Le chapitre III est le développement du modèle d'optimisation du dimensionnement en définissant les fonctions bi-objectifs (fiabilité et coût) et l'optimalité de Pareto parallèlement à l'algorithme classique de gestion de l'énergie avec l'outil d'optimisation. Il donne ensuite les résultats de l'optimisation et les différentes configurations du système (PV seulement, batterie seulement) avec la discussion multi-source. En outre, une analyse de la sensibilité économique du système pour mettre en évidence l'influence du coût est réalisée.

Le chapitre IV est consacré aux modèles de panne de réseau, avec un rappel du profil de panne du client. Deux modèles présentant les particularités d'une mise en oeuvre minimale sans nécessiter de données détaillées sur les pannes et la capacité de prendre en compte des scénarios extrêmes dans les pannes, ce qui nous intéresse particulièrement, sont développés.

De plus, une analyse de test de corrélation et de périodicité est faite pour expliquer la pertinence des modèles proposés. Enfin, il donne les résultats et les comparaisons statistiques des profils de pannes simulés par les modèles.

Le chapitre V présente les performances du système de stockage PV-batterie développé sur un profil de panne important du chapitre IV en utilisant un système optimal de référence sélectionné sur la base des résultats du chapitre III. Il décrit la comparaison des deux modèles d'interruption proposés ainsi que le système exposé à plusieurs profils d'interruption. En résumé, la démarche scientifique suivie dans cette thèse est de développer une méthodologie d'optimisation de dimensionnement global d'un système multi-source en intégrant le PV et le stockage par batterie sur un réseau de distribution radial faible. Dans le but de fournir une alimentation électrique adéquate pendant les pannes et de remplacer la solution diesel conventionnelle en raison de ses nombreux impacts négatifs, nous proposons dans ce manuscrit une étude de cas pour développer les approches méthodologiques. Nous étudions un bâtiment universitaire situé à Addis Abeba, en Éthiopie, qui est alimenté par un réseau stochastique faible, avec des pannes fréquentes et de longue durée. Sur la base des données disponibles sur les pannes du réseau et des mesures réelles (profil de charge et de panne) effectuées par l'auteur, l'approche de dimensionnement suit une gestion globale de l'énergie basée sur des règles et sur le choix des ressources renouvelables, c'est-à-dire le panneau photovoltaïque et le stockage par batterie Li-ion. Ainsi, l'optimisation de dimensionnement multi-objectif (MOSO) de ce travail, diffère du faible nombre de travaux connexes existants, avec une application particulière et l'inclusion d'un réseau stochastique réel dans notre méthodologie. La MOSO offrira également la flexibilité d'explorer l'amélioration Résumé viii de la fiabilité avec différentes combinaisons PV-batterie pour un niveau de fiabilité souhaité.

En outre, la troisième contribution consiste à développer une modélisation stochastique des pannes de réseau basée sur des données historiques de pannes, ce qui permet également d'évaluer l'efficacité et de valider les résultats optimaux de MOSO, en plus d'étudier l'état d'une situation de panne de réseau pour un cas considéré. Enfin, les simulations de grands profils de pannes à travers ces modèles qui nous permettent d'évaluer les performances, permettront de montrer la faisabilité et le bénéfice de la solution. Pour cela, nous avons proposé deux modèles de panne de réseau en fonction de leurs avantages et de leurs capacités.

Enfin, les conclusions finales du travail de thèse, les travaux scientifiques réalisés et les principaux résultats résument ce manuscrit. Il présente ensuite les limites de ce travail et les perspectives pour souligner l'élargissement et la poursuite future de ce travail.

Praise to be the almighty GOD and the blessed holy mother, virgin Mary! I want to take the opportunity to gratefully thank institutions and individuals who have been involved and backing me in the long journey of this Ph.D. It has been challenging yet enabled me to endure and got me here. My first thanks to the French Embassy in Addis Ababa and the Ethiopian Ministry of Education for financing this study.

Accomplishing this work would have not been possible without the unreserved continuous quality supervision. My thanks then start during when they accepted me to work with them and give me wisdom with trust. I greatly thank to M. Salvy Bourguet, my thesis director, for all the scientific insights, consistent and rigorous direction, endless encouragement and be there in my difficult times. A huge thanks also to M. Jean-Christophe Olivier, the co-director of this work, guiding me in the right direction and opening my eyes to the scientific developments, his comments, and for the support that allowed me to progress my work. Their regular comments and suggestion throughout the process of writing, humanly support and sympathy during when times were tough, and their all-rounded advice meant a lot to me. I am forever indebted to their unreserved constant support.

Secondly, I would like to thank all the jury members for accepting to examine my thesis work and devoting their time and availability during the defense day. I am honored and want to express my gratitude for Mme. Delphine Riu, president of the jury, for accepting to chair and for the judicious discussions. Many thanks to M. Philippe Dessante and M. Dhaker Abbes for being the rapporteurs of the thesis work, for their thorough reading and valuable comments and remarks in their pre-defense report, and the constructive remarks during the defense. I am grateful to M. Mohamed Machmoum for his insightful and pragmatic exchanges of my work in the defense. All these questions, remarks, and exchanges helped me to think twice and open up future perspectives in the continuation of the research work. I appreciate and thank Campus France and personnel from the French embassy for all the coordination to have a smooth stay in France through all these years and for quick response whenever needed. I am honored to work at the Institut de Recherche en Énergie Électrique Nantes Atlantique (IREENA), and very thankful for all the personnel in charge of the lab, the administrative and technical support with all their kindness and charming welcome. Regarding other institutions, the data provision from the Ethiopian Electric Power and Utility, ii measurement device, and technical support from the Ethiopian Energy Authority is also appreciative.

The lively 2 nd floor of the CRTT building at which I have shared various discussions, so enriching and pleasant moments both in the lab and of course outside deserve many thanks, merci beaucoup! In particular, I want to thank the former Ph.D. student and my office colleague Alioune Seck as well as to Anthony Roy for all the various discussions we had. I am also thankful for another former Ph.D. student, Ramzi Saidi, for the support and friendly discussions we shared. Thanks to all the Ph.D. students and members of the lab to make my stay in France so fruitful. I then want to express my gratitude for the collegial encouragement and support from all colleagues at Addis Ababa Institute of Technology-Addis Ababa University (particularly the Center for Renewable Energy), and also for all the pleasant moments we shared. Now, despite not being sufficient to express my gratitude and thankfulness to my family and parents in this piece of paper, my special thanks is for you from the bottom of my heart. Even with the unimaginably difficult years, the flawless and all-time advice, prayers, inspiration, and financial support from all of you throughout these years is immense to mention. I thank you a lot for all your irreplaceable support being my strengths for my scientific and professional success without whom I could never have reached where I am now.

At last, I wish you a pleasant read for the anonymous reader, and thank you for your time.

Résumé

Dans les pays en développement comme l'Afrique subsaharienne (ASS) et l'Asie du Sud-Est, les consommateurs sont habitués à des coupures fréquentes et de longue durée, ce qui pose un problème insupportable de fiabilité de l'alimentation électrique. De telles interruptions et coupures de courant, qui sont de natures très aléatoire et souvent imprévisibles, et qui durent le plus souvent de quelques minutes ou heures à plusieurs jours dans le pire des cas, affectent grandement un large éventail de consommateurs, des ménages aux grandes entreprises et industries. Cela accroît l'utilisation de générateurs diesel chez de nombreux clients des services publics, ce qui est l'option la plus inefficace, polluante et coûteuse. Contrairement aux pannes des réseaux modernes des pays développés, qui sont généralement dues à des conditions météorologiques extrêmes ou à des opérations de maintenance planifiées, les pannes dans les pays en développement sont dues à la faiblesse du réseau, à son architecture radiale, à l'insuffisance de la redondance des composants du réseau, à l'ancienneté des composants, à la demande excessive et à l'insuffisance de l'offre, ainsi qu'à l'utilisation d'infrastructures médiocres et non conformes aux normes. Par conséquent, cette thèse propose un système solaire photovoltaïque et de batterie multi-source intégré à un réseau faible basé sur des données de pannes stochastiques enregistrées et un profil de charge. L'objectif de dimensionnement est de trouver le meilleur compromis entre les contraintes technicoéconomiques afin d'assurer la continuité de l'approvisionnement (réduction des pannes) et la minimisation du coût de l'énergie, ce qui met en oeuvre une optimisation de dimensionnement multi-objectifs pour obtenir un ensemble de solutions Pareto optimales. Il comprend la modélisation énergétique des dispositifs photovoltaïques (PV) et de stockage par batterie, la demande de puissance électrique d'un client connecté à un réseau stochastique réel où les données de pannes enregistrées et la charge mesurée du client sont utilisées. L'analyse de diverses configurations (PV seul, batterie seule, hybride) et de la sensibilité économique permet de mettre en évidence la capacité à fournir de l'énergie pour une meilleure fiabilité et l'influence du coût respectivement.

Pour étudier les pannes de réseau stochastiques fréquentes et de longue durée qui posent un problème délicat de fiabilité de l'alimentation électrique, deux méthodes différentes sont développées afin de modéliser et de simuler les pannes de réseau possibles, en fonction de données enregistrées. Nous avons d'abord effectué une analyse de corrélation et de périodicité/tendance sur les données historiques des pannes. Le modèle de Markov (matrice de Résumé vi transition de Markov) qui capture les répartitions jour-nuit démontrées dans le test de périodicité, est capable de simuler les pannes comme un processus stochastique à partir des données historiques. Il est aussi particulièrement important lorsque nous disposons de petites données d'enregistrement des pannes et en même temps lorsque les ensembles de données sont des paramètres globaux. De plus, le modèle de Weibull est construit de telle manière qu'il est capable de prendre en compte de manière précise les événements extrêmes tels que des durées d'interruption longues mais moins fréquentes, ce que ne fait pas le modèle de Markov. Les résultats sont utilisés pour évaluer la performance et l'efficacité des systèmes d'énergie renouvelable multi-sources.

Cette thèse est donc structurée et divisée en cinq chapitres présentant les aspects de l'étendue et de la profondeur des pannes de courant en particulier dans les pays en développement, les solutions de pointe pour leur atténuation et l'approche méthodologique pour l'optimisation du dimensionnement multi-source et, enfin la modélisation des pannes de réseau, résumant brièvement le contenu des chapitres comme suit.

Le chapitre I est une revue détaillée des problèmes de pannes pour illustrer leur gravité et les implications financières associées, en se concentrant particulièrement sur les réseaux faibles des pays en développement, caractérisés par des pannes de courant aléatoires et récurrentes, ainsi que leurs causes. Les caractéristiques particulières des pannes de ces réseaux et les différentes solutions sont ensuite présentées. L'état de l'art pour l'intégration de solutions multi-sources d'énergie renouvelable et les optimisations de dimensionnement sont discutés.

Enfin, le chapitre résume le positionnement et l'objectif de la thèse ainsi que l'ensemble des travaux et contributions scientifiques qui en découlent.

Le chapitre II présente les modèles énergétiques des systèmes multi-sources et l'architecture des dispositifs photovoltaïques et de stockage. Les modèles économiques et de fiabilité qui permettent l'optimisation du dimensionnement comme fonctions objectives sont ensuite développés. Ce chapitre présente également un modèle de générateur diesel pour le comparer à son homologue renouvelable. Enfin, l'étude de cas est décrite avec un client sélectionné afin de mettre en évidence les phénomènes de panne et d'incorporer le réseau stochastique réel, puis la localisation géographique. L'étude de cas comprend des données historiques sur les pannes provenant d'un opérateur de services publics et des profils de demande de charge et de pannes mesurés pour le client sélectionné.

List of Figures xi

List of Figures

List of Tables

General Introduction

In recent economies, electrical power supply plays a crucial role in all sectors, whereby there are no economic activities without energy services. This necessitates the need for access to a reliable system with greater continuity of supply at all times and an affordable price. The global electricity access in 2019 was about 90%. Although the majority of the developed countries such as Europe, Central Asia, North America, and others have achieved full access, a large population in other parts of the world such as sub-Saharan Africa (SSA) and Southeast Asia is living without access. For instance, the access rate in the SSA in the same year was around 47%. Here what is more challenging in lowand middle-income countries is that, even with such a low access rate, the power supply also experiences frequent power interruption and blackout which is very random in nature and often unpredictable. Despite the fact that power outages also rarely occur in developed countries' power systems, they are usually extreme-weather-caused or for maintenance tasks. However, factors such as the use of weak network with radial grid architecture having insufficient redundancy of network components, comprised of aged components, excess demand with an inadequate supply, and use of poor and substandard infrastructure are the most common causes for grid outages in developing countries. More frequent and longduration outages in developing countries increase the use of diesel generators among many utility customers, which is the most inefficient, polluting, and more and more costly option.

These outages while lasting more often from a few minutes or hours to several days in worst cases, greatly affect a wide range of consumers from households to large businesses and industries.

Most EU member countries rigorously monitor the continuity of supply and related interruptions or outages. Even if there are disparities among countries, they usually report outages of any duration, be it long, short or transient durations of outages. In developing countries on the other hand, it is usually difficult to monitor and report outages, hence posing challenges to studying the detailed situation of continuity of power supply and constraints the process of developing solutions for outage reductions and reliable supply. This is where the reliability of electricity supply is even more crucial, particularly for those with an already lower access rate and critically poor reliability level.

To address the problem of outages, solutions such as the integration of distributed generation (DG) usually from renewable resources exist. On modern grids, the knowledge of the outage causes and availability of sufficient outage information and data, enables the development of other outage reduction solutions too, like grid modernization and outage prediction models. Outage models take into account weather information, historical extreme weather, other environmental parameters with related historical outages as well as grid components' failure history for predictions. The application of these approaches seems to be challenging in case of weak grids such as in developing countries. First off, the causes are not often linked to the weather. Secondly, the unavailability of comprehensive outage data poses a limitation not only for outage modeling but also for integrating DG systems accounting for the existing real grid outages, which are stochastic and recurrent. Electric power outage impacts both the customers and utility operators as well entailing related costs.

Currently, it is one of the interesting options to use renewables-based multi-source systems, particularly photovoltaic systems, for various applications. This is due to the favorable price decrease in the last few decades, significantly very low emissions throughout their lifecycle, minimal O&M requirement for optimal performance, and ease of their use and integration. Despite the benefits, they are usually coupled with auxiliary systems to complement their intermittency and to ensure reliable supply. The auxiliary systems, such as storages, also help to compensate for their intermittent nature, either by charging them in case of surplus or later supply the demand during production deficit through the storages discharging. The integration of electrochemical storages is able to supply in states where the production is not sufficient and outage reduction, but also for flexibility and added degree of freedom for the energy management. For this work, battery storage devices are chosen to be integrated with the PV and the frail grid network. Now, taking the continuity of power supply with sufficient capacity of the multisource devices into account, the challenge is then to have optimal sizing and cost with the objective to reduce or avoid outages. The sizing then concerns the PV generator and the storage capacity with the associated costs to satisfy the load demand of consumers considering the real outage scenario and to improve the reliability. The sizing of the PV can either be the required PV surface area or peak power required and; that of the storage device can be the number of battery storages or capacity needed.

One of the difficulties is integrating the real stochastic grid with the load demand and then evaluating the effectiveness of the optimal solution. In addition, considering such a grid also is able to account for the intermittent nature of renewable systems, such as solar resource variability. In this work, a two-objective sizing optimization that takes the reliability and system cost into account is proposed. In the common applications of single objective optimization problems, minimization of the total cost of the system or the cost of energy produced is the objective. Here, the contradicting objectives of reliability and cost of energy are used implying the best solutions by the trade-off between the continuity of supply (load satisfaction without outage) and energy cost. Thanks to the concept of the Pareto front, this gives the set of non-dominated solutions, for both the utilities and consumers.

Moreover, a sufficiently large outage profile, as to the load profile and resources data corresponding to the geographical location considered, is necessary to achieve reasonable optimality, again helping to study the outage phenomena with a fairly thorough breadth of the challenges and impact of outage.

In this work, new models approach without any predicting parameters and without linking to any outage-causing factors are proposed. The idea is to generate a large outage profile from a limited available data of a weak grid. The models are able to incorporate any existing attributes, if there exist, and bring their own advantages. Several research works demonstrate the ability of the Markov Transition Matrix (with transition probabilities of the grid on-off states) to effectively represent stochastic processes like random outages.

The MTM is derived from the concept that the probability of occurrence of the next outage state is only dependent on the present state of the grid, without any history and predictive nature. The Markov model has the advantage of its minimalism in historical data requirement, and we have adapted to be able to capture existing periodicity in the outage occurrences, but limited to delineating extreme events with rare occurrences due to limited variability. The two-parameter Weibull model on the other hand is able to capture the extreme phenomena with sufficient diversity as it is devised for such failure analysis of extreme events. These also help to test the effectiveness and performance of the designed multi-source (PV-battery) system on a large number of outage profiles of potential feasible scenarios over the life span of the renewable system. Furthermore, this work also proposes to evaluate the models' characteristics by comparing with measured outage data. First off, the Markov and Weibull models are developed from existing historical data, in such a way that, the Markov probability matrix is calculated from more limited data. On the other hand, the Weibull model scale and shape parameters are derived from a relatively large dataset of 5 years. This thesis is therefore structured and divided into five chapters presenting aspects of the scope and depth of power outages, particularly in developing countries, the state-ofthe-art solutions for its mitigation, and the methodological approach for the multi-source sizing optimization and, lastly grid outage modeling as follows.

Chapter I is a review of the detailed outage problems to illustrate their severity and associated cost implications, especially focusing on weak grids in developing countries, characterized by random and recurrent power outages with the causes as well. The peculiar outage characteristics of such grids and various solutions are then presented. The state-ofthe-art for the integration of renewable energy multi-source solutions and sizing optimizations are discussed. At last, it summarizes the thesis positioning and objective with its subsequent overall scientific works and contribution.

Chapter II presents the energetic models of the multi-source systems and architecture for the photovoltaic and storage devices. The reliability and economic models which allow for the sizing optimization as the objective functions are then developed. This chapter also gives a model for a diesel generator for comparison with its renewable counterpart system. At last, the case study description with also a selected customer in order to highlight the outage phenomena and incorporate the real stochastic grid, and later the geographical location is described. The case study includes historical outage data from a utility operator and measured load demand and outage profiles for the selected customer.

Chapter III is the development of the sizing optimization model by defining the biobjective functions (reliability and cost) and Pareto optimality alongside the classical energy management algorithm with the optimization tool. It then gives the optimization results and various system configurations (PV only, battery only) with the multi-source discussion. In addition, an analysis of the system's economic sensitivity to highlight the influential cost component is achieved.

Chapter IV is all about the grid outage models, with a recap of the customer outage profile. Two models with peculiarities of minimal implementation without requiring detailed outage data and the ability to take into account extreme scenarios in outages, which we particularly are interested in, are developed. That is to say weak grids with stochastic outages are insufficiently studied due to lacking comprehensive and adequate data, such as for modeling their attributes to study and developing solutions like outage reduction and reliability upgrade. On top of that, a correlation and periodicity test analysis is done to explain the suitability of the proposed models. It then lastly gives the results and statistical comparisons of the model-simulated outage profiles.

Chapter V provides the performance of the developed PV-battery storage system on a large outage profile from chapter IV using selected reference optimal systems based on the results from chapter III. It depicts the comparison of the proposed two outage models and also the system exposed to several outage profiles. In advent of the technological advancements, rapidly emerging environmental concerns, and encouraging economic feasibility, renewable energy resources are playing an important role in the power sector. Recently amongst various proposed solutions, many commonly contemplate the use of multi-source systems mainly comprising renewable energy sources for lessening the negative impacts of outages and improving reliability of electrical grids. Outages may contextually be classified into two categories, those on modern grids basically in the developed countries [START_REF] Ceer | 6th CEER benchmarking report on the quality of electricity and gas supply[END_REF]- [START_REF] Balducci | Electrical Power Interruption Cost Estimates for Individual Industries, Sectors, and U.S. Economy[END_REF] and others on weak grids in the developing and least developed countries [START_REF]World Energy Outlook 2019[END_REF]- [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF]. On modern electricity grids, factors causing most of the outages are well studied, which also leads to the development of knowledge on how to manage and cope with outages. Commonly, the causes of outages may either be extreme-weather related or planned maintenance. There are also already existing sophisticated technologies and techniques to help either avoid their occurrence or minimize the impact if they happened. On top of this, outages are mostly short durations and very infrequent [START_REF] Ceer | 6th CEER benchmarking report on the quality of electricity and gas supply[END_REF]. Conversely, the reasons that trigger power outages in developing countries could be inadequate generation, substandard power system infrastructure, weak architecture mainly in the distribution grid networks, and other uncontrolled faults and errors. For these later, on weak grids, meager has been done for worst-case scenarios of outages in which power consumers in several developing countries are accustomed to frequent and long-duration outages [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF]. They are also unpredictable hence heightening the challenge and damage to users. This is because customers would have reduced outages' consequences if they were at least scheduled, and cognizant of their occurrence beforehand to the consumers.

In Ethiopia, one of the developing countries facing unbearable reliability issues, increased interruption and repeated customer complaints are observed. This is an impediment on top of an already poor electricity access rate, leaving more than half of the country's population without access to it, which for example in 2016 stood at about 56% [START_REF] Schwaegerl | Active Distributed Systems and Distributed Energy Resources[END_REF]. The outages while lasting more often from a few minutes or hours to several days in worst cases, greatly affect a wide range of consumers from households to large businesses.

Due to this, several business owners, manufacturing firms, and some residential consumers rely on backup generators run on diesel fuel in case of outages. Additionally, the added cost of diesel operation, economic and market competitiveness hurdle, etc. is to be blamed due to these interruptions. This further calls for a reliable energy supply system with sufficient security and quality of supply. A summary of the power sector profile and detailed outage status in Ethiopia will be presented in the next chapter, which is then the case considered for this study.

In an attempt to address this problem, the subsequent sections are dedicated to elucidate and summing up the scope of the outage problem and presenting a summary of existing approaches and solutions. For a quick grasp, a brief electrical power system theoretical background is proposed to enable an easier understanding of outage susceptible parts and overall reliability theory. The breadth of the current state of outage on a global scale, its cost impacts, and broad implications are then detailed. Following this, a review of the various state-of-the-art solution options, particularly focusing on renewable sources in addressing the issues of reliability because of outages is presented. Finally, a literature survey on the sizing optimization of multi-source systems especially aiming to minimize outages and improve reliability is established.

The problem of grid interruption/outage and supply reliability

In modern economies, access to electricity and its continuity of supply to customers is imperative from both the electric utilities' and customers' point of view. Once a country has access to electricity, it has to be reliable and of greater quality with less interruptions of supply as possible. As to the former, global electricity access rose from around 83% in 2010 to 90% in 2019 [START_REF]Access to electricity (% of population) -Sub-Saharan Africa | Data[END_REF]. While the majority of the developed countries such as Europe, Central Asia, North America, and others having achieved full electricity access, this index remains very low for other parts of the world. For example, the Sub-Saharan Africa (SSA) region still has only 47% access in 2019, from a mere 34% in 2010. In another study, a closer look shows up countries with exceptionally acute electricity access rate problems (Nigeria, DR Congo, Ethiopia, Tanzania and Uganda) with a large proportion of people that do not have access [START_REF] Schwaegerl | Active Distributed Systems and Distributed Energy Resources[END_REF].

On the other hand, monotonous power interruptions and blackouts even in modern electrical power systems are customary throughout the world from which both the power utility operators and customers are suffering a lot. It is at this point where one can see the severe state of power supply in general, especially in these developing and lower-and middle-income countries such as the SSA. In fact, in these regions, even if a country has such an access rate, even with a lower figure, it again suffers due to power supply interruption and outagesan irony of power supply status which is detailed later in section I. 1.4. A World Bank survey showed that a typical firm in this region could experience at least 9 power outages in 2020, as compared to less than one outage in the European Union in every 2 years [START_REF]Power outages in firms in a typical month (number) | Data[END_REF]. This is where the reliability of the electricity supply is even more crucial, particularly for those with an already lower access rate and critically poor reliability level. Owing to this in recent economies, there are virtually no economic or any other activities in any country without energy services, i.e., predominantly, no activities to be carried out under power interruptions. Largely, grid outages are crucial with huge negative impacts on both consumers and suppliers.

For all this in a power grid, from the plant to the customer, distribution networks are the constituent elements of the power system, which make up the major outages and interrupting events worldwide. They account for more than 80% [START_REF] Kaduru | Distribution System Reliability with Distributed Generation Based on Customer Scattering[END_REF], [START_REF] Al-Muhaini | The reliability of power distribution systems as calculated using system theoretic concepts[END_REF] of customer reliability issues, and could usually even reach 90% [START_REF] Brown | [END_REF].

I.1.1. Electrical Power Systems: The basics with respect to outage

The electrical power systems hierarchical chain consists of the generation, transmission, distribution, and end-use consumers in the ladder. Their basic function is to generate sufficient power to fulfill customer load demand, convey this bulk-generated power over a long distance, and deliver this power to customers [START_REF] Brown | [END_REF]. Distribution systems are those responsible for delivering electricity from the high voltage (typically 35 to 400 kV, but even higher) long-running transmission and subtransmission lines to the customers (usually 120/240V). They can be classified into two levels based on the voltage ranges: primary (ranging from 600V to 35 kV) and secondary (customer-level voltages like 120/240V, 120/208V, or 277/480V) distribution circuits [START_REF] Short | Distribution Handbook[END_REF].

It is at this point where the secondary distribution circuit delivers power to customers from the distribution transformers. Of course, there could be variations in the voltage levels from one to another utility companies, and also based on configuration variations.

The distribution circuit can be composed of either overhead lines, underground cables, or the combination of both. It is the case in most distribution systems and utility operators like in Europe and North America.

In any way, electrical energy should be delivered to the customers with acceptable standards by ensuring the expected voltage level, with continuous supply, and apparently with low cost and high reliability, i.e., without interruption.

Distribution circuit configurations can typically be [START_REF] Brown | [END_REF], [START_REF] Short | Distribution Handbook[END_REF]:

 Primary or radial type: known for its easy configuration and generally low cost, whereby customers are connected to a single feeder (a power line emanating from substations in the medium voltage range). In this arrangement, when a fault occurs, it will leave all downstream customers without a power supply until the fault is cleared out. It is used where the load density is sparse like in rural applications.

 Loop or ring systems: With varieties of arrangements, it is where customers can optionally be supplied through available feeders in open or closed loop mode. These systems may have a single feeder with more than one source. These configurations can sometimes be considered as a variance of the radial type with a different operational arrangement.

 Network or mesh type: They have sufficiently redundant distribution system components for maximum reliability for areas with high load density, requiring higher reliability or critical load points. It is the most expensive arrangement by which load interruptions may occur only due to simultaneous fault occurrence of the different feeders supplying the networked distribution.

Broadly speaking, causes of interruption can be due to:  Weather (rain, wind, lightning, severe storms, snow, heat/wildfires, hurricane, thunderstorm, etc.),  Human actions or anthropogenic (maintenance and rehabilitation, operational faults/errors, traffic accidents, vandalism, theft, etc.),  Other natural causes (animals, trees, birds, etc.),  Component or equipment failures (transformers, overhead lines and cables, poles, circuit breakers, surge arresters, switches, disconnectors, insulators, etc.), and  Operation (overload, short circuit, etc.).

Whichever causes have instigated interruptions and are deteriorating the reliability of the distribution systems, an electric utility should invest in improving its service reliability to customers. Hence, the cost of interruption should first be known by utilities for optimal investment for reliability improvement. In other words, as shown in Figure I-1, the level of reliability to customers is acceptable when the cost of reliability improvement surpasses the cost of interruption to these customers. For this, utilities need to have accurate outage recording and survey techniques and systems to capture the reliability level and estimate associated costs. Important indicators for an interruption like interruption start time, restoration time, interruption cause, number of interrupted customers, type of customers, etc. are at least the very first parameters that need to be registered accurately [START_REF] Brown | [END_REF] for a good reliability study and investment by distribution operators for reliability improvement. Many utility providers and different authors have claimed that it is costly for a utility to have such a robust outage management system and collect as much detailed interruption information as possible. It is also eminent that the cost of interruption and reliability can typically be assessed and modeled using surveys, which of course also are difficult and costly. This data scarcity entails a huge challenge for comprehensive outage study and advancing an effective solution design for outage reduction and ensuring continuity of supply in such grid utility systems. In the operating zone of period one, the failure rate is relatively high due to design and manufacturing flaws, damages during transportation or installation, or could be due to lack of proper installation and use of the component. Whereas in the second period during the component's useful or normal operating life, it is assumed to have a constant failure rate which is random in nature. Finally, when the component has approached its operational lifetime, the failure rate is modeled by an exponential function which is due to the aging of the component [START_REF] Brown | [END_REF], [START_REF] Gönen | Electric Power Distribution System Engineering[END_REF].

As explained earlier, in studying a distribution system's reliability, the number of outages and outage durations during the study period, usually per year, are at least the initial key figures to consider.

I.1.2. What is the notion of interruption or outage?

In simple terms, though it has different meanings to many, it is the distribution service interruption for the concerned customers. Whereas reliable power, the state of averting interruption, describes the continuous supply of power without interruption to customers.

Interruption is the loss of power delivery service where a customer is totally de-energized.

Despite differences among different distribution operators, customer interruptions may be classified as momentary and sustained. Interruptions lasting from a few seconds to a few minutes, commonly 5 minutes, can be considered momentary and more than 5 minutes are sustained interruptions [START_REF][END_REF]. Moreover, interruptions could also have an extra feature of either being planned, unplanned or unscheduled; which are based on intentional disruption of the power supply according to a predefined time in case of planned interruptions and which otherwise is the case for unplanned. For the latter, they are usually unpredicted and unavoidable types of grid power outages. Component failures, operational faults, human errors, and other natural factors contribute to the unplanned outages [START_REF] Brown | [END_REF]. However, an intended power cut by utility grid operators for activities, such as to perform routine maintenance and operation, contributes to the planned outages.

In general, many authors have defined outage as the state of a distribution system component at which it is de-energized and is not available for service [START_REF] Brown | [END_REF], [START_REF] Gönen | Electric Power Distribution System Engineering[END_REF]. This may or may not cause service interruption based on the system's configuration. Then for the interruption as defined above, the total loss of power supply service to any connected customer is counted as an interruption.

However, in this dissertation, the general electricity supply service outage or interruption concept is considered and the terminologies are used interchangeably despite the delicate difference in their meaning. Alternately, the term blackout is also used to refer to the same state of dysfunction and cut off power. Moreover, as it has an interrupting event, one minute and above interruption duration is taken as an outage or interruption to be considered in a detailed system modeling. This can also be justified by the fact that in recent modern societies, customer electric-consuming devices and equipment have become very sensitive and cannot even tolerate a brief and very short outage or so [START_REF] Vieira Pombo | Multiobjective formulation of the integration of storage systems within distribution networks for improving reliability[END_REF]. Moreover, most industrial customers like process industries need highly stable and uninterruptible supply.

I.1.3. Brief overview of global power supply outage and related reliability

Due to the fact that most, if not all, economic and social well-being are extremely dependent on electricity supply, reliability is a major concern that needs to be addressed globally. In modeling and data presentation for the interruption, it is a common practice to categorize as planned and unplanned interruptions. In the developed region like Europe, the planned interruption is strictly to mean that the consumers should be notified in advance for which a planned or scheduled work on the power system shall then be executed [START_REF] Ceer | 6th CEER benchmarking report on the quality of electricity and gas supply[END_REF]. In the contrary, in developing countries, despite scheduling maintenance activities on the network, these scheduled outages are not communicated to customers. This entails a twofold challenge for the consumers due to the actual outage and lack of knowledge about its occurrence causing loss, which could otherwise be minimized.

The Council of European Energy Regulators (CEER) benchmarking report [START_REF] Ceer | 6th CEER benchmarking report on the quality of electricity and gas supply[END_REF] encompassing the 28 EU states and several other countries shows that the EU is characterized by high reliability, but can still be facing some interruptions. For the planned interruptions, utilities in all the countries have to inform their consumers, albeit the variation in the notice time in advance for informing the interruption schedule. In some countries like Denmark and Greece, the consumers should be notified 48 hours in advance according to the report. In addition, even some further strict rules are in place, like in Belgium, the interruption should get customers' approval. The report also showed that, during the wider reporting period 2002 -2014, countries with resilient power networks with less than 50 minutes of interruption duration per year for a customer were Denmark, Germany, Luxembourg, Netherlands and Switzerland. Similarly, countries such as Austria, France, Spain, Great Britain, etc. experienced 50 to 200 minutes interruptions per year.

Other networks like Latvia, Croatia and Romania had close to 1300, 1200, and 1100 minutes interruption occurrences in 2010, 2006, and 2008 respectively. In this same report with regard to the interruption numbers per year per customer, Denmark, Germany, Great Britain, Luxembourg, Netherlands, and Switzerland had the most stable network with less than just one interruption per year. On the other end, Croatia had the highest interruptions but with an improving downward trend from 8.5 per year in 2008 to 6 per year in 2014.

Similarly, the average interruption duration per year for a customer in the U.S. in 2016 and 2017 were about 4 h and 8 h (470 minutes) including major interrupting events.

Additionally, in 2017 a customer faced 1.4 interruptions per year [START_REF]Average U.S. electricity customer interruptions totaled nearly 8 hours in 2017 -Today in Energy -U.S. Energy Information Administration (EIA)[END_REF].

Conversely, recurrent power interruption from distribution networks is overwhelmingly becoming an everyday problem in developing countries [START_REF]Enterprise Surveys -Ethiopia 2015 Country Profile[END_REF]. The outage of power or interruption is more serious in these regions as reported in [START_REF] Farquharson | Sustainability implications of electricity outages in sub-Saharan Africa[END_REF], due to the fact that most interruption occurs out of the control of utility operators. Likewise, particularly these distribution systems are weak due to the common use of radial grid architecture and aged components. In sub-Saharan African countries, the power outage could range from 50 hours in South Africa to 4,600 hours per year in Nigeria. It is shown that most sub-Saharan countries experience 100's of hours of interruption every year.

Contrary to the weak electrical networks in developing countries, outages in the highly stable networks of the developed world, even when happened, are usually related to extreme weather and similar causes. As highlighted from the outset earlier, a summary of the level of outages and associated impacts in particular for developing countries is presented in the next section.

I.1.4. Power supply interruption status quo: survey and implications

It is an agreed fact that a reliable power supply is the key to the economic and human development of a nation. As stated in the UN sustainable development goals (SDGs), it is imperative to achieve this purpose by explicitly pinpointing to "ensure access to affordable, reliable, sustainable and modern energy for all" in the agenda [START_REF]Ensure access to affordable, reliable, sustainable and modern energy for all[END_REF].

Even with the lower overall electricity access rate globally, the unreliability of its The challenges from outages span from the economic cost directly related to the outage, to increased fuel cost, emissions as well as the rise in fossil fuel use and associated negative impacts. Authors in [START_REF] Farquharson | Sustainability implications of electricity outages in sub-Saharan Africa[END_REF] showed the challenges due to the use of diesel backup generators to overcome outages in SSA countries. In terms of consumer costs, their analysis identifies the huge gap between the existing grid electricity cost and diesel backup cost.

For instance, Zambia has an average backup generation fuel cost of 0.59 $/kWh (without the O&M cost) as compared to the 0.06 $/kWh grid electricity cost. The number of outages in a typical month was from about one outage in South Africa to 33 in Nigeria. Besides, the outage duration was ranging from around 1.8 h in Senegal to 13.5 h in Angola in the same typical month period. The region encountered, even with such wide disparity, roughly an average around of 785 h of outages in a year.

In a similar literature, the eminent influence and impact of economic loss and business reliance from power outages are pointed out for African firms [START_REF] Cole | Power outages and firm performance in Sub-Saharan Africa[END_REF]. They also found out that there is a higher tendency of backup diesel generator ownership to overcome outages. While it varies widely among countries, the continent has an average backup diesel generator ownership level of around 52%, which again gives the picture of the selfgenerated electricity share. For example, Mozambique possesses the lower generator ownership rate at only 10%, while this number is more than 70% in Angola and Burundi.

As to the continental share of electricity coming from self-generation, it stands at about 26%. This clearly shows one of the challenges and negative impacts of outages on endusers in every economic sector when they are pressed to look for alternative sources of power supply. It can be noted that the self-generation electricity share depends on the total duration of outages and the economic ability to own a generator, such as with the GDP per capita [START_REF] Cole | Power outages and firm performance in Sub-Saharan Africa[END_REF], [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF].

The measure of a reliable supply of power has a substantially different interpretation in a well-developed and modern grid network and for those with a weak network. For example, the reliability measure on modern systems such as in the U.S. is the count of 'nines' after a decimal point in the metric which gives the level of reliability. Utilities provide electricity with almost 100 percent reliability (such as with one 'nine' as in 99.9%), which portrays a downtime of approximately 8.75 h annually [START_REF] Balducci | Electrical Power Interruption Cost Estimates for Individual Industries, Sectors, and U.S. Economy[END_REF]. This gives a much more impression as compared to the various numbers of outages in fragile power infrastructures discussed in the previous sections.

Access to reliable power, as the most fundamental infrastructure in every country's economy, power users, or people in general, are affected either directly or indirectly by outages, as depicted in Figure I-4. Likewise, the economic costs and implications of outage and on which parts of the economy can be summarized as in Figure I-5. Despite the fact that the cost components differ from the utility and end-users' perspective, which we briefly described the balance between them in section I.1.1, we actually focus on the end-users perspective. The impact ranges from the direct cost to the indirect costs, which the latter are inherently difficult to quantify easily, and to the general environment, as adapted from [START_REF] Balducci | Electrical Power Interruption Cost Estimates for Individual Industries, Sectors, and U.S. Economy[END_REF], [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF], [START_REF] Šljivac | Survey on Customer Power Supply Interruption Costs and Calculation of Expected Customer Damages[END_REF].

Outages

Direct impact

End-users

Figure I-5 Some of the direct and indirect negative implications due to outages on the different economic sectors and end-users (source: adapted from [3], [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF], [START_REF] Šljivac | Survey on Customer Power Supply Interruption Costs and Calculation of Expected Customer Damages[END_REF]).

Rentschler et al [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF], in their survey of infrastructures disruptions on firms from more than 130 low-and middle-income countries, elaborated on the direct monetary cost consequence of outages on firms (see Figure I-6). It can be seen that the outage duration (OD) in some countries such as Nigeria and Central African Republic reaches about 250 h annually. Furthermore, CA Republic and Pakistan experience a hefty amount of sales loss due to power blackouts, which averages close to 25%. In spite of the unrelated trend in the results, it is implicitly well-understood that longer outage durations result in higher losses.

Here, we could imagine other factors like the nature of business and economic differences among surveyed countries as contributing points. Assessments of outage costs and overall impacts on the various consumer categories such as residential, commercial, and industrial in several countries are carried out. In these interruption cost assessments, customer surveys or interruption customer cost estimations and models are the basic approaches utilized as a valuation methodology. In this regard, endeavors to quantify outages costs and impacts can be found in several works of literature such as: for the various sectors mainly residential, commercial, public, and industrial sectors collectively in the U.S. [START_REF] Lacommare | Improving the estimated cost of sustained power interruptions to electricity customers[END_REF], Greece [START_REF] Dialynas | Interruption cost analysis for the electrical power customers in Greece[END_REF], Portugal [START_REF] Castro | The cost of electricity interruptions in Portugal: Valuing lost load by applying the production-function approach[END_REF], Spain [START_REF] Linares | The costs of electricity interruptions in Spain: Are we sending the right signals?[END_REF], and Austria [START_REF] Reichl | The value of supply security: The costs of power outages to Austrian households, firms and the public sector[END_REF]. Similarly, the interruption impact with a particular focus on one consumer group, for industrial sector in [START_REF] Panya | Economic impact of power outage in Thailand: Industry perspectives[END_REF]- [START_REF] Pasha | The economic cost of power outages in the industrial sector of Pakistan[END_REF] and the residential sector [START_REF] Woo | Average residential outage cost estimates for the lower 48 states in the US[END_REF], [START_REF] Shaalan | Impact of Power Outages on Residential Consumers in Riyadh City[END_REF] in different countries have been studied. On the other hand, for sectors that appear to be more sensitive to power outages such as the healthcare system and facilities, authors tried to study its harmful consequences in the sector [START_REF] Koroglu | Effect of power outages on the use of maternal health services: Evidence from Maharashtra, India[END_REF], [START_REF] Dominianni | Health impacts of citywide and localized power outages in New York City[END_REF]. Despite their scarce availability, related impact assessments on Internet infrastructure [START_REF] Cowie | Impact of the 2003 Blackouts on Internet Communications: Preliminary Report[END_REF], indirect associated costs to outage [START_REF] Hu | Analyzing regional economic impact and resilience: A case study on electricity outages caused by the 2008 snowstorms in southern China[END_REF], and continental average such as for Africa [START_REF] Nduhuura | Power Outages in Africa -An Assessment Based on Regional Power Pools[END_REF] are some of the literature we have found in our review. In summary, the aftermath of interruptions highly differs with consumer types such as for hospitals or residences, time of outage occurrence, and outage durations [START_REF] Shivakumar | Estimating the socio-economic costs of electricity supply interruptions[END_REF]. For instance, direct costs such as investing in diesel backup generators lead consumers to pay as much as 40% of their energy bills [START_REF] Natarajan | Statistical Analysis of Cost of Energy due to Electricity Outages in Developing Countries[END_REF]. Moreover, even if not vast in the literature as compared to the developed region, a direct outage cost assessment based on the willingness-to-pay (WTP) index in developing countries is conducted in Ethiopia as a case [START_REF] Meles | Impact of power outages on households in developing countries: Evidence from Ethiopia[END_REF]. The WTP for an avoided 1-hour outage for households is estimated at $0.03, in addition to their electricity bill. This figure is nearly equal to the current kWh price. In a similar study on household consumers in Ghana, the suggested WTP index is $0.09 per kWh served [START_REF] Amoah | Electricity outages in Ghana: Are contingent valuation estimates valid?[END_REF]. In [START_REF] Carlsson | Cost of power outages for manufacturing firms in Ethiopia: A stated preference study[END_REF], it is also examined to find out the WTP in manufacturing firms' consumer sector in Ethiopia. It results in $0.01 and $0.014 for a reduced one-unit number of outages and 1-hour outage duration respectively.

The previous sections summarize the detrimental impacts of outages on different sectors and countries. Outage costs and impacts are determined either by the direct monetary estimates of outage cost or by the values of lost load (for each unsupplied energy unit). It is worth noting that intraday fluctuations, sectoral differences, and characteristics of the outage events are some of the important variables to quantify outage costs.

Apparently, whichever method is used to estimate the outage cost, it leads to a huge economic loss for both the consumers and utility companies.

Recalling the causes of power interruption, in robust and modern grids are usually severe weather impacts, environmental hazards such as untrimmed trees, and integration of renewables in the generation mix, which have inherent variability and which eventually could lead to issues of power disruption.

Nowadays every economic activity and value-chains are greatly interdependent and complicated which again highly rely on electricity. Major technological advances and shifts to electricity, such as the transport sector, heating, and cooling, further heighten the adverse impacts caused by outages. As outage has a high direct and indirect economic impact on customers and on a country at large, different researchers have examined different solutions. As explained earlier, the cost of interruptions also highly differs from the power supplying-utility and consumers' perspective (see Figure I-1). In any case, the cost is usually treated as direct (such as economic and social) and indirect costs caused by the interruption [START_REF]Enterprise Surveys -Ethiopia 2015 Country Profile[END_REF], [START_REF] Masrur | Optimal and economic operation of microgrids to leverage resilience benefits during grid outages[END_REF]. It must be recalled that the indirect costs, as its name, are generally not trivial to quantify and account for.

The first step to improvnig power supply reliability is the study and detailed assessment of such reliability issues which is essential for the judicious evaluation of mitigation solutions and investment decisions. Studies are made on reliability evaluation with respect to incessantly growing and changing power demands, supply imbalances, and weather-caused interruptions. Before diving into different available solutions, let us once more recap the common outage causes. Apart from the severe weather effect, causes of outages could include operational (overvoltage, short circuit, maintenance, etc.), aging and component failures, human actions, weather, and natural causes, inadequate maintenance, other natural factors, and so on.

Under all these outage costs and impacts on the economic system and a nation as a whole, with a heightening concern and an already established understanding of power as a lifeline for one's economy, it is eminent to have feasible solutions to deal with outages.

The power interruption has adverse effects on customers' productivity, daily routine activities, and on the overall economic aspects. All that being the case, many attempts have been made to mitigate the perturbing power supply situation and hence to improve its reliability.

I.2. State-of-the-art I.2.1. Current solutions and approaches for reducing outages

In order to respond to and address the challenges of power outages and their undesirable cost implications, the existing approaches are summarized from the literature.

Apart from a direct deployment/investment in diesel backup systems to meet power demands during outages, a review of the different solutions to date shows a range of propositions for outage reduction and better reliability according to different contexts.

Methodically in our view, the techniques and approaches with the aim of outage reduction or reliability improvement for customers can be broadly grouped into three issues grid modernization, outage prediction techniques, and distributed generations, see Figure

I-7.
In modernizing the grid, issues to be dealt with include reliability, resilience, and security of grid infrastructures and supply for improvement. Likewise, some of the purposes, contributing factors, and characteristics to consider in proposing solutions for outage mitigations are listed as indicated in the figure.

Several grid modernization techniques for reliability improvement are explored such as the placement of disconnect switches [START_REF] Dat | Evaluation and Improvement Distribution actual network Reliability using ETAP[END_REF], automated operation of sectionalizing circuit breakers [START_REF] Fomin | Improvement of power supply reliability by means of remote control of the automatic repeated switchingon of sectionalizing circuit-breakers[END_REF], a review underscores network reconfiguration like with distribution automation [START_REF] Sultana | Review on reliability improvement and power loss reduction in distribution system via network reconfiguration[END_REF], reliability enhancement and loss reduction through radial grid reconfiguration with Distributed Generations (DGs) [START_REF] Pavani | Reconfiguration of radial distribution networks with distributed generation for reliability improvement and loss minimization[END_REF]- [START_REF] Pombo | A multiobjective placement of switching devices in distribution networks incorporating distributed energy resources[END_REF]. Further studies in this category of reliability improvement are also available. Here when saying reconfiguration, it is where the grid components' operational topology, sequence, and states are manipulated to achieve the intended objective, reliability in this case. For instance, the objective in [START_REF] Tsai | Network reconfiguration to enhance reliability of electric distribution systems[END_REF] was to minimize the system average interruption duration and frequency. A statistical probability concept is implemented for a transmission and distribution system reliability evaluation and enhancement [START_REF] Zheng | Investigation of Power System Blackouts and Reliability Improvement for Power Distribution Systems[END_REF]. Monte Carlo simulation of reliability indices (system average interruption frequency index and system average interruption duration index -SAIFI, SAIDI) based on the system topology is used, and adding automatic switches for its enhancement is examined.

Proposing the placement of reclosers and switches optimally has been explored for reliability improvement [START_REF] Junior | Optimal allocation of maneuver devices in distribution networks for reliability improvement[END_REF]- [START_REF] Rahmawati | Optimal switch placement in radial distribution system based on reliability worth analysis[END_REF]. In [START_REF] Karpov | Integral indicators improvement (SAIFI) of power supply reliability in electric distribution systems based on reclosers placement optimization[END_REF], a genetic algorithm technique search for the

Outage reductionreliability improvement

Grid modernization: reconfiguration, automation, switches/disconnects placement, etc.

-Reliability -Resilience -Security and safety.

Prediction techniques:

-Weather-based -Components failure -Maintenance planning, -Failure response and outage management.

Distributed generations:

-Sizing and operation/management -Location.

optimal location for the reduction of the number of outages in a distribution is suggested.

Reliability improvement by way of the reduction in the system's average interruption frequency has been reported. With a reliability improvement objective, authors of [START_REF] Junior | Optimal allocation of maneuver devices in distribution networks for reliability improvement[END_REF], [START_REF] Rahmawati | Optimal switch placement in radial distribution system based on reliability worth analysis[END_REF] determined the ideal location of distribution network switches to minimize the reliability cost, where the latter implemented a metaheuristic technique.

In a different manner, various studies in the literature, such as in [START_REF] Yue | A Bayesian approach-based outage prediction in electric utility systems using radar measurement data[END_REF]- [START_REF] Eskandarpour | Machine Learning Based Power Grid Outage Prediction in Response to Extreme Events[END_REF], developed system outage prediction to estimate the probable operational or damaged states of components under weather-caused interruptions. This is because extreme and severe weather impacts such as hurricanes, storms, floods, tornados, and so on are the predominant causes of distribution system outages in countries such as the U.S. having advanced systems [START_REF] Chen | Fuzzy Logic Approach to Predictive Risk Analysis in Distribution Outage Management[END_REF], [START_REF] Eskandarpour | Improving power grid resilience through predictive outage estimation[END_REF]. The authors reported that the input data for the predictive models are principally the weather information, historical extreme weathers, other environmental parameters, or historical outages with their associated factors, such as seasonal birds. Some of the recommended purposes of the prediction include either facilitating grid restoration for critical customers, locating fault areas, or planning maintenance. The authors essentially try to predict the probability of grid components faults in response to extreme weather like hurricanes and strong thunderstorms, based on historical data of the components and geospatial characteristics of the hurricane. When possible, it is also attempted to correlate those two features of components failures and weather data. It is also reported that the prediction can help in informed energy resources scheduling and outage handling in severe weather-causing outages. In addition, authors in [START_REF] Watson | Improved quantitative prediction of power outages caused by extreme weather events[END_REF] proposed data-intensive weatherrelated outage prediction, by which the data include the extreme weather information (such as storms), environmental variables, and grid infrastructure data. Statistical processing was applied to develop a machine learning prediction model.

On the other hand, DGs with or without Energy Storage Systems (ESS) have the capability to attenuate utility level reliability issues ranging from cases of power quality to sufficiently supplying power in cases of an outage, as is the main consideration in this work.

Existing renewable energy-based DG solutions for reliability improvement

To date, a lot of research and academic studies have been conducted for reliability metrics improvement under different circumstances, of which DGs have attracted research focus since long ago.

Recently, introducing DGs into distribution systems for addressing several objectives such as voltage variation, power loss, reliability, system flexibility, etc. has essentially increased their share of power supply [START_REF] Azizivahed | A hybrid evolutionary algorithm for secure multi-objective distribution feeder reconfiguration[END_REF], [START_REF] Singh | A survey on enhancement of power system performances by optimally placed DG in distribution networks[END_REF]. They are usually relatively small capacity power generation units installed in the power consumers' vicinity or near to load points in distribution networks. The contribution of DGs integrated to distribution system reliability is demonstrated in [START_REF] Falaghi | Distributed Generation Impacts on Electric Distribution Systems Reliability: Sensitivity Analysis[END_REF]- [START_REF] Ahmad | Impact of Distributed Generation on the Reliability of Local Distribution System[END_REF], while using system-level reliability indices such as SAIDI and SAIFI to quantify the performance. The studies in [START_REF] Falaghi | Distributed Generation Impacts on Electric Distribution Systems Reliability: Sensitivity Analysis[END_REF], [START_REF] Ahmad | Impact of Distributed Generation on the Reliability of Local Distribution System[END_REF] further tried to examine the location, capacity, and number of DGs to install and check the resulting improvement.

Distribution system data like feeder length, number of customers supplied, average load, etc. are required to calculate the system reliability indices. Furthermore, integrating renewable energy sources can be done to answer many different issues, such as reliability improvement, reduction of utility energy bills and CO2 emission, peak demand shaving, grid voltage stability, and supply of electric power for offgrid areas. The effects of DGs size and location on distribution systems reliability are presented in [START_REF] Falaghi | Distributed Generation Impacts on Electric Distribution Systems Reliability: Sensitivity Analysis[END_REF]- [START_REF] Ahmad | Impact of Distributed Generation on the Reliability of Local Distribution System[END_REF].

The reliability study principally deals with the interruption duration and the number of outages of distribution systems in the main grid. Hence, multi-source distributed generations comprising renewable sources such as solar, wind, etc. as well as the inclusion of diesel generators for many applications has been the growing knowledge base and main area of research in this application. But as these renewables are not available all the time, due to the large part of uncertainty on the outage instant and duration, these multi-sources DGs include, in addition to renewable sources, auxiliary power units and storage systems solutions.

The multi-sources distributed generation is then another step or way of improving the reliability of the overall system. Before providing an extended review of renewable DGs applying to outages, it is proposed to recap and demonstrate the concept of multi-source systems.

Multi-source power systems

The use of different energy technologies from renewable resources (solar, wind, marine power, hydro, biomass, fuel cell, etc.), diesel generators, and energy storage systems (batteries, supercapacitors, etc.) has been explored extensively for vast application schemes, one of which is to aid existing grid supply in case of shortage and sparsely to grid unavailability or outages. With this in mind, the multi-source system with rigorous selection to fit the intended purpose is formulated. Figure I-8 illustrates a multi-sources power system, which includes grid, (PV, wind turbine), diesel groups, storage, and power conditioning components to supply a load. On this figure, note that the outage is also integrated at the grid side supply level. Furthermore, the development of multi-source systems mainly incorporating renewable energy sources to warrant the supply of power to consumers during outages requires further reserve supply for continuity. In this aspect, storage solutions are notably the pivotal options, despite attempts to include diesel generators in cases of low production and outages. The advances in their technology, reduction in costs, and avoiding issues of environment and fuel use, as in the case of diesel, storage solutions are the primary constituents of multi-source systems. They not only ensure reliable supply but also provide more flexibility in energy management.

For instance, solutions are suggested such as in a large grid-utility scale, for enhanced grid integration due to the solar intermittency and resilient supporting of peak loads [START_REF] Yang | Florida State University Libraries Optimization of Battery Energy Storage Systems for PV Grid Integration Based on Sizing Strategy[END_REF].

In the same way, for offsetting grid electricity purchase and carbon reduction, a PV-battery system is proposed for a commercial building application in the UK [START_REF] Mariaud | Integrated optimisation of photovoltaic and battery storage systems for UK commercial buildings[END_REF].

2 www.homerenergy.com

The amount of energy not supplied in the distribution network is used to assess the reliability and recommend the use of a PV-diesel system [START_REF] Myint | Integrating the PV-Diesel Hybrid System for Reliability Improvement in Distribution System[END_REF]. A specified penetration level of these distributed energy resources is used to optimally size their capacities to be connected to the distribution grid. Moreover, the reliability is based on the annual failure rates of the different load points in the network. A similar solution for conventional distribution networks with outages is proposed in [START_REF] Adefarati | The Impacts of PV-Wind-Diesel-Electric Storage Hybrid System on the Reliability of a Power System[END_REF], where the authors test the improvement impacts of various configurations with PV, wind turbine (WT), diesel generators, and various energy storage technologies. They concluded that the PV-WT-ES configuration leads to higher improvement.

A hybrid PV-diesel and PV-only system with a pre-set unmet load for a village in Uganda is proposed in a scheduled outage context [START_REF] Murphy | Analysis of the cost of reliable electricity: A new method for analyzing grid connected solar, diesel and hybrid distributed electricity systems considering an unreliable electric grid, with examples in Uganda[END_REF]. The authors simulated exponentially random grid outages based on global data of uptime and downtime, which is considered uniform for weekdays and weekends. A diesel-only system for improved reliability is reported, with a projected change of cost in the future and a reduction in GHG emissions, they concluded the optimality of a PV-diesel-battery system.

The power supply for only critical loads of industrial parks in Ethiopia connected to the unreliable grid is undertaken using a grid-tied PV-wind-diesel system [START_REF] Azerefegn | Techno-economic analysis of grid-integrated PV/wind systems for electricity reliability enhancement in Ethiopian industrial park[END_REF]. As in [START_REF] Murphy | Analysis of the cost of reliable electricity: A new method for analyzing grid connected solar, diesel and hybrid distributed electricity systems considering an unreliable electric grid, with examples in Uganda[END_REF],

the authors modeled the unreliable grid using an annual average number of outages and duration from a uniform random generator.

Some of the other solutions also focus primarily on replacing an existing diesel generator formerly acting as an outage solution. Solar PV system with battery storage to reduce or eliminate the use of diesel generators in Nigeria is proposed in [START_REF] Babajide | Solar PV systems to eliminate or reduce the use of diesel generators at no additional cost: A case study of Lagos, Nigeria[END_REF]. The authors promote the context of the country, hardly hit by recurrent power outages. It is shown that nearly 4.5 million people, close to 12% of households, run diesel generators on top of the already fossil fuel dominated national power system which provides 82% of the generation.

In the contrary, a scenario case analysis in [START_REF] Masrur | Optimal and economic operation of microgrids to leverage resilience benefits during grid outages[END_REF] examined the use of a PV-diesel-battery microgrid solution for a one-time grid outage case (with varying outage durations of 6 h, 12 h, and 24 h) throughout the life of the project.

In [START_REF] Yahaya | Reliability assessment of load points including solar and wind DGs[END_REF], the effect of integrating solar, wind, and combined DG systems to improve power system failure times and frequency is proposed. The authors tested the proposed systems on loads of three consumer types (residential, commercial, and industrial) from Roy Billinton Test Systems (RBTS). With DGs size equal to the peak loads, managed to achieve a reduction of failure duration and also the Energy Not Supplied (ENS) for a total system outage. It seems imperative that the load variation from the peak will highly affect the performance and sizing of this system.

Likewise, in [START_REF] Khoury | Optimal sizing of a residential PV-battery backup for an intermittent primary energy source under realistic constraints[END_REF], a residential PV-battery backup system is presented for reliability improvement in a weak grid supply facing scheduled grid outages. A predetermined outage in terms of the fraction of the total simulation is investigated. Two conditions, harsh and moderate, are defined with a corresponding 52% and 34% outage time from the total simulation period with a 1-hour time step. A rural household in Lebanon is studied, with such a blackout schedule that seems to be rather severe, and also perhaps could lead to a slightly oversized system. A hybrid PV-wind-battery system can also help to support the deficiency of power due to seasonal effects [START_REF] Masih | Optimal Economical Sizing of Grid-Connected Hybrid Renewable Energy System[END_REF]. A variance in the reliability measure is the system's momentary interruption of temporary ones. In [START_REF] Vieira Pombo | Multiobjective formulation of the integration of storage systems within distribution networks for improving reliability[END_REF] it is proposed to integrate storage systems for reducing these temporary interruptions with optimal battery sizes and locations on the network. They also considered minimizing the system's cost by using a genetic algorithm.

As an additional review, a summary of various resource combinations to provide power and improve the challenges of outages is presented in Table I-1, highlighting the type of technology, capacity, and associated cost. With the objective to reduce the outage from the customer perspective and the case study considered, solar PV with battery storage solutions proved to be used in several circumstances among others. The application of other resources such as wind and bioenergy-based sources like biogas is very customer-and location-specific and their usage is rare due to low availability and operation issues. Apart from diesel integration as a main source in proposed solutions as discussed early, some other studies also attempt to include diesel groups with an aim of flexibility, sometimes meant to replace storage devices. Without considering the disadvantages like gas emissions, fuel use impacts, demanding maintenance, noise pollution, and initial cost, the use of diesel generators in multi-source hybrid systems could eventually be for supply stability and to cancel the stochastic behavior of renewable resources, assisting demand response and sometimes to reaching optimal technology mix.

The inclusion and integration of storage systems such as battery storage in renewable energy applications like in solar photovoltaics as mentioned earlier play an important role.

Extending this review, the literatures highlight the use of grid-connected renewable sources and battery storage devices integration to maximize grid reliability, and sometimes by incorporating conventional diesel groups, (see Table I-1).

Optimal energy storage systems are examined for improving distribution network reliability in reducing the unsupplied energy for a radial distribution network [START_REF] Saboori | Reliability improvement in radial electrical distribution network by optimal planning of energy storage systems[END_REF].

Moreover, a reliable power supply for a scheduled grid outage mitigation by a residential PV-battery system is examined [START_REF] Alramlawi | Optimal operation of hybrid PV-battery system considering grid scheduled blackouts and battery lifetime[END_REF]. They attempted to reduce a blackout caused by the supply capacity shortage and thus meet the demand where the customers are facing to a scheduled outage and aware of it a priori.

A known time of outage in a weak grid is supplemented with a PV-battery microgrid for enhancement of the reliability evaluated from the energy not supplied (ENS) index [START_REF] Ecike | Using microgrids featuring PV panels and batteries connected to the grid to improve the reliability of a low-voltage feeder in Kinshasa[END_REF].

The loss of power in these times of outage, during which peak loads are observed, is shaved by the microgrid including the battery with specified autonomy to make the supply available for the consumers in the study. The considered network consists of households, businesses, and communal lighting. The work in [START_REF] Colmenar-Santos | Microgrids as a mechanism for improving energy resilience during grid outages : A post COVID-19 case study for hospitals[END_REF] assesses the suitability of PV, battery storage, and diesel generator microgrid applied for one of the most critical consumer groups, a hospital facility in the US. The authors introduced a 24h outage on a month of lower irradiation and a day with a higher load. They supported the solution with its NPC, which is able to sustain the outage.

In [START_REF] Sanni | Analysis of backup power supply for unreliable grid using hybrid solar PV/diesel/biogas system[END_REF], a load demand for an abattoir in Nigeria with an assumption of an average 7 h daily grid availability, a grid-PV-biogas, and grid-PV-diesel options are investigated.

Both configurations considered battery storage. The unreliable grid is able to supply only 13% and 12% for each of these cases respectively. Their results show that the PV-biogas system makes it possible to reduce energy cost and emissions. An existing PV-diesel micro-grid demonstrated for a community application in Tanzania, [START_REF] Bastholm | Techno-economic study of the impact of blackouts on the viability of connecting an off-grid PV-diesel hybrid system in Tanzania to the national power grid[END_REF], an Egyptian PV-wind-diesel residential application [START_REF] Elkadeem | Optimal Design of Low-cost and Reliable Hybrid Renewable Energy System Considering Grid Blackouts[END_REF], a grid-connected PV-battery system sized using the HOMER tool (a commercially available hybrid power system optimization model) for a household in Pakistan [START_REF] Rehman | Feasibility Study of a Grid-Tied Photovoltaic System for Household in Pakistan: Considering an Unreliable Electric Grid[END_REF] and a hybrid PV-WT-fuel cell system for a resort application for an unreliable grid in Egypt by a PSO (Particle Swarm Optimization) and Firefly/harmony search techniques [START_REF] Samy | Optimal economic study of hybrid PVwind-fuel cell system integrated to unreliable electric utility using hybrid search optimization technique[END_REF] in case of blackouts, are among the rare propositions that are applied to developing countries. Solar PV-battery systems experienced numerous application schemes and demonstrated success stories, yet their prospect for stochastic blackouts in weak grids is very rarely studied. This envisages the need to give much attention to their techno-economic applicability in the area, i.e., to be part of the solution in tackling erratic power outages.

Based on the disparities in the composition of the hybrid systems, the energy costs, and the power demands (see Table I-1), it seems rational to link each work to the nature of the considered outages, how these outages are integrated into the energy management and perhaps what are the proposed economic assumptions.

Considering the reviewed works in Table I-1 to illustrate more, the proposed objectives or constraints, the proposed solutions simulation consider various parameters and aspects, hence obviously shall bear deviations due to those aspects such as:

o Consideration of deferrable loads, the amount of energy drawn from or sent to the grid, and associated particular cost [START_REF] Rehman | Feasibility Study of a Grid-Tied Photovoltaic System for Household in Pakistan: Considering an Unreliable Electric Grid[END_REF]; the presence of excess generation (if not loss of load) [START_REF] Bastholm | Techno-economic study of the impact of blackouts on the viability of connecting an off-grid PV-diesel hybrid system in Tanzania to the national power grid[END_REF], [START_REF] Elkadeem | Optimal Design of Low-cost and Reliable Hybrid Renewable Energy System Considering Grid Blackouts[END_REF];

o Inclusion of different energy management and control schemes [START_REF] Alramlawi | Optimal operation of hybrid PV-battery system considering grid scheduled blackouts and battery lifetime[END_REF];

o Different assumptions/values such as for interest rate, and other economic factors (eg. bank loan/other economic details) [START_REF] Khoury | Optimal sizing of a residential PV-battery backup for an intermittent primary energy source under realistic constraints[END_REF]; different project lifetime (20, 25 yrs), cost assumptions (for component, O&M, replacement); consideration of salvage value [START_REF] Samy | Optimal economic study of hybrid PVwind-fuel cell system integrated to unreliable electric utility using hybrid search optimization technique[END_REF];

o Considering gas emissions [START_REF] Sanni | Analysis of backup power supply for unreliable grid using hybrid solar PV/diesel/biogas system[END_REF], [START_REF] Elkadeem | Optimal Design of Low-cost and Reliable Hybrid Renewable Energy System Considering Grid Blackouts[END_REF];

o Inclusion of power conditioning components (eg. converter, etc.) [START_REF] Murphy | Analysis of the cost of reliable electricity: A new method for analyzing grid connected solar, diesel and hybrid distributed electricity systems considering an unreliable electric grid, with examples in Uganda[END_REF], [START_REF] Khoury | Optimal sizing of a residential PV-battery backup for an intermittent primary energy source under realistic constraints[END_REF]; etc.

Faced with the severity of outages in developing countries and the different solutions in the literature, it can be concluded that outage occurrences are random, unpredictable, and stochastic, which requires to actually include it in multi-source systems solutions and outage modeling. In this regard, apart from the previously described heterogeneity in the literature, this review demonstrates that meager attempts are available to include outage characteristics which are commonly based on generic assumptions from data values. This inevitably limits the ability of methods in considering the real nature of outages which is explicitly stochastic, hence, potentially resulting in suboptimal solutions.

It is possible to summarize the challenges and issues of outages in weak grids, which intrinsically is the case in developing countries.

 Firstly, the outages in weak networks are not necessarily linked to the weather. The main causes are usually shortage to supply the increasing demand due to factors such as economic developments and societal advancements as well as urbanization.

Mostly the distribution network components are very old and lack sufficient maintenance as well. Other factors such as vandalism, accidents, and illegal power connections also contribute to outage incidents. The nature of outages is then very erratic and large durations of outage are very common in the aforementioned countries' context. What is more difficult is also these outages are forced ones in which they are usually out of the control of utility operators.

 The reason being, reliability improvement solutions such as the prediction methods are not suitable. Then for a particular application in emerging countries, considering stochastic modeling for the grid is certainly a good choice.

 Regarding the network architectures, they are predominantly composed of overhead lines with little progress in deploying underground cables for the major distribution circuits. Additionally, what is more challenging and make them prone to interruptions is that, their topology is mostly radial type, which lacks sufficient redundancy. One of the shortcomings is the failure of a single component would lead power supply cut to all customers downstream of the failed component. In such events, there may not be a secondary supply feeder and necessary isolating switches to try to power up, at least, a certain group of customers.

 For a detailed study of network component-level solutions, it is simple to understand that this level of operation and control, data logging, and monitoring is, first, costly to implement for developing countries. Secondly, the technology unavailability poses further difficulty. This necessitates a profound infrastructure and study on the existing situation of distribution grid networks and to be able to establish a registry of such exhaustive datasets to further come up with possible solutions.

 Even with the technological advancements and a huge cost decrement, options such as solar PV are not well studied and recommended for outage remedies in countries with weak networks. Hence, still results in recurrent interruptions and power shedding. Moreover, in the scant available literature which considers PV and other sources, there are again very few studies evaluating the performance of the proposed solutions on a realistic grid to measure the effectiveness facing to uncertainties.

Such studies would make it possible to model such a stochastic grid to characterize it probabilistically, and also to evaluate how sufficiently the solution supplies power without interruption and validate its adequacy.

 Finally, this review also highlights there is a dire lack of attempts to implement real stochastic outage data in solutions meant to be for outage alleviation. The modest works so far dealing with outages use average values from historical data and hypothetical outage profiles, which compromise aspects of outages such as the real stochastic nature, existence of trends, cyclic variations, etc.

With all the real context of outages in developing countries, the different solutions based on multi-source systems and a considered case having a certain outage and climatic characteristics, it is now important to define the choice of components for our study. Based on the various works discussed, the next sections deal with this subject followed by an approach for the proper sizing via the evaluation of the best balance of sizing metrics.

Renewables for outages: Selection of the multi-source components

Resource availability for the considered case, technical feasibility and practicality of The use of diesel groups could have a comparably equivalent cost impact as the outage [START_REF] Farquharson | Sustainability implications of electricity outages in sub-Saharan Africa[END_REF], [START_REF] Kupzig | Power Outages , Firm Productivity , and Generator Ownership in East Africa[END_REF] and their sustainability downside [START_REF] Farquharson | Sustainability implications of electricity outages in sub-Saharan Africa[END_REF]. Much of the earlier reviewed literature showed the advantages of solar photovoltaics and wind turbines. Being a renewable resources-based, the current technological developments and effective market penetration and commercialization, and, not to mention with no CO2 emission, make them the most preferable technologies. Based on the literature-wide applicability and suitability of their application in customer-oriented outage solutions, the solar PVs are found to be more suitable with the added advantage of no rotating parts and requiring very low maintenance as compared to other counterparts.

 Solar photovoltaic: The solar energy harnessed by the PVs is not constant due to the daily cycles, seasonal changes, weather, and geographic constraints. However, the sun availability during the day timescale, fortunately, coincides with the increase of load demand for consumer groups such as commercial sectors and buildings. We can link this to the interrelation between increased load demand and outage, as shown in the considered load and outage profiles in [START_REF] Bastholm | Techno-economic study of the impact of blackouts on the viability of connecting an off-grid PV-diesel hybrid system in Tanzania to the national power grid[END_REF], [START_REF] Elkadeem | Optimal Design of Low-cost and Reliable Hybrid Renewable Energy System Considering Grid Blackouts[END_REF]. The solar PV is the most widely used technology and favors its suitability for powering customers in the event of outages.

 Wind turbines: Like solar, the wind resource has also variability with the season, particular time of interest, etc. with shorter or extended time cycles. It is apparent that outages highly affect urban consumers due to dynamic economic activity and high dependence on electricity supply with large consumption. Resource unavailability and wind profile disturbance due to big city infrastructures like buildings, wind turbines noise and visual pollution in and near cities, space requirements, etc. are some of the factors that could lead to their possibility of use in question and making it often impractical for many solutions.

 Energy storage:

The energy storage means are required for an added degree of freedom for short-term outages (few minutes) to long-term outages (hours or even days) and for smoothing out the load demand during outages with the available production. This means, with an added freedom degree they assist the demand change during an outage but insufficient or absence of solar production, which counts for a multi-sources solution. Among various techno-economic factors for the type of storage technology selection particularly in outage applications [START_REF] Zobaa | Energy Storage -Technologies and Applications[END_REF] (see The electrochemical batteries with moderate power output, stored energy application, and time availability in service are the characteristics that make them preferable. Considering an unreliable grid with an outage of a few minutes to several hours (even days), load demands ranging from residential, for instance about 3 kWp [START_REF] Khoury | Optimal sizing of a residential PV-battery backup for an intermittent primary energy source under realistic constraints[END_REF], to community load of around 56 kW [START_REF] Murphy | Analysis of the cost of reliable electricity: A new method for analyzing grid connected solar, diesel and hybrid distributed electricity systems considering an unreliable electric grid, with examples in Uganda[END_REF] show their operation ranges suitability. For their efficient utilization, the right sizing and appropriate energy management strategy for the circumstance under study is necessary.

The magnitude and rationalization of the problem caused due to outages, and possible solutions are established. The choice of technologies and components of the system for this study are also defined. The size of the hybrid system components differs mainly based on the outage characteristics, load profile, and fluctuations of the sources such as from seasonal variations. However, this problem can be approached from the elements of the sizing that have been controlled over to either confidently consider or achieve optimal solutions in our sizing techniqueoutage profile due to stochastic grid, cost, and reliability to come up with the best possible solution sizes via optimization.

I.2.2. Sizing optimization of multi-source systems considering outages

While the use of PVBESS (PV-battery energy storage system) will be practically an effective solution to lessening outages' impacts and improving reliability, the best possible sizing dictates the economic viability. Attributable to the ever-growing power outages caused by several factors and power shedding, particularly in developing countries, such a promising alternative solution for the reliable supply of power to consumers is essential.

The appropriate sizing requires an optimal solution both from the reliability measure and cost perspectives, and a reasonably validated solution subjected to effective performance on long-term realistic random grid outage.

In [START_REF] Adefarati | The Impacts of PV-Wind-Diesel-Electric Storage Hybrid System on the Reliability of a Power System[END_REF], the expected energy not supplied (EENS) and expected interruption cost (ECOST), featuring the total unserved energy and cost of unsupplied load respectively, were used as curtailment functions for optimality. The Net Present Cost (NPC) and cost of energy (CoE) in [START_REF] Elkadeem | Optimal Design of Low-cost and Reliable Hybrid Renewable Energy System Considering Grid Blackouts[END_REF] are taken as the objective functions. The authors used a load following and cycle charging strategy in the sizing, the approach prioritizes the generator operation to either run at a level in meeting the primary load (leave charging for renewables) or at full power where only surplus production goes to tasks such as charging. In a similar way, as both have used the HOMER tool [START_REF] Energy | HOMER -Hybrid Renewable and Distributed Generation System Design Software[END_REF], the optimal system is prioritized by the NPC [START_REF] Bastholm | Techno-economic study of the impact of blackouts on the viability of connecting an off-grid PV-diesel hybrid system in Tanzania to the national power grid[END_REF], [START_REF] Elkadeem | Optimal Design of Low-cost and Reliable Hybrid Renewable Energy System Considering Grid Blackouts[END_REF]. The system cost is also used as a fitness value for the optimization process by evaluating loss of power supply probability (LPSP) to select system choices [START_REF] Khoury | Optimal sizing of a residential PV-battery backup for an intermittent primary energy source under realistic constraints[END_REF] implementing a genetic algorithm and PSO solver.

In a different context than outages or blackouts, for grid-connected hybrid systems, optimization methods considering lifecycle cost (LCC) [START_REF] González | Optimal sizing of a hybrid grid-connected photovoltaic-wind-biomass power system[END_REF], system operating costs [START_REF] Maleki | Optimal operation of a grid-connected hybrid renewable energy system for residential applications[END_REF],

probability of grid electricity purchase [START_REF] Nacer | Optimal sizing method for grid connected renewable energy system under Algerian climate[END_REF], and both the LCC and LPSP [START_REF] Masih | Optimal Economical Sizing of Grid-Connected Hybrid Renewable Energy System[END_REF] as an indicator are reported. While the majority of the papers used genetic algorithm, a comparison with particle swarm optimization [START_REF] Maleki | Optimal operation of a grid-connected hybrid renewable energy system for residential applications[END_REF] is tested. In addition, classical energy management for sizing in standalone systems taking LPSP [START_REF] Yahiaoui | Optimal sizing and control strategy of renewable hybrid systems PV-diesel generator-battery: Application to the case of djanet city of Algeria[END_REF] and in grid-connected applications with a similar reliability measure (LPSP) [START_REF] Masih | Optimal Economical Sizing of Grid-Connected Hybrid Renewable Energy System[END_REF] is evaluated.

To ensure the satisfaction of all the load demand during an outage indeed needs a proper energy management strategy through the key aspectsthe sources, the storage devices, the considered load, and the associated outage profile, which is the heart of an optimal sizing solution. This strategy not only deals with the outage but also the intrinsic fluctuations of the renewable sources, such as solar power.

As explained, the study and hence system design for improvement of distribution reliability is either to prevent or reduce power supply interruptions. In this sense, the duration and number of interruptions/outages are the basic and crucial parameters to deal with. The cost of interruption characterized by these two is primarily dependent on the kind of power consumers, the time of interruption, and the length of interruption duration. Any further reliability measurement indices are usually formulated based on these two.

Concisely, addressing the outage problem and improving the reliability is the main objective, which is considered in many optimization approaches. This being the crucial technical design, the economical aspect should be incorporated for its practicability. Thus, the system cost linked with the served energy due to the outage is reflected by the cost of energy, another objective to achieve in the optimization process. Unlike the single objective optimization which often is the system cost and by respecting minimum values such as for loss of power supply [START_REF] Samy | Optimal economic study of hybrid PVwind-fuel cell system integrated to unreliable electric utility using hybrid search optimization technique[END_REF], we here use both the reliability and cost metrics.

I.3. Conclusion and thesis context

The previous sections in this chapter demonstrate the gravity of grid outage problems and associated cost implications globally, with a particular focus on developing countries.

They are characterized by recurrent power outages due to weak grids which commonly use radial distribution grid architecture and aged components. Grid supply reliability improvement techniques and methods for either modern grid applications or, despite the low number of studies, for weak grid applications are discussed. Existing hybrid power systems considering outages use coarse data (mostly scheduled outages, some are based on real data but with gross and average values, etc.), which is not agreeable and often fail to promptly achieve the intended level of reliability in avoiding real stochastic outages. The oddities here are a tendency to fall to either have an oversized system with the expense of increased cost or not be able to achieve an acceptable reliability.

Secondly, the majority of a multi-source hybrid solution for outages that exist in the literature lack to test the system performance, its effectiveness, and certainty, and hence validate their proposed approaches, particularly in a more realistic context. Such studies would make it possible to model a stochastic grid to characterize it probabilistically, and also to evaluate how the solution sufficiently supply power without interruption and validate its adequacy.

This latter helps to propose an approach to modeling the haphazard outages that are quite common in weak grid networks and seem to be insufficiently covered in the literature.

Therefore, this thesis proposes a solar photovoltaic and battery system for a weak grid network based on real stochastic outage data, which implements a multi-objective global sizing optimization approach to mitigate outages. The system while aimed at wholly avoiding outages and achieving a highly reliable solution, unlike other existing approaches, also gives sufficient flexibility to choose a range of reliability levels for end-users.

Moreover, based on historical (5-years) outage data from a predominantly radial distribution grid, we have to develop grid outage models which are used to simulate the probable outage states of the grid. The simulated long-term outage profiles from these models are then used to test the performance of the proposed PV-battery solution for effectiveness and validation, with a representation of more realistic outages of operating situations.

To sum up, the scientific steps followed in this thesis are, to develop a global sizing optimization methodology of a multi-source system by integrating PV and battery storage on a weak radial distribution grid, see Figure I-1. With the aim to adequately supply power during outages and replace the conventional diesel solution due to the several negative impacts, in this manuscript, we propose a case study to develop the methodological approaches. A university building located in Addis Ababa, Ethiopia, which has a power supply from a weak stochastic grid with frequent and long-duration outages is studied.

Based on the grid outage available data and actual measurement (load and outage profile) conducted by the author, the sizing approach follows a causal rule-based energy management based on the choices of renewable resources, i.e., the photovoltaic panel and Li-ion battery storage. Thus, the multi-objective sizing optimization (MOSO) of this work, differs from the existing little number of related works, with a particular application and inclusion of a real stochastic grid in our methodology. The MOSO will also give the flexibility of exploring the reliability improvement with various PV-battery combinations for a desired reliability level taking into account the two objectives as equally important.

In addition, the third contribution is to develop a stochastic grid outage modeling based on historical outage data, which also enables the effectiveness evaluation and validation of the MOSO optimal results, apart from studying the state of a grid outage situation for a case considered. Finally, the large outage profile simulations through these models which allows us to evaluate the performance will enable us to show the feasibility and benefit of the solution. For these, we proposed two models of grid outage based on their advantages and capabilities, which are detailed in Chapter IV:.

Chapter II to V: Chapter II: Modeling of the Multi-Source Energy System

II.1. Introduction

The main work in this thesis deals with the sizing optimization of a multi-source system (solar PV and battery energy storage connected to an unreliable grid) to improve the electrical power supply continuity and reliability of a consumer facing recurrent grid outages. The system modeling for optimal sizing for this particular application is presented in this chapter, see Figure II-1. This chapter then presents the energetic modeling of the various devices (energy conversion and storage devices) which will be used for the rest of the thesis. It includes the economic analysis and reliability model of the multi-source system integrated with the grid.

Furthermore, a description of the selected case study that gives an overview of gridside supply with outages and a more detailed examination of a customer is presented in the last section of the chapter. The energy demand and outage profile are important to be used in the sizing of the multi-source system.

II.2. Architecture of the multi-source system

One of the most suitable to implement and highly competing technology is the solar photovoltaic panels. Photovoltaic panels are one of the renewable energy technologies with several advantages such as wide application, easy installation, and low CO2 emissions in their life cycle. The PV industry is one of the fastest growing that exhibits a huge growth in global installed capacity and sharp cost decline in the last decades. The global total PV capacity spiked by about 4,300% by the end of 2017 in just 10 years [START_REF] Schmela | Global Market Outlook for Solar Power /2018 -2022[END_REF]. It has shown a rapid price decrease amounted to more than 80% between 2008 and 2016 [START_REF] Jäger-Waldau | PV Status Report 2018[END_REF], [START_REF] Schopfer | Economic assessment of photovoltaic battery systems based on household load profiles[END_REF].

Moreover, it is also silent with no sound pollution as it does not have rotating parts, and has lower O&M costs compared to other renewable counterparts. Thus, the PV technology option is retained as the main source for our study bearing these and the merits discussed in the previous chapter.

Strongly linked to the photovoltaics and renewable sources in general is their intermittency, to overcome this, at the same time as the grid outage, the grid and PV must be coupled with storage devices.

The energetic models of the energy conversion, the storage devices, and the grid characteristics are developed, which are relatively simple and have low computation time compared to more detailed and complex dynamic models [START_REF] Bouabdallah | Méthodologie de dimensionnement et d'optimisation des systèmes multisources à énergies renouvelables pourl'habitat[END_REF], [START_REF] Roy | Gestion optimale d'un système multi-sources pour un site isolé en mer[END_REF]. In the last part of this chapter, the cost evaluation using the cost of energy production per kWh and the system performance indicator (reliability) is developed.

The modeling of the multi-source system that enables the sizing optimization and system simulation in the following sections will be according to the schematic diagram in Whenever the grid is available, i.e., no outages, the grid fully supply the building loads and the PV will charge the battery eventually assisted by the grid if PV alone is insufficient. This implies that the overproduction will be normally used to charge the storage. During outages, the PV fulfills the demand. On the other hand, the supply system will be islanded from the grid in order to avoid power flow to the faulty grid in case of outages. If the PV power is inadequate, the storage batteries will flexibly be able to complementarily backup the solar power. Whenever there is no PV production, the storage supplies the load until it gets to the operational limit, which will be discussed in the storage models later. It is then the second case where the load will be islanded after the storage reaches the minimum capacity.

What is developed here is a methodology for modeling and simulating the grid state in scrutinizing its outage phenomena. Compared to available literature, a high-resolution 1-minute sampled actual stochastic grid interruption rather than hypothetically synthesized interruption data is considered in our study, for a consumer shown in Table II-

Sample load and outage profile for the client considered (AAiT)

For an ordinary university building with conventional functions (classes, offices, labs, halls, cafeteria, etc.), a sample of weekday and weekend days' electrical load profile from our measurement is presented in Figure II-4.

The load profiles show a higher demand curve during the weekdays and a reduced load demand during the weekend as expected; the mean peak demand appears in the working hours window of 8:00 AM to 5:00 PM.

Sizing characteristics and models

The load power measurement and outage periods are the physical parameters that are recorded from a utility grid client from the case selected. For the entire system simulation and energy balance, we have the electrical power production from the PV to meet the load and also energy exchange from the storage (rate of power) during the power deficit. The following models will enable us to extract the power available from the site-specific solar radiation and correspond to the PV model.

The sizing models in the next sections also add up to the following rules:

 For this work, unlike the widely used simulation time step of an hour, we opted for a lower sampling time (minute), more adapted to the possible outage durations.

 This sampling is chosen to accurately capture the electrical power outage in a reasonable time span. Then, it also helps for accurate design of the hybrid system.

 The electrical energy (the approximate integral of rate of power, P) of the system served during the simulation year, T, follows:

𝐸 𝑡𝑜𝑡 = ∫ 𝑃 𝑔𝑒𝑛 𝑑𝑡 𝑇 𝑡=1 (II.1)
The power exchange between the different devices in the multi-source system (as shown in Figure II-2) is detailed in chapter III, which will give the model representing the power flow.

II.2.1. Solar Photovoltaic Model

The geographic and climatic dependent solar PV power is based on the solar radiation received on the plane of the solar panels [START_REF] John | Solar Engineering of Thermal Processes[END_REF], [START_REF] Bugler | The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation[END_REF], at that particular location under investigation. This is because, at meteorological/weather stations, the global solar radiation is generally measured onto the horizontal surface plane.

The power generated from the photovoltaic panel is defined by the cell temperature as a function of the location's ambient temperature and the available solar irradiation. The PV power is used to supply the load directly in case of outages, and/or to charge the battery bank. The maximum available power from the PV with maximum power point tracking (MPPT) for a given surface area of the panel, Apv, and the total irradiation hitting the PV surface, Gtot, is mathematically modeled by widely used model with peak PV power [START_REF] Thomas | Optimal design and technoeconomic analysis of an autonomous small isolated microgrid aiming at high RES penetration[END_REF]:

𝑃 𝑝𝑣 = 𝑃 p 𝑓 𝑝𝑣 ( 𝐺 𝑡𝑜𝑡 𝐺 𝑠𝑡𝑐 ) [1 + 𝛽 𝑝 (𝑇 𝑐𝑒𝑙𝑙 -𝑇 𝑠𝑡𝑐 )] (II-1)
with 𝜂 𝑝𝑣 = 𝑓(𝑇 𝑎𝑚𝑏 , 𝐺 𝑡𝑜𝑡 ) where:

• Ppv: the generated peak PV power [W];

• fpv: PV derating factor (due to soiling, wiring losses, shading, snow cover, aging, etc.); The panel efficiency is a function of its temperature and manufacturer's characteristic parameter and ambient temperature, and is given by [START_REF] Kaabeche | Sizing optimization of gridindependent hybrid photovoltaic/wind power generation system[END_REF], [START_REF] Bouabdallah | Safe sizing methodology applied to a standalone photovoltaic system[END_REF]:

𝜂 𝑝𝑣 = 𝜂 𝑚 [1 -𝛽 𝑝 (𝑇 𝑐𝑒𝑙𝑙 -𝑇 𝑟𝑒𝑓 )] (II-2) 𝑇 𝑐𝑒𝑙𝑙 = 𝑇 𝑎𝑚𝑏 + (𝑁𝑂𝐶𝑇 -20)
𝐺 𝑡𝑜𝑡 800

(II-3)
where:

• ηm: the module reference efficiency [%];

• βp: the photovoltaic panel power temperature coefficient [%/°C];

• Tcell: the cell temperature [°C];

• Tamb: the ambient temperature [°C];

• Tref: is the reference temperature at standard reference conditions [°C], which is 25°C;

• NOCT: the nominal operating cell temperature [°C];

• GNOCT: the NOCT irradiance, is represented by the 800 W/m 2 value in the above equation.

Optionally an equivalent simplified energetic model for the photovoltaic panel can be found in our paper [START_REF] Kebede | Photovoltaic System integration in Unreliable Power Distribution System[END_REF] explained in detail. The basic difference is the PV array sizing space for the optimization, represented in terms of the array area and the array peak power.

The module efficiency and temperature coefficient are variables depending on the selected module and be derived from the manufacturers' datasheet. In our study, the selected photovoltaic module from one of the manufacturers in the technology, LG electronics [START_REF] Electronics | LG Neon[END_REF], is considered. The details of the technical parameters and their values for the considered PV panel are given in Table II

II.2.2. Storage Battery Model

Battery storage is also included for added degree of freedom to the system in the energy management, which also increases the system flexibility for outage reductions.

During grid outages and in cases where there is no PV power or with a deficit in the solar power production, the battery bank supplies the consumer load demand. Considering this key system component to work in connection with the existing grid, a universal sizing method and energy management according to the charging-discharging rule is implemented, which is widely used in [START_REF] Masih | Optimal Economical Sizing of Grid-Connected Hybrid Renewable Energy System[END_REF], [START_REF] Kaabeche | Sizing optimization of gridindependent hybrid photovoltaic/wind power generation system[END_REF], [START_REF] Maleki | Optimal sizing of autonomous hybrid photovoltaic/wind/battery power system with LPSP technology by using evolutionary algorithms[END_REF], [START_REF] Sharafi | Multi-objective optimal design of hybrid renewable energy systems using PSO-simulation based approach[END_REF] mainly for lithium-ion batteries. From the different battery technologies available in the market, we opted for this battery type in our study. It takes into account the grid availability and compares the total (grid + PV) power with the load demanded at the instantaneous simulation time step. Thus, depending on the grid availability together with the total available power and the instantaneous battery stored energy, the battery storage system can be modeled by its State of Charge (SoC) at any time instant, t, and given by (II-4) for the charging, and (II-5) for the discharging operations.

For chargingwhen the total system power output from grid and PV (and/or no outage) is greater than the load demand, the state of charge of the battery in its charging state will be calculated by: In this case, the system assures to supply the energy demanded and the surplus energy will be stored in the battery until it reaches the maximum operating boundary of the battery.

𝑆𝑜𝐶(𝑡) =
When it reaches this state, the surplus energy will be fed to the grid, as the algorithm checks the grid availability a priori.

For dischargingwhen the total output is less than the demand, the battery will be in discharging state:

𝑆𝑜𝐶(𝑡) = 𝑆𝑜𝐶(𝑡 -∆𝑡) -(𝑃 𝑡𝑜𝑡 (𝑡) - 𝑃 𝑙𝑜𝑎𝑑 (𝑡) 𝜂 𝑖𝑛𝑣 ) 1 𝐶 𝑏𝑎𝑡 ∆𝑡 𝑖𝑓 (𝑃 𝑡𝑜𝑡 (𝑡) - 𝑃 𝑙𝑜𝑎𝑑 (𝑡) 𝜂 𝑖𝑛𝑣 ) < 0 (II-5)
where:

• Ptot: the total available power from the grid and PV [W];

• Pload: the load demand [W];

• 𝜂inv: power conversion efficiency/inverter efficiency [%];

• 𝜂bat: the battery charge-discharge efficiency [%];

• Cbat: is the battery storage nominal capacity [kWh];

• ∆t: is the simulation time step increment [min.]; recap: it is one minute in this study.

Here, note that the total power during outages is only the PV power, as the grid power is zero; and if in case there is a power deficit, the battery power is available in discharge mode used to ensure all the load power until it reaches the minimum operating boundary. It should also be noted that even with the availability of higher production compared to the demand at any instant, t, no power will be sent to the grid during an outage state.

The storage battery ensures the power supply during outages in its allowable operational boundaries of states, SoCmin and SoCmax, to maximize their lifetime and efficiency, which also protects the battery from overcharging and deep discharging.

The operating battery state of charge is constrained by its operational range of states:

𝑆𝑜𝐶 𝑚𝑖𝑛 ≤ 𝑆𝑜𝐶(𝑡) ≤ 𝑆𝑜𝐶 𝑚𝑎𝑥 (II-6)
where:

• SoCmin: the minimum state of charge of the battery [%];

• SoCmax: the maximum state of charge [%].

Additionally, for a more rigorous model of the storage, a maximum allowable charging and discharging power (Pmax,ch and Pmax,disch, respectively) are considered in the energy management model. Hence, in summary, the battery energy exchange is governed by the operational constraints of SoC, see (II-6), and the manufacturer's charge-discharge power limits. Furthermore, the techno-economic representation of batteries takes into account their lifetime provided by the manufacturer.

Estimation of battery lifetime

Various approaches for calculating the battery performance degradation and its lifetime are adopted in recent works as summarized in [START_REF] Kebede | Photovoltaic System integration in Unreliable Power Distribution System[END_REF]. Based on the technical and economic metrics, the lifetime of the storage battery is evaluated by estimating the battery aging as:

𝐴𝑔𝑖𝑛𝑔(𝑡) = 𝐴𝑔𝑖𝑛𝑔(𝑡 -∆𝑡) + |𝑆𝑜𝐶(𝑡) -𝑆𝑜𝐶(𝑡 -∆𝑡)| 𝑁 𝑐𝑦𝑐𝑙𝑒,𝑚𝑎𝑥

(II-7)

where:

Ncycle,max: the maximum number of charge-discharge cycles enduring the DOD.

Based on the ratio of the sum of the yearly simulation time to that of the last aging value at the end of the simulation, it is then compared to the manufacturer's datasheet battery lifetime. The minimum lifetime value will be taken to consider battery replacement for the entire project lifetime modeling period. The battery life with the estimated cycle operation is used, where other aging models such as calendar aging are not considered. For this typical application, we know that the temperature at which the storage could operate is not high, with an annual average in Addis Ababa around 15-18° C. Related to the outages, they occur randomly and from a few minutes to hours in most cases, leaving the battery often at higher SoC in most of the operation times. This means, as the degradation tends to be higher in lower SoC due to internal resistance, the aging in higher SoC will then be expected to be very minimal. Moreover, calendar aging increases with increasing temperature and long idle storage time. Here then, we barely will have long idle storage time as outages are recurrent, leading to very low capacity fade (for higher SoC storing, short storage time, and average storage temperature).

In this modeling, a lithium-ion battery from [START_REF] Electronics | LG Neon[END_REF] is considered, Table II-3 presents the technical values related to the selected battery. The Li-ion battery has a high cycle life such as roughly from 2,000 to 5,000 cycles as compared to 300 to 500 (limited cycle count)

to that of lead acid batteries [START_REF] Murden | Lead Acid vs. Lithium Batteries -Which One Utilize the Better Technology -Eco Tree Lithium[END_REF]. In this same summarized comparison, similarly with their technological advancement and cost reduction evolution, they can handle deep discharge (like 80-90% easily, and even more), whereas the lead acids can go only up to 50% beyond which it risks the lifespan. They also have a good performance and higher efficiency in charging which efficiently utilizes the PV power available. The low cost of lead batteries is for the short term as they have a shorter life, whereas the lithium could be cost-competitive for long-term applications. The low depth of discharge (lead acid) raises the risk of unavailability if a subsequent outage is occurring before it is charged, hence the lithium battery can be advantageous for long period of usage. Even after an outage and insufficient PV production, the lithium battery shall get charged fast to be ready to be used for the next coming unpredicted outage. They require little time to charge from a few hours (3 h) to even minutes; compared to about 10h lead acid.

II.2.3. Modeling a Diesel Generator

To compare the economic practicability of the renewable-based system with conventional diesel backup for outages, the CoE for the PV-BSS is compared with the cost of energy for a diesel-only system by considering a generator for the studied building. With a similar analogy as above, the cost of energy can be computed by: 𝐶𝑜𝐸 𝑔 = 𝐶𝐶 𝑔,𝑎𝑛𝑛 + 𝑂𝑀𝐶 𝑔,𝑎𝑛𝑛 + 𝑓𝑢𝑒𝑙𝐶 𝑔,𝑎𝑛𝑛 𝐸 𝑡𝑜𝑡 (II-8)

where:

• CoEg: the cost of energy generated from the diesel generator [€/kWh];

• CCg,ann: the annualized capital cost of the diesel generator accusation [€];

• OMCg,ann: the annual operation and maintenance cost of the diesel generator

[€];

• fuelCg,ann: the annual cost of diesel fuel [€];

• Etot: the total served diesel energy [kWh].

Based on the daily maximum load of 48.4kW recorded during the data measurement, a 48kW rated power generator is considered in this study. The fuel consumption characteristic of the diesel generator is given by its fuel curve and the efficiency performance [105], [START_REF] Das | Performance analysis of a PV/Diesel hybrid system for a remote area in Bangladesh: Effects of dispatch strategies, batteries, and generator selection[END_REF] which are given by (II-9) and (II-10) as a function of the generator's output power.

𝐿𝑃𝑆𝑃 = ∑ 𝐿𝑃𝑆(𝑡)

𝑇 𝑡=1

∑ 𝑃 𝑙𝑜𝑎𝑑 (𝑡) 𝑇 𝑡=1

(II -11) where:

• LPS: the loss of power supply [W], the power that cannot be met during a sample time ∆t, even by the PV-battery system during outages;

• T: the simulation time [min.], minutes in one year, in this case.

This power unbalance leading to total deficit is then simply given by the difference of power between the load and the power system: where,

𝐿𝑃𝑆(𝑡) =
Pbat: is the battery power [W].

The LPSP is normally in the range of [0, 1], in which 0 means the load is fully satisfied and 1 means the load can never be met at all. The loss of power supply probability (LPSP) approach is able to quantify the reliability both from the grid outage and the intermittency of the solar power, and hence used to measure the performance. This technique is a powerful tool to quantify the lost energy which cannot be supplied by the combined PVbattery system, and hence to be minimized as a primary objective in designing such systems.

On the contrary, when there is surplus electrical energy production (for instance, no outage or outage occurred but surplus PV generation), battery charging will be prioritized Finally, the annualized cost of energy (CoE), which is the average cost (in euros) for the energy actually generated and served (in Wh), is then defined by:

𝐶𝑜𝐸 = 𝐶 𝑡𝑜𝑡,𝑎 ∑ 𝑃 𝑔𝑒𝑛 ∆𝑡 𝑁 𝑡=1
(II- [START_REF] Vieira Pombo | Multiobjective formulation of the integration of storage systems within distribution networks for improving reliability[END_REF] where:

Pgen: the total generated power [kW].

The details of the technical and economic parameters of the system components are sum up in Table II-4 for commercially available PV panels and storage batteries, taken from literature, see details in [START_REF] Kebede | Photovoltaic System Sizing for Reliability Improvement in an unreliable Power Distribution System[END_REF].

II.3. Description of the case study

Considering intermittent electrical grid supply with random power outages, the objective of this work is to design a multi-source energy system composed of photovoltaic panels integrated with storage batteries to develop a reliable electrical power supply system.

In order to achieve the proper sizing optimization and energy management of the different energy sources, we need to define the application case study for both available corresponding resources and the demanded load profile to be supplied in the event of the electrical grid outages. For this reason, based on the acute electrical grid outage problem discussed in chapter I, a utility power grid in Ethiopia is elaborated below.

a. The Ethiopian power sector profile in brief

Ethiopia's energy sector, currently with an installed electricity generating capacity of more than 4,200 MW is experiencing a shortage of power. The national electrical power supply is heavily dependent on hydropower, which is around 90% of the total installed capacity. Wind constitutes only around 8% followed by an approximate 2% of diesel and the remaining from geothermal plant [113]. The major causes include highly unreliable electrical network due to interruption or outages and huge electricity annual demand growth estimated at about 30% [START_REF]Scaling-Up Renewable Energy Program (Ethiopia Investment Plan-Draft Final)[END_REF]. Its economic development, the transition from an agricultural to an industrial economy, population growth, and urbanization are some of the factors in raising this modern energy demand. The sector's entire supply chain, generationtransmission-distribution-sale, is run by the state-owned power and utility companies, formerly a single power company known as 'Ethiopian Electric Power Corporation'.

Nowadays in Ethiopia, even in urban areas like its capital Addis Ababa, the electric power interruptions are increasingly common. Such incidents are leaving many customers without power for a few minutes up to several hours and days in a single interruption event.

In this context, many customers installed backup diesel generators in order to maintain the supply during the outages. These systems imply high maintenance and operating cost and the use of non-environmentally friendly diesel groups.

b. Study of power distribution grid: A case of Addis Ababa city

Addis Ababa city, located at an altitude between 2,300 m and 2,600 m (geographical locations, see Table II-1), is the capital and the seat for quite a lot of international organizations, embassies, etc.

Mainly the city is interconnected with 400kV, 230kV, and 132kV transmission networks with a sub-transmission system of 66kV, 45kV, and 33kV lines. Moreover, the 15kV distribution system has 28 Bulk Supply Points (BSPs) or substations [START_REF] Brinckerhoff | Addis Ababa Distribution Master Plan Study -Modelling and Analysis of Existing Network[END_REF].

Customer-level voltages are 380V for a three-phase and 220V for single phase. Generally speaking, the distribution system has mostly an overhead radial topology, lacking sufficient redundancy whereby in case of an outage of major circuit or transformer, clients connected to the network way from these faulty devices will have power cut-off.

Addis Center Distribution Substation Outage Survey

For a detailed survey, Addis Centre substation (ADC) is selected; which is one of the The causes of outage as reported and considered by the utility operators are categorized as listed below:

 Earth fault (EF)which includes both permanent and temporary faults,  Overload (OL)this can be either the distribution line, transformer or system overload,  Operational (OPE)is to mean an intended interruption of power for maintenance, load transfer and expansion or upgrading works,  Short circuit (SC), and,  Under frequency (UF). Further outage records on one of the feeders (ADC-9) emanating from ADC showed relatively higher outages. Considering an average electricity sales price of 0.0321 €/kWh [START_REF]Ministry of Water Irrigation and Electricity of Ethiopia[END_REF], it leads to an approximate monthly sales revenue loss of about €8,500 for this single feeder for one month. It thus seems obvious that these outages have a significant impact on the economy as well and that solutions must be considered.

Customer outage survey: A case for AAiT building

Further outage and load profile case study has been made on a university campus located in Addis Ababa. The institute's electrical grid power is supplied by the sole government-owned power utility company. It has also installed a 650 kVA standby diesel generator to sustain outages, for the whole of the institute. An Automatic Transfer Switch (ATS) will turn on the generator when an outage is detected and turn off if the grid is back (Figure II-9). (Note, the load demand here is supplied under normal operation where the grid is available.)

In case of an outage when the ATS detects power loss, the ATS automatically transfers the load supply from the diesel backup system; and then when the grid comes back, the ATS shifts back the load supply to the primary supply, which is the grid.

Client's load measurement and power outage recording

Fluke 435 Series II three-phase power quality and energy analyzer is used to measure the load demand and record the outage events. The device is installed at one of AAiT's buildings with regular functions like offices, classrooms, laboratories, and conference rooms. Even with the standby diesel generator, the power supply experiences interruption dips due to total outage when the diesel is not yet supplying (due to the time to switch between the grid and the generator). Again, this can also occur either when the generator is out of fuel or faulty ATS operation. Based on these sequences of loss of power, it becomes possible to automatically detect the start and end of an outage. The data measurement was taken for about one month from October to November 2018. To capture the power supply switching between the grid and generator, the time step was set to one second. After that, for the sizing analysis, a minutely averaged load and outage data is synthesized to perform the energy systems modeling and optimization. Because the building is a university campus, it can be seen that the consumption is higher during the first five days (from Monday to Friday) than during the weekend. The profile is on the load side, for which the power comes from the grid or diesel. A more detailed client's outage phenomena with measurement and voltage profile will be presented and described in Chapter IV: under section IV.2. The location with a yearly average ambient temperature of about 16.5°C (with the warmest and coldest months mean of 18.5°C in May and 15.5°C in July) [START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF] and the load demand is expected to be moderately constant throughout the year. Electrical demand for heating, cooling, air conditioning, fan, and hot water will then be not required for these types of consumers in this sample location. In fact, due to the relatively insignificant temperature variance and a temperature with such level of comfort, it will not have summer and winter load distinction. Hence, the measured consumption profile is concatenated to

Solar resource data

The PV power output directly depends on the global solar irradiation incident on the PV panel surface and the ambient temperature of the location. The radiation can be determined by having the knowledge of the site characteristics of geographical location, climatology (either sunny or cloudy), and astronomical (time of the day and the sun's angle towards the earth). In addition to this, when considering its PV application, it is the radiation hitting the PV panel's surface according to the inclination angle from the ground and orientation of the panel.

For a fixed axis panel, maximum yearly energy production can be obtained by installing using a yearly optimum tilt angle for the site selected. In literatures, studies propose a rule of thumb for optimal tilt considered to be equal to the local latitude and facing south (for northern hemisphere locations) [START_REF] Muzathik | Estimation of global solar irradiation on horizontal and inclined surfaces based on the horizontal measurements[END_REF], or latitude plus 15° [START_REF] Venkatachalam | Dataset of solar energy potential assessment for Adama city (Ethiopia)[END_REF]. Sitespecific studies and databases indicated an optimum tilt angle for the case location to be about 16° [START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF], [START_REF] Jacobson | World estimates of PV optimal tilt angles and ratios of sunlight incident upon tilted and tracked PV panels relative to horizontal panels[END_REF].

For the site considered in the case study in Ethiopia (9.04°N latitude and 38.8°E longitude), the used solar irradiation resource and ambient temperature data are from [START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF], [START_REF] Huld | A new solar radiation database for estimating PV performance in Europe and Africa[END_REF] and given by Figure II-11. The dished shape irradiation and the temperature minimums in mid-year is due to the rainy season in the country.

II.4. Conclusion

In this chapter, we presented the multi-source system architecture and the detailed energetic models of the energy conversion components (solar PV and storage battery), the measure of performance of the system (reliability model), and the economic models for properly sizing the system as a whole. The PV array area and battery storage capacity are the parameters for defining the sizing. This system then produces sufficient power for a stochastically unreliable power grid in Ethiopia, considered as a case.

Thus, the developed models allow us to base the sizing of the components on a selected case study of an electric utility client in Addis Ababa. The solar resources data, and load demand and grid outage profiles are presented. Here, the basis for the developed system will be the outage profile (integrated into the load) of the grid electricity, which is different from the classical daily load profile under normal operation.

Remember that sizing alone results in either power deficit to supply the load demanded or surplus power which could lead to an oversized system than required. For this case, these simplified energy models, and techno-economic models will be used for optimization in the next chapter for optimal sizing of the whole system. This ensures examining the reliability of load supply and economic soundness.

Chapter III: Techno-Economic Sizing Optimization based on

Measurement

III.1. Introduction

During the previous chapter's discussion, we presented the modeling of a multisource energy system that combines the photovoltaics and battery storage devices connected to the weak electrical network. It is clear that multi-source systems include one or more power sources with or without storages. As we demonstrated for the case study, the storages here are important to cover the energy during either the grid outages or when there is a gap between the production and demand. Recalling that the techno-economical models include the solar PV generator, the battery storage, the load demand, the random grid unavailability profile due to outages, solar radiation, and ambient temperature data as an input and for the mathematical formulations. In order to fulfill all the demanded load during outages, the solar PV and storage batteries required are generally large to satisfy the load demand at any instant, which actually results in high investment costs. Then, here in this chapter, the sizing optimization enables us to consider the trade-off between the total cost of the system in terms of energy cost and the improvement of the grid satisfaction ratio (i.e. the reduction of the outage issues). This can be achieved using the dimensioning optimization of the energy and economic modeling of the aforementioned system elements developed in section II.2.

Furthermore, an energy management to ensure the power flow or energy balance in meeting the reliability criteria with an optimal cost is applied as will be discussed in this chapter. The rule-based energy management, which is simple to apply yet effective for the global sizing optimization of our two-objective problem, is implemented using the evolutionary search algorithm, NSGA-II (Non-dominated Sorting Genetic Algorithm). The power supply reliability and hence satisfaction rate of a power consumer and the cost of energy comprise our bi-objective, also called multiobjective optimization (MOO) problem.

The MOO problem definition includes the design variable vectors (Ai, Cb) and the competing objective functions (LPSP, CoE), whereby the sizing aim is to achieve the continuity of power supply or the consumer demand satisfaction, and the optimality of cost of the system or energy produced. Our result will then lead to an equally good set of solutions, at which one of the objectives cannot be improved without sacrificing or deteriorating the other. Here, the solar radiation profile, load demand and outage profiles, and grid side supply for active grid are assumed to be the input variables which are known to the designer.

The MOO allows a designer to develop a multi-criteria optimal solutions options, which other single objective approaches or tools (such as HOMER 7 ) cannot do. As an example, such tools take one objective for minimization (or maximization based on its application), for instance for HOMER, it lets the user to minimize the NPC with other parameters put as input and constraints [START_REF] Bastholm | Techno-economic study of the impact of blackouts on the viability of connecting an off-grid PV-diesel hybrid system in Tanzania to the national power grid[END_REF], [START_REF] Rehman | Feasibility Study of a Grid-Tied Photovoltaic System for Household in Pakistan: Considering an Unreliable Electric Grid[END_REF].

Moreover, we will also present different configurations of the system to test the ability to supply the demanded power, during grid outages, and their economic optimality and the corresponding reliability. The configuration options are photovoltaic only, battery only, and PV-battery systems, which we also compared with a classical backup generator for the continuity of power supply during outages.

At last, the chapter concludes by showing the sensitivity of the developed optimal multi-source system to variations of components' costs and grid electricity price changes.

III.2. Rule-Based Energy Management for Optimization

As the core of this study, the LPSP is the primary objective criterion to be satisfied with an optimal associated cost. It measures the ability of the proposed power system to satisfy the demanded load and quality of the overall power production. The sizing of the PV and battery are the decision variables governed by the energy flow and management. This is a generalized energy management system (EMS) for the power flow and control in the multi-source system to best satisfy the consumer demand in events of an outage, based on the different possible scenarios of grid outage occurrence and energy availability from different sources (PV, batteries).

7 https://www.homerenergy.com/

The energy management rule governs the balance of energy based on the availability of the grid (outage, or available grid), the PV production, and the state of charge (SoC) of the battery storage. Here, we have considered a maximum subscribed grid power (Pgrid,max)

and SoC operational constraints of SoCmin and SoCmax in the algorithm.

On the other hand, the cost of energy (CoE) is the measure of the practicability of the system in achieving a zero LPSP or any other possible solution. It is at this point where the optimization is important due to the conflicting nature of these two objective variables, The energy management considers the following strategies in the sizing optimization process:

LPSP
 The grid covers the loads and assists in battery charging, when the grid is available;

 At any instant, t, the management strategy checks the grid signal (outage, or available grid) and for the balance of power from grid and PV (Ptot), and also the SoC of battery (SoC(t));  During an outage, no power is sent back to the grid;  The algorithm should satisfy the SoC limits (SoCmin and SoCmax) and battery charge-discharge power constraints (Pmax,ch and Pmax,disch);  The storage battery charging process starts whenever the total power generation exceeds the load demand, i.e. overproduction;  Pgrid means the power 'drawn from' or 'sent to' the electrical grid network.

The advantage of the rule-based global energy management strategy is to ensure the power balance, hence mitigate the outage and ensure continuity of supply, and protect the storage battery from overcharging and over-discharging. This also increases the storage life and efficiency. The rules ensure that the load demand is met, particularly during grid outages, whenever possible. Based on the grid signal (whether outage or available grid), the total power produced will supply the load, charge the battery or be sent to the grid. The battery charging is again based on its instantaneous SoC and surplus power availability after supplying the demand. For the case, if there is an outage and PV production is not sufficient, then the management strategy will activate battery discharging, such that it does not violate the SoC thresholds. Furthermore, the power to and from the battery should follow the maximum allowable charge and discharge powers, respectively. The SoCs are calculated at each time step as to the power flow computations. In the end, if there is an outage, insufficient PV production, and battery reached at the SoCmin, then the power deficit (loss of power supply) is computed. This is then the basis to quantify the level of reliability in mitigating the outages, which leads to the LPSP. Then, based on the energy management algorithm, the economic evaluation using the CoE will be measured.

The EMS algorithm computes the instantaneous power from the different sources and the battery state of charge at every time instant, t (Figure III-2). The multi-source weak grid-PV-battery system power flow is presented by: Grid + Outage Building Load P PV (t) P bat (t) P l (t) G state P g (t) The grid power is composed of two components: one is the normal grid power without outage (normal operation), and the binary state grid outage sequence in case of grid outages. In this latter case, the grid status or outage profile, which is a binary state sequence, is given as: Referring the power exchange in Figure III-2, the total power balance among the different system devices can be given as follows . The total power generated is the power from the grid (booth under normal operation and with an outage if happened). Note here that the maximum power to be drawn from the grid if there is no an outage is equal to the maximum subscribed grid power (Pgrid,max).

Rule-based energy management algorithm [P grid (t), P PV (t), P l (t), P bat (t), SoC(t), SoC(t-1)]

𝑃 𝑡𝑜𝑡 (𝑡) = 𝑃 𝑃𝑉 (𝑡) + 𝑃 𝑔𝑟𝑖𝑑,𝑜𝑢𝑡𝑎𝑔𝑒 (𝑡)

(III-1)

III.3. Defining the Multiobjective Optimization Problem

As one of the prime objectives for an acceptable robust system configuration, the PVbattery sizing process is formulated as an optimization problem [START_REF] Maleki | Optimal sizing of autonomous hybrid photovoltaic/wind/battery power system with LPSP technology by using evolutionary algorithms[END_REF], [START_REF] Kebede | Photovoltaic System Sizing for Reliability Improvement in an unreliable Power Distribution System[END_REF]. In this problematic as explained above, there are two conflicting objectives that will have a set of possible solutions called the Pareto frontier. For these non-dominated sets of solutions, for instance, one of the objectives will never be any better without making the other objective worse. This tradeoff will give the flexibility for consumers or power utilities to choose any of the combinations of the hybrid system based on their required level of reliability.

The 2-objectives function considered is the minimization of the cost of energy (CoE) which is composed of the system and associated costs as well as grid charging cost, and the minimization of the LPSP which is the measure of the reliability improvement by reducing or fully overcoming the grid outages. The optimization algorithm evaluates the area of PV panels (Apv) and the number of batteries (Nbat) to be installed either to avoid outages or for minimization. Hence, the area of the PV panels and the number of storage batteries are the sizing parameters in the optimization.

min { 𝐿𝑃𝑆𝑃 (𝑓(𝐴 𝑝𝑣 , 𝑁 𝑏𝑎𝑡 )) [%] 𝐶𝑜𝐸 (𝑓(𝐴 𝑝𝑣 , 𝑁 𝑏𝑎𝑡 )) [€/𝑘𝑊ℎ] (III-2)
Then, all the possible solutions (Apv, Nbat) of the two decision variables will be evaluated on real load demand and outage profile recorded at the university building (AAiT) in Ethiopia. The system evaluation comprises the Apv and Nbat in wider sizing boundaries to a maximum range of 600 m 2 PV panels and 50 batteries, respectively, projected to the project's lifetime of 25 years. Theoretically, the 𝐿𝑃𝑆𝑃 ranges in [0, 1] inclusive, represent fully supplying the load as 0% and not able to supply at all as 100%.

But here the ceiling of the 𝐿𝑃𝑆𝑃 is the grid state during outages at which no backup system is considered, which is 5.9%.

These cost functions are subject to a feasible search space of the PV power produced and battery capacity respectively ranging in:

0 ≤ 𝑃 𝑝𝑣 ≤ 𝑃 𝑝𝑣,𝑝𝑒𝑎𝑘 (III-3) 0 < 𝐶 𝑏𝑎𝑡 ≤ 𝐶 𝑏𝑎𝑡,𝑚𝑎𝑥 (III-4)
These are the integer design variables used for the optimization of the two contradicting objectives (LPSP and CoE) to decide and arrive at the optimal solutions.

Here it can be observed that the upper boundaries either can be set open in the algorithm or maximum values which are able to fulfill the energy balance criteria can be selected. Due to the small number of decision variables, we are able to test the possible set of solutions in these boundaries, which gives us the possible best solutions.

Constraints:

The constraints and operating conditions of the optimization problem include the following:

1. The power balance needs to fulfill that at any time instant, t, the net production (PV and grid) should satisfy the required demand and battery charge/discharge power.

2. The battery state of charge (SoC) is limited by its predefined maximum and minimum operational limits as discussed in chapter II, section II.2.2:

𝑆𝑜𝐶 𝑚𝑖𝑛 ≤ 𝑆𝑜𝐶(𝑡) ≤ 𝑆𝑜𝐶 𝑚𝑎𝑥
3. The battery charging and discharging power at t should respect the predefined maximum values of the corresponding power (Pmax,cha and Pmax,discha).

4. Whenever the signal of the grid status is on 'outage', there should not be any power that will be sent to the grid.

Concept of Pareto front

Recalling the optimization problems in eq. (III-2) one by one, we can represent the set of all possible solutions (Pareto front) in terms of the objective functions and design variables as:

𝒇 𝑜𝑏𝑗,𝟏 (𝑃 𝑝𝑣 , 𝐶 𝑏𝑎𝑡 ) = min(𝐿𝑃𝑆𝑃, %) . . . reliability;

𝒇 𝑜𝑏𝑗,𝟐 (𝑃 𝑝𝑣 , 𝐶 𝑏𝑎𝑡 ) = min(𝐶𝑜𝐸, €/𝑘𝑊ℎ) . . . cost of energy.

In this concept (Figure III-4), the red line is composed of the optimal solutions, which are the blue circlesmaking the Pareto front (optimization with the MOO); and the light-bluesub-optimal or non-optimal dominated solutions. From these, either of the two objectives, the f1(x) or f2(x) functions, could be the results of single objective optimization (SOO), at which one of them will be reached its desired best solution, but deteriorating the other objective function leading to the worst solution. This latter case is not our intention as we already have two conflicting objective functions to be reached at their optimal solution sets. That means, we only either have cost optimal or reliability optimal solution, but not both at a time. Moreover, the region which is to the left of the Pareto front covers the infeasible solutions, but with the extreme point located at the origin, said to be the utopian point, representing an ideal and unachievable solution. The optimal solutions are then the non-dominated solutions of the decision variables where the objective functions (LPSP and CoE) values will no longer be any better than these optimal points. These Pareto solutions result in the cost function value and corresponding positions of the optimization variables, in this case, the PV area and the number of batteries.

The evolutionary algorithm, Non-dominated Sorting Genetic Algorithm (NSGA-II)

with the parameters given in Annex C-parameters of the algorithm, was used for the optimization of the PV-battery system using the aforementioned monthly load profile and outage data. This metaheuristic, which is a very popular optimization algorithm, implemented the Pareto optimality concept has less complexity of use yet is efficient. In its basic principle in the evolutionary cycle, the optimization starts with a random initial population evaluating the objective functions. The solutions will be retained based on the sorted population which are defined by their rank and distance. A more detailed algorithm cycle can be referred to in [START_REF] Roy | Gestion optimale d'un système multi-sources pour un site isolé en mer[END_REF], [START_REF] Afrouzy | A metaheuristic approach supported by NSGA-II for the design and plan of supply chain networks considering new product development[END_REF]. The optimization Pareto front (non-dominated solutions) lets users choose the possible combination of LPSP and associated CoE depending on the acceptable reliability degree.

III.4. Sizing Optimization Results

Based on the real load demand and outage measurement, a techno-economically optimal solution is developed. The proposed sizing optimization in minimizing the LPSP and CoE enables to achieve greater reliability improvement. The frequency of outage occurrence and outage durations are the two crucial factors dictating the size of the power system components. Longer outages occurring during the night entail bigger battery capacity, conversely, such similar outages during sunny days necessitate larger PV area.

The proposed optimal photovoltaic-battery system sizing analysis was implemented in MATLAB.

The sizing algorithm calculates the desired quality of service measured by the LPSP with the corresponding global energy cost, which are set to be minimized. The nondominated solutions result in the cost function value and corresponding positions of the optimization variables, in this case, the PV area and the number of batteries. This solutions' front is the best optimal set from which any of the two objective variables, LPSP or CoE, cannot be further improved without degrading the other. The optimization algorithm is run repeatedly to ensure the algorithm led to these best results of the Pareto, resulting in similar values.

Figure III-5 illustrates the modeling objective targeted at a reduced or zero loss of power is achieved by optimizing the LPSP and CoE for all the possible combinations of the system's PV area and storage battery capacity. Note here that, we are dealing with small LPSPs (with the reference equal to 5.91%), hence the optimization result is shown for smaller LPSPs for a closer look. A zero LPSP can be achieved with relatively high battery capacity and PV array, at a cost of energy of 17c€ per kWh, which corresponds to 182m 2

PV and 39 batteries. The Pareto front set of all the possible solutions indicates that lower costs with an acceptable choice of LPSP than zero shall give the freedom to choose other PV-battery combinations, for users to make an informed decision in implementing such systems. A customer can select the intended level of comfort (LPSP level) as a primary objective with the best possible value of the corresponding CoE.

Longer outage duration surges the LPSP, demanding larger system sizes and hence the energy cost. Except for the likelihood of slightly enhancing the reliability, the LPSP barely responds to PV area and battery sizes after certain values. A further increase in PV sizes with 10 or more number of batteries, notably does not further improve the LPSP. This reliability and system sizing optimization result indicated that there is an important sensitivity of the LPSP index to the PV size at low-cost solution with higher LPSP values.

This shows the role of PV in low-cost systems. Increasing the number of batteries from 5

to 20 leads to a 1% LPSP improvement. At lower LPSPs for improved comfort, the batteries play an important role with higher system cost as expected. However, the LPSP insensitivity beyond this higher battery value highlighted the outage that occurred during non-PV generation has dictated the sizing process. This means, the compensation for the longest outages, or those which occur during the night, is provided by the batteries. System sizing would have been primarily ruled by outages if they occurred during a non-zero PV power generation. Regarding the battery SoC, while it mostly remains at its full charge, the two consecutive stressing outages draw power until it reaches the SoCmin, whereby the customer experiences a brief loss of power leading to unsupplied energy of about 30kWh.

III.4.1. Comparison of Sizing Solutions

It is apparent to realize that the reliability can be highly improved using the hybrid PV/battery arrangement, at a reasonable cost compared to a classical PV-only or storageonly systems. that the prior arrangement will not be able to achieve zero LPSP, due to the effect of the outage occurrence which essentially can be when the solar radiation is not available.

On the other hand, an increased battery capacity can enable the latter configuration to wholly avoid any outages, of course with higher costs, as shown in Figure III-8. In the battery-only system, a zero LPSP can be achieved at a cost of energy for 33c€/kWh, which is more than an 80% increase as compared to the hybrid system. A comparative evaluation pertaining to their corresponding LPSPs of these three options was carried out; which is then assessed based on the related energy costs. Additionally, the best LPSP that the PV-only system can attain is close to 1% (but higher) with a cost of about 10c€/kWh, which we can see such reliability level with the hybrid and battery-only systems can be realized at a 6c€/kWh and 20c€/kWh 𝐶𝑜𝐸 respectively.

In Figure III-9, the system performances of the three configurations are presented.

The battery-only system demonstrates a higher cost than the other two. With a limited maximum capability of reliability improvement, the PV-only system shows a comparable CoE with the hybrid system up until the reliability level of 1.7%. It is clear that for further boost of a steady power system, that is approaching to zero LPSP, the PV/battery system excels the other two. For a recap, the 5.9% LPSP indicated in Figure III-9 is the grid side state of supply without the inclusion of any power system.

The Diesel option -comparison:

Based on the diesel generator model presented under section II.2.3, the CoE for an ideally operated generator without delay to achieve a zero LPSP based on (II-8) is 0.1054€/kWh, this can correspond with the PV-BSS's LPSP of 0.64%. Despite this slightly low cost, in the range of 50 to 60% less than its PV counterparts, issues such as the emission concern, instability, and rise of imported diesel price, spiking maintenance hours and cost, etc. are yet to be considered for such investment.

III.4.2. Sensitivity analysis of the PV-Battery system to cost uncertainties

The various optimal results of photovoltaic panels and storage batteries resulting in the Pareto front solutions are based on the economic values of the different components discussed in chapter II. However, these values could be impacted due to different reasons such as the data source disparity, component advances and their cost variations, etc. Despite the commonly predicted data analytics for the PV technology evolution in the last decades showing a sharp decrease in cost, reports and data recently show an estimated PV cost increase of around 57% in the years 2020-21 [START_REF]Higher PV module prices may point to stable demand and more sustainable pricing trends[END_REF]. On the other hand, IEA showed a decrease of about 50% from 2019 to 2020 in PV costs [START_REF]Evolution of solar PV module cost by data source, 1970-2020 -Charts -Data & Statistics -IEA[END_REF]. Due to this reason, we propose to present the impact of the change in the economic aspect on the overall system cost based on the energy cost. For this, the impacts of the capital investment cost of the photovoltaics, batteries, and the grid electricity cost variations are done and presented below. The results (Table III-1) show that the cost of energy is highly sensitive and influenced due to the battery capital cost variations. The increase in the battery capital cost greatly increases the cost of energy. For example, a 50% decrease or increase in its cost leads to a 41% decrease/increase in the cost of energy for the optimal solution. On the other hand, the CoE is less sensitive to the change in the PV costs while moderately affected by the increase of the grid electricity tariff. As can be seen from the results, a 50% variation in the cost of PV panels leads to a variation of CoE between -3 % and +3% only.

III.5. Conclusion

This chapter presents the sizing optimization of the multi-source system to supply the electrical load of a consumer connected to a highly unreliable grid with frequent and random outages. The optimization takes into account the two conflicting objectives, the load satisfaction rate or reliability and the cost of energy served. It is clear that to fully satisfy the load demand for the consumer in cases of grid outages, the capacities of the power sources (photovoltaics and battery) should be sufficient enough which actually leads to increased total system cost (higher cost of energy). On the contrary, if one wants to have a reduced overall cost, then the reliability and continuity of electrical load supply will face an increase in dissatisfaction rate. Thus, this phenomenon is formulated into the multiobjective (bi-objective) optimization problem and solved using the evolutionary genetic algorithm -NSGA-II taking the PV and battery sizes as the design decision variables. The optimization based on the global energy management strategy then enables us to obtain the Pareto optimal set of solutions.

The results show the advantage of using the multi-source system instead of a photovoltaic or battery-only system: on the one hand, it ensures the satisfaction rate and is able to achieve a zero LPSP, which is not the case for the PV-only option. Secondly, it reduces the cost of energy in comparison to the battery-only option.

It can also be observed that the optimal set of solutions enables consumers, as the one case considered in this study, to choose the desired level of satisfaction rate (LPSP) with the corresponding cost of energy.

Based on the results of the influence of the components' cost variations on the energy cost, it is evident that the cost of energy demonstrated its sensitivity to the variations in the battery cost. The explanation for this is, due to a nighttime occurrence which in turn dictates the sizing problem, the batteries are able to supply the demand. In addition, the comparatively high cost of the battery to that of the PV panel exposes the system's versatility to variations in cost of devices.

On the other hand, the system seems to be resilient to the change in cost of PV and grid electricity, which is in the range of ±2.5% for PV cost and about +6.5% for grid electricity price changes. Note here that, the sensitivity for the grid cost is only for the impact of the increment of the cost to the system.

In summary, this chapter detailed the sizing optimization for the techno-economic optimality of the multisource system (PV, grid with outage, and battery) to improve customers' satisfaction in terms of grid reliability. The global energy management enables us to use the models developed in chapter II and implement the multiobjective optimization to solve the problem. Introduction

The main challenge for the sizing of such auxiliary power systems is the stochastic nature of the outages, which represent the main part of the energy to supply when installed on weak distribution systems, such as the case considered in this work from developing countries. Thus, faced with this problem of modeling network outages, this work proposes to link a sizing optimization approach, carried out on the basis of a sample of measurements, and to verify the effect of the random nature of the outages on the power system quality. In the end, the idea is that for a given sizing it becomes possible to predict its behavior from a statistical point of view. Then, two different methods are developed in order to model and simulate possible grid outages, according to short historical recorded data. To develop and evaluate these models, long-time outage profiles (from five years of measurement) are used, which will later permit a comparative performance analysis of the PV-battery system with projected profiles from the actual historical outage data of a real radial distribution network.

From the developed modeling approaches, the first is a Markov model, which can minimally be implemented with limited outage data available. The second is a Weibull model, which is commonly used to describe extreme phenomena and failure analysis. This latter is more faithful to reproduce the dispersion of the events, both on their duration and on their frequency of occurrence. But it requires a larger initial data set.

In developing the outage models, the fundamental aspects of the modeling objective in general and other model-specific aspects should be carefully examined. The main point includes the objective and scope of the model, historical data representing the model property, and parameters and their relationships. In this chapter, we present the modeling methods which are random and show no predictive information on the causes of outages.

As discussed in the state-of-the-art, these types of outages are quite common and mostly the cases in developing countries attributed to the reasons discussed. Recall that the widely used outage modeling and prediction in the literature are on the basis of spatio-temporal environmental and weather causes as well as seasonal variations [START_REF] Yue | A Bayesian approach-based outage prediction in electric utility systems using radar measurement data[END_REF], [START_REF] Jaech | Real-Time Prediction of the Duration of Distribution System Outages[END_REF].

In this case, the most common structural causes such as widely used radial grid (NB: the structural causes and links are our summary from the measurements made on our case study, and the common causes for extreme weather are synopsized from literature, such as, [START_REF] Yue | A Bayesian approach-based outage prediction in electric utility systems using radar measurement data[END_REF], [START_REF] Jaech | Real-Time Prediction of the Duration of Distribution System Outages[END_REF]).

Thus, in order to develop the models, we have done correlation and periodicity/trend tests on the available historical outage data. The tests for trendenable us to verify the independence of outage in time, otherwise to take into account the effect if it exists.

Furthermore, the correlation is also among the important parameters of outages, if there exist; it then helps to integrate into the simulation or rule-out the dependence.

The Markov model is based on considering a day-night Markov Transition Matrix (MTM), which in this case demonstrates the link with the change of the daily load profile and is adapted accordingly to capture this relationship. On the other hand, the Weibull model is able to describe the disparity of outage durations ranging from a few minutes to hours, which both are presented in this chapter. Finally, from both Markov and Weibull models, outage profiles are depicted by their corresponding LPSP profiles.

IV.2. Outage profile of the selected customer -Recap

In studying electrical grid outages and devising improvement solutions, the power outage/interruption durations and the number of interruptions are mainly the key areas.

With this, the appropriate outage reduction and reliability enhancement solutions, as discussed in section I.2.1, can be carried out.

Outage prediction models usually relate to season, interference of birds, extreme weather (hurricanes, storms, floods, tornados, lightning, heat waves, etc.), or to component aging. The approaches then try to predict according to models developed/trained with these historical data and then based on their associated predictions (of weather, extreme events, inventory of power system components with their aging trends, etc.) to a specific time window of interest. A different scenario is where electric supply outages are not weatherinduced or difficult to have detailed component failure records. In all cases, they all affect and deteriorate electric supply reliability. In this latter case, which is common in developing countries, an approach or model adapted to the particular outage problem and hence characterizing outages needs to be devised. Recalling the case we considered, the outage models are presented and discussed in the subsequent sections.

Thanks to such models, they enable us to characterize and model to further the behavior of the outages, and to be used for testing the effectiveness of the multi-source system in the next chapter. Regarding the outage models, the case dataset for the feeder described in chapter II is used, and later the modeled outage profiles will be used on the case customer, which the outage duration and sequence is as shown below in Moreover, for unbiased modeling and simulation of outages based on the measured data, it is essential to identify and take into account whether there is a trend, interdependencies between outage variables, factors, and their relations affecting the outage records, etc. Hence, we investigated such peculiarities and outage behaviors, whether there exist in the historical datasets.

IV.3. Correlation and periodicity tests

In the first step, whether there exists a seasonal variation and other outage periodicity impacts were tested. In addition, other possible periodicities such as due to regular maintenance activities, building activities variation, day-night cycles, and owing to mild summer-winter seasonal dependences, as is the case in the Ethiopian climate, are tested for their periodic effect, if there exist. For instance, it can be observed a clear seasonal load difference according to seasonal variation during winter and summer in studies [START_REF] Khoury | Optimal sizing of a residential PV-battery backup for an intermittent primary energy source under realistic constraints[END_REF], [START_REF] Alramlawi | Optimal operation of hybrid PV-battery system considering grid scheduled blackouts and battery lifetime[END_REF].

For this, the frequency representation of the outage sequence data using the Fast It is quite clear that there are no correlations between the TBO and the OD. It is also interesting to see that the lag between two consecutive outages has a large dynamic and can range from a few minutes to a few hours.

The periodicity check showed that the distribution outage is totally random without

IV.4. Distribution Grid Outage Modeling and Simulation

One of the other key parts of this work and the main objective of this section, which proposes to evaluate the influence of the stochastic nature of the outage apparition and duration, is the modeling of this stochastic process. Modeling distribution grid outage needs long-term profiles to be accurate and faithful. It then permits to study the nature of outage in depth for further analysis and reliability improvement solutions. Additionally, it becomes highly important to evaluate the performance of the optimal energy system over a long period of time for the techno-economic feasibility. Having in mind that, due to a substandard power system infrastructure in developing countries, the existing fragile power supply service experiences many power outages which greatly affects the service quality [START_REF] Rentschler | Underutilized Potential: The Business Costs of Unreliable Infrastructure in Developing Countries[END_REF]. Hence, the modeling technique is very important in that power distribution system measurements, data acquisition, and storage in real-time and for a long time is very costly, and even hardly possible to find such datasets for most infrastructures in developing countries.

Markov model has been used in several power distribution system problems such as in the evaluation and assessment of interruptions taking into account their stochastic nature.

It is a probabilistic mathematical model which can sufficiently capture system failures and failure rates in the form of state transitions for most dynamic systems with greater flexibility [START_REF] Al-Muhaini | The reliability of power distribution systems as calculated using system theoretic concepts[END_REF], [START_REF] Kovač | Modelling of power system reliability assessment[END_REF].

Alternatively, the Weibull distribution approach is widely used to characterize grid network interruptions. It is one of the statistical approaches used to denote interruption durations or component failure durations as well as the chain states of interrupting grid components [START_REF] Ortmeyer | Characterization of Distribution System Interruption Duration[END_REF]- [START_REF] Van Casteren | Reliability assessment in electrical power systems: the Weibull-Markov stochastic model[END_REF].

Hence in this thesis, the Markov and Weibull models are applied for the simulation of grid interruptions which can respectively characterize the random probabilistic outages and also be able to signify rare outage occurrences which could be overlooked with a classical approach.

IV.4.1. Markov model

It may perhaps be customary to use the historical annual number of outages and corresponding duration data to characterize the distribution grid reliability and further utilize it for various applications, like grid management, outage prediction, and adding hybrid systems. To have a trustworthy reliability study, long-term historical data is usually required. Hence, it is clear that sometimes small data will not be able to adequately describe the distribution system reliability. For each year, the sequence of grid availability and outage may not be found but just some global parameters such as the total number of hours of outage, the number of outages in the year, the average duration of an outage, and the average time between two successive outages, can be given.

Here, considering the case where a small amount of data is available, it is proposed to use a relatively simple to implement Markov model with the transition probabilities of the grid outage with the binary state Markov transitions, from/to 'ON' and 'OFF' states.

The Markov model enables the calculation of the probability of system outage states using the analytic matrix computations. This again gives the durations and frequency of outages.

The Markov model is important due to its computational minimalism and the capability to be built from global data to develop a full representation of stochastic systems, such as a standalone photovoltaic system sizing based on a stochastic cloud cover represented by possible state values of clearness index [START_REF] Bouabdallah | Safe sizing methodology applied to a standalone photovoltaic system[END_REF]. Solar irradiation measurement observations were used to label the possible states of the clearness index parameter. The principal indices of the distribution system outage, i.e., outage duration and frequency, are used for the model and subsequently, any desired length of time can be simulated.

To characterize the stochastic grid outage as a Markov process, some steps are followed. Firstly, the state transitions from state to state are enumerated (see Here, the power grid description can be limited to only two states, which are 'ON' for a working grid, and 'OFF' when an outage occurs. There are then four possible state transition probabilities to calculate, given by:

𝑀𝑇𝑀 = Current state, 𝑆𝑡𝑎𝑡𝑒 𝑁 S𝑡𝑎𝑡𝑒 𝑁 ⃗ ⃗ + 1 [ 𝑃 𝑂𝐹𝐹→𝑂𝐹𝐹 𝑃 𝑂𝐹𝐹→𝑂𝑁 𝑃 𝑂𝑁→𝑂𝐹𝐹 𝑃 𝑂𝑁→𝑂𝑁 ] (IV-1)
When current states are listed in the column and future states in line. An easy way to build the transition matrix is to count the total number of each transition.

IV.4.2. Weibull Model

In the previous section, the Markov Transition Matrix has been proposed to describe the outage occurrence. It has been shown that this matrix is easily built from global annual data, such as average and total outage durations. The Markov approach also permits to consider the daily periodicity through the use of two different MTMs, used respectively during the day and the night time. As opposed to this, MTM is very limited to take into account the shape of the probability distribution for both parameters that are the time between outages and the outage duration, because of the limited number of setting parameters with this approach. Especially, it is not possible to take into account the longest outages without changing the step time or the order of MTM. Then, a second model to address this difficulty is proposed, which is more adapted to approximate the extreme outage events. This time, the outage duration (OD) and time between outages (TBO) are directly based on statistical distributions.

The Weibull distribution is able to closely fit with the 5-year outage data, consequently, for the case study of this work, the standard two-parameter Weibull distribution is used to model the distribution grid outage. Outage durations or repair times of a component can be represented by this distribution [START_REF] Tao | Power quality & reliability assessment of distribution system considering voltage interruptions and sags[END_REF], [START_REF] Van Casteren | Reliability assessment in electrical power systems: the Weibull-Markov stochastic model[END_REF], and its cumulative distribution function (cdf) for a random variable x can be given as: and TBO. This once again results in building the grid outage sequence, the grid being in up state or down state, for the simulation time frame considered.

𝑐𝑑𝑓 = 𝑓(𝑥|𝑎, 𝑏) = 1 -𝑒 -( 𝑥
We have selected the 2015 yearly distribution grid outage as a reference year, which has relatively average outage durations, number of outages, and LPSP values. For calculating the LPSPs for each of the 5-years, recall that we considered the monthly load measurement to be the same in the 12 months. This customer building is assumed to be connected to the 11kV feeder under consideration, sections II.3 and IV.2.

IV.5. Predictive Outage Modeling Results: Markov, Weibull

Contrary to the weather conditions that cause distribution systems outages elsewhere in the world, in this work's context, there is no wide variation in season and weather conditions. This can be also seen from the two major decisive factors of the PV system design, global irradiation, and ambient temperature data (see Figure .

For the Markov method, the resulted MTM of probabilities for the daytime and nighttime are: The transition probabilities apparently show a higher chance of staying in the 'off' or 'on' state, if the grid is already in those respective states initially. For example, in the daytime MTMd (1,1), the probability of the grid remaining in its initial position of 'off' state It is obvious that the Weibull distribution, with only two parameters, is able to accurately reproduce the measurements results. On the contrary, the Markov Matrix approach, although it is simpler to implement and it can differentiate the day and night phases, it does not allow the measured distributions to be accurately followed. Especially, long and unlikely or rare events are very difficult to be captured by the Markov approach. This confirms the idea that using a Markov process to describe grid outages is a simple, accessible, and advantageous method when the data (including the type of data required for the model and sufficiently large datasets with relevant details) are not available, but that it will not allow to be accurate to the actual outage profiles. Weibull is able to demonstrate a more realistic simulated profile as compared to Markov, by also capturing rarely occurring outage events. Similarly, the Markov model shows lesser distribution while Weibull shows better spread appearing to better cover the data profiles. The outage data without any power system in the years 2012-2016, i.e., the grid side power supply state without any backup or standby system is shown in broken vertical lines superimposed on these model simulation probability curves.

𝑀𝑇𝑀 𝑑𝑎𝑦 = [ 0.

Comparison of simulated outage profiles

For both models, their mean LPSPs are closer to the 5-year average, which is 2.53%, with marginal dispersion (Table IV-1), showing the effectiveness of the models, noting the Markov median equal to the data average and again the Weibull standard deviation showing better variability. This ensures the proposed simulation models tend to yield a comparable practical outage scenario. Integrating hybrid systems like in this study, better outage management, system reconfiguration, system planning and so on can then be examined and reliability will effectively be addressed based on such sufficient system profile.

Finally, the 1000-year simulated outage profiles from both models are tested for their performance with the optimal system sizing. This will enable both to evaluate the outage models results as well as the performance of the optimal hybrid system, which discusses this aspect and is presented in the next chapter.

IV.6. Conclusion

This chapter presents techniques to model and develop outage profiles taking into account the grid outages which are very common in developing countries with peculiar characteristics different from modern grids such as in developed countries.

In modeling the grid outages, we take into account two key aspects based on the outage data and electric supply systems in developing countries. Sufficiently detailed and large outage data is required to develop and train models. However, it is hardly possible to have detailed outage data, for example, the number of connected customers, the load connected, the location of failed components of the network, and component-based failure and repair history. It is costly for utility operators to have such a robust outage management system and collect as much detailed interruption information as possible for electricity infrastructures in developing countries. On the other hand, the causes and power system failures resulting in supply outages are random and not associated with any deterministic factors, with a higher unreliability and frequent random outages. Thus, for these two reasons, grid outage models are developed taking into account these cases. To this end, the models also help to explain the situation of the outage and also to simulate the state of outage in future time.

Markov model which captures the day-night repartitions as is demonstrated in the periodicity test is able to simulate outages as stochastic process from historical data. It is also particularly important when we have small outage record data and at the same time when the datasets are global parameters. Moreover, the Weibull is constructed in such a way that it is able to accurately take into account extreme outage events such as long but less frequent outage durations, which the Markov does not.

From the model results, it has been shown that the transition probabilities from the Markov model apparently show a higher chance of staying in the 'off' or 'on' state, if the grid is already in those respective states initially. The Weibull distribution, with only two parameters, is able to accurately reproduce the measurement results. On the contrary, the Markov approach, although it is simpler to implement and can differentiate the day and night phases, it does not allow the measured distributions to be accurately followed.

Lastly, for both models, we have done a test to check whether there exists a seasonal variation and other outage periodicity impacts. Possible periodicities and trends such as due to regular maintenance activities, day-night cycles, and owing to mild summer-winter seasonal dependences were tested. Such periodicities and even correlated failures or even dependencies on factors like temperature could be incorporated in the model, if they had existed, but found to be non-existent. This testing gives us the privilege to incorporate the only existing daily periodicity within the Markov model.

sequences are considered in order to test the performance and the robustness of the proposed PV-battery system. To do that, the proposed idea is to test an optimal solution, obtained for a limited measurement data set (a few days or few weeks), on a large number of simulated outage scenarios obtained from both Markov Transition Matrices and Weibull distribution. The methodology is illustrated in Figure V-1. Based on the measurement sequence, optimization is achieved and the set of optimal solutions is obtained, on the base of CoE-LPSP objectives.

In parallel, the measurement data are used to fit the MTM and Weibull distribution parameters. It must be noted that the MTM approach also makes it possible to find the parameters of the transition matrix only from annual global data, as given by equation To illustrate this methodology with the proposed case study, PV-battery system optimal solutions are adopted, with 0.0%, 0.1%, 0.5%, 1.0%, 1.5%, and 2% LPSPs as reference systems based on the 2015 optimal design. The PV and battery sizes and corresponding energy costs for the reference system are given below,in Table V-1. values, the performance of the PV-battery system ensures greater reliability under these circumstances. For example, the performance at LPSP = 0.5%, the optimal size line graph dominantly satisfies load demand under such simulated outages, i.e., the Pareto graph (solid line) crosses above the 3 rd quartile (well above the median) of the corresponding boxplot.

Note, the red lines inside the boxes are the median.

Nevertheless, the Weibull shows the Pareto line crossing through the median in the 0.5% LPSP case. This phenomenon is reversed for higher LPSP values. Taking the 1.5% value, it can be seen that the point at which the front crosses the boxplots is vice versa to the above-explained case. variables. The longer the outage duration and/or occurring at night when there is no PV production, the higher the battery size. And also the higher the load demand during the day, the larger the PV area.

Furthermore, the outage modeling by the Markov and Weibull techniques shows a very good estimate closer to historical data. The latter with minimal computational requirement than its counterpart gives a dispersed outage profile weighing in reference to the historical average LPSP. Conversely, the reduced dispersion in Markov offers an advantage for the optimal system to be able to satisfy the load-demand-outage profile to a higher degree of reliability. This is because the simulated outage profiles tend to be situated well below the historical average metric. Its requirement of crude outage data for the modeling is an advantage, but with slightly higher computations than the Weibull.

V.3. Conclusion

In the case of modern grids, outages resulted from weather-related causes, planned maintenance and even if other causes induce supply outages, they are predominantly known before they occur which enables utilities for better outage minimization strategies, preventive measures and even avoiding them. From the consumers' perspective, they will be made aware of coming outages for planning and for optional supply options such as selfgenerations.

But in the case of developing countries whereby the power system is prone to unpredicted and severe outages caused by substandard infrastructures, supply deficiency, lack of preventive maintenance hence aged power system components (especially the distribution sections), etc. are exposed to frequent failures and causing long duration outages and other unpredicted causes. Hence, for developing solutions to either avoid or reduce outages and also to better understand the outage phenomena, we proposed outage modeling which later is then used for performance and effectiveness analysis of the renewable power systems and also evaluate their behavior.

For frail distribution systems in developing countries, it is imperative that the use of smart control and monitoring devices and logging real-time power system data is not readily available. This further entails the lack of data for detailed modeling and assessment.

For this, an outage model is proposed which can be developed and able to simulate outages with minimal data requirement. The Markov outage model is able to account for the day and night repartition and with minimal data requirement. On the other hand, another outage model is the Weibull model which requires more data but is able to represent the wide outage variation characteristics as it is commonly used to capture extreme phenomena in statistical analyses. This feature is an addition as compared to the Markov, in which the behavior is based on the transition probabilities of states of the power supply (either 'on' or 'off' states).

Moreover, the comparative analysis of the various configurations (PV only, battery only and the hybrid solution) connected to the weak grid demonstrates the advantage of the multi-source PV-battery storage system over the counterparts. Although the battery storage-only can be able to satisfy the load, it is the costly option. But this is not true for the PV only, as this is an option which is unable to fully achieve reliable supply. Also, the economic sensitivity of the proposed system illustrates that an increase or decrease on the storage cost highly influences the system. A 50% battery cost variation leads to an economic sensitivity of the system by more than 40%. The system is less sensitive to PV cost variations and grid electricity cost. Here, it can be inferred that the storage sizing is more pronounced due to outages that occurred at night or during cloudy days with less PV production.

On the other hand, outage evaluation and reliability study usually demand detailed information about outage-causing factors, sufficiently high data records, and component/ subsystem level contributing elements. Nevertheless, it is unlikely to have those datasets for distribution grids dealing with recurrent outages such as in developing countries. Hence, we developed a methodology to model and extend longtime outage simulations as needed, using global outage parameters and a small amount of data.

A new approach to outage modeling using the existing Markov stochastic and Weibull statistical models has been introduced. The proposed models are able to generate large outage profiles with sound computational effort and effectiveness. These profiles can then either be used for various applications, including hybrid system inclusion to a weak grid for outage reductions and hence reliability improvement. Likewise, as above, real outage data from a stochastic grid in a developing country is applied for the two modeling approaches. The Markov and Weibull models have the advantages of being able to capture the day-night repartition, which matches with the load profile and the ability to take less frequent extreme outages into account, respectively. The results show, such sufficiently disperse outage simulation using the Weibull model will lead to a notable realistic scenario in order to avoid oversized hybrid systems. On the contrary, the PV-battery system on Markov simulated outage profiles showcased the reference system seems to perform above the expected reference LPSP. The effectiveness of the proposed methodology for the optimal system is validated on a large set of simulated outage profiles with these two models. Regardless of the effort to verify the absence of any correlation among the outage parameters and any periodicities, factors such as weather conditions, changes in the load demand, future grid maintenance routines and so on may affect and change the expected LPSP.

Perspectives

Further on these approaches, additional demand side management techniques and load monitoring such as load prioritization or modifying load profile could be explored to further improve the cost optimality and increase power supply security during outages. The load prioritization can lead to a lower cost of energy (CoE) through reduced sizing of the multi-source devices which allows supplying the critical loads only identified by the load prioritization (perhaps load shifting could also be an option, such as moving less critical loads to the times of high production). Besides, considering other consumer groups, as only one type of power consumer group (in a commercial building) load and outage profile is used in this work, allows to take into account the load variation for practicable solution.

Moreover, there could be thermal demands for consumers such as the hotel industries or cold storage, hence posing the possibility to include other multi-source devices and technologies. Besides, the application of the methodology for other geographical locations can also be envisaged where other renewable resources could be explored and study their impacts on the sizing optimization.

Regarding the power system devices, only battery storage devices are considered for the system flexibility and to ensure load supply for the times when PV production is not sufficient and in cases of outage occurrence. The prospect here is, other types of storage can be integrated by considering the various length of outage durations and the number of outages, as well as the load demand at the time of outage occurrence. This can help to introduce an appropriate technology and size of storage which enables integrating the storage that matches the load size and operation times with the outage characteristics. For instance, for shorter durations (like momentary outages), storages that are capable to store energy for shorter times, such as flywheels or supercapacitors can be modeled as an option).

In such cases with more storage options, due to relatively cheaper grid costs in the case study, it can be considered to charge these storages from the grid at off-peak hours.

In the outage modeling, relatively smaller datasets are used, only 5-years distribution grid outage data. More accurate outage model outputs can be achieved through measuring and collecting more outage data, hence better modeling applicability. As to the individual models, the Markov and Weibull grid outage models assume a constant probability of outage occurrence and are based on statistical knowledge of historical outages,

respectively. An extended work by appraising the designs that are able to account for particular characteristics such as grid network degradation and other future time behavior of the grid can also be one potential area.

Likewise, the use of a different optimization tool and experimental validation in a lab-scale system will permit to support the work of this thesis and extend the conclusions drawn. This also allows examining the adequacy of such proposed systems to guarantee the continuity of power supply under outages and reinforce the findings before real-life applications.

On top of these, an extensive study on the existing backup/standby generators' impact on air pollution and emissions, and an extended study on quantification of the emission from the self-generation using diesel generators in cases of outage can give a favorable insight for the utility customers to consider renewable resource-based systems such as this. It can also highlight the suitability of the systems.

With that in mind, many research perspectives of this work can be envisaged as discussed above, at which more grid outage data and respective load demand data availability can eventually help to a more practical advancement of such solutions. Titre : Amélioration de la fiabilité de l'alimentation électrique par le développement d'une solution multisources sur un réseau électrique faible Mots clés : production distribuée, panne de réseau, modélisation, optimisation de dimensionnement multiobjectif, système hybride, énergie renouvelable, photovoltaïque-batterie, gestion de l'énergie.

Nomenclature

Résumé : Les pannes de réseau stochastiques, fréquentes et de longue durée, sont très courantes dans les pays en développement, ce qui pose un problème insupportable de fiabilité de l'alimentation électrique. Les clients sont souvent obligés d'utiliser des générateurs diesel pour surmonter ces pannes imprévisibles et récurrentes. Cette thèse propose un système multi-source solaire photovoltaïque et batterie intégré à un réseau faible basé sur des données de pannes stochastiques enregistrées et un profil de charge. L'objectif de dimensionnement est de trouver le meilleur compromis entre les contraintes technico-économiques, d'assurer la continuité de l'approvisionnement (réduction des pannes) et la minimisation du coût de l'énergie, ce qui met en oeuvre une optimisation de dimensionnement multi-objectifs pour obtenir un ensemble de solutions Pareto optimales.

Ce travail porte également sur la modélisation et la simulation stochastiques des pannes de réseau. Deux modèles présentant les avantages d'une mise en oeuvre minimale sans nécessiter de données détaillées sur les pannes et la capacité de prendre en compte les événements extrêmes dans les pannes sont développés. Le modèle de Markov (matrice de transition de Markov) est capable de capturer les répartitions jour-nuit démontrées dans le test de périodicité. Le modèle de Weibull est capable de prendre en compte de manière précise les événements extrêmes des pannes tels que les durées de pannes longues mais moins fréquentes. Les résultats nous permettent également de tester l'efficacité et la performance du système multisource (PV-batterie) conçu sur un grand nombre de profils de pannes de scénarios potentiels réalisables pendant la durée de vie du système renouvelable.

Title : Improving the Reliability of the Electrical Supply by the Development of a Multi-source Solution on a Weak Electrical Network Keywords : distributed generation, grid outage, modeling, multiobjective sizing optimization, hybrid system, renewable energy, photovoltaic-battery, energy management.

Abstract : Frequent and long-duration stochastic grid outages are very common in developing countries, thus posing unbearable power supply reliability problem. Customers are often forced to use diesel generators to overcome these unpredictable and recurrent outages. This thesis proposes a solar photovoltaic and battery multisource system integrated to a weak grid based on a real recorded stochastic outage data and load profile. The sizing objective is to find the best trade-off between the techno-economic constraints of ensuring the continuity of supply (outage reduction) and minimization of the cost of energy which implements a multi-objective sizing optimization to obtain a set of Pareto optimal solutions. This work also deals with stochastic grid outage modeling and simulation, at which two models with advantages of minimal implementation without requiring a detailed outage data and ability to take into account extreme events in outages are developed. The Markov model (Markov Transition Matrix) is able to capture the day-night repartitions demonstrated in the periodicity test. The Weibull model is able to accurately take into account extreme outage events such as long but less frequent outage durations. The results also allow us to test the effectiveness and performance of the designed multi-source (PV-battery) system on a large number of outage profiles of potential feasible scenarios over the life span of the renewable system.

Fitsum

  ሆይ፥ የአባትህን ምክር ስማ፥ የእናንትህንም ሕግ አትተው፤ ለራስህ የሞገስ ዘውድ ለአንገትህም ድሪ ይሆንልሃልና።» መጽሐፈ ምሳሌ ፩፥፰-፱ iv Résumé v

I. 1 .

 1 INTRODUCTION ............................................................................................................. 7 I.1.1. Electrical Power Systems: The basics with respect to outage ................................. 9 I.1.2. What is the notion of interruption or outage? ....................................................... 12 I.1.3. Brief overview of global power supply outage and related reliability .................. 13 I.1.4. Power supply interruption status quo: survey and implications ........................... 15 I.2. STATE-OF-THE-ART .................................................................................................... 21 I.2.1. Current solutions and approaches for reducing outages ...................................... 21 I.2.2. Sizing optimization of multi-source systems considering outages......................... 35 I.3. CONCLUSION AND THESIS CONTEXT ........................................................................... 36 CHAPTER II: MODELING OF THE MULTI-SOURCE ENERGY SYSTEM ............. 40 II.1. INTRODUCTION ........................................................................................................... 40 II.2. ARCHITECTURE OF THE MULTI-SOURCE SYSTEM ........................................................ 41 II.2.1. Solar Photovoltaic Model .................................................................................. 45 II.2.2. Storage Battery Model ....................................................................................... 47 II.2.3. Modeling a Diesel Generator ............................................................................ 51 II.2.4. Reliability Model ................................................................................................ 52 II.2.5. Economic Model ................................................................................................ 54 II.3. DESCRIPTION OF THE CASE STUDY .............................................................................. 55 II.4. CONCLUSION .............................................................................................................. 63 CHAPTER III: TECHNO-ECONOMIC SIZING OPTIMIZATION BASED ON MEASUREMENT.................................................................................................................. 66 III.1. INTRODUCTION ........................................................................................................... 66 III.2. RULE-BASED ENERGY MANAGEMENT FOR OPTIMIZATION ........................................ 67 Table of Contents x III.3. DEFINING THE MULTIOBJECTIVE OPTIMIZATION PROBLEM ........................................ III.4. SIZING OPTIMIZATION RESULTS ................................................................................. III.4.1. Comparison of Sizing Solutions ......................................................................... III.4.2. Sensitivity analysis of the PV-Battery system to cost uncertainties ................... III.5. CONCLUSION .............................................................................................................. CHAPTER IV: GRID OUTAGE MODELING .................................................................. IV.1. INTRODUCTION ........................................................................................................... IV.2. OUTAGE PROFILE OF THE SELECTED CUSTOMER -RECAP ........................................... IV.3. CORRELATION AND PERIODICITY TESTS...................................................................... IV.4. DISTRIBUTION GRID OUTAGE MODELING AND SIMULATION ..................................... IV.4.1. Markov model .................................................................................................... IV.4.2. Weibull Model .................................................................................................... IV.5. PREDICTIVE OUTAGE MODELING RESULTS: MARKOV, WEIBULL .............................. Comparison of simulated outage profiles ......................................................................... IV.6. CONCLUSION .............................................................................................................. CHAPTER V: PERFORMANCE EVALUATION AND VALIDATION OF THE MULTI-SOURCE SYSTEM............................................................................................... V.1. INTRODUCTION ......................................................................................................... V.2. OPTIMAL SYSTEM PERFORMANCE ON SIMULATED OUTAGES ................................... V.3. CONCLUSION ............................................................................................................ CONCLUSIONS AND PERSPECTIVES ......................................................................... ANNEXES ............................................................................................................................ ANNEX A: THE DIESEL GENERATOR EFFICIENCY CURVE. ..................................................... ANNEX B: SIMPLIFIED LAYOUT OF ADDIS CENTER SUBSTATION WITH FEEDERS. ................ ANNEX C: PARAMETERS OF THE NSGA-II ALGORITHM. ..................................................... ANNEX D: SAMPLE GRID OUTAGE SEQUENCE SIMULATION USING MARKOV IN COMPARISON WITH THE 2015 MEASURED DATA. ....................................................................................... NOMENCLATURE ............................................................................................................. VARIABLES ......................................................................................................................... UNITS .................................................................................................................................. REFERENCES .....................................................................................................................

Figure I- 1

 1 Figure I-1 Reliability and Cost Interdependence [14]. ............................................................

Figure I- 2

 2 Figure I-2 Bathtub Curve [12], [14]. ........................................................................................

Figure I- 3

 3 Figure I-3 Electricity outage in selected regions of the world, World Energy Outlook 2019 [4]. ..................................................................................................................................................

Figure I- 4

 4 Figure I-4 Common types of impacts caused by outage (source: adapted from [3], [6], [20]). ..................................................................................................................................................

Figure I- 5

 5 Figure I-5 Some of the direct and indirect negative implications due to outages on the different economic sectors and end-users (source: adapted from [3], [6], [20]). ...................................

Figure I- 6

 6 Figure I-6 Power outage durations and percentage share of loss in sample firms due to outage in low-and middle-income countries [6]. ................................................................................

Figure I- 7

 7 Figure I-7 Grouped approaches for outage reduction and reliability improvement. ...............

Figure I- 8

 8 Figure I-8 Sample illustrative diagram of a multi-source system ............................................

Figure I- 9

 9 Figure I-9 Classifications of energy storage technologies based on system power output, storage capacity and availability time [84]. .............................................................................

Figure I- 10

 10 Figure I-10 Thesis framework and contents of subsequent chapters. ......................................

Figure II- 1

 1 Figure II-1 The overview of techno-economic modeling of the multi-source system. ...........

Figure II- 2

 2 Figure II-2 Power flow schematic of the considered system. ..................................................

Figure II- 3

 3 Figure II-3 Geographical location of the selected consumer. ..................................................

Figure II- 4

 4 Figure II-4 Load demand profile of the building illustrating the weekdays and weekend load behavior, down sampled at 1h step time. .................................................................................

Figure II- 5 A

 5 Figure II-5 A typical-day grid side electrical supply (Oct 31, 2018). ......................................

Figure II- 6

 6 Figure II-6 Causes contributions to the outage durations (top) and number (bottom) for the 2012 -2016 records. ................................................................................................................

Figure II- 7

 7 Figure II-7 Monthly numbers of outages and durations in year 2016. ....................................

Figure II- 8

 8 Figure II-8 Feeder ADC-9 monthly total outage duration and monthly average load.............

Figure II- 9

 9 Figure II-9 Simplified schematic of the diesel backup system operated through an ATS. .....

Figure II- 10

 10 Figure II-10 One week total average load/power demand of the building in kW (Nov 12 -18, 2018). .......................................................................................................................................

Figure II- 11

 11 Figure II-11 Solar irradiation and ambient temperature, during one year (2015), in Addis Ababa, Ethiopia........................................................................................................................

Figure III- 1 70 Figure III- 3 70 Figure III- 4 76 Figure III- 6 77 Figure III- 7 78 Figure III- 8 78 Figure III- 9 79 Figure IV- 1 85 Figure IV- 2 88 Figure IV- 4 92 Figure IV- 6 93 Figure IV- 7 96 Figure IV- 8 97 Figure V- 1 104 Figure V- 2 106 Figure 0 - 1 115 Figure 0 - 2

 170370476677778878979185288492693796897110421060111502 Figure III-1 Rule-based energy management strategy flowchart. ...........................................

  .................................................................................................................................................. 51 Table II-4 The economic parameters and values of the system considered. ........................... 55 Table II-5 Distribution grid outage 5-years data. ..................................................................... 57 Table III-1 Sensitivity results of the cost of energy (CoE) with the variation in capital costs of components and grid electricity price. ..................................................................................... 80 Table IV-1 Summary statistics of LPSPs for the two models. ................................................ 97 Table V-1 Reference optimal systems' PV and battery sizes with their corresponding CoE, based on measurements data of the year 2015. ...................................................................... 104 Table 0-1 The Parameters of the NSGA-II algorithm. .......................................................... 117

Finally

  , the general conclusions of the thesis work, the scientific works carried out and the main results summarize this manuscript. It then frames the limitations in this work and perspectives to underscore the broadening and future continuation of this work. The following communications of publications are from this work -International peer-reviewed journal: F.S. Kebede, J.-C. Olivier, S. Bourguet, M. Machmoum, "Reliability Evaluation of Renewable Power Systems through Distribution Network Power Outage Modelling", Energies, 14, 3225, 2021. International Conference: F. S. Kebede, S. Bourguet, J. -C. Olivier, "Photovoltaic System Sizing for Reliability Improvement in an unreliable Power Distribution System", 2020 Fifteenth International Conference on Ecological Vehicles and Renewable Energies (EVER), pp. 1-8, Monaco, 2020. National Conference: F. S. Kebede, J. -C. Olivier, S. Bourguet, "Photovoltaic System integration in Unreliable Power Distribution System", Symposium de Génie Électrique (SGE 2020),

Figure I- 1

 1 Figure I-1 Reliability and Cost Interdependence [14].

  If the interruption is caused due to an outage of a component or combination of components, which is usually the case, the reliability of the component/s comprising the system is represented by the bathtub curve, FigureI-2. It shows that the failure rate and hence its reliability is a function of time, and then gives the operation life of a component.

Figure I- 2

 2 Figure I-2 Bathtub Curve [12], [14].

  supply in developing countries due to poor infrastructure, aged components, demandsupply imbalance, and other factors worsened the situation. Here we try to detail the situation of interruptions or outages and the associated consequences it could cause on consumers and utility companies. In lower-and middle-income countries which are fundamentally challenged due to unreliability, unscheduled power outages or supply shedding are quite common, see Figure I-3[START_REF]World Energy Outlook 2019[END_REF].

Figure I- 3

 3 Figure I-3 Electricity outage in selected regions of the world, World Energy Outlook 2019 [4].

Figure I- 4

 4 Figure I-4 Common types of impacts caused by outage (source: adapted from [3], [6], [20]).

Firms: 

  Lost production  Spoiled products  Damaged products  Damaged equipment  Idle labor and facility  Production process restart  etc. Infrastructure:  Health  Water  Transport  Telecommunication, internet services.  Basic building function (elevators, heating/cooling)  etc.

Figure I- 6

 6 Figure I-6 Power outage durations and percentage share of loss in sample firms due to outage in low-and middle-income countries [6].

Figure I- 7

 7 Figure I-7 Grouped approaches for outage reduction and reliability improvement 1 .

Figure I- 8

 8 Figure I-8 Sample illustrative diagram of a multi-source system (diagram made on HOMER 2 ).

  the selected technology, space availability and access, and techno-economic evaluations assist in the selection and development of such solutions to supply during outages. Other important details like the daily and seasonal variations of the available resources, climatic characteristics, and outage occurrence behaviors in relation to day-night and seasonal alternations facilitate the technology selection and their sizing. Having all these, the selection and sizing of components highly depend on the resources through geographical location, the techno-economic soundness, and the outage profiles.

Figure I- 9

 9 Figure I-9):

Figure I- 9

 9 Figure I-9 Classifications of energy storage technologies based on system power output, storage capacity and availability time [84]. Where, CAES: Compressed air energy storage, FBES: Flow batteries energy storage, FES: Flywheel energy storage, HES: Fuel-cells hydrogen energy storage, PHS: Pumped hydro storage, SES: Supercapacitors energy storage, SMES: Superconducting magnetic energy storage, TES: Thermal energy storage.

Figure I- 10

 10 Figure I-10 Thesis framework and contents of subsequent chapters.

Figure II- 1 Figure II- 2

 12 Figure II-1 The overview of techno-economic modeling of the multi-source system.

Figure II- 2

 2 Figure II-2 in its simplified form.

  1 (and Figure II-3).

Figure II- 3

 3 Figure II-3 Geographical location of the selected consumer.

Figure II- 4

 4 Figure II-4 Load demand profile of the building illustrating the weekdays and weekend load behavior, down sampled at 1h step time.

Figure II- 5 A

 5 Figure II-5 A typical-day grid side electrical supply (Oct 31, 2018).

•

  Gstc: the incident solar irradiation at STC (standard test conditions) [W/m 2 ]; • Gtot: global solar radiation received on the plane-of-array (PoA) / tilted panel surface [W/m 2 ]; • ηpv: the PV panel overall efficiency [%]; • t: simulation time step [min.].

Figure II- 6

 6 Figure II-6 also shows the contributions of the causes for the outage in the same period. On top of that, the average outage duration sustaining in a single event (i.e. minutes per one outage event) is shown at the top of the bar graphs. In all the years, the operational outage has the highest durations, and of course, it has the highest number of events.

  Figure II-8 shows the total monthly outage duration and the corresponding average load for the year 2016. To give an economic perspective during the longer outage duration in June, the total energy not supplied (ENS) is about 264 MWh.

Figure II- 6

 6 Figure II-6 Causes contributions to the outage durations (top) and number (bottom) for the 2012 -2016 records.

Figure II- 7

 7 Figure II-7 Monthly numbers of outages and durations in year 2016.

Figure II- 8

 8 Figure II-8 Feeder ADC-9 monthly total outage duration and monthly average load.

Figure II- 9

 9 Figure II-9 Simplified schematic of the diesel backup system operated through an ATS.

Figure II- 10

 10 Figure II-10 shows the load profile for one week with a time step of one minute.

Figure II- 10 

 10 Figure II-10 One week total average load/power demand of the building in kW (Nov 12 -18, 2018).

  load curve for the sizing of the multi-source installation. A more detailed climatological data is presented below.

Figure II- 11

 11 Figure II-11 Solar irradiation and ambient temperature, during one year (2015), in Addis Ababa, Ethiopia.

Figure III- 1

 1 Figure III-1 Rule-based energy management strategy flowchart.

Figure III- 2

 2 Figure III-2 Multi-source system model and respective power flow.

Figure III- 3

 3 Figure III-3 Binary state grid sequence.

Figure III- 4

 4 Figure III-4 Pareto solutions 8 .

Figure III- 5

 5 Figure III-5 Photovoltaic-battery system Pareto optimal sizing.

Figure III- 6

 6 Figure III-6 Power flow dispatch of the selected system on a one-month time.

Figure III- 7

 7 Figure III-7 and Figure III-8 demonstrate the reliability indicators of the performance of the other two system configuration variances: PV-only and battery-only. It is obvious

Figure III- 7

 7 Figure III-7 Power system performance (LPSP [%]) with PV-only configuration.

Figure III- 8

 8 Figure III-8 Power system performance (LPSP [%]) with Battery-only configuration.

Figure III- 9

 9 Figure III-9 System comparison of the three configurations.

  topology lacking sufficient redundancy, weak grids and time-worn network components, demand-supply imbalance, and human factors (see Figure IV-1), require a technique mainly formulated for such scenarios.

Figure IV- 1

 1 Figure IV-1 Power outage causes and links.

  Figure IV-2.

Figure IV- 2

 2 Figure IV-2 Outage sequence (upper) and corresponding outage durations (lower) during the measurement period -Oct 24 to Nov 21, 2018.

Fourier

  Transform (FFT) of the autocorrelation function is used. What this basically does is analyze and extract the frequency components of the outage data and present it as a function of the time period for observation. This can help either to select an appropriate model for outage simulation or modify currently existing methods. It can also help which DG technologies to consider depending on resource availability and then integrate any periodicity into the rule-based energy management.

Figure IV- 3

 3 Figure IV-3 Correlation diagram between (a) Outage Duration-OD and Time Between Outages-TBO, (b) between two consecutive outages, and (c) frequency analysis to highlight the outage periodicities.

  any long time periodicities, concluding there is no cyclic nature with the season as well as neither in yearly, monthly nor weekly times. Characteristics of outages such as effects of periodic preventive maintenance (PM), correlated failures or even dependencies on factors like temperature could be incorporated into the model. It could also have the possibility to show failures related to seasonal changes, despite severe weather impacts not considered to contribute to grid electricity supply failures. For such cases, if demonstrated they do exist, outage models (failure and reliability), could have given the knowledge to apply timevarying approaches or by incorporating the failure-causing variables. Hence, this result shows the failure leading to supply outages is random and no deterministic factors (such as PM) exist, which essentially leads to the need for a stochastic outage model. In contrast, a clear daily periodicity is visible. FigureIV-4shows the time of occurrence of the failure during the day. Most outages occur during the day and seem to be linked to the global load demand. In other words, we can say that the probability of outage occurrence is positively correlated to the power requirements of grid users.

Figure IV- 4

 4 Figure IV-4 Histogram of 5-years data of outage occurrence during the day.

Figure IV- 5 )

 5 to define the size of the Transition Matrix. Secondly, Markov Transition Matrix (MTM) is built from the recorded data according to all the transitions encountered. Once this MTM is obtained, simulated scenarios of grid outage can be generated on the base of the MTM probabilities. This Markov probability model assumes the next outage event is absolutely dependent on the current state only, which is widely used in reliability modeling[START_REF] Kim | Reliability Modeling and Simulation in Power Systems With Aging Characteristics[END_REF].

Figure IV- 5 A

 5 Figure IV-5 A schematic representation of MTM transition states.

(IV- 3 )

 3 To capture the daily occurrence and periodicity (Figure IV-3 (c)), more adapted Markov transition matrices (MTM) are repartitioned to day and night MTMs, based on the corresponding actual outage occurrences data (see Figure IV-4). Hence, the above approach enables the MTMs for better simulated prediction and are calculated from the global data (2012-2016); the historical outage data enables to adapt the MTM simulation model using the said global parameters.

Figure IV- 6

 6 Figure IV-6 Typical time series of the state of a power grid.

a

  and b are the scale and shape parameters; a, b > 0, and are derived from the data. By fitting a Weibull cdf to the set of historical data, the values of a and b are determined; these characteristic parameters permit to get a calibrated two-parameter Weibull distribution with the actual distribution grid outage data. For clarity, the grid outage and restoration sequence is constructed from the simulated TBO and OD, see Figure IV-6 for grid sequence depiction. Based on those two global parameters, OD and TBO, from the 5-years distribution grid operating history, a uniformly distributed pseudorandom number generator in the interval (0,1) were used to build the distribution of simulated OD

  is 0.978, refer Annex D-sample grid outage sequence for the simulation based on 5-years data with these transition probability matrices compared to actual data. With regard to the Weibull process, the estimated a and b values with a 95% confidence interval through the maximum likelihood estimates technique are: for TBO: a = 1620 (min), b = 0.77, and for OD: a = 36 (min), b = 0.56.

Figure IV- 7

 7 Figure IV-7 Cumulative distribution function of outage duration and time between outages, for measurements (5-years data), Markov transition matrices (day and night) and Weibull distribution.

Figure IV- 7

 7 Figure IV-7 shows the cumulative distribution functions result for both OD and TBO.

Figure IV- 8

 8 Figure IV-8 Cumulative distributions of the simulation of grid outage sequences, without power system, by using Markov and Weibull models.

((V- 1 )Figure V- 1

 11 Figure V-1 Proposed methodology for the Reliability test of the sizing solution.

Figure V- 2

 2 Figure V-2 Power system performance on predictive outage models: Weibull (top); Markov (bottom) cumulative distribution functions of LPSP.

Figure V- 3

 3 Figure V-3 Optimal PV-battery power system based on 2015 data: performance test on 4years measured grid outage data represented by blue symbols and performance on

Table I -

 I 1 Summary of applications of multi-source systems in the context of grid outage. ... 29 Table II-1 The consumer considered as a case. ....................................................................... 43 Table II-2 Technical specifications of the photovoltaic panel considered. ............................. 47 Table II-3 Technical specifications and related parameters of the storage battery considered.

Table I -

 I 1 Summary of applications of multi-source systems in the context of grid outage.

	Paper	Year	Country	Outage type	Load considered	Proposed solution/ components	Cost	Remark
	C.Bastholm et al. [79]	2018 Tanzania	408 h/yr: every weekend on Apr, Sep; 408 h/yr: Mon-Fri on Apr; 365 h/yr: everyday; 1460 h/yr: 4 h everyday	Daily avg.: 1.09 kWh AC, 0.35 kWh/d Yearly: 522 kWh	Solar PV: 655 Wp Diesel: 12 kW Battery: 12 kWh PV-diesel-battery-grid	1.17 $/kWh (normalized)	Existing PV-diesel microgrid-different configurations. Off-grid PV-D-MG as base case
	M.R.Elkadeem et al. [80]	2019 Egypt	Feb: 4 h (08:00-12:00) May: 4 h (12:00-16:00) Sep: 2 h (04:00-06:00)	Avg. load: 17 MWh/d/yr, Peak: 1.771 MW	PV: 700 kW Grid-Wind-PV-Diesel WT: 96 kW Diesel: 900 kW	0.154 $/kWh
	M.Alramlawi et al. [75]	2018 Palestine	8-12 h everyday	Peak: 1.5 kW	PV: 3 kWp Battery: 9.6 kWh Grid: 5 kW (max.)	0.265 $/kWh (winter); 0.207 $/kWh (summer) 3
	L.Khoury et al. [72]	2015 Lebanon	52% of time ('harsh blackout schedule')	31 kWh/d	PV: 2.59 kW PV-Battery-Grid Battery: 27 kWh	0.25 $/kWh 4, 5
	P.M.Murphy et al. [67]	2013 Uganda	10.1 h avg. outage (exponential pseudorandom outage profile, 83.4% grid availability -1450 h outage)	Daily peak: 56.4 kW Annual avg.: 618 kWh/d	Grid-Diesel-Battery (40 kW, 222 kWh) Grid-PV-Battery (100 kW, 1110 kWh)	0.25 $/kWh 0.325 $/kWh	Synthesized data (load, outage), LCOE
						Grid-PV-Biogas
	S.O.Sanni et al. [78]	2021 Nigeria	7 h grid availability (double in rainy months)	Daily peak: 10.5 kW Monthly peak: 18.3 kW	(10 kW, 20 kW) Grid-PV-Diesel	0.164 $/kWh 0.280 $/kWh	Existing diesel backup
						(40 kW, 20 kW)
	S.U.Rehman et al. [81]	2018 Pakistan	6 h/d (weekdays) on Jan -Dec.	Peak: 2.27 kW (13.72 kWh/d)	Grid-PV-Battery (2 kW, 14.4 kWh)	0.191 $/kWh
	M.M.Samy et al. [82]	2021 Egypt	0.5% LPSPmax 6	Peak: 391 kW (4.82 MWh/d)	Grid-PV-WT-FC (58 kW, 200 kW, 60 kW)	0.0888 $/kWh	A preset reliability level objective

Table II -

 II [START_REF] Ceer | 6th CEER benchmarking report on the quality of electricity and gas supply[END_REF] The consumer considered as a case.

	Consumer/case building	Lat. (°N)	Long. (°E)	Altitude (m)	Grid connection	Bldg. function
	Addis Ababa Institute of Technology (AAiT)	9.04	38.76	2355	15kV/230V University

Table II -

 II [START_REF] Balducci | Electrical Power Interruption Cost Estimates for Individual Industries, Sectors, and U.S. Economy[END_REF] Technical specifications and related parameters of the storage battery considered.

	Parameter	Cbat	Pmax,cha / Pmax,discha	𝜂bat	SoCmin SoCmax DOD	tbat	Ncycle,max
	Value	7 kWh	3.5 kW	> 97 %	0.2	0.9	0.8	6 yrs	2000

  𝑃 𝑙𝑜𝑎𝑑 (𝑡) -[𝑃 𝑡𝑜𝑡 (𝑡) + 𝑃 𝑏𝑎𝑡 (𝑡)]𝜂 𝑖𝑛𝑣

						(II-12)
	s.t.
	{	(𝑆𝑜𝐶(𝑡) -𝑆𝑂𝐶 𝑚𝑖𝑛 )	𝐶 𝑏𝑎𝑡 𝜂 𝑏𝑎𝑡 ∆𝑡	<	𝑃 𝑙𝑜𝑎𝑑 (𝑡) 𝜂 𝑖𝑛𝑣	-𝑃 𝑡𝑜𝑡 (𝑡)
	𝑃 𝑙𝑜𝑎𝑑 (𝑡) -[𝑃 𝑡𝑜𝑡 (𝑡) + 𝑃 𝑏𝑎𝑡 (𝑡)]𝜂 𝑖𝑛𝑣 > 0

Table II -

 II [START_REF]World Energy Outlook 2019[END_REF] The economic parameters and values of the system considered.

	PV module	Values
	Ccap,pv [€/m 2 ]	200
	CO&M,pv, [€]	1% of Ccap,pv
	PV lifetime [yrs.]	25
	Storage battery
	Ccap,bat [€/kWh]	550
	Battery lifetime [yrs.]	6
	System
	N [yrs.]	25
	i [%]	2
	Max. grid subscribed power [kW]	50
	Current grid elec. cost [€/kWh]	0.0321

  BSPs with wide area coverage and a large number of customers (see Annex B-the simplified radial topology of the substation with the corresponding feeders covering large supply area). It is also known to experience frequent outages. It is supplied by a single 132 kV overhead line with 11 outgoing 15 kV feeders. It has two 25/31.5 MVA transformers and another 12 MVA transformer[START_REF] Brinckerhoff | Addis Ababa Distribution Master Plan Study -Modelling and Analysis of Existing Network[END_REF].

		2012	2013	2014	2015	2016
	Outage duration (hr.)	714.82 1799.08 1955 1873.65 1270.9
	Number of outages (№)	1306	1610	2305	1847	1528

Figure II-6 shows the outage records and its severity from 2012 to 2016 [116]. For instance, the duration and number of outages grew by more than 150% and 23% from 2012 to 2013 respectively. In contrast, the outage trend has shown a decrease from 2015 to 2016 by nearly 32% and 17%. The 5-year summary of outage duration and number of outage events can also be seen below in Table II-5. Table II-5 Distribution grid outage 5-years data.

  and CoE. That is, with decreasing LPSP (improved reliability), the CoE will increase and vice versa. Below Figure III-1 shows the rule-based energy management strategy implemented in the sizing algorithm based on the grid availability status and the power available from the PV and battery SoC with other constraints.

				Start
				t = 0
			Load and outage profiles, PV
			and battery parameters,
			constraints and weather data
			t = t 0 + t
			Y	P tot > P load	N
	P bat = 0 P grid = P ne t	Y	SoC SoC max		SoC SoC min	Y	P bat = 0 P grid = P grid,ac tual
	SoC = SoC max		N		N		SoC = SoC min LPS (Eq. II-12)
			P bat P cha,max		P bat P discha,max
			P grid = P ne t		P grid = P ne t
				SoC
			(Eqs. II-4, II-5)
			N	t > T max
				Y
				End
	P ne t : Net power after supplying load and/or battery charging/discharging
	P grid : Power to / from grid
	P grid,ac tual : Grid power with outage

Table III -

 III [START_REF] Ceer | 6th CEER benchmarking report on the quality of electricity and gas supply[END_REF] Sensitivity results of the cost of energy (CoE) with the variation in capital costs of components and grid electricity price.

	Scenarios	CoE [€/kWh]	∆CoE, %	PV = 182; Nbat = 39; LPSP = 0%
	Base-case (optimal solution)	0.1728	-	Cpv = 200€/m 2 Cbat = 550€/kWh CCgrid = 0.0321 [€/kWh
	∆Cpv = -50%	0.1684	-2.55	100
	∆Cpv = +50%	0.1771	+2.49	300
	∆Cbat = -50%	0.1018	-41.09	275
	∆Cbat = +50%	0.2437	+41.03	825
	∆CCgrid = +50%	0.1838	+6.37	0.04815
	∆CCgrid = +100%	0.1949	+12.79	0.0642

  It is interesting to see here that the MTM parameters can be deduced from global data. Indeed, it is common that published data on outages are annual and only the total number of outages and the total duration of the outages for one year are given. For instance, in[116], the average durations of the outages can be deduced per year, as for the total annual outage duration. It is easy to see that these data are enough to build the equivalent MTM. If the total annual outage duration is noted Toutage, the average outage duration Tavg and the duration of one year Tyear (8760 h), the MTM can be rewritten as:

		𝑀 𝑂𝐹𝐹 -𝑁	𝑁
	𝑀𝑇𝑀 = [	𝑀 𝑂𝐹𝐹 𝑁		𝑀 𝑂𝐹𝐹 𝑀-𝑀 𝑂𝐹𝐹 -𝑁	] with 𝑀 𝑂𝐹𝐹 = ∑	𝑁 𝑛=1	𝑚 𝑂𝐹𝐹 [𝑛]	(IV-2)
		𝑀-𝑀 𝑂𝐹𝐹	𝑀-𝑀 𝑂𝐹𝐹
		1 -	1 𝑇 𝑎𝑣𝑔		1 𝑇 𝑎𝑣𝑔
	𝑀𝑇𝑀 =	1 𝑇 𝑎𝑣𝑔	𝑇 𝑜𝑢𝑡𝑎𝑔𝑒	𝑇 𝑦𝑒𝑎𝑟 -𝑇 𝑜𝑢𝑡𝑎𝑔𝑒 (1 +	1 𝑇 𝑎𝑣𝑔	)
	[ 𝑇 𝑦𝑒𝑎𝑟 -𝑇 𝑜𝑢𝑡𝑎𝑔𝑒		𝑇 𝑦𝑒𝑎𝑟 -𝑇 𝑜𝑢𝑡𝑎𝑔𝑒	]
								Suppose a
	measured historical global data with a series of M samples times and these two possible
	states. The MTM with a total number of outages N, a total number of samples in OFF state
	Moff, and deducing the total number of samples in ON state M-Moff gives (see Figure IV-6):

Table IV -

 IV 1 Summary statistics of LPSPs for the two models.The Markov model takes into account a piecewise constant transition probability depending on the hour of the day to develop the outage sequences. Regarding the Weibull model, it gives a slightly more variability and diverse LPSP probability (see TableIV-1)pertaining to its capability to take into account lower frequency outage occurrences or extreme events.Once again considering outage simulation without any power system, the results (Figure show that the LPSPs are less than or equal to the 5-year average (2.53%) for the Weibull and Markov for about 72% and 55% respectively. Moreover, both the models predictively modeled the outages with nearly the same performance for about 90%.Numerically, the LPSPs are less than or equal to 2.76% and 2.78% for Weibull and Markov models respectively. At the same time, it can also be observed from the result that the Weibull outperforms with LPSP up until around 2.6%. Note that this value is almost equal to the yearly measured data for the years 2014-2016, see the broken lines in Figure

	Statistics parameter Markov (%) Weibull (%)
	Min.	1.62	1.51
	Max.	3.33	3.67
	Mean	2.512	2.38
	Std.	0.221	0.323
	Median	2.514	2.36

Table V -

 V 1 Reference optimal systems' PV and battery sizes with their corresponding CoE, based on measurements data of the year 2015. The next step of the proposed methodology consists in testing the performance of these sizing on the 1000 years' simulated outage profiles. It gives a cloud of points of LPSP and CoE which are the two objectives of the optimization problem. These clouds of points are analyzed as statistical distributions of both parameters and are presented in Figure V-2.Considering a completely reliable reference system with LPSPref = 0%, this value is well equal to zero for about 92% and 32% of simulations for Markov and Weibull respectively. Comparatively, for a relatively higher LPSP reference system, such as 2%, this metric shows an LPSP less than or equal to this value (i.e., 2%) for 51% and 71% in Markov and Weibull methods. Summing up these figures, for all the different solutions, both Markov and Weibull lead to distributions centered around the reference LPSP values and it can also be deduced that the Markov approach leads to more optimistic results compared to the Weibull distribution results. But, as seen previously, the Weibull distribution approach is more faithful to reality and therefore offers more reliable indicators. It can also be seen that the Weibull model performs well for higher reference LPSPs (poor reliability systems), and conversely, the Markov simulates and behaves closer to the reference system in lower values (better reliable systems); referring to Figure V-2, observe the right and left-hand side of the graphs for Weibull and Markov respectively.

	LPSPref (%)	Nbat (№)	Apv (m 2 )	CoE (€/kWh)
	0.0	39	151	0.176
	0.5	304	7	0.063
	1.0	284	4	0.05
	1.5	260	1	0.039
	2.0	179	0	0.033

  MoffTotal number of grid state samples in 'OFF' state, i.e., outages MTMday, MTMnight MTM for the day time, and MTM for the night time POFFOFF Probability of grid state transition (remaining in its initial 'OFF' state; technically, transition from 'OFF' to 'OFF' state) POFFON Probability of grid state transition from 'OFF' state to 'ON' state) PONOFF Probability of grid state transition from 'ON' state to 'OFF' state) PONON Probability of grid state transition (remaining in its initial 'ON' state; technically, transition from 'ON' to 'ON' state)

	fuelCg,ann SoCmax	The annual cost of diesel cost [€] Maximum SoC value [%]
	GNOCT SoCmin	The NOCT solar irradiation [W/m 2 ] Minimum SoC value [%]
	Gstc t Variables Gtot T	Solar irradiation at STC [W/m 2 ] simulation time-step/t th time interval Total solar irradiation on tilted surface [W/m 2 ] Simulation year [min.]
	∆Cbat i Tamb	Change in battery capital cost interest rate [%] Ambient temperature [°C]
	∆CCgrid LHVfuel Tavg	Change in grid electricity cost Lower heating value of the diesel fuel [kJ/kg] Average grid outage duration [hr.]
	∆Cpv LPS Tavg	Change in PV capital cost Loss of power supply [W] Average outage duration [h]
	a LPSP tbat	Scale parameter (of the Weibull function) [min.] Loss of power supply probability [%] Battery life [yr]
	Apv LPSPref Tcell	PV surface area [m 2 ] Reference Loss of power supply probability [%] PV-cell temperature [°C]
	Apv,max M Toutage	Sizing upper boundary/ maximum PV area [m 2 ] Series of grid state samples times Total annual grid outage duration [hr.]
	b Toutage	Shape parameter (of the Weibull function) Total annual outage duration [h]
	Cbat Tref	battery storage nominal capacity [Wh] Reference temperature [°C]
	Cbat,max N Tyear	Sizing upper boundary/ maximum storage capacity [Wh] Project lifetime [yr] Total hours in a year, 8760hr.
	Ccap N Tyear	Capital cost of the system [€] Total number of outages The duration of one year (8760 h)
	Ccap,bat Nbat βp	Initial capital cost of the battery bank [€] Number of batteries PV module power temperature coefficient [/°C]
	Ccap,pv Ncycle,max ηbat	Initial capital cost of the PV [€] The maximum charge-discharge cycle of the battery Battery storage efficiency [%]
	CCg,ann NOCT ηgen	Annualized capital cost of the diesel generator [€/yr] Nominal operating cell temperature [°C] Generator efficiency [%]
	Ccha OMCg,ann ηinv	Battery storage charging cost from grid [€] Annualized operation and maintenance cost of the diesel generator [€/yr] Inverter efficiency [%]
	CO&M Pcha,max ηm	Operation and maintenance cost [€/yr] Maximum allowable battery charging power [W] PV model module efficiency [%]
	CO&M,pv Pdischa,max ηpv	Annual operation maintenance cost of the PV [€/yr] Maximum discharge power from the battery [W] PV panel efficiency [%]
	CoE Pgen ρfuel	Cost of energy generated [€/kWh] Total generated power [kWh/yr] Diesel fuel density [kJ/m 3 ]
	CoEg Pgrid,max	Cost of energy generated from diesel generator [€/kWh] Subscribed grid maximum power [W]
	Crep Pload Units (only selected) Replacement cost [€/yr] Load demand [W]
	CRF	Capital recovery factor
	Ctot,a h	Annualized total system cost [€/yr] Hour
	DOD h/d	Depth of discharge (of the battery) [%] Hour per day
	dt h/yr	Change in time step [min.] Hour per year
	Etot kWh/d	Total served electrical energy [Wh] Kilowatt-hour per day
	Etot l/h	Total served diesel energy [Wh] Liter per hour
	F MWh/d/yr f(obj,1), f(obj,2) Ppv Pp yr	Megawatt-hour per day per year Diesel fuel consumption at any time interval [l/h] Generated power output of the PV [W] Objective functions 1 and 2 Module nominal power [W] Year
	F0 Prated	Fuel curve intercept coefficient [l/h/kW] Generator rated capacity [W]
	F1 Ptot	Fuel curve slope [l/h/kW] Total available power from grid and PV [W]
	fpv SoC	PV derating factor [%] Battery storage state-of-charge [%]

The three icon figures, from top to bottom respectively, are from www.progressiverecruitment.com, www.engerati.com and www.simson-maxwell.com.

CoE is derived from the paper given as the total cost for the proposed strategy to the energy served in the simulation timeframe.

We derived this CoE based on the total system cost for the project life and assuming a uniform yearly cost, then a weekly energy served from the system cumulated to one year.

Converted currency, 1 euro = 1.19 USD, for uniformity of comparison.

Loss of Power Supply Probability, maximum.

Adapted from: Klammer, Martin; Nikolaj Dybowski, J.; Hoffmann, Daniel; Schaab, Christoph (2015): Example of a Pareto front in a minimization problem.Figure. https://doi.org/10.1371/journal.pone.0128542.s001

Acknowledgments i

Acknowledgements

Chapter IV: Grid Outage Modeling IV.1.

Abbreviations

AAiT

Addis Ababa Institute of Technology AC Alternating current (representing the AC bus)

CHAPTER II

Modeling of the Multi-Source Energy System

𝐹 = 𝐹 0 𝑃 𝑟𝑎𝑡𝑒𝑑 + 𝐹 1 𝑃 𝑔𝑒𝑛 (II-9)

𝜂 𝑔𝑒𝑛 = 3600𝑃 𝑔𝑒𝑛 𝐹𝜌 𝑓𝑢𝑒𝑙 𝐿𝐻𝑉 𝑓𝑢𝑒𝑙 (II -10) where:

• F: the fuel consumption at any time interval [l/h];

• F0: the fuel curve intercept coefficient [l/h/kW];

• F1: the fuel curve slope [l/h/kW];

• Prated: the rated capacity [kW];

• Pgen: generated power [kW];

• 𝜂gen: its efficiency [%];

• 𝜌fuel: the density [kg/m 3 ];

• LHVfuel: the lower heating value [kJ/kg] of diesel fuel.

A generator such as given in [107] is selected and economic parameters as given in [START_REF] Kuznetsov | Economic Feasibility Analysis of PV/Wind/Diesel/Battery Isolated Microgrid for Rural Electrification in South Egypt[END_REF] are considered. Furthermore, with an operation only during grid outage hours, the lifetime of the generator can be assumed to be equivalent to the project life without significant error. The diesel generator efficiency (see Annex A-efficiency curve) is calculated based on the fuel consumption profile with the fuel curve slope and intercept coefficients as given above.

II.2.4. Reliability Model

In this work, the reliability criteria, which is the major metric is modeled such that, the portion of the power that cannot be supplied by the PV-battery system during outages is the reliability measure considered in this study. Apart from this main fraction due to the outage, it also inherently incorporates the intermittency of solar production. This whole portion of power can effectively be described by the widely used parameter, loss of power supply probability (LPSP) [START_REF] Masih | Optimal Economical Sizing of Grid-Connected Hybrid Renewable Energy System[END_REF], [START_REF] Kaabeche | Sizing optimization of gridindependent hybrid photovoltaic/wind power generation system[END_REF], [START_REF] Bouabdallah | Safe sizing methodology applied to a standalone photovoltaic system[END_REF], [START_REF] Maleki | Optimal sizing of autonomous hybrid photovoltaic/wind/battery power system with LPSP technology by using evolutionary algorithms[END_REF], [START_REF] Sharafi | Multi-objective optimal design of hybrid renewable energy systems using PSO-simulation based approach[END_REF], [START_REF] Ofry | The Loss of Power Supply Probability as a Technique for Designing Stand-Alone Solar Electrical (Photovoltaic) Systems[END_REF] in modeling and optimization of renewable energy systems, as given by equation .

over supplying back to the grid until SoCmax. The details will be discussed in chapter III under the energy management.

II.2.5. Economic Model

Various economic analysis approaches for renewable energy systems can be found in the literature [START_REF] Masih | Optimal Economical Sizing of Grid-Connected Hybrid Renewable Energy System[END_REF], [START_REF] Bouabdallah | Méthodologie de dimensionnement et d'optimisation des systèmes multisources à énergies renouvelables pourl'habitat[END_REF], [START_REF] Kaabeche | Sizing optimization of gridindependent hybrid photovoltaic/wind power generation system[END_REF], [START_REF] Bouabdallah | Safe sizing methodology applied to a standalone photovoltaic system[END_REF], [START_REF] Maleki | Optimal sizing of autonomous hybrid photovoltaic/wind/battery power system with LPSP technology by using evolutionary algorithms[END_REF], [START_REF] Iannone | Hybrid photovoltaic and hybrid photovoltaicfuel cell system: economic and environmental analysis[END_REF]. The system cost and cost of energy produced can be marked as the frequently used economic indicators to optimize the multisource system sizing and to study the practicability and profitability of such systems. With the objective of improving the LPSP, the economic evaluation index considers the total system present cost containing the sum of the capital cost, O&M and replacement costs of the different components, and grid charging cost which is calculated as [START_REF] Bouabdallah | Safe sizing methodology applied to a standalone photovoltaic system[END_REF], [START_REF] Olatomiwa | Optimal configuration assessments of hybrid renewable power supply for rural healthcare facilities[END_REF]:

where:

• Ctot,a: the total annualized system cost [€];

• Ccap: the sum of the capital cost of the PV panels and the storage batteries [€];

• CO&M: the operation and maintenance cost of the PV [€];

• Crep: the battery replacement cost [€];

• Ccha: the cost of grid energy utilized during the battery energy exchange [€];

• CRF: the capital recovery factor.

Note that here the capital cost covers all the components, the O&M cost is considered for the PV, and the replacement cost for the storage battery. The system capital recovery factor (CRF) is given by [START_REF] Kaabeche | Sizing optimization of gridindependent hybrid photovoltaic/wind power generation system[END_REF]:

where:

• i: the interest rate [%];

• N: the lifetime of the system [yr], considered as 25 years. Based on these two modeling approaches, effectiveness and performance of the PVbattery system connected to the unreliable grid are then verified on a large set of simulated outages (equivalent to one thousand profiles), to try to estimate the real performance of the optimized design. Comparisons of the performances of the optimal sizing are then possible and open the discussion on the best way to consider outages in a sizing process.

CHAPTER III

Techno-Economic Sizing Optimization based on Measurement

CHAPTER IV

Grid Outage Modeling

CHAPTER V

Performance Evaluation and Validation of the Multi-Source System

Now that the proposed models and methods to generate outage profiles are presented, and the safe sizing methodology is developed. On the basis of optimal sizing obtained for a reference year (2015), the method consists in testing it for a large set of simulated outage profiles and testing its result in the objective plan, which are the electrical reliability and the cost of the solution in this study. Then, the obtained LPSP results (there are then as many LPSP result data points as there are years simulated) are statistically compared to the 5-year average LPSP value, obtained without any power system (LPSP = 2.53%).

V.2. Optimal System Performance on Simulated Outages

The performance analysis helps to quantify how the optimal system is able to supply under possible future outages under stochastic uncertainty for improving the knowledge of the outage behavior and system performance. For the robustness criteria, long-term outage

Conclusions and Perspectives

Conclusions and Perspectives

Nowadays, intermittent power interruptions and long outages are very common, especially from distribution systems in developing countries that are aging, with poor maintenance practices, substandard infrastructures, capacity shortages, and so on. The outages in weak networks are not necessarily linked to weather, although this is often the cause in grids of developed countries in addition to interruptions such as due to maintenance routines. This consequently affects utilities and diverse groups of consumers, with substantial negative effects for critical consumers such as hospitals.

In this work, we presented an optimal hybrid PV-battery system optimization methodology to improve or wholly achieve the expected reliability in the context of an unreliable grid. Literature showed the integration of hybrid power systems in weak grids often uses hypothetical outages or real outages with coarse data (average and gross values of outage parameters, and scheduled outages). What we are more interested in is considering realistic grid outages as in the aforementioned contexts in the power system sizing methodology. The sizing optimization is done with the classical bi-objective approach taking into account the reliability and cost of energy criteria (LPSP %, CoE €/kWh), so that it integrates the load demand outage profiles of a customer connected to a grid with frequent unpredictable outages. This renewable-based system will also substitute the commonly used diesel backups, which are harmful to the environment and relatively costly options. For practicality, we have used real load consumption and outage data, in Ethiopia, as a case study for the sizing optimization using the NSGA-II algorithm.

Moreover, unlike what was used in several works, a high-resolution simulation time step of 1 minute is used, which enables us to consider the power supply dynamics and the outages with reasonably better accuracy. It was proven that the system can successfully reduce the outage effects and entirely supply the load. The set of optimum solutions gives the freedom for consumers and utility operators to choose a safe system size with the desired reliability index. For instance, consumers connected to a weak distribution network with critical load requirements can have a zero LPSP with 0.17€/kWh. Conversely, consumers with tolerable load types such as households, may go for the system with higher LPSP and perhaps, shall afford partial outages. Let's say with about 0.11€/kWh, the system could give a 0.5% reliability compromise.

Annexes

Annexes Annex A: The diesel generator efficiency curve.

The diesel generator efficiency curve: -running at around 35% efficiency close to its optimal full loading, and losses from heat and friction, as in any other most mechanical devices.