Reliability Analysis Algorithm for Multiple Microgrids in Distribution

Systems Based on Complex Network Mathematical Theory

FABIAN RODRIGUEZ, SERGIO RIVERA*

Electrical and Electronics Engineering Department

Universidad Nacional de Colombia

Carrera 30 Número 45-03, Bogotá

COLOMBIA

Abstract: - It exists a great potential in microgrids connected to distribution systems of being taken into

advantage reconfiguration possibilities with the purpose of achieving the quality of service regulatory

requirements that become more demanding each day. In addition, it is possible to optimize the

network operators’ income by increasing the incentives for upgrading the quality indexes. In this

paper, it's proposed an evaluation algorithm of the connection points of multiple microgrids in a

distribution system that upgrades the reliability of the system as a whole, being based in complex

network analysis (CNA), a perspective of power systems that allows the evaluation of an electrical

system as a graph. For that, a model of a trial system is made from the CNA point of view utilizing the

MATLAB software and afterwards, as validation of the proposal of this work, the system's reliability

is evaluated by connecting multiple microgrids into critical nodes provided by the CNA making use of

the NEPLAN tool of power systems simulation.

Key-Words: - Complex network mathematical theory; operation in uncertain environments; optimization;

reliability; smart microgrid

Received: March 15, 2021. Revised: December 23, 2021. Accepted: January 20, 2022. Published: February 11, 2022.

1 Introduction

Historically, the importance of the reliability

analysis of the performance of power systems is

proportional to the tension level of the operation and

the main focus was in the generation stage and in

the energy transmission. However, approximately

80% of all the interruptions that the clients suffer

occur due to failures in the distribution networks

[1]. Due to the above, the performance evaluation in

respect to the reliability (or quality of the technical

service) of distribution systems has been widely

studied since the decade of 1930 as a key aspect in

the planning of electrical energy systems [2], and

since then countless studies that develop new

techniques, models and reliability analyses

applications in these systems have been published

[2], [3].

Despite the great quantity of studies and

publications and the progress made in new analysis

techniques thanks to the increasing development of

computing tools that allow advanced analyses to be

made, studies remain to be developed and

investigation is still being made in this field,

focusing in networks with high penetration of DER

(Distributed Energy Systems) [4], NCRES (Non-

Conventional Renewable Energy Sources) [5],

microgrids and in the management of assets and

maintenance [6]-[11]. So, it's evident that the

reliability analyses still hold their importance in the

planning of power systems and distribution

networks.

Taking into account that in the past years the

regulation of the Colombian electrical sector has

implemented different mechanisms to promote and

incentivize the connection of “small scale auto

generators (SSAG)”, “NCRES (Non-Conventional

Renewable Energy Sources)” and “DER

(Distributed Energy Systems)” in the “National

Interconnected System (NIS)” and in the “Not

Interconnected Zones (NIZ)”, through different

norms [12]-[15].

The microgrids have the capacity of improving

the tension regulation, the energy quality, the

protection schemes, diminish losses from the system

and the emissions due to the nature of the used

technologies and substantially improve the system's

reliability thanks to its reconfiguration capability

before events and to operate isolated from the

network [7], [9], [16]. In such virtue, it exists a great

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Volume 17, 2022

potential in taking advantage of the microgrids to

achieve the quality of service regulatory

requirements and also optimize the network

operators’ incomes by increasing the incentives for

the improvement of quality indexes.

The reliability analyses are based mainly in two

aspects: the individual parameters of each one of the

elements that conform the electrical system such as

the failure rate, availability, repair time, average

failure time, average time between failures, etc., and

in how every one of these elements are connected,

that is to say, the topology of the network. However,

due to the complexity of the future energy

distribution networks, it exists an urgent necessity of

upgrading these traditional reliability evaluation

methods into new techniques [7].

One of the new tools for the electrical systems

analysis that has been growing in interest is the

complex network theory [17]. Taking the above into

account, the complex network theory is shown as a

new tool that can assist not only the efficient

processing in the calculation, but also in giving a

new perspective in the problem’s analysis. Some of

the studies developed from the point of view of the

complex networks and focused in the power

system’s analysis are the following:

In [18] a vulnerability and recognition analysis

of key nodes in the electrical networks from the

complex networks’ perspective is made. Using the

centrality of the nodes from the complex network

theory, the key nodes are identified and the system's

behavior is analyzed by modifying the network’s

parameters to correct the centrality of the vulnerable

nodes. Indicators are introduced as the netability and

the vulnerability index to describe the transfer and

performance capacity in a normal functioning and to

evaluate the vulnerability of the electrical system in

case of waterfalling failures.

In [19] it's studied how a waterfalling failure in

electrical networks is produced and the correlation

between the key parameters is searched using the

complex network theory to improve the sturdiness

of electrical networks. In [20] a methodology for

evaluating the stability of a Smartgrid that includes

microgrids is proposed. For this evaluation an index

called Intermediation Index that is based in the

theory of complex networks is taken. An improved

intermediation index is proposed, since it’s

considered the real charge flow through the

transmission lines along the network. This work is a

starting point in the investigation area of complex

systems to evaluate the stability of power systems.

In [21] an indicator that is based in the

impedance of the transmission lines as a criterion to

measure the vulnerability of the system is proposed.

This indicator, obtained through the analysis of

complex networks, can identify the critical

transmission lines of the network, whether by its

position in the system or by the power transmitted

along the network.

In the work developed in [22] a systematic

method based in the complex network theory is

stablished to propose that, in normal conditions,

every modern system of distributed generation with

variable topology and bounded control entries, can

be represented as a Hamiltonian stable system. With

the prior premise, it’s analyzed a microgrid driver to

evaluate the proposed method.

A methodology for the optimal localization of

microgrids in electrical distribution systems using

complex network analysis [23]. The optimal

localization in this work is acknowledged as the

localization that gives as a result a greater resilience

in the network, a reduction in the energy losses an in

the lines’ chargeability, a better voltage stability and

the supply to the critical charges during a blackout

[24]-[32]. The criteria used to select the optimal

localization of the microgrids were based in the

centrality analysis taken from the complex network

theory [33]-[41].

The work in [42] showed that the network’s

structure influences in a very important way the

reliability of the system. Additionally, in the same

article it was observed that the reliability of the

microgrids is very sensible to many other factors;

such as the system’s demand, the network structure

and the coupling method proposed. The literature

revision [7], [43], [44] shows that there are many

factors that must be taken into consideration to

determine in an exhaustive way the reliability of the

microgrids. Some of the factors include the type of

failure, the weather and the network’s structure.

Taking into account that the microgrids have

the property of improving the reliability of the

distribution networks, as evaluated in [9], where the

SAIDI and SAIFI indicators were used for the

evaluation, it exists a great potential for the

microgrids to be taken into advantage to fulfill the

quality of service regulatory requirements and to

optimize the income of the network operators by

increasing the incentives for improving the quality

indexes.

In this way, by taking into account that the

reliability analysis is based fundamentally in the

topology of the network, in this work is proposed

the use of complex networks to analyze, from other

perspective, the topological parameters of the

network that allow the identification of nodes in

which the microgrids connection achieves an

improvement in the general reliability of the system.

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2 Materials and Methods: Complex

Network Analysis in Electrical

Systems

In the formality of network analysis, a network is a

group of elements called vertexes or nodes, with

different interconnections between them, (these

connections can be power lines or also

communication interconnections), called edges. The

systems that take the form of networks (or graphs)

are plenty in the world. The examples include:

internet, social networks, organizational networks,

networks for commercial relationships between

companies, public transportation networks, and for

our case, electrical distribution networks, and many

others [45], [46].

The different fields’ researchers have stablished,

along the past years, a wide array of mathematical,

computing and probabilistic tools, destined to study,

model, and comprehend different network systems.

The study of the network science stablished its basic

foundations in the development of the graph theory,

that was explored for the first time by Leonhard

Euler in 1736, when he published the article Seven

Bridges of Königsberg [47]. The solution to the

problem of the seven bridges of Königsberg is

generally considered as a starting point for the graph

theory and the network science [46], [48].

2.1 Complex Networks

The complex network theory is a new discipline that

has as main focus to analyze different static

topological characteristics, as well as dynamic

behaviors in interconnected systems at great scale.

The theoretical work in complex networks came to

be from the graph theory and the network science

[45], [46].

In the context of the network theory, a complex

network could be defined as a graph that is made up

from many nodes related between them [49]. It

could also be defined as a network that has non-

observable topological characteristics that don’t

surface in simple networks such as random ones, but

that often occur in real systems’ graph models [50].

Without taking into account the intrinsic dynamic of

every component, a complex network can be

described simply as a graph or a G network, that can

be defined as a finite group of ordered pairs G = (V,

E), where V is a subgroup of non-empty elements

called nodes or vertexes and E a subgroup formed

by ordered pairs of elements different from V,

called borders or edges [51].

In Figure 1 ([51]) a non-directed graph is

exemplified, G=(V, E), where V={v1, v2, …, v5}

and E = {v1v2, v2v3, v3v4, v3v5, v2v5, v4v5}.

Fig. 1: Example of a 5 vertexes non-directed graph.

A group of vertexes joined together by borders

is the most simple of network types; but generally,

the real networks are much more complex than this.

For example, there can be more than just one

different type of vertex in a network, or more than

one type of edge. And the vertexes or edges can

have a variety of properties, numerical or others,

associated to them. A graph is considered directed

when a direction is assigned to the edges that form

them, like the graph shown in Figure 2(d), in which

the directed graph D = (V, E) has ordered pairs (vi,

vj) ∈ E, where vi is the end of the edge and vj the

start.

If to every one of the graph’s edges is associated a

value or w cost, it’s resulted in a graph Gw = (V, E,

w) called graph with weights like the one shown in

Figure 2(c). In these graphs, there can be considered

short or geodesic paths between vertexes, since the

longitude notion is introduced, defined as the sum of

the weights along a trajectory between vertexes

[46].

(a)

(b)

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(c)

(d)

Fig. 2: Examples of network types (a) non-directed graph with simple vertexes and edges; (b) graph with

different types of discrete vertexes and edges; (c) graph with weights of vertexes and edges; (d) directed graph

where each edge has a direction.

2.2 Power Systems from the Point of View of

Complex Networks

A power system is also formed by vertexes (nodes,

substations, derivations, barrages) and edges

(transmission or distribution lines, power

transformers) between them as a complex network

[52], thus, the electric energy distribution systems

can be qualified as complex networks and be studied

through the optic of complex network analysis

(CNA) [20].

To study an electrical distribution network using

the complex network analysis, the first step is to

model the network as a graph. In the context of

complex networks and for the present analysis, the

electrical nodes, substations, generation centers and

microgrid connection points with the distribution

system correspond to the vertexes, while the

distribution lines and transformers correspond to the

edges.

In Figure 3 is shown the original unilinear diagram

of the IEEE trial system with 30 nodes that will be

used as case of study. This system counts with 30

nodes connected through 41 impedances (lines or

transformers).

Fig. 3: General unifiliar diagram of the IEEE trial system with 30 nodes. Source: Figure taken from

http://labs.ece.uw.edu/pstca/pf30/pg_tca30bus.htm.

In Figure 4 is shown the IEEE system with 30

nodes as a non-directed graph and in Figure 2-5 is

shown the topology mapping of the system. The

next aspect to take into consideration for the

network’s analysis as a graph is to formulate the

weight matrix of the graph (adjacency matrix with

weights). The traditional focus of the complex

network analysis only considered the physical

connection [17], [53]. This model does not reflect

one of the fundamental aspects of the electrical

energy system as it is the impedance, that plays an

important role in the power flow, the losses and

stability of the system.

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Fig. 4: IEEE system with 30 nodes as a non-directed

graph.

Fig. 5: Topology mapping of the IEEE system with

30 nodes.

The focus to find the adjacency matrix with

weights is based in the admittance matrix [54]. In

this case, the weight matrix can be found from the

elements outside the admittance matrix’s diagonal.

For a system of n nodes, the voltage equation is

written in matrix form as:

(1)

(2)

Where Ybus is the admittance matrix. The

diagonal elements of the admittance matrix

correspond to the sum of the impedances of the lines

connected to each one of the system’s nodes. Given

that the diagonal elements are not included in the

weight matrix, for this analysis the impedances

between the nodes and the ground are not

considered. The elements outside the diagonal are

equal to the negative of the admittance equivalent

between the nodes. So, in this case, the element ij of

the weight matrix [A(Gw)] can be found from wij =

Yij. It’s evident that the Ybus matrix is a

symmetrical matrix, that is to say that, Yij = Yji,

thus, by being equal the impedance in both ways,

the power flow’s directionality is not considered in

this model and is considered a non-directed graph.

For the system’s modelling as a complex network,

some considerations like the following are made:

It is assumed that the system is balanced.

The power transformers and transmission lines are

modeled as edges with weights, being the weight

equal to each one of the components in p.u.’s

admittance.

The parallel lines between two substations or

nodes are considered as a single equivalent

transmission line with the purpose of simplifying

the graph.

2.3 Centrality Measures

From the complex network analysis perspective

numerous measures or indexes that can define

certain characteristics in the network knowing the

network’s structure have been developed. For

example, the social scientists have used some

centrality indexes to better explain the impact a

person has inside of a network [55]-[57]. Between

these centrality indexes, the more used in electrical

energy systems are the grade centrality and the

betweenness centrality.

2.3.1 Grade Centrality

The simplest one of the centrality measures is the

grade centrality. It's defined as the number of edges

possessed by a node that connects it to the others

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[45]. The grade centrality can be obtained from the

matrix’s grade diagonal. In complex networks, most

of the nodes have a low grade, but a hierarchy of

higher-grade nodes called “hubs” exists, which have

an important role inside the network. In the

electrical networks, these hubs correspond to

important transformation or generation substations

and is called system’s hub to the vertex with the

highest grade. In the case of the considered 30 node

system, the hub node corresponds to the node 7 that

has 6 grade and the average system’s grade is 2.73.

In Figure 6 is shown the grade distribution for the

system under study.

2.3.2 Betweenness Centrality

The betweenness centrality could be defined as the

number of shortest paths between the vertexes

(nodes) that pass through this vertex. The

betweenness centrality measures how many times a

vertex is found in the direct path between any other

pair of vertexes [45].

A high betweenness value indicates that a vertex

can reach other vertexes in relatively short

trajectories, or that a vertex is found in a

considerable fraction of the shortest trajectories that

connect pairs of other vertexes. The betweenness

can be calculated through the Equation (6).

(3)

Fig. 6: Grade distribution for the IEEE trial system with 30 nodes.

Where nst(v) is the number of shortest paths

between the vertexes s and t that pass through the

vertex v, and Nst is the total number of shortest

paths between the vertexes s and t.

2.3.3 Closeness Centrality

The closeness centrality could be defined as the

average of shortest routes between a specific node

and the rest of the network’s nodes. A high value in

the closeness centrality of a node indicates how

close it is from the other nodes. The shortest

electrical route is the one that has the minimum

impedance possible, thus, the edges’ weights are

chosen in function of the impedance. The closeness

centrality can be calculated using the Equation (2-

7).

(4)

Where vi is the number of close vertexes from the

vertex i (without including i), N is the number of

vertexes in the G graph and Ci is the total sum of the

distances between the vertex i to any nearby

vertexes. If a path from the vertex i to the nearby

vertexes does not exist, then c(i) is zero.

2.3.4 Clustering

The clustering is an index that evaluates how

interrelated the neighboring nodes inside of a vertex

are. The local clustering CC(i) of a node calculates

the average connections of its nearby nodes. The

C(i) for each node and the total CC(G) of the

network can be calculated with the equations (2-8)

and (2-9), respectively:

(5)

1234567

Nodos

Grado del nodo

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(6)

Where, Nv is the number of edges between the

neighbors of the node v and Kv is the node v’s

grade.

From the point of view of electrical systems, a node

with a high CC(v) could indicate that, if separated,

the power flow will have alternative routes to the

closest nodes.

In other words, a high node coefficient CC(v) could

indicate a less central node. The total CC(G) of the

IEEE system with 30 nodes is 0.2348. The CC(G)

indicator is equal to 1 in a graph if every vertex in

the graph are connected to each other in the same

way.

In Table 1 are summarized some of the centrality

measures applied in the IEEE trial system with 30

nodes.

Table 1. Summary of the IEEE network with 30 nodes’ measures from the CNA perspective.

Number of nodes

Edges

Maximum grade

Average grade

2.73

Clustering

0.2348

Average betweenness

33.433

Average betweenness with weights

(admittance)

40.4

3 Reliability Analysis of Multiple

Microgrids

3.1 Case of Study

Taking as a base the IEEE trial circuit with 30 nodes

[58], [59], a case of study was developed making

use of the power systems simulation tool NAPLAN.

In Figure 6 is shown the unifiliar diagram of the

simulated system and the connection points of the

microgrids for the based case.

The original IEEE trial model with 30 nodes does

not have assigned longitudes for the lines, the

characterization of the lines is made only with the

total electrical parameters of resistance, reactance

and susceptance of each one of the lines. The

longitude parameter of the lines is fundamental for

the reliability analysis in distribution networks, in a

way that it was estimated the longitude taking

typical reactance values per unit of longitude with

the purpose of assigning the longitude of each one

of the model’s lines and apply the reliability

parameters according with the IEEE 493-2007 norm

[60] per unit of longitude for each one of the lines.

For the analysis is considered that all the elements

connected through nodes (edges, in this case lines

and transformers of the system) count with cut

elements and protection in both ends. The charges,

capacitances, generators and the microgrids in the

connection points with the distribution system also

have elements of cut and protection.

The system was modified locating in the node 1

(slack node with a generation unit in the original

model) an infinite bar, with the purpose of

simulating the connected network to an

interconnected network of great capacity with an

ideal reliability. The original model’s generators of

the IEEE network with 30 nodes were replaced by

microgrids to simulate multiple microgrids

connected to the system. In a way that the modified

system counts with 5 microgrids.

The maximum capacities of the elements were

modified so that in the network’s initial conditions

there aren’t any overcharged lines and transformers,

also that the tension in every node is between 0.9

and 1.0 p.u., because of the contrary this condition

would be taken as a contingency and in the original

IEEE model with 30 nodes there is an overcharged

line to the charge flow channel. The range criterion

of allowable tensions was taken from the Code of

Networks in its section Operation Code [61].

The reliability parameters for the elements that

form the system, such as the failure rate, repair time,

MTTF and MTTR were taken from the IEEE

standard 493-2007 [60]. The reliability parameters

for the microgrid were taken with a base in the

article [62].

It's considered that the microgrids are connected

to the distribution system of intermediate tension

through an impedance that represents the IC

(Interlinking Converter between the MG and the

distribution network) in the case of microgrids with

DC bus or the coupling transformer in case of

microgrids of AC bus.

The relation between the internal generation and

the charge (Generation-Load Ratio GLR) of the

microgrids is important in the evaluation on the

reliability of the distribution system to which

they’re connected. A microgrid’s GLR can be

minor, greater or equal to 1; it’s minor when the

microgrid does not satisfy every necessity of the

consumer’s charge in its interior, although it has the

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capacity to do so (for example, for economical

reasons), and it’s greater than one when it can

export energy to the distribution network [9].

Taking into account then that it's fundamental to

considerate the GLR in the evaluation of the

reliability of the system, since the backup that can

give the microgrid to the system before events of

failure from the network’s elements depends on it.

The microgrids are modeled with the VPP concept

(Virtual Power Plant) [63], through a model of

distributed generation and a charge and the

parameters of these elements are varied with the

purpose of simulating the GLR of the microgrid.

The VPP concept is employed as a tool to

facilitate the reliability evaluation in the active

distribution network with multiple microgrids. The

concept consists in considering every one of the

microgrid’s components in a single entity to offer a

simplified equivalent model and to be used in the

distribution network. When it’s integrated in a

distribution network, the functionality of a

microgrid is to interchange energy with the

distribution network, this way the microgrid

becomes an energy source if it outputs more energy

than the local charge and it becomes into a charge

when the charge exceeds the available output. Thus,

in a similar way to a conventional generation plant,

a VPP will be represented by a model of multiple

states in the reliability analysis [63].

For the analyzed case, the generation capacities

of the microgrids are considered equal for all five

subsystems and are assumed as 40 MVA (34 MW,

21 MVAr). The microgrid’s charge is variable

according to the GLR as exposed beforehand. For

the present analysis the variable charge will be

considered in function to the GLR but with a

constant power factor of 0.85.

The microgrid’s contribution to the reliability

tends to be greater with higher GLR values.

However, the impact that the GLR has in the

reliability depends in the microgrid’s localization in

the system and in the localization of the failure for a

specific case [9].

3.2 System’s Reliability Evaluation

By using as a NEPLAN software as a simulation

tool, the SAIDI, SAIFI and CAIDI reliability

indexes were calculated for the configuration of the

modified IEEE system with 30 nodes that is shown

in Figure 6.

With the purpose of verifying the impact that

the GLR has in the reliability of the system, the

reliability indexes were evaluated for different

values of the GLR and one case without including

the microgrids in the system. The results are shown

in Table 2.

Table 2. Summary of the reliability indicators for different cases with MG for the analyzed IEEE system with

30 nodes.

Indicator

Unit

Without MG

With MG,

GLR=2

With MG,

GLR=1.5

Clients

SAIFI

1/yr.

2.307

0.442

0.184

SAIDI

min/yr.

1254

320.327

243.748

CAIDI

9.061

12.092

22.118

MW/yr.

750.35

149.914

211.018

MWh/yr.

5250.5

1640.39

2280.141

Note: P corresponds to the total power that was stopped from supply in a year and W corresponds to the total energy that

was stopped from supply in a year.

Fig. 7: Behavior of the SAIDI for different values of GLR of the MGs.

1000

2000

3000

4000

5000

6000

7000

8000

9000

Sin MG

Con MG,

GLR=2

Con MG,

GLR=1.5

Con MG,

GLR=1.2

Con MG,

GLR=1.0

Con MG,

GLR=0.9

Con MG,

GLR=0.8

SAIDI (min/yr)

MGs' operation configuration

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The probability of system’s failure grows

exponentially by diminishing the GLR of the

microgrids. This shows that effectively the backup

that the microgrids can give to the system is

fundamental in the reliability analysis. It's also

evidenced that the system without including the

MGs has a similar reliability to the system when the

MGs are included with a GLR=1.0, that is to say

that even though the MGs don’t contribute power to

the system they don’t impact the reliability of the

system but actually improve it when they can

provide a backup to the system. It's important to

note that the general reliability of the system

depends on the reliability of the microgrids. In this

work, reliability data was taken for the microgrids’

components of the work done in [62].

Fig. 8: Behavior of the SAIFI for different values of GLR of the MGs.

Fig. 9: Behavior of the CAIDI for different values of GLR of the MGs.

The problem of the reliability analyses with multiple

microgrids in a distribution system is a high

complexity problem. Particularly if is wanted to

analyze a way to optimize the reliability of the

system, not only are the economical and technical

aspects of the systems taken into account, but also

the operative conditions of the microgrids, without

mentioning that these three aspects are a function of

the localization of the DG’s resources of the

microgrids.

4 Results: Complex Network

Application to Improve the Reliability

4.1 Trial Network’s Centrality Measures

Making use of the MATLAB software, the

centrality measures for the modified IEEE trial

system with 30 nodes were calculated. In Figures

10, 11, and 12 are graphically shown the centrality

measures of closeness, betweenness and

intermediation considering the chargeability limits

of the lines and transformers. The centrality

measures of the graph with weights are the ones that

Sin MG

Con MG,

GLR=2

Con MG,

GLR=1.5

Con MG,

GLR=1.2

Con MG,

GLR=1.0

Con MG,

GLR=0.9

Con MG,

GLR=0.8

SAIFI (1/yr)

MGs' operation configuration

500

1000

1500

2000

2500

3000

Sin MG

Con MG,

GLR=2

Con MG,

GLR=1.5

Con MG,

GLR=1.2

Con MG,

GLR=1.0

Con MG,

GLR=0.9

Con MG,

GLR=0.8

CAIDI (h)

MGs' operation configuration

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Fabian Rodriguez, Sergio Rivera

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Volume 17, 2022

are taken into account for the selection of plausible

nodes for the connection of multiple microgrids in the considered system.

Fig. 10: Closeness centrality for each node.

Fig. 11: Betweenness centrality for each node.

Fig. 12: Intermediation centrality for each node considering the chargeability limits of the elements.

0,00E+00

1,00E-02

2,00E-02

3,00E-02

4,00E-02

5,00E-02

6,00E-02

123456789101112131415161718192021222324252627282930

Closeness centrality

Node

100

150

200

250

12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Betweenness Centrality

Node

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Intermediation centrality (limits)

Node

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Fabian Rodriguez, Sergio Rivera

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Volume 17, 2022

4.2. Reliability Analysis based in Complex

Network Theory

Based on the results obtained from the complex

network analysis is proposed in this work to select

nodes with centrality measures that allow the

identification of which are the critical nodes where

the microgrids’ connection is optimal from a

system’s reliability point of view.

Selected the connection nodes based on the

given measures by the complex network theory, the

system’s reliability is calculated, without altering

any other variables in the network, in a way that the

impact in the reliability can be compared. The

reliability analysis is made with a GLR of 1.2 for

every microgrid.

Localization of the MMG based in closeness

centrality.

Taking as reference the first 5 nodes

hierarchically classified with the greater closeness

centrality indexes, the nodes to assign the

microgrids are selected. In Table 3 are shown the 5

nodes with the highest closeness index that

considers the graph’s weights.

Table 3. Nodes with the highest closeness centrality.

Node

Closeness centrality

Without weights

With weights

0.015152

0.052748

0.013333

0.051342

0.010870

0.049101

0.012048

0.048605

0.013889

0.047808

Table 4: Results of the reliability calculation for the reconfiguration of the MGs based in closeness.

Indicator

Unit

Value

SAIFI

1/yr.

2.253

SAIDI

min/yr.

1298.347

CAIDI

9.607

MW/yr.

880.35

MWh/yr.

7977.943

As indicated in Table 4, the SAIDI and the SAIFI

increased with respect to the base case analyzed by

changing the connection points of the MG based in

the closeness centrality measure obtained through

the complex network analysis.

Localization of the MMG based in the betweenness

centrality

Taking as reference the first 5 nodes

hierarchically classified with the greatest

betweenness centrality indexes, the nodes are

selected to assign the microgrids. For this centrality

measure are considered the chargeability limits for

the lines and transformers.

Table 5. Nodes with the highest betweenness centrality.

Node

Betweenness Centrality

Without weights

With weights

With weights considering

limits

56.417

121.67

176.58

208

102.33

115.67

161

101.17

87.5

125

97.5

Table 6. Results of the reliability calculation for the reconfiguration of the MGs based in betweenness.

Indicator

Unit

Value

SAIFI

1/yr.

0.875

SAIDI

min/yr.

630.178

CAIDI

12.003

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Fabian Rodriguez, Sergio Rivera

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Volume 17, 2022

MW/yr.

392.891

MWh/yr.

4442.803

As shown in Tables 5 and 6, the SAIDI and SAIFI

values for the case in which the system is

reconfigured, connecting the MGs in the nodes with

highest betweenness are minor than in the base case.

Localization of MMG focused in reliability based in

complex networks

Taking into account that the reliability

indicators calculated for the reconfigurations of the

MGs in the distribution system of 30 nodes show

that the quality indicators were improved using the

betweenness centrality, but worsen while using the

closeness centrality, up next are shown cases in

which the nodes with the lowest closeness centrality

are taken, also lowest betweenness centrality and a

case with random node selection to make the same

analysis.

Table 7. Nodes with the lowest closeness centrality.

Node

Closeness centrality

Without Weights

With weights

0.007874

0.017861

0.008265

0.019412

0.008265

0.020425

0.010638

0.024411

0.010101

0.025163

Table 8. Results of the reliability calculation for the reconfiguration of the MGs based in the lowest closeness.

Indicator

Unit

Value

SAIFI

1/yr.

0.856

SAIDI

min/yr.

995.359

CAIDI

19.388

MW/yr.

409.636

MWh/yr.

6840.687

As shown in the results of the Tables 7 and 8, the

reliability keeps being greater than the calculated for

the base case, however, the quality of service

indicators doesn’t get too much worse when

compared to the selection case of nodes based in the

nodes with lowest closeness centrality.

For the case of the lowest betweenness centrality,

there are 8 nodes with a value of zero, thus, the five

nodes indicated in the Table 9 are selected.

Table 9. Nodes with the lowest betweenness centrality.

Node

Betweenness centrality

Without weights

With weights

considering limits

Table 10. Results of the reliability calculation for the reconfiguration of the MGs based in the lowest closeness.

Indicator

Unit

Value

SAIFI

1/yr.

0.875

SAIDI

min/yr.

752.052

CAIDI

15.773

MW/yr.

428.8878

MWh/yr.

5562.6351

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Volume 17, 2022

According to the indicators resulted in the Table 10

is observed that the node selection with the lowest

betweenness has a negative impact in the reliability

indicators. For this case is noted that some of the

nodes with the lowest betweenness correspond to

the microgrids connection nodes in the base case.

This corresponds that to the microgrids connection

nodes in the base case correspond to external nodes

of the system, that is to say radial nodes with grade

1 centrality.

5 Discussion and Conclusion

With this work is proposed a selection alternative of

nodes for connection of multiples microgrids in an

electrical distribution system based in the analysis

of complex networks. This perspective allows the

realization of a quick and efficient selection of the

connection nodes of the microgrids that improve the

reliability of the system.

The reliability of a distribution network was

evaluated including multiple microgrids through the

analysis of an electrical network modeled as a

complex network. From this perspective, is expected

that the nodes with the highest betweenness

centrality, a measure provided from the complex

network theory, are the nodes that when connected

the microgrids they improve the reliability of the

whole system more than if they were connected in

other nodes. Besides this it can’t be concluded that

the localization of the microgrids using this unique

criterion based in betweenness centrality will be the

most optimal localization that minimizes the quality

of service indicators (SAIDI and SAIFI) since for

this specific case every possible combination of

arrays of the available microgrids must be evaluated

in every node of the system. Additionally, to

perform the localization optimization of the

microgrids simulations of Monte Carlo must be

made to calculate the reliability and find the

minimum of the quality of service indicators

through optimization algorithms, such as PSO

(Particle Swarm Optimization) or genetical

optimization algorithms.

The proposed method in this work for the

selection of connection nodes of microgrids can be a

useful tool, since to perform an optimization with

the fore mentioned methods it requires a high

computational consumption and long processing

times. The method in which the complex network

theory can proportionate a quick, efficient and

computationally low costly analysis.

Based on the above, not only it's convenient, but

also will be essential that the studies in reliability

analyses from the NO in the electrical sector take

into account these new technologies of reliability in

the planning stage of the distribution systems and

take advantage of its potential to fulfill the quality

of service regulatory requirements.

It was analyzed the relation that can exist

between the closeness and betweenness centrality

measures of the system obtained from the complex

network analysis with the reliability indicators used

in electrical distribution networks (SAIDI, SAIFI)

that are measured to evaluate the given service by

the NO according to the Colombian regulatory

framework. It was observed that the betweenness of

closeness does not provide a guide to determine

nodes that have an impact in the reliability of the

system by connecting multiple microgrids, however,

it can exist some relation between the system’s

betweenness centrality measure since by calculating

the reliability of the reorganized system the

microgrids in the distribution network based in the

nodes with the highest betweenness index low

values are obtained in the quality of service

indicators, which indicates an improvement in the

reliability.

Due to the reliability analysis that involves

multiple microgrids in an electrical system

involving many variables and it’s a very hard

problem to confront, for the analysis made in this

work several simplifications like modeling the

microgrids with the VPP concept were made,

assuming a constant GLR for all the microgrids and

a power generation and equal consumption for all,

as well as a failure rate equal for every microgrid.

The generation costs according to the type of DER

resource aren’t taken into account as well as the

charges variability. Considering all the elements to

find the connection nodes of the microgrids that

optimize the reliability, that is to say they make the

quality indicators better (nodes that minimize the

SAIDI and SAIFI) is a very complex optimization

problem that would require advanced optimization

algorithms such as PSO, genetic optimization

algorithms, Customer Scattering, etc. These could

be applied to the optimization of the reliability

calculation through Monte Carlo methods. The

study of the complex network analysis can continue

with a focus to verifying that it's a quick, simple and

very efficient analytic method to find optimal nodes,

verifying it with optimization algorithms like the

ones mentioned above. The prior is not only for the

reliability analyses, but it can be also used in other

power systems’ analyses.

The complex network theory provides a

completely different perspective for the power

systems’ analysis. The concepts that are applied

from the complex network theory to the electrical

networks can help us to better comprehend the

topology, the characteristics, the behavior and for

this particular case the reliability of an electrical

distribution network from other focus point.

However, in the present the application of the

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Volume 17, 2022

complex network theory to the electrical networks is

still in a theoretical level and requires greater depth

in the researches that lead to being able to apply

these concepts and allow the development of ways

to improve the reliability, safety, stability and

efficiency of the electrical networks.

References:

[1] A. A. Chowdhury and D. O. Koval, Power

Distribution System Reliability. Hoboken, NJ,

USA: John Wiley & Sons, Inc., 2009.

[2] R. Billinton and R. N. Allan, Reliability

Evaluation of Power Systems, 2nd ed. Boston,

MA: Springer US, 1996.

[3] R. N. Allan, R. Billinton, A. M. Breipohl, and C.

H. Grigg, “Bibliography on the application of

probability methods in power system reliability

evaluation,” IEEE Transactions on Power Systems,

vol. 14, no. 1, pp. 51–57, 1999, doi:

10.1109/59.744483.

[4] P. Jahangiri and M. Fotuhi-Firuzabad, “Reliability

assessment of distribution system with distributed

generation,” in 2008 IEEE 2nd International Power

and Energy Conference, Dec. 2008, pp. 1551–

1556, doi: 10.1109/PECON.2008.4762728.

[5] M. Makandar, C. S. R. Atla, and S. Velamuri,

“Reliability assessment of distribution system with

renewable Distributed Generation,” in 2016

Biennial International Conference on Power and

Energy Systems: Towards Sustainable Energy

(PESTSE), Jan. 2016, pp. 1–5, doi:

10.1109/PESTSE.2016.7516365.

[6] S. Xin, C. Yan, Z. Xingyou, and W. Chuanzhi, “A

novel multi-microgrids system reliability

assessment algorithm using parallel computing,” in

2017 IEEE Conference on Energy Internet and

Energy System Integration (EI2), Nov. 2017, pp.

1–5, doi: 10.1109/EI2.2017.8245363.

[7] M. A. Al-Shehri, Y. Guo, and G. Lei, “A

Systematic Review of Reliability Studies of Grid-

Connected Renewable Energy Microgrids,” in

2020 International Conference on Electrical,

Communication, and Computer Engineering

(ICECCE), Jun. 2020, no. June, pp. 1–6, doi:

10.1109/ICECCE49384.2020.9179273.

[8] Z. Bie, P. Zhang, G. Li, B. Hua, M. Meehan, and

X. Wang, “Reliability Evaluation of Active

Distribution Systems Including Microgrids,” IEEE

Transactions on Power Systems, vol. 27, no. 4, pp.

2342–2350, Nov. 2012, doi:

10.1109/TPWRS.2012.2202695.

[9] P. M. Costa and M. A. Matos, “Assessing the

contribution of microgrids to the reliability of

distribution networks,” Electric Power Systems

Research, vol. 79, no. 2, pp. 382–389, Feb. 2009,

doi: 10.1016/j.epsr.2008.07.009.

[10] F. Tooryan and E. R. Collins, “Optimum size and

placement of distributed generators in microgrid

based on reliability concept,” in 2018 IEEE Power

and Energy Conference at Illinois (PECI), Feb.

2018, vol. 2018-Janua, pp. 1–6, doi:

10.1109/PECI.2018.8334992.

[11] T. Adefarati and R. C. Bansal, “Reliability

assessment of distribution system with the

integration of renewable distributed generation,”

Applied Energy, vol. 185, pp. 158–171, Jan. 2017,

doi: 10.1016/j.apenergy.2016.10.087.

[12] Comisión de Regulación de Energía y Gas CREG,

Ministerio de Minas y Energía, and Republica de

Colombia, Resolución CREG No. 015 de 2018.

Colombia, 2018, p. 239.

[13] M. Castaño-Gómez and J. J. García-Rendón,

“Análisis de los incentivos económicos en la

capacidad instalada de energía solar fotovoltaica

en Colombia,” Lecturas de Economía, no. 93, pp.

23–64, Jul. 2020, doi:

10.17533/udea.le.n93a338727.

[14] Comision de Regulacion de Energia y Gas CREG;

Ministerio de Minas y Energía; Republica de

Colombia, Resolución CREG No. 030 de 2018.

Colombia, 2018, p. 13.

[15] Comisión de Regulación de Energía y Gas CREG,

Ministerio de Minas y Energía, and Republica de

Colombia, Resolución CREG No. 038 de 2018.

Colombia, 2018.

[16] R. Kaduru and N. Srinivas GONDLALA,

“Distribution System Reliability with Distributed

Generation Based on Customer Scattering,” no. 13,

2015, doi: 10.15598/aeee.v13i2.1025.

[17] Ke Sun, “Complex Networks Theory: A New

Method of Research in Power Grid,” in 2005

IEEE/PES Transmission & Distribution

Conference & Exposition: Asia and Pacific,

2005, vol. 2005, pp. 1–6, doi:

10.1109/TDC.2005.1547099.

[18] B. Liu, Z. Li, X. Chen, Y. Huang, and X. Liu,

“Recognition and Vulnerability Analysis of Key

Nodes in Power Grid Based on Complex Network

Centrality,” IEEE Transactions on Circuits and

Systems II: Express Briefs, vol. 65, no. 3, pp. 346–

350, 2018, doi: 10.1109/TCSII.2017.2705482.

[19] Y. Nurdin, “Understanding the Cascading Failures

in Indonesian Power Grids with Complex Network

Theory,” in 2019 IEEE International Conference

on Cybernetics and Computational Intelligence

(CyberneticsCom), Aug. 2019, pp. 50–55, doi:

10.1109/CYBERNETICSCOM.2019.8875659.

[20] A. B. M. Nasiruzzaman and H. R. Pota, “Transient

stability assessment of smart power system using

complex networks framework,” 2011, doi:

10.1109/PES.2011.6038970.

[21] A. Dwivedi, X. Yu, and P. Sokolowski,

“Identifying vulnerable lines in a power network

using complex network theory,” in 2009 IEEE

International Symposium on Industrial Electronics,

Jul. 2009, no. ISlE, pp. 18–23, doi:

10.1109/ISIE.2009.5214082.

[22] A. T. Alexandridis and P. C. Papageorgiou, “A

complex network deployment suitable for modern

power distribution analysis at the primary control

level,” IFAC-PapersOnLine, vol. 50, no. 1, pp.

9186–9191, 2017, doi:

10.1016/j.ifacol.2017.08.1272.

[23] M. Saleh, Y. Esa, N. Onuorah, and A. A.

Mohamed, “Optimal microgrids placement in

electric distribution systems using complex

WSEAS TRANSACTIONS on POWER SYSTEMS

10.37394/232016.2022.17.3

Fabian Rodriguez, Sergio Rivera

E-ISSN: 2224-350X

Volume 17, 2022

network framework,” 2017 6th International

Conference on Renewable Energy Research and

Applications, ICRERA 2017, vol. 2017-Janua, pp.

1036–1040, 2017, doi:

10.1109/DISTRA.2017.8191215.

[24] EPRI, “Strategic Insights on Security, Quality,

Reliability, and Availability,” 2005. [Online].

Available:

https://www.epri.com/research/products/00000000

0001008566.

[25] R. Billinton and R. N. Allan, “Power-system

reliability in perspective,” Electronics and Power,

vol. 30, no. 3, p. 231, 1984, doi:

10.1049/ep.1984.0118.

[26] R. Billinton and R. N. Allan, Reliability

Evaluation of Engineering Systems, 2nd ed., vol.

43, no. 4. Boston, MA: Springer US, 1992.

[27] M. A. Bucher, M. Vrakopoulou, and G.

Andersson, “Probabilistic N-1 security assessment

incorporating dynamic line ratings,” 2013, doi:

10.1109/PESMG.2013.6672679.

[28] EPRI, “Value Modeling for Reliability of

Distribution and Transmission Systems,” 2006.

[Online]. Available:

https://www.epri.com/research/products/00000000

0001012501.

[29] G. C. Loehr, “The ‘good’ Blackout: The Northeast

Power Failure of 9 November 1965 [History],”

IEEE Power and Energy Magazine, vol. 15, no. 3.

Institute of Electrical and Electronics Engineers

Inc., pp. 84–96, May 01, 2017, doi:

10.1109/MPE.2017.2659379.

[30] J. McCalley et al., “Probabilistic security

assessment for power system operations,” in 2004

IEEE Power Engineering Society General

Meeting, 2004, vol. 1, pp. 212–220, doi:

10.1109/pes.2004.1372788.

[31] W. Li and J. Zhou, “Probabilistic reliability

assessment of power system operations,” Electric

Power Components and Systems, vol. 36, no. 10,

pp. 1102–1114, Oct. 2008, doi:

10.1080/15325000802046868.

[32] Y. Sun, L. Cheng, X. Ye, J. He, and P. Wang,

“Overview of power system operational

reliability,” in 2010 IEEE 11th International

Conference on Probabilistic Methods Applied to

Power Systems, PMAPS 2010, 2010, pp. 166–171,

doi: 10.1109/PMAPS.2010.5528993.

[33] W. Li, “Evaluating Mean Life of Power System

Equipment with Limited End-of-Life Failure

Data,” IEEE Transactions on Power Systems, vol.

19, no. 1, pp. 236–242, Feb. 2004, doi:

10.1109/TPWRS.2003.821434.

[34] R. Billinton and W. Li, Reliability Assessment of

Electric Power Systems Using Monte Carlo

Methods, 1st ed. Boston, MA: Springer US, 1994.

[35] R. Billinton and A. Jonnavithula, “Variance

reduction techniques for use with sequential Monte

Carlo simulation in bulk power system reliability

evaluation,” in Canadian Conference on Electrical

and Computer Engineering, 1996, vol. 1, pp. 416–

419, doi: 10.1109/ccece.1996.548125.

[36] Y. Wang, C. Guo, Q. Wu, and S. Dong, “Adaptive

sequential importance sampling technique for

short-term composite power system adequacy

evaluation,” IET Generation, Transmission and

Distribution, vol. 8, no. 4, pp. 730–741, 2014, doi:

10.1049/iet-gtd.2013.0279.

[37] Q. Chen and L. Mili, “Composite power system

vulnerability evaluation to cascading failures using

importance sampling and antithetic variates,”

IEEE Transactions on Power Systems, vol. 28, no.

3, pp. 2321–2330, 2013, doi:

10.1109/TPWRS.2013.2238258.

[38] P. Jirutitijaroen and C. Singh, “Comparison of

simulation methods for power system reliability

indexes and their distributions,” IEEE

Transactions on Power Systems, vol. 23, no. 2, pp.

486–493, May 2008, doi:

10.1109/TPWRS.2008.919425.

[39] R. H. Lasseter, “Smart distribution: Coupled

microgrids,” in Proceedings of the IEEE, 2011,

vol. 99, no. 6, pp. 1074–1082, doi:

10.1109/JPROC.2011.2114630.

[40] N. D. Hatziargyriou, A. Dimeas, A. G. Tsikalakis,

J. A. Pecas Lopes, G. Kariniotakis, and J.

Oyarzabal, “Management of microgrids in market

environment,” in 2005 International Conference on

Future Power Systems, 2005, vol. 2005, doi:

10.1109/fps.2005.204225.

[41] P. M. Costa and M. A. Matos, “Economic analysis

of microgrids including reliability aspects,” 2006,

doi: 10.1109/PMAPS.2006.360236.

[42] A. Abdulkarim et al., “Reliability Study of Stand-

alone Hybrid Renewable Energy Microgrids,”

Iranian Journal of Science and Technology -

Transactions of Electrical Engineering, vol. 43, no.

1, pp. 411–425, Jul. 2019, doi: 10.1007/s40998-

018-0119-8.

[43] E. Zio, “Reliability Analysis of Complex Network

Systems: Research and Practice in Need,” IEEE

Transactions on Reliability, vol. 57, no. 3, pp. 1–4,

2008.

[44] J. Huang, Y. Feng, and S. Zhang, “Research of

complex system theory application on reliability

analysis of network system,” Proceedings of 2009

8th International Conference on Reliability,

Maintainability and Safety, ICRMS 2009, pp.

1141–1145, 2009, doi:

10.1109/ICRMS.2009.5270062.

[45] M. Newman, Networks: An Introduction. Oxford

University Press, 2010.

[46] M. E. J. Newman, “The Structure and Function of

Complex Networks,” SIAM Review, vol. 45, no.

2, pp. 167–256, Jan. 2003, doi:

10.1137/S003614450342480.

[47] R. Shields, “Cultural Topology: The Seven

Bridges of Königsburg, 1736,” Theory, Culture &

Society, vol. 29, no. 4–5, pp. 43–57, Jul. 2012, doi:

10.1177/0263276412451161.

[48] R. Albert and A.-L. Barabási, “Statistical

mechanics of complex networks,” Reviews of

Modern Physics, vol. 74, no. 1, pp. 47–97, Jan.

2002, doi: 10.1103/RevModPhys.74.47.

[49] J. Kim and T. Wilhelm, “What is a complex

graph?,” Physica A: Statistical Mechanics and its

Applications, vol. 387, no. 11, pp. 2637–2652,

2008, doi: 10.1016/j.physa.2008.01.015.

WSEAS TRANSACTIONS on POWER SYSTEMS

10.37394/232016.2022.17.3

Fabian Rodriguez, Sergio Rivera

E-ISSN: 2224-350X

Volume 17, 2022

[50] A. B. M. Nasiruzzaman and H. R. Pota, “Critical

node identification of smart power system using

complex network framework based centrality

approach,” 2011, doi:

10.1109/NAPS.2011.6025194.

[51] M. Mesbahi and M. Egerstedt, Graph Theoretic

Methods in Multiagent Networks. Princeton

University Press, 2010.

[52] L. A. Machuca Moreno, “Análisis de estabilidad

transitoria basado en teoría de redes complejas y el

fenómeno de percolación,” 2017.

[53] G. Zhang, C. Wang, J. Zhang, J. Yang, Y. Zhang,

and M. Duan, “Vulnerability assessment of bulk

power grid based on complex network theory,” 3rd

International Conference on Deregulation and

Restructuring and Power Technologies, DRPT

2008, no. April, pp. 1554–1558, 2008, doi:

10.1109/DRPT.2008.4523652.

[54] G. Chen, Z. Y. Dong, D. J. Hill, and G. H. Zhang,

“An improved model for structural vulnerability

analysis of power networks,” Physica A: Statistical

Mechanics and its Applications, vol. 388, no. 19,

pp. 4259–4266, Oct. 2009, doi:

10.1016/j.physa.2009.06.041.

[55] I. Rajasingh, B. Rajan, and I. D. Florence,

“Betweeness-centrality of grid networks,” in

ICCTD 2009 - 2009 International Conference on

Computer Technology and Development, 2009,

vol. 1, pp. 407–410, doi:

10.1109/ICCTD.2009.230.

[56] M. Girvan and M. E. J. Newman, “Community

structure in social and biological networks,”

Proceedings of the National Academy of Sciences

of the United States of America, vol. 99, no. 12,

pp. 7821–7826, Dec. 2001, doi:

10.1073/pnas.122653799.

[57] L. C. Freeman, “Centrality in social networks

conceptual clarification,” Social Networks, vol. 1,

no. 3, pp. 215–239, Jan. 1978, doi: 10.1016/0378-

8733(78)90021-7.

[58] R. Christie, “Power Systems Test Case Archive -

30 Bus Power Flow Test Case,” 1993.

http://labs.ece.uw.edu/pstca/pf30/pg_tca30bus.htm

(accessed Jan. 23, 2021).

[59] The Grainger College of Engineering and

University of Illinois at Urbana-Champaign,

“Illinois Center for a Smarter Electric Grid

(ICSEG). IEEE 30-Bus System,” 2013. [Online].

Available: https://icseg.iti.illinois.edu/ieee-30-bus-

system/.

[60] “IEEE Recommended Practice for the Design of

Reliable Industrial and Commercial Power

Systems,” IEEE Std 493-2007 (Revision of IEEE

Std 493-1997), pp. 1–383, 2007, doi:

10.1109/IEEESTD.2007.380668.

[61] Comision de Regulacion de Energia y Gas CREG;

Ministerio de Minas y Energía; Republica de

Colombia, Resolución CREG No. 025 de 1995.

Colombia, 1955.

[62] S. Peyghami, M. Fotuhi-Firuzabad, and F.

Blaabjerg, “Reliability Evaluation in Microgrids

With Non-Exponential Failure Rates of Power

Units,” IEEE Systems Journal, vol. 14, no. 2, pp.

2861–2872, Jun. 2020, doi:

10.1109/JSYST.2019.2947663.

[63] Z. Bie, P. Zhang, G. Li, B. Hua, M. Meehan, and

X. Wang, “Reliability evaluation of active

distribution systems including microgrids,” IEEE

Transactions on Power Systems, vol. 27, no. 4, pp.

2342–2350, 2012, doi:

10.1109/TPWRS.2012.2202695.

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WSEAS TRANSACTIONS on POWER SYSTEMS

10.37394/232016.2022.17.3

Fabian Rodriguez, Sergio Rivera

E-ISSN: 2224-350X

Volume 17, 2022