Security of Electricity Supply in Power Distribution System
Optimization Algorithms for Reliability Centered Distribution System Planning
SANJA DUVNJAK ŽARKOVIĆ
Licentiate Thesis
Stockholm, Sweden 2020
TRITA-EECS-AVL-2020:47 ISBN 978-91-7873-648-5
KTH School of Electrical Engineering and Computer Science SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av licentiatexamen i elektro- och system- teknik tisdagen den 20. oktober 2020 klockan 14.00 i videokonferenssession (zoom link: https://kth-se.zoom.us/j/61121599309) och Mötesrum 1411, Teknikringen 33, Kungliga Tekniska högskolan, Stockholm.
© Sanja Duvnjak Žarković, October 2020
Tryck: Universitetsservice US AB
Abstract
The importance of electricity in everyday life and demands to improve the reliability of distribution systems force utilities to operate and plan their networks in a more secure and economical manner. With higher demands on reliability from both customers and regulators, a big pressure has been put on the security of electricity supply which is considered as a fundamental requirement for modern societies.
Thus, efficient solutions for reliability and security of supply improvements are not just of increasing interest, but also have significant socio-economic relevance.
Distribution system planning (DSP) is one of the major activities of distribution utilities to deal with reliability enhancement.
This thesis deals with developing optimization algorithms, which aim is to min- imize customer interruption costs, and thus maximize the reliability of the system.
This is implemented either by decreasing customer interruption duration, frequency of customer interruptions or both. The algorithms are applied on a single or multi- ple DSP problems. Mixed-integer programming has been used as an optimization approach.
It has been shown that solving and optimizing each one of the DSP problems contributes greatly to the reliability improvement, but brings certain challenges.
Moreover, applying algorithms on multiple and integrated DSP problems together leads to even bigger complexity and burdensome. However, going toward this inte- grated approach results in a more appropriate and realistic DSP model.
The idea behind the optimization is to achieve balance between reliability and the means to achieve this reliability. It is a decision making process, i.e. a trade-off between physical and pricing dimension of security of supply.
iii
Sammanfattning
Elens betydelse i vardagen och kraven på att förbättra distributionssystemens till- förlitlighet tvingar verktyg att driva och planera sina nät på ett säkrare och mer ekonomiskt sätt. Med högre krav på tillförlitlighet från både kunder och tillsyns- myndigheter har ett stort tryck lagts på elförsörjningssäkerheten som anses vara ett grundläggande krav för moderna samhällen. Effektiva lösningar för tillförlitlig- het och försörjningssäkerhetsförbättringar är således inte bara av ökande intresse utan har också en betydande socioekonomisk relevans. Distributionssystemplane- ring (DSP) är en av distributionsverktygens viktigaste aktiviteter för att hantera tillförlitlighetsförbättringen.
Den här avhandlingen behandlar utvecklingsoptimeringsalgoritmer, som syftar till att minimera kundavbrottskostnaderna och därmed maximera systemets tillför- litlighet. Detta implementeras antingen genom att minska kundavbrottets längd, frekvensen för kundavbrott eller båda. Algoritmerna tillämpas på en eller flera DSP-problem. Blandad heltalsprogrammering har använts som optimeringsmetod.
Det har visats att lösning och optimering av vart och ett av DSP-problemen i hög grad bidrar till förbättringen av tillförlitligheten, men ger vissa utmaningar.
Dessutom leder algoritmer på flera och integrerade DSP-problem tillsammans till ännu större komplexitet och tyngre. Att gå mot denna integrerade strategi resulterar emellertid i en mer lämplig och realistisk DSP-modell.
Tanken bakom optimeringen är att uppnå balans mellan tillförlitlighet och me- del för att uppnå denna tillförlitlighet. Det är en beslutsprocess, det vill säga en avvägning mellan fysisk och prissatta dimension av försörjningstryggheten.
v
Acknowledgements
This thesis is written as part of the Ph.D. project "Security of Electricity Supply in Power Distribution System" at KTH Royal Institute of Technology. The project is funded by Swedish Centre for Smart Grids and Energy Storage - SweGRIDS. I would like to thank Vattenfall AB and Ellevio AB for supporting the project.
This thesis could never be completed without Docent. Patrik Hilber and Dr.
Ebrahim Shayesteh, my supervisors, to whom I am immensely grateful for the guidance, discussions and support.
I would like to thank Prof. Rajeev Thottappillil, the previous head of my department at KTH, for supporting the research, internally reviewing my licentiate thesis and counseling my work.
Prof. Martin Norgren, the current head of my department at KTH, for all workshops and effort to make EME department an amazing place to work at.
Docent. Nathaniel Taylor for assisting me in getting the access to "supercom- puter" and keeping my SweGRIDS page up to date.
Docent. Daniel Månsson for guiding my academic duties as a teaching assistant.
Carin Norberg and Brigitt Högberg for the administrative work support.
it-support@kth.se for all IT support one PhD student can ask for.
The members of the QED reference group for the valuable input to my work, data and discussions on the development of my research.
I would like to thank all former and current members of QED Asset Management group for all insights, knowledge, live and zoom fikas and unforgettable discussions about who makes the best coffee and how to pronounce exotic words from other languages.
The current and former members of the EME and EPE department and all my friends at Teknikringen 31/33. We were always there for each other when someone would forget an access card to enter the corridor.
All my amazing friends and big family, especially my mom and my siblings for keeping my spirit up and believing in me.
And last but not the least, I am grateful to my incredible husband Marko and my adorable son Teodor for fulfilling my life with never-ending love and inspiration.
Sanja Duvnjak Žarković, Stockholm, 2020
vii
List of Papers
The following publication are included in the thesis:
Paper I
S. Duvnjak Zarkovic, P. Hilber and E. Shayesteh, "On the Security of Electricity Supply in Power Distribution Systems," in 2018 IEEE International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), 2018.
Paper II
S. Duvnjak Zarkovic, S. Stankovic, E. Shayesteh and P. Hilber, "Reliability Im- provement of Distribution System through Distribution System Planning: MILP vs. GA," in 2019 IEEE Milan PowerTech, 2019.
Paper III
S. Duvnjak Zarkovic, E. Shayesteh and P. Hilber, "Reliaiblity Centered Distribu- tion System Planning - Cable Routing and Switch Placement," Submitted to IEEE transactions on power systems, 2020.
Paper IV
S. Duvnjak Zarkovic, E. Shayesteh and P. Hilber, "Onshore Wind Farm - Re- liability Centered Cable Routing," Submitted to Electric Power Systems Research, 2020.
Paper V
M. Z. Habib, S. Duvnjak Zarkovic, N. Taylor, P. Hilber and E. Shayesteh, "Re- liability Centered Planning of a Resonant-earthed Distribution System with Focus on Fault Location Methods," Submitted to Electric Power Systems Research, 2020.
I am the main author of Papers I-IV. The idea of Paper I belongs to my super- visors, P. Hilber and E. Shayesteh; I have conducted a detailed background study on the topic of security of electricity supply, identified the important works and reviewed them. The idea for Paper II has been proposed by me; S. Stankovic has
ix
x LIST OF PAPERS
developed Genetic Algorithm for the proposed problem, everything else has been carried by me; P. Hilber and E. Shayesteh have supervised the work. Paper III has been developed through discussion between me, P. Hilber and E. Shayesteh;
the study has been carried by me; E. Shayesteh helped to practically implement optimization algorithm. The idea of Paper IV came after discussion with Daniel Fahlgren from Ellevio AB; all details were developed by me, E. Shayesteh and P.
Hilber; the practical work was carried by me. The idea and the main work of Paper V has been conducted by M. Z. Habib; I have developed the optimization algorithm for FPI (fault passage indicator) placement and helped in writing the paper; N. Taylor, P. Hilber and E. Shayesteh have supervised the work.
The following publication is not included in the thesis:
I M. Z. Habib, M. T. Hoq, S. Duvnjak Zarkovic, N. Taylor, "Impact of the fault lo-
cation methods on SAIDI of a resonant-earthed distribution system," in 2020 IEEE
PES POWERCON, 2020.
List of Acronyms
DAS Distributed automated system
DG Distributed generator
DSP Distribution system planning
FPI Fault passage indicator
GA Genetic algorithm
MIP Mixed-integer programming
MILP Mixed-integer linear programming MINLP Mixed-integer nonlinear programming
POC Point of connection
xi
Contents
Abstract iii
Sammanfattning v
Acknowledgements vii
List of Papers ix
List of Acronyms xi
Contents xii
1 Introduction 1
1.1 Background and Motivation . . . . 1
1.2 Research Objectives . . . . 1
1.3 Research Contributions . . . . 2
1.4 Research Ethics . . . . 3
1.5 Thesis Organization . . . . 3
2 Security of Electricity Supply 5 2.1 Defining Security of Electricity Supply . . . . 5
2.1.1 Long-term Security of Supply . . . . 6
2.1.2 Short-term Security of Supply . . . . 6
2.2 Evaluating Security of Electricity Supply . . . . 7
2.2.1 Customer Outage Cost . . . . 7
2.2.2 Regulation of Energy Networks . . . . 9
3 Improving Security of Supply in Distribution System 11 3.1 DSP Approaches for Improving Security of Supply . . . . 11
3.1.1 Optimal Feeder Routing . . . . 11
3.1.2 Optimal Switch Placement . . . . 12
3.1.3 Network Reconfiguration . . . . 12
3.1.4 Automation . . . . 12
xii
CONTENTS xiii
3.1.5 Distributed Generation . . . . 13
3.2 Optimization Algorithms . . . . 13
3.3 MILP vs. GA . . . . 14
3.3.1 Optimality Gap . . . . 14
3.3.2 Simulation Results . . . . 15
4 Interruption Frequency Improvement 19 4.1 Cable Routing - Base Case . . . . 19
4.1.1 Algorithm . . . . 19
4.1.2 Application . . . . 21
4.2 Different Cable Options . . . . 21
4.2.1 Algorithm . . . . 21
4.2.2 Application . . . . 23
4.3 Cable Routing and Switch Placement . . . . 25
4.3.1 Algorithm . . . . 25
4.3.2 Application . . . . 27
4.4 Willingness to Rebuild Cables . . . . 27
5 Interruption Duration Improvement 29 5.1 Tie-switch Placement . . . . 29
5.1.1 Algorithm . . . . 29
5.1.2 Application . . . . 31
5.2 Fault Passage Indicators (FPI) Placement . . . . 32
5.2.1 Algorithm . . . . 32
5.2.2 Application . . . . 34
6 Conclusion 35
7 Future Work 37
Bibliography 39
Chapter 1
Introduction
1.1 Background and Motivation
Electricity is an essential good and it plays a vital role in everyday life. Today’s society is critically dependent on a power system that is able to provide electric- ity to its consumers in a reliable and economical way. However, the structure of power systems has changed significantly over the past few years, resulting in new challenges for power systems’ planners and operators. As a result of deregulated electricity market scenarios, rapidly increasing demand for electricity and rise in levels of supply volatility due to the significant growth of electricity generation from renewable energy, distribution systems are being operated under very stressed con- ditions. One of the biggest concerns nowadays is the security of electricity supply.
Security of electricity supply has become a fundamental requirement for modern societies. With higher demands on reliability from both customers and regulators, efficient solutions for reliability improvements are of increasing interest. Moreover, demands to improve the reliability of distribution systems force utilities to oper- ate and plan their networks in a more secure and economical manner. Therefore, to study and improve methods for the security of supply has a very high social, economical and energy relevance.
1.2 Research Objectives
This thesis deals with apparatus for improving security of supply of mainly local power systems. The objective is to minimize customer interruption costs and other corresponding measures of reliability performance.
The thesis focuses on developing algorithms that take into account system reli- ability through distribution system planning (DSP). DSP problems and challenges that are addressed in this licentiate are:
1
2 CHAPTER 1. INTRODUCTION
• Cable routing and optimal network layout
• Optimal placement and number of sectionalizing and tie switches
• Effects of distributed automated systems
• Distributed generation
1.3 Research Contributions
Research contributions of this thesis can be summarized as follows:
• Defining the concept of Security of Electricity Supply (Paper I) - Sections 2.1, 2.2.
• Identifying approaches within distribution system planning for reliability and security of supply improvement (Paper I) - Sections 3.1, 3.2.
• Justifying use of mixed-integer programming (MIP) as an optimization algo- rithm over meta-heuristic algorithm (Paper II) - Sections 3.3, 4.1.
• Applying MIP on a single DSP problem with the aim to minimize customer interruption cost by improving interruption frequency (Paper II, IV) - Sec- tions 4.1, 4.2.
• Applying MIP on multiple DSP problems with the aim to minimize customer interruption cost by improving interruption frequency (Paper III) - Sections 4.3, 4.4.
• Applying MIP on a single DSP problem with the aim to minimize customer interruption cost by improving interruption duration (Paper III, V) - Sec- tions 5.1, 5.2.
Therefore, this thesis presents successfully developed algorithms for reliability and security of electricity supply improvement, that deal with decreasing the fre- quency of customer interruptions and customer interruption duration per fault, using mixed-integer programming optimization.
The aim of optimization is not to achieve perfect reliability no matter what, but
rather to make decisions in order to allow the network business to maximize long-
term profits, while delivering high service levels to the customers with acceptable
and manageable risks. Balance between reliability and price is the key in this
decision making.
1.4. RESEARCH ETHICS 3
1.4 Research Ethics
Security of Electricity Supply is a subject that significantly affects society, econ- omy and energy. The ethical aspect of this thesis is taken into account as far as the research leads towards minimizing customer interruption costs and other cor- responding measures of reliability performance, using monitoring control devices.
Moreover, optimization models are designed and modelled with certain simplifica- tions and assumptions in order to respect power utility data privacy requirements and customer information. Research is conducted through computer simulations, causing no harm to the environment.
1.5 Thesis Organization
This thesis is organized in seven chapters as follows:
• Chapter 1 provides the background, motivation, objective, scope and main contributions of the research work.
• Chapter 2 defines Security of Electricity Supply and means to evaluate it.
• Chapter 3 reviews DSP approaches for improving Security of Supply and ap- plied optimization algorithms. A base case of MIP algorithm that deals with Security of Supply improvement is presented in this chapter. The proposed MIP is also compared to one of the meta-heuristic algorithms to highlight its effectiveness within DSP.
• Chapter 4 presents refined versions of the developed algorithm for decreasing the frequency of customer interruptions within the cable routing problem.
• Chapter 5 focuses on decreasing the customer interruption duration per fault by considering different DSP approaches for Security of Supply improvement.
• Chapter 6 concludes the work.
• Chapter 7 outlines the future work.
Chapter 2
Security of Electricity Supply
2.1 Defining Security of Electricity Supply
According to [1], security of electricity supply is the ability of the electrical power system to provide electricity to end-users with a specified level of continuity and quality in a sustainable manner, relating to the existing standards and contractual agreements at the points of delivery.
From a broader perspective, security of electricity supply covers a diverse range of areas. First of all, security of supply is associated with the supply chain. The supply chain consists of all activities ranging from the production/extraction of raw materials to the delivery of the desired product to the final consumer [2]. The more elements in the supply chain, the higher the risk that some element may not work properly; hence, the supply chain becomes vulnerable. Therefore, security of supply simultaneously requires the physical availability of primary energy carriers, sufficient conversion and reliable transport of electricity, as well as distribution sys- tems whenever needed [3]. Technical risks such as system failure due to weather or obsolescence and poor maintenance of the energy system also affect security of supply. Additionally, security of supply is influenced by issues such as geopolitical conflicts, the threat of global terrorism and cyber terrorism. The scarcity of con- ventional energy sources, as well as the increasing global energy demands due to the growth of the world population and industrialization are other challenges related to energy security [3]. Finally, security of supply is linked to climate changes, global security issues and environmental risks such as natural disaster, oil spills, nuclear accidents, pollution, and greenhouse gas emissions, which all may lead to a supply limitation [2].
To narrow it down, security of electricity supply depends on the extent to which electricity consumption and electricity generation can be balanced and on the elec- tricity grid’s capacity to transfer the electrical energy and handle faults [4]. Thus, we can distinct two aspects of security of supply: long-term and short-term security.
5
6 CHAPTER 2. SECURITY OF ELECTRICITY SUPPLY
Security of Electricity Supply
Long-term Security of Supply
Short-term Security of Supply
System adequacy
Generation adequacy
Grid adequacy
System security (Continuity of supply)
Figure 2.1: Security of Electricity Supply.
2.1.1 Long-term Security of Supply
If a supply is disrupted because the existing supply capacity is unable to meet de- mand, we have an adequacy of supply problem or a long-term problem. Long-term security requires investment to maintain the supply chain (production, transport, and transformation infrastructures) and a supply capacity large enough to meet demand during normal conditions [2]. It is associated with static conditions which do not include system disturbances [5].
System adequacy can be subdivided into generation adequacy and grid adequacy [4]. Generation adequacy is the availability of generating (and import) capacity to meet demand. Grid adequacy is defined as the systems’ ability to transport suf- ficient electricity from generation site to consumption site, i.e. the availability of network infrastructure to meet demand. This infrastructure covers transmission and distribution system as well as the cross-border interconnections. A lack of sys- tem adequacy typically leads to announced disconnections of consumers in limited areas, i.e. brownouts. A brownout is a precaution taken to protect against a black- out in a large area. Brownouts are serious incidents, yet less severe than blackouts [4].
2.1.2 Short-term Security of Supply
On the other hand, if a supply capacity is adequate, but supply is disrupted due
to events such as technical failures, extreme weather conditions, terrorist attacks,
or strikes, we have a continuity of supply problem or a short-term problem. Short-
2.2. EVALUATING SECURITY OF ELECTRICITY SUPPLY 7
term or system security of electricity supply focuses on the system’s robustness, i.e. the ability of the energy system to react promptly due to sudden changes in the network. Continuity of supply is related to the actual delivery of electricity, and includes the ability to overcome short-term failures of individual components.
A lack of system security leads to a blackout, i.e. a complete breakdown of the electricity system. The consequence may be substantial installation damage and long restoration times for the electricity supply [4].
2.2 Evaluating Security of Electricity Supply
The concept of supply security itself encompasses two dimensions: the physical dimension (the availability, reliability and adequacy of energy supply and the related infrastructure), and the pricing dimension (the affordability and validity of market- determined prices) [2].
Reliability is a very important physical aspect of security of supply. Quite often, reliability and security of supply are identified as the same concept: quantifying re- liability means quantifying security of supply. Reliability represents the probability that an electric power system can perform a required function under given condi- tions for a given time interval; it quantifies the ability of the electric power system to provide adequate electric service on a nearly continuous basis to its consumers [6].
Reliability of electricity supply is primarily concerned with duration and fre- quency of service interruptions. Thus, reliability of supply is a customer-oriented quantity that does not consider the origin or the causes of interruptions [7].
The probability of supply being disrupted for any reason can be reduced by increased investment during the planning phase, operating phase, or both [5]. The- oretically, there is no limit to how much security of supply can be improved. How- ever, this would require extreme resources. Overinvestment can lead to unnecessary expenditures. On the other hand, underinvestment leads to the unacceptable level of continuity and unreliable supply.
Security of supply is not only a matter of advanced planning and prioritizing, but more importantly it is a matter of compromising between reliability and cost to achieve this reliability. Balance between physical and pricing dimensions is the key concept in defining security of supply.
2.2.1 Customer Outage Cost
The more developed societies are, the more vulnerable they are to supply inter- ruptions [7]. Outages can seriously affect industries and companies. For example, production losses and costs involved in restoring production can be substantial.
Individuals can suffer as well, since the lack of electricity involves significant effort
and resources to maintain tolerable living conditions. Thus, certain costs can be
associated with supply interruptions.
8 CHAPTER 2. SECURITY OF ELECTRICITY SUPPLY
Optimal Level of Supply Security Costs
Level of Supply Security Total Costs
Cost for ensuring uninterrupted Electricity Supply Damage
Costs
Figure 2.2: Optimal Level of Supply Security presented in Paper I.
According to [8], the optimal level of security is reached when the cost for ensur- ing security of supply equals the expected damages due to the supply disruption.
However, this is rarely the case. Usually, these two individual costs are added and the total cost is obtained. The optimal level of supply security is then achieved when the total cost is minimized, as shown in Figure 2.2 [5].
The damage costs or consequences of interruptions can be measured in terms of customer outage costs on an aggregated system level. The outage costs represent the economical losses for the customers that are exposed to electricity supply inter- ruptions. The size of these economical losses depends largely on the composition of the customers that experience interruptions [7].
The outage cost of the whole system is used as a measure of total system relia- bility performance [9]. The outage cost of the whole system is obtained by summing estimated outage cost at every load point i in the system (i = 1, 2, ...N ), as formu- lated in 2.1:
OutageCost =
N
X
i=1
λ
N(i) · L
a(i) · (k(i) + c(i) · r
N(i)) (2.1)
where,
λ
N(i) denotes the average failure rate of load point i
L
a(i) denotes the annual average load demand at load point i
k(i) denotes the average interruption cost rate for interrupted power at load point i
c(i) denotes the average interruption cost rate for energy not supplied at load point i
r
N(i) denotes the average interruption duration of load point i.
2.2. EVALUATING SECURITY OF ELECTRICITY SUPPLY 9
2.2.2 Regulation of Energy Networks
The optimal level of security of supply is a challenging and demanding problem.
Rules are needed to manage scarcity situations cheaply, safely, and to the benefit of all customers. Furthermore, energy networks and appropriate regulations are necessary to adequately address and handle this problem. The electricity network regulatory methods can differ between countries. Nevertheless, the goal of the regulation is to secure electricity supplies at acceptable levels of quality and at reasonable tariffs [7].
In liberalized energy systems, the main purpose of network regulations is to
attract sufficient investment, promote adequate maintenance of existing facilities,
promote the efficient operation of network infrastructure and ensure adequate re-
wards for innovation and technological progress, if necessary through regulatory
incentives [10].
Chapter 3
Improving Security of Supply in Distribution System
With new sources of production and patterns of consumption, distribution networks are changing towards a new kind of active networks, where power flow is becoming more complex and the electrification of society places major demands on supply quality. These new active networks have many advantages such as a reduction of power losses, a better voltage profile, a greater flexibility and ability to cope with the load growth. However, without careful planning strategies, some technical aspects can get worse - a more complex planning and operation, which consequently involve a higher cost [11]. The importance of electricity in everyday life and demands to improve the reliability of distribution systems force utilities to operate and plan their networks in a more secure and economical manner. Therefore, innovation and technological progress are crucial for the long-term efficiency and flexibility of the entire energy supply system.
3.1 DSP Approaches for Improving Security of Supply
Distribution companies use many different ways to improve security of supply and reliability of the system. DSP is one of the major activities of distribution utilities to deal with reliability enhancement. DSP is a decision making process that is usually carried out through the reinforcement, reconstruction or installation of new components in the distribution power system, with the aim to provide a reliable and cost effective service to consumers [12].
3.1.1 Optimal Feeder Routing
The optimal feeder routing aims to find optimal connections or routes between nodes in the system in order to achieve the best network outline so that demand is met. Total feeder cost should be minimized, while satisfying technical and physical
11
12
CHAPTER 3. IMPROVING SECURITY OF SUPPLY IN DISTRIBUTION SYSTEM constraints of the considered system. Choosing certain feeder type (underground cable or overhead line), as well as considering different line options (with differ- ent capacities and prices) can be part of optimal feeder routing problem and can significantly affect reliability of the system.
3.1.2 Optimal Switch Placement
Normally-closed switches are installed on feeders to provide sectionalizing capability and improve reliability. These switches allow fault isolation and restoration of certain customers during line repair as well as switching flexibility for maintenance.
Sectionalizing switches can also be remotely controlled. Automated sectionalizing switches reduce the time required to detect and locate faults, increasing the speed of isolating faulted equipment, and providing faster load restoration above and below a faulted feeder zone [13]. Different feeders are connected through normally-open tie switches, allowing certain interrupted customers to be temporarily transferred to adjacent feeders during maintenance and repair.
The proper number, location and type of sectionalizing and tie switches, and protective devices in general, play an important role in the reliability of distribu- tion systems by minimizing the impact of interruptions. The restorable customers’
interruption duration may be reduced by either increasing the number of switches, or changing the manual ones to automatic [14]. However, the cost associated with installation and maintenance of the switches is quite significant. If the distribution company does not pay attention to the cost incurred for switch investment, it will not provide any benefits for the company. Increasing the number of switches more than it is optimal will not provide the best reliability solution - the cost will be in- creased, but the reliability might even be reduced. Switch cost minimization can be obtained through improved operational practices and efficient planning techniques which can determine the minimum number of switches and their optimum locations [15].
3.1.3 Network Reconfiguration
Network reconfiguration is a complex, combinatorial, non-linear and discrete opti- mization problem that improves the performance of the system but requires a little incremental investment [16]. By definition, network reconfiguration is a method of alternating the configuration of the distribution network by changing the status of normally closed sectionalizing switches and normally opened tie-switches to accom- plish particular objectives [17]. Reconfiguration can help to faster restore supply to some feeder branches due to faults occurring in an upstream branch.
3.1.4 Automation
Distributed automated systems (DASs) have been introduced to augment the ef-
ficiency and improve the reliability of distribution systems. DASs reinforce the
3.2. OPTIMIZATION ALGORITHMS 13
distribution system by installing equipment such as automatic or remote-controlled switches, fault detectors, remote-controlled protective relays and remote terminal units (RTUs) to perform fault location, isolation and service restoration processes (FLISR), as well as automatic network reconfiguration. DASs can provide a reli- able and self-healing distribution network that is able to rapidly react to real-time events by taking appropriate actions [14].
3.1.5 Distributed Generation
Security of supply is also influenced by the installation of distributed generators (DG). Distributed generators can be loosely defined as sources of energy connected to distribution systems. They are much smaller than traditional central station generators, ranging from several kWs to approximately 10 MW [18]. If DG units are installed and coordinated properly, they can have positive impacts on distribution system reliability. However, power quality issues, protection coordination, voltage regulation, voltage flicker and short circuit levels are among some of the troublesome impacts of DG. The main advantage of DG units, however, is their close proximity to the loads they serve. They provide backup generation (serving the customers when a utility supply interruption occurs), peak shaving (reduction of overall feeder loading) and net metering (a billing mechanism that credits DG system owners for the electricity they add to the grid) [18].
3.2 Optimization Algorithms
DSP problems are complex nonlinear problems that involve a large set of variables (both continuous and discrete), whose complexity increases with the number of nodes in the system. To solve DSP problems, optimization algorithms are used [19].
Generally, optimization algorithms can be classified by the solution methods that they employ into four groups: conventional optimization algorithms, blend of heuristics and optimization algorithms, pure heuristic algorithms and meta- heuristic algorithms [20].
In large distribution systems, using conventional optimization methods can be
very inefficient regarding the computational time and real-time control [21]. The
combinatorial nature of the optimization problems led to investigation of purely
heuristic techniques. By using heuristic methods, the number of computations,
even for a large system, becomes more manageable and faster. These methods
do not result in an optimal solution and they are dependent on the initial system
configuration, but they provide near-optimal results and reduce time necessary to
obtain these results [21]. In the beginning of 90s, the new optimization methods,
so called meta-heuristic, have started to be exploited and have been gaining pop-
ularity since. These algorithms are based on principles from physics, biology or
ethology. They are used commonly in solving non-differentiable, nonlinear objec-
tive functions. They can provide optimal or near-optimal solutions through direct
14
CHAPTER 3. IMPROVING SECURITY OF SUPPLY IN DISTRIBUTION SYSTEM searching. Although the use of meta-heuristic based methods has proven to be valuable in a wide variety of applications, they do not represent the best solution in every implementation [20].
3.3 MILP vs. GA
In order to justify specific algorithm for achieving optimality when solving DSP problems, strong need to compare conventional optimization and meta-heuristic algorithm has emerged. Paper II presents a comparison between two algorithms, mixed-integer linear programming (MILP) and genetic algorithm (GA), applied to the DSP problem: construction of cables/feeders in radial distribution power system from scratch (feeder routing), with the aim to minimize total investment and outage costs and maximize reliability of the system. The essence is to indicate which method provides better results, regarding optimality, complexity and computation time. The developed algorithms have been tested on an actual distribution model.
The proposed simulation for MILP is formulated in GAMS software, while GA is formulated in MATLAB R2017b.
Moreover, Paper II proposes the base case algorithm for cable routing. The base case algorithm has been refined and expanded in chapters 4 and 5, in order to develop a more appropriate and realistic DSP model.
The idea behind the base case cable routing algorithm is to simultaneously update the failure rate of every node while deciding on the cable outline. By updating the failure rate, the outage cost of every node is updated as well, thus taking into account system reliability. The important input parameters of the algorithm are the geographic coordinates of all nodes (the supply node and sink nodes) and load demand for every sink node. The optimization needs to ensure the delivery of demanded load to all nodes, radial connectivity and power transfer boundaries of the cables.
3.3.1 Optimality Gap
According to [22], general problems are often extremely difficult to solve and prov- ing that the found solution is indeed the best possible solution can use enormous amounts of resources.
In GAMS there is the optCR setting which represents a tolerance for the relative optimality gap in a MIP or other discrete model [22]. This option specifies a relative termination tolerance for a global solver. The solver will stop as soon as it has found a feasible solution proven to be within optCR, i.e. the optimality gap falls below optCR.
Default value of optCR is 0.1 meaning that the objective value will be within
10% of the true objective value. However, it can be set to some other value. Even
though setting this option can help in reducing the solution time, it may cause
the true integer optimum to be missed. Therefore the reported solution could be
3.3. MILP VS. GA 15
the best, but it is guaranteed only to be within the tolerance of the true optimal solution.
3.3.2 Simulation Results
The results of compared optimization studies are presented in Table 3.1. The costs are given in SEK
1.
Table 3.1: Optimal Solution Results
MILP GA
Total Cable Lenght [m] 15.854,7 16.531,4 Total Cost [SEK] 4, 40 · 10
94, 59 · 10
9Cable Installation Cost [SEK] 4, 39 · 10
94, 58 · 10
9Outage Cost [SEK] 4, 43 · 10
64, 58 · 10
6Execution Time [sec] 10.000, 01 139, 24
As it can be seen, MILP provides better results regarding total cable lenght and therefore, total cost. Both cable installation and outage cost are lower in the MILP approach, which implies better reliability level than in GA approach. However, ex- ecution time for GA simulation is significantly lower than for the MILP simulation.
It is worth noting that the MILP simulation is time limited and the result presented in Table 3.1 reached optimal solution with a 1.7% optimality gap. To get better understanding how the MILP results develop over time, MILP is run for different execution times. Results are presented numerically and graphically in Table 3.2 and Figure 3.1, respectively. It can be seen that the optimal result presented in Table 3.1 can be reached even sooner, for execution time of 2000,01s with optimality gap of 2,193%. More importantly, the solution obtained after 100,01 sec (corresponding to optimality gap 4,054%) is still better than the solution obtained by GA.
Table 3.2: MILP Solution Results
Execution Time [sec] Optimality Gap [%] Total Cost [SEK]
46, 18 9, 988 4, 675 · 10
9100, 01 4, 054 4, 413 · 10
9200, 01 3, 729 4, 411 · 10
9500, 01 3, 139 4, 408 · 10
91000, 01 2, 699 4, 408 · 10
92000, 01 2, 193 4, 403 · 10
95000, 01 1, 841 4, 403 · 10
910000, 01 1, 707 4, 403 · 10
9GA, on the other side, provides reasonably good results in relatively short time.
In Figure 3.1 final GA solution is graphically highlighted as a comparison with the
1
Swedish Krona; 1 e≈ 10SEK
16
CHAPTER 3. IMPROVING SECURITY OF SUPPLY IN DISTRIBUTION SYSTEM
Figure 3.1: Total Cost according to Execution Time, for both MILP and GA pre- sented in Paper II.
Figure 3.2: GA convergance results over time presented in Paper II.
3.3. MILP VS. GA 17
MILP approach, regarding objective value and computation time. This solution represents the best fitness value after certain number of iterations during one exe- cution. Progress of GA convergance results over time is plotted in Figure 3.2. To assist a comparison between optimization methods regarding computation time, the number of iteration in the GA algorithm is successively increased. However, no difference in the final solution result is reported. That means, once the GA converges, the optimal value does not change regardless of time. Therefore, even after 10 000,01 sec the optimal GA value would still be 4, 59 · 10
9SEK, as it is after 139,24 sec.
Capability of GA to solve network optimization problems is highly dependent on the representation of the network and design of its genetic operators. Therefore, an implementation of GA might substantially influence its performance. Additionally, GA has difficulties handling constraints in optimization problems. On the other hand, once the optimization objective is clear, the constraint implementation in GAMS is very simple.
Having an accurate and fast optimization algorithm is always desirable. How-
ever, DSP is a planning and not a realtime problem. Therefore, getting a more
optimal results is much more important than getting a fast answer.
Chapter 4
Interruption Frequency Improvement
According to [14], better reliability may be achieved either by decreasing the cus- tomer interruption duration per fault or by decreasing the frequency of customer interruptions. This section focuses on decreasing the frequency of customer inter- ruptions. DSP problem through which is this improvement implemented is feeder routing.
Feeder routing is considered as a mixed integer nonlinear combinatorial prob- lem. Optimization model needs to decide how and where to build cables/routes between nodes (whose positions are fixed) while respecting technical and econom- ical constraints. Usually feeder routing aims to minimize total cable length and the associated costs. The novelty in this study is the calculation of the outage cost which is directly linked to the proposed cable layout. Moreover, based on the suggested outline of the network, the failure rate as well as the outage cost of every node is dynamically updated.
4.1 Cable Routing - Base Case
4.1.1 Algorithm
The integer or binary decision variable is described as follows:
x(i, j) =
1, if there is a direct link from node i to node j 0, otherwise
The idea behind reliability centered cable routing is to simultaneously update failure rate of every node while deciding on the cable outline. The optimization ensures the delivery of demanded load to all nodes, radial connectivity and power transfer boundaries of the cables.
19
20 CHAPTER 4. INTERRUPTION FREQUENCY IMPROVEMENT
λ
L(i, j) = λ
0· distance(i, j) · sf (4.1)
λ
N(s) = λ(s) (4.2)
λ
N(j) =
N
X
i=1
x(i, j) · (λ
N(i) + λ
L(i, j)) (4.3)
With known coordinates, Euclidean distances between all nodes in the system (supply node s and sink nodes) can be estimated. To address the real terrain between the nodes, these Euclidean distances are multiplied with a safety factor, sf . Failure rate of every cable depends on the distance between nodes i and j, and it is described by the Equation (4.1), where λ
0represents statistical failure rate of the cable in f/yr.km.
As a starting point, failure rate of the primary substation, i.e. supply node s, needs to be known as well (Equation (4.2)). However, failure rate of every other sink node is calculated based on the proposed configuration (Equation (4.3)).
The algorithm is moving downstream, and it updates the failure rate of every node j by summing the failure rate of its upstream nodes λ
N(i) and upstream cables λ
L(i, j). Therefore, every downstream node j has a higher failure rate than its upstream node i. The assumption here is that in the case of fault, all downstream nodes from the place of fault will be disconnected until the fault is repaired.
x(i, i) = 0 (4.4)
N
X
i=1
x(i, j) = 1, ∀j 6= s (4.5)
x(i, j) + x(j, i) ≤ 1, i 6= j (4.6)
N
X
j=1
x(s, j) ≥ 1 (4.7)
Equations (4.4), (4.5), (4.6) represent constraints that are ensuring radial supply to each node and hence only one upstream node, without any loops made [23].
Equation (4.4) ensures that there is no connection leading from and to the same node. In equation (4.5) it is assured that every node (except supply node) needs to have only one upstream connection. If there is a connection from node i to node j (i being the upstream node), equation (4.6) prevents possible connection from j to i, i.e. creating a loop in the system between nodes i and j. The supply node is assumed to be capable of serving the power demand to all sink nodes. The link to the supply node, i.e. at least one connection downstream of the supply node, is confirmed by Equation (4.7).
LC =
√
3V I (4.8)
4.2. DIFFERENT CABLE OPTIONS 21
N
X
l=1
LF (l, i) = L
a(i) +
N
X
j=1
LF (i, j), ∀i 6= s (4.9)
LF (i, j) ≤ x(i, j) · LC(i, j) · 0.5 (4.10)
The power transfer capacity of the cables is calculated according to the formula (4.8), where V denotes the voltage level in the feeders and I denotes the current capacity of the cables. Equation (4.9) represents power balance at node i. In other words, equation (4.9) assures that the input power flow to node i (i.e. the summation of power flow entering the node i from the upstream node l) is equal to the demand at node i (L
a(i)) plus the output power flow from node i (i.e. the summation of power flow exiting to the downstream node(s) j). Equation (4.10), on the other hand, guarantees that the power flow at each line is smaller than the power capacity of the line. As suggested by the utility practice, a safety margin of 50% usability of the current capacity is adopted for normal operation conditions [24].
4.1.2 Application
This presented model has been used in Paper II. The objective in Paper II is to minimize the total cost which is comprised of cable installation cost (Equation (4.11)) and outage cost defined in section 2.2.1.
CableInstallationCost =
N
X
i=1 N
X
j=1
x(i, j) · CCost · distance(i, j) · sf (4.11)
where CCost represents the cable installation cost per km given in SEK/km.
4.2 Different Cable Options
4.2.1 Algorithm
In the previous section, the cable routing algorithm that takes into account system reliability is defined. In this section, the algorithm is refined by taking into account the possibility of installing cables with different capacities and different costs. These capacities are based on the maximum current that the cable can withstand and the voltage level of the network.
Thus, in this model, more than one cable option is possible. Number of possible cable options is C (C ∈ {1, 2...}). It should be noted that this does not mean that C parallel cables are constructed. Instead, there are C binary decision variables, corresponding to C cables with different costs and capacities, described as follows:
x
c(i, j) =
1, if cable c is constructed from node i to j
0, otherwise
22 CHAPTER 4. INTERRUPTION FREQUENCY IMPROVEMENT
C
X
c=1
x
c(i, j) ≤ 1 (4.12)
It is important to notice that these binary variables exclude one another, i.e. if one of them is 1, all others are 0 for the same pair of nodes (i, j) (Equation (4.12)).
λ
L(i, j) = λ
0· distance(i, j) · sf (4.13)
λ
N(s) = λ(s) (4.14)
λ
N(j) =
N
X
i=1 C
X
c=1
x
c(i, j)(λ
N(i) + λ
L(i, j)) (4.15)
Again, failure rate of every cable is given by the Equation (4.13) while the Equation (4.15) describes the failure rate of each node. However, the difference here comparing to the base case is the possibility for different cable options which is given through P
Cc=1
x
c(i, j).
C
X
c=1
x
c(i, i) = 0 (4.16)
N
X
i=1 C
X
c=1
x
c(i, j) = 1, ∀j 6= s (4.17)
C
X
c=1
(x
c(i, j) + x
c(j, i)) ≤ 1, i 6= j (4.18)
N
X
j=1 C
X
c=1
x
c(s, j) ≥ 1 (4.19)
Equations (4.16), (4.17), (4.18), (4.19) are respectively equivalent to (4.4), (4.5), (4.6), (4.7). The only difference is possibility for more cable options.
In this model the power transfer capacity of the cables is calculated according to the following formula:
LC
c=
√
3V I
c(4.20)
where V denotes the voltage level in the feeders and I
cdenotes the current capacity of the cable c. Therefore, every cable type has its own power transfer capacity.
N
X
l=1
LF (l, i) = L
a(i) +
N
X
j=1
LF (i, j), ∀i 6= s (4.21)
LF (i, j) ≤
C
X
c=1
x
c(i, j) · LC
c· 0.5 (4.22)
Equation (4.21) represents power balance at node i. Equation (4.22), on the
other hand, guarantees that the power flow at each line is smaller than the power
capacity of the line for every cable type.
4.2. DIFFERENT CABLE OPTIONS 23
4.2.2 Application
This model was applied in Paper IV in order to find the optimal cable layout for an onshore wind farm with slight modifications.
First of all, all sink nodes are identified as wind turbines, while supply node is identified as point of connection (POC) - point at which wind farm is connected to the distribution system, and the flow of power is reversed.
Initial failure rate λ
initof every node is given - every wind turbine has its own failure rate λ
W T, while the initial failure rate of POC is assumed to be 0.
λ
init(i) =
λ
W T, node representing wind turbine 0, node representing POC
λ
N(j) = λ
init(j) +
N
X
i=1 C
X
c=1
x
c(i, j)(λ
N(i) + λ
L(i, j)) (4.23)
The failure rate of every node j is updated by summing its initial failure rate λ
init(j), failure rate of its upstream nodes λ
N(i) and upstream cables λ
L(i, j).
Therefore, every downstream node j has a higher failure rate than its upstream node i. Here the assumption is that in the case of fault, all upstream wind turbines from the place of fault will be disconnected. The closer the fault is to the POC, more wind turbines are disconnected and the cost of lost energy is bigger.
N
X
j=1 C
X
c=1
x
c(i, j) = 1, ∀i 6= P OC (4.24)
N
X
i=1 C
X
c=1
x
c(i, P OC) ≥ 1 (4.25)
Equation (4.17) is replaced with equation (4.24) where it is assured that every node (except POC) have only one downstream connection.
The POC node is assumed to be capable of receiving the power supply from all WT nodes. The link to the POC, i.e. at least one connection upstream of the POC node, is confirmed by Equation (4.25).
N
X
j=1
LF (i, j) = P (i) +
N
X
l=1
LF (l, i), ∀i 6= P OC (4.26)
Equation (4.21) is replaced with (4.26), that assures that the output power flow from node i (i.e. the summation of power flow exiting the node i to the downstream node j) is equal to the supply at node i (active power of a wind turbine P (i)) plus the input power flow to node i (i.e. the summation of power flow entering the node i from the upstream node(s) l).
The interconnection of the onshore wind farm to the distribution grid is often
regarded as a potential source of power quality disturbances. In order to analyze
24 CHAPTER 4. INTERRUPTION FREQUENCY IMPROVEMENT
power quality requirements, the maximum steady-state voltage variation (%) at the POC is evaluated using the following simplified relation [25]:
≈ 100 V
2· (R ·
N
X
i=1
P (i) + X ·
N
X
i=1
Q(i)) (4.27)
In practical situations, Equation (4.27) will yield a voltage increase, due to the active power flow on the resistive part of the network impedance. For this reason, slightly inductive power factor values are usually preferred (Q<0, where Q is the reactive power of the wind turbine) [25].
R and X are series line resistance and reactance at POC, respectively, and they depend on the cable type (every cable type c has its own electrical parameters, R
cand X
c).
R =
N
X
i=1 N
X
j=1
(
C
X
c=1
R
c· x
c(i, j)) · distance(i, j) (4.28)
X =
N
X
i=1 N
X
j=1
(
C
X
c=1
X
c· x
c(i, j)) · distance(i, j) (4.29)
Instead of outage cost, cost of lost energy defined in Equation (4.30) is used as a measure of reliability.
LostEnergyCost(i) = λ
N(i) · P (i) · et(i) · r
N(i) · f (i) (4.30)
where P (i) is an active power of a wind turbine i, et(i) is the electricity tariff associated with the production of turbine i and f is a capacity factor of a wind turbine i.
Thus, the objective of Paper IV was to minimize the total cost which is com- prised of cost of lost energy production (Equation (4.30)) and cable installation cost (Equation (4.31)):
CableInstallationCost =
N
X
i=1 N
X
j=1 C
X
c=1
x
c(i, j) · CCost
c· distance(i, j) · sf (4.31)
where CCost
crepresents the installation cost per km of the cable type c given in SEK/km.
Simulation results show that by introducing more cable options, total costs are
decreasing. Moreover, cables with bigger capacity contribute to lower values of
voltage increase at POC. Additionally, these cables have the ability to support
supply from more turbines, creating longer feeders. However, these longer feeders
increase certain risk - in case the failure happens on a feeder, more turbines on
that feeder can be disconnected, thus negatively affecting system reliability and
therefore cost of lost energy production.
4.3. CABLE ROUTING AND SWITCH PLACEMENT 25
4.3 Cable Routing and Switch Placement
4.3.1 Algorithm
This section reviews new refinement of the base case algorithm: simultaneous ca- ble routing and switch placement. The optimization model investigates the best possible combination of node connections and switch positions.
Beside already defined x(i, j) binary variable, there is an additional binary de- cision variable y(i, j) described as follows:
x(i, j) =
1, if there is a direct link from node i to node j 0, otherwise
y(i, j) =
1, if there is a switch directly between nodes i and j 0, otherwise
Derived binary variables, q(i, j) and z(i, j), are described by Equations (4.32) and (4.33).
q(i, j) =
1, if there is a direct link from i to j without a switch 0, otherwise
z(i, j) =
1, if there is a direct link from i to j with a switch 0, otherwise
q(i, j) = x(i, j) · (1 − y(i, j)) (4.32)
z(i, j) = x(i, j) · y(i, j) (4.33)
This model is subjected to the following constraints:
λ
L(i, j) = λ
0· distance(i, j) · sf (4.34)
Failure rate of every cable depends on the distance between nodes i and j, and it is described by the Equation (4.34).
λ
00N(s) = λ(s) (4.35)
x(i, j) · λ
00N(i) ≤ x(i, j) · (λ
00N(j) − λ
L(i, j)) (4.36) λ
0N(i) ≥ λ
00N(i) +
N
X
j=1
q(i, j) · λ
L(i, j) (4.37)
q(i) · λ
0N(i) =
N
X
j=1
q(i, j) · λ
0N(j) (4.38)
q(i) ≥ q(i, j) (4.39)
z(i, j) · λ
0N(i) = z(i, j) · (λ
0N(j) − λ
L(i, j)) (4.40)
λ
N(s) = λ
0N(s) (4.41)
λ
N(j) =
N
X
i=1
x(i, j) · (λ
N(i) + y(i, j) · λ
L(i, j)) (4.42)
26 CHAPTER 4. INTERRUPTION FREQUENCY IMPROVEMENT
Equations (4.35)-(4.42) describe the process of calculating the failure rate of each node. The concept of the suggested algorithm is to analyze the network, move up- and downstream through it, and update failure rate of every node based on the proposed cable layout and switch placement.
The algorithm starts going downstream through the network, with the assump- tion that there is a switch between every two connected nodes, i and j. Considered switch is a recloser with radial topology, which is able to interrupt current flow after a fault is detected and to isolate a faulted part from a healthy part, inter- rupting all downstream customers [26]. No additional restoration is possible until the fault is repaired. That means the failure rate of every node λ
00N(j) is affected only by its upstream part of the network (failure rate of upstream node λ
00N(i) and upstream cable λ
L(i, j)). Any failure that happens downstream of that node is automatically isolated by the switch. Consequently, every downstream node j has higher failure rate than its upstream node i; hence the auxiliary failure rate of every node λ
00Nis calculated (Equation (4.36)). Once the algorithm reaches the bottom of the network, it starts moving upstream. By doing that, now it actually checks where did optimization propose switches and updates failure rate of every node (λ
0N). Any line not secured by the switch automatically affects failure rates of its upstream nodes (Equations (4.37), (4.38)). Equation (4.39) is a constraint assuring that the binary variables are in consistence with one another, where q(i) represents an auxiliary binary variable which is equal to 1 if there is at least one direct link from node i without a switch, and 0 otherwise. Equation (4.40) assures that every node i protected with the switch is not affected by the failure rate of its downstream node j. Once the algorithm reaches the top, it updates the failure rate of the supply node (Equation (4.41)) and moves again downstream for the final check and update (Equation (4.42)).
x(i, i) = 0 (4.43)
N
X
i=1
x(i, j) = 1, ∀j 6= s (4.44)
x(i, j) + x(j, i) ≤ 1, i 6= j (4.45)
N
X
j=1
x(s, j) ≥ 1 (4.46)
LC = √
3V I (4.47)
N
X
l=1
LF (l, i) = L
a(i) +
N
X
j=1
LF (i, j), ∀i 6= s (4.48)
LF (i, j) ≤ x(i, j) · LC(i, j) · 0.5 (4.49)
N
X
i=1 N
X
j=1
y(i, j) ≥ 0 (4.50)
y(i, i) = 0 (4.51)
4.4. WILLINGNESS TO REBUILD CABLES 27
y(i, j) ≤ x(i, j) (4.52)
Beside already defined radiality and power transfer capacity constraints, there is a new set of switch placement constraints (Equations (4.50), (4.51) and (4.52)) that ensure placement of switches in the distribution network, and prevent placing the switch between two nodes that are not connected.
4.3.2 Application
The proposed algorithm was used in Paper III as a part of the Stage 1 algorithm.
The main optimization objective was to minimize the total cost which is comprised of three elements: outage cost (Equation (2.1)), cable installation cost (Equation (4.11)) and switch installation cost defined as follows:
SwitchInstallationCost =
N
X
i=1 N
X
j=1
y(i, j) · SwgCost (4.53)
where SwgCost is the cost of installing the recloser on the line between two connected nodes.
The biggest challenge with the algorithm was the time of execution. Com- bination of optimal cable routing and switch placement made the problem more demanding, execution time longer and the convergence slower.
For the proposed algorithm, the optimality gap defined in Section 3.3.1 could not drop below 20%, even when running simulation for 100 000 s (≈ 28h). Therefore, certain relaxation was introduced. Namely, the lower bound in (4.50) was loosen, setting the minimum number of reclosers in the system to be different than 0.
N
X
i=1 N
X
j=1
