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Optimizing the Structure of Distribution Smart Grids with Renewable Generation against Abnormal Conditions: A Complex Networks Approach with Evolutionary Algorithms

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In this work, we describe an approach that allows for optimizing the structure of a smart grid (SG) with renewable energy (RE) generation against abnormal conditions (imbalances between generation and consumption, overloads or failures arising from the inherent SG complexity) by combining the complex network (CN) and evolutionary algorithm (EA) concepts. We propose a novel objective function (to be minimized) that combines cost elements, related to the number of electric cables, and several metrics that quantify properties that are beneficial for SGs (energy exchange at the local scale and high robustness and resilience). The optimized SG structure is obtained by applying an EA in which the chromosome that encodes each potential network (or individual) is the upper triangular matrix of its adjacency matrix. This allows for fully tailoring the crossover and mutation operators. We also propose a domain-specific initial population that includes both small-world and random networks, helping the EA converge quickly. The experimental work points out that the proposed method works well and generates the optimum, synthetic, small-world structure that leads to beneficial properties such as improving both the local energy exchange and the robustness. The optimum structure fulfills a balance between moderate cost and robustness against abnormal conditions. Our approach should be considered as an analysis, planning and decision-making tool to gain insight into smart grid structures so that the low level detailed design is carried out by using electrical engineering techniques.
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energies
Article
Optimizing the Structure of Distribution Smart Grids
with Renewable Generation against Abnormal
Conditions: A Complex Networks Approach with
Evolutionary Algorithms
Lucas Cuadra 1,*, Miguel del Pino 1, José Carlos Nieto-Borge 2and Sancho Salcedo-Sanz 1
1Department of Signal Processing and Communications, University of Alcalá, Alcalá de Henares,
28805 Madrid, Spain; mdelpino45@hotmail.com (M.d.P.); sancho.salcedo@uah.es (S.S.-S.)
2Department of Physics and Mathematics, University of Alcalá, Alcalá de Henares, 28805 Madrid, Spain;
josecarlos.nieto@uah.es
*Correspondence: lucas.cuadra@uah.es; Tel.: +34-91-885-66-98
Academic Editor: Guido Carpinelli
Received: 16 May 2017; Accepted: 19 July 2017; Published: 26 July 2017
Abstract:
In this work, we describe an approach that allows for optimizing the structure of a smart
grid (SG) with renewable energy (RE) generation against abnormal conditions (imbalances between
generation and consumption, overloads or failures arising from the inherent SG complexity) by
combining the complex network (CN) and evolutionary algorithm (EA) concepts. We propose a novel
objective function (to be minimized) that combines cost elements, related to the number of electric
cables, and several metrics that quantify properties that are beneficial for SGs (energy exchange at the
local scale and high robustness and resilience). The optimized SG structure is obtained by applying
an EA in which the chromosome that encodes each potential network (or individual) is the upper
triangular matrix of its adjacency matrix. This allows for fully tailoring the crossover and mutation
operators. We also propose a domain-specific initial population that includes both small-world and
random networks, helping the EA converge quickly. The experimental work points out that the
proposed method works well and generates the optimum, synthetic, small-world structure that
leads to beneficial properties such as improving both the local energy exchange and the robustness.
The optimum structure fulfills a balance between moderate cost and robustness against abnormal
conditions. Our approach should be considered as an analysis, planning and decision-making tool to
gain insight into smart grid structures so that the low level detailed design is carried out by using
electrical engineering techniques.
Keywords: robustness; abnormal conditions; smart grid; complex network; evolutionary algorithm
1. Introduction
Motivation: The growing importance of renewable energy (RE) sources in the current energy mix
is essential to decrease the economic and geopolitical dependence on fossil fuels and to reduce the
emission of CO
2
, one of the causes of climate change [
1
] and global warming [
2
]. Efficiently integrating
distributed RE generation systems [
3
5
] is a key research topic because the most used renewable
energies—photovoltaic (PV) solar energy [
6
,
7
], wind energy [
8
,
9
] and marine energy [
10
]—are
intermittent and more difficult to store [
11
] and integrate without affecting the quality of the electrical
network [
12
] or the electricity prices [
13
]. The current proliferation of small-scale urban PV in
buildings [
14
] and urban wind generators [
15
] can help home electricity consumers become also
producers (“prosumers”) [
16
] using the smart grid (SG) [
17
,
18
] and micro-grids (
µ
Gs) [
19
] concepts.
Energies 2017,10, 1097; doi:10.3390/en10081097 www.mdpi.com/journal/energies
Energies 2017,10, 1097 2 of 31
On the one hand,
µ
Gs exhibit the potential to cost-efficiently increase the use of RE along with
the power supply reliability, as studied by Wang and Huang [
20
22
]. Specifically, [
20
] focuses on an
optimization methodology that can efficiently integrate distributed energy resources by leveraging
complementary resources, such as, solar and wind REs and energy storage. Using real data, [
21
]
proposes a framework for planning micro-grid systems that aims at increasing the use of RE along
with the reliability of power supply. Even more, the method proposed in [
22
] aims at enabling the
bidirectional exchange of power among interconnected micro-grid, increasing the global efficiency.
As long as the increasing penetration of distributed RE resources is one of the driving forces for
micro-grids deployments [
20
22
], other catalysts for change are some new loads such as electric vehicles
(EV) [
23
], data centers [
24
] and home RE-prosumers [
25
]. In this context, distribution systems (DSs)
involve complex issues such as modeling their sensitivity with respect to distributed RE sources [26],
the efficient control of distributed generation [4] or scheduling problems [27].
On the one hand, the SG paradigm is a relatively novel conception of the electric power network
that, based on hi-tech monitoring, control and communication technologies [
28
30
], aims not only
to efficiently integrate RE sources, but also to supply reliable and safe electric power. As mentioned,
thanks to the efficient integration of distributed REs via the SG [
31
], electricity consumers can also
become prosumers. The SG approach allows for the bidirectional exchange of electric energy at the
local scale, which is very positive because it stimulates the local production (small-scale photovoltaic
systems and small-wind turbines) and consumption, helping end-users obtain economic benefits by
selling the energy generated in excess [
30
]. Integrating small-scale renewable energies is thus one
of the driving forces that is fueling the evolution of conventional grids to smart grids. The second
driving force, inter-related with the RE integration, is the pressure for unbundling the energy sector
(as occurred in access telecommunication networks). Ideally, unbundling the electric sector would
allow everyone to generate electricity, becoming a seller on a free energy market [
30
]. The distribution
medium and low voltage parts of the power grid are the best candidates for unbundling the electric
market. In this respect, smart grids are now becoming the enabling technology for not only the
unbundling of electric sector through the integration of small-scale renewable energies, but also for the
efficient integration of electric vehicles [
23
], which are increasingly important in the effort of reducing
air pollution in big cities [1].
In this complex context, abnormal operating conditions in SGs with RE generation can be caused
by the occurrence of: (1) random failures (such as imbalances between generation and consumption,
the presence of overloads or failures arising from the inherent SG complexity [
32
], which can cause
cascading failures); and (2) targeted or intentional attacks [28,29,33].
The vulnerability to abnormal operating conditions can be studied from different viewpoints
that include methods from both the Electrical Engineering (EE) and the complex network (CN) fields.
In turn, vulnerability in power grids using CN concepts is a broad research area that involves two
different approaches [
33
]. The first one is based solely on “topological” concepts and use metrics
such as the mean path length, the clustering coefficient or the betweenness centrality, among many
others [
33
]. Aiming at enhancing the topological approach, the second “hybrid” methodology consists
of introducing concepts arising from EE into the CN framework and takes advantage of novel electric
metrics, such as those belonging to the “extended topological model” [
34
]. Regarding the first
topological approach, there is a controversy [
34
,
35
] about whether or not it is able to give physical
insights into all aspects of real power grids. The CN community argues that its approach does not aim
to focus on the detailed operation, but to find out the unexpected emergence of collective behavior (for
instance, the synchronization in smart grids [
36
]). Conversely, part of the EE community asserts that
this leads to an unreasonable simplification [
33
35
]. This controversy, not yet resolved and recently
discussed in [
33
], is the reason why we devote Section 5to clarifying this and other issues, after
introducing the necessary background.
Regardless of this debate, smart grids have been studied very recently by
Pagani et al. [16,30,35,37,38]
, on the basis of real data extracted from low and medium voltage
Energies 2017,10, 1097 3 of 31
power grids. These works propose successful strategies to evolve the already deployed conventional
grids into smart grids. Instead of grid evolution, the research line explored in [
39
41
] has adopted the
different approach of generating synthetic smart grid structures.
Purpose and contributions: Within the aforementioned context, the two-fold purpose of this paper
consists of: (1) modeling the topological structure of distribution SGs with RE generation using
CN concepts; and (2) minimizing the negative effects of abnormal events by maximizing the grid
robustness by using an evolutionary algorithm (EA) tailored for this goal. The SG is represented by
a graph, a set of nodes (generators and loads) that are connected to each other by means of links
(equivalently, electric cables). With this in mind, the contributions of our paper are:
1.
We model a smart grid with RE generators and loads (prosumers) as an undirected graph
G
so
that each link allows for the bidirectional exchange of electric energy.
2.
We propose an objective function to be optimized that combines cost elements (related to the
number and average length of links and also to the number of nodes with many links) and
several properties that are beneficial for the SG (such as energy exchanges at local scale and high
robustness and resilience). Our optimization problem includes some restrictions used in [
30
]
and also others that help our EA find optimal synthetic structures for the SG, starting from
scratch. This is a “greenfield” strategy, used by companies in those zones where they do not have
infrastructure, deploying thus the new grid starting from scratch. This is another difference when
compared to [
30
], in which the authors have just adopted a “brownfield” approach aiming at
evolving the conventional low voltage power grid into a smart grid.
3. We use an EA with a problem representation in which the chromosome cG, which encodes each
potential graph
G
(or individual), is the upper triangular matrix of its “adjacency matrix”,
AG
.
In this formulation,
AG
is a square, symmetric and binary matrix in which any element
aij
encodes
whether node
i
is linked to node
j
(
aij =
1) or not (
aij =
0) [
42
]. Since there is no self-connected
node, the adjacency matrix has zeros on its main (principal) diagonal (
aii =
0). These are the
reasons why the connection information in graph
G
is stored by its upper triangular matrix
TG
. Thus, chromosome
cG=TG
encodes in a compact form the graph
G
. As will be shown in
detail in Section 2, this encoding is different from others found in the literature using EAs on
graphs, such as, for instance, a chromosome formed by a one-dimensional array with
N
elements
(the number of graph nodes) [
43
],
N
-length chromosome of two-dimensional elements [
44
]
(where a node is specified by its location in the graph) or a set of vectors in which each allele
(or gene value) represents a community [
45
]. The mutation and crossover operators are fully
adapted to our encoding. This approach could be generalized by considering the strength of the
connection between node iand jin terms of its link weight wij .
Paper Positioning: There are several research works that have applied EA and CN concepts to smart
grids problems, which are partially related to our proposal and whose detailed discussion we postpone
to Section 2for clarity. There are also many research papers that focus on studying the smart grid from
the point of view of CNs and graph theory and others that study graph problems using evolutionary
computation (EC) techniques, in general, and EAs, in particular. However, a combined EA-CN
approach to optimize the topology of smart grids, based on a variety of design constrains, has not yet
been carried out to the best of our knowledge.
Practical perspectives: Our approach should be considered as a high level analysis, planning and
decision-making tool to gain insights into how to design robust structures for smart grids and does
not attempt and cannot replace the well-founded techniques of EE. Because of its importance and to
make this paper stand by itself, we devote Section 5, as mentioned before, to justify the consistency
of our proposal. The synthetic structure provided by our EA can be taken as a starting point to test
whether or not it fulfills all of the electrical requirements. In this sense, our approach can be considered
Energies 2017,10, 1097 4 of 31
as a complementary high-level tool, so that the low level detailed design is carried out by using
EE techniques.
Paper organization: The rest of this paper is organized as follows: Section 2reviews those works
that are related to our approach to a greater or lesser extent. Sections 3and 4introduce, respectively,
topological and hybrid CN concepts that will assist us in better explaining our method, while Section 5
discusses to what extent these CN approaches are useful in power grids. Sections 6and 7state,
respectively, the SG topology optimization problem and the particular EA we propose to solve it.
Section 8discusses the experimental work we have carried out. Finally, Section 9summarizes the key
findings and conclusions.
For the sake of clarity, Table 1lists the symbols used in this paper.
Table 1. List of symbols used in this work.
Symbol Definition or Meaning
AGAdjacency matrix of graph G.
aij
Element of the adjacency matrix
AG
that encodes whether node
i
is linked to node
j
(
aij =
1) or not
(aij =0).
¯
b1Mean value of betweenness b1or multi-scale vulnerability of order 1.
¯
b2Mean value of of the multi-scale vulnerability of order 2.
bp
lBetweenness centrality of link l.
bp(G)Multi-scale vulnerability of order pof a graph G. It is defined by Equation (9)
C=¯
CMean clustering coefficient of a network. It is defined by Equation (4).
CSet of all chromosomes.
CB(v)Betweenness centrality of node v. It quantifies how much a node vis found between the paths linking
other pairs of nodes. It is defined by Equation (5).
cGChromosome that encodes the graph G.
Ci
Clustering coefficient of node
i
. It is defined as the ratio between the number
Mi
of links that exist
between these kivertices and the maximum possible number of links (Ci.
=2Mi/ki(ki1).
CRG Clustering coefficient of a random graph
DNode degree matrix: diag(k1,· · · ,kN). It is the diagonal matrix formed from the nodes degrees.
dE(ni
,
nj)Euclidean distance between any pair of nodes niand njin a spatial network.
dij
Distance between two nodes
i
and
j
. It is the length of the shortest path (geodesic path) between them,
that is, the minimum number of links when going from one node to the other.
b1Coefficient of variation for betweenness. It is defined by Equation (10).
fOBJ(G) = fζ(G) = objective function to be minimized. It is defined by Equation (11).
¯
fζMean value of the objective function fζ.
GGraph representing a network.
GSet of all possible connected graphs Gwith Nnodes and M=Nllinks.
GSet containing all of the candidate graphs.
GζOptimum graph that solves the objective function with combination parameter ζ.
hkiAverage node degree: hki=1
NN
i=1ki.
ki
Degree of a node
i
. It is the number of links connecting
i
to any other node. It is defined by Equation (2).
kMAX Maximum node degree.
`
Average path length of a network. It is the mean value of distances between any pair of nodes in the
network. It is defined by Equation (3).
LSet of links (edges) of a graph.
LGLaplacian matrix (or Kirchhoff matrix) of graph G. It is defined by Equation (14).
`RG Average path length of a random graph.
λ2(G)Algebraic connectivity of graph G.
MSize of a graph G= (N,L). It is the number of links in the set L. It is defined by Equation (1).
NSet of nodes (or vertices) of a graph.
N
Order of a graph
G= (N
,
L)
. It is the number of nodes in set
N
, that is the cardinality of set
N
:
N=|N|card(N).
P(k)Probability density function giving the probability that a randomly selected node has klinks.
pcross Crossover probability.
pmut Mutation probability.
pselec Selection probability.
pij Normalized weight of the link between nodes iand j:pi j .
=wij
jwij .
Energies 2017,10, 1097 5 of 31
Table 1. Cont.
Symbol Definition or Meaning
Psize Population size.
¯
SAverage entropic degree.
SiEntropic degree of node idefined by Equation (6).
σb1Standard deviation of betweenness.
TGUpper triangular matrix of graph G.
Tsize Tournament size.
WSet of weight elements wij .
wij Weight of link lij. It models the strength of the connection between node iand j.
ζ
Parameter that controls the linear combination between components with opposing trends in the
objective function to be minimized given by Equation (11).
2. Related Work
For the sake of clarity, we have divided this section into two subsections. Section 2.1 discusses
the research papers that focus on studying the smart grid from the viewpoint of complex networks
and graph theory. Section 2.2 reviews those articles that tackle graph problems using evolutionary
computation techniques, and EAs, in particular. To the best of our knowledge, there is no work
combining both branches of knowledge for the problem of optimizing the structure of smart grids.
2.1. The Smart Grid as a Complex Network: Related Work
The SG paradigm has been modeled very recently as a complex network in a series of papers by
Pagani et al. [
16
,
30
,
35
,
46
53
]. The approach adopted in these works is based on the need for improving
the low voltage power grid, motivated in Section 1, and aims to analyze and adapt the already
deployed distribution power grids on the basis of complex network approaches [
30
]. The ultimate
goal of such a series of papers consists of putting into practice a decision support system to guide
operators, utilities and policy makers to evolve the current grid to a more efficient smart grid.
These works cover a significant variety of research topics appearing in smart grids: the study of
the potential of the SG to generate big data [
48
], the search for topological vulnerabilities [
50
], the study
of the optimal spatial distribution [
49
], the use of agent-based systems for deregulated smart grids [
52
],
the modeling of dynamic prices [
51
], the study of the last mile of the SG [
47
] and other technological
aspects of SGs as CNs [
16
,
30
,
46
]. With regard to the latter, the authors carried out a topological analysis
of some representative medium and low voltage power grids [
46
] and obtained a set of metrics based
on topological properties in the effort of including their influence on the cost of electricity distribution.
In the next step in this line of research, [
16
] has studied complex models to evaluate to what extent these
allow for local electricity exchange, finding out that: (1) increasing the connectivity from the current
value of the average degree
(hki ≈
2
)
to higher values is beneficial; and (2) the small-world complex
network with average degree
hki ≈
4 fulfills a feasible balance between performance enhancement and
cost. The key, more recent work [
30
] explores a variety of feasible evolution strategies towards the SG
and applies them to the real Dutch distribution system. An interesting finding in [
30
] is that increasing
the connectivity leads to a topology that could lead to a more efficient and reliable electric grid.
While the aforementioned research line [
16
,
30
,
35
,
46
53
] proposes successful strategies to evolve
the already deployed conventional grids into smart grids, the research line [
39
41
] has a different
approach that consists of generating synthetic structures for smart grids. In these papers, the so-called
“RT-nested-small world” model, based on analyses of real grid topologies and their electrical properties,
has been proposed to create a large number of synthetic power grid test structures, with scalable size
and the similar small-world topology and electrical features found in some real power grids [40].
As will be shown later on, our method has some aspects in common with both approaches. On the
one hand, we state an optimization problem with some of the design constraints proposed in [
30
]
(and with additional ones that we propose in this paper). On the other hand, our method generates
Energies 2017,10, 1097 6 of 31
synthetic structures, similar to the approach [
39
41
], but with the difference that we find the optimum
synthetic structure by using an evolutionary algorithm.
2.2. Evolutionary Computation in Graph Approaches: Related Work
The use of evolutionary computation in the broad field of graphs has been carried out with
different purposes and approaches. Aiming at discussing these within the goal of this paper, we use
the encoding strategy adopted in each work as the comparative criterion. The reason is that the
way candidate solutions are represented in EAs has a crucial influence on the design of evolutionary
operators and on the algorithmic efficiency.
One of the most fertile areas of research combining graph theory and EC is focused on clustering
in graphs. Cluster analysis is a key technique used for splitting a set of
N
objects into a number
k
of comparatively-homogeneous groups (clusters) based on their similarities [
54
]. The number
of different attainable partitions in a clustering problem is given by the Stirling number of second
kind,
S(N
,
k)
, and is known to be NP-hard when
k>
3 [
54
,
55
]. For instance, in a relative small
graph with 50 nodes and five clusters,
S(
50, 5
)
7.4
×
10
32
, which makes it convenient to use
approximate methods such as those of evolutionary computation. In this respect, the approach in
the recent work [
43
] consists of converting probabilistic datasets into probabilistic graphs with the
purpose of clustering by using an EA. The genetic representation of this problem in this research
work is a one-dimensional array having
N
elements, the number of graph nodes. This representation
has been found to be feasible to extract neighborhood node information and uses such information
to process probabilities on their links [
43
]. Graph partitioning via a multi-objective EA has been
studied in [
44
] by encoding the problem with a chromosome that is a set of
N
(number of nodes)
two-dimensional elements, in which any node is represented by its location in the graph. Instead of
using EAs, the problem of optimal non-hierarchical clustering has been tackled in [
54
] by using a
novel algorithm that combines differential evolution and
k
-means algorithms. Centered on social
networks, the problem of community detection using graph-based information is tackled in [
45
]
by applying graph clustering algorithms based on its topology information. Since any candidate
solution should contain a group of communities, a chromosome encoding an individual in [
45
] is
a set of vectors of binary values in which each allele represents a community composed by a set
of binary values, one for each node in the social network. Partially related to the latter work, the
paper [
56
] explores the feasibility of a spectrum optimization algorithm for community detection
based on selecting those links to be removed by minimizing the algebraic connectivity metric. In the
effort of making it applicable to large-scale networks, a greedy heuristic method has been tested
to get the lower bound of optimal value. The problem of community detection, either disjoint
or overlapping, has been tackled using a multi-objective EA [
57
] in which an individual consists
of two components: the first one is a permutation of all nodes, while the second component is
the set of communities. Another recent method that is gaining impulse is spectral clustering [
58
],
which builds a similarity graph and applies spectral analysis to retain the data continuity in the cluster.
The approach in [
58
] has proposed a novel algorithm inspired by the spectral clustering algorithm, the
co-evolutionary multi-objective genetic graph-based clustering algorithm, which includes a variable
number of clusters [
58
]. The encoding here is a simple label-based representation [
58
] inspired by
the conventional integer encoding of genetic algorithms. Each individual is a
q
-dimensional vector
(
q
being the number of data examples) with integer values between one and the number of clusters of
the sub-population to which it is assigned. Very recently, reference [
59
] has explored a multi-objective
EA for detecting overlapping communities. This differs from most other articles (focus on disjoint
communities) in which each gene of the chromosome is an integer number that encodes the community
label of the corresponding clique node of the maximal-clique graph. Also within the approach of
maximum clique-finding problem, reference [
60
] explores an EA able to tackle the problems such as
maximum independent set, set packing, set partitioning, set cover, minimum vertex cover, subgraph
and double subgraph isomorphism. In the proposed approach, each problems is first mapped onto the
Energies 2017,10, 1097 7 of 31
maximum clique-finding problem, which is later tackled by an evolutionary strategy that represents
each subgraph with a binary string.
Another important research field that combines CNs and EC consists of analyzing and/or
generating CN structures [
61
63
]. On the one hand, reference [
61
] focuses on optimizing the structure
of complex networks based on a memetic algorithm. Its problem encoding consists of an array
containing the node number and the number of the node with which such a node is connected. On the
other hand, the problem of automatically generating complex network models is tackling using genetic
programming (GP) [
62
]. In this proposal, the goal is to find out those more appropriate network
measures that capture as much as possible their structure, and the used tool is a GP that generates
automatically CNs on which such measures can be tested. Similarly, the feasibility of a GP for the
automatic inference of graph models for complex networks has also been explored in [63].
The graph coloring problem has also recently been tackled using an EA [
64
,
65
]. It aims at finding
the minimum number of colors where each node dominates at least one non-empty color class and is
an NP-complete for general graphs. In [
65
], the EA approach makes use of an encoding in which an
individual is represented by a two-dimensional array with
k
columns,
k
being the number of colors
used to color the nodes.
An interesting, partially related to our approach is the work carried out in [
66
], which focuses
on a hybrid evolutionary graph-based multi-objective algorithm for the layout optimization of truss
structures. The encoding of each candidate solution is composed of three matrices (the adjacency
matrix of the simple graph model, the adjacency matrix corresponding to the weighted graph model
and the coordinate matrix of nodes) along with two Boolean vectors (representing restricted nodes,
which cannot be left out, and those movable ones, respectively). This approach is related to a
certain degree with [
67
,
68
], which use a matrix representation based on the graph concept in truss
topology optimization.
Finally, the main conclusion of the reviews carried out in Sections 2.1 and 2.2 is that, to the best
of our knowledge, there is no work combining complex network/graph theory and evolutionary
algorithms in the effort to optimize the structure of smart grids starting from scratch, which is useful
not only from a modeling and theoretical perspective, but also from the practical viewpoint as a high
level tool for analysis, planning and decision-making.
3. Background: Complex Networks Concepts
The purpose of this section is to introduce basic definitions (Section 3.1) along with the
concept of “small network” that, as will be shown, has important advantages for the network
robustness (Section 3.2).
3.1. Some Useful Definitions in Complex Network
Any network can be mathematically represented by using a graph,
G= (N
,
L)
, where
N
represents the set of nodes (or vertices) and
L
denotes the set of links (edges) [
42
]. The interested
reader is referred to [
42
]; a very lucid and comprehensive description of complex networks, with many
examples of their existence in a great variety of natural and artificial systems, can be found
in [
42
]. The following list contains only those definitions that are important to understand our
work [33,42,69,70]:
An “undirected” graph is a graph for which the relationship between pairs of nodes are symmetric,
so that each link has no directional character (unlike a “directed graph”). Unless otherwise stated,
the term “graph” is assumed to refer to an undirected graph.
A graph is “connected” if there is a path from any two different nodes of
G
. A disconnected graph
can be partitioned into at least two subsets of nodes so that there is no link connecting the two
components (“connected subgraphs”) of the graph.
A “simple graph” is an unweighted, undirected graph containing neither loops nor multiple edges.
Energies 2017,10, 1097 8 of 31
The “order” of a graph
G= (N
,
L)
is the number of nodes in set
N
, that is the cardinality of set
N, which we represent as |N|. We label the order of a graph as N,N=|N|card(N).
The “size” of a graph
G= (N
,
L)
is the number of links in the set
L
,
|L|
, and can be defined
(.
=) as:
M.
=
i
j
aij =Nl, (1)
where
aij =
1 if node
i
is linked to node
j
and
aij =
0 otherwise. As mentioned before,
aij
are the
matrix elements of the adjacency matrix.
The “degree” of a node iis the number of links connecting ito any other node and is simply:
ki.
=
N
j
aij (2)
The node degree is characterized by a probability density function
P(k)
giving the probability
that a randomly-selected node has klinks.
A “geodesic path” is the shortest path through the network from one nodes to another; or in other
words, a geodesic path is the path that has the minimal number of links between two nodes.
Note that there may be and often is more than one geodesic path between two nodes [42].
The “distance” between two nodes
i
and
j
,
dij
, is the length of the shortest path (geodesic path)
between them, that is the minimum number of links when going from one node to the other.
The “average path length” of a network is the mean value of distances between any pair of nodes
in the network [42]:
`.
=1
N(N1)
i6=j
dij , (3)
where dij is the distance between node iand node j.
The “clustering coefficient” is a local property capturing the density of triangles in a network.
That is, two nodes that are connected to a third node are also directly connected to each other.
Thus, a node
i
in a network has
ki
links that connects it to
ki
other nodes. The clustering coefficient
of node
i
is defined as the ratio between the number
Mi
of links that exist between these
ki
vertices
and the maximum possible number of links (
Ci.
=
2
Mi/ki(ki
1
)
. The clustering coefficient of
the whole network is [33]:
C.
=1
N
i
Ci, (4)
that is, for a given node, we compute the number of neighboring nodes that are connected to each
other and average this number over all of the nodes in the network.
The “betweenness centrality” quantifies how much a node
v
is found between the paths linking
other pairs of nodes, that is,
CB(v)≡ Bv.
=
s6=v6=t∈V
σst(v)
σst , (5)
where
σst
is the total number of shortest paths from node
s
to node
t
and
σst(v)
is the number
of those paths that pass through
v
. A high
CB
value for node
v
means that this node, for certain
paths, is critical to support node connections. The attack or failure of
v
would lead to a number of
node pairs either being disconnected or connected via longer paths.
3.2. Small-World Property and Its Importance in Robustness
There is a property of some complex networks that has been found to be especially beneficial for
smart grids [
16
,
30
]: “small world”. Some properties of small-world networks that are interesting for
the purpose of this paper are:
Energies 2017,10, 1097 9 of 31
A small-world network is a complex network in which the mean distance or average path length
`
is small when compared to the total number of nodes
N
in the network:
`=O(log N)
as
N
.
That is, there is a relatively short path between any pair of nodes [
71
,
72
]. The term “small-world
networks” is often used to refer Watts–Strogatz (WS) networks, first studied in [
72
]. It can be
generated by the “rewiring” method shown in Figure 1a: Link
l13
, which was connecting Node
1 to Node 3, is disconnected (from Node 3) and rewired to connect Node 1 to Node 9. In the
resulting network, going from Node 1 to Node 9 only requires one jump via the rewired link
(and thus,
dnew
1,9 =
1). However, in the original regular network, going from Node 1 to Node
9 through the geodesic or shortest path (1
3
5
7
9) involves four links (
d1,9 =
4).
This leads to networks with small average shortest path lengths between nodes
`
, and high
clustering coefficient
C
. Figure 1b shows the aspect and
P(k)
of a WS we have generated with
N=
100 nodes and “rewiring probability”
p=
0.2. It has a short mean distance,
`'
6.04,
and high clustering,
C ≈
0.274. Most of the small-world networks have exponential degree
distributions [73].
Figure 1b (
N=
100 and
p=
0.2) also illustrates that the architecture of real small-world
networks is extremely heterogeneous: the vast majority of the elements are poorly connected,
but simultaneously, few have a large number of connections [
74
]. The robustness of small-world
network has been explored in [
75
,
76
] leading to the conclusion that, in a non-sparse WS
network (
M
2
N
), simultaneously increasing both rewiring probability and average degree
(hki=1
NN
i=1ki)improves significantly the robustness of the small-world network.
An interesting variation of the WS model is the one proposed by Newman and Watts [
77
]
(NW small-world model) in which one does not break any connection between any two nearest
neighbors, but instead, adds with probability
p
a connection between a pair of nodes. It has
been found that for sufficiently small
p
and sufficiently large
N
, the NW model is basically
equivalent to the WS model [
78
]. At present, these two models are together commonly termed
small-world models.
Figure 1b also helps us introduce the concept of network robustness (or its inverse concept,
vulnerability). It is related to the degree to which a network is able to withstand an unexpected event
without degradation in its performance. It quantifies how much damage occurs as a consequence of
such unexpected perturbation. Intuitively, the random failure of the marked link in Figure 1b does
not affect the network functionality, while the targeted attack on the marked node in Figure 1c will
make the network disintegrate in many unconnected parts before recovery. Figure 1b,c represent,
respectively, a very robust network and another very fragile one. In particular, note in Figure 1c
that, as most nodes have only a few connections and only a few nodes (“hubs”) have a high number
of links, then the network is said to have no “scale” [
79
]. This is why they are called “scale-free”
networks. A scale-free network can be generated by progressively adding nodes to an existing network
by introducing links to nodes with “preferential attachment” [
80
,
81
] so that the probability of linking
to a given node
i
is proportional to the number of existing links
ki
of the node. This is the so-called
Barabási and Albert (BA) model.
As mentioned in Section 1, there are however some authors arguing that this topological approach
should be enriched by adding electrical concepts. The following section introduces only those concepts
that will assist us in explaining our proposal. For a more complete and in-depth discussion, which is
beyond the scope of this paper, the interested reader is referred to [33,34].
Energies 2017,10, 1097 10 of 31
Rewiring
123
l13
4
5
6
7
8
9
123
4
5
6
7
8
9
10
Number of links,
k
!
!
!
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
0 20 40 60 80 100
0.0
0.1
0.2
0.3
0.4
0.5
=6.04434
C=0.274038
An attack on this
hub will disconnect
the network
An attack on this
node almost does
not affect the
network
Node
Link
Node Link
l19
Robust network
Fragile network
(a)
( )
(b)
(c)
Figure 1.
(
a
) First step in the creation of a small-world Watts–Strogatz (WS) network; (
b
) example of a
WS network and its node degree distribution; (
c
) scale-free network. See the main text for further details.
4. Background: Hybrid Approaches Combining Complex Networks and Electric
Engineering Concepts
As shown in detail in the review paper [
33
], there are some selected works [
34
,
37
] that emphasize
that the topological approach may lead to inaccurate results because it does not capture some of
the properties of power grids described by Kirchoff’s laws. Regarding this, there are some basic
concepts that compel engineers to include electrical power engineering concepts [
33
]. The first one,
in contrast to general purpose CNs, is that a power grid is a flow-based network in which electric power
flowing between two nodes can involve many links. From the EE viewpoint, the topological distance
metric in CN theory should be substituted by an “electrical distance”, involving line impedances [
34
].
The second cause is that, in conventional CN analysis, all elements are usually identical. This is not
often the case in power transmission grids because of the existence of different types of nodes such as
generation and load buses. Additionally, in power grids, transmission lines have flow limits. Based on
these concepts, reference [
34
] argues that, when applying to power grids, the graph must be weighted
(impedance, maximum power) and directed (since electric power flows from generators to loads).
However, since smart grids are bidirectional; the corresponding graphs are undirected.
Thus, aiming at overcoming the mentioned limits of pure topological approaches,
hybrid approaches combine CN and EE concepts [
33
]. An interesting research line belonging to
hybrid approaches is the extended topological approach [
34
,
82
85
]. It includes in the CN methodology
Energies 2017,10, 1097 11 of 31
novel metrics such as the “entropy degree”. The entropic degree of a node
i
, denoted as
Si
, aims at
including three elements in the topological definition of node degree when computed over a weighted
network [
82
]: (1) the strength of the connection between node
i
and
j
in terms of link weight
wij
;
(2) the number of links connected with such node; and (3) the distribution of weights among the links.
The entropic degree of node iis defined as [82]:
Si.
= 1
j
pij log pi j!
j
wij , (6)
where pij .
=wij
jwij is the normalized weight of the link between nodes iand j.
The question arising here is whether or not applying CN approaches on the power grid is useful.
5. Discussion: Is the CN Approach Useful in Power Grids?
There are some selected papers [
33
,
46
,
86
90
] that point out that CN science is a unifying, powerful
technique that enables one to analyze, within the same conceptual framework, a great variety of very
different systems whose constituent elements are organized in a networked way. The CN community,
including part of the electrical engineering community [
91
], argue that the CN approach does not aim
to reflect the detailed operation of a given grid, but to discover the possible emergence of a systemic or
collective behavior, beyond that of its single components. This is supported by a number of high-impact
works [
86
,
92
97
]. An interesting example of its feasibility is the appearance of synchronization in smart
grids [
36
]. However, the opposing community asserts that the pure topological CN approach loses
the details of the physics behind Kirchhoff’s laws and fails at predicting important aspects of power
grids. In this respect, as mentioned, hybrid approaches include concepts from EE [
82
84
,
96
,
98
102
].
Nonetheless, the CN approach with purely topological analysis (or even with extended ones to take into
account minimal electrical information [
34
]) has been found to be useful to detect critical elements and
to assess topological robustness [
35
,
97
,
103
]. Specifically, Luo and Rosas-Casals have recently reported
studies [
97
,
103
] that aim at correlating electric-based vulnerability metrics (based on the extended
topological approach) with real malfunction data corresponding to some European power transmission
grids (Germany, Italy, France and Spain). The results, validated and proven by empirical reliability
data, are statistically significant (Kolmogorov–Smirnov test) and suggest the existence of a relationship
between structure (described by extended topological metrics) and dynamical empirical data.
Although much research must be carried out, the evidence in [
97
,
103
] opens a research line to find
a more meaningful link between CN-based metrics and the real empirical data of power grids. The CN
approach could be useful to make vulnerability assessment and to design specific actions to reduce
topological fragility [
34
]. The analyses of [
33
,
35
,
97
,
103
] suggest that there is a connection between
the topological structure and operation performance in a power grid because the structural change
could disturb its operational condition and, as a consequence, degrade its operation performance. As a
result, there is an increasing interest in analyzing structural vulnerability of power grids by means
of the CN methodology. A deeper discussion on these issued can be found in [
33
] or in [
35
], where a
lucid introduction to complexity science in the context of power grids is provided.
Regardless of which of the two confronting options is more accurate (or useful, depending
on their purpose), there are some important practical issues related to whether or not there is a
predominant power grid structure (and, in particular, whether there is an optimal topology for smart
grids; Section 5.1) and whether or not it is better to model them with weighted graphs (Section 5.2).
5.1. Power Grids: Is There a Dominant Topology?
There are several graph structures aiming at abstracting the real power grid topology. For instance,
the research in [
72
] points out that the U.S. western power grid seems to have a small-world network.
However, the works by Cotilla-Sanchez et al. [
104
] and Hines et al. [
100
] show that (a) the explored
power grid does not exhibit a small-world nature and that (b) a spatial approach to connectivity
Energies 2017,10, 1097 12 of 31
and distance fails in setting up a graph model representing the electrical properties of the grid.
Furthermore, the research [
80
] suggested that the degree distribution of the power grid seemed to
be scale-free following a power law distribution function, although not all of the subsequent works
have agreed on this [
33
]. In this respect, some other works have also found that there are exponential
cumulative degree functions, for instance in the Californian power grid [
73
] and in the whole U.S.
power grid [
105
]. This notwithstanding, on the other hand, reference [
106
] has shown that the
topologies of the North American eastern and western electric grids can be analyzed based on the
Barabasi–Albert network model, with good agreement with the values of power system reliability
indices previously obtained from standard power engineering methods. This suggests that scale-free
network models are applicable to estimate aggregate electric grid reliability [
34
]. In addition to [
72
],
there are also several works that report on power grids with small world nature: the Shanghai
Power Grid (explored with a hybrid CN Direct Current (DC) and Alternating Current (AC) power
flow models) [
107
], the Italian 380-kV, the French 400-kV and the Spanish 400-kV grids [
108
] or the
Nordic power grid [
109
]. Rosas-Casals et al. [
110
], using data from thirty-three different European
power grids, found that, although the different explored grids seem to have an exponential degree
of distributions and most of them lack the small-world property, these grids showed however a
behavior similar to scale-free networks when nodes are removed, concluding that this behavior is not
unique to scale-free networks. This could suggest similar topological constraints, mostly associated
with geographical restrictions and technological considerations [
34
]. Thus, the existence of several
topologies in high-voltage transmission power grids suggest that there is no predominant structure,
except for the fact that many grids have a heterogeneous nature [
34
] and that they are vulnerable to
fails/attacks on the most connected nodes and robust against random failure.
However, in the particular case of smart grids, the small-world model seems to be beneficial.
As pointed out in [
16
], the exchange of electric energy at the local scale could be very positive because
it stimulates the local production and consumption of renewable-based electric energy (small-scale
photovoltaic systems and small-wind turbines), helping the end-user obtain economic benefit by selling
the energy produced in excess. Using real data from Dutch grids, and within the CN framework,
the key contribution of [
16
] is to propose the use of CN theory (combined with global statistical
measures) as a design tool to synthesize the best smart grid structures, in terms of performance and
reliability (for a local energy exchange) and cabling cost. The authors in [
16
,
30
,
35
,
46
53
] made the
conclusion that the small-world model seems to have many feasible features, not only structural, but
also economic, related to electricity distribution.
Finally, and although not conclusive, power grids with a small network structure seem to be
the most robust (except random networks) or, at least, seem to be those with the highest potential to
improve robustness in a feasible way. In particular, the research work [
75
] has compared the robustness
of random networks (ER), small-world (WS) and scale free (BA). Random networks are the most
robust and scale-free the most vulnerable. Among the structures found in power grids (scale-free,
small-world, as mentioned before), non-sparse small-world networks (
M
2
N
) are the most robust.
According to [
76
], non-sparse small-world networks have also the beneficial property of increasing
easily their robustness by a feasible method that consists of simultaneously increasing both the rewiring
probability and the average degree
(hki=1
NN
i=1ki)
, which improves significantly the robustness of
the small-world network.
5.2. Unweighted and Weighted Graphs: Which Is the Best?
An interesting point of discussion is whether the graph representing the particular grid under
consideration uses either weighted or unweighted links. The review [
33
] points out that many
works [46,86,89,90,105,106,108112]
have in common that each power grid has been represented using
the simplest graph model: undirected and unweighted. This is because these approaches do not
include any characterization of the link weights. Unweighted graphs are by far the most used
representation in the group of references that tackle robustness in power grids from the pure topological
Energies 2017,10, 1097 13 of 31
CN viewpoint. On the contrary, most of the hybrid approaches, which include power flow models
and/or electric-based metrics, made use of weighted graphs [
33
]. A deeper insight into the role of
weighted links is given in [
34
], where it is noted that in power grids, transmission lines have power
flow limits, which must be represented by weights
wij
standing for the flow limit on line
lij l(i
,
j)
linking nodes
i
and
j
. The authors in [
34
] argue that when applying CN analysis to power grids, the
electrical power grid must be represented as a weighted and directed network graph
G= (N
,
L
,
W)
,
where Wis the set of weight elements wij .
6. Proposal: Metrics, Objective Function and Problem Statement
6.1. Metrics to Construct the Objective Function
We have mentioned that in a smart grid made up of prosumer nodes, there should be a
bidirectional interchange of energy at the local scale. Aiming at achieving this goal, the following
features are beneficial:
1.
It is necessary for the SG to have a structure with reduces losses in the electric cables used to
transport electric power from one node to another. This electrical restriction can be modeled
using the condition:
`6log N, (7)
which, as pointed out in [
30
], is related to giving a reduced path when moving from one node to
another in a general purpose complex network. In the particular case of a smart grid, this may
lead to a topology with limited losses in the circuits used to transfer electricity from one node to
another. That is, it is a requirement related to the efficiency of the network. Along with the high
clustering coefficient, this is also one of the properties of small-world networks, in which the
mean distance or average path length
`
is small when compared to the total number of nodes
N
in the network:
`=O(log N)
as
N
. A small value of
`
is also important from the economic
viewpoint since it may lead to smaller cost.
2.
The entropic degree of a node
i
,
Si
, defined by Equation (6), has the advantage of providing
a quantitative measurements of the importance of buses [
82
] in power grids by including the
involved link weights and their number and distributions.
3.
Since the node degree
ki
of a node
i
is the number of links connecting
i
to any other node,
its maximum value gets an upper limit related to the maximum power that a node can support:
max(ki)kMAX. (8)
The value of
kMAX
is related to the maximum power that a node is able to support and is directly
related to its economic cost. In [
30
], average degree values
hki
ranging from
3 to
4 lead to a
good balance between performance and cost.
4.
The clustering coefficient defined by Equation (4) of a smart grid
CSG
should be higher than
that of the corresponding random network (RN) with the same order (number of nodes) and
size (number of links). This aims at assuring a local clustering among nodes because it is more
likely that electricity exchanges occur in the neighborhood in a scenario with many small-scale
distributed RE generators [30].
5.
We measure the network vulnerability by using the concept of multiscale vulnerability of order
p
of a graph G[113,114],
bp(G).
= 1
Nl
Nl
l=1
bp
l!1/p
, (9)
where
bp
l
is the betweenness centrality of link
l
. The multi-scale vulnerability
bp
of a graph
G
measures the distribution of shortest paths when links are failing (or attacked) [
114
] and is
Energies 2017,10, 1097 14 of 31
very useful when comparing the vulnerability of networks because it helps distinguish between
non-identical although very similar network topologies [
113
]. As shown in [
114
], if we want to
distinguish between two networks with graphs
G
and
G
, one first computes
b1
. If
b1(G) = b1(G)
,
then one takes
p>
1 and computes
bp
until
bp(G)6=bp(G)
. Using this approach, we have
considered
b2(G)
. A network represented by a graph
G
is less vulnerable (more robust) than
another Gif b2(G)<b2(G). Please see [113,114] for further details.
6. A coefficient of variation for betweenness [30],
b1=σb1
¯
b1, (10)
where
σb1
is the standard deviation of betweenness (
b1
) and
¯
b1
is the mean value of betweenness.
Distributions with
b1<
1 are known as low-variance ones. This requirement leads to network
resilience by providing distributions of shortest paths that are more uniform among all nodes.
See [30] for further details.
6.2. Proposed Objective Function
We propose an objective function (to be minimized), which is a combination of different functions
related to topological and hybrid CN metrics mentioned before in Lists 1–6 in Section 6.1. The objective
function is:
fOBJ(G) = fζ(G) = ζ·(Nl+`+1
¯
C) + (1ζ)·(b2+σb2
¯
b2), (11)
where Nlis the number of links, `is the average path length, ¯
Cis the mean clustering coefficient and
b2
is the multi-scale vulnerability of order 2. The rest of the components have already been defined
before. Note that fOBJ is only one of the possible functions among several ones that aim at:
Reducing
Nl
in the effort of decreasing the economic cost and the electric losses in the links used
to transport electricity from one node to another. Reducing
Nl
makes the network less robust.
This is because the minimum value of
b2
,
b2,min =
1 [
113
], is reached for the “fully-connected
network” or “completely-connected graph” in which any node is connected with all of the others.
As the number of links decreases, the network becomes increasingly fragile and
b2>
1. Reducing
Nl
to a great extent leads to an inexpensive, but very fragile structure (
b2
1 [
113
,
114
]). Thus,
the decrease of the number of links and the increase of the robustness have opposite tendencies.
This is why we propose a balance between
Nl
and
b2
via the weight parameter
ζ
, which controls
the linear combination between constituents with opposing trends.
Reducing
b2
(approaching one from above) to increase robustness and also
σb2
¯
b2
to
improve resilience.
Reducing `along with maximizing ¯
Cleads to a small-world structure.
Increasing
¯
C
aiming to stimulate the local electricity exchanges in scenarios with many small-scale
distributed RE generators.
6.3. Problem Statement
Let
N
be the “order” or the number of nodes (generators, loads) of the graph representing a smart
grid. The number of links (
M=Nl=
network size) to connect the
N
nodes and the specific way in
which these nodes connect to each other are two of the aspects to be determined. Let
G
be the set of all
possible connected graphs
G
with
N
nodes and
M=Nl
links. A graph is connected if there is at least
a path between every pair of nodes.
The problem consists of finding the topological structure (the network, or equivalently,
the optimum graph b
G) that minimizes the objective function fOBJ stated by Equation (11),
b
G=arg min
G∈GfOBJ (G), (12)
Energies 2017,10, 1097 15 of 31
subject to a the condition that the graph Gmust be connected,
λ2(G)>0. (13)
Parameter
λ2
is called “algebraic connectivity” (or the Fiedler eigenvalue) [
113
] and, in graph
theory, is one of the available parameters that can be used to mathematically measure to what extent
a graph is connected. The algebraic connectivity
λ2
is a positive real number whose magnitude
quantifies the level of connectivity in the graph. Larger values of algebraic connectivity represent
higher robustness against efforts to break the graph into isolated parts. In the opposite limit,
λ2=
0
means that the network has been broken into several disconnected parts. The algebraic connectivity
λ2
is computed as the second smallest eigenvalue of the “Laplacian matrix” of a graph
G
,
LG
. The
Laplacian matrix, sometimes also called the admittance matrix or Kirchhoff matrix, is an
N×N
symmetric matrix defined by: [113]
LG=DGAG, (14)
where
DG=diag(k1
,
· · ·
,
kN)
is the node degree matrix, which is the diagonal matrix formed from
the nodes degrees, and AGis the adjacency matrix of graph G.
7. Proposed Evolutionary Algorithm
7.1. Basic Concepts
An EA is an optimization, population-based algorithm, inspired by the principles of natural
selection and genetics, which is able to tackle complex problems [
115
,
116
] such as the one formulated.
Among these advantages, EAs do not require derivative information and are able to optimize functions
with a large number of continuous or discrete variables, finding the global solution for multi-local
extrema problems [
117
]. As discussed in Section 2.2, although GA and EA are sometimes used
interchangeably in the reviewed works, in this paper, we prefer to use the term EA since, as will
be explained in Section 7.2.1; we have encoded each feasible solution as a binary triangular matrix,
instead of a bit-string. For further details about this, the interested reader is referred to [118].
The underlying concepts of EAs and the way they are computationally implemented are inspired
by the way Nature finds out solutions to extremely complex problems, such as the “survival of the
fittest” individual in a evolving ecosystem [
115
,
117
]. Aiming at better explaining our approach, it
is convenient to introduce here two biological phenomena from which EAs are inspired: (1) the
external characteristics (“phenotype”) of living beings are encoded (represented) using genetic
material (“genotype”); and (2) evolution is the result of the interaction between the random
creation of new genetic information and the selection of those living beings that are best adapted to
the environment [117].
7.1.1. Genotype-Phenotype Relationship
As mention, in natural evolution, genotype is the genetic information that encodes and
causes the phenotype (all external characteristics) of a living being (or “individual”). Specifically,
each characteristic is encoded by a “gene”, a “chromosome” being the set of these genes [
117
].
Each gene is located at a particular position on the chromosome and can exhibit different
values (“allele”).
7.1.2. Natural Evolution
The random creation of new genetic information in Nature may lead to a better (or sometimes,
worse) ability to survive. The better a living being is adapted to its environment, the higher its
probability of survival is. This is called “survival of the fittest”. In turn, the longer the individual’s life
is, the higher its probability of having descendants. In the procreation process, the parent chromosomes
Energies 2017,10, 1097 16 of 31
are crossed or combined (“recombination”) to generate a new chromosome (which encodes the
offspring). With very small probability, “mutations” (or random variations in genes) can occasionally
occur, caused by external factors (for instance, radiation) or simply by unavoidable errors when copying
genetic information. This leads to offspring with new external properties, which are different from
those of their predecessors. If such arising external characteristic makes the offspring better adapted
to the environment, its probability of survival and having descendants increases. In turn, part of the
offspring can inherit the mutated genes (and thus the corresponding external characteristic), which
can be passed from generation to generation. These natural processes make the population evolve,
resulting in the emergence of individuals better adapted to the environment and in the extinction
of those less fitted. For deeper details about the main similarities and differences between natural
evolution and evolutionary algorithms, the interested reader is referred to [115].
7.2. Evolutionary Algorithm Used
The analogy of our problem with the biological metaphor described is that we are looking for the
“best graph”
b
G
that minimizes the objective function
fOBJ
in Equation (12). In this search, a very large
number of possible graphs
G
has to be evaluated aiming at computing the corresponding value
fOBJ(G)
.
Each possible graph is a candidate, trial solution or “individual”. The complete set of individuals
is called the “population”. The extent to which a candidate solution is able to solve our problem is
the “fitness of the individual”. The smaller the
fOBJ
value of an individual, the better the fitness of
the individual.
Just like in natural evolution, each individual or candidate solution is encoded using a
chromosome, a kind of representation that eases the problem solution because it transforms the
real search space into another in which working is much easier. The population is evolved via
the application of genetic operators that mimic the natural processes of reproduction, mutation
and selection.
7.2.1. Encoding Method
In our problem, the chromosome
cG
, which encodes each potential graph
G
(or individual), is the
upper triangular matrix of its adjacency matrix
AG
. In this formulation,
AG
is a square, symmetric
and binary matrix whose elements encode whether a node is linked (
aij =
1) to another adjacent one
in the graph or not (
aij =
0). Since there is no node self-connected, the adjacency matrix has zeros
on its main (principal) diagonal ((
aii =
0)). These are the reasons why all of the information of link
connections of graph
G
is stored by the upper triangular matrix
TG
. Thus, chromosome
cG=TG
encodes in a compact form the information of the adjacency matrix AGof graph G(or individual).
For illustrative purposes, Figure 2a shows a simple random graph with 10 nodes and 20 links,
while Figure 2b,c represents its corresponding adjacency matrix (AG) and its upper triangular matrix
(TGor chromosome cG=TGencoding the information of individual G), respectively.
In the discussions that follow, the terms “individual” and “chromosome” are used interchangeably
because each chromosome uniquely represents each solution in the actual search space. This strategy
can be considered as transforming the actual search space into another in which the computational
working is much easier. From a mathematical point of view, if
G
is the set containing all of the
candidate graphs and
C
is the set of chromosomes that encodes each of them (
cG
), this representation
is equivalent to defining a bijection,
Ξ:GC(15)
so that any candidate graph is represented by a unique chromosome Ξ(AG) = cG=TG.
Energies 2017,10, 1097 17 of 31
0111101001
1001111010
1001011101
1110000100
1100011100
0110100000
1110100000
0011100000
0100000000
1010000000
0111101001
0001111010
0001011101
0000000100
0000011100
0000000000
0000000000
0000000000
0000000000
0000000000
n1
n2
n3
n4
n5
n6
n7
n8
n9
n10
n1
n10
n7
n9
n2
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n5
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n4
individual), is the upper triangular matrix of its adjacency matrix AG. In this
is a square, symmetric, and binary matrix whose elements
=
. Thus chromosome cG=TG
the information of the adjacency matrix AGof graph
, which encodes each potential graph G
G
(a)
(b)
(b)
(c)
Figure 2.
Simple example illustrating the encoding process. (
a
) Small random graph
G
(or individual)
with 10 nodes and 20 links; (
b
) adjacency matrix
AG
of graph
G
; (
c
) upper triangular matrix
TG
or
chromosome cG=TGencoding the information of individual G.
7.2.2. Initial Population
The size of the initial population (number of chromosomes),
Psize
, is a crucial parameter for EA
performance [
118
]. On the one hand, a large population could cause more diversity of candidate
solutions (and thus, a higher search space), leading to a slower convergence. On the other hand, too
small a population leads to reduced diversity: only a limited part of the search space will be explored.
This increases the risk of prematurely converging to a local extreme. In our specific problem, after a
number of experiments, the initial population has been chosen as
Psize =
50 individuals, as a tradeoff
between computational complexity and performance.
As important as the population size is the way in which such an initial population is generated.
Usually, the initial population is initialized at random. This strategy is appropriate for those problems
in which there is no information about how the solution will be. However, there are problems in
which a non-random, domain-specific initial population is more suitable [
119
]. This is the case of our
problem since we have information about a suitable (although non-optimum) solution: small-world
networks have been found to exhibit beneficial properties in some smart power grids. See [
30
] for a
more detailed explanation. In our preliminary work, we have found that the EA works better if the
initial population is generated as follows:
Fifty percent of
Psize
are Watts–Strogatz random graphs (with small-world properties, including
short average path lengths and high clustering) with rewiring probability ranging from 10
2
to one.
Fifty percent of Psize are Erd˝os–Rényi (ER) random graphs with Nnodes and N×5 links.
Figure 3shows some examples of four graphs belonging to the initial population.
An important point is to ensure that any graph
G
in the initial population is connected by checking
that it fulfills the condition λ2(G)>0 [113].
This approach to generate an SG domain-specific aims to reduce the number of searches within
the solution space and to assist operators in finding the global minimum quickly.
Energies 2017,10, 1097 18 of 31
Gi
Gm
Gy
Gn
(a)
(b)
(b)
Figure 3.
Examples of four graphs belonging to the initial population. (
a
) Watts–Strogatz random
graphs; (b) Erd˝os–Rényi (ER) random graphs.
7.2.3. Implementation of Evolutionary Operators
Selection Operator
Selection operators can be basically classified into two classes [
118
]: fitness proportionate selection
(such as roulette-wheel selection and stochastic universal selection) and ordinal selection (tournament
selection and truncation selection) [
118
]. After a number of experiments, we have selected as the
selection operator the tournament selection. This strategy is one of the most widely-used selection
operators in EAs since it performs well in a broad variety of problems, is susceptible to parallelization
and can be implemented efficiently [
118
,
120
]. A very clear description of its key concepts and further
details can be found in [120].
Tournament selection basically aims at selecting individuals based on the direct comparison
among their fitness. In our problem, a candidate solution, a graph
G
, encoded by chromosome
cG
is
more fit than another, cH, if the corresponding objective function fOBJ is better (lower):
fOBJ(cG)<fOBJ (cH). (16)
The simplest tournament selection operator consists of picking out at random two individuals
(contenders) from the population and carrying out a combat (tournament) to elucidate which one
will be selected. In particular, each combat involves the generation of a random real number
ntour [0, 1]R
to be compared to a prearranged selection probability,
pselec
. If
ntour pselec
, then the
stronger (fitter or best) candidate is selected, otherwise the weaker candidate is selected. The probability
parameter
pselec
gives a suitable strategy for adjusting the selection pressure. To favor best (fittest)
candidates, pselec is usually set to be pselec >0.5 [120].
This simplest implementation of tournament with only two competitors (tournament size
=
2)
can be generalized to involve more than two individuals. As shown in [
120
], the selection pressure
can be adjusted by changing the tournament size. If the tournament size increases, weak individuals
have a smaller probably of being selected. That is, the more competitors, the higher the resulting
selection pressure.
Energies 2017,10, 1097 19 of 31
Regarding this, the tournament selection operator we have implemented has a tournament size of
Tsize =Psize =
50 contenders (that is, all individuals are fighting each other) and a selection probability,
pselec =
0.8. As mentioned,
pselec >
0.5 favor best (fittest) candidates [
120
]. The individual that
accumulates the most wins is selected as the one that pass to the next generation in the selection process.
Crossover Operator
The crossover operator works as follows:
1. Select at random (pcross) two individuals from the population (father and mother).
2. Select at random the same row in the parents.
3.
Exchange the selected rows between the father and the mother, which leads to two
child chromosomes.
Mutation Operator
Mutation operators are designed to generate diversity in each generation and aim at exploring
the whole search space by introducing local changes with very small probability. Specifically,
the implemented mutation operator selects at random an individual with a given probability
pmut
.
The mutation operator then picks out at a random row (of the upper triangular matrix representing
such an individual). Note that row “
i
” encodes how node
i
is connected to others: element
aij =
1
means that there is a link between nodes
i
and
j
. The next step that the operator makes is to select at
random two elements of the row and to perform a permutation. This is equivalent to rewiring the links
of node
i
to other nodes and ensuring that: (1) node
i
is not disconnected from the rest of the network
and (2) that the degree of node iremains unchanged, despite having made the mentioned rewiring.
8. Experimental Work
8.1. Methodology
The EA is stochastic as it begins with a population randomly generated (see Section 7.2.2), and then
evolutionary probabilistic operators are applied to the population in each generation. The result gets
better (
fOBJ
is reduced) quickly with the first iterations (generations) until it ends up stagnating,
converging to a near-optimal result. As the EA is stochastic, obtaining statistical values is compulsory.
This is the reason why the EA has been repeated 20 times, which have been found long enough.
The values for the EA parameters that we have considered in the experimental work described
below are:
Tsize =Psize =
50 graphs (50% being WS small-world graphs, with rewiring probability
ranging from 10
2
to 1, and 50% being ER random graph with
N
nodes and
N×
5 links),
pselec =
0.8,
pmut =0.09 and pcross =0.2.
For illustrative purposes, Figure 4shows the mean value (a) and variance (b) of
fOBJ =f0.7
as a
function of the number of generations.
Energies 2017,10, 1097 20 of 31
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