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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(ki−1).

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

N∑N

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(N−1)∑

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

N∑N

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

P(k)

!

!

!

!

!

!

!

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

0 20 40 60 80 100

0.0

0.1

0.2

0.3

0.4

0.5

ℓ=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

N∑N

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 [46,86,89,90,105,106,108–112]

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 G∗if 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) + (1−ζ)·(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=DG−AG, (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:G←→ C(15)

so that any candidate graph is represented by a unique chromosome Ξ(AG) = cG=TG.

Energies 2017,10, 1097 17 of 31

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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

n6

n5

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n3

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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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