Keywords: Distributed control, direct current microgrids, current sharing, average voltage regulation, stateof-charge balancing, consensus-like integral actions, real-time hardware-in-the-loop implementation

ACRONYMS xii

List of Symbols

R

-the set of real numbers.

R >0

-the set of positive real numbers.

⊗

-the Kronecker product of two matrices.

I n -identity matrix of the size (n × n).

A T -transpose of a matrix A.

1 n -vector of dimension n with all components equal 1.

0 m×p -zero matrix of the size (m × p).

∅ -empty set.

Ker(A)

-kernel of a matrix A.

(A) [i 1 ,i 2 ]×[j 1 ,j 2 ]
-submatrix formed by the entries (i, j) of A between range i 1 ≤ i ≤ i 2 and j 1 ≤ j ≤ j 2 .

(x 1 , x 2 , • • • , x n ) -a column vector as an n-tuple whose entries x i can be also column vectors or Un microgrid peut être situé dans différents endroits, par exemple dans des petites maisons, des quartiers intelligents, des îles, etc. En outre, les microgrids peuvent fonctionner en mode connecté au réseau ou en mode autonome (déconnecté). Cette capacité des microgrid à fonctionner de manière autonome offre la possibilité d'électrifier des zones isolées sans avoir besoin de grandes lignes de transmission.

equivalently (x 1 , x 2 , • • • , x n ) = x T 1 x T 2 • • • x T

Les problèmes de commande des Microgrids

Lors de la conception de systèmes de contrôle pour les microgrids DC autonomes, plusieurs défis opérationnels doivent être relevés afin de garantir la fiabilité et les avantages de la génération distribuée. [START_REF] Olivares | Trends in microgrid control[END_REF]. Dans cette section, nous présentons quelques défis liés aux microgrids DC que nous considérons dans cette thèse.

• Profil de charge inconnu : La coordination entre les différentes DGU pour maintenir l'équilibre entre la demande et l'offre est nécessaire pour un fonctionnement économique et résilient des des microgrids DC. Cependant, sous les incertitudes du profil de charge et l'intermittence de plusieurs ressources énergétiques dans le microgrid (par exemple, les panneaux photovoltaïques), l'équilibre entre la demande et l'offre devient plus difficile [2].

• Problèmes de stabilité: Un microgrid stable doit être capable de maintenir l'alimentation électrique de la charge sans dynamique indésirable (par exemple, oscillations de tension ou de flux de puissance).

Les microgrids DC peuvent être sensibles aux problèmes de stabilité à grand signal et à petit signal.

La stabilité des grands signaux fait référence à la capacité du système à résister à des perturbations très importantes, telles que des changements brusques de la demande de puissance ou des défauts dans le système. La stabilité des petits signaux se réfère à la stabilité du système sous de petites perturbations, par exemple le bruit et les changements de paramètres [START_REF] Chakraborty | Power Electronics for Renewable and Distributed Energy Systems A Sourcebook of Topologies, Control and Integration[END_REF].

• Partage du courant (CS): Il s'agit d'une caractéristique hautement souhaitable pour le maintien du bon fonctionnement des microgrids. Elle vise à partager, de manière proportionnelle, la demande de courant entre les différents DGU, en tenant compte de leur capacité de puissance. En d'autres termes, les DGU doivent être capables de partager correctement les courants de charge en fonction de leur capacité de génération. De plus, le CS permet d'éviter les situations où un DGU ne peut pas fournir le courant demandé pour sa charge locale et peut être amenée à délivrer au-delà de son courant maximum, ce qui peut détériorer l'unité et causer des dommages au système [START_REF] Meng | Review on control of dc microgrids and multiple microgrid clusters[END_REF], [START_REF] Tucci | Stable current sharing and voltage balancing in dc microgrids: A consensus-based secondary control layer[END_REF].

• Régulation de tension moyenne (AVR): La régulation de la tension est également l'un des paramètres de qualité importants de l'alimentation en courant continu. Une condition nécessaire au bon fonctionnement du microgrid est que les tensions restent proches de leurs valeurs nominales, quelle que soit la perturbation. En effet, une déviation incontrôlée de la tension par rapport à sa référence peut provoquer une défaillance de la charge puisque celle-ci a été conçue pour être alimentée avec une tension de niveau nominal. Réaliser le CS et la régulation de la tension est une tâche ardue et généralement impossible car le CS nécessite une déviation des tensions par rapport à leurs valeurs de référence. Par conséquent, une alternative consiste à fournir une régulation de tension moyenne (AVR), c'est-à-dire que la valeur moyenne des tensions aux points de couplage communc (PCC) est égale à la moyenne de leurs références [START_REF] Trip | Distributed averaging control for voltage regulation and current sharing in dc microgrids[END_REF], [START_REF] Tucci | Line-independent plug-and-play controllers for voltage stabilization in dc microgrids[END_REF].

• Equilibrage des états de charge (SB): Pour un microgrid doté d'unités de stockage (SU), un déséquilibre de l'état de charge entre ces unités peut affecter leur durée de vie en provoquant un phénomène de surcharge ou de surdécharge, une surchauffe, une détérioration, etc. [START_REF] Huang | Energy sharing control scheme for state-of-charge balancing of distributed battery energy storage system[END_REF]. Lorsque le SB est assuré entre les SU, le SU dont l'état de charge est le plus faible absorbera plus de puissance que les autres pendant le cycle de charge et, par conséquent, celui dont l'état de charge est le plus élevé fournira plus de puissance que les autres pendant le cycle de décharge [START_REF] Li | Voltage scheduling droop control for state-of-charge balance of distributed energy storage in dc microgrids[END_REF]. Cela permet de prolonger la durée de vie des batteries et de les maintenir dans un bon état de fonctionnement. Notez qu'un équilibrage de l'état de charge est également nécessaire entre les cellules de batterie connectées en série qui composent le SU. Ce dernier est effectué par le système de gestion de la batterie (BMS), qui sort du cadre de ce manuscrit.

Noter que les microgrids peuvent confronter d'autres types de défis qui sortent du cadre de cette thèse, comme les problèmes de cyber-sécurité, d'optimisation des coûts énergétiques, de transition entre les modes de fonctionnement autonomes et connecté, etc.

Techniques de Contrôle des Microgrids

Pour préserver les avantages des microgrids, les défis déjà mentionnés doivent être relevés par des stratégies de contrôle efficaces qui tiennent compte de la complexité du système et exploitent les capteurs et les actionneurs largement distribués. En fonction de leur échelle temporelle, les techniques de contrôle des microgrids peuvent être classées en trois niveaux de contrôle, à savoir le contrôle primaire, secondaire et tertiaire. Dans chacun de ces niveaux, le contrôleur peut être mis en oeuvre de manière centralisée, décentralisée et distribuée.

Dans les techniques centralisées, une seule unité centrale contrôle tous les éléments du microgrid. Malgré les avantages de ces techniques en termes d'observabilité et de contrôlabilité élevées, elles présentent de nombreux inconvénients tels que le point de défaillance unique (c'est-à-dire que toute défaillance de l'unité centrale entraîne l'arrêt de l'ensemble du microgrids et la nécessité d'une communication étendue entre le contrôleur central et les unités contrôlées. En outre, la centralisation des données peut poser des problèmes de confidentialité et de sécurité. Par conséquent, cette architecture de contrôle convient aux microgrids de petite taille où les informations à transmettre sont limitées [START_REF] Meng | Review on control of dc microgrids and multiple microgrid clusters[END_REF].

Dans les techniques décentralisées, les composants du microgrid sont contrôlés par des contrôleurs indépendants sans avoir besoin de communiquer entre eux. Bien que ces techniques présentent une excellente fiabilité et flexibilité, elles souffrent de plusieurs inconvénients tels qu'une mauvaise réponse transitoire et une dépendance à la charge nécessitant un contrôle secondaire. De plus, en raison du fort couplage entre les différents composants du microgrid, un niveau minimum de coordination est nécessaire. Ceci ne peut être atteint en utilisant uniquement des variables locales, ce qui rend impossible une approche entièrement décentralisée.

Les architectures distribuées comprennent un schéma dans lequel chaque élément du microgrid possède son propre contrôleur et échange des informations avec les autres éléments du système via un réseau de communication afin d'atteindre des objectifs communs de manière coopérative. Les stratégies les plus connues de cette technique sont les architectures de contrôle basées sur le consensus et sur les systèmes multi-agents [START_REF] Yazdanian | Distributed control techniques in microgrids[END_REF], [START_REF] Anand | Distributed control to ensure proportional load sharing and improve voltage regulation in low-voltage dc microgrids[END_REF]. Ces derniers sont utilisées pour réduire le problème centralisé à plusieurs sous-problèmes locaux.

Ainsi, les composants du microgrid peuvent être considérés comme des agents qui interagissent entre eux et appliquent un algorithme de contrôle distribué afin d'atteindre les objectifs globaux souhaités, ce qui apporte plus de flexibilité, d'extensibilité et de fiabilité au microgrid.

Pour revenir aux niveaux de contrôle, le contrôle primaire est la première couche et la plus rapide du schéma de contrôle hiérarchique [START_REF] Prabaharan | 6 -an overview of control techniques and technical challenge for inverters in micro grid[END_REF]. Il se compose de deux parties principales, la première est appelée boucle interne et concerne le contrôle des convertisseurs de puissance (par exemple, les convertisseurs DC/DC) afin de fournir une interface contrôlable entre les charges et les sources [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF], [START_REF] Sferlazza | Min-type control strategy of a dc-dc synchronous boost converter[END_REF]. La seconde partie est responsable du contrôle de la tension locale et du courant local. En outre, un partage de courant préliminaire peut également être réalisé à ce niveau.

Le contrôle secondaire, également appelé niveau de gestion, est utilisé pour la gestion de la qualité de l'énergie délivrée aux consommateurs et l'amélioration des objectifs du contrôle primaire [START_REF] Anand | Distributed control to ensure proportional load sharing and improve voltage regulation in low-voltage dc microgrids[END_REF], [START_REF] Lu | An improved droop control method for dc microgrids based on low bandwidth communication with dc bus voltage restoration and enhanced current sharing accuracy[END_REF], [START_REF] Adhikari | Secondary control for dc microgrids: A review[END_REF], [START_REF] Nasirian | Distributed cooperative control of dc microgrids[END_REF], [START_REF] Wang | An improved distributed secondary control method for dc microgrids with enhanced dynamic current sharing performance[END_REF], [START_REF] Nasirian | Distributed adaptive droop control for dc distribution systems[END_REF], [START_REF] Mehrizi-Sani | Potential-function based control of a microgrid in islanded and gridconnected modes[END_REF], [START_REF] Guerrero | Hierarchical control of droop-controlled ac and dc microgrids-a general approach toward standardization[END_REF]. L'implémentation de la commande secondaire peut se faire de manière centralisée [START_REF] Mehrizi-Sani | Potential-function based control of a microgrid in islanded and gridconnected modes[END_REF], [START_REF] Guerrero | Hierarchical control of droop-controlled ac and dc microgrids-a general approach toward standardization[END_REF], de manière distribuée avec un réseau communication complet (c'est-à-dire, deux unités quelconques sont directement connectées) [START_REF] Lu | An improved droop control method for dc microgrids based on low bandwidth communication with dc bus voltage restoration and enhanced current sharing accuracy[END_REF], [START_REF] Wang | An improved distributed secondary control method for dc microgrids with enhanced dynamic current sharing performance[END_REF] ou avec une communication voisin-voisin seulement (c'est-à-dire un réseau de communication creux) [START_REF] Trip | Distributed averaging control for voltage regulation and current sharing in dc microgrids[END_REF], [START_REF] Tucci | Line-independent plug-and-play controllers for voltage stabilization in dc microgrids[END_REF], [START_REF] Nasirian | Distributed cooperative control of dc microgrids[END_REF], [START_REF] Nasirian | Distributed adaptive droop control for dc distribution systems[END_REF], [START_REF] Tucci | A decentralized scalable approach to voltage control of dc islanded microgrids[END_REF], [START_REF] Cucuzzella | A robust consensus algorithm for current sharing and voltage regulation in dc microgrids[END_REF].

Le niveau de contrôle tertiaire est le niveau le plus élevé et sa dynamique est la plus lente. Il est utilisé à plusieurs fins : coordination entre plusieurs microgrid, optimisation du coût de l'énergie, coordination avec le gestionnaire du réseau de distribution et la gestion de l'énergie des éléments de stockage à long terme [START_REF] Olivares | Trends in microgrid control[END_REF], [START_REF] Fei | Primary and secondary control in dc microgrids: a review[END_REF], [START_REF] Prabaharan | 6 -an overview of control techniques and technical challenge for inverters in micro grid[END_REF].

Objectives et Contributions

L'objectif de ce travail est de développer une nouvelle approche de contrôle distribué aux niveaux primaire et secondaire pour les microgrids DC en mode de fonctionnement autonomes assurant un partage de courant proportionnel entre les unités de production, une régulation de la tension moyenne des lignes et un équilibrage simultané des états de charge des éléments de stockage. La méthodologie de contrôle proposée est basée sur la séparation entre les problèmes du partage du courant et la régulation de la tension moyenne d'une part et le problème de l'équilibrage des état de charge d'autre part ; les principaux outils sont : le consensus dans les systèmes multi-agents, la passivité, la commande par retour d'état, les actions intégrales distribuées, la stabilité de Lyapunov et les inégalités matricielles linéaires, etc.

La nouveauté dans ce travail est l'utilisation de trois actions intégrales distribuées de type consensus pour atteindre simultanément les trois objectifs considérés même en cas de perturbations constantes. Nous montrons qu'à l'intérieur de l'ensemble des équilibres, les objectifs de contrôle sont atteints. La preuve de la Hardware-In-the-Loop (HIL) en temps réel (Fig. 2).

Plan de la thèse

w j I j Soc j I i ω j γ j φ i γ i V i Soc i θ i V i I i ω i ω i ω i γ i ω i I i V ref i Soc ref (
Les résultats de de chacun de ces scénarios montrent que la méthodologie de contrôle proposée permet un SmartGrids. Generally, a traditional power system includes generation, transmission, distribution and loads where the flow of energy is unidirectional, from producers to consumers. However, this actual concept faces the problems of environmental pollution because of the massive use of fossil fuel resources, and the ever-increasing energy demand requiring huge investment. Moreover, the uncertainty of energy demand is, traditionally, managed by adjusting the supply. Nevertheless, because of the massive emergence of renewable energy resources of a volatile and unpredictable nature, such as solar panels and wind turbines, the uncertainty on the production side needs to be addressed as well.

Chapitre 4: Conclusion et Perspectives

These challenges, for which the existing power systems are no longer sufficiently adapted, are leading to the development of smartgrid, a more flexible, eco-friendly and bi-directional paradigm in which a consumer can also be a producer and vice versa. Moreover, this new concept is characterized by the vast integration of modern information and communication technologies allowing for better coordination between the generation and the end user.

Microgrids (MGs) represent one of the efficient and sustainable key components of smartgrids. Generally, a MG is a cluster of several interconnected power supplies, storage systems and loads interconnected through power lines. A MG can be located in different places e.g., small houses, smart neighborhoods, islands, etc.

Moreover, MGs can operate in grid-connected mode or in standalone (autonomous) modes. This ability of MGs to operate autonomously gives the possibility to electrify isolated areas with no needs for large transmission lines. However, the issues of stability, energy management, power sharing, voltage regulation, cyber-security, energy cost optimization, etc. are more critical for islanded MGs.

INTRODUCTION

To preserve the advantages of MGs, these challenges should be addressed with effective control strategies that take into account the complexity of the system and exploit the widely distributed sensors and actuators.

Because of the single point of failure and the needs for extensive communication, centralized control approaches are no more adapted to the distributed nature of modern power systems. Multi-agent systems can be used to tackle these issues by reducing the centralized problem to several local sub-problems. Hence, the components of the MG can be considered as agents that interact between each other and apply a distributed control algorithm in order to achieve the desired global objectives providing more flexibility, expandability and reliability for the MG.

Structure of the thesis

The thesis is structured as follows.

Chapter 1. The first chapter presents the concept of Microgrid, its definition, importance and challenges with an overview of the existing control techniques of DC-MGs. It contains a presentation of the concept of consensus in linear multi-agent systems with some mathematical preliminaries required for a good understanding of the manuscript.

Chapter 2. The second chapter presents the theoretical contributions of the thesis where the design of the proposed distributed controllers is detailed. The chapter is divided into four parts. In the first part, the dynamic model of the studied DC-MG with a meshed topology is presented. The considered power network is composed of Distributed Generation Units (DGUs), Storage Units (SUs), loads and power lines.

The second part is dedicated to the problems of CS and AVR for a DC-MG composed of only DGUs with resistive power lines and loads. The third part addresses the problem of SB for a DC-MG that includes only SUs with resistive power lines and loads through a leader-follower approach. The fourth part combines the results in the previous parts to achieve simultaneously CS, AVR, and SB for a DC-MG that includes both DGUs and SUs; first, by considering resistive power lines and then resistive-inductive power lines.

Chapter 3. In the third chapter, a case study is considered, and the proposed control approach is tested in different scenarios with Matlab/Simulink simulations and real-time digital Hardware-In-the-Loop (HIL)

experiments.

Chapter 4. The fourth chapter presents the conclusions of the thesis and some suggestions for future research.

ULHyS Project

This thesis is part of Université de Lorraine Hydrogen Sciences and technologies (ULHyS) project, which is one of the impact projects of the Lorraine Université d'Excellence (LUE) initiative. ULHyS aims at reinforcing the excellence of basic and applied research by connecting several actors in the hydrogen sector.

It seeks to introduce education programs at master level in energy carrier management focusing specifically on hydrogen. Moreover, ULHyS is composed of 10 laboratories and over 50 researchers presenting a high level in their respective disciplines. ULHyS is organized into 5 interconnected thematic Work-Packages (WPs) to cover the entire supply chain, from hydrogen production methods to the end user, the impact on the local economy and territorial deployment (Fig. 3). This work is part of WP 3 that concerns Microgrids, Multi-source, Multi-vector with a special focus on the control of DC-MGs. Chapter 1

Background and Preliminaries

CHAPTER 1. BACKGROUND AND PRELIMINARIES

This chapter provides the preliminaries needed for the further developments presented in this thesis. First, general definitions of Microgrids (MGs) and their concept are provided with a particular focus on Direct Current (DC) MGs and their challenges. Next, an overview of the existing control techniques to overcome the challenges related to DC-MGs is given. Then, we recall some definitions of graph theory and consensus in linear multi-agent systems. In Section 1.5, we emphasize the fact that several challenges in DC-MGs can be represented as consensus problems of multi-agent systems. Finally, some fundamental results of stability, passivity and linear parameter-varying techniques are recalled.

Microgrids

Several definitions of MGs exist in the literature, one of them is given by the U.S. Department of Energy's Microgrid Initiative

Definition 1.1. [27]

A microgrid is a group of interconnected loads and distributed energy resources within clearly defined electrical boundaries that acts as a single controllable entity with respect to the grid. A microgrid can connect and disconnect from the grid to enable it to operate in both grid-connected or island mode (Fig. 1.1).

Another definition from [START_REF] Meng | Review on control of dc microgrids and multiple microgrid clusters[END_REF] and [START_REF] Olivares | Trends in microgrid control[END_REF] is as follows Definition 1.2. A microgrid is a cluster of several interconnected power supplies, loads and energy storage system which coordinate between each-other to reliably provide energy, it can be connected to the main-grid or operates independently.

In these two definitions, one can remark that they include three main points:

• interconnection and distribution,

• cooperation and control and • independence from the main-grid. These three points make microgrids one of the critical components of modern power systems. Interconnection and distribution properties give the ability to simplify the integration of renewable and eco-friendly [START_REF] Banerji | Microgrid: A review[END_REF]. Finally, independence from the main-grid allows providing energy for isolated places.

A MG can be installed in several places starting from a small building to a smart neighborhood, from urban places to islands and even in elevators, air-crafts and ships [START_REF] Sguier | Development of a photovoltaic low voltage dc microgrid for buildings with energy storage systems[END_REF], [START_REF] Pham | Dissipated energy minimization for an electro-mechanical elevator of a dc microgrid[END_REF], [START_REF] Buticchi | On-board microgrids for the more electric aircraft-technology review[END_REF], [START_REF] Hirsch | Microgrids: A review of technologies, key drivers, and outstanding issues[END_REF].

Components

MGs are mainly composed of Distributed Generation Units (DGUs), Storage Units (SUs), loads, power electronic converter and control units.

• DGUs : They can be various types of renewable energy, e.g., fuel cell, photovoltaic (PV) and wind turbines; or thermal energy sources e.g., natural gas or biogas. In this thesis, we consider that the DGU is composed of the energy source with its connection interface to the grid i.e. power-electronic converters, filters, etc.

• SUs: They can provide broad benefits to MGs and include all of the hydrogen, chemical, gravitational, flywheel and heat storage technologies. They can perform multiple functions, e.g., decreasing losses and increasing reliability, providing backup power for the MG and playing a crucial role in cost optimization. Storage can be combined with non-dispatchable renewable energy sources (e.g., wind turbines and PV panels) to turn them into dispatchable units enabling a large-scale of their integration. SUs are used to maintain power balance between generation and demand, peak-shaving and helping to maintain power quality by contributing additional energy at peak hours and absorbing energy at off-peak hours [START_REF] Faisal | Review of energy storage system technologies in microgrid applications: Issues and challenges[END_REF]. Usually, a SU is composed of several battery cells and is connected to the MG through a power-electronics bidirectional interfacing converter.

• Loads: They represent the elements that consume energy in the grid, e.g., single devices, electrical vehicles, lighting, heating system, etc. Some of the loads are controllable allowing to manage the energy demand in the MG [START_REF] Dahmane | Coordinated charging of large electric vehicle fleet in a charging station with limited transformer power[END_REF].

• Power electronic converters: They are used for four types of power conversion, i.e., (1) DC to DC, (2) AC to DC, (3) DC to AC and (4) AC to AC. DC to DC converters are used to step-up or step-down a DC voltage (e.g., boost and buck converters).

AC to DC converters (e.g., rectifiers) are used to supply DC loads, such as DC motors, using AC power supply. DC to AC conversion is ensured by inverters, it gives the possibility to supply AC loads using DC sources, such as batteries. An example of AC to AC converters is back-to-back converters used to connect wind turbines to the power grid.

Power electronics converters are mainly composed of linear passive elements such as resistances, inductors and capacitors and nonlinear passive elements (switches) such as diodes, MOSFET, Thyristors, etc. Some of the switches can be controlled (e.g, Thyristors) and others are not (e.g., diodes). The controllable ones represent the control input of the power electronic converters. The switches can be either on or off and their switching frequency can be very high giving a fast time response for the converter.

The use of power electronic converters has enhanced the integration of renewable energy. However, they can cause serious problems notably the injection of harmonics [START_REF] Blin | A comparison of harmonic modeling methods with application to control of switched systems with active filtering[END_REF], [START_REF] Gutierrez | Performance analysis of rl damper in ganbased high-frequency boost converter[END_REF].

• Control units: They constitute the control system of the MG components. The control algorithms can be implemented in several electronic solutions as microcontrollers and programmable logic controllers.

Types of Microgrids

MGs can be classified into three types based on the nature of the output voltage fed to the load. There are (i) DC-MG, (ii) AC-MG and (iii) AC/DC hybrid MG.

In AC-MGs, the components are interconnected through AC power lines. DGUs and SUs are connected to the AC power line via an inverter (see Fig. inverter is required to supply AC loads [START_REF] Fusheng | Chapter 2 -composition and classification of the microgrid[END_REF]. However, most storage systems, renewable energy resources and modern loads (e.g., laptops, tablets, computers, etc.) are inherently DC requiring more integrated inverters [START_REF] Hirsch | Microgrids: A review of technologies, key drivers, and outstanding issues[END_REF], [START_REF] Kumar | A review on overall control of dc microgrids[END_REF].

In DC-MGs, the components are interconnected through DC power lines (see Fig. 1.3). Therefore, DC-MGs are more natural interfaces to many types of renewable energy resources and energy storage systems. They have received increasing interest in the power system control engineering community due to its efficiency, simplicity and a wide range of applicability [START_REF] Dragicevic | Dc microgrids-part i: A review of control strategies and stabilization techniques[END_REF].

In AC/DC hybrid MGs, the components are interconnected through DC and AC power lines. The AC and

Operation Modes

MGs can have two modes of operation either grid-connected or island mode. In Grid-connected operation mode, the MG is connected to the main-grid and can exchange energy with it. The grid-connected mode is further divided into power-matched operation when there is no exchange of energy between the MG and the main-grid and power-mismatched operation when there is an exchange of energy between the MG and the main-grid.

In island mode, the MG is disconnected from the main-grid and operates autonomously. In this mode, the local demand is only supplied by the local generation which is generally small. MGs in island mode have more challenging control issues. In this thesis, we consider a DC-MG operating in island mode.

Examples of Microgrids Project

One of the first villages in Europe to be totally using MG is Hoogkerk, a suburb of the Groningen city in the north of the Netherlands (Fig. 1.5). The Hoogkerk MG includes 25 interconnected houses and is part of the SmartHouse/SmartGrid project (see this Video).

In France, the Grenoble city has already launched several projects for energy self-sufficient districts or buildings such as the LearningGrid project launched by Schneider Electric and the Grenoble CCT 1 . This project was installed on the campus of the industrial school IMT 2 in Grenoble and aims to reduce the campus' A low-voltage DC-MG project has also been developed in the ADREAM building of the LAAS3 laboratory in Toulouse, France (Fig. 1.6). The building is 1200 m 2 and includes PV panels, energy storage system, more than 7000 sensors, etc. The objective of this project is to transform the ADERAM building to a net-zero emission one [START_REF] Sguier | Development of a photovoltaic low voltage dc microgrid for buildings with energy storage systems[END_REF].

DC-MG Challenges

When designing control systems for MGs, several operational challenges need to be addressed so as to guarantee the reliability and the benefits of distributed generation [START_REF] Olivares | Trends in microgrid control[END_REF]. In this section, we will present some DC-MGs challenges where some of them will be studied in this thesis.

• Bi-directional power flows: Initially, distributed generators are designed for unidirectional power flows.

However, in DC-MGs, DGUs can operate at low voltage levels. This might cause power flows in the reverse direction and leads to undesirable power flows patterns. Hence, bi-directional power flows in MGs requires new technical strategies, protecting devices and control techniques [START_REF] Olivares | Trends in microgrid control[END_REF].

• Transition between island and grid-connected operation modes: The ability to operate in both island and grid-connected modes is one of the desirable features of MGs. Fast islanding detection is required to ensure a smooth transition between these two modes and to adjust the control strategy accordingly -since different control strategies might be used for each mode of operation [START_REF] Karimi | Islanding Detection and Control of an Islanded Electronically-coupled Distributed Generation Unit[END_REF].

• Cyber-security: Control system networks implemented for MGs may have cyber vulnerabilities allowing an attacker to jeopardize the confidentiality, stability, or availability of the MG [START_REF] Veitch | Microgrid cyber security reference architecture[END_REF], [START_REF] Zhong | Cyber security in smart dc microgrid operations[END_REF].

• Unknown load profile: Coordination between the several DGUs to maintain the balance between demand and supply in DC-MGs is required for the economical and resilient operation of DC-MGs. However, under the uncertainties in the load profile (current demand) and the intermittency of several energy resources in the MG (e.g., PV panels), demand-supply balance becomes more challenging [2].

• Stability issues: A stable DC-MG must be able to maintain the power supply to the load without undesirable dynamics (e.g. voltage or power flow oscillations). DC-MG may be susceptible to large-signal and small-signal stability concerns. Large signal stability refers to the ability of the system to withstand larger disturbances such as step changes in the power demand or faults in the system. Small signal stability refers to the stability of the system under small disturbances, e.g., noise and parameter changes [START_REF] Chakraborty | Power Electronics for Renewable and Distributed Energy Systems A Sourcebook of Topologies, Control and Integration[END_REF].

• Current Sharing (CS): It is a highly desirable feature for maintaining the proper operation of DC-MGs. It aims to share, proportionally, the current demand between the different DGUs, taking into consideration their power capacity. In other words, DGUs must be able to distribute (share) correctly the load currents according to given constants, e.g., their rated currents. Moreover, CS helps to avoid 1.3. LITERATURE REVIEW OF DC MICROGRIDS CONTROL situations where a DGU cannot provide the requested current for its local load and might be led to deliver beyond its maximum rated current (overloading), which can spoil the unit and cause system damage [START_REF] Meng | Review on control of dc microgrids and multiple microgrid clusters[END_REF], [START_REF] Tucci | Stable current sharing and voltage balancing in dc microgrids: A consensus-based secondary control layer[END_REF].

• Average Voltage Regulation (AVR): Voltage Regulation is also one of the important quality parameters in DC-MG supply. A necessary condition for proper operation of the MG is that voltages remain close to their nominal values under any perturbation. Indeed, uncontrolled deviation of the voltage from its reference can cause failure to the load device since the latter was designed to be supplied with a nominal level voltage. Achieving CS and voltage regulation is an arduous and generally impossible task as CS requires voltages deviation from their reference values. Therefore, an alternative is to provide an Average Voltage Regulation (AVR), i.e., the average value of voltages at the points of common coupling is equal to the average of their references [START_REF] Trip | Distributed averaging control for voltage regulation and current sharing in dc microgrids[END_REF], [START_REF] Tucci | Line-independent plug-and-play controllers for voltage stabilization in dc microgrids[END_REF].

• State-of-charge Balancing (SB): For a MG with Storage Units (SUs), unbalanced state-of-charges can affect the life of the batteries by causing over-charge or over-discharge phenomenon, overheating, deterioration, etc. [START_REF] Huang | Energy sharing control scheme for state-of-charge balancing of distributed battery energy storage system[END_REF]. When the SB is ensured between the SUs, the SU with lower state-of-charge will absorb more power than the others during the charging cycle and accordingly, the one with higher state-of-charge will provide more power than the others during the discharging cycle [START_REF] Li | Voltage scheduling droop control for state-of-charge balance of distributed energy storage in dc microgrids[END_REF]. This will prolong the batteries' life and maintain them in a good functional state. Note that a state-of-charge balancing is also required between the series-connected battery cells that compose the SU. The latter is performed by the Battery Management System (BMS), which is out of the scope of this manuscript.

In this thesis, the following challenges will be considered: (i) current sharing and (ii) average voltage regulation between the DGUs, (iii) state-of-charge balancing between the SUs, (iv) stability and (v) unknown load profile.

Literature Review of DC Microgrids Control

To overcome the aforementioned challenges, different control schemes for DC-MG have been proposed in the literature. Generally speaking, they can be categorized into three levels of control (hierarchies): primary, and the need for extensive communication between the central controller and the controlled units. Also, data centralization can introduce confidentiality and security issues [START_REF] Simmhan | An analysis of security and privacy issues in smart grid software architectures on clouds[END_REF]. Therefore, this control architecture is suitable for small size DC-MGs where the information to be transmitted is limited [START_REF] Meng | Review on control of dc microgrids and multiple microgrid clusters[END_REF].

In decentralized techniques (Fig. Primary control. Going back to the control levels, primary control is the first and fastest layer in the hierarchical control scheme [START_REF] Prabaharan | 6 -an overview of control techniques and technical challenge for inverters in micro grid[END_REF]. It consists of two main parts, the first part is called inner-loop and concerns the control of power electronic converters (e.g., DC/DC converters) to provide a controllable interface between loads and sources [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF], [START_REF] Sferlazza | Min-type control strategy of a dc-dc synchronous boost converter[END_REF]. The second part is responsible for local voltage and local current control.

Moreover, a preliminary CS can also be achieved at this level.

LITERATURE REVIEW OF DC MICROGRIDS CONTROL

In [START_REF] Rajagopalan | Modeling and dynamic analysis of paralleled dc/dc converters with master-slave current sharing control[END_REF] and [START_REF] Mazumder | Master-slave current-sharing control of a parallel dc-dc converter system over an rf communication interface[END_REF], a master-slave control approach is used to achieve CS among parallel converter system. In this method, one converter is selected as the master to stabilize the DC bus voltage, while all the other converters become slaves. The master module has both current and voltage loops. However, the slave modules have only the current loop. Then, the output signal of the voltage loop in the master converter is used as a reference for the current loop of all the slave modules so as to reach CS among the slaves and balance the load current. The main disadvantage of this control strategy is the fact that it is less flexible and expandable since it is centralized.

To overcome the drawbacks of the master-slave control, other techniques are used at the primary level such as droop control. In droop-controlled DC-MG, the CS is achieved by linearly reducing the local output voltage of the DGU as the local output current increases. This strategy is decentralized since it does not require any communication between the controllers. Even if droop-control has been widely applied on DC power networks [START_REF] Batarseh | Investigation of the output droop characteristics of parallel-connnected dc-dc converters[END_REF], [START_REF] Perkinson | Current sharing of redundant dc-dc converters in high availability systems-a simple approach[END_REF], [START_REF] Kim | A novel droop method for converter parallel operation[END_REF], it has several limitations. In [START_REF] Haileselassie | Impact of dc line voltage drops on power flow of mtdc using droop control[END_REF], it is shown that if the power line resistance is not negligible, the efficiency of the droop control is reduced, and the CS accuracy is degraded. To overcome this problem, the value of the droop coefficients can be increased so the power line resistance becomes negligible. However, this approach is not always efficient since for low voltage DC-MG the power lines are mostly resistive and the CS accuracy will still be degraded. Moreover, by increasing the droop coefficient, the voltage deviation will also increase. A detailed explanation of the conventional droop control method and its drawbacks can be found in [START_REF] Perkinson | Current sharing of redundant dc-dc converters in high availability systems-a simple approach[END_REF] and Section 2 of [START_REF] Lu | An improved droop control method for dc microgrids based on low bandwidth communication with dc bus voltage restoration and enhanced current sharing accuracy[END_REF]. Several modified droop-control characteristics have been proposed in [START_REF] Chen | Investigation of nonlinear droop control in dc power distribution systems: Load sharing, voltage regulation, efficiency, and stability[END_REF], [START_REF] Chen | A nonlinear droop method to improve voltage regulation and load sharing in dc systems[END_REF], [START_REF] Prabhakaran | Novel nonlinear droop control techniques to overcome the load sharing and voltage regulation issues in dc microgrid[END_REF] to resolve the trade-off between the accuracy of CS and the voltage deviation. Most of these approaches are based on the idea of CHAPTER 1. BACKGROUND AND PRELIMINARIES increasing the droop coefficient when load increases to achieve better CS at high-load operation mode and tighter voltage regulation at low-load operation mode. In [START_REF] Prabhakaran | Novel nonlinear droop control techniques to overcome the load sharing and voltage regulation issues in dc microgrid[END_REF], linear and nonlinear droop characteristics (control laws) are combined, and conditions of switching between them are defined based on the operation mode. The main inconvenience of this technique is when we are near the point of switching between the droop-control laws (i.e., linear and nonlinear characteristics), because of sensor errors, the switch can not take place exactly at the desired point. Hence, one DGU can operate with nonlinear droop-control and another DGU with linear one that will not ensure the accuracy of the CS.

MG (c)

Moreover, [START_REF] Chen | Investigation of nonlinear droop control in dc power distribution systems: Load sharing, voltage regulation, efficiency, and stability[END_REF] presents an investigation of the different nonlinear droop control methods proposed in the literature. Even if the use of nonlinear droop techniques can reduce the impact of the CS accuracy degradation and voltage deviation, the complexity of the MG requires more cooperative techniques.

Secondary control. Also called management level, is used for energy management and enhancing the objectives of primary control [START_REF] Anand | Distributed control to ensure proportional load sharing and improve voltage regulation in low-voltage dc microgrids[END_REF], [START_REF] Lu | An improved droop control method for dc microgrids based on low bandwidth communication with dc bus voltage restoration and enhanced current sharing accuracy[END_REF], [START_REF] Adhikari | Secondary control for dc microgrids: A review[END_REF], [START_REF] Nasirian | Distributed cooperative control of dc microgrids[END_REF], [START_REF] Wang | An improved distributed secondary control method for dc microgrids with enhanced dynamic current sharing performance[END_REF], [START_REF] Nasirian | Distributed adaptive droop control for dc distribution systems[END_REF], [START_REF] Mehrizi-Sani | Potential-function based control of a microgrid in islanded and gridconnected modes[END_REF], [START_REF] Guerrero | Hierarchical control of droop-controlled ac and dc microgrids-a general approach toward standardization[END_REF]. The implementation of the secondary control can be done in a centralized fashion [START_REF] Mehrizi-Sani | Potential-function based control of a microgrid in islanded and gridconnected modes[END_REF], [START_REF] Guerrero | Hierarchical control of droop-controlled ac and dc microgrids-a general approach toward standardization[END_REF], distributed fashion with all-to-all communication (i.e., any two units are directly connected) [START_REF] Lu | An improved droop control method for dc microgrids based on low bandwidth communication with dc bus voltage restoration and enhanced current sharing accuracy[END_REF], [START_REF] Wang | An improved distributed secondary control method for dc microgrids with enhanced dynamic current sharing performance[END_REF] or with a neighbor-to-neighbor communication (i.e., a sparse communication network) [START_REF] Trip | Distributed averaging control for voltage regulation and current sharing in dc microgrids[END_REF], [START_REF] Tucci | Line-independent plug-and-play controllers for voltage stabilization in dc microgrids[END_REF], [START_REF] Nasirian | Distributed cooperative control of dc microgrids[END_REF], [START_REF] Nasirian | Distributed adaptive droop control for dc distribution systems[END_REF], [START_REF] Tucci | A decentralized scalable approach to voltage control of dc islanded microgrids[END_REF], [START_REF] Cucuzzella | A robust consensus algorithm for current sharing and voltage regulation in dc microgrids[END_REF].

In [START_REF] Lu | An improved droop control method for dc microgrids based on low bandwidth communication with dc bus voltage restoration and enhanced current sharing accuracy[END_REF], a distributed secondary control is proposed where the primary level is ensured by droop control.

First, local controllers are implemented for each element of the MG (i.e., DGUs in closed-loop with droop control). Each controller receives information from all the other DGUs using a low bandwidth communication network. Then the output of each local controller is used as a voltage-shifting term to adjust the voltage set-point of the droop mechanism. This helps to reduce the voltage deviation and improve the current sharing accuracy. Even if this approach is distributed (since all the calculations are done locally), the use of an all-to-all communication network still a serious drawback since any failure in a communication link between two units will influence the whole control system.

In [START_REF] Nasirian | Distributed cooperative control of dc microgrids[END_REF], a consensus-based 4 approach is proposed where each agent (unit) exchanges information with only its neighbors. Then, the collected information will be used to update the control variables of the agent. Each local controller generates two correction terms that are sent to the droop mechanism. The first term is used to rectify the voltage deviation while the second one is used to ensure the CS accuracy.

SB is also considered in the secondary level [START_REF] Li | Voltage scheduling droop control for state-of-charge balance of distributed energy storage in dc microgrids[END_REF], [START_REF] Lu | State-of-charge balance using adaptive droop control for distributed energy storage systems in dc microgrid applications[END_REF], [START_REF] Lu | Double-quadrant state-of-charge-based droop control method for distributed energy storage systems in autonomous dc microgrids[END_REF], [START_REF] Morstyn | Multi-agent sliding mode control for state of charge balancing between battery energy storage systems distributed in a dc microgrid[END_REF], [START_REF] Wu | Multiagent based distributed control with timeoriented soc balancing method for dc microgrid[END_REF], [START_REF] Hu | Distributed secondary control for state of charge balancing with virtual impedance adjustment in a dc microgrid[END_REF], [START_REF]Hierarchical distributed control for decentralized battery energy storage system based on consensus algorithm with pinning node[END_REF]. In [START_REF] Li | Voltage scheduling droop control for state-of-charge balance of distributed energy storage in dc microgrids[END_REF], a centralized technique based on droop control is proposed to balance the state-of-charge where the central controller computes the average of all the state-of-charge and generates a voltage shifting term. This correction term 1.3. LITERATURE REVIEW OF DC MICROGRIDS CONTROL is then sent to the local droop-control to adjust the voltage setpoint of the interfacing converter of the SUs.

In [START_REF] Lu | State-of-charge balance using adaptive droop control for distributed energy storage systems in dc microgrid applications[END_REF], the SB is achieved through a decentralized droop-based method by adjusting the droop coefficient using only the local state-of-charge. However, only the discharge operation mode was taken into account.

In [START_REF] Lu | Double-quadrant state-of-charge-based droop control method for distributed energy storage systems in autonomous dc microgrids[END_REF], a similar approach is proposed but this time, both charge and discharge process were considered.

In [START_REF] Wu | Multiagent based distributed control with timeoriented soc balancing method for dc microgrid[END_REF], a consensus-based observer is proposed to estimate the average of the state-of-charge. Then, the difference between the local state-of-charge and the average one is sent to the local control. The output of the latter is used as a voltage shifting term that will be added to the local droop control to balance the stateof-charge. Even if this control paradigm uses a sparse communication network for data exchange, the global convergence is not proved. In fact, stability is analyzed through a centralized method, i.e., the root locus.

In [START_REF] Tucci | Line-independent plug-and-play controllers for voltage stabilization in dc microgrids[END_REF], [START_REF] Tucci | A decentralized scalable approach to voltage control of dc islanded microgrids[END_REF], [START_REF] Nahata | A passivity-based approach to voltage stabilization in dc microgrids with zip loads[END_REF], only voltage stabilization is taken into account but with proof of global stability. In [START_REF] Cucuzzella | A robust consensus algorithm for current sharing and voltage regulation in dc microgrids[END_REF], a sliding-mode approach is proposed by designing a manifold that couples CS and AVR with global stability assurance. However, proper initialization of the controller is needed. In [START_REF] Trip | Distributed averaging control for voltage regulation and current sharing in dc microgrids[END_REF], a control approach, in an ad-hoc manner, is proposed with global convergence independently of the MG parameters and the initial conditions but only CS and AVR were considered.

Tertiary control. Also known as the grid level control, is the highest level in the control hierarchy and has the slowest dynamics. It is used for several purposes: coordination and optimal operation between several MGs (cluster of MGs), to carry out the power flow in grid-connected mode, optimization of the system efficiency such as optimal pricing, coordination with the distribution system operator and long-term energy storage management [START_REF] Olivares | Trends in microgrid control[END_REF], [START_REF] Fei | Primary and secondary control in dc microgrids: a review[END_REF], [START_REF] Prabaharan | 6 -an overview of control techniques and technical challenge for inverters in micro grid[END_REF].

Contribution of the thesis

• In this thesis, we propose a new distributed control methodology in the primary and the secondary levels for a more general DC-MG model including not only DGUs but also SUs interconnected through resistive-inductive power lines, and where the control objectives are CS, AVR and SB.

• The novelty in this work is the use of three consensus-like distributed integral actions to achieve simultaneously the three aforementioned objectives. We show that inside the set of equilibria, SCB, CS and AVR objectives are achieved. The proof of the global exponential convergence to this set is provided despite the unknown variation of the load and the initial conditions of the MG and the controller state.

• The consensus algorithm that we propose to achieve AVR at the equilibrium is a new modified version of [START_REF] Spanos | Dynamic consensus for mobile networks[END_REF] and its dynamics are considered in the proof of convergence, which is not the case in [START_REF] Morstyn | Multi-agent sliding mode control for state of charge balancing between battery energy storage systems distributed in a dc microgrid[END_REF] where the consensus algorithm used to compute the average voltage is assumed to converge for a voltage step variation.

• Also, the SB is achieved using a leader-follower strategy that gives the ability for high-level monitoring by adjusting the state-of-charge reference sent to the leader.

• The controllers do not need any information about the topology of the power network, neither the parameters and currents of the power lines. The topology of the communication network does not need to be the same as the topology of the DC-MG.

• Furthermore, the proposed control approach is LMI-based, which makes it attractive numerically.

• Finally, we provide Matlab/Simulink simulation and real-time Hardware-in-the-loop (HIL) implementation tests where the results show the effectiveness of the proposed controllers.

In the next part of this chapter, we will recall fundamental definitions and results of some mathematical tools needed for the modeling and control of DC-MGs.

Graph Theory

Usually, graph theory is used to describe the structural properties (topology) of a network. It gives an abstraction of how the information is exchanged between agents in a network. This high-level description is done in terms of objects referred to as vertices and edges. In this section, we recall some definitions and properties from [START_REF] Mesbahi | Graph Theoretic Methods in Multiagent Networks[END_REF] and [START_REF] Godsil | Algebraic Graph Theory[END_REF].

A finite graph G is composed of a finite set of elements that we call the vertex set and denote it by V. The vertices represent the elements of the graph; the set V can be represented as

V = {1, 2, • • • , n}.
The interconnection between the vertices is described by a subset of V × V called the edges set; denoted by E.

Definition 1.3. (Neighborhood)

The neighborhood N i ⊆ V of the vertex i is defined by the set {j ∈ V | (i, j) ∈ E}.

1.4. GRAPH THEORY Definition 1.4. (Undirected Graph) A graph G is undirected if for all i, j ∈ V, (i, j) ∈ E =⇒ (j, i) ∈ E.
Otherwise the graph is directed. For example the graph in Fig. 1.9 is connected and the graph in Fig. 1.10 is disconnected. Graphs can be presented not only by graphical representation but also using matrices. For an undirected graph G, the degree of a given vertex i ∈ V is the cardinality of the neighborhood set N i , that is, it is equal to the number of vertices that are adjacent to the vertex i. Thus, for the graph in Fig. 1.9, the degrees of the CHAPTER 1. BACKGROUND AND PRELIMINARIES vertices are

v 2 v 3 v 1 v 4 v 5
d(1) = 2, d(2) = 4, d(3) = 2, d(4) = 1, d(5) = 1.
Definition 1.6. (Degree Matrix) The degree matrix D ∈ R n×n of a graph G with n vertices, is the diagonal matrix, containing the vertex-degrees of G on the diagonal, that is,

D =         d(1) 0 • • • 0 0 d(2) • • • 0 . . . . . . . . . . . . 0 0 • • • d(n)         . (1.1)
For example the degree matrix of the graph in Fig. 1.9 is

D =            2 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1            . (1.2)
Definition 1.7. (Adjacency Matrix) The adjacency matrix A ∈ R n×n is the symmetric matrix encoding the adjacency relationship in the graph G, that is,

A ij =      1 if (i, j) ∈ E, 0 otherwise.
(1.3) Going back to the example in Fig. 1.9, the corresponding adjacency matrix is The laplcian matrix of the graph in Fig. 1.9 is

A =            0 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0            . ( 1
L =            2 -1 -1 0 0 -1 4 -1 -1 -1 -1 -1 2 0 0 0 -1 0 1 0 0 -1 0 0 1            . (1.6)
Remark 1.2. Let 1 ∈ R n×1 be the vector of all ones, i.e.,

1 =      1 . . . 1      .
Note that by construction, for every connected undirected graph G, the vector of all ones is the eigenvector associated with the zero eigenvalue of L, i.e.:

1 ∈ Ker(L).

(1.7)

Now we can define a graph as Definition 1.9. (Graph) A graph is a triple (V, E, L), where V = {1, 2, ..., n} is the node set, E ⊆ V × V is the edge set and L ∈ R n×n is the Laplacian matrix.

Remark 1.3. The following properties will be used in the rest of the thesis:

• The Laplacian matrix of an undirected graph is symmetric and positive semidefinite. Thus, its eigenvalues are real and can be ordered as follows:

λ 1 (L) = 0 ≤ λ 2 (L) ≤ • • • ≤ λ n (L). (1.8)
• The graph G is connected if and only if λ 2 (L) > 0.

CHAPTER 1. BACKGROUND AND PRELIMINARIES

• If for a given X ∈ R n , we have LX = 0, implies X = α1 n , with α ∈ R.

• There exists an orthonormal matrix T ∈ R n×n that verifies T T = T T = I n such that: MAS has become an essential tool for understanding inter-elemental interactions in MGs. Indeed, DGUs and SUs can be considered as agents that interact with each other through physical coupling (power lines).

T LT = diag(λ 1 (L), λ 2 (L), • • • , λ n (L)).

Consensus in Linear Multi-Agent Systems

Moreover, due to the distributed nature of the generation in MGs, agents (DGUs or SUs) could have a partial representation of the power network, e.g., a SU knows only its state-of-charge and may receive information about the state-of-charge of some other SUs, but it does not know what is happening in the whole network.

Therefore, several decisions can be made locally and coordination between the agents is needed to achieve an agreement regarding a certain quantity of interest, i.e., a consensus. This requires a certain level of autonomy for the agents and a certain rule of interaction that specifies how information is exchanged between the agents, i.e., a consensus algorithm. The latter introduces another type of interaction, the soft coupling 1.5. CONSENSUS IN LINEAR MULTI-AGENT SYSTEMS (communication links), making the analysis of the system more complicated [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF].

Our interest here is to present the formulation of some consensus problems of MAS. This will be useful when formulating the considered control problems in Chapter 2. Moreover, we aim to highlight the difficulties associated with the analysis of stability and the design of distributed control laws for DC-MGs.

The most commonly used models to study MAS are decoupled single integrator and double integrator models [START_REF] Ren | Distributed Consensus in Multi-vehicle Cooperative Control: Theory and Applications[END_REF]. However, in DC-MGs, agents may be governed by more commonly coupled inherent linear dynamics.

Consider a physical linear MAS represented by a connected and undirected graph G phy = (V, E phy , L phy ) composed of N linear systems interconnected with static physical coupling of the form:

ẋi = Ax i + B u u i -B z j∈N phy i α ij (z i -z j ), z i = M x i , (1.9
)

with x i ∈ R n ,u i ∈ R m , z i ∈ R q , α ij ∈ R and A, B u , B z , M
α ij (z i -z j )
represents the physical coupling between the agents.

Let us consider that there exist certain output variables of interest for which the agents wants to reach an agreement. These variables can be defined as follows

y i = Cx i , i ∈ V, (1.10)
for simplicity we consider y i as scalar i.e., y i ∈ R and C ∈ R 1×n .

Let us assume that agents can exchange the value of the variable of interest through a communication network represented by a connected and undirected graph G com = (V, E com , L com ) whose topology is not necessarily the same as that of the physical network, i.e. L com = L phy (e.g., Fig. 1.12). Let N com i be the set of neighbors of agent i with respect to G com . Each agent i ∈ V receives the following measurements:

• Internal state measurement x i .

• External state measurements (relative to other agents) y j , j ∈ N c i .

Agents can achieve several types of agreement on the output variables of interest (1.10) such as, weighted output consensus, leader-follower consensus and output averaging.

Weighted Output Consensus Problem

Before presenting the weighted output consensus problem, we need to recall the following definitions Definition 1.10. (Invariant set) A set Ω ⊆ R n is invariant, with respect to a dynamical system ẋ = f (x), if for every trajectory x, Let us associate to each agent i a weight parameter ω i ∈ R >0 . The weighted output consensus of MAS (1.9) means designing a distributed control laws u i , i ∈ V, (i.e., a consensus algorithm) based on the local information obtained by each agent, such that all the agents reach an agreement on the output variables of interest (1.10) by negotiating with their neighbors. For DC-MGs, these variables might represent currents.

x(t) ∈ Ω =⇒ x(τ ) ∈ Ω f or all τ ≥ t. ( 1
Definition 1.12. (Weighted output consensus) The state of system (1.9) converges to a weighted output consensus if it converges to an invariant set A (the agreement set) with reference to (1.9), defined equivalently as

• A = {y = (y 1 , • • • , y N ) ∈ R N : ω i y i = ω j y j , ∀(i, j) ∈ V × V}. • A = {y = (y 1 , • • • , y N ) ∈ R N : L com W y = 0, W = diag(ω 1 , • • • , ω N ).} 1.5. CONSENSUS IN LINEAR MULTI-AGENT SYSTEMS • A = {y = (y 1 , • • • , y N ) ∈ R N : y = βW -1 1 N , β ∈ R}.
Remark 1.4. The weights ω i , i ∈ V in Definitions 1.12 and 1.14 can be used to give to some agents more influence (weight) on the result of the agreement than other agents in the same graph. In the context of DC-MGs, the weights can be related to the generation capacity or rated current of the DGUs .

Leader-follower Consensus Problem

The leader-follower consensus of the MAS (1.9) means to design a distributed control laws u i , i ∈ V, (i.e., a leader-follower algorithm) based on the local information obtained by each agent, such that the values of the output variables of interest (1.10) converge to a common value defined by one agent (called leader) by negotiating with their neighbors. For DC-MGs, these variables might represent the state-of-charge.

Definition 1.13. (Leader-follower consensus problem) The state of system (1.9) converges to a leaderfollower consensus if it converges to an invariant set A with reference to (1.9), defined equivalently as

• A = {y = (y 1 , • • • , y N ) ∈ R N : y i = y j = y i l , ∀(i, j) ∈ V × V, i l is the index of the leader}. • A = {y = (y 1 , • • • , y N ) ∈ R N : y = y i l 1 N }.

Output Averaging Problem

Consider that each output variable of interest (1.10) has its reference y ref i for which we associate a weight parameter ω i ∈ R >0 . The output averaging consensus problem of the MAS (1.9) means to design a distributed control laws u i , i ∈ V, (i.e., an output averaging algorithm) based on the local information obtained by each agent, such that the weighted average value of the output variables of interest (1.10) of all the agents converge to the weighted average value of their references by negotiating with their neighbors. For a MG, these variables might represent the voltages at the points of common coupling.

Definition 1.14. (Output averaging problem)

The output averaging problem of system (1.9) is solved if the state of the system converges to an invariant set A with reference to (1.9), defined equivalently as

• A = {y = (y 1 , • • • , y N ) ∈ R N : 1 N i∈V ω i y i = 1 N i∈V ω i y ref i , ∀(i, j) ∈ V × V}. • A = {y = (y 1 , • • • , y N ) ∈ R N : 1 N W y = 1 N W y ref i , W = diag(ω 1 , • • • , ω N )}.

CHAPTER 1. BACKGROUND AND PRELIMINARIES

To summarize this part of the chapter, several objectives in DC-MGs can be formulated as a weighted output consensus problem, leader-follower problem and output averaging problem. Among the difficulties when controlling a DC-MG in a distributed fashion and analyzing its stability are the existence of physical coupling (power lines) and the availability of only partial information (no agent has access to the overall state of the MG).

Stability and Passivity of Interconnected Linear Systems

In this section, some basic definitions of stability of Linear Time-Invariant (LTI) systems are presented.

Moreover, we provide two application examples to show how Linear Parameter-Varying (LPV) techniques and passivity are useful for the stability analysis of linear MAS.

Lyapunov Stability

When studying the properties of a dynamical control system, one of the important questions is whether the system is stable or not. The most useful method for studying the stability of a control system is the theory introduced by the russian mathematician and engineer Alexandr Mikhailovich Lyapunov in the late 19th century [START_REF] Jean-Jacques | Applied nonlinear control[END_REF], [START_REF] Hassan | Nonlinear systems[END_REF].

Consider the following autonomous LTI system:

ẋ(t) = Ax(t).
(1.12)

We call x e an equilibrium point of (1.12) if Ax e = 0. The stability (in the sense of Lyapunov) means that solutions that start nearby an equilibrium point remain close enough to it. Moreover, an equilibrium point is unstable if it is not stable.

Definition 1.15. An equilibrium point x e of system (1.12) is said to be stable if for all > 0, there exists

δ > 0, such that ||x(0) -x e || < δ =⇒ ||x(t) -x e || < , ∀t > 0.
Asymptotic stability means that solutions that start close enough to equilibrium point not only remain close enough but also eventually converge to it.

any initial condition x(0) ∈ R n , lim t→+∞ x(t) = x e , (∃β, γ ∈ R >0 , ||x(t) -x e || ≤ β||x(0) -x e ||e -γt , ∀t > 0).
The stability of an equilibrium point can be ascertained by testing the rate of change of the system's energy, i.e.

if the total energy of the system is continuously dissipating, then the system tends to return to its equilibrium.

Lyapunov's direct method is a generalization of this idea. It gives sufficient conditions for the stability of an equilibrium point by constructing an "energy-like" scalar function for the system and examining its time variation [START_REF] Jean-Jacques | Applied nonlinear control[END_REF].

Note that by shifting the origin of system (1.12), we can assume that the equilibrium point of interest occurs at

x e = 0. If several equilibrium points exist, one needs to analyze the stability of each of them by appropriately shifting the origin.

Theorem 1. Consider system (1.12) with x e = 0 as equilibrium point and V (x) : x ∈ R n → R a scalar function called Lyapunov function.

• x e is stable if V (x) is positive definite and -V (x) is positive semidefinite 6 .

CHAPTER 1. BACKGROUND AND PRELIMINARIES

• x e is asymptotically stable if V (x) is positive definite and -V (x) is positive definite.

It is noteworthy that one can look at V (x) as a generalized energy function and -V (x) as the corresponding generalized dissipation function.

Remark 1.5. The notions of exponential and asymptotic stability are equivalent for linear time-invariant systems.

Usually, for an LTI system, a standard Lyapunov function candidate is the quadratic function of its state, e.g.,

V (x) = x P x where P is a symmetric positive definite matrix.

Theorem 2. The equilibrium point x e of system (1.12) is asymptotically stable if and only if, for all Q = Q > 0, there exists a matrix P = P > 0 such that

A P + P A + Q ≤ 0. (1.13) Equation (1.13) is called Lyapunov equation.
The analysis of stability becomes more challenging when considering systems with nonlinear dynamics such as LPV systems.

Stability of Linear Parameter-Varying System

Consider an autonomous LPV system [START_REF] Scherer | Linear Matrix inequalities in Control[END_REF]:

ẋ = A(θ)x, (1.14)
where the matrix A depends on the vector of the varying parameters θ

= [θ 1 , • • • , θ k ], with
θ i ≤ θ i ≤ θi , 1 ≤ i ≤ k. (1.15)
Generally, when no information is given about the dependence of the matrix A on θ, the condition of stability in Theorem 2 should be verified for each value of the parameters θ i , 1 ≤ i ≤ k, which results an infinite number of LMI to verify.

STABILITY AND PASSIVITY OF INTERCONNECTED LINEAR SYSTEMS

A polytop is a convex7 set limited by a finite number of points called vertices (Fig. 1.13). For polytopic LPV system, the matrix A is a linear convex combination of several matrices: A p 1 , ..., A p m , i.e.,

A(θ) = m i=1 α i A p i , α i ∈ [0 1], m i=1 α i = 1,
such that there exists a function f : R k R m mapping every value of the vector of parameters θ to a unique value of the vector of coefficients

α = [α 1 , • • • , α m ]. Moreover, A p i , 1 ≤ i ≤ m
, are the value of the matrix A(θ) at the vertex i of the polytop.

In case of polytopic LPV system, it is sufficient that the conditions of stability are verified at the vertices of the polytop, with a common Lyapunov function, to be verified inside the whole polytop. Theorem 3. If there exist P = P > 0, Q = Q > 0 such that:

A pT i P + P A p i + Q ≤ 0, 1 ≤ i ≤ m. (1.16)
Then for all the values of the vector θ that respect (1.15), the following LMI condition is verified:

A(θ) P + P A(θ) + Q ≤ 0.
(1.17)

Example 1.2. Consider the following polytopic LPV system:

ẋ =   -θ 1 0.3 θ 1 + 2 -2   x, (1.18)
where x ∈ R 2 and 1 ≤ θ 1 ≤ 2. Thus, the polytop is a straight line that has two vertices, i.e.,

A(θ) = α 1 A p 1 + α 2 A p 2 , where 0 ≤ α 1 , α 2 ≤ 1, α 1 + α 2 = 1, and
A p 1 =   -1 0.3 3 -2   , A p 2 =   -2 0.3 4 -2   .
If there exist P = P > 0, Q = Q > 0 such that:

A p 1 P + P A p 1 + Q ≤ 0, A p 2 P + P A p 2 + Q ≤ 0, (1.19) (1.20)
then System (1.18) is asymptotically stable for all values of

θ 1 ∈ [1 2].
The result in Theorem 3 can be useful when studying the stability of heterogeneous linear MAS. To illustrate this utility, consider the following decoupled MAS:

ẋi = A i x i , (1.21)
where x i ∈ R 2 and

A i =   α i a 12 a 21 a 22   , (1.22)
and where the elements a 12 , a 21 , and a 22 have the same values for all the agents and the value of α i can change from an agent to another within a certain interval, i.e.,

α ≤ α i ≤ ᾱ, 1 ≤ i ≤ n.
One can conclude about the stability of all the agents by only analyzing the stability of two agents (corresponding to α i = ᾱ and α i = α, respectively) with a common Lyapunov function. In another words, the 1.6. STABILITY AND PASSIVITY OF INTERCONNECTED LINEAR SYSTEMS MAS (1.21) can be seen as an LPV system and the agents corresponding to the maximum and minimum values of α i as the vertices of the polytop. However, this technique does not handle the case when there is an interconnection between the agents.

When studying interconnected systems as MGs, the analysis of stability becomes more complex. The question is, does there exist any property that gives us the possibility to conclude about the stability of a whole system by analyzing its interconnected components separately. Here comes the utility of passivity.

Passivity

Passivity is part of a broader and general theory of dissipativeness [START_REF] Willems | Dissipative dynamical systems Part II: Linear systems with quadratic supply rates[END_REF]. It is a notion directly related to energy that was first introduced in circuit analysis and since then has been successfully applied in various fields, such as stability [START_REF] Zhang | Passivity analysis of bam nns with mixed time delays[END_REF].

Consider the following linear minimal-realization time-invariant system:

ẋ = Ax + Bu, y = Cx, (1.23)
where the input u and the output y have the same dimension.

Definition 1.19. [START_REF] Hassan | Nonlinear systems[END_REF] Consider a continuously differentiable positive semidefinite function S(x), system (1.23) is said to be

• passive if u y ≥ Ṡ = δS δx f (x, u), ∀(x, u) ∈ R n × R p .
• strictly passive if there exists a positive definite function φ(x) such that

u y ≥ Ṡ + φ(x), ∀(x, u) ∈ R n × R p . • lossless if u y = Ṡ, ∀(x, u) ∈ R n × R p .
The function S(x) is called Storage function.

Note that the main point of passivity is that "The feedback interconnection of passive system yields a passive system". The following lemmas show two algebraic characterizations of the strict passivity property of System (1.23) [START_REF] Hassan | Nonlinear systems[END_REF].

CHAPTER 1. BACKGROUND AND PRELIMINARIES Lemma 1.1. System (1.23) is said to be strictly passive if and only if there exist matrices P = P > 0 and

Q = Q > 0 such that A P + P A + Q ≤ 0, P B = C .
(1.24)

(1.25)

Lemma 1.2. System (1.23) is said to be strictly passive if and only if there exist matrices P = P > 0 and

Q = Q > 0 such that   A T P + P A + Q P B -C T B T P -C 0   ≤ 0.
(1.26)

The following example illustrates the utility of passivity for interconnected systems.

Example 1.3. Consider a coupled MAS composed of two agents whose dynamics are as follows:

ẋ1 = A 1 x 1 + B 1 (y 2 -y 1 )
,

y 1 = C 1 x 1 , (1.27)
and ẋ2 = A 2 x 2 + B 2 (y 1 -y 2 ),

y 2 = C 2 x 2 , (1.28)
where the matrices A 1 , A 2 , B 1 , B 2 , C 1 and C 2 are of appropriate dimension. The overall system is represented by a connected and undirected graph G = (V, E, L), where V = {1, 2}, E = {(1, 2)} and

L =   1 -1 -1 1   .
The overall dynamics can be written as follows

ẋ = Ax -BLy, (1.29)
where

x = [x 1 x 2 ] , y = [y 1 y 2 ] , A = diag(A 1 , A 2 ) and B = diag(B 1 , B 2 ).
Let us consider that systems (1.27) and (1.28) are strictly passive, i.e., there exist symmetric positive definite

1.7. CONCLUSION matrices P 1 , P 2 , Q 1 , Q 2 such that A 1 P 1 + P 1 A 1 + Q 1 ≤ 0, P 1 B 1 = C 1 , (1.30)
and

A 2 P 2 + P 2 A 2 + Q 2 ≤ 0, P 2 B 2 = C 2 .
(1.31)

Now, we will show how by ensuring the strict passivity of each agent separately, we can conclude about the stability of the whole system. The stability of System (1.29) can then be guaranteed by considering the following Lyapunov function:

V (x) = 1 2 x 1 x 2   P 1 0 0 P 2     x 1 x 2   . (1.32)
Indeed, by derivation of V (x) along the trajectories and using the conditions of passivity (1.30) and (1.31), we get:

V (x) = x P Ax -x P BLy, = x 1 P 1 A 1 x 1 + x 2 P 2 A 2 x 2 -x P BLCx, ≤ -x 1 Q 1 x 1 -x 2 Q 2 x 2 <0 -x C LCx ≤0 . (1.33) (1.34)
(1.35)

Since, the graph G is connected and undirected, L is positive semidefinite and x C LCx ≥ 0. Hence, we can conclude that,

V (x) < 0, x = 0. (1.36)
Thus, System (1.29) is asymptotically stable.

Conclusion

MGs represent the future of power systems. They are mainly composed of distributed generation units, storage units, loads and power lines. They can be classified into three categories: DC, AC and DC-AC MGs and have two operation modes, grid-connected and island modes. Moreover, several challenges of DC-MGs have been addressed in the literature using several control architectures. Depending on their time scale, they can be categorized into three control levels, namely primary, secondary and tertiary control. In each of these CHAPTER 1. BACKGROUND AND PRELIMINARIES levels, the control can be implemented in a centralized, decentralized and distributed manner. Graph theory as well as consensus dynamics, stability, passivity, etc., are essential tools for studying DC-MG. Finally, several control objectives in DC-MGs can be formulated as a consensus problem of multi-agent systems.

In the next chapter, we consider the challenges of current sharing, average voltage regulation and state-ofcharge balancing for a DC-MG where the control laws will be designed in a distributed fashion.

CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING

In this chapter, we study the problem of Current Sharing (CS), Average Voltage Regulation (AVR) and Stateof-charge Balancing (SB) for DC-MGs composed of DGUs, SUs, resistive-inductive power lines and loads.

The chapter is divided into four parts. In Section 2.1, we present the considered DC-MG model. Then in Section 2.2, we consider the problem of CS and AVR for a DC-MG composed of only DGUs, resistive power lines and loads. To achieve the control objectives, the model is augmented with two distributed consensuslike integral actions. A change of variable will be proposed to transform the system into several decoupled subsystems (DGU augmented with local integral actions). We show that if there exists a unique feedback gain that stabilizes the subsystems under a passivity condition, the global exponential convergence of the nominal model is ensured. At the end of the section, we give an LMI formulation for the design of the controller gain.

In Section 2.3, a DC-MG composed of only SUs with resistive power lines and loads is considered where the problem of SB is studied. To achieve the control objectives, a leader-follower consensus algorithm is proposed. The design steps of the controllers are similar to those in Section 2.2.

In Section 2.4, we show that the designed controllers in the previous sections can be used to reach Objectives 2.1-2.3 at the same time for a DC-MG comprises both DGUs and SUs and even resistive-inductive power lines. Finally, Section 2.5 concludes the chapter.

DC Microgrid Model

In this work, we consider a DC-MG comprising m Distributed Generation Units (DGUs) and p Storage Units (SUs) interconnected through q resistive-inductive (RL) power lines; N = m + p is the number of agents (DGUs and SUs) in the MG.

The DC-MG is represented by a graph G pow = (V pow , E pow ) where V pow is the set of all the agents and For simplicity, we will first assume that the dynamics of power lines are very fast comparing to the DGUs and SUs dynamics. Thus, by considering the QSL-approximation1 (i.e., l l ij = 0, ∀(i, j) ∈ E pow ) and applying the Kirchhoff's current and voltage laws, the dynamics of the agent i (i.e., DGU i or SU i) can be described by the following generic equation:

E pow = V pow × V pow represents the set of power lines. Moreover, V pow = V d ∪ V s where V d = {1, • • • , m} and V s = {1, • • • ,
V dc i r i I i l i r l i j I l ij l l ij r l ij I l ij l l ij c i + - V i r L i PCCi I Li
l i İi = -r i I i -V i + u i , c i Vi = I i -I Li - V i r Li - j∈N pow i 1 r l ij (V i -V j ), (2.1)
where r l ij = r l ji and N pow i denotes the set of agents neighbors of the agent i.

The state-of-charge (Soc) dynamics at each SU is given as follows: means that the power lines are considered mainly resistive. This assumption will be relaxed in Section 2.4 and resistive-inductive power lines will be reconsidered. Load current near Agent i.

Ṡ oc i = -k soc i I i ∀i ∈ V s , (2.2

Remark 2.1. (Very fast converter)

The DC/DC dynamics is considered very fast regarding the other dynamics of the power network. In fact, this is a mild assumptions since modern converters can switch at very high frequency [START_REF] Cucuzzella | Design and Analysis of Sliding Mode Control Algorithms for Power Networks[END_REF].

Remark 2.2. (Kron reduction)

The topology of the considered DC-MG model is such that the loads are connected to the PCC of each agent (Fig. 2.2). If the loads are connected elsewhere, the Kron reduction method can be used to map it to the PCC [START_REF] Tucci | Scalable control of islanded microgrids[END_REF], [START_REF] Dorfler | Kron reduction of graphs with applications to electrical networks[END_REF].

The overall diagram of the proposed controller is illustrated in Fig. the local current I i and voltage V i , will define the augmented state of the DGU i (X d i ). Then, the augmented state is multiplied by a feedback gain K to generate the local controller u d i .

w j I j Soc j I i ω j γ j φ i γ i V i Soc i θ i V i I i ω i ω i ω i γ i ω i I i V ref i Soc ref (for the leader only) X s i X d i
The second module includes a distributed integral action to ensure the SB. The state-of-charge is estimated locally. Each local controller receives from its neighbors the state-of-charge Soc j and generates a control variable θ i . Note that if the SU i is the leader, its local controller will also receive the Soc ref from a high-level monitor (e.g., a tertiary controller). The control variable θ i with the local current I i , the voltage V i and the state-of-charge Soc i will define the augmented state of SU i (X s i ). Then, the augmented state is multiplied by a feedback gain F to generate the local controller u s i .

In what follows, we aim to design the distributed controller gains (K and F ) with the corresponding integral actions. However, the existence of physical coupling between the Agents (the term

j∈N pow i 1 r l ij (V i -V j )
in (2.1)) makes the design difficult. To relax this difficulty, the controller gains of the DGUs and SUs will be designed separately. First, a DC-MG composed of only DGUs is considered where the model is augmented with m integral actions, and the gain K is synthesized. Then, we consider a model of the MG that includes only SUs where the state is augmented with p integral actions, and the gain F is designed.

Current Sharing and Average Voltage Regulation

In this section, we consider a DC-MG consists only of DGUs, i.e., V pow = V d where the overall MG system can be written from (2.1), compactly, as:

L İ = -RI -V + u, C V = I -(R -1 L + L pow )V -I L , (2.3)
with I, V , I L , u ∈ R m . As well, L, C, R, R L ∈ R m×m are positive definite diagonal matrices, e.g,

L = diag(l 1 , • • • , l m ).
The Laplacian matrix L pow ∈ R m×m represents the topology of the power network.

Problem Formulation

In this section, we present the considered control objectives. At steady state (2.3) becomes:

0 = -RI e -V e + u e , 0 = I e -(R -1 L + L pow )V e -I L .
(2.4a)

(2.4b) By left-multiplying (2.4b) by 1 we get

1 I e = 1 R -1 L V e + 1 I L , (2.5)
that is, the sum of the generated current is equal to the total current demand 3 . The latter should be shared proportionally and not necessarily equally. If this requirement is not taken into account, the module with low power may be driven to deliver its maximum rated current and sometimes beyond. This situation can lead to overloading and temperature stresses which can spoil the module.

CURRENT SHARING AND AVERAGE VOLTAGE REGULATION

Objective 2.1. (Current sharing) At steady state, currents have to reach an equilibrium I e (depending on the total current demand) s.t.:

ω i I e i = ω j I e j ∀i, j ∈ V d ,
where the weights ω i , ∀i ∈ V d , are given parameters.

In fact, ω -1 i can be chosen as the corresponding DGU i rated current. Hence, a relatively small value of ω i corresponds to a relatively large generation capacity of DGU i and Objective 2.1 can be read as follows: At steady state all the DGUs generate the same percentage of their rated currents. Note that Objective 2.1 can be re-written as:

I e = W -1 1i * , (2.6)
where

W = diag(ω 1 , • • • , ω m ) and i * = 1 (R -1 L V e + I L ) 1 W -1 1 ∈ R.
Another requirement for a good operation of the MG is that the voltages stay near to its nominal values under any perturbation. Generally, achieving Objective 2.1 does not permit to attend an equilibrium voltage V e i = V ref i , ∀i ∈ V d , at the same time. Indeed, from (2.4b) we have 4 :

V e = (R -1 L + L pow ) -1 (I e -I L ). ( 2
I e = W -1 1 1 T W -1 1 1 (R -1 L (V e + 1v * ) + I L ), = W -1 1 (1 R -1 L V e + 1 I L ) 1 T W -1 1 + W -1 1 1 R -1 L 1v * 1 T W -1 1 , = W -1 1i * + W -1 1 1 R -1 L 1v * 1 T W -1 1 , = W -1 1(i * + 1 R -1 L 1v * 1 T W -1 1
).

(

(2.10)

(2.11) Since 1 R -1 L 1v * 1 T W -1 1
is a scalar, it is obvious that Objective 2.1 is still reached.

Consequently, we will consider an average voltage regulation, where the controller aims to shift the voltages with the same suitable value such that the weighted average value of V e i , i ∈ V d , equal to the weighted average value of the desired reference voltages V ref i , i ∈ V d and the second control objective can be stated as Objective 2.2. (Average voltage regulation) At steady state, voltages have to reach an equilibrium V e s.t.:

1 T m W -1 V e = 1 T m W -1 V ref , where W = diag(ω 1 , • • • , ω m ), ω i > 0, ∀i ∈ V d .

CURRENT SHARING AND AVERAGE VOLTAGE REGULATION

The choice of the weights for voltages as ω -1 i is a standard practice motivated by the fact that the DGU i with the highest capacity should impose the voltage of the MG.

Distributed Controller Design

In this section, a solution to Control Problem 1 is provided. Going back to equations (2.4a) and (2.4b), for a given constant load current I L , these equations define 2m linear independent equations with 3m unknown variables (I e , V e , u e ). Thus, the equilibrium exists with m degrees of freedom to fix it. Our purpose is then to determine a controller including m integral actions in order to solve Control Problem 1. The proposed control approach exploits a communication network linking the DGUs and fulfilling the following Assumption. Let us introduce an augmented state X = (I, V, φ, γ) whose dynamics are given by the following equations:

Σ d                  L İ = -RI -V + u, C V = I -(R -1 L + L pow )V -I L , φ = W T L com W I, γ = -W T L com W γ + τ (V -V ref ), (2.12a) (2.12b) (2.12c) (2.12d)
where L com is defined in Assumption 2.1, τ ∈ R >0 and the ith line of (2.12c)-(2.12d) is given by:

φi = ω i j∈N com i α ij (ω i I i -ω j I j ), γi = -ω i j∈N com i α ij (ω i γ i -ω j γ j ) + τ (V i -V ref i ),
0 = I e -(R -1 L + L pow )V e -I L , 0 = W T L com W I e , 0 = -W T L com W γ e + τ (V e -V ref ).} (2.14a) (2.14b) (2.14c) (2.14d)
In what follows the following assumption will be considered:

CURRENT SHARING AND AVERAGE VOLTAGE REGULATION

Proof. Let us introduce the following change of coordinates a :

( Ĩ, Ṽ , φ, γ) = (I 4 ⊗ T T )(I, V, φ, γ), (2.15)
where T ∈ R m×m is a unitary matrix s.t. Lcom = T T W T L com W T = diag(0, λ 2 , . . . , λ m ) where

λ i , ∀i ∈ V d , represent the eigenvalues of W T L com W s.t. λ i < λ j , ∀ i < j. The matrix T exists because W T L com W is symmetric. Moreover, λ 1 = 0, since the graph G com d
is undirected and connected, and because the vector

W -1 1 m satisfies W T L com W (W -1 1 m ) = 0.
In this new basis, Σ d (2.12) can be rewritten as follows:

Σd                                            L İ = -R Ĩ -Ṽ + ũ, C V = Ĩ -(R -1 L + Lpow ) Ṽ -ĨL , φ =         0 0 • • • 0 0 λ 2 • • • 0 . . . . . . . . . . . . 0 0 • • • λ m         Lcom Ĩ, γ = -Lcom γ + τ ( Ṽ -Ṽ ref ), (2.16a) (2.16b) (2.16c) (2.16d)
where Lpow = T T L pow T and:

(ũ, ĨL , Ṽ ref ) = (I 3 ⊗ T T )(u, I L , V ref ).
(2.17)

Note that the matrices L, C, R and R L remain unchanged by Assumption 2.2. Let us define the set Sd ( ĨL , Ṽ ref ) of all the equilibrium points of Σd as (2.18c) and using the diagonal form of Lcom , one gets:

Sd ( ĨL , Ṽ ref ) = { X e = ( Ĩe , Ṽ e , φe , γe ) ∈ R 4m , ũe ∈ R m s.t.: 0 = -R Ĩe -Ṽ e + ũe , 0 = Ĩe -(R -1 L + Lpow ) Ṽ e -ĨL , 0 = Lcom Ĩe , 0 = -Lcom γe + τ ( Ṽ e -Ṽ ref ).} (
Ĩe i = 0 for i = 2, • • • , m. (2.20)
Moreover, the first line of (2.18d) gives:

Ṽ e 1 = Ṽ ref 1 . ( 2
Ṽ e [2,m] = -H(( Lpow ) [2,m]×[1] Ṽ ref 1 + ĨL[2,m] ), (2.22)
where

H = (R -1 L + Lpow ) -1 [2,m]×[2,m] . Note that one can easily show that the matrix H is well de- fined since (R -1 L ) [2,m]×[2,m] is diagonal positive definite and ( Lpow ) [2,m]×[2,m
] is positive semidefinite. Furthermore, reporting (2.20)-(2.22) in (2.18b) leads to:

Ĩe 1 = r -1 L Ṽ ref 1 + ( Lpow ) [1]×[1,m]
Ṽ e + ĨL1 .

(2.23)

Since Ĩe and Ṽ e are now fixed, it follows from (2.18a) that ũe is also defined by

ũe 1 = R Ĩe 1 + Ṽ ref 1 , ũe i = Ṽ e i for i = 2, • • • , m.
(2.24)

(2.25)

Since φ1 (t) = φ1 (0), ∀t ≥ 0, it is obvious from (2.18c)-(2.18d) that:

φe 1 = φ1 (0), γe i = λ -1 i τ ( Ṽ e i -Ṽ ref i ), for i = 2, • • • , m.
Finally, any value for γe 1 and φe i for i = 2, • • • , m is allowed. W I e = α1 m , or equivalently ω i I e i = ω j I e j , ∀i, j ∈ V d . Consequently, Objective 2.1 is achieved. Since

rank(W T L com W ) = m -1,
(2.12c) is equivalent to introduce m -1 integrators. Accordingly, only (m -1) degrees of freedom are required to attend the first objective and the mth one will be used to achieve Objective 2.2 with an additional distributed integral action. Indeed, left-multiplying (2.14d) with 1 T m W -1 leads to:

0 = 1 T m W -1 (V e -V ref ), (2.27)
since 1 T m L com = 0. Thus, it follows that, inside the set S d , Objective 2.2 is achieved.

CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING

To summarize, system (2.3) was augmented with the additional state (φ, γ) so as to introduce m distributed integral actions to the controller. Then, we showed that the augmented system defined by (2.12) has at least one equilibrium point since S d = ∅. Finally, it was proved that if the couple (X , u) of the augmented system converges to one of its equilibrium points, Objectives 2.1-2.2 are achieved thanks to (φ, γ). Now, our goal is to design local controllers u i that depend only on local variables

x i = (I i , V i , φ i , γ i ),
∀i ∈ V d . Hence, the gain matrix K (see (2.26)) should be restricted to the form:

K = (K I , K V , K φ , K γ ), (2.28)
where

K I , K V , K φ , K γ are diagonal matrices of dimension m × m.
The main difficulty to find a gain matrix of this form for the augmented system (2.12) is the existence of physical (L pow ) and communication (L com ) coupling terms.

Proposition 2.3. (Unique gain)

If there exists a controller:

ũ = -K X , (2.29)
where X = ( Ĩ, Ṽ , φ, γ), K = (K ⊗ I m ) and

K = k I k V k φ k γ ∈ R 1×4 s.t
u = -(K ⊗ I m )X (2.30) converges asymptotically to the set S d (V ref , I L ). Moreover, K = (K ⊗ I m ) satisfies (2.28). Proof. As u = T ũ, X = (I 4 ⊗ T T )X (see (2.15)), K = (K ⊗ I m ) = (k I I m , k V I m , k φ I m , k γ I m ),
and since T is a unitary matrix, it follows:

u = -T k I T T k V T T k φ T T k γ T T X , = -(K ⊗ I m )X .
(2.31)

(2.32)

CURRENT SHARING AND AVERAGE VOLTAGE REGULATION

Since the change of coordinates (2.15) and (2.17) is bijective, the asymptotic convergence of ( X , ũ) to Sd implies that (X , u) converges asymptotically to the set S d (V ref , I L ).

Finally, K = (K ⊗ I m ) is of the required form (2.28) since it can be rewritten as

K = (k I I m , k V I m , k φ I m , k γ I m ).
Proposition 2.4. (Unicity of the equilibrium point) For a given reference voltage V ref and an unknown load current I L , the equilibrium point of system (2.12) is uniquely determined by the feedback :

u = -(K ⊗ I m )X .
Proof. As it has been shown in the proof of Proposition 2.1 after a change of coordinates, only some components of φ and γ have remained undetermined. Using the expressions ũe = -(K ⊗ I m ) X e and X e = ( Ĩe , Ṽ e , φ, γ), a simple calculation leads to:

γe 1 = -1 k γ (ũ e 1 + k I Ĩe 1 + k V Ṽ e 1 + k φ φe 1 ), φe i = -1 k φ (ũ e i + k I Ĩe i + k V Ṽ e i + k γ γe i ), i = 2, • • • , m.
Recalling that φe 1 = φ1 (0) and from the proof of Proposition 2.1, it can be concluded that the equilibrium is uniquely determined in the set S d . Remark 2.5. By Propositions 2.1 and 2.4, for a given V ref and I L the set Sd is equivalent to a straight line parametrized by the initial values of the variable φ1 .

By considering a second change of coordinates of the form

x = M d X , (2.33)
where where

M d =         I m ⊗ [1 0 0 0] I m ⊗ [0 1 0 0] I m ⊗ [0 0 1 0] I m ⊗ [0 0 0 1]         . ( 2
• x = (x 1 , • • • , xm ), xi = ( Ĩi , Ṽi , φi , γi ), • d i = -(0, 1 c ĨLi , 0, τ Ṽ ref i ), • B p = 0 c -1 0 0 T , C p = 0 1 0 0 ,
• l pow i,j for 1 ≤ i, j ≤ m denotes the elements of Lpow ,

•

A cl i = A o i -B u K
where

A o i =      A 0 2×2 λ i 0 0 0 0 τ 0 -λ i      , B u =   1 l 0 3×1   and A =    - r l - 1 l 1 c - 1 cr L    .
(2. [START_REF] Dahmane | Coordinated charging of large electric vehicle fleet in a charging station with limited transformer power[END_REF] In this basis, it can be noticed that the local variables xi = ( Ĩi , Ṽi , φi , γi ), ∀i ∈ V d , are only coupled by the term B p j=i, j∈N i l pow i,j C p xi (related to the matrix Lpow ). The next theorem shows how it is possible to determine the gain matrix K in (2.30) using some passivity arguments.

Theorem 2.1. If there exists a static state feedback

K = k I k V k φ k γ s.t. the triple (A cl i , B p , C p ) for i = 2, • • • , m and ( Ȃcl 1 , B1 , C1
) are strictly passive where:

Ȃcl 1 =   A 0 2×1 0 τ 0   -   1 l 0 2×1   k I k V k γ , B1 = 0 c -1 0 T , C1 = 0 1 0 , ( 2
ẋ = (A -B Lpow C)x + d, (2.38)
where

• A = diag(A cl 1 , • • • , A clm ), • B = I m ⊗ B p , • C = (I m ⊗ C p ) and • d = (d 1 , • • • , d m ).
From the expression of Lcom in (2.16c), it is clear that φ1 = 0. Thus, this variable is not controllable and acts as a constant disturbance like the terms d. Thus, for the purpose of stability analysis, φ1 can be removed from systems Σd 1 . Accordingly, it is sufficient to design the gain K such that the matrices A cl i , i = 2, • • • , m and Ȃcl 1 are Hurwitz (see (2.36) and (2.37)).

In other words, if x denotes the vector x whose mentioned constant component ( φ1 ) have been removed, then it is sufficient to prove the asymptotic stability of the following reduced system:

ẋ = ( Ȃ -B Lpow C)x + d, (2.39)
where

• Ȃ = diag( Ȃcl 1 , A cl 2 , • • • , A clm ), • B = diag( B1 , (I m-1 ⊗ B p )), • C = diag( C1 , (I m-1 ⊗ C p )), • d = ( d1 , d 2 , • • • , d m ), d1 = -(0, 1 c ĨL1 , τ Ṽ ref 1 )
.

By assumption, the triple (A cl i , B p , C p ) for i = 2, • • • , m and ( Ȃcl 1 , B1 , C1 ) are strictly passive, i.e, there exist matrices

P i = P T i > 0 and i > 0 such that for i = 2, • • • , m A T cl i P i + P i A cl i ≤ -i P i , P i B p = C T p , (2.40a) (2.40b)
and for i = 1

ȂT cl 1 P 1 + P 1 Ȃcl 1 ≤ -1 P 1 , P 1 B1 = CT 1 . (2.41a) (2.41b)
The stability of system (2.39) can then be guaranteed by considering the Lyapunov function V d (x) = xT P x, with

P = diag(P 1 , • • • , P m ). (2.42)
Indeed, by derivation of V d along the trajectories, we get:

V d (x) = xT (( Ȃ -B Lpow C) T P + P ( Ȃ -B Lpow C))x, (2.43)
it follows:

V d (x) = 2x T P ( Ȃ -B Lpow C)x, (2.44)
using (2.40)-(2.41), (2.44) can be written as:

V d (x) = m i=1 2x T i P i Ȃcl i xi -2x T CT Lpow C x, ≤ - m i=1 2 i xT i P i xi -2x T CT Lpow C x, (2.45a)
(2.45b)

CURRENT SHARING AND AVERAGE VOLTAGE REGULATION

since CT Lpow C ≥ 0, we can conclude that:

V d (x) < 0, x = 0. (2.46)
Consequently, system (2.39) is asymptotically stable and the state x converges asymptotically to the equilibrium point xe = -( Ȃ -B Lpow C) -1 d. Hence, the state x converges asymptotically to the equilibrium point xe given by:

xe i = xe i , for i = 2, • • • , m, xe 1 = xe 1 1 xe 1 2 φ1 (0) xe 1 3 .
Since X = Mx, where M is given by (2.34), the state X converges asymptotically to the equilibrium point X e = Mx e . Moreover, it is obvious that the couple ( X e , ũe = -(I m ⊗ K) X e ) verifies (2.18).

Thus, the couple ( X , ũ = -(I m ⊗ K) X ) converges asymptotically to the set Sd ( Ṽ ref , ĨL ).

By Propositions 2.3-2.4, the couple (X , u = -(I m ⊗K)X ) converges asymptotically to the set S d (V ref , I L ).
Furthermore, by Proposition 2.2, Objectives 2.1-2.2 are reached. Finally, since the system is linear the convergence is exponential and Theorem 2.1 is proved.

LMI Computational Algorithm

Now, we show how to compute the controller gain matrices K so as to comply with the conditions of Theorems 2.1. By considering Assumptions 2.2, one can remark that the only parameter values that change between matrices A o i , ∀i ∈ V d , (see (2.36)) are the values of the eigenvalues λ i , ∀i ∈ V d . Thus, the design of the controllers can be addressed by considering, a problem of robust stability for polytopic systems. A polytope with two vertices is considered, where the first and second vertices correspond to the highest and smallest eigenvalues of W T L com W , respectively. Therefore, at each vertex we have:

A o =      A 0 2×2 λ max 0 0 0 0 τ 0 -λ max      , A o =      A 0 2×2 λ min 0 0 0 0 τ 0 -λ min     
,

where λ min = min(λ i ), λ max = max(λ i ); i = 2, • • • , m.
Thus, all matrices A o i lie in the straight segment between this two vertices, i.e., the change zone is a convex combination of vertices A o and A o :

∀i = 2, • • • , m, ∃α ∈ [0, 1] s.t. A o i = αA o + (1 -α)A o .
Hence, the gain K can be computed by solving the following LMI: there exist a constant > 0 and matrices M 1 = M T 1 > 0 and M 2 of appropriate dimensions s.t.

A o M 1 + M 1 A T o -B u M 2 -M T 2 B T u ≤ -M 1 , A o M 1 + M 1 A T o -B u M 2 -M T 2 B T u ≤ -M 1 , B p = M 1 C T p , (2.47a) (2.47b) (2.47c)
where B u , B p and (2.47) verify the following LMI: there exist a constant 1 > 0 and a matrix P = P T > 0 of appropriate dimension s.t.:

C p are given in (2.35)-(2.36). Then, K = [k I k V k φ k γ ] = M 2 M -1 1 . Using, [k I k V k γ ] computed from
ȂT cl 1 P + P Ȃcl 1 ≤ -1 P , P B1 = CT 1 , (2.48a) (2.48b)
where Ȃcl 1 , B1 and C1 are given in (2.37).

To resume this part of the chapter, a DC-MG composed of only DGUs has been considered with the problems of CS and AVR. The objectives have been achieved by means of two distributed integral actions. The proof of the convergence of the augmented system in closed-loop with the proposed static state feedback controller was based on two ideas, a change of coordinates to overcome the communication coupling between the agents, and a passivity condition during the design of the feedback gain to ensure the convergence despite the physical coupling. Finally, an LMI algorithm has been given to simplify the design of the controller gain.

The next step will be the design of controllers for SB objective.

STATE-OF-CHARGE BALANCING

State-of-charge Balancing

In this part, we study the problem of SB, for a DC-MG that includes only SUs. We start first by presenting the considered model and formulating the control problem. Next, we augment the model with distributed leaderfollower consensus-like integral actions. By considering similar assumptions as in the previous section, we show that all the states-of-charge converge to the state-of-charge reference. Finally, we give an LMI formulation for the design of the controller gain.

Grid Network

Consider a DC-MG composed of only SUs, i.e., V pow = V s . Using the generic equation (2.1) and the state-of-charge dynamics (2.2), the overall MG system for all SUs can be written as follows:

L İ = -RI -V + u, C V = I -(R -1 L + L pow )V -I L , Ṡ oc = -K soc I, (2.49)
with I, V , I L , u, Soc ∈ R p . As well, L, C, R, R L , K soc ∈ R p×p are positive definite diagonal matrices, e.g,

L = diag(l 1 , • • • , l p ).
The Laplacian matrix L pow ∈ R p×p represents the topology of the power network.

Problem Formulation

Unbalanced state-of-charge between the SUs can affect the life of the batteries by causing over-charge or overdischarge phenomenon. Thus, balancing the states-of-charge between the SUs is very important to prolong its life and avoid its deterioration.

Objective 2.3. (State-of-charge balancing)

At steady state, the state-of-charge have to reach an equilibrium Soc e s.t.:

Soc e i = Soc ref ∀i ∈ V s ,
where the scalar Soc ref represents the state-of-charge reference given by a high-level controller.

Distributed Controller Design

In this section, a solution to Control Problem 2 is provided. Before proceeding to the design of the distributed controllers, similarly to Section 2.2, the following assumptions are considered Let us introduce an augmented state X = (I, V, Soc, θ) whose dynamics are given as follows:

c i = c, r i = r, r Li = r L , k soc i = k soc , ∀i ∈ V s , with l, c, r, r L , k soc ∈ R >0 represent the nominal values.
Σ s                  L İ = -RI -V + u, C V = I -(R -1 L + L pow )V -I L , Ṡ oc = -K soc I, θ = (L com + J )Soc -Soc r , (2.50a) (2.50b) (2.50c) (2.50d)
where the ith line of (2.50d) is given by: θi =

j∈N com i α ij (Soc i -Soc j ) + β i (Soc i -Soc ref ), (2.51) 2.3.

STATE-OF-CHARGE BALANCING

where N com i is the set of SUs connected to the SU i via the communication network with the edge weights

α ij = α ji ∈ R >0 , β i = 0, ∀ i = i l ,
and β i l > 0 (i l is the index of leader). Equation (2.50d) represents a leader-follower dynamics where the state-of-charge reference Soc ref is only available for SU i l . Moreover, L com is defined in Assumption 2.5 and J ∈ R p×p is a diagonal matrix of the form

J i,i =      β i l > 0, i = i l , 0, otherwise, (2.52)
and Soc r ∈ R p is s.t.

Soc ri      β i l Soc ref , i = i l , 0, otherwise.
(2.53)

Lemma 2.1. The matrix (L com + J ) is invertible and its inverse is symmetric and verifies:

[(L com + J ) -1 ] i,i l = 1 β i l , 1 ≤ i ≤ p. (2.54)
Proof. First, we prove that (L com + J ) is invertible by proving that it is positive definite. Consider a variable Z = (z 1 , ..., z p ) ∈ R p , we have

Z T (L com + J )Z = (i,j)∈E com α ij (z i -z j ) 2 + β i l z 2 i l ,
where α ij ∈ R >0 are defined in (2.51). Since the graph G com s is connected and undirected, (L com + J )

is symmetric and (i,j)∈E com α ij (z i -z j ) 2 = 0 implies z i = z j , ∀i, j ∈ V s , (consensus).
Hence, one can easily prove that Z T (L com + J )Z is positive definite function i.e.:

Z T (L com + J )Z > 0 ∀Z = 0 p . (2.55)
Thus, (L com + J ) is invertible. Now, to prove (2.54), consider the following linear system of equation:

(L com + J )Z = [I p ] * ,i l , (2.56)
CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING with Z ∈ R p and where [I p ] * ,i l is the i l th column of the identity matrix I p .

Since (L com + J ) is invertible, a vector Z solution of (2.56) exists and its unique. Now, since L com is a Laplacian matrix we have:

(L com + J ) 1 β i l 1 p = J 1 β i l 1 p , (2.57)
substituting (2.52) in (2.57) yields:

(L com + J ) 1 β i l 1 p = [I p ] * ,i l . (2.58)
Thus, Z = 1 β i l 1 p is the unique solution of (2.56). Moreover, we have:

(L com + J )(L com + J ) -1 = I p , (2.59)
it follows:

(L com + J )[(L com + J ) -1 ] * ,i l = [I p ] * ,i l , (2.60)
where [(L com + J ) -1 ] * ,i l is the i l th column of the matrix (L com + J ) -1 .

Since (L com + J ) is symmetric, (2.58) and (2.60) yields

[(L com + J ) -1 ] * ,i l = 1 β i l 1 p . (2.61)
Finally, (2.54) is proved.

Yet again, it is noteworthy that in (2.51), α ij = 0 if SU j is not neighbor of SU i. Hence, only local information are exchanged between the SUs augmented with the additional states (2.51). Thus, if a state feedback controller depending only on the local state at each SU, i.e, (I i , V i , Soc i , θ i ), ∀i ∈ V s , is considered, u will be distributed. Therefore, our goal is to design a distributed controller of the form:

u = -F X , (2.62)
where the gain matrix F should be restricted to the form:

F = (F I , F V , F soc , F θ ), (2.63)
where F I , F V , F soc , F θ are diagonal matrices of dimension p × p. Moreover, Objective 3 is achieved at that equilibrium.

Proof. At the equilibrium, (2.50) yields to the following conditions:

0 = -RI e -V e + u e , 0 = I e -(R -1 L + L pow )V e -I L , 0 = -K soc I e , 0 = (L com + J )Soc e -Soc r . (2.64a) (2.64b) (2.64c) (2.64d)
From (2.64c) we have:

I e = 0. (2.65)
Replacing (2.65) in (2.64b) yields:

V e = (R -1 L + L pow ) -1 I L , (2.66)
note that (R -1 L + L pow ) -1 is well defined because R -1 L is positive definite matrix and L pow positive semidefinite matrix. By Lemma 2.1, (2.64d) gives:

Soc e = (L com + J ) -1 Soc r .
(2.67) Substituting (2.65) in (2.64a) gives:

u e = V e . (2.68)
Since in closed-loop u e = -F I I e -F V V e -F soc Soc e -F θ θ e (see (2.62)-(2.63)), (2.68) yields:

θ e = -F -1 θ ((F V + I p )V e + F soc Soc e ). (2.69)
Thus, the equilibrium exists and is unique. Moreover, using Lemma 2.1, (2.67) gives:

Soc e = 1 p Soc ref , (2.70)
it follows that, at the equilibrium, Objective 3 is achieved.

Similarly to Theorem 2.1, the next theorem shows how it is possible to determine the gain matrix F in (2.62).

Theorem 2.2. If there exists a state feedback gain

F = f I f V f soc f θ s.t. the triple (A cl i , B p , C p ) for i = 1, • • • , p are strictly passive, where B p = 0 c -1 0 0 T , C p = 0 1 0 0 , A cl i = A o i -B u F, B u = l -1 0 3×1 T A o i =      A 0 2×2 -k soc 0 0 0 0 0 λ i 0      , A =    - r l - 1 l 1 c - 1 cr L    and
where λ i , i ∈ V s , are the eigenvalues of the matrix (L com + J ) then the equilibrium point of the augmented system (2.50) in closed loop with the state feedback control law

u = -(F ⊗ I p )(I, V, Soc, θ) (2.71)
is globally exponentially stable and Objective 2.3 is satisfied.

Proof. Let Σs be the system (2.50) after applying the following change of coordinates:

( Ĩ, Ṽ , Soc, θ) = (I 4 ⊗ T T )(I, V, Soc, θ), (2.72)
(ũ, ĨL , Soc r ) = (I 3 ⊗ T T )(u, I L , Soc r ), (2.73)
where T ∈ R p×p is a unitary matrix s.t.

( Lcom + J ) = T T (L com + J )T = diag(λ 1 , . . . , λ p ),

where λ i , ∀i ∈ V s , represent the eigenvalues of (L com + J ) s.t. λ i < λ j , ∀ i < j.

Remark 2.6. Note that, differently from (2.16), in Σs λ 1 = 0; hence, the design of the controller will be easier since system Σs does not include any uncontrollable constant state as φ1 in system (2.16).

We denote by X e the equilibrium point of system Σs . Similar to the proof of Theorem 2.1, it is sufficient to prove the asymptotic stability of the equilibrium point X e .

By applying a second change of coordinates of the form

x = M s X , (2.74)
where

M s =         I p ⊗ [1 0 0 0] I p ⊗ [0 1 0 0] I p ⊗ [0 0 1 0] I p ⊗ [0 0 0 1]         , (2.75)
another state space representation of system Σs with the controller (2.71) can be written as:

ẋ = (A -B Lpow C)x + v, (2.76)
where

• x = (x 1 , • • • , xp ), xi = ( Ĩi , Ṽi , Soc i , θi ), • v = (v 1 , • • • , v p ), v i = (0, - 1 c ĨLi , 0, Soc ri ), • A = diag(A cl 1 , • • • , A clp ), • B = (I p ⊗ B p ), • C = (I p ⊗ C p ).
By assumption, the triple (A cl i , B p , C p ) for i = 1, • • • , p are strictly passive, i.e, there exist matrices

Q i = Q T i > 0 and µ i > 0 such that for i = 1, • • • , p A T cl i Q i + Q i A cl i ≤ -µ i Q i , Q i B p = C T p .
(2.77a)

(2.77b)

Similarly to Theorem 2.1, the proof of Theorem 2.2 can be done by considering the Lyapunov function

V s (x) = x T Qx, with Q = diag(Q 1 , • • • , Q p ).
(2.78)

LMI Computational Algorithm

The computation of the gain F can be done similarly to K by solving the following LMI: there exist a constant µ > 0 and matrices M 3 = M T 3 > 0 and M 4 of appropriate dimensions s.t.

A o M 3 + M 3 A T o -B u M 4 -M T 4 B T u ≤ -µM 3 , A o M 3 + M 3 A T o -B u M 4 -M T 4 B T u ≤ -µM 3 , B p = M 3 C T p , (2.79a) (2.79b) (2.79c)
where B u , B p and C p are given in Theorem 2 and

A o =      A 0 2×2 -k soc 0 0 0 0 0 λ max 0      , A o =      A 0 2×2 -k soc 0 0 0 0 0 λ min 0     
, where λ min = min(λ i ),

λ max = max(λ i ), i = 1, • • • , p. Then, F = M 4 M -1 3 .

DC-MG with DGUs and SUs

In this part, we show that using the designed controllers in the previous parts, Objectives

DC-MG with Resistive Power Lines

In this section, we consider a DC-MG composed of m DGUs, p SUs and resistive power lines, the overall compact DC-MG model is as follows:

L   İd İs   = -R   I d I s   -   V d V s   +   u d u s   , C   V d V s   =   I d I s   -(R -1 L + L pow )   V d V s   -   I d L I s L   , Ṡ oc = -K soc I s , (2.80)
where I d , V d , u d , I d L ∈ R m are related to the DGUs and I s , V s , Soc, u s , I s L ∈ R p are related to the SUs with L, C, R, R L ∈ R N ×N , K soc ∈ R p×p are diagonal matrices.

Note that the physical coupling is not only between DGU-DGU and SU-SU but also between DGU-SU, since the matrix L pow is of the following form:

L pow =   L pow dd L pow ds L pow sd L pow ss   , ( 2
φ = W T L com d W I d , γ = -W T L com d W γ + (V d -V ref ), θ = (L com s + J )Soc -Soc r .
(2.82) Let X be the state of system Σ defined as

X = (X d , X s ), (2.83)
where X d = (I d , V d , φ, γ) and X s = (I s , V s , Soc, θ). The following theorem gives a solution to Control Problem 3. Theorem 2.3. Consider system Σ, let Assumptions 2.1-2.5, if there exists a state feedback controller

u = -[K ⊗ I m F ⊗ I p ]X , (2.84)
Remark 2.8. Note that for simplicity and without loss of generality, we have changed the power line index to k rather than ij. Furthermore, by considering a non-zero inductance for power lines, they are no longer static, i.e., we have more dynamics in the system.

Similarly to (2.80), the overall model of the DC-MG with resistive-inductive power lines can be written from (2.86), (2.87) and (2.2) as

L   İd İs   = -R   I d I s   -   V d V s   +   u d u s   , C   V d V s   =   I d I s   -R -1 L   V d V s   + HI l -   I d L I s L   , Ṡ oc = -K soc I s , L l İl = -R l I l -H T V, (2.88)
(2.89)

where

I l = [I l 1 , • • • , I lq ] ∈ R q and R l = diag(r l 1 , • • • , r lq ), L l = diag(l l 1 , • • • , l lq ) ∈ R q×q .
The topology of the network is now described by the incidence matrix H ∈ R N ×q . The ends of the power line k are arbitrarily labeled with a + and a -, and the entries of H are given by

H ik              +1 if i
ẋ = Ax + BHI l + D, L l İl = -R l I l -H C x.
(2.91)

(2.92)

where

• x = xd , xs , • A = diag(A d , A s ), • C = diag(C d , C s ), • B = diag(B d , B s ), • D = d v , • xd , A d , C d ,
V (x, I l ) = 1 2 x Λx + 1 2 I l L l I l , (2.93)
where

Λ =   P 0 0 Q   ,
and where P and Q are positive definite matrices defined by (2.42) and (2.78). By differentiating V (., .)

along the trajectories we get: 2.5. CONCLUSION Remark 2.11. Assumptions 2.2 and 2.4 can be viewed as conservative. In practice, the values of the parameters of the agents are prescribed within a fixed range that notably depends in particular on the power of the DGU and SU. Moreover, we will see in the next chapter that the ability of the proposed controllers to reject the discrepancies between nominal and real parameters is noticeable. Although it is not the scope of the thesis, additional stability results for the MG with the controller can be formulated using a polytopic approach.

V (x, I l ) = xd P A d xd + xs QA s xs + x ΛBHI l -I l R l I l -I l H C x, (2
This approach gives the possibility to check the stability of the proposed controllers when parameters are within a prescribed range of values [START_REF] Daafouz | Stability analysis and control synthesis for switched systems: a switched lyapunov function approach[END_REF].

Conclusion

Under the assumption that the agents (DGUs or SUs) have the same physical parameter values (Assumptions 2.2 and 2.4). A distributed-based static-state-feedback control law, including integral actions to achieve, provably and simultaneously, proportional current sharing, average voltage regulation and state-of-charge balancing in DC power networks has been proposed.

First, the problems of CS and AVR were considered for a DC-MG composed of only DGUs with resistive power lines and loads. The system is augmented with two distributed consensus-like integral actions. The existence of equilibrium points for the augmented system was proved (Proposition 2.1) and a distributedbased static state feedback control architecture was proposed (Proposition 2.3). We provided proof of global convergence of the augmented system state to a set of equilibrium for which the two objectives are achieved (Theorem 2.1).

Second, the problem of SB was addressed for a DC-MG composed of only SUs with resistive power lines and loads. The system is augmented with a distributed leader-follower-like integral action where the information about the state-of-charge reference Soc ref is provided to only one SU, that we called the leader. Similarly to the previous part, the proofs of the existence of equilibrium points for the augmented system (Proposition 2.5) and the global exponential stability (Theorem 2.2) were provided.

Finally, we combined the previous results to achieve simultaneously CS, AVR, and SB for a DC-MG that includes both DGUs and SUs. First, by considering resistive power lines (Theorem 2.3) and then resistiveinductive power lines (Theorem 2.4). LMI computational algorithms were given to simplify the design of the controllers.

In the next chapter, several tests using both Matlab/Simulink simulation and real-time hardware-in-the-loop experiment will be performed to evaluate the performance of the proposed controllers. In this chapter, the controllers proposed in Chapter 2 are tested. In Section 3.1, we present the considered DC-MG case study with its topology, parameter values, the controller gains values, etc. In section 3.2, a Matlab/Simulink simulation scenario is considered to show the performance of the controller in case of resistive and resistive-inductive power lines. In this scenario, the agents (DGUs or SUs) have the same parameter values. In Section 3.3, more scenarios are considered for HIL experiments e.g., current demand variation, voltage reference variation, state-of-charge reference variation, etc. In each scenario of this section, the efficiency of the controller is assessed in two cases i.e. when the agents have the same or different physical parameter values.

Case Study

We consider a MG composed of 3 DGUs and 3 SUs as shown in Fig. 3 Remark 3.1. Note that, the use of • ∈ {d, s} in Tables 3.1-3.3 is to distinguish the parameters and variables power lines, the topology of the DC-MG in Fig. 3.1 is represented by a Laplacian matrix given by:

L com d = 28.10 4      1 -1 0 -1 2 -1 0 -1 1      , L com s =      10
L pow =               26
              . 45
In the other case the topology of the MG is represented by the following incidence matrix:

H =               -1 0 0 0 0 1 1 -1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 1 -1 0 0 0 1 -1 0 0               .

Matlab Simulation

In this section, the DC-MG is implemented in Matlab/Simulink environment. To investigate the performance of the proposed control approach, two cases are considered with the same scenario. In the first case, we consider that the power lines are only resistive. However, in the second one, resistive-inductive power lines are considered. Moreover, in both of the cases, the proposed controller is tested with the nominal parameters of DC-MG (see Table 3.1).

The system is initially at the steady state with I L (0), V ref The results illustrate the ability of the controller to practically maintain the same performance in the two considered cases and its robustness under the change of the unknown load current.

Scenario 1: load current variation near DGUs

In this scenario, we investigate the effectiveness of the proposed controllers under a load current variation near the DGUs. The system is initially at the steady state with I L (0), V ref (0) and Soc ref (0). Then:

Scenario 2: load current variation near SUs

In the second scenario, the performance of the proposed controllers is tested under a load current variation near the SUs. The system is initially at the steady-state with I L (0), V ref (0) and Soc ref (0). Then:

• At the time instant t = t 1 the load current I s L1 near SU 1 is stepped up with ∆I s L 1 = 15 A.

• At the time instant t = t 2 the load current I s L2 near SU 2 is stepped up with ∆I s L 2 = 15 A.

• At the time instant t = t 3 the load current I s L3 near SU 3 is stepped up with ∆I s L 3 = 15 A.

• At the time instant t = t 4 the load current I s L1 near SU 1 is stepped down with -∆I s L 1 .

• At the time instant t = t 5 the load current I s L2 near SU 2 is stepped down with -∆I s L 2 .

• At the time instant t = t 6 the load current I s L3 near SU 3 is stepped down with -∆I s L 3 .

As we can see in Fig. 3.9 and 3.10, Objectives 2.1 and 2.2 are still respected. The state-of-charge and the currents of the SUs are illustrated in Fig. 3.11. One can note how the state-of-charge react to load current differently than Scenario 1 (see Fig. 3.8). Indeed, in this scenario, the state-of-charge are more reactive to the load current i.e. when the load current increases the state-of-charge decrease more than Scenario 1 and take more time to go back to their reference. This aims to compensate the load current variation because when the state-of-charge decrease the generated currents of the SUs increase.

Scenario 3: high-load operating mode

In the third scenario, the performance of the proposed controllers is assessed in the case of high-load operation mode. The system is initially at the steady state with I L (0), V ref (0) and Soc ref (0). Then:

Scenario 4: voltage reference variation

In the fourth scenario, we investigate the performance of the proposed controllers under a voltage reference variation of the PCC near the DGUs. The system is initially at the steady state with I L (0), V ref (0) and Soc ref (0). Then:

Conclusion

In this chapter, the distributed control approach proposed in Chapter 2 was assessed. The considered DC-MG was composed of three DGUs, three SUs, resistive inductive power line and loads. Several scenarios were considered and implemented in Matlab/Simulink (Section 3.2) and real-time HIL platform (Section 3.3). The results of each of these scenarios show that the proposed control methodology provides precise current sharing, average voltage regulation and state-of-charge balancing even in case of resistive or resistive-inductive power lines, agents with the same or different parameter values, current demand and references variations, etc. The results of Section 3.3.6 also show that if no state-of-charge reference is defined by the user (or if the SU leader losses the Soc reference due to a communication problem), the state-of-charge will converge to the average of their initial values providing more flexibility for state-of-charge balancing.

Chapter 4

Final Conclusions

The purpose of this work was the design of a new distributed control approach for DC-MGs to achieve three main objectives, Current Sharing (CS), Average Voltage Regulation (AVR) and State-of-charge Balancing (SB). The studied DC-MG was composed of Distributed Generation Units (DGUs), Storage Units (SUs) and loads interconnected through power lines. Moreover, consensus in multi-agent systems, passivity, Lyapunov stability, Linear Matrix Inequalities (LMI) and state-feedback control were the main tools used for the design of the controllers.

We started with a brief introduction to the theoretic concept, a literature review of the existing control strategies for DC-MG and preliminaries related to this work (Chapter 1). Then, we turn our attention to the design of a new distributed-based static-state-feedback control law including distributed consensus-like integral actions to reach the considered objectives (Chapter 2). The controllers exploit two sparse communication networks, one for the DGUs and another for the SUs. Under the assumption that the agents (DGUs or SUs)

have the same physical parameter values, we have proved that the state of the closed-loop system converges exponentially and globally to the set of all equilibrium points. The proof was based on the passivity of each agent in closed loop with the local controller, and the inherent passivity of power lines. In the proposed control paradigm, the DGUs exchange relative information with their neighbors about weighted (per-unit) generated currents and control variables, and accordingly adjust the local controllers to reach CS and AVR.

Analogously, the SUs share their state-of-charge with their neighbors and update their local controllers to reach SB.

FINAL CONCLUSIONS

The main contribution of the thesis was the proof of global exponential convergence of the studied DC-MG in closed-loop with the proposed controller despite its meshed topology, the communication coupling between its components, the unknown variation of the load, and the fact that the three objectives should be achieved simultaneously. In addition, the convergence is independent of the initial conditions of the power network and the controller state. The proposed control strategy includes a leader-follower approach which ensures the convergence of all the state-of-charge to a common reference state Soc proposed approach can also be an interesting extension of this work. Regarding the communication network, other issues can also be considered such as transmission delay, switching topology and optimization of the communication cost [START_REF] Ben Rejeb | Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems[END_REF]. Finally, more tests can be performed by applying the proposed control approach on a real DC-MG prototype.

everyMotivation

  1 , ..., A n ) -block diagonal matrix having the matrices A 1 to A n on the diagonal and 0 Au cours des dernières années, les systèmes électriques traditionnels connaissent une transition rapide vers un nouveau concept appelé SmartGrids. En général, un système électrique traditionnel comprend la production, la transmission, la distribution et la consommation où le flux d'énergie est unidirectionnel, des producteurs aux consommateurs. Cependant, ce concept actuel est confronté aux problèmes de pollution de l'environnement en raison de l'utilisation massive de ressources en combustibles fossiles, et l'augmentation constante de la demande d'énergie qui nécessite d'énormes investissements. En outre, l'incertitude de la demande d'énergie est, traditionnellement, gérée en ajustant l'offre. Néanmoins, en raison de l'émergence massive de ressources énergétiques renouvelables de nature volatile et imprévisible, comme les panneaux solaires et les éoliennes, l'incertitude du côté de la production doit également être traitée. Ces défis, pour lesquels les systèmes électriques existants ne sont plus suffisamment adaptés, conduisent au développement du smartgrid, un paradigme plus flexible, plus écologique et bidirectionnel dans lequel un consommateur peut également être un producteur et vice versa. En outre, ce nouveau concept se caractérise par une vaste intégration des technologies modernes de l'information et de la communication permettant une meilleure coordination entre la production et l'utilisateur final. Les micro-réseaux (microgrid) représentent l'un des composants clés efficaces et durables des réseaux électriques intelligents. En règle générale, un micro-réseau est un groupe de plusieurs alimentations électriques, systèmes de stockage et charges interconnectés par des lignes électriques. Un microgrid est principalement composées d'unités de production distribuées (DGU), d'unités de stockage (SU), de charges électrique, de 1 LIST OF SYMBOLS convertisseurs électroniques de puissance et d'unités de contrôle. Les microgrids peuvent être divisés en trois types, à savoir les microgrids à courant continu (DC), à courant alternatif (AC) et les microgrids hybrides DC/AC. Les microgrids DC ont suscité un intérêt croissant au sein des communautés de contrôle automatique et d'ingénierie électrique. Cet intérêt croissant est dû à leur efficacité, leur simplicité et leur large gamme d'applicabilité. En fait, la plupart des systèmes de stockage, des ressources énergétiques renouvelables et des charges modernes, par exemple les ordinateurs portables, les tablettes, les téléphones, etc. sont par nature à courant continu.

Figure 2 :

 2 Figure 2: Plateforme expérimentale utilisé pour les tests HIL.

Figure 4 :

 4 Figure 4: Structure of the thesis.

Contents 1 . 1 1 . 3 1 . 5 1 . 6

 11131516 Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Types of Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Operation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Examples of Microgrids Project . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 DC-MG Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review of DC Microgrids Control . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consensus in Linear Multi-Agent Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Weighted Output Consensus Problem . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Leader-follower Consensus Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Output Averaging Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability and Passivity of Interconnected Linear Systems . . . . . . . . . . . . . . . . 1.6.1 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Stability of Linear Parameter-Varying System . . . . . . . . . . . . . . . . . . . . 1.6.3 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1

 1 Figure 1.1: Microgrids.

20 1Figure 1 . 2 :

 2012 Figure 1.2: Structure of an AC-MG with single power line.

Figure 1 . 4 :

 14 Figure 1.4: Structure of an AC/DC hybrid MG.

Figure 1 . 5 :

 15 Figure 1.5: A smart-district in the Hoogkerk village, the Netherlands [44].

Figure 1 . 6 :

 16 Figure 1.6: ADREAM building developed in the LAAS-CNRS laboratory [31].

  secondary and tertiary control (Fig.1.7). At each of these levels, the controller can be implemented in several ways, centralized, decentralized and distributed architecture (Fig.1.8).CHAPTER 1. BACKGROUND AND PRELIMINARIESIn centralized techniques (Fig.1.8-(a)), a central unit controls all elements of the MG. Despite the advantages of these techniques in terms of high observability and controllability, they suffer from many drawbacks like the single point of failure (i.e., any failure of the central unit will cause the breakdown of the whole MG)

1 . 8 -

 18 (b)), the components of the MG are controlled by independent controllers without the need to communicate with each other. Although these techniques have excellent reliability and flexibility, they suffer from several disadvantages such as poor transient response and load dependence requiring secondary control. Moreover, due to the strong coupling between the different components of the MG, a minimum level of coordination is required. This cannot be achieved by using only local variables, making a fully decentralized approach not possible.The distributed architectures include a scheme in which, each element of the MG has its own controller and exchanges information with other elements of the grid over a communication network to cooperatively achieve common objectives (Fig.1.8-(c)). The most known strategies of this technique are consensus-based and agent-based control architectures[START_REF] Yazdanian | Distributed control techniques in microgrids[END_REF],[START_REF] Anand | Distributed control to ensure proportional load sharing and improve voltage regulation in low-voltage dc microgrids[END_REF].

Figure 1 . 7 :

 17 Figure 1.7: Illustration of the DC-MG control hierarchy (inspired from[START_REF] Fei | Primary and secondary control in dc microgrids: a review[END_REF]).

Figure 1 . 8 :

 18 Figure 1.8: Illustration of the three main control architectures in MGs. From the left, centralized, decentralized and distributed control (• element of the MG and controller).

Remark 1 . 1 .Example 1 . 1 .

 1111 All the graphs considered in this thesis are assumed to be without self loops, i.e., (i, i) / ∈ E Fig.1.9 shows an example of a graph where the vertex set is V = {1, 2, 3, 4, 5} and the adjacency set is E = {(1, 2), (1, 3), (2, 3),(2,[START_REF] Meng | Review on control of dc microgrids and multiple microgrid clusters[END_REF], (2, 5)}.

5 Figure 1 . 9 :

 519 Figure 1.9: Example of an undirected graph with 5 vertices and 5 edges.

Figure 1 . 10 :

 110 Figure 1.10: Example of an undirected graph with 5 vertices and 5 edges.

. 4 )

 4 Definition 1.8. (Graph Laplacian Matrix) An important matrix representation of a graph is the graph 1.4. GRAPH THEORY Laplacian matrix L defined as L = D -A.(1.5)

A

  Multi-Agent System (MAS) consists of a set of dynamical systems that we call agents, interacting with each other to form a network. Agents can have the same dynamical system (i.e., homogeneous MAS) or a possibly different one to another (i.e., heterogeneous MAS). Several problems and approaches of MAS are inspired by nature e.g., flocking of birds, schooling of fishes, the swarm of bacteria, etc. (see Fig.1.11),5 and have many applications in several fields as in MGs[START_REF] Dorri | Multi-agent systems: A survey[END_REF].

Figure 1 . 11 :

 111 Figure 1.11: Collective behaviours of animals, from the left: flocking of birds and schooling of fishes [70].

Figure 1 . 12 :

 112 Figure 1.12: An illustration of MAS with physical and communication coupling.

. 11 )

 11 Definition 1.11. (Convergence to a set) If d(x, y) denotes a distance in a metric space, the distance of a point x to a set S is defined by: d(x, S) = inf y∈S d(x, y). A trajectory x(•) is said to converge asymptotically to a set S if lim t→+∞ d(x(t), S) = 0.

Figure 1 .

 1 Figure 1.13: A 3-dimensional convex polytop.

  p} define the sets of DGUs and SUs, respectively. A simple electrical scheme of the considered model is shown in Fig. 2.1. We refer the reader to Fig. 2.2 for a representative diagram of an example of DC-MG topology. The generic energy source of each DGU and SU is modeled as a DC voltage source V dc i that supplies a local load through a DC-DC converter. The local load is connected to the Point of Common Coupling (PCC) through an RLC (low-pass) filter. Furthermore, two types of local load are considered, resistive load r L i and unknown constant current source I Li (see Fig. 2.1).

Figure 2 . 1 :

 21 Figure 2.1: The considered electrical scheme of agent i and power lines.

Figure 2 Figure 2 . 2 :

 222 Figure 2.1

2 . 3 .

 23 The control scheme consists of two modules. The first module includes two distributed integral actions to guarantee the CS and AVR. Each local controller receives the voltage reference V ref i and, from its neighbors, the weighted (per-unit 2 ) current ω j I j and the control variables ω j γ j . Then it generates two control variables φ i and γ i . These two variables, with 2

Figure 2 . 3 :

 23 Figure 2.3: Structure of the proposed distributed cooperative controller.

Example 2 . 1 .

 21 Consider a DC-MG composed of four DGUs with a total current demand equal to 96 A.

Assumption 2 . 1 .

 21 (Communication network) The controllers exchange information through a communication network modeled as an undirected graph G com d = (V d , ε com , L com ) where L com ∈ R m×m is symmetric positive semidefinite Laplacian matrix which allows to exchange information between the DGUs. CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING

( 2

 2 .13) where N com i is the set of DGUs connected to the DGU i via the communication network with the edge weights α ij = α ji ∈ R >0 . The consensus dynamics in (2.12c)-(2.12d) are used to attend Objectives 2.1-2.2, respectively, this will be detailed in the proof of Proposition 2.2. Definition 2.1. (Set of equilibrium points) For a given reference voltage V ref and an unknown load current I L , the set of all the equilibrium points is defined by S d (I L , V ref ) = {X e = (I e , V e , φ e , γ e ) ∈ R 4m and u e ∈ R m s.t.: 0 = -RI e -V e + u e ,

Assumption 2 . 2 .Proposition 2 . 1 .

 2221 (Nominal model) All the DGUs have the same nominal values of parameters, i.e., l i = l, c i = c, r i = r, r Li = r L , ∀i ∈ V d , with l, c, r, r L ∈ R >0 represent the nominal values. (Existence of equilibrium points) For a given reference voltage V ref and an unknown load current I L , the set S d (I L , V ref ) is not empty.

Assumption 2 . 4 .

 24 (Nominal model) All the SUs have the same nominal values of parameters, i.e., l i = l,

Assumption 2 . 5 .

 25 (Communication network) The controllers exchange information through a communication network modeled as an undirected graph G com s = (V s , ε com , L com ) where L com ∈ R p×p is symmetric positive semidefinite Laplacian matrix which allows to exchange the states-of-charge Soc i , ∀i ∈ V s , between the SUs.

Proposition 2 . 5 .

 25 (Equilibrium point) For a given state-of-charge reference Soc ref and an unknown load current I L , system (2.50) with a stabilizing control input of the form (2.62) has a unique equilibrium point.

Definition 2 . 2 .

 22 (Set of equilibrium points) For a given reference voltage V ref , an unknown load current I L = (I d L , I s L ) and a state-of-charge reference Soc ref the set of all the equilibrium points is defined by S r = {X e ∈ R 4N and u e ∈ R N s.t.: Ẋ = 0}.

Remark 2 . 9 .

 29 .94) by considering the passivity conditions defined by (2.40) and (2.77), (2.94) becomesV (x, I l ) = xd P A d xd + xs QA s xs + x C HI l -I l R l I l -I l H C x, = xd P A d xd + xs QA s xs -I l R l I l , ≤ -xd P xd -µx s Qx s -I l R l I l , < 0, (2.95) where = diag( 1 , • • • , m ) and µ = diag(µ 1 , • • • , µ p ).Consequently, the state vector x I l T converges globally exponentially to the equilibrium point xe I e l whose value depends on the constant vector D in (2.91). (Independent communication networks) The communication networks of the DGUs and SUs are independent, i.e., it can be separated and do not need to have the same topology or the same Laplacian matrix. Furthermore, the topology of the communication network does not necessarily have to be the same as the topology of the physical network. Remark 2.10. (Power lines infromation) The controller proposed in this thesis does not need to measure the power line current I l k and the load current I L or to know the power line parameters L l k , R l k .

1 84 3. 2 3 . 3 3 . 4

 18423334 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matlab Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Real-Time Hardware-In-the-Loop Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3.1 Scenario 1: load current variation near DGUs . . . . . . . . . . . . . . . . . . . . 92 3.3.2 Scenario 2: load current variation near SUs . . . . . . . . . . . . . . . . . . . . . 96 3.3.3 Scenario 3: high-load operating mode . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.4 Scenario 4: voltage reference variation . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.5 Scenario 5: state-of-charge reference variation . . . . . . . . . . . . . . . . . . . 108 3.3.6 Scenario 6: state-of-charge balancing without leader . . . . . . . . . . . . . . . . 112 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 3 . 1 :

 31 Figure 3.1: MG with 3 DGUs, 3 SUs, power lines and two independent communication networks.

  (0) and Soc ref (0). Then, at the time instant t = t 1 the load current I L is stepped up with ∆I L = 15 10 -5 7.5 5 -2.5 A .As we can see in Fig.3.2, in the two cases, the weighted currents converge to the same consensus value 3.2. MATLAB SIMULATION achieving Objective 2.1 and the generated currents converge to the desired steady-state, and all the DGUs are at the same percentage of their rated current. Moreover, Objective 2.1 is still maintained during the transient phase. The voltages at the PCC, in the two cases, are illustrated in Fig.3.3. Note that the weighted average voltage converges to the weighted average value of the reference voltages (see Objective 2.2). Furthermore, the voltages at the PCC converge, without oscillations, to a steady-state near to the reference voltage. The state-of-charge and currents of the SUs are illustrated in Fig.3.4. One can appreciate that, after the load current variation, the state-of-charge converge to their reference (Objective 2.3) by pursuing their leader Soc 1 . However, their convergence is very slow since their time constants are very big regarding the other time constants of the system.

Figure 3 . 4 :

 34 Figure 3.4: Matlab Simulation: from the top, states-of-charge of the SUs; generated current of SUs in case of resistive and resisitive-inductive power lines, respectively.

Figure 3 . 14 :

 314 Figure 3.14: Real-Time tests -Scenario 3: from the top, states-of-charge of the SUs; generated current of SUs in case of the nominal and real models, respectively.

Figure 3 . 16 :Figure 3 . 17 :Figure 3 . 18 :

 316317318 Figure 3.16: Real-Time tests -Scenario 4: from the top, weighted average voltage at the PCC near the DGUs together with the weighted average voltage reference value (dashed line); voltage at the PCC of each DGU and SU together with the reference value (dashed line) in case of the nominal and real models, respectively.

Figure 3 . 20 :Figure 3 . 21 :

 320321 Figure 3.20: Real-Time tests -Scenario 5: from the top, states-of-charge of the SUs; generated current of SUs in case of the nominal and real models, respectively.

Figure 3 . 22 :Figure 3 . 23 :

 322323 Figure 3.22: Real-Time tests -Scenario 6: from the top, weighted average voltage at the PCC near the DGUs together with the weighted average voltage reference value (dashed line); voltage at the PCC of each DGU and SU together with the reference value (dashed line) in case of the nominal and real models, respectively.

  ref available only for one SU (called the leader). This gives the possibility for high-level monitoring of the states-of-charge through an energy management strategy, which defines the Soc ref value e.g. it can send a high Soc ref in case of low-load operation mode, and small Soc ref in case of high-load operation mode. Moreover, the controllers do not need any information about the topology of the MG or the currents and parameters of the resistiveinductive power lines. The topology of the power network does not need to be the same as the topology of the communication network. Furthermore, we applied the proposed controllers to a DC-MG composed of three DGUs, three SUs, resistive-inductive power lines, loads and two communication networks (Chapter 3).The DC-MG was implemented in two environments, Matlab/Simulink software and Real-Time Hardware-In-the-Loop (HIL) platform. Several scenarios were considered where the results showed the efficiency of the proposed control methodology and where the control objectives were successfully achieved. The results shows also that SB can be achieved without defining a state-of-charge reference. In this case, all the state-of-charge converge to the average of their initial values, which illustrates the flexibility of the proposed technique.The results of the work developed in this thesis lead to various possible extensions and new researches. For instance, this approach can be applied for AC-MGs by considering dq transformation and already synchronized MG[START_REF] Tucci | A scalable, line-independent control design algorithm for voltage and frequency stabilization in ac islanded microgrids[END_REF]. Even though the HIL tests (Section 3.3) showed that the proposed controllers work in the case when the DGUs and SUs have different physical parameter values, it could be of interest to consider relaxing Assumptions 2.2 and 2.4 in the proof of the global convergence. Another possible enhancement of the proposed control approach can be the use of observers to estimate the value of the local voltage or the local current rather than measuring it. Looking at the load side, it may be interesting to consider Constant Power Load (CPL) with proof of global convergence. The plug-and-play functionality of DC-MGs with the

  LIST OF SYMBOLSConvergence Exponentielle Globale (CEG) vers cet ensemble est fournie, sous l'hypothèse que tous les DGU (respectivement, tous les SU) ont les mêmes valeurs de paramètres physique, malgré la variation inconnue de la charge et les conditions initiales du microgrid et de l'état du contrôleur. De plus, l'équilibrage des état de charge est réalisé en utilisant une stratégie leader-suiveur qui donne la possibilité d'une surveillance de haut niveau (contrôleur tertiaire) en ajustant la référence d'état de charge envoyée au leader. Les contrôleurs n'ont besoin d'aucune information sur la topologie du réseau électrique, ni sur les paramètres et les courants des lignes électriques. La topologie du réseau de communication n'a pas besoin d'être la même que la topologie du microgrid. En outre, l'approche de contrôle proposée est basée sur les inégalités linéaires matricielles, ce qui la rend intéressante sur le plan numérique. En outre, le contrôleur proposé est robuste en cas de perturbations constantes à l'entrée de commande. Enfin, nous fournissons une simulation Matlab/Simulink et

des tests Hardware-in-the-Loop (HIL) en temps réel dont les résultats montrent l'efficacité des contrôleurs proposés.

Commande distribuée pour le partage du courant, la régulation de la tension moyenne et l'équilibrage des états de charge

  Ce chapitre contient les principaux résultats théoriques de la thèse. Nous présentons la méthodologie de contrôle proposée pour atteindre les objectives de contrôle considérés pour les microgrids DC composés de DGU, de SU, de lignes électriques résistives-inductives et de charges. Ce chapitre est divisé en quatre parties.Dans la première partie, nous présentons le modèle du microgrid considéré. Ensuite, dans la deuxième partie, nous considérons le problème du CS et de AVR pour un microgrid composé uniquement de DGU, de lignes électriques résistives et de charges. Pour atteindre les objectifs de contrôle, le modèle est augmenté de deux actions intégrales distribuées de type consensus. Un changement de variable est proposé pour transformer le

				LIST OF SYMBOLS
	Réseau de
	Communication 1
	Consensus
	Action Intégrale -CS	-K	u d i	DGU i
	Action intégrale -AVR
	Estimateur du Soc	-F	i u s	SU i
	Leader-follower
	Action intégrale -SB
	Réseau de
	Communication 2
	Le deuxième module comprend une action intégrale distribuée pour assurer le SB. L'état de charge est estimé
	localement. Chaque contrôleur local reçoit de ses voisins l'état de charge Soc j et génère une variable de

Chapitre 1: Préliminaires

Le chapitre 1 fournit les préliminaires nécessaires aux développements présentés dans cette thèse. Tout d'abord, nous présentons des définitions générales des microgrids et les trois propriétés principales sur lesquelles ils s'appuient, à savoir : l'interconnexion et la distribution, la coopération et la commande et enfin une certaine indépendance du reste du réseau. Puis, nous présentons les différents types de microgrids avec un accent particulier sur les microgrids DC et leurs défis. Ensuite, un aperçu des techniques de contrôle existantes pour surmonter les défis liés aux microgrids DC est donné. Ensuite, nous rappelons quelques définitions de la théorie des graphes et du consensus dans les systèmes multi-agents linéaires. Nous soulignons le fait que plusieurs défis des microgrids DC peuvent être représentés comme des problèmes de consensus des systèmes multi-agents. Enfin, nous rappelons quelques résultats fondamentaux de la stabilité, de la passivité et des techniques d'analyse de stabilité des systèmes à paramètres variants. Chapitre 2: système en plusieurs sous-systèmes découplés (DGU augmenté d'actions intégrales locales). Nous montrons que s'il existe un gain unique de retour d'état qui stabilise les sous-systèmes sous une condition de passivité, la CEG du modèle nominal est assurée. À la fin de cette partie, nous donnons une formulation LMI pour la conception du gain du contrôleur. Dans la troisième partie, on considère un microgrid composé uniquement de SU avec des lignes électriques et des charges résistives où le problème de SB est étudié. Pour atteindre les objectifs de contrôle, un algorithme de consensus leader-follower est proposé. Les étapes de conception des contrôleurs sont similaires à celles de la partie précédente. Dans la dernière partie du chapitre, nous montrons que les contrôleurs conçus dans les parties précédentes peuvent être utilisés pour atteindre les trois objectifs de contrôle en même temps pour un microgrid comprenant à la fois des DGU et des SU et même des lignes électriques résistives-inductives. Le schéma global du contrôleur proposé est illustré dans la Fig. 1. Il se compose de deux modules, le premier module comprend deux actions intégrales distribuées pour garantir le CS et l'AVR. Chaque contrôleur local reçoit la référence de tension V ref i et, de ses voisins, les courant pondérés (par unité 1 ) ω j I j et les variables de commande ω j γ j . Il génère ensuite deux variables de commande φ i et γ i . Ces deux variables, avec le courant local I i et la tension locale V i , définiront l'état augmenté du DGU i. Ensuite, l'état augmenté est multiplié par un gain de de retour d'état K pour générer le contrôleur local u d i .

Étude de Cas: Simulation Matlab et mise en oeuvre des tests HIL en temps réel

pour le leader seulement) Figure 1: Structure de la commande distribuée proposée. contrôle θ i . Il convient de noter que si le SU i est le leader, son contrôleur local recevra également le Soc ref d'un moniteur de haut niveau (par exemple, un contrôleur tertiaire). La variable de contrôle θ i ainsi que le courant local I i , la tension V i et l'état de charge Soc i définiront l'état augmenté de SU i. Ensuite, l'état augmenté est multiplié par un gain de retour d'état F pour générer le contrôleur local u s i .

Chapitre 3:

Dans le chapitre 3, l'approche de commande distribuée proposée au chapitre précédent est évaluée. Le microgrid considéré est composé de trois DGU, trois SU, des lignes électriques résistives-inductives et des charges. Plusieurs scénarios sont envisagés et mis en oeuvre dans Matlab/Simulink et dans un plateforme

  Furthermore, MGs can be divided into three types, namely Direct Current (DC), Alternating Current (AC), and DC/AC microgrids. DC-MGs have received an increasing interest within the automatic control and

electrical engineering communities. This growing interest is due to its efficiency, simplicity and wide range of applicability. In fact, most storage systems, renewable energy resources and modern loads, e.g. laptops, tablets, computers, etc. are inherently DC.

The purpose of the present work is to develop a novel distributed control approach for islanded DC-MGs to provably achieve Current Sharing (CS), Average Voltage Regulation (AVR) and State-of-charge Balancing (SCB) at the same time. Resolving these issues is critical for the safe and reliable operation of the MG. The proposed control methodology is based on the separation between the CS and AVR problems on one side and the SB problem on the other; the main tools are: consensus in multi-agent systems, passivity, state feedback, distributed integral actions, Lyapunov stability and Linear Matrix Inequalities (LMIs).

  1.6. STABILITY AND PASSIVITY OF INTERCONNECTED LINEAR SYSTEMS Definition 1.16. x e is asymptotically stable, if it is stable and there exists δ > 0, such that ||x(0) -x e || < δ =⇒ lim Exponential stability is a form of asymptotic stability which requires an exponential rate of convergence. Definition 1.17. x e is exponentially stable if there exist three positive real constants α, β and γ such that||x(0) -x e || ≤ α, ||x(t) -x e || ≤ β||x(0) -x e ||e -γt .Definitions 1.15-1.17 give information about local stability, i.e., when the system starts nearby its equilibrium point. The following definition gives information in the case where the trajectory of the system can start from any point in R n .

t→+∞ x(t) = x e . Definition 1.18. (Global stability) x e is globally asymptotically (exponentially) stable if it is stable and for

  >0 . Although it is out of the scope of this work, some details about the model used for the state-of-charge estimation can be found in Appendix B.

	CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND
	STATE-OF-CHARGE BALANCING
	with k soc
	The notations used in Fig. 2.1 and (2.1) are described in Table 2.1. The use of QSL-approximation in (2.1)
	)

i ∈ R

Table 2 .

 2 1: Description of the used notations.

	State variables
	I i	Generated current of Agent i.
	V i	Voltage at the PCC near Agent i.
	I l ij	Current of power line ij.
	Parameters
	r i	Output filter resistance of Agent i.
	l i	Output filter inductance of Agent i.
	v i	Output filter capacitor of Agent i.
	r Li	Local resistive load near Agent i.
	r l ij	Power line resistance.
	l l ij	Power line inductance.
	Inputs
	u i	Output voltage of the converter
		(Control input of Agent i ).
	I Li

Table 2 .

 2 2 shows the rated current I r i , the generated current I i and the associated weight of each DGU when Objective 2.1 is achieved. As we can see in Table2.1 all the DGUs converge to 80% of their rated current.

		DGU 1 DGU 2 DGU 3 DGU 4
	I r i (A)	10	20	40	50
	ω i	0.1	0.05	0.025	0.02
	I i (A)	8	16	32	40
	ω i I i	0.8	0.8	0.8	0.8

Table 2 . 2

 22

	This can causes several problems as mentioned in Section 1.2.
		DGU 1 DGU 2 DGU 3 DGU 4
	I r i (A)	10	20	40	50
	ω i	0.1	0.05	0.025	0.02
	I i (A)	18	26	22	30
	ω i I i	1.8	1.3	0.55	0.6
	Table 2.3: Generated current of a DC-MG without current sharing objective.

: Generated currents of a DC-MG with current sharing objective. Table

2

.3 illustrates a case when Objective 2.1 is not considered. We can see clearly that the DGU with the lower power generates beyond of its nominal capacity whereas the one with the higher power provides less than its nominal capacity (DGU 1 with 180% of its rated current, and DGU 2 with 60% of its rated current).

Table 2 .

 2 Example 2.2. Consider a DC-MG as in Example 2.1. The following table shows the voltages at PCC with their references when Objective 2.2 is achieved. 4: Voltages at the PCC of a DC-MG with AVR controller. Note that CS is always obtained when the voltages are shifted with the same constant, because the variation in current demand (R -1 L 1v * in (2.11)) caused by the voltage shift is also shared proportionally between the DGUs. Now, we can state the first control problem as : For a given reference voltage V ref and an unknown load current I L , design a distributedbased control law u s.t. the state of system (2.3) in closed-loop converges globally and exponentially to a set of equilibrium points whose elements satisfy Objectives 2.1-2.2.

	It is noteworthy that the voltages stay near to their references while achieving Objective 2.2.
		DGU 1 DGU 2 DGU 3 DGU 4
	V ref i	380	380	380	380
	ω i	0.1	0.05	0.025	0.02
	V i	378.1 379.3 380.2 380.5
	Remark 2.3. Control Problem 1.

  Sd ( ĨL , Ṽ ref ), then (X e , u e ) ∈ S d (I L , V ref ) since ( X e , ũe ) = (I 5 ⊗ T T )(X e , u e ).(2.19) Thus, it is sufficient to prove that Sd ( ĨL , Ṽ ref ) = ∅, i.e., system (2.18) has at least one solution. From

	CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND
	STATE-OF-CHARGE BALANCING
	Obviously, if ( X
	2.18a)
	(2.18b)
	(2.18c)
	(2.18d)

e , ũe ) ∈

  Thus, system (2.18) has an infinite number 2.2. CURRENT SHARING AND AVERAGE VOLTAGE REGULATION of equilibrium points parameterized by (I L , V ref ). See Appendix A for the definition of the Kronecker product denoted by ⊗. It is worthy to note that only relative information are exchanged between the DGUs augmented with the additional states (2.13) (notice that α ij = 0 if DGU j is not neighbor of DGU i). Hence, if a state feedback controller that uses only the local state at each DGU, i.e., (I i , V i , φ i , γ i ), ∀i ∈ V d , is considered, u will be distributed.

	Remark 2.4. Proposition 2.2. (Augmented system with state feedback) If there exists a state feedback u of the form:
	u = -KX ,	(2.26)

a where X = (I, V, φ, γ) and K ∈ R m×4m s.t., in closed-loop, the couple (X , u) of system (2.12) converges asymptotically to the set S d (I L , V ref ) then Objectives 2.1-2.2 are always achieved in this set S d .

Proof. If there exists a state feedback control (2.26) s.t. the couple (X , u) converges, in closed-loop, asymptotically to the set S d (I L , V ref ), then (2.14c) implies that I e ∈ Ker(W T L com W ). As L com is a symmetric positive semidefinite Laplacian matrix for which the kernel is provided by the set Ker(L com ) = {z ∈ R m : ∃α ∈ R, z = α1 m } (see Remark 1.3) and as the weighting matrix W is diagonal strictly positive definite, it follows that there exists a scalar α s.t.:

  Σd with the control law ũ given by (2.29) can be written as m interconnected subsystems of the form: ∀i ∈ V d ,

	CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND
		STATE-OF-CHARGE BALANCING
	Using (2.34) system Σd i : ẋi = A cli xi + d i -B p	l pow i,j C p xi ,	(2.35)
	j=i, j∈N pow i
			.34)

  CURRENT SHARING AND AVERAGE VOLTAGE REGULATION the augmented system (2.12) converges exponentially to an equilibrium (X e , u e ) ∈ S d (V ref , I L ) for which the control objectives 2.1-2.2 are satisfied. Proof. Using Proposition 2.3, it is sufficient to prove the convergence of the couple ( X , ũ = -(I m ⊗ K) X ) to the set Sd ( Ṽ ref , ĨL ).A state space representation of system Σd with the controller ũ in Proposition 2.3 can be written from subsystems defined by (2.35) as:

.37)

and where A cl i , B p , C p and A are given with subsystems (2.35), then the couple (X , u = -(K ⊗ I m )X ) of 2.2.

  The latter can use an energy management strategy to define the Soc ref value, e.g. it can send a high Soc ref in case of CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING low-load operation mode, and small Soc ref in case of high-load operation mode. We assume that Soc ref is available only for one SU i l (called hereafter leader), where i l denotes the index of the leader. Now, we are able to state the second control problem as : Control Problem 2. For a given state-of-charge reference Soc ref and an unknown load current I L , design a distributed-based control law u s.t. the state of system (2.49), in closed-loop, converges globally and asymptotically to an equilibrium for which Objective 3 is satisfied.

	Assumption 2.3. (States-of-charge leader)

  .81) where L pow dd ∈ R m×m , L pow ss ∈ R p×p , L pow ds ∈ R m×p and L pow sd ∈ R p×m . In Sections 2.2 and 2.3, for the sake of simplicity, generic equation and, mostly, the same notations were used to present the design of the controllers for the DGUs and SUs. However, in what CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING follows, for more clarity, when it is necessary the indices 'd' and 's' are used for the parameters and variables related to the DGUs and SUs, respectively, as in (2.80). Soc ref , design a distributed-based control law u = (u d , u s ) s.t. the state of system (2.80) in closed-loop converges globally and exponentially to a set of equilibrium points whose elements satisfy Objectives 2.1-2.2 for the DGUs and Objective 2.3 for the SUs.

	Control Problem 3. Consider Assumptions 2.1-2.5, for a given reference voltage V ref , an unknown load current I L = (I d L , I s L ) and a state-of-charge reference Similarly to Sections 2.2 and 2.3, we denote by Σ system (2.80) augmented with the following additional Remark 2.7. (Notation) At this stage, we consider the following problem: state:

  is the positive end of power line k -1 if i is the negative end of power line k Note that the physical coupling between the agents is no more static and the term -L pow V in (2.80) is replaced by the term HI l . The next theorem shows that the distributed controllers proposed in Theorem 2.3 can solve Control Problem 3 even if the power lines are resistive-inductive. The triple (X ,I l ,u) of system (2.88)-(2.89) augmented with (2.82), and in closedloop with the state feedback controller (2.84) that respects the conditions of Theorem 2.3, converges globally exponentially to the set of all equilibrium points S rl = {X e ∈ R 4N , I l ∈ R q and u e ∈ R N s.t.: Ẋ = 0 and İl = 0}, The proof of Theorem 2.4 is based on the passivity of System (2.88) augmented with(2.82) and the inherent passivity of the power lines (2.89). Hence, the proposed Lyapunov function will be composed of the Lyapunov functions used in the proofs of Theorems 2.1 and 2.2 and the storage function of the power lines.

	2.4. DC-MG WITH DGUS AND SUS
	Theorem 2.4. (main result) where Objectives 2.1-2.3 are satisfied.
	Proof. By applying the change of variables (2.17), (2.15), (2.33), and (2.72)-(2.74) on System (2.88)-(2.89)
	augmented with (2.82) in closed-loop with the control input (2.84) we get:
	(2.90)
	0 otherwise.

  B d and d are defined in (2.38), • xs , A s , C s , B s and v are defined in (2.76). CHAPTER 2. CURRENT SHARING, AVERAGE VOLTAGE REGULATION AND STATE-OF-CHARGE BALANCING Now, the global exponential convergence of the triple (X , I l , u) to the set S rl can be concluded by the same reasoning in the proof of Theorems 2.1 and 2.2, i.e., first considering x as the vector x for which the uncontrollable variable ( φ1 ) has been removed and by considering the following Lyapunov function:

  .1, where also two independent communication networks are depicted. The nominal and real parameters of each DGU and SU, load current and power lines parameters are reported in Tables 3.1-3.4. In this example, we consider that the information about the Soc ref is available only for SU 1 (the leader). The computed controller gains are

	K = 1501.96 1.11 0.47 68.49 ,	(3.1)
	F = 02.98 0.168 -537.18 -0.09 .

The parameter τ in (2.12d) is set to 400 and the gain β i l in (2.51) is set to 10. The Laplacian matrices of the power and communication networks are, respectively, as follows:

•

  At the time instant t = t 1 the load current I d L1 near DGU 1 is stepped up with ∆I d L 1 = 15 A. • At the time instant t = t 2 the load current I d L2 near DGU 2 is stepped up with ∆I d L 2 = 15 A. • At the time instant t = t 3 the load current I d L3 near DGU 3 is stepped up with ∆I d L 3 = 15 A. • At the time instant t = t 4 the load current I d L1 near DGU 1 is stepped down with -∆I d L 1 . • At the time instant t = t 5 the load current I d L2 near DGU 2 is stepped down with -∆I d L 2 . • At the time instant t = t 6 the load current I d L3 near DGU 3 is stepped down with -∆I d L 3 .The currents and the weighted current of the DGUs are illustrated in Fig.3.6. One can appreciate that the weighted currents converge to the same consensus value achieving Objective 2.1 and the generated currents converge to the desired steady-state, and all the DGUs are at the same percentage of their rated current.

	3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS 3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS
	60.005		6.65		10 4
	1.3 60.004		6.6
	1.2 60.003
	1.1 60.002		6.55
	60.001		6.5
	0.9
	60		6.45
	0.8
	0.7 59.999		6.4
	0.6 59.998		6.35
	0.5 59.997
	0.4 59.996		6.3				t 1 t 1		t 1	t 2 t 2		t 2	t 3 t 3		t 3					t 4 t 4		t 4	t 5 t 6 t 5 t 6	t 5 t 6
	0.3 59.995	0	6.25	0		2	2	2		4	4	4		6	6	6	8 8	8	10 10	10	12 12		12	14 14	14	16 16	16	18 18	18	20	20
	6			400
	4			395
				390
	2
				385
	0			380
	-2			375
				370
	-4
				365				t 1 t 1		t 1	t 2 t 2		t 2	t 3 t 3		t 3					t 4 t 4		t 4	t 5 t 6 t 5 t 6	t 5 t 6
	-6 60.005 0		360 6.65	0	10 4	2 2		2		4 4		4		6 6		6	8 8	8	10 10	10	12 12		12	14 14	14	16 16	16	18 18	18	20	20
	1.3 60.004		6.6
	1.2 60.003
	1.1 60.002		6.55
	60.001		6.5
	0.9 60		6.45
	0.8 59.999
	59.997 0.6 Moreover, note that Objective 2.1 is still maintained during the transient phase. Finally, Objectives 2.2 and 6.35 6.4 0.7 59.998
	2.3 are also still reached (Fig. 3.7 and 3.8). 2 4 6 8 t 1 t 2 t 3 0 2 4 6 6.3 t 1 t 2 59.996 0.3 0.4 0.5 t 1 t 2 t 3 t 3 59.995 6.25 0 2 4 6 8 400 6	8	10 10	10	12 12	t 4 t 4	12	14 t 4 14	t 5 t 6 14 t 5 t 6	16 t 5 t 6 16	16	18 18	18	20	20
	4			395
				390
	2
				385
	0			380
	-2			375
				370
	-4
	-6	0		360 365	0		2 2	t 1 t 1	2	t 1	4 4	t 2 t 2	4	t 2	6 6	t 3 t 3	6	8 8 t 3	8	10 10	10	12 12	t 4 t 4	12	14 14 t 4	t 5 t 6 14 t 5 t 6	16 16 t 5 t 6	16	18 18	18	20	20
	Figure 3.6: Real-Time tests -Scenario 1: from the top, weighted generated currents and generated currents of the DGUs in case of the nominal and real models, respectively. Figure 3.8: Real-Time tests -Scenario 1: from the top, states-of-charge of the SUs; generated current of SUs Figure 3.7: Real-Time tests -Scenario 1: from the top, weighted average voltage at the PCC near the DGUs together with the weighted average voltage reference value (dashed line); voltage at the PCC of each DGU in case of the nominal and real models, respectively.
		and SU together with the reference value (dashed line) in case of the nominal and real models, respectively.

  Figure 3.10: Real-Time tests -Scenario 2: from the top, weighted average voltage at the PCC near the DGUs together with the weighted average voltage reference value (dashed line); voltage at the PCC of each DGU and SU together with the reference value (dashed line) in case of the nominal and real models, respectively.

	3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS 3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS CHAPTER 3. CASE STUDY: MATLAB SIMULATION AND REAL-TIME HIL IMPLEMENTATION
	60.01		6.65
	60.008		6.6
	60.006
				6.55
	60.004
	60.002		6.5
	60		6.45
	59.998		6.4
	-2 59.996
				6.35
	59.994
	-4 59.992		6.3				t 1 t 1	t 1	t 2 t 2	t 2	t 3 t 3	t 3					t 4 t 4	t 4	t 5 t 6 t 5 t 6	t 5 t 6
	-6 59.99	0	6.25	0	2	2		2	4	4		4	6	6		6	8 8	8	10 10	10	12 12		12	14 14	14	16 16	16	18 18	18	20	20
	6			400
	4			395
				390
	2
				385
	0			380
	-2			375
				370
	-4
				365				t 1 t 1	t 1	t 2 t 2	t 2	t 3 t 3	t 3					t 4 t 4	t 4	t 5 t 6 t 5 t 6	t 5 t 6
	-6	0		360	0	2 2			2	4 4			4	6 6			6	8 8	8	10 10	10	12 12		12	14 14	14	16 16	16	18 18	18	20	20
	60.01		6.65
	60.008
				6.6
	60.006
	60.004		6.55
	60.002		6.5
	60		6.45
	59.998
				6.4
	-2 59.996
	59.994		6.35
	-6 -4 59.992 59.99 6	0	6.3 6.25 400	0	2	2	t 1 t 1	2	4 t 1	4	t 2 t 2	4	6 t 2	6	t 3 t 3	6	8 t 3 8	8	10 10	10	12 12	t 4 t 4	12	14 t 4 14	t 5 t 6 14 t 5 t 6	16 t 5 t 6 16	16	18 18	18	20	20
	4			395
				390
	2
				385
	0			380
	-2			375
				370
	-4
	-6	0		360 365	0	2 2		t 1 t 1	2	4 4 t 1		t 2 t 2	4	6 6 t 2		t 3 t 3	6	8 8 t 3	8	10 10	10	12 12	t 4 t 4	12	14 14 t 4	t 5 t 6 14 t 5 t 6	16 16 t 5 t 6	16	18 18	18	20	20
	Figure 3.11: Real-Time tests -Scenario 2: from the top, states-of-charge of the SUs; generated current of Figure 3.9: Real-Time tests -Scenario 2: from the top, weighted generated currents and generated currents of the DGUs in case of the nominal and real models, respectively. SUs in case of the nominal and real models, respectively.

•

  At the time instant t = t 1 the load current I d L near the DGUs is stepped up with ∆I d L = 40 40 40 A. • At the time instant t = t 2 the load current I d L near the DGUs is stepped down with -∆I d L A. Fig. 3.12-3.14 illustrate the ability of the MG in closed loop with the proposed controllers, to manage an instantaneous transition between high-load and small-load operating modes (±120A).Figure 3.13: Real-Time tests -Scenario 3: from the top, weighted average voltage at the PCC near the DGUs together with the weighted average voltage reference value (dashed line); voltage at the PCC of each DGU and SU together with the reference value (dashed line) in case of the nominal and real models, respectively. 3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS

	3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS
	2.5 60.04		7.2		10 4
	60.03		7
	60.02
			6.8
	1.5 60.01		6.6
	60
			6.4
	59.99
	59.98		6.2
	0.5
	59.97		6							t 1 t 1	t 1									t 2 t 2	t 2
	59.96	0	5.8	0		2	2	2	4	4	4	6	6	6	8 8	8	10 10	10	12 12	12	14 14	14	16 16	16	18 18	18	20	20
	30
	20
	10
	0
	-10
	-20
	-30									t 1 t 1	t 1									t 2 t 2	t 2
	-40
	2.5 60.04 0		6.9	0	10 4	2 2		2	4 4		4	6 6		6	8 8	8	10 10	10	12 12	12	14 14	14	16 16	16	18 18	18	20	20
	60.03		6.8
	60.02		6.7
	60.01		6.6
	1.5
			6.5
	60
			6.4
	59.99
			6.3
	59.98
	0.5		6.2
	59.97		6.1							t 1 t 1	t 1									t 2 t 2	t 2
	59.96	0	6	0		2	2	2	4	4	4	6	6	6	8 8	8	10 10	10	12 12	12	14 14	14	16 16	16	18 18	18	20	20
	30
	20
	10
	0
	-10
	-20
	-30									t 1 t 1		t 1									t 2 t 2	t 2
	-40
	0			0		2 2		2	4 4		4	6 6		6	8 8	8	10 10	10	12 12	12	14 14	14	16 16	16	18 18	18	20	20
	Figure 3.12: Real-Time tests -Scenario 3: from the top, weighted generated currents and generated currents
	of the DGUs in case of the nominal and real models, respectively.

  • At the time instant t = t 1 the voltage reference V ref is stepped up with ∆V ref = 5 5 5 V .One can appreciate that Objectives 2.1 and 2.3 are still achieved (see Figs.3.15 and 3.17). The voltages at the PCC are illustrated in Fig.3.[START_REF] Adhikari | Secondary control for dc microgrids: A review[END_REF]. We can see clearly that the average of the voltages at the PCC near the DGUs converge the average voltage reference (Objective 2.2). Note that even if the voltages of the SUs storage units are not controlled, all the voltage at the PCC is within the range of 385 ± 2V during the steady-state and 385 ± 4V during the transient phase, implying that the voltage deviations are less than 0.5% of the reference value V ref = 385 V during the steady state and less than 1% of the reference value V ref = 385 V during the transient phase. 3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS Figure 3.15: Real-Time tests -Scenario 4: from the top, weighted generated currents and generated currents of the DGUs in case of the nominal and real models, respectively.

		6.65		10 4
	1.3
	1.2	6.6
	1.1
		6.55
	0.9
		6.5
	0.8
	0.7	6.45
	0.6
	0.5	6.4
	0.4			t 1	t 1
	0.3	6.35
			0		5	5	10	10	15	15	20	20	25	25	30	30
		390
		388
		386
		384
		382
		380
		378
		376
		374
		372		t 1	t 1
		370
		6.65	0	10 4	5	5	10	10	15	15	20	20	25	25	30	30
	1.3
	1.2	6.6
	1.1
		6.55
	0.9
		6.5
	0.8
	0.7	6.45
	0.6
	0.5	6.4
	0.4			t 1	t 1
	0.3	6.35
			0		5	5	10	10	15	15	20	20	25	25	30	30
		390
		388
		386
		384
		382
		380
		378
		376
		374
		372		t 1	t 1
		370
			0		5	5	10	10	15	15	20	20	25	25	30	30

  CASE STUDY: MATLAB SIMULATION AND REAL-TIME HIL IMPLEMENTATION Figure 3.19: Real-Time tests -Scenario 5: from the top, weighted average voltage at the PCC near the DGUs together with the weighted average voltage reference value (dashed line); voltage at the PCC of each DGU and SU together with the reference value (dashed line) in case of the nominal and real models, respectively.

	3.3. REAL-TIME HARDWARE-IN-THE-LOOP TESTS
	6.55	10 4
	6.5
	6.45
	6.4
			t 1	t 1
	6.35
			1000	1000	2000	2000	3000	3000	4000	4000	5000	5000	6000	6000	7000	7000	8000	8000	9000	9000	10000	10000
	3
	2
	1
	0
	-1
	-2
			t 1	t 1
	-3
	6.55	10 4	1000	1000	2000	2000	3000	3000	4000	4000	5000	5000	6000	6000	7000	7000	8000	8000	9000	9000	10000	10000
	6.5
	6.45
	6.4
			t 1	t 1
	6.35		1000	1000	2000	2000	3000	3000	4000	4000	5000	5000	6000	6000	7000	7000	8000	8000	9000	9000	10000	10000
	3
	2
	1
	0
	-1
	-2
			t 1	t 1
	-3
			1000	1000	2000	2000	3000	3000	4000	4000	5000	5000	6000	6000	7000	7000	8000	8000	9000	9000	10000	10000

  CASE STUDY: MATLAB SIMULATION AND REAL-TIME HIL IMPLEMENTATION

	6.55	10 4
	6.5
	6.45
	6.4
		t 1
	6.35
			500	1000	1500	2000	2500	3000	3500
		t 1
	6.55	10 4	500	1000	1500	2000	2500	3000	3500
	6.5
	6.45
	6.4
		t 1
	6.35
			500	1000	1500	2000	2500	3000	3500
		t 1
			500	1000	1500	2000	2500	3000	3500

les poids ωi peut être défini comme l'inverse du courant nominal du DGU i

Laboratoire d'analyse et d'architecture des systèmes

The consensus techniques will be detailed in Section 1.5.

For more examples, with animation, about the collective behavior in nature and its application, the reader is invited to see the following videos: Video 1, and Video 2)

See Definition A.2 in the appendix for the definition of positive and positive semidefinite functions.

A set E is convex if and only if ∀x1, x2 ∈ E, ξ ∈ [0 1] =⇒ ξx1 + (1 -ξ)x2 ∈ E.

Quasi Stationary Line

We will see later that the weight ωi can be defined as the inverse of the rated current of DGU i

The total current demand is the sum of all the currents of the resistance loads (R -1 L V e ) and all the load currents (IL)

The matrix (R -1 L +L pow ) is invertible because R -1L is symmetric positive definite and L pow is symmetric positive semidefinite.

LIST OF SYMBOLS d'observateurs pour estimer la valeur de la tension locale ou du courant local plutôt que de la mesurer. Du côté de la charge, il peut être intéressant de considérer les charges à puissance constante (CPL) dans la preuve de convergence globale. La fonctionnalité plug-and-play des microgrids DC avec l'approche proposée peut également être une extension intéressante de ce travail. En ce qui concerne le réseau de communication, d'autres questions peuvent également être considérées comme le retard de transmission, la topologie variante et l'optimisation du coût de communication [START_REF] Ben Rejeb | Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems[END_REF]. Enfin, d'autres tests peuvent être réalisés en appliquant l'approche de contrôle proposée sur un prototype réel de microgrids DC.

where K, F respect the conditions of Theorems 2.1 and 2.2, respectively. Then, the couple (X ,u) of system Σ, in closed-loop, converges globally exponentially to the set of all equilibrium points S r where Objectives 2.1-2.3 are satisfied.

Proof. Let Σ represents system Σ after applying the same changes of coordinates given by (2.15), (2.17), (2.72)

where

and where a φ1 is uncontrollable because it is a constant since φ1 = 0.

DC-MG with Resistive-Inductive Power Lines

In this section, we consider that the power lines are resistive-inductive. Hence, the dynamics of the agents (2.1) becomes:

The dynamics of the power line k, k ∈ ξ pow i is given by:

where ξ pow i is the set of power lines connected to agent i . In what follows, the power line can be considered as resistive or resistive-inductive. In the case of resistive

Real-Time Hardware-In-the-Loop Tests

In this section, we aim to validate the proposed controller by Real-Time hardware-in-the-loop tests. The MG was emulated using National-Instruments (NI) PXI system (PXIe-1082 chassis with NI PXIe-8135, FPGA and Input/Output modules). A MicroLabBox of dSPACE with DS1202 baseboard is used to receive the information and send the commands.

Differently from Section 3.2, we consider more scenarios to assess the proposed control methodology. In each scenario, two cases are considered. The first one corresponds to a DC-MG with nominal parameters (Table 3.1) while the second one correspond to the DC-MG with real parameters (Table 3.2) both with resistive-inductive power lines.

Scenario 5: state-of-charge reference variation

In the fifth scenario, we investigate the effectiveness of the proposed controllers under a variation in the stateof-charge reference. The system is initially at the steady state with I L (0), V ref (0) and Soc ref (0) = 60%.

Then, at the time instant t = t 1 the Soc ref is stepped up with 10%. As we can see in Fig. 3.20, the statesof-charges of the SUs converges to the new Soc ref asymptotically while Objectives 2.1-2.2 still achieved (Fig. 3. 18-3.19). Furthermore, one can appreciate that the current generated by the SUs converges to zero after being negative in order to charge the batteries. The results illustrate the robust performance of the proposed controllers under the change in the state-of-charge reference.

Scenario 6: state-of-charge balancing without leader

In the last scenario, we investigate the behavior of the system in closed-loop with the proposed controllers when there is no state-of-charge reference for the SU, i.e., the SUs do not have any leader and the parameter

The system is initially at the steady state with I L (0), V ref (0), Soc 1 (0) = 50%, Soc 2 (0) = 20% and Soc 3 = 80%. Then at the time instant t = t 1 we activate the controllers without any leader. Note that in Finally, one can appreciate that in all the scenarios, even if the system is implemented in a near realistic environment, and with parametric discrepancies from the nominal parameter values i.e. time constants for the interconnection RLC filters that differ significantly from their nominal values: e.g.,

the performances of the proposed control approach are not practically affected.

Appendix B

State-of-charge Dynamics

The state of charge (SOC) of a SU can be defined as the ratio between its available capacity and its maximum available capacity in percentage [START_REF] Maharjan | State-of-charge (soc)-balancing control of a battery energy storage system based on a cascade pwm converter[END_REF].

Soc = T he available capacity T he maximum available capacity

The estimation of the state-of-charge can be done by different techniques. This techniques are based on the measurement of the voltage, current or any other convenient parameter that varies with the Soc [START_REF] Piller | Methods for state-of-charge determination and their applications[END_REF]. In this work we consider that the dynamic of Soc i , i ∈ V s , is given by coulomb counting method [START_REF] Shafiee | Distributed consensus-based control of multiple dc-microgrids clusters[END_REF] 1 :

where C dc i and I dc i are the capacity and the output current of the SU. Let P e i and P o i be the input and output power of the DC/DC converter (Fig. B.1). By neglegting the power losses in the converter, we get:

we can write:

1 The details explanations of the SoC estimation techniques is out of the scope of this thesis [START_REF] Piller | Methods for state-of-charge determination and their applications[END_REF].

where V dc i is the voltage at the input side of the DC/DC converter, yields:

Replacing (B.5) in (B.2) gives:

Generally, the output voltage of a battery (especially Lithium batteries) is almost constant [START_REF] Lu | Double-quadrant state-of-charge-based droop control method for distributed energy storage systems in autonomous dc microgrids[END_REF]. Moreover, since the resistances of the power lines are very low, the deviation between the voltages at the points of common coupling is very small as well (∆V = r l I l ). Hence the output voltage of the interfacing converter can also be considered as constant. Thus, without loss of generality (B.6) becomes:

where Abstract: In recent years, the power grid has undergone a rapid transformation with the massive penetration of renewable and distributed generation units. The concept of microgrids is a key element of this energy transition. Microgrids are made up of a set of several distributed generation units (DGUs), storage units (SUs) and loads interconnected by power lines. A microgrid can be installed in several locations, for example in houses, hospitals, a neighborhood or village, etc. and operates either in connected mode to the main grid or in isolated (autonomous) mode.

Microgrids are facing several challenges related to stability assurance, cyber-security, energy cost optimization, energy management, power quality, etc. In this work, we focus our attention on the control of islanded direct current microgrids. The main contribution is the design of a new distributed control approach to provably achieve current sharing, average voltage regulation and state-of-charge balancing simultaneously with global exponential convergence. The main tools are consensus in multi-agent systems, passivity, Lyapunov stability, linear matrix inequalities, etc.

The thesis is divided into three parts. The First part presents the concept of microgrids, a literature review of their control strategies and the mathematical preliminaries required throughout the manuscript.

The second part deals with the design of the proposed distributed control approach to achieve the considered objectives. The system is augmented with three distributed consensus-like integral actions. The existence of equilibrium points for the augmented system is proved and a distributed-based static state feedback control architecture is proposed. Starting from the assumption that the agents (DGUs or SUs) have the same physical parameters, we provide proof of global exponential convergence of the system state to the set of equilibria where the control objectives are achieved.

Moreover, the proposed control approach is distributed, i.e., each agent exchange relative information with only its neighbors through sparse communication networks. A leader-follower approach is used to ensure the convergence of all the state-of-charge to a common reference available only for one SU which acts as a leader. This gives the possibility for high-level monitoring of the state-of-charge through an energy management strategy. The proposed controllers do not need any information about the parameters of the power lines neither the topology of the microgrid. The control objectives are reached despite the unknown load variation.

In the third part, the proposed distributed controllers are assessed in different scenarios through Matlab/Simulink simulation and real-time Hardware-in-the-Loop implementation tests. The results show that the control objectives are successfully achieved, illustrating the effectiveness of the proposed control methodology.