV North America
[56] ∼400 ∼700 Synthetic
[57] ∼900 ∼1300 Synthetic and real HV South-East US
[59] ∼60 ∼110 Real HV India
a
The values for nodes and lines in this table refer only to a snapshot of Shanghai Power Grid.
lines composing the Grid (second and third column)
1
; the type of sample considered either a real Grid or synthetic samples,
for instance, coming from IEEE literature such as IEEE Bus systems (fourth column); the type of Grid analyzed (fifth column)
in belonging either to the transmission part (High Voltage) or to the distribution part (Medium and Low Voltage); another
essential information deals with the geography of the Grid (last column).
The data are in the most cases extracted from real samples, that is, they represent real electric infrastructures deployed.
Other works in addition to real Power Grids consider synthetic models as shown in Fig. 1. Most of these studies investigating
synthetic samples use IEEE blueprint networks such as IEEE Bus systems (a representation of the various IEEE Bus models
used in the surveyed articles is shown in Fig. 2), while very few concentrate only on other synthetic networks (e.g., non-IEEE
models, small-world models, random graphs). The number of synthetic models used is shown in Fig. 3 and almost 85% of
them consider IEEE literature. Almost all samples belong to the High Voltage end of the Power Grid. High Voltage contains
the lines used for long range transmission to which big power plants are attached too; the only exception is our study [36]
that is focused on the distribution part of the Grid (i.e., Medium and Low Voltage network).
From a geographical perspective the samples are mainly localized in the United States or in Europe with some studies
that consider Chinese High Voltage samples; a map of the countries whose Grids are analyzed is represented in Fig. 4, while
the number of Grids analyzed for a given country is shown in Fig. 5. Another main commonality is to treat the Grid as an
undirected graph where each substation or transformer represents a node and each line transporting electricity is an edge.
3.2. Statistical global graph properties
The main characteristics from a graph and Complex Network Analysis perspective of the Grids under analysis are
summarized in Table 2. Several aspects of comparison are considered: the order (N) and size (M) of the graph (second and
third column) corresponds to the number of nodes (order) and number of lines (size) actually in the Power Grids. The average
1
Notice that the numbers in the second and third column are not the exact numbers, but they are an approximation to give the idea of the importance
of the sample.
G.A. Pagani, M. Aiello / Physica A 392 (2013) 2688–2700 2691
Fig. 1. Number of studies that consider real Power Grid samples or synthetic models.
Fig. 2. IEEE literature bus model analyzed.
Fig. 3. Number of models used coming from the IEEE literature compared to other models.
113
Fig. 4. Map of the Power Grid infrastructure studied using CNA approach.
