配電管理系統自動區段開關之多目標最佳配置研究

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Graduate Institute of Industrial Engineering College of Engineering

National Taiwan University Master Thesis

ଛႝᆅ౛س಍Ծ୏୔ࢤ໒ᜢϐӭҞ኱ന٫ଛ࿼ࣴز Multi-objective Optimal Placement of Automatic Line Switches

in Power Distribution Networks

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Logrono Vargas Diego Orlando

ࡰᏤ௲௤;ֆЎБ!റγ!

Advisor: Wu, Wen-Fang, Ph.D.

ύ๮҇୯ 212 ԃ 9 Д!

August 2012

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Acknowledgements

I wish to express my sincere gratitude to my advisor, Prof. Wen-Fang Wu, for his guidance and encouragement during this stage of my life which have made possible the culmination of the present research work.

Special thanks to professors and fellow students from NTU-GIIE, for providing me such a wonderful environment for my academic preparation and personal growth.

My deepest gratitude is to my parents, Jaime and Susana, for their precious support and affection.

From the bottom of my heart I would like to thank my dear Elsa, for her understanding and always standing by my side with love and care. Thank you very much for everything you do for me and I am glad we are travelling together through this path of happiness.

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Abstract

In modern power distribution utilities, there is a growing demand for an improved system response in case of outages. In order to address that demand, automatic line switches can be installed in distribution networks to reduce the number and durations of power interruptions.

However, automatic switching devices involve an increased investment cost. For distribution utilities, obtaining a high level of reliability at a relatively low cost becomes a multi-objective optimization problem. To solve the problem, a computational procedure based on Elitist Nondominated Sorting Genetic Algorithm (NSGA-II) is developed in the present study.

Following the proposed methodology, we are able to obtain a set of optimal trade-off solutions identifying the number and placement of automatic switches in a distribution network for which we can obtain the most reliability benefit out of the utility investment. To determine the effectiveness of the procedure, two case studies were carried out. For comparison purposes, one of the cases corresponds to a previous study of an actual distribution system belonging to Taipower Company. The result of both tests indicates the improvement in system reliability indices due to the addition of a certain degree of automation investment in the distribution network, and demonstrates the present methodology is able to satisfy the system requirements in a better way than the mentioned previous study.

Keywords: Power distribution systems, automatic line switches, multi-objective optimization, NSGA-II, optimal placement, case study simulation.

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๏ᆛၡ΢ޑԾ୏୔ࢤ໒ᜢёԖਏ෧ϿႝΚυᘋޑԛኧᆶਔ໔ǶฅԶǴԾ୏୔ࢤ໒ᜢး࿼ޑ

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Κଛ๏ޑଯё᎞ࡋԋࣁ΋ঁӭҞ኱ന٫ϯୢᚒǶࣁှ،೭ኬޑୢᚒǴҁፕЎว৖΋ᅿ୷ܭ ᆒम฼ౣߚЍଛ௨ׇᒪ໺ᄽᆉݤޑᄽᆉኳԄǶਥᏵፕЎ܌ගޑБݤǴךॺёளډ΋ಔ࿶ၸ

៾ᑽࡕޑှ،БਢǴ،ۓ΋੝ۓଛႝᆛၡԾ୏୔ࢤ໒ᜢး࿼ޑኧໆᆶӼးଛ࿼ǴբࣁႝΚ Ϧљӧ΋ۓԋҁΠޑനଯё᎞ࡋୖԵǶҁࣴزаٿঁਢϩ݋ᔠᡍᄽᆉݤޑԖਏᆶցǴࣁၲ

ډٿ࣬ჹྣޑਏ݀Ǵځύ΋ঁਢϩ݋ࢂჹᔈܭӃ߻Ѡ᡼ႝΚϦљ੿ჴଛႝᆛၡޑᜪ՟ࣴزǶ

ঁਢϩ݋ޑ่݀ᡉҢǴჹܭଛႝᆛၡޑ΋٤Ծ୏ϯ׫ၗዴёԖਏගϲё᎞ࡋࡰ኱Ǵ໔ௗᡍ

᛾ፕЎ܌ගᄽᆉݤ࣬ၨܭӃ߻ࣴزǴ׳ૈԖਏᅈىس಍ሡ؃Ƕ!

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ᜢᗖຒǺႝΚଛ๏س಍ǵԾ୏୔ࢤ໒ᜢǵӭҞ኱ന٫ϯǵߚᓬ༈௨ӈᒪ໺ᄽᆉݤǵന٫ϯ ଛ࿼ǵঁਢϩ݋ኳᔕ!

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List of Figures

Figure 2-1 Overview of Taipower System ... 10

Figure 2-2 Radial Distribution Feeder and its Line Switches ... 14

Figure 2-3 Operation of Automatic Sectionalizing and Tie-point switches in Distribution Networks... 18

Figure 3-1 Illustration of a general multi-objective optimization problem (Tran, 2006) ... 27

Figure 3-2 Pareto-optimal front including Pareto-optimal solutions and a non-optimal solution . 28 Figure 3-3 Creation of Mating Pool from six hypothetical solution fitness using Tournament Selection ... 34

Figure 3-4 Elitist Selection Mechanism of NSGA-II (Tran, 2006) ... 39

Figure 4-1 Chromosomal Representation for a Feasible Solution ... 43

Figure 4-2 Procedures performed in the proposed NSGA-II ... 44

Figure 4-3 Distribution Network having 2 neighbor feeders, 11 load points and 10 possible switch locations... 45

Figure 5-1 Optimal Solution (using Max-Min) in One-line Diagram for Case 1 ... 58

Figure 5-2 Scatter of the Pareto-optimal set obtained by the integer NSGA-II ... 59

Figure 5-3 (a) SAIFI vs TCOST (b) SAIDI vs TCOST (c) SAIFI vs SAIDI ... 60

Figure 5-4 Scatter of the Constrained Pareto-optimal Solutions with Max-Min... 63

Figure 5-5 Pareto-optimal Solutions & Max-Min solution for Case 2 ... 65

Figure 5-6 Optimal Placement of Line Switches using Max-Min approach ... 66

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List of Tables

Table 4-1 Random Generation of 6 Feasible Solutions ... 45

Table 4-2: Fitness Evaluation ... 48

Table 4-3: Crossover operation... 50

Table 4-4: Mutation Operation ... 52

Table 5-1: NSGA-II Parameter Settings ... 57

Table 5-2: Distribution Feeder Parameters... 57

Table 5-3: Impact of Automatic Line Switches in Distribution Networks ... 61

Table 5-4 Results of the Proposed Study ... 67

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List of Symbols

ܰܵܩܣ െ ܫܫ Elitist Nondominated Sorting Genetic Algorithm

ܵܣܫܨܫ System Average Interruption Frequency Index

ܵܣܫܦܫ System Average Interruption Duration Index

ܶܥܱܵܶ Total Investment Cost

ߣ_௜௦ Permanent failure rate of load point i due to failure in section s

ݎ_௜௦ Average time per interruption of load point i due to outages in section s ߣ_௦ Permanent failure rate of section s

ݎ_௥௦ Average repair time of the fault ݎ_௦௪ Average switching time of the devices

ܰ_௜ Number of customers at load point i n Number of load points

m Number of sections

ܵ_௦ Set of all sections connecting the power source and section s ܮ_௜ Set of the section path that links power source and load point i ܦ Set of available automatic sectionalizing switches

ܶܲ Set that contains the available automatic tie-point switches

ܰݑ݉_஽ൈ ܥ_஽ Total cost due to the investment on ܰݑ݉_஽ auto sectionalizing sw

ܰݑ݉_்௉ൈ ܥ_்௉ Total cost due to the investment on ܰݑ݉_்௉ auto tie-point sw

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Chapter 1: Introduction

1.1. Background and Motivation

The distribution system is a vital component of any electric power system. It constitutes the final linkage between bulk power source and end customers. However, distribution is also one of the most susceptible to failures within the power system (Brown, 2008a). Therefore, power quality and continuity have become among the most important objectives that distribution utilities have concentrated significant efforts on in order to satisfy system load and energy requirements as economically as possible.

Distribution system reliability can be improved by reducing the frequency of occurrence of faults and by reducing the repair time by means of maintenance strategies (Zheng et al., 2011).

The addition of switches along the distribution network contributes to reduce the number and duration of interruptions; however, this involves investment costs. The two aspects of obtaining high level of reliability at a relatively low cost are often in direct conflict due to the fact that providing a higher reliability will cost utilities more capital.

The above statement drives a motivation to emphasize on the multi-objective optimization of utility investment costs and reliability benefits, the result of which will be a set of trade-off solutions that optimize both objectives so that the decision-maker can choose from.

Two types of line switches are normally installed along the distribution feeders: sectionalizing switch (normally closed switch) and tie-point switch (normally open switch). The former is a device that isolates a faulted section from the system so that the healthy sections upstream can

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still be electrically supplied. The latter is a device that restores power to the disconnected loads downstream the failure by transferring them to a neighbor distribution feeder without violating operational and engineering constraints.

Whenever a fault has been identified at any point of a network, acting as soon as possible may result in a minimum affected area. The process of restoring a feeder from a fault, (Bernardon et al., 2011), can be stated in the following steps:

x Identify the exact fault position,

x Isolate the faulted section by opening normally closed switches,

x Restore power supply to customers upstream and downstream of the isolated block, x Correct the problem,

x Re-operate the switches to get back to normal network status.

Automation of distribution systems significantly contributes to reduce the time to perform the service restoration procedure and utterly minimize the impact of power interruptions. With the installation of automatic or remote-controlled line switches in the network, we can experience a faster response to isolate the fault without maintenance personnel even having to physically be at the location. New regulation policies have allowed automatic sectionalizing switches to operate faster, more efficient and reliable than traditional manual switches (Romero, Wesz da Silva, &

Mantovani, 2011). Automatic switches have also shown to be an economically viable solution due to the emergence of a large number of automation equipment suppliers and new communication technologies.

A frequent topic currently discussed is how the electric power distribution systems of the future will be. In this sense the term “smart grid” has arisen to describe how the new distribution

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systems will behave, this is in a “smart” or “intelligent” manner. A deeper analysis about Smart Grid will be presented in Chapter 2, but as an introduction we can mention that among the features of the Smart Grid are the ability to carry out maneuvers in automatic mode (self- reconfiguration) and high reliability, all with low operation and maintenance costs.

The selection of an efficient methodology to determine the optimum location and number of automatic line switches is essential for utilities, since that procedure is closely related to the restoration time and consequently associated to the system reliability indices. The optimization methodology constitutes not an easy task because it is a combinatorial¹ constrained problem described by a nonlinear and nondifferentiable objective function and its solution can be challenging to solve (Tippachon & Rerkpreedapong, 2009).

Different approaches have proposed solutions for the problem of switch placement in distribution networks. Some studies develop optimization methodologies for a single objective function (mono-objective) such as minimizing economic cost or reliability indices (Chen et al., 2006), (Bernardon, et al., 2011). Other researches are focused on covering the impact of automatic switches, without considering its optimal allocation (Zheng, et al., 2011). Finally, some important studies about multi-objective allocation in distribution networks do not consider automatic switches after all (Tippachon & Rerkpreedapong, 2009), (Ferreira, Bretas, & Cardoso, 2010). The multi-criteria methodology for optimal placement of automatic line switches has not been included.

The multi-objective optimal placement of switches in distribution networks allows better operation and improvement on the reliability of the system (Ferreira, et al., 2010). Moreover, the

1 Combinatorial optimization consists, in a sentence, on finding the optimal solution among a finite set of solutions (Schrijver, 2003).

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reliability indices most commonly used to quantify the quality of the utilities services are related to sustained interruptions (interruptions longer than 5 minutes): System Average Interruption Frequency Index (SAIFI), and System Average Interruption Duration Index (SAIDI). These two indices highly depend on network topology and location of automatic switching devices.

Therefore, our optimization task will be driven to design a methodology for optimal placement of automatic line switches in distribution networks that simultaneously minimizes cost expenditures and maximizes system reliability (by minimizing SAIFI and SAIDI).

As more study on this new trend of power distribution technologies has extended around the world, Taiwan is also taking steps to become a major force in Smart Grid. According to MOEA’s Bureau of Energy, the Taiwanese government plans to invest an additional US$4.6 billion in smart power grid infrastructure starting from first quarter 2012 (Wu, 2011). This amount includes NT$123.7 billion for improvement of power grid efficiency, NT$10.1 billion for the promotion of smart grid industry and NT$ 6.1 billion for technological research and development.

It is expected that this project will create an output value of NT$1 trillion (Wu, 2011) in smart grid industry by 2030, making Taiwan an output country for global smart grid industry and equipment manufacturing.

Taiwan has a long-term experience in the ICT (Information and Communications Technology) industry which represents a solid foundation for developing the smart grid industry. However, the expertise with individual components will have to lead to research on abilities for system integration. That constitutes another of the motivations for developing the present research topic, giving the need of research in terms of network automation that will allow self-regulation, including automatic reconfiguration in the event of failures, threats, or disturbances.

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1.2. Research Purpose

This study aims to the development of a computational algorithm to address the automatic line switch allocation problem on the basis of a Nondominated Sorting Genetic Algorithm (NSGA-II), in order to improve the reliability of the distribution systems and minimize costs expenditures.

This method has proven to be effective in solving multi-objective optimization problems and promoting satisfactory solutions belonging to the Pareto-optimal front. The algorithm can be conﬁgured according to the needs of the utilities and help the network designer in the decision making process. Therefore, the proposed methodology will indicate where the utility should invest resources for switching automation in order to improve the reliability of the system, which constitutes an important decision support tool for planning and operating the distribution networks.

The proposed approach was tested in actual distribution feeders belonging to Taipower Company and its effectiveness, on the specified portion of the real system, was seen in the sense of the improvement in the reliability indices (SAIFI and SAIDI) and the different trade-off possibilities for reliability that we can expect depending on the degree of automation investment, letting one conclude the relevant economic benefits obtained by providing a set of optimal solutions to the decision-maker so that he/she can decide the most appropriate alternative based on his/her own professional experience.

Consequently, the main contributions of this study are highlighted as follows:

1) A new algorithm to assess the impact on reliability due to the installation of automatic line switches in distribution networks.

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2) A new multi-objective optimization methodology for solving the automatic switch allocation problem using a modified integer version of NSGA-II.

1.3. Research Procedure

In this study we propose a methodology that specifies the optimal number and location of automatic line switches in distribution networks by following the steps below:

1) Understanding the function of line switches in distribution networks and how their number and allocation can impact system reliability.

2) Propose a mathematical model that represents the behavior of distribution networks due to the installation of automatic line switches.

3) Development of a computational algorithm based on integer-coding NSGA-II in order to search for the best combinations of line switch number and locations that optimizes the objective functions.

4) Simulation of the prosed algorithm for two case studies. One of them, an actual distribution system of Taipower Company. Additionally, results discussion for every case study.

5) Research study conclusions.

1.4. Chapter Outline

The remainder of this thesis dissertation is organized as follows:

In Chapter 2 we introduce some basic definitions of distribution systems and the problem formulation for the placement of line switches in distribution networks.

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In Chapter 3, the algorithm foundations are presented concerning to multi-objective optimization theory, genetic algorithms and the concepts behind Elitist Nondominated Sorting Genetic Algorithm (NSGA-II).

Chapter 4 is dedicated to explain the proposed algorithm and its working mechanism.

In Chapter 5 we perform the simulation of the implemented algorithm in case studies and present the results discussion.

Finally, we outline the conclusions of this study in Chapter 6.

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Chapter 2: Literature Review and Problem Formulation

2.1. Power Distribution Systems

Since distribution systems account for up to 90% of all customer supply interruptions and reliability problems (Brown, 2008a), improving distribution reliability is key in order to improve customer reliability. For succeeding in this task, a basic outline of power distribution systems will be stated. The following sections present fundamental concepts and terminology that will provide foundation for further reliability analysis.

2.1.1. Generation, Transmission, Distribution

Generation plants consist of one or more generating units that convert mechanical energy into

electricity by turning a turbine coupled to an electric generator. Most turbines are driven by steam produced in a boiler fired by coal, oil, natural gas, or nuclear fuel. Others may be driven by non-thermal sources such as hydroelectric dams and wind farms. Typically, generators produce line– to–line voltages between 11 kV and 30 kV, but since this is not a sufficiently high voltage to transport electricity long distances, generation substations step up voltages to transmission levels (typically between 115 kV and 1100 kV).

Transmission systems transport electricity over long distances from bulk power generation

facilities to substations that serve sub-transmission or distribution systems. Most transmission lines are overhead but there is a growing trend towards the use of underground transmission cable.

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To increase flexibility and improve reliability, transmission lines are interconnected at transmission switching stations and transmission substations. This improves overall performance, but makes the system vulnerable to cascading failures².

Distribution systems deliver power from bulk power systems to retail customers. To do this,

distribution substations receive power from sub-transmission lines and step down voltages with power transformers to utilization levels. Distribution systems consists on distribution transformers (which supply distribution feeders and contain a main 3ϕ trunk, 2ϕ and 1ϕ laterals), feeder interconnections, and distribution feeder.

In order to illustrate the above definitions, Figure 2-1 presents an overview of Taipower System – based on (F. Lin, 2011) – where we can distinguish the three stages that take place before power is delivered to the final customers. After being generated at a power station, the power is firstly steeped up to 345KV in order to be transported long distances. In transmission substations, the power is steeped down to 161KV. For the case of Taipower, there is a series of distribution transformers that converts 161 KV to 69 KV and 11.4 KV since there is specific customers that need those power levels. Finally, distribution power transformers lower down 11.4 KV to utilization levels (220-110V).

2 Cascading failure is a type of failure in a system of interconnected parts in which a failure of one part can trigger the failure of successive parts.

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Figure 2-1 Overview of Taipower System

It is important to note that the key elements and principles of operation for interconnected power systems were established before the 1960s (Massoud Amin & Wollenberg, 2005), which means before the emergence of extensive computer systems and communication networks. Nowadays, computation is heavily applied throughout all levels of power network planning and optimization as well as local control of the equipment and data processing.

The incorporation of Smart Grid technologies allows us to have a two-way communication interaction to get access to a better real time network control and data acquisition which permits

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an easier task fulfillment. However, the new technologies require improvements in the approaches for solving existing problems in automation applications and research on new ones (Farhangi, 2010).

We present next a more formal definition to what Smart Grid involves and the philosophies and technologies within.

2.1.2. Concepts in Smart Grid

Our current electric grid was conceived around a century ago when electricity needs were simpler. Power generation was localized and built around the communities; in addition, most homes only had small energy demands. The grid was designed for utilities to deliver electricity to consumer homes and then bill them for the service every month. However, this limited one- way interaction makes it difficult for the grid to respond to the always changing and rising energy demands of the 21th century. The Smart Grid introduces a two-way communication where electricity and information can be exchanged between the utility and its customers.

According to (Litos, 2008) Smart Grid is a developing network of communications, controls, automation and new technologies and tools working together to make the grid more efficient, more reliable, more secure and greener.

Some of the benefits associated with the Smart Grid include:

x More efficient transmission of electricity

x Self-reconfiguration and quicker restoration of electricity after power disturbances

x Reduced operations and management costs for utilities, and ultimately lower power costs for consumers

x Enabling active participation by consumers in demand response

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x Reduced peak demand, which will also help lower electricity rates x Increased integration of renewable energy systems

x Improved security

Today, an electricity disruption such as a blackout can have a domino effect—a series of failures that can affect banking, communications, traffic, and security. A smarter grid will add resiliency to the electric power system and make it better prepared to address emergencies such as severe natural disasters. Because of its two-way interactive capacity, the Smart Grid will allow automatic rerouting when equipment fails or outages occur. This will minimize outages and minimize the effects when they do happen.

Distribution automation (DA) is one the most research-active areas in the field of Smart Grid (Brown, 2008b). It refers to monitoring, control, and communication functions located out on the feeder. From a design perspective, the most important aspects of distribution automation are in the areas of protection and switching (Brown, 2008b).

Automatic switches allow detection of a fault event and containment of it before it becomes a large-scale interruption. This technology also help ensure that electricity resumes quickly and strategically after an interruption occurs: automatically routing electricity to sections downstream the isolated faulty area, for instance.

Consequently, the optimal placement of automatic switches in Smart Grid networks plays a significant role that will contribute to future networks to have better operation and improved reliability and also permits new technology to be incorporated on the network system.

2.1.3. Power Distribution Networks

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The function of power distribution networks is to deliver electricity to each end-customer, transforming it to a suitable voltage when necessary. However, it has been estimated (Falaghi, Haghifam, & Singh, 2009) that most of the supply interruptions to customers are because of failures in the distribution networks.

The frequency of interruptions can be reduced by improving the network failure rates, while the duration can be reduced by decreasing the restoration time. An effective way to reduce the frequency of interruption and restoration time is the installation of automatic line switches in the feeders of distribution systems (Chao-Shun Chen et al., 2006).

Sectionalizing and tie-point switches work together to identify faults, automatically isolate problem areas and reconfigure the controlled feeders in order to restore power to un-faulted customer as soon as possible from the main or alternative sources. This reduces the number and length of electric system outages, and minimizes the impact to customers.

The effectiveness of this process strongly depends on the number and location of sectionalizing and tie-point switches. Therefore, an algorithm that addresses the automatic switch optimal allocation problem has significant importance since it allows Smart Grids to embrace advanced controls, monitoring and innovative metering systems.

2.2. Distribution Feeder Model

An illustration of a radial feeder of a distribution system is presented in Figure 2-2. It consists of one main and several lateral sections. The main feeder CB-1 is interconnected to an adjacent feeder (CB-2) by the existence of normally open (N.O.) tie-point switch, through which power can be supplied to the main feeder when needed.

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Figure 2-2 Radial Distribution Feeder and its Line Switches

Each portion of feeder in the above figure has a unique end node. The end node of section j also constitutes the physical connection point for customers to the distribution feeder. Any device allocated in section j of the distribution system is also identified by j.

Let s(i) be the immediate predecessor of section i. We define ݏ_௜ as the section path which contains all sections belonging to the path that connects the power source to section i. For example, for Figure 2-2, let us suppose a failure has occurred in section 17. The section path to the failure event corresponds to the expression: ݏ_ଵ଻ ൌ ሼͳǡ͵ǡͺǡͳʹǡͳͷǡͳ͸ǡͳ͹ሽǤ

Similarly, the load point path ܮ_௜ is the pathway containing all sections that connects the power source to the load point i. For the same illustration, the load point path for customer 7 is:

ܮ_଻ ൌ ሼͳǡ͵ǡͶǡ͸ǡ͹ሽ.

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Switches and protective devices shown in Figure 2-2 play a vital role for reliability improvement of the distribution system (Tippachon & Rerkpreedapong, 2009). Each type of devices has unique functionalities. Following we give more details about them:

Circuit Breakers have switching and protective properties, which are used to handle permanent

and temporary faults. Circuit breakers are located in transmission substations and they protect a distribution feeder from damage caused by overload or short circuit. Its basic function is to detect a fault condition and, by interrupting continuity, immediately disconnect electrical flow to the entire distribution network. A circuit breaker can be reset (trip-reclose function), either manually or automatically, to resume normal operation.

Fuses have only protective function, which means no switching capability. It separates a fault by

melting its fuse-link. A fuse can only perform open-circuit function, and is not able to clear the momentary fault by itself. Fuses are not allowed to be installed on the main feeder.

Switches can be of two types: sectionalizing or tie-point switches. Both types, they cooperate to

isolate faulted sections of the network and restore power to customers in healthy areas through the main or neighbor feeders. This reduces the number of customers affected by the interruption and the duration of it, which is beneficial to the overall reliability of the system.

Sectionalizing switches and tie-point switches installed in the distribution feeder are grouped into sets D and TP, respectively. In other words, for Figure 2-2, the sets of switching devices can be expressed: ܦ ൌ ሼͺǡͳʹǡͳͷሽ, ܶܲ ൌ ሼʹͲሽ.

2.3. Line Switches in Distribution Networks

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After a fault is cleared by the circuit breaker, the system needs to be reconfigured to isolate the fault and restore power for the remaining customers. This reconfiguration is performed by sectionalizing and tie-point devices, most of which are manually operated. Thus, the customers have to wait for a crew to drive to the location and manually switch these devices on or off to execute the reconfiguration. In order to enhance reliability and improve customer satisfaction, electric utilities have begun to install automatic switching systems in their distribution systems.

Automatic switching systems are usually composed of several automatic switches with the capability to isolate the faulted circuits, and these switches “talk” with one another through communication equipment to determine the status of the portion they cover. The selection of the location of an automatic switch should guarantee that the adjacent feeders, to which the load will be transferred, have sufficient capacity to pick up the customers affected by the fault on their main feeder.

2.3.1. Automatic Sectionalizing Switches and Tie-point Switches

The installation of automatic sectionalizing switches benefits distribution network reliability, above all, by reducing the outage duration time when a fault occurs, given that the fault is isolated in a period of time equal to the switching time of the device –in automatic switches, less than five minutes. With reduction of outage duration time, the unsupplied amount of energy to the customers is also decreased.

An automatic sectionalizing switch typically employed by utilities in distribution systems is the SF6 Gas Insulated Automatic Sectionalizing Switch for overhead lines operating at a voltage up to 25.8kV. The full specification datasheet for the mentioned device is presented in Appendix1.

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Automatic tie-point switches in distribution networks contribute to the improvement of interruption frequency on the system. Tie-switches are able to restore electrical service to sections of feeder within the order of minutes avoiding that sustained interruption events are accounted for those sections and hence affect network reliability.

2.4. Problem Formulation for Placement of Automatic Line Switches

As discussed previously, distribution automation in terms of installation of automatic or remote- controlled switches provides major benefits to distribution utilities. However, its implementation requires economic justification, and the contribution to the performance should be also quantified. Benefits of automated sectionalizing and tie-point switches can be calculated in terms of reduced duration of outage and reduced number of customers affected during permanent faults by fast restoration of power to un-faulted customers.

The main purpose of this study is to find the optimal number and location of automatic switches in order to minimize System Average Interruption Frequency Index (SAIFI), System Average Interruption Duration Index (SAIDI), in conjunction with line switch capital investment costs (TCOST).

We present Figure 2-3 in order to investigate the benefits of implementing automatic line switches in a typical distribution feeder. In this figure we have a main radial feeder connected to a neighbor network through an automatic tie-point. Additionally, two automatic sectionalizing switches will be allocated in each feeder, in sections 2 and 5. This analysis will allow us to differentiate the benefits of the installation of automatic switches.

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Figure 2-3 Operation of Automatic Sectionalizing and Tie-point switches in Distribution Networks

Let us consider a case where a fault³ occurs on the main feeder in section 1 of Figure 2-3 (a).

Firstly, the circuit breaker operates and de-energizes all the downstream load points. Following, the automatic sectionalizing switch located in section 2 will open and “tell” the automatic tie- point in section 4 that there is a fault in section 1. Then, after the tie-point switch receives the information and makes sure that the automatic switch in section 1 has isolated the fault, it will close and transfer the load downstream the open sectionalizing switch to the neighbor feeder. At this point, the loads in the main feeder located between the two automatic switches will experience an interruption equal to the switching time of the devices (less than five minutes).

Therefore, as seen in Figure 2-3 (b), customers in load points LP2 and LP3 will suffer no impact

3 In this study, fault is defined as the occurrence of a permanent disturbance on a feeder section or transformer or occurrence of a sustained interruption due to the failure of any component of the system.

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on SAIFI given they did not experience any sustained interruption, and a considerably reduction in SAIDI because of the automatic switching time of the devices. However, customers in load point LP1 will experience outage duration equal to the repair time of feeder section 1, which is much longer than the switching time.

Mathematically, the benefits of automatic line switches in improving the reliability of distribution network can be determined by considering all possible switch placement combinations and evaluating all possible contingencies utilizing the objective functions for SAIFI and SAIDI. On the other hand, the impact on investment cost for utilities due to the installation of line switches in distribution networks can be quantified by the objective function TCOST.

2.4.1. Assumptions

Before we state to the formulation of the objective functions it is worth mentioning the assumptions under which they will be estimated:

x The network under study is radially operated;

x A fault is repaired before a subsequent fault occurs;

x The power substation is assumed to be fully reliable. So that we can calculate our reliability indexes independently from another subsystems.

2.4.2. Objective Functions

This study deals with a multi-objective optimization algorithm to obtain the best possible distribution system reliability while simultaneously minimizing investment costs as a result of acquisition of automatic switches.

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There exist several reliability indices that are used to assess the performance of distribution systems (Billinton & Jonnavithula, 1996). The most common indices used by distribution utilities are System Average Interruption Frequency Index (SAIFI), and the System Average Interruption Duration Index (SAIDI) (Tippachon & Rerkpreedapong, 2009). They are used to calculate the impact on reliability of power outages in terms on number of interruptions and interruption duration, respectively.

We select three objectives to be minimized: SAIFI, SAIDI and TCOST. The last corresponds to the total investment cost that the distribution utility has to incur due to switch automation purchase and installation.

Based on the study performed by (Tippachon & Rerkpreedapong, 2009), we present a modification to that original work in order to obtain the mathematical models that fit our described problem allowing us to achieve the three mentioned objectives; the ones that are defined as follows:

1) SAIFI, ݂_ଵሺ࢞ሻ: system average interruption frequency index. It represents the average frequency of sustained interruptions per customer. This index can be calculated using the following equation:

ܵܣܫܨܫ ൌσ^௡_௜ୀଵሺσ^௠_௦ୀଵߣ_௜௦ሻܰ_௜

σ^௡_௜ୀଵܰ_௜ ݐ݋ݐ݈ܽ݊ݑܾ݉݁ݎ݋݂ܿݑݏݐ݋݉݁ݎ݅݊ݐ݁ݎݎݑ݌ݐ݅݋݊ݏ

ݐ݋ݐ݈ܽ݊ݑܾ݉݁ݎ݋݂ܿݑݏݐ݋݉݁ݎݏݏ݁ݎݒ݁݀ ሺ݅݊ݐǤȀܿݑݏݐǤ െݕ݁ܽݎሻ

݁ݍǤ ሺͳሻ where ܰ_௜ is the number of customers at load point i, n identifies the number of load points and ݉ the number of sections. ߣ_௜௦ is the permanent failure rate of load point i due to failure in section s. It depends on the topology of the system and location of the

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switching devices. From the characteristics of automatic line switches and the model of the distribution networks in sections 2.3.1 and 2.2, respectively, ߣ_௜௦ can be estimated as follows:

ߣ_௜௦ ൌ ቐ

ߣ_௦݂݅ܵ_௦ת ሺܦ ׫ ܶܲሻ െ ܮ_௜ת ሺܦ ׫ ܶܲሻ ൌ ׎

ܽ݊݀ܵ^ᇱ_௦ת ሺܦ ׫ ܶܲሻ െ ܮ^ᇱ_௜ת ሺܦ ׫ ܶܲሻ ൌ ׎

Ͳ݋ݐ݄݁ݎݓ݅ݏ݁

݁ݍǤ ሺʹሻ where ߣ_௦ is the permanent failure rate of section s. ܵ_௦ is the set of all sections connecting the power source and section s. Analogically, ܮ_௜ is the set of the path that links power source and load point i. ܵ^ᇱ_௦ and ܮ^ᇱ_௜ are the complement of ܵ_௦ and ܮ_௜, and are defined as the pathway that connects, either s or i, to the neighbor feeder power source.

The purpose of the above set operations is to identify if there is any switching device between the faulted section and the load point and also verify if there is an alternative way to restore power through a tie-point. Based on those parameters we are able to decide ߣ_௜௦ accordingly.

2) SAIDI, ݂_ଶሺ࢞ሻ: system average interruption duration index. It is referred to as the average time that a customer is interrupted per year. The following equation is employed to calculate this index:

ܵܣܫܦܫ ൌ σ^௡_௜ୀଵሺσ^௠_௦ୀଵߣ_௜௦ݎ_௜௦ሻܰ_௜

σ^௡_௜ୀଵܰ_௜ σ݅݊ݐ݁ݎݎݑ݌ݐ݁݀ܿݑݏݐ݋݉݁ݎݏ ൈ ݅݊ݐ݁ݎݎݑ݌ݐ݅݋݊݀ݑݎܽݐ݅݋݊

ݐ݋ݐ݈ܽ݊ݑܾ݉݁ݎ݋݂ܿݑݏݐ݋݉݁ݎݏݏ݁ݎݒ݁݀ ሺ݉݅݊ǤȀܿݑݏݐǤ െݕݎǤ ሻ

݁ݍǤ ሺ͵ሻ where ݎ_௜௦ is the average time per interruption of load point i due to outages in section s. It also depends on the topology of the system and location of the switching devices. Using

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the same concepts as mentioned before, ݎ_௜௦ can be calculated with the following expression:

ݎ_௜௦ ൌ ەۖ

۔

ۖۓͲ݂݅ܵ_௦ת ܶܲ െ ܮ_௜ת ܶܲ ് ׎

݋ݎܵ^ᇱ_௦ת ܶܲ െ ܮ^ᇱ_௜ת ܶܲ ് ׎

ݎ_௥௦݂݅ܵ_௦ת ܦ െ ܮ_௜ ת ܦ ൌ ׎

ܽ݊݀ܵ^ᇱ_௦ת ܦ െ ܮ^ᇱ_௜ ת ܦ ൌ ׎

ݎ_௦௪݋ݐ݄݁ݎݓ݅ݏ݁

݁ݍǤ ሺͶሻ ݎ_௥௦ is the average repair time of the fault. On the other hand, ݎ_௦௪represents the switching times of the devices.

3) TCOST, ݂_ଷሺ࢞ሻ: total cost. It is the objective function that accounts for the summation of the total expenses associated to the investment on automatic line switches. It can be computed by the equation:

ܶܥܱܵܶ ൌ ܰݑ݉_஽ൈ ܥ_஽൅ ܰݑ݉_்௉ൈ ܥ_்௉ሺܷܵ̈́ െ ݕ݁ܽݎሻ

݁ݍǤ ሺͷሻ where ܰݑ݉_஽ accounts for the number of sectionalizing switches and ܰݑ݉_்௉ for the number of tie-point switches to be installed. ܥ_஽ and ܥ_்௉ are the total costs including purchase and installation of sectionalizing and tie-points, respectively.

The possible switch placement locations will be grouped together into a set of possible combinations and they will be considered as feasible solutions for the optimization problem. For this task, a decision variable will be associated to every section and its value (0, 1 or 2) represents the cases in which: 0 – no device is assigned to that section, 1 – a sectionalizing switch is assigned, or 2 – a tie-point switch is assigned. By doing so, we are able to the generate feasible combinations for placement of switches in the distribution network; the ones that a

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search algorithm will compare looking for closeness to the minimal objective functions in order to obtain the set of optimal solutions.

2.4.3. Constraints

To guarantee that the proposed methodology does not violate technical restrictions, the following operation constraints are considered in this study:

x Only one automatic tie-point switch can connect two neighbor feeders. In other words, only one section for every feasible combination is allowed to be given a value of 2 as its decision variable;

x When performing the load transfer for service restoration, no overloading should be introduced to the power transformers. In Taipower distribution system, the rated levels for transformers are 450A.

In conclusion, SAIFI, SAIDI and TCOST represent the three objective functions for the multi- objective optimization problem to be addressed by the search algorithm that will be presented in Chapter 4. Meanwhile, in the following Chapter 3 we will provide the algorithm background we need before formulating the proposed algorithm approach.

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Chapter 3: Algorithm Background

This chapter is imperative for our analysis since it will present the three most important concepts we need in order to implement the proposed methodology for the switch allocation problem.

They are termed as follows: Multi-Objective Optimization, Genetic Algorithm, and Elitist Nondominated Sorting Genetic Algorithm (NSGA-II).

In order to better understand each concept, a review concerning their basic approaches will also be presented in the way of subsections.

3.1. Multi-objective Optimization

Optimization is a procedure of finding and comparing feasible solutions until no better solution can be found (K. Deb, 2001). Solutions are considered good or bad in terms of an objective function, which is often the cost of fabrication, product reliability, efficiency of a process, or

other factors. A significant amount of research efforts in the optimization field are carried in terms of a single objective, although most real-world problems involve more than one objective (Engelbrecht, 2005).

Real-world optimization problems habitually involve simultaneous optimization of multiple and often conflicting objectives (such as simultaneously minimizing cost of fabrication and maximizing product reliability). In a multi-objective optimization problem, it is not always possible to find a solution that is the best with respect to all objectives. A solution may be optimal regarding to one objective, but at the same time be inferior regarding to another

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objective. Typically, the goal is to find a set of optimal trade-off solutions known as Pareto- optimal set.

Solutions belonging to the Pareto set are optimal in a broader sense given that no other solutions in the search space are better than them when all objectives are considered. Additionally, since no any solution in the Pareto-optimal set can be said to be absolutely better than any other solution in the same set with respect to all objectives, all solutions belonging to the set are recognized as acceptable solutions for the optimization problem. The decision-maker is able to select one solution over the others based on his/her previous knowledge about the problem and professional experience.

3.1.1. Multi-objective Optimization Problem

The purpose of the multi-objective problem is to minimize or maximize a number of objective functions. Those objective functions are subject to constraints which any feasible solution, including the optimal solution set, must satisfy. Based on (K. Deb, 2001), the multi-objective optimization problem (MOOP), in its general form, can be expressed using the following structure:

ܯ݅݊݅݉݅ݖ݁Ȁܯܽݔ݅݉݅ݖ݂݁_௠ሺ࢞ሻǡ ݉ ൌ ͳǡʹǡ ǥ ǡ ܯǢ ݏݑܾ݆݁ܿݐݐ݋݃_௝ሺ࢞ሻ ൒ Ͳǡ ݆ ൌ ͳǡʹǡ ǥ ǡ ܬǢ ݄_௞ሺ࢞ሻ ൌ Ͳǡ ݇ ൌ ͳǡʹǡ ǥ ǡ ܭǢ

ݔ_௜^{௅௢௪௘௥} ൑ ݔ_௜ ൑ ݔ_௜^{௎௣௣௘௥}ǡ ݅ ൌ ͳǡʹǡ ǥ ǡ ݊Ǥ

A solution ࢞ is a vector of ݊ decision variables ࢞ ൌ ሺݔ_ଵǡ ݔ_ଶǡ ǥ ǡ ݔ_௡ሻ^். The terms ݃_௝ሺ࢞ሻ, ݄_௞ሺ࢞ሻ are the inequality and equality constraints, respectively. Additionally, ݔ_௜^{௅௢௪௘௥} ൑ ݔ_௜ ൑ ݔ_௜^{௎௣௣௘௥} is also a constraint called variable bounds. These bounds establish a decision variable space D or

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decision variable, simply, and they are the ones that restrict each decision variable ݔ_௜ to take a value within ݔ_௜^{௅௢௪௘௥} and ݔ_௜^{௎௣௣௘௥}.

A solution ࢞ that satisfies all the ሺܬ ൅ ܭሻ constraints and is allocated within the ʹܰ variable bounds is known as feasible solution. Moreover, the set of all feasible solutions is called feasible region S or search space.

We have ܯ objective functions ݂ሺ࢞ሻ ൌ ሺ݂_ଵሺ࢞ሻǡ ݂_ଶሺ࢞ሻǡ ǥ ǡ ݂_ெሺ࢞ሻሻ^்and each one of them can be either minimized or maximized. However, for our case only minimization of objective functions is allowed in the search algorithm.

In multi-objective optimization the objective functions constitute a multi-dimensional space called objective space Z. For each solution ࢞ in the decision space there exist a point in the objective space and the mapping takes place between a ݊-dimensional solution vector and a ܯ- dimensional objective vector.

Figure 3-1 shows us the feasible decision space in the left and the feasible objective space in the right. Every feasible solution in the decision space can be mapped to a solution in the feasible objective space. This correspondence shows us the different trade-off solutions between the two objectives.

3.1.2. Pareto-optimality

In this section we are going to provide a number of definitions that are needed when talking about multi-objective optimization. Those definitions include dominance, Pareto-optimal set, Pareto-optimal front and others.

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Figure 3-1Illustration of a general multi-objective optimization problem (Tran, 2006)

We can observe in Figure 3-1 that in some cases if we pick and compare some pair of solutions from the feasible objective space, one of the solutions is better than the other in both objectives.

For another pair of solutions, one is better than the other in one of the objective but worse in the second objective. In order to determine which solutions are optimal with respect to both objectives we are going to introduce the concept of dominance.

In Figure 3-2 we illustrate a number a solutions coming from the feasible objective space.

Solution 1 provides the lowest value for objective 1 but the highest value for objective 2, while solution 4 offers the minimum value for objective 2 but under the highest sacrifice of objective 1.

None of these two solutions can be said to be better than the other when considering both objectives. The same reassembles when considering solutions 2 and 3, no superiority of any of the solutions can be established if the two objectives are equally important. When this happens we can call them non-dominated solutions.

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All non-dominated solutions are jointed together using a curve. The solutions lying on that curve are special in terms of multi-objective optimization and are called Pareto-optimal solutions. The curve that contains these solutions is called Pareto-optimal front. We can distinguish the mentioned curve in Figure 3-2. It is interesting to note that for optimization problems in which the minimization of the two objective functions is required, this curve is located in the bottom- left corner.

Figure 3-2 Pareto-optimal front including Pareto-optimal solutions and a non-optimal solution

We can deduce that the feasible objective space is constituted by Pareto-optimal solutions and by solutions that are non-optimal, in fact, the total feasible objective space can be divided into Pareto-optimal set, or non-dominated set, and non-optimal set. If we consider solutions 3 and 5 in Figure 3.2, for instance, we can realize that solution 3 is better than solution 5 in both objectives. Thus, we can say that solution 3 dominates solution 5. There always exists at least one solution in the Pareto-optimal set which is better than any member of the non-optimal

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solution set. A formal definition for dominance will be stated below assuming minimization problems.

3.1.3. Domination

Definition 3.1: Domination (K. Deb, 2001): A solution ࢞_૚ is said to dominate the other solution ࢞_૛ (denoted by ࢞_૚ ط ࢞_૛), if and only if:

x ࢞૚ is not worse than࢞_૛ in all objectives, i.e. ݂_௞ሺ࢞_૚ሻ ൑ ݂_௞ሺ࢞_ଶሻǡ ׊݇ ൌ ͳǡ ǥ ǡ ܯ , and x ࢞૚ is strictly better than ࢞_૛ in at least one objective, i.e. ׌݇ ൌ ͳǡ ǥ ǡ ܯ ׷

݂_௞ሺ࢞_૚ሻ ൏ ݂_௞ሺ࢞_ଶሻ.

If any of the above conditions is not satisfied, the solution ࢞_૚ does not dominate solution ࢞_૛.

We can perform all possible pair-comparisons for a finite set of solutions, and find which solutions dominate which and the solutions that are non-dominated with respect to each other. In the end, we will obtain a set of non-dominated solutions which any of its elements will dominate any solution outside of this set.

Definition 3.2: Non-dominated set (K. Deb, 2001): Among a set of solutions ܲ, the non- dominated set of solutions ܲ^ᇱ are those that are not dominated by any member of the set ܲ.

If the set P is the entire search space, then P=S, and the obtained non-dominated set ܲ^ᇱ is called the Pareto-optimal set.

From the above discussion we can conclude that the Pareto-optimal set is the non-dominated set.

But there may be some Pareto-optimal sets containing some Pareto-optimal solutions and some non-Pareto optimal solutions. The task of finding the true Pareto-optimal solutions is usually computational prohibitive (Engelbrecht, 2005). Therefore, it is important to realize that the non-

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dominated solutions found by an optimization algorithm need not to represent the true Pareto- optimal set but rather an approximation such that:

1. The set of solutions represent as close as possible the true Pareto-optimal front.

2. The set of non-dominated solutions, Pareto-optimal set, is as diverse as possible.

The first goal represents the desirable near-optimality property the Pareto-optimal set must have.

The second goal let us know that being converged close to the true Pareto-optimal front is not enough, only a diverse set of solutions can guarantee adequate Pareto-front coverage.

Therefore, the population-based search algorithm must adequately emphasize the non-dominated set of a given population to ensure the two goals mentioned above are satisfied. We are now interested in a computational efficient procedure to identify the non-dominated set from a population of feasible solutions.

In this study we propose a computational algorithm based on Nondominated Sorting Genetic Algorithm (NSGA-II). The tests we perform on the mentioned algorithm have showed that this methodology ensures a very good approximation to the true Pareto-front and diversity of the solutions is also guaranteed, all of this while being computationally efficient when performing the procedure of finding the non-dominated set from a population of feasible solutions.

Before get into more detail with such algorithm, we first need to know how genetic algorithms perform their search technique, since that constitute the basis for the former algorithm.

3.2. Genetic Algorithms

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Genetic Algorithms (GAs) are one of the main types of Evolutionary Algorithms (EA). In general, EA indicate any population-based stochastic search algorithm that uses mechanisms inspired by biological evolution and genetic operators such as reproduction, mutation, crossover, natural selection and survival of the fittest.

GAs for computer simulation were mainly developed by Holland in the 1960s and published in his book (Holland, 1975). Over the years, Holland’s original GA has evolved into many forms Multi-objective GA (MOGA), Nondominated Sorting GA (NSGA), Niched-Pareto GA (NPGA), Elitist Nondominated Sorting GA (NSGA-II), and others. However, the general framework remains the same as in the basic GA.

A simple GA attempts to find a good solution to some problem (finding the minimum of a function, for instance) by generating a random population of candidate solutions and then manipulating those solutions using genetic operators of reproduction, crossover and mutation.

Once the initial population of solutions has been generated, the GA begins to evaluate and rank each candidate solution based on the on its fitness to the objective function. Solutions with higher fitness value will have better chances of survival and reproduction according to the evolutionary process. Hence, the best fitted solutions will be selected to produce the next generation of candidate solutions using genetic operators. Finally, when a terminating condition has been satisfied, usually number of generations, the most excellent solution, which is the most evolved one, constitutes the optimal to the optimization problem.

3.2.1. Solution Representation

In GAs, each feasible solution can be denoted by a chromosome, which is a coding representation. In order to perform this task, first we need to code the decision variables of the

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optimization problem using finite-length character strings, from phenotype (decimal representation) to genotype (binary or real representation). There are two main ways for coding a variable:

1) Binary-valued representation (binary coding) consisting of 0 and 1. One character (0 or 1) in the binary strings is called a gene.

2) Real-valued representation. In this case the variable is coded using real numbers instead.

The set of chromosomes is called population, and each member evolves in every generation toward better solutions. The number of chromosomes in a population is called the population size.

In our proposed algorithm for the optimal placement of line switches in distribution networks, we will apply integer-representation for the coding of the decision variables, which is an special case of real coding. This way we are able to reduce computational complexity, because variables are used directly without any string coding, and also due to the fact that this type of representation is ideally suited for solving combinatorial problems (K. Deb, 2001) and when solutions are composed of many variables.

3.2.2. Fitness Assignment

Fitness is an indicator for measuring a solution quality for survival. All solutions are evaluated and ranked based on their fitness values at each generation. The fitness is similar to the objective function in conventional optimization problems. Thus, solutions having higher fitness are good ones. During the evolution process, therefore, relatively good solutions reproduce, and relatively bad solutions with lower fitness die in each generation. Finally, the solution having maximum fitness is obtained as an optimal solution.

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One of the reasons why we decided on a GA-based methodology in this study is due to the fact that GAs can deal with a wide range of objective functions (nonlinear, non-differentiable, constrained, or discontinuous) without the need of additional requirements for the fitness evaluation.

3.2.3. Genetic Operators

There are mainly three types of genetic operators in GAs: reproduction and selection, crossover, and mutation. The population of solutions is modified by genetic operators and a new (hopefully better) population is created.

3.2.3.1. Reproduction and Selection

The main purpose of the reproduction operator is to make duplicates of good solutions and eliminate bad ones from the population, while keeping the population size constant.

There exist several alternatives to achieve this task, one of the most common methods is tournament selection, and since it is the one we are going to apply in our proposed algorithm, we will provide more details about it next.

In binary-tournament selection, tournaments are played between two solutions in terms of ranked fitness. The better solution is chosen and placed in a mating pool. Two other solutions are picked and another slot of the mating pool is filled with the better one. If this methodology is performed accordingly, each solution will participate in exactly two tournaments. The best solution in the population will win both rounds; therefore, two copies of it will be present in the new population. On the other hand, the worst solution will lose both tournaments and will be eliminated from the population. Thus, each solution will have zero, one or two copies of it in the new population.

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In Figure 3-3, we find six different solutions ranked according to their fitness with the objective function. Tournaments are played among these solutions and each one gets to participate in two rounds. The two solutions for each tournament are chosen at random, and for the first tournament we have the solution with a hypothetical fitness of 12 being better and hence a copy of it is placed in the mating pool. The same process is followed for the rest of tournaments, and the mating pool is formed. It is interesting to note better solutions (with minimum values) have handled to have multiple copies in the mating pool and worse solutions have gotten discarded.

Figure 3-3 Creation of Mating Pool from six hypothetical solution fitness using Tournament Selection

3.2.3.2. Crossover

It is clear that the reproduction operator does not create any new solutions in the population, it just make copies of good solutions and deletes not so good ones. By proliferating good solutions

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26 15

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26

15

27

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26 12 15

12 21

Mating Pool

12 12

First Round Second Round

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in the mating pool we ensure better chances of crossover and mutation for them than those with lower fitness values, this is how we stimulate the creation of more excellent solutions in each generation.

The crossover operator combines the features of two different solutions (parents) into two new solutions (offspring). It operates by picking two random solutions belonging to the mating pool and exchanging some portion of their coded strings in order to create two new strings. Like reproduction operator, there are many ways to perform the crossover task: single-point crossover, multi-points crossover, uniform, intermediate crossovers and so on, but almost all operators have the same above notion.

The crossover rate, ݌_௖, is used to determine that 100݌_௖% of strings will experience crossover and the rest 100(1-݌_௖)% of the population is simply copied to the new population. Crossover rate ranged from 0.6 to 0.8 is usually used in order to allow new offspring to be created sufficiently.

The higher the crossover rate, the more excellent created individuals will be. If the crossover rate is too low, the searching process may deteriorate due to lack of new strings with better performance.

3.2.3.3. Mutation

Mutation randomly alters one or more genes of a solution string to generate a new mutated solution. The mutation operation increases the variability of the population and helps to prevent premature convergence to local optima in the evolution process.

The number for mutation events depends on the mutation rate, ݌_௠. The mutation rate in natural evolution is usually very small. Therefore, the mutation rate is traditionally given by low values at the range of 0.01 – 0.1. In each generation, 100݌_௠% solutions undergo mutation. The lack of

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mutation induces poorer performances in evolution. Therefore, if mutation rate is too low, the possibility to fall into local optima increases. Relatively high mutation rates up to 0.4 or 0.6 have been found beneficial (Morimoto, 2006). It is to be noted, however, that a significantly high mutation rate leads to an essentially random search.

The three genetic operators (reproduction, crossover, mutation) are straightforward. The reproduction operator selects good strings and makes copies of them, crossover operator recombines good string sections from two solutions in order to obtain a hopefully better string, and mutation operator alters a string locally to hopefully create a better solution.

Since none of these operations are performed deterministically, these claims are not guaranteed, nor explicitly tested, during the GA generation (K. Deb, 2001). Nevertheless, parents that undergo crossover and mutation are not any two arbitrary random strings. These strings have survived tournaments played with others solutions during the reproduction operator. Therefore, it is expected that if bad solutions are created, then the reproduction operator will eliminate them in the next generations and when good solutions are created, they will be emphasized to ensure next generation will contain better solutions.

In conclusion, GAs are powerful search techniques that have proven to be successful in many optimization applications (Morimoto, 2006). Their capacity to handle virtually any objective function with no requirement of derivatives or other knowledge, and their ability to reach global (or at least near global ones) optimal solutions makes GA-base algorithms a major tool when considering complex optimization problems.