They worked all the life without spared efforts or sacrifice to put each day a new stone in our home thinking always in other following stone. This goal is dedicated to my parents, José and Doris.

She is today the coauthor of my achievements and weaknesses, and in this case, it is her love, patience and constancy, the pillars of this building. This goal is dedicated to my wife Viviam.

The highest cost I paid for this trip has been to be far away from his hugs, his cries, his voice and his life. This goal is especially dedicated to my son Nicolás.

I wish to express my immense gratitude and my deepest admiration to professors Germain Garcia and Luis Martinez Salamero whose are the promoters of this project. It is really nostalgic for me remembering all the moments shared with them which have meant that this achievement is a reality today. Among the things that I have learned, beyond of the knowledge, there are many lessons of these two admirable people that have inspired me to be better. I want to offer my acknowledgements to the institutions in Colombia which have participated to the funding of this project through the program of doctoral formation "Metas 2019": ASCUN (Asociación Colombiana de Facultades de Ingeniería), COLFUTURO (Fundación para el future de Colombia), Embassy of France in Colombia, Ministerio de Educación de Colombia and Universidad de Ibagué. I wish to express my gratefulness to the Universidad de Ibagué, particularly to William Alexander Londoño Marín, Gloria Piedad Barreto Bonilla, Luis Alfonso Muñoz Hernández and Helga Patricia Bermeo Andrade, who believed in my abilities and appreciated my work promoting the initiative that has brought me here. I am grateful for the constant support, the aid and the encouragement words to realize this thesis. I want to express my gratitude to the Laboratoire d'Analyse et d'Architecture des Systèmes (LAAS-CNRS), the Université de Toulouse, the Institute National des Sciences Appliquées (INSA), the Ecole Doctorale de Systèmes (EDSYS) and to the Equipe MAC, for the place and opportunity given to me in France, the support to realize this dissertation and the divulgation of the associated results, and especially by integrate me as an student member of the group.

I wish to express my appreciation to the Group GAEI of the Universitat Rovira i

Virgili for open the doors receiving me in their laboratory, giving me the space and support to obtain experimental results, contributing with ideas, discussions, experience and critics during the development of this thesis. I express my thanks to the Group ISGE in LAAS, especially to professor Corine Alonso, Youseff ElBasri and Lionel Seguier, to support with patience my presence in the laboratory during the experimental tests and providing me all the required equipment, aid and support.

I offer my thanks to the students of the "research hotbed SICEP" of the Electronics Engineering Program in the Universidad de Ibagué, specially to Tomás E. Sierra, David F. González, Juan C. Ávila, David F. Zambrano and Yamel A. Moreno. I wish to thank to my colleague and friend Tomás Urrego to offer to me his collaboration to make more easy and familiar my stay in Toulouse, reviving the camaraderie and friendship of those times when the table was brimming of MOSFET corpses in the Universidad Distrital Francisco José de Caldas. I offer my thanks to my friends in LAAS, especially to Carlos Quintero, Josu Doncel, Maialen Larranaga, Carlos Bartual, Aloña Berasategui, Isidre Mateu, Marti Bassas, and Youssef ElBasri, to share the coffee breaks, weekly dinners, birthdays and several conversations.

Abstract

These last years, a growing interest in power electronic systems has been motivated by the emergence of distributed renewable energy resources and their interconnection with the grid. In this context, the need of low power topologies fed by a few photovoltaic modules avoiding the use of transformers opens the study of special converters and the associated control strategies ensuring stability, reliability and high efficiency. A resulted generic device known in the commercial and scientific literature as "microinverter" or "module integrated converter" performs a plug and play product together with the PV module called an "AC module". This work is devoted to the study of a transformer-less single-phase double-stage gridconnected microinverter. The proposed topology has a non-isolated high-gain boost type DC-DC converter and a non-isolated buck type DC-AC converter connected in cascade through a DC bus. The DC-DC converter permanently extracts the maximum power of the PV module ensuring at the same time a good performance coping with power changes introduced by the change in the environmental conditions. The DC-AC stage injects the power extracted by the DC-DC stage into the grid ensuring a high level of power quality. The research efforts focus on the involved control functions based on the sliding mode control theory, which leads to a simple implementation with a comprehensive theoretical description validated through simulation and experimental results.

After giving the state-of-the-art in the first chapter, the manuscript is divided into four chapters, which are dedicated to the Maximum Power Point Tracking (MPPT), the DC-DC stage and its control, the DC-AC stage and its control and the complete microinverter. A new Extremum Seeking Control (ESC) MPPT algorithm is proposed.

The single-switch quadratic boost converter is studied operating as a Loss-Free-Resistor (LFR) obtaining a high DC output voltage level with a safe operation. The full-bridge converter is controlled as a Power Source Inverter (PSI) using a simple sliding-mode based tracking law, regulating the voltage of the DC bus and then ensuring a high power quality level in the grid connection. Finally, the three building blocks are merged to obtain a sliding mode controlled microinverter constituting the main result and contribution of the work.

Résumé

Ces dernières années, un intérêt croissant pour les systèmes électroniques de puissance a été motivé par l'émergence de sources d'énergie distribuées et renouvelables raccordées aux réseaux électriques. Dans ce contexte, la nécessité de topologies de faibles puissances alimentées par quelques modules photovoltaïques, en évitant l'utilisation de transformateurs, a ouvert l'étude de convertisseurs spéciaux et l'étude des stratégies de commande associées afin d'assurer la stabilité, la fiabilité et un rendement élevé du dispositif. Une possible solution est d'utiliser un dispositif générique connu dans la littérature scientifique et commerciale comme « micro-onduleur » ou «convertisseur intégré au module » qui avec le module photovoltaïque définit un produit « plug and play » appelé "module AC".

Ce travail est consacré à l'étude d'un micro-onduleur monophasé avec deux étapes sans transformateur raccordée au réseau. La topologie proposée est composé d'un convertisseur DC-DC non isolé élévateur avec un gain quadratique et un onduleur réducteur lié au réseau connectés en cascade. Le convertisseur DC-DC extrait en permanence la puissance maximale du module photovoltaïque malgré les changements dans les conditions environnementales. L'étape DC-AC injecte la puissance extraite par l'étape DC-DC dans le réseau et assure un niveau élevé de qualité de l'énergie. Les efforts de recherche de ce travail sont concentrés sur la mise au point de commandes utilisant comment base, la théorie de contrôle par mode de glissement, qui conduit à une mise en oeuvre simple avec une description théorique complète validée á partir de simulations et expérimentations.

Après avoir décrit l'état de l'art dans le premier chapitre, le manuscrit est divisé en quatre chapitres, qui sont dédiés respectivement à l'algorithme de recherche du point de puissance maximale (MPPT), á l'étape de conversion DC-DC, á l'étape de conversion DC-AC et finalement au micro-onduleur complet. Un nouvel algorithme de recherche extrémal du point de puissance maximale est développé (SM-ESC). Pour la étape DC-DC, le convertisseur élévateur quadratique avec seulement un interrupteur contrôlé est étudié utilisant le concept de résistance sans perte par mode de glissement (de l'acronyme anglais : Sliding-Mode Loss-Free-Resistor -SM-LFR) afin d'obtenir un gain de tension élevé avec un fonctionnement sûr et compatible avec l'algorithme MPPT. Pour la étape DC-AC, le convertisseur de pont complet est contrôlé comme un onduleur de source de puissance (de l'acronyme anglais : Power Source Inverter -PSI) en utilisant une commande par mode de glissement qui poursuit une référence sinusoïdale de courant de sortie. Cette commande est complétée par une boucle de régulation de la tension du bus DC qui assure une haute qualité d'énergie injectée dans le réseau. Enfin, les trois étapes constitutives sont fusionnées pour obtenir un microonduleur complètement contrôlé par la technique de mode de glissement, ce qui constitue le principal résultat et contribution de cette thèse.

Résumé long

Ces dernières années, un intérêt croissant pour les systèmes de conversion d'énergie a été motivé en part pour l'émergence du concept de génération distribue, le quel consiste en l'intégration des sources renouvelables d'énergie avec le réseau électrique. D'entre eux, les générateurs photovoltaïques peuvent être composés d'une grande quantité de panneaux solaires et un seul convertisseur centralisé de haute puissance ou d'un arrangement de quelques panneaux solaires utilisant une chaine avec diffèrent convertisseurs combines. Une solution plus récente consiste d'utiliser un dispositif générique connu dans la littérature scientifique et commercial comme « microonduleur » ou «convertisseur intégré au module » qui rejoint avec le module photovoltaïque assemblent le produit appelé « module AC ». Ce dispositif est développé pour travailler avec faibles niveaux de puissance et également avec faibles niveaux de tension d'entrée. Par conséquence, il y a diffèrent défis liés avec le développement des micro-onduleurs qui justifient la recherche pour assurer la stabilité, la robustesse, la fiabilité, la qualité d'énergie, la performance dynamique et le rendement élevé. D'entre le plusieurs objectives possibles à proposer, c'est importante trouver des configurations du dispositif en évitant l'utilisation de transformateurs car elles peuvent réduire les couts et augmenter la densité de puissance sans augmenter la complexité. Ainsi c'est importante à définir un schéma de control pour fournir une performance adéquate de toutes les fonctionnes de contrôle sans s'introduire en solutionnes qui peuvent être inutilement complexes.

Pour ce que concerne l'électronique de puissance, cette thèse est consacrée à l'étude du micro-onduleur monophasé sans transformateur à deux étapes de conversion. La topologie proposée est composé par un convertisseur DC-DC non isolé avec un gain d'élévation quadratique et un convertisseur DC-AC réducteur, lesquels sont connectés en cascade. Le convertisseur DC-DC extrait en permanence la puissance maximale du panneau photovoltaïque pour générer le niveau de tension DC nécessaire pour alimenter l'onduleur. L'étape DC-AC injecte le pourcentage le plus élevé possible de la puissance délivrée par l'étape DC-DC dans le réseau. D'un point de vue de l'automatique et le commande appliquée, cette thèse est dédié à l'étude des fonctionnes de contrôle nécessaires pour l'opération des étapes constitutifs du micro-onduleur. Le contrôle par mode de glissement a été choisi comme le fondement théorique des travaux afin d'obtenir une mise en oeuvre simple, un fonctionnement robuste, et une justification mathématique claire. Chacun des fonctions de commande est étudiée de façon indépendante et postérieurement le micro-onduleur est analysé comme un système unique.

Le manuscrit est divisé en cinq chapitres, qui sont dédiés respectivement à l'introduction général de l'application, l'algorithme de recherche du point de puissance maximal, l'étape de conversion DC-DC et son commande, l'étape de conversion DC-AC et son commande et finalement au micro-onduleur complet. Toutes les hypothèses théoriques qui sont présentées dans cette thèse sont aussi validées utilisant des résultats de simulation et expérimentation. L'algorithme proposé utilise la commande par mode de glissement pour obtenir la valeur de la conductance que permettre piloter le système permanentent vers le point de puissance maximum indépendamment des changements d'irradiation solaire. Comme cela est illustré dans la figure 2, l'algorithme utilise deux fonctionnes non linéaires de deux niveaux (u and v) pour produire les quatre possibles structures du système lesquelles sont obtenu dans une séquence spécial et évoluent en fonction de l'erreur de puissance obtenu de la comparaison entre la mesure instantanée de la puissance du panneau (Pp) et le référence auto-généré pour l'algorithme (Pp ref ). L'algorithme assure au même temps une bonne performance dynamique pour répondre aux changements provoqués par les conditions environnementales.

Compte tenu que l'algorithme transforme le problème de recherche du point de puissance maximum en un problème d'optimisation et que la variable à manipuler est la conductance, une transformation de variables est appliquée pour obtenir la courbe de puissance du panneau solaire comme une fonction de la conductance. De cette manière, est illustre l'obtention de la valeur de la dérive maximum de cette fonction, laquelle est utilisé pour le réglage des paramètres de l'algorithme. L'étude théorique de l'algorithme a permis d'obtenir les relations entre les paramètres qui garantirent la stabilité de et la convergence permanent vers le point de puissance maximal. A la fin du chapitre, le fonctionnement de l'algorithme a été validé utilisant des résultats obtenus par simulation et résultats expérimentaux montrant sa robustesse.

Le troisième chapitre est dédié à l'étape de conversion DC-DC laquelle utilise un convertisseur élévateur quadratique avec seulement un commutateur contrôlé.

Initialement, le choix du convertisseur élévateur quadratique est justifié utilisant une introduction qui décrit les différentes topologies de convertisseurs utilises pour obtenir grandes gains de tension et explique les avantages du convertisseur proposée. Le circuit du convertisseur élévateur quadratique est montré dans la figure 3. A partir de cette représentation, le fonctionnement du convertisseur utilisant un modulateur de largueur de pulse, et dans autre cas, un comparateur d'hystérésis est distingué en raison des restrictions que son imposées pour chaque méthode de commutation.

Finalement, le convertisseur est commandé en mode de conduction continu, utilisant une loi de commande par mode glissante laquelle est implémenté utilisant un comparateur d'hystérésis.

Le commande proposé consiste en suivre une référence de courant laquelle est traite initialement comme une valeur constante mais qui va être imposé finalement pour la boucle de commande externe constitué pour l'algorithme « MPPT ». La stabilité du convertisseur avec ce loi de contrôle est démontré à travers d'une fonction de Lyapunov. Postérieurement, la surface de glissement est modifié de façon que la référence de courant est obtenu à partir du concept du LFR (en anglais: Loss Free Resistor) lequel calcule la référence de courant à partir de la valeur de la tension mesuré dans l'entrée du convertisseur et une valeur donnée de conductance. Cette problème de control est traite considérant uniquement une source de tension d'entrée et un charge de sortie composé par une résistance et une source de courant. La stabilité du convertisseur est montrée utilisant la méthode du contrôle équivalent pour obtenir une représentation linéaire du système opérant autour un point d'équilibre. Ensuite, le rendement du convertisseur est estimé considérant les particularités du convertisseur élévateur quadratique et la variation de fréquence introduit par l'implémentation utilisant un comparateur d'hystérésis. A la fin du chapitre, le fonctionnement de l'algorithme est validé utilisant des résultats obtenus par simulation et résultats expérimentaux qui montrent la validité des résultats théoriques.

Pour la étape DC-AC, dans le quatrième chapitre, l'onduleur de pont complet de caractéristique réductrice est traite considérant une source de puissance à l'entrée et le réseau électrique a la sortie comme charge. L'entrée du convertisseur est étudié comme une source de puissance considérant la caractéristique donnée à l'étape DC-DC à travers la loi de commande de LFR et pris en compte que la tension de entrée minimum pour l'onduleur doit être plus grande que l'amplitude de la tension de la réseau. La dénomination d'onduleur de source de puissance est utilisée pour surligner les particularités de ce mode d'opération de l'onduleur. Le diagramme du circuit correspondant est montré dans la figure 6. L'onduleur est contrôlé utilisant la commande par mode de glissement pour suivre une référence sinusoïdale dans le courant de sortie. La forme d'onde de la courant est obtenu à partir d'un system de synchronisation basée en un PLL (en anglais : Phase-Locked-Loop) lequel donne un signal sinusoïdale d'amplitude normalisé synchronisé en fréquence et phase avec le réseau. L'amplitude de la référence est donnée pour une boucle externe de régulation de la tension du bus CC qui maintient la balance de puissance dans la chaine de conversion. Au même temps, la boucle de régulation assure une haute qualité d'énergie dans le raccordement au réseau évitant l'influence de l'ondulation naturelle de la tension du bus CC dans la référence de courant. Le schéma fonctionnel du control de l'onduleur est montré dans la figure 7. Ainsi, un modèle linéaire pour le convertisseur est obtenu considérant la valeur de la conductance, l'amplitude de la source de courant qui représente le panneau solaire et une source de courant qui représente l'onduleur comme une charge connectée à la sortie du convertisseur. The distributed generation (DG) of energy can be associated with a decentralized way to process electrical power from energy resources such as photovoltaic (PV), wind, fuelcells, geothermal, hydroelectric, waste-to-energy turbines and others [1]- [5]. These generators allow covering the user's consumption leading to economic savings and slowly replacing the centralized sources that threaten the environmental integrity of the planet. The DG is intimately related with the micro-grid [6]- [7] and the smart-grid concepts [8]- [9] present in many of the current developments of the energy systems.

Among the available alternative energy resources involved in a distributed generation, the photovoltaic solar energy (PSE) together with hydro and wind power have an important role , they motivating an intensive research work in recent years [10]- [14].

The photovoltaic generators can be introduced in many applications depending on particular situations which induce associated constraints. For example, a photovoltaic generator used in a place where other energy sources are absent must have the ability to provide the required power with the specific voltage characteristics such as frequency and amplitude. In other cases, the photovoltaic generator must work in a cooperative way with other generators connected to a grid system providing only a portion of the required power. It is then possible to classify the photovoltaic generators in "standalone" or "grid-connected" (GC) systems. The "stand-alone generators" are used in isolated applications and are in general complemented by a storage system like battery arrays. Thus, these systems can feed a load while a portion of the energy is accumulated in batteries in order to guarantee an autonomy level during the absence of solar radiation [15]- [16]. The "GC generators" are used in interconnected applications which work with a power grid. This type of generators can produce and inject true power into the grid reducing the consumption from the main power source [17]- [20]. Commercially, these devices are also presented as self-consumption generators. A third possibility appears when the properties of both systems are combined. This type of "hybrid" or "universal" generator must have the capability to permanently inject the power into the grid and also to operate eventually in an isolated mode [21]- [23]. It goes without saying that the control of this last type of generators is more complex because of the need of islanding control and power flow configuration algorithms [24]- [27]. This dissertation focuses on a kind of "GC generators" and hence, the literature review is concentrated on this family of power converters.

CLASSIFICATION OF THE PV GC GENERATORS

Some different ways can be used to classify the PV GC generators, in particular, taking into account the advantages, drawbacks, application possibilities and limitations.

A detailed classification, presented below, is used to introduce the main motivations of our research work.

According to the installation size

The size of the installation differentiates three kinds of PV GC generators [17].

Commonly, in this classification, it is possible to give to a generator the attribute of "oriented to" complemented with a description of the installation size using the words "central", "array" or "module". Then, in the first group, one finds the "central inverters"

or "centralized inverters" working with many photovoltaic modules connected in seriesparallel arrays in order to have high levels of DC voltage and hundreds of kilowatts or even of megawatts. This type of installation uses a centralized inverter which can cover the whole system power requirement [28]- [30]. In a second group, we have the "array oriented generators" or "string oriented generators" working with series connected arrays of some photovoltaic modules covering only the requirement to have a high DC voltage level working with amounts of power in the order of dozens of kilowatts. This kind of installation can use two different configurations of power converters which have been denominated in the literature as "string inverters" [31]- [34] or "multi-string inverters" [35]- [37]. The multi-string inverters are a generalization of the string inverters which uses an output connection of DC-DC converters whose inputs are independently connected to a separate array of PV modules centralizing the power in only one inverter. Finally, in the third group, we have "module oriented converters" which work with the low voltage level of one or two PV modules generating powers of only a few hundreds of watts. This same concept has been developed using the denomination "module integrated converter (MIC)" [38]- [40], "microinverter" [41]- [44] or "AC module" [45]- [47], which consists on connecting a low power converter to each photovoltaic module. The figure 1.1 shows the different generator topologies classified according to the installation size.

According to the number of stages

The photovoltaic generators are in general a type of DC-AC converters. These converters can be composed by one, two or more stages. Then, we can have either a single-stage topology or a dual-stage topology or a multi-stage topology [3], [48]- [49].

The single stage topology has only a DC-AC stage which is directly connected to the PV module at the DC side and connected to the grid at the AC side [50]. The dual stage topology is performed using a DC-DC stage and a DC-AC stage connected in cascade obtaining a conversion chain of the type DC-DC-AC [51]. In this case, the DC-DC converter is connected to the PV module and the DC-AC converter is connected to the grid. The point of connection of both converters commonly defines a DC bus. An alternative way to constitute a dual stage topology can be obtained using a series output connection of DC-DC converters , each one being fed by a PV module [52]- [54]. Then, the DC voltage obtained as the sum of the output voltage of each converter feeds an inverter connected to the grid. Finally, a multi-stage topology can be obtained from the connection of multiple converters preserving a DC side at the input port and an AC side at the output port. For example, it is possible to have a three -stage topology in such a way that the conversion chain is defined as DC-AC-DC-AC. In this case, we have a DC-AC stage, an AC-DC stage and a DC-AC stage. The first stage (an inverter) is connected to the PV module at the input port and connected to the second stage (a rectifier) at the output port. The second stage is connected to the first stage at the input port and to the third one (an inverter) at the output port. The third stage is connected to the second stage at the input port and connected to the grid at the output port. Using this configuration it is possible to configure a resonant converter in the first stage allowing galvanic isolation through a high-frequency transformer feeding a high-frequency rectifier which in turn feeds the output inverter [3].

Some authors have discussed the topology which can offer the best performance.

However, the selection of a topology depends on both power and voltage levels. In this context, the dual stage topology (figure 1.1c) is advantageous in terms of control because the control functions can be distributed between both stages [2]- [4], [48].

According to the DC-link configuration

In [55], the MIC topologies are classified in either converters with DC link, converters with a pseudo DC link and converters without DC link. Converters with DC link produce a DC voltage level higher than the peak of the grid voltage, this ensuring the correct operation of a buck type inverter stage. Converters with a pseudo DC-link produce a rectified current from a first DC-DC stage to unfold subsequently this signal using a low frequency bridge inverter. Converters without DC link use a high frequency AC conversion involving transformers or coupled inductors using forced-commutated cycloconverters [56].

According to the isolation

The galvanic isolation of a grid-connected inverter can be determinant in high power applications. In fact, in many industrial applications, a low frequency transformer is used because it gives robustness and an inherent protection to the system. However, in low power applications the existence of this type of transformer is increasingly less acceptable because of the size and weight of the final solution. Therefore, if isolation is needed a converter topology with a high frequency transformer can be used obtaining a reduced size at the expense of some complexity. But, if isolation is not required, many topologies can be used and better performances can be attained. Then, we can differentiate either transformer-based [57]- [58] or transformer-less topologies [59]- [60].

The more typical configurations of transformer-based topologies employ the fly-back topology [61]- [62] or the interleaved fly-back topology in more recent works [63]- [64].

The transformer-less topologies use existing topologies of DC-DC converters or interconnection of them to look for a solution satisfying a DC-gain requirement [65]- [72]. Moreover, multilevel topologies have been proposed in order to improve the power quality of the microinverters, avoiding at the same time, the use of transformers [73]- [75].

In microinverter applications using a double-stage topology, a high-gain DC-DC converter is needed because of the low voltage levels at the input port. This is a real challenge when no transformer-based topology is used.

According to the type of the generated power

A power converter connected to the grid can generate either "active power", "reactive power" or a combination of them. In this context, there are power conditioners or generator-conditioners. The control of the first group consists in enforcing the inverter to inject all of the available active power into the grid preserving a unitary power factor with a reduced harmonic content. The control of the power conditioners is more complex because the system must compensate the displacement of power factor giving reactive power or compensating a portion of the harmonic content generated by a load increasing the power factor and reducing the harmonic distortion to admissible levels [76]- [77]. Moreover, the control of active power is also necessary in power conditioners because of the power losses of the converter. An attractive solution can be the implementation of a system with two modes, having a first mode to inject the maximum power produced by the PV module into the grid and a second mode to operate as a conditioner when the solar radiation is lower.

REVIEW OF COMMERCIAL MICROINVERTERS

Technical characteristics

The name "microinverter" is given commercially to inverters which work with low power levels (less than 500 W) [78]. However, a really important difference exists between this generic product and a photovoltaic microinverter because the photovoltaic application requires both the extraction of the maximum power from the DC side and a grid connection. The industrial development of this technology offers a wide availability of that device as a product. Some manufacturers have gained popularity because of the presence of their microinverters in the market. The main electrical characteristics of a few commercial microinverters are listed in table 1.1 and table 1.2. As it can be noted in table 1.1, only the microinverters of the manufacturers "Siemens" [91] and "AE Conversion" [93] work with the voltage levels corresponding to a 24 VDC PV module while the others work with voltage levels corresponding to a series connection of two or three PV modules. Further, efficiency levels are around 95% being the microinverter of the manufacturer "Emphase" [81] the device with the best performance. The MPPT efficiencies are reported always higher than 99%. The power factor is always higher than 0.95, but it is not clear if all manufacturers report the power factor or the displacement power factor. The best lower THD (2%) is offered by the manufacturer "Sparq" [87] besides the better MPPT efficiency.

As it is observed in table 1.2, the majority of the microinverter manufacturers offer warranties of more than 20 years. This fact implies that converter topologies do not use electrolytic capacitors since they suffer degradation in less than 10 years. The best trade-off power-weight is found for the microinverter of the manufacturer "Aurora" [79]. In general, it is possible to identify a trend to increase the lifetime of the microinverters in order to ensure warranty times around 25 years. It is possible to observe that the best maximum efficiency of 96.5% is offered by the manufacturers "Emphase" and "ABB" with very different power densities: 0.3 and 0.11 W/cm 3 respectively. However, only the price can be determinant because a low difference in size or weight is acceptable considering the dimension of the PV modules. The AC module offered by "SunSine" [88] does not exhibit much information because it considers the microinverter and the PV module as an integrated product.

It is possible to conclude that the microinverter is an interesting commercial device which leads to wide developments in future years in order to adapt these technologies to the specific conditions and electrical standards of many countries around the world. For instance, it is common to find microinverters covering the standards of 110 V at 60 Hz and consisting in an adaptation of a microinverter developed to work at 220 V with an autotransformer to obtain the required voltage levels. Naturally, this solution increases the price and reduces the performance. Moreover, in several microinverters, because the converter topology is not known, it is not possible to conclude on the reliability or the stability of them, so that they can be categorized as a black box technology. Hence, it is necessary to develop a technology providing the most appropriate answers to wellposed technical problems of control, stability, assessment and reliability.

Standards and regulations

Some standards and regulations related to microinverters are listed below:

-Electromagnetic Compatibility (EMC)-Part 3-2: Limits for Harmonic Current Emissions (Equipment Input Current Under 16 A per Phase) [94].

-IEEE 1547. Systems [START_REF] Cid-Pastor | Synthesis of loss-freeresistors based on sliding-mode control and its applications in power processing[END_REF].

-IEC 61727 (1995-06) Photovoltaic Systems -Characteristics of the Utility Interface [98].

-DS/EN 61000-3-2 (2001) EMC, Limits for harmonic emissions (equipment input current up to and including 16 A per phase) [99].

Challenges and opportunities

The microinverter concept appears as an interesting solution to many present challenges in PV systems. We can highlight that PV installations were not popular in the past for the reasons listed below:

-Short lifetime of the converters:

The need of high capacitances to reduce the voltage and current ripples in the PV modules forces the use of electrolytic capacitors which are very sensitive to the temperature and have a reduced lifetime in comparison with the PV module and other electronic components.

-Need of storage elements:

The grid-connected concept was only applied in large-size installations while little installations with a few PV modules were only used in isolated stand-alone applications requiring storage of energy. The use of batteries increases the cost of the system and reduces the reliability in general. Moreover, the supervision or maintenance increases the cost to sustain the operation of the system.

-Difficult reproducibility:

The need of different elements or devices to constitute a PV installation and their size leads to a difficult industrial production process because of the prohibitive assembly times and the required infrastructures.

-Problems with partial shadowing:

Depending on the size of the PV installation, the effect of the partial shadowing can have more or less impact on the general performance of the system. This is mainly due to the series connection of modules and the utilization of a MPPT algorithm per each series array.

-Limited scalability:

The cost of the PV system including PV modules and power converters increases with the number of PV modules in the system. Then, the initial investment for the realization of a PV system is important and requires a long amortization period (years or decades).

Considering the reduced life time of the converters, the system requires a replacement before the amortization of the initial investment. For all the previous reasons, many users considered other type of electric generators.

Taking into account all the points mentioned above, it is necessary to think in a low power solution in which the power converter can be integrated directly with the PV modules constituting a single device which permanently operates maximizing the power harvesting. Since each PV module has its own converter, the effect of partial shadowing is reduced. In addition, the resulting power device can be more easily assembled and reproduced leading to a flexible plug-and-play product. Users can built their own installations in non-used surfaces in their homes, using a step by step planning while enjoying this technology in the consumption but also contributing to the utilization of clean energies. In the future, roofs and windows can be used as areas for installing AC modules generating improvements in the developments of the photovoltaic cells. Figure 1.2 shows an example in which the microinverter concept is explained together with complementary products assembling a complete self-consumption solution. The current interest in microinverters has prompted some manufacturers of digital devices to develop specific applications in the microinverter field [99]- [101]. The competition in the development of digital devices generates intensive works which give prospectively a central role to the microinverter in the next generation of photovoltaic installation around the world, this satisfying all the conditions needed in the context of electrical grids [102].

RESEARCH TRENDS IN MICROINVERTERS

As it was introduced in the state-of-the-art of the grid-connected PV generators, a trend to work with low power levels and distributed generators can be easily identified.

Many research groups around the world consider the "microinverter" as an attractive subject in the fields of power electronics and control. The motivations of that research have been reported in [102]- [105] and are summarized in the following paragraphs.

Efficiency

Although the efficiency of some topologies have reached levels higher than 95% (see table 1.1), looking for improvements in the efficiency of other topologies can offer other advantages and change the current industrial preferences [START_REF] Sira-Ramirez | On the Generalized PI Sliding Mode Control of DC-to-DC Power Converters: A Tutorial[END_REF]. Because of the reduced efficiency of the commercial PV modules, any additional watt potentially exploitable is a significant contribution in the context of the microinverter-based distributed generation technology. However, it is worth noting that the reported efficiencies correspond to the peak efficiencies which are not the efficiencies in all operational points. Hence, the weighted efficiency (expression 1.1) is a more accurate measure of the global efficiency.

𝜂 𝑒𝑢 = 0.03𝜂 5% + 0.06𝜂 10% + 0.13𝜂 20% + 0.1𝜂 30% + 0.48𝜂 50% + 0.48𝜂 100% (1.1)

Semiconductor devices advances

The development of the technologies for power semiconductor devices has led to an increment of the power density in several power electronics applications. The use of SiC or GaN in fast recovery diodes and MOSFETs has increased the efficiency levels and reduced many undesired phenomena [START_REF] Sira-Ramirez | Control Design Techniques in Power Electronic Devices[END_REF]. Moreover, it has been possible to increase the break-down voltages of the semiconductor devices improving their potentialities.

Diagnosis and fault detection

The reliability is another key aspect in the development of the microinverter system. It can be improved from the design of more stable systems providing robustness through the control. However, the diagnosis and fault detection of the device become necessary because the fault probability increases as the number of connected microinverters increases [START_REF] Utkin | Sliding mode control in electromechanical systems[END_REF]. In the same direction as the anti-islanding algorithms which prevents erroneous operation of the system in the absence of the grid, the supervision of variables such as temperature, the injected DC current could be desirable to prevent faults in the system.

Life-time and reliability

The life-time of the microinverters is also a key factor in the development of the photovoltaic technology. In fact, one of the more relevant problems in photovoltaic systems in the last decades is the reduced life-time of the power converters in comparison with the PV modules [START_REF] Golnaraghi | Automatic Control Systems[END_REF]. The main problem is caused by the required large capacitance to decouple the PV modules, which can be only solved using electrolytic capacitors. In this context, the reduced life-time of these elements, mainly due to the temperature, voltage level and current ripple affect the microinverter. The advances in the improvement of capacitors less sensitive and more efficient allow replacement of the electrolytic capacitors by thin film capacitors [110]- [START_REF] Frattini | AC Module Design Employing Low Capacitance Values[END_REF], increasing the life-time of the photovoltaic inverters. It is apparent that this technology imposes to work with lower capacitances implying high voltages and low powers.

Leakage and ground currents

Another important task is related to the compatibility between the grid and the PV modules which depends on the isolation of the inverter. In transformer-less topologies, the leakage and ground currents are more important because there is no isolation element which can decouple them [112]- [START_REF] Lopez | Leakage current evaluation of a single phase transformer-less PV inverter connected to the grid[END_REF]. These currents are caused by oscillatory voltage components produced between the PV modules and the grid. They circulate through the parasitic capacitances introduced by the PV modules and the length of the interconnection cables. As it can be expected, this problem is proportionally accentuated with a high number of PV modules.

Communications

The intensive work in the smart-grid and micro-grid fields introduces some special constraints for involved devices. Because of the need of centralized supervision of power converters, power meters and protection systems, some interactivity is necessary and then the role of communications in these systems is central [START_REF] Choi | High-efficiency grid-connected photovoltaic module integrated converter system with high-speed communication interfaces for small-scale distribution power generation[END_REF]. They offer some opportunities of monitoring and supervision which have been extended for example to even smart phone applications (apps) [START_REF]REbus DC Microgrid, PV Link[END_REF] (see figure 1.3). The system modeling and control theories also play a fundamental role in power electronics. It is well known that the majority of the applications requires at least of a feed-back control system. Many efforts have been made in the last decades to improve control techniques applicable to power conversion. Controllers for high frequency DC-DC, DC-AC and AC-DC converters have been involved in the associated basic control problems. However, complex structures with more than one conversion stage or special functionalities require a special attention to solve the related control problems. In the case of the photovoltaic generators these problems remain as an important challenge [START_REF] Jiang | Grid-Connected Boost-Half-Bridge Photovoltaic Micro Inverter System Using Repetitive Current Control and Maximum Power Point Tracking[END_REF]- [START_REF] Ho | An Integrated Inverter With Maximum Power Tracking for Grid-Connected PV Systems[END_REF].

THE PROPOSED MICROINVERTER 1.5.1 General description

A possible way to contribute to the development of the low power DC-AC conversion for photovoltaic generation is to work on the development of a microinverter whose control problems are studied and solved by means of specific techniques. In our case, the selected technique is the sliding-mode control because of its inherent robustness and simple implementation [START_REF] Utkin | Sliding mode control in electromechanical systems[END_REF]- [START_REF] Sira-Ramirez | Sliding motions in bilinear switched networks[END_REF].

Proposed research and contributions

As mentioned above, the sliding-mode control technique is central in our work. First, we develop a new sliding-mode based MPPT algorithm that can be classified within the family of the sliding-mode Extremum Seeking Control MPPTs [START_REF] Levron | Maximum Power Point Tracking Employing Sliding Mode Control[END_REF]- [START_REF] Alqahtani | Self-optimization of photovoltaic system power generation based on sliding mode control[END_REF]. This algorithm does not employ the duty cycle as the manipulated variable but it uses instead the conductance to obtain impedance matching for maximum power transference. This constitutes a first contribution which in fact is not limited to our microinverter application.

The use of a high DC voltage (400 VDC) reduces the coupling capacitance allowing the use of thin film capacitors increasing the lifetime of our device. Then, considering the need to have a transformer-less converter topology with a high DC gain, we select the quadratic boost converter. This converter has been recently proposed in the literature and hence some important aspects such as discontinuous conduction modes, efficiency, dynamic performance together with stability and control have still to be investigated in detail [START_REF] Leyva-Ramos | Switching regulator using a quadratic boost converter for wide DC conversion ratios[END_REF]- [START_REF] Lopez-Santos | Efficiency analysis of a sliding-mode controlled quadratic boost converter[END_REF]. The need of a simple control law orients our choice to a Sliding-Mode Loss-Free-Resistor (SM-LFR) design , this providing simultaneously the impedance matching with the PV module and the input voltage step-up to the required levels [START_REF] Singer | Realization of loss-free resistive elements[END_REF]- [START_REF] Cid-Pastor | Synthesis of loss-freeresistors based on sliding-mode control and its applications in power processing[END_REF].

The DC-AC conversion stage is focused on the traditional H-bridge as grid-connected inverter. Using the SM-LFR properties allows considering together the PV module and the quadratic boost converter as a power source. Therefore, by using this kind of source to feed the inverter, we extended the group of the grid-connected inverters adding the class of "Power Source Inverter (PSI)" to the well-known Voltage Source Inverter (VSI)

and Current Source Inverter (CSI). The sliding-mode control is applied to the inverter which tracks the current reference when both unipolar and bipolar commutation techniques are employed [137]- [START_REF] Kim | Sliding mode controller for the single-phase grid-connected photovoltaic system[END_REF]. A deep study is done for the bipolar commutation applying the harmonic balance technique to the ideal sliding dynamic of the converter.

Moreover, a linear controller is designed to regulate the voltage in the DC bus obtaining an appropriate equilibrium for the power [140]- [START_REF] Karimi-Ghartemani | A Systematic Approach to DC-Bus Control Design in Single-Phase Grid-Connected Renewable Converters[END_REF]. The influence of the controller in the power quality is studied in the controller design. A feed-forward compensation technique is also proposed as an alternative to maximize the power quality in the grid connection.

By assembling all the above described building-blocks, we show that the sliding-mode control theory is a good candidate for the control requirements of a grid connected photovoltaic applications [START_REF] Lopez-Santos | Sliding-mode control of a transformer-less dual-stage grid-connected photovoltaic micro-inverter[END_REF].

CONCLUSION

In this chapter, the state-of-the-art of module integrated converters, a review of commercial microinverters and AC modules and the main objectives of our work have been presented. The study of an entirely sliding-mode controlled single-phase transformer-less dual-stage grid-connected microinverter has been proposed by describing its main components, namely, a MPPT algorithm, a DC-DC converter and a DC-AC converter.

1.7

Chapter 2

MPPT Algorithm:

Sliding-Mode Extremum Seeking Control

INTRODUCTION

Among the different control functions that integrate the photovoltaic generators, the controller which defines the instantaneous power extracted from the PV modules can be considered as one of the most influential in the overall performance of the system.

Therefore, the proper operation of this controller consists in enforcing the system to always work at the maximum power. To solve this important problem, the controller must have the ability not only to find a maximum power point but also to permanently track the maximum power point when unpredictably changes appear in the power as a consequence of changes in the environmental variables (irradiance and temperature).

Hence, a solution of the problem can be obtained using tracking algorithms or optimization methods. In the control community, the problem has been largely investigated and is known as the "Maximum Seeking Control". In the context of energy conversion, the problem is the well-known "Maximum Power Point Tracking" (MPPT)

and it has been extensively studied in the literature [1]- [9]. Among the existing methods, some of them use the characteristic of the power source. Also, there can be either parametric-based or heuristic-based algorithms. The first group uses a parametrical model and works with direct or indirect measurements of the power source [10]- [12]. The second group is composed of methods that use measurements of the electrical variables in order to determine whether the system is at the maximum power point or not. Recently, an interesting MPPT integrating both parametric and heuristic methods has been proposed [13].

From a widespread and historically point of view, the heuristic MPPT algorithms can be classified as part of the Extremum Seeking Control (ESC) techniques which have been studied since the 1950s. A first group involves algorithms which apply an additive disturbance or introduce an oscillation in one or more variables of the system observing the evolution of some variables in order to determine the appropriate action of the control. The perturbation can be introduced in either the electric variables (current and voltage) or in the control variables such as frequency [14]- [16] or duty cycle [17]- [21].

Some of these methods are named Perturb and Observe (P&O) algorithms which are widely used in academic studies and commercial products because of its simplicity and low cost implementation [22]- [28]. However, despite of the volume of latest works in P&O algorithms, this research field still retains the attention of researchers who try to improve their performances [29]- [34]. A second group involves algorithms which apply slope regulation, because a necessary condition to reach the maximum of a function consists in having a null derivative. These methods also introduce a disturbance in the system to evaluate the derivative of the power for determining the action of the control.

In the context of photovoltaic applications with this philosophy, we can cite the Incremental Conductance (INC) [35]- [42] and the Incremental Resistance [43]- [44].

These methods are also the source of constant research focused on the improvement of their performances [45]- [47]. A third group considers the algorithms which define an error signal between the maximum or a fictitious maximum and the actual operational point in order to ensure a trajectory toward the real maximum of the function. These are named peak-holding, peak-searching or peak-detection algorithms and are widely involved in many engineering areas. This group includes the fuzzy-logic [48]- [52], the predictive [53] and the random search approaches [54]. Other methods such as Ripple

Correlation Control (RPC) can be considered as an ESC method [55], because they use a natural oscillatory component and the knowledge of its behavior around the optimum of the function to drive the system to a single desired operational point [56]- [59]. Many different proposals are named and grouped specifically under ESC [60]- [70].

Regarding the tracking of the maximum power point as an optimization problem, some methods use a sliding-mode strategy as the way to ensure that the maximum is reached. These MPPT techniques are known in the literature as the Sliding-Mode Extremum Seeking Control (SM-ESC) methods [71]- [75], and can be combined in some cases with a perturb and observe algorithm [76]. An interesting approach to solve the problem applying sliding-modes is presented as "one-dimensional self-optimization" by professor V. Utkin in [77] (published in 1981). Recently, an attempt to describe and analyze this algorithm working in a PV system has been presented in [78]. It has to be pointed out that this preliminary work, introducing the Utkin's algorithm into photovoltaics applications has captured our interest and motivated the work presented in this chapter. The main characteristics of the proposed method can be summarized as follows:

-The implementation is easy.

-The performances of the method can be justified theoretically.

-The knowledge of an upper bound of the panel characteristics derivative and the measurement of the extracted power are the only needed information.

-Four tuning parameters have to be selected. They must satisfy simple inequality constraints and, because their impacts on the performance of the method are clearly identified, they can be easily determined.

The MPPT algorithm must be compatible with the proposed control in the DC-DC stage (SM-LFR). In that sense, the current control loop requires a conductance reference which must be given by the MPPT algorithm. This constitutes a first difference with the other sliding-mode methods: the output variable of the MPPT is the conductance not the duty cycle. The first part of this chapter describes an asymptotic approximation of the I-V and P-V characteristics in order to extract a P-G characteristic where G represents the equivalent conductance of the DC-DC converter determining the limits of the slopes, and thus avoiding a complex modelling of the PV module Then, the new algorithm is studied in detail in terms of sliding-mode techniques, namely, attraction and transversal conditions of the sliding surface using the equivalent control method are developed. The dynamic responses of the algorithm produced by a sudden positive or negative change due to variations in irradiance are also studied.

Simulation and experimental results illustrate the potentialities of the new algorithm.

ASYMPTOTIC MODELING OF THE PV MODULE

The Current-Voltage (I-V) and Power-Voltage (P-V) characteristics of the solar panel are crucial in the MPPT problem [79]. Because of the complexity of the mathematical models which define these characteristics, some methods work with approximations in order to reduce the complexity of the controller synthesis [80]. In this paragraph, the objective is to asymptotically approximate the I-V and the P-V characteristics of a PV module in order to determine a P-G characteristic curve.

As shown in figure 2.1a, the I-V characteristic curve of a PV module can be approximated by two asymptotes. They are defined using the short circuit current (𝐼 𝑠𝑐 ), the open circuit voltage (𝑉 𝑜𝑐 ) and the voltage corresponding to the maximum power (𝑉 𝑚𝑝 ).

Using this approximation, the I-V characteristic is defined by:

𝑖 𝑝 (𝑣 𝑝 ) = { 𝐼 𝑠𝑐 0 < 𝑣 𝑝 < 𝑉 𝑚𝑝 -𝐺 0 (𝑣 𝑝 -𝑉 𝑜𝑐 ) 𝑉 𝑚𝑝 < 𝑣 𝑝 < 𝑉 𝑜𝑐 (2.1)
where 𝐺 0 = 𝐼 𝑠𝑐 /(𝑉 𝑜𝑐 -𝑉 𝑚𝑝 ). The P-V characteristic can be deduced multiplying the previous approximation by 𝑣 𝑝 . The two asymptotes are transformed into the curves depicted in figure 2.1b. In order to improve the approximation, a third asymptote corresponding to the maximum attainable power (𝑃 𝑚𝑝 ) is introduced (in red in figure 2).

An analytical expression for the derived approximation follows and is given by: In order to obtain the P-G characteristic, the variable G, which is defined as 𝐺 = 𝑖 𝑝 /𝑣 𝑝 is introduced in the previous approximation. Remarking that 𝑣 𝑝 2 = 𝑝 𝑝 /𝐺 and replacing in the previous approximation leads to: The maximum derivatives of the two asymptotes representing the curve 𝑝 𝑝 (𝐺) are evaluated for 𝐺 = 𝐺 1 and 𝐺 = 𝐺 2 , obtaining:

𝑝 𝑝 (𝑣 𝑝 ) = { 𝐼 𝑠𝑐 𝑣 𝑝 0 < 𝑣 𝑝 <
𝑝 𝑝 (𝐺) = { 𝐺 0 2 𝑉 𝑜𝑐 2 𝐺 (𝐺 + 𝐺 0 ) 2 0 < 𝐺 < 𝐺 1 𝑃 𝑚𝑝 𝐺 1 < 𝐺 < 𝐺 2 𝐼 𝑠𝑐 2 𝐺 𝐺 2 < 𝐺 < ∞ (2.
𝑑𝑃 𝑝 𝑑𝐺 (𝐺 1 ) = 𝐺 0 2 𝑉 𝑜𝑐 2 (𝐺 0 -𝐺 1 ) (𝐺 1 + 𝐺 0 ) 3 (2.3) 𝑑𝑃 𝑝 𝑑𝐺 (𝐺 2 ) = - 𝐼 𝑠𝑐 2 𝐺 2 2 (2.4)
where the maximum computed value using (2.3) and 2.4 is always above that the maximum real derivative.

Usually, in a MPPT algorithm, the power versus duty cycle characteristic is used.

Here, because the high-gain of the converter, the duty cycle could also be high (greater than 0.75). Then, the conditioning of the P-G characteristic is better than the one of power versus duty cycle. The main reason, as the previous analysis revealed, is that the high duty cycles are located between small values of conductance and the values of conductance around the maximum power 𝑃 𝑚𝑝 . Also, it is worth noting that this transformation allows an objective function which has a domain of conductance from zero to infinite solving the problem that is present in the P-V characteristic when the domain is defined between zero and the open circuit voltage. These facts explain why the P-G characteristic is preferred. In addition, the DC-DC converter will be controlled through its input conductance to extract the maximum power of a PV module.

A NEW SLIDING-MODE MPPT METHOD

Fundamentals of the method

The proposed method for tracking the MPPT is summarized in figure 2

Reachability conditions

First, the conditions of reachability to the sliding surface are analyzed. This analysis allows the determination of constraints on constants 𝑀, 𝐾 1 and 𝐾 2 guaranteeing that the sliding surface is always reachable. Consider the Lyapunov function:

𝑽(𝜀) = 1 2 𝜀 2 (2.13)
Reachability is ensured if 𝜀𝜀̇< 0. We have: The above conditions show that the algorithm attains the sliding surface at least asymptotically. In practice, it is necessary to attain the sliding-surface in a finite time.

𝜀𝜀̇= 𝜀𝑃 𝑝 [- 1 2
The appropriate stability concept in such a situation is the "finite time stability concept" whose definition can be found, for example, in [81]. Let us define:

𝐶 1 = [𝑀 -𝐾 2 + 𝐾 1 | 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥 ] 𝑃 𝑝 (0 + ) > 0 𝐶 2 = [𝐾 2 -

Trajectory towards the maximum power point

To analyze the evolution of 𝑃 𝑃𝑟𝑒𝑓 (𝑡), it has to be remarked first that nonlinearities 𝑢 and 𝑣 are practically implemented as shown in figure 2.5. As reported in sliding mode control approaches, a hysteresis band 𝛥 is introduced in the switching functions. Here only 𝑣 is modified accordingly while 𝑢 is maintained ideal. The resulting waveforms are presented in figure 2.6. The dynamical equations describing the motion of 𝑃 𝑝 𝑟𝑒𝑓 (𝑡) and 𝐺(𝑡) for each time interval can be easily calculated, recalling that:

𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝑡 = 𝐾 2 𝑃 𝑝 + 𝑀𝑃 𝑝 𝑣 (2.22) 𝑑𝐺 𝑑𝑡 = 𝐾 1 𝑃 𝑝 𝑢 (2.23)
They are summarized in the following table.

𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝑡 | (1,0) = 𝐾 2 𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (1,0) = 𝐾 2 𝐾 1 𝑑𝐺 𝑑𝑡 | (1,0) = 𝐾 1 𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝑡 | (1,-1) = [𝐾 2 -𝑀]𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (1,-1) = 𝐾 2 -𝑀 𝐾 1 𝑑𝐺 𝑑𝑡 | (1,-1) = 𝐾 1 𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝑡 | (-1,-1) = [𝐾 2 -𝑀]𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (-1,-1) = 𝑀 -𝐾 2 𝐾 1 𝑑𝐺 𝑑𝑡 | (-1,-1) = -𝐾 1 𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝑡 | (-1,0) = 𝐾 2 𝑃 𝑝 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (-1,0) = - 𝐾 2 𝐾 1 𝑑𝐺 𝑑𝑡 | (-1,0) = -𝐾 1 𝑃 𝑝
Remarking that:

𝜀 = 𝑃 𝑝 𝑟𝑒𝑓 -𝑃 𝑝 and 𝑑𝜀 𝑑𝑡 = 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝑡 - 𝑑𝑃 𝑝 𝑑𝑡 = [ 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 - 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑑𝐺 𝑑𝑡 (2.24)
The associated differential equations describing the motion of the error for each time interval can also be easily determined. They are listed in the following table.

𝑑𝑡 | (1,0) = [ 𝐾 2 𝐾 1 - 𝑑𝑃 𝑝 𝑑𝐺 ] 𝐾 1 𝑃 𝑝 = [𝐾 2 -𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝 𝑑𝜀 𝑑𝑡 | (1,-1) = [ 𝐾 2 -𝑀 𝐾 1 - 𝑑𝑃 𝑝 𝑑𝐺 ] 𝐾 1 𝑃 𝑝 = [𝐾 2 -𝑀 -𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝 𝑑𝜀 𝑑𝑡 | (-1,-1) = [ 𝑀 -𝐾 2 𝐾 1 - 𝑑𝑃 𝑝 𝑑𝐺 ] (-𝐾 1 𝑃 𝑝 ) = [𝐾 2 -𝑀 + 𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝 𝑑𝜀 𝑑𝑡 | (-1,0) = [- 𝐾 2 𝐾 1 - 𝑑𝑃 𝑝 𝑑𝐺 ] (-𝐾 1 𝑃 𝑝 ) = [𝐾 2 + 𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝
From the conditions of reachability reproduced below

𝐾 2 > | 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥 𝐾 1 and 𝑀 > 𝐾 2 + | 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥 𝐾 1 (2.25)
It can be derived

𝑑𝜀 𝑑𝑡 | (1,0) > 0, 𝑑𝜀 𝑑𝑡 | (1,-1) < 0, 𝑑𝜀 𝑑𝑡 | (-1,-1) < 0, 𝑑𝜀 𝑑𝑡 | (-1,0) > 0 (2.26)
This is in accordance with the waveform of figure 2.6. In practical applications, Δ is usually small and then, 𝑃 𝑝 and

𝑑𝑃 𝑝 𝑑𝐺
do not significantly vary inside intervals of lengths Δ.

Consequently, we can consider that they are practically constant in such intervals.

Taking into account the equations describing the motion of , we deduce that:

𝑡 1 -𝑡 0 ≅ ∆ [𝐾 2 -𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝 , 𝑡 2 -𝑡 1 ≅ ∆ [𝑀 -𝐾 2 + 𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝 (2.27) 𝑡 3 -𝑡 2 ≅ ∆ [𝑀 -𝐾 2 -𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝 , 𝑡 4 -𝑡 3 ≅ ∆ [𝐾 2 + 𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] 𝑃 𝑝
Having in mind, the differential equations describing the error motion (see table 2.2), the variation of 𝑃 𝑃𝑟𝑒𝑓 can be easily evaluated in the interval [𝑡 0 , 𝑡 4 ] expressed as:

𝑃 𝑃𝑟𝑒𝑓 (𝑡 4 ) ≅ 𝑃 𝑃𝑟𝑒𝑓 (𝑡 0 ) + 𝐾 2 𝑃 𝑝 (𝑡 1 -𝑡 0 ) + (𝐾 2 -𝑀)𝑃 𝑝 (𝑡 2 -𝑡 1 ) + (𝐾 2 -𝑀)𝑃 𝑝 (𝑡 3 -𝑡 2 ) + 𝐾 2 𝑃 𝑝 (𝑡 4 -𝑡 3 ) = 𝑃 𝑃𝑟𝑒𝑓 (𝑡 0 ) + 𝐾 2 𝑃 𝑝 (𝑡 4 -𝑡 0 ) -𝑀𝑃 𝑝 (𝑡 3 -𝑡 1 )
(2.28)

After some simple algebraic calculations, we obtain:

𝐾 2 𝑃 𝑝 (𝑡 4 -𝑡 0 ) -𝑀𝑃 𝑝 (𝑡 3 -𝑡 1 ) = ∆ [ 2𝐾 2 2 𝐾 2 2 -𝐾 1 2 ( 𝑑𝑃 𝑝 𝑑𝐺 ) 2 - 2(𝑀 -𝐾 2 ) 2 (𝑀 -𝐾 2 ) 2 -𝐾 1 2 ( 𝑑𝑃 𝑝 𝑑𝐺 ) 2 ] = ∆[𝑓(𝐾 2 ) -𝑓(𝑀 -𝐾 2 )] (2.29)
where 𝑓(𝑥) is defined by:

𝑓(𝑥) = 2𝑥 2 𝑥 2 -𝑎 2 𝑤𝑖𝑡ℎ 𝑎 = 𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 (2.30)
The derivative of 𝑓(𝑥)is given by: can be evaluated from the panel power characteristic curve. To summarize, the expected properties of the algorithm will be guaranteed if the parameters 𝑀, 𝐾 1 𝑎𝑛𝑑 𝐾 2 are selected in order to satisfy the following constraints:

𝑑𝑓(𝑥) 𝑑𝑥 = 4𝑥(𝑥 2 -𝑎 2 ) -2𝑥 2 (2𝑥) (𝑥 2 -𝑎 2 ) 2 = -4𝑎 2 𝑥 (𝑥 2 -𝑎 2 ) 2 < 0 𝑖𝑓 𝑥 > 0 𝑎𝑛𝑑 𝑥 ≠ 𝑎 (2.31) If 𝑓(𝐾 2 ) -𝑓(𝑀 -𝐾 2 ) > 0 ,
𝑀 > 2𝐾 2 𝑎𝑛𝑑 𝐾 2 > | 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥 𝐾 1 (2.32)
From the above analysis, 𝑃 𝑃𝑟𝑒𝑓 (𝑡) increases until it reaches 𝑃 𝑚𝑝 and remains in its vicinity where it imposes the condition 𝑃 𝑝 ≅ 𝑃 𝑚𝑝 . Then:

𝑑 𝑃 𝑃𝑟𝑒𝑓 𝑑𝑡 ≅ 𝑑 𝑃 𝑝 𝑑𝑡 ≅ 𝑑𝐺 𝑑𝑡 ≅ 0 (2.33)
Invoking now the expression of the derivative of the error on the sliding surface, we can deduce the equivalent controls defined by:

𝑣 𝑒𝑞 = - 𝐾 2 𝑀
> -1 and 𝑢 𝑒𝑞 = 0 (2.34)

Influence of parameters 𝚫, 𝑴, 𝑲 𝟏 𝐚𝐧𝐝 𝐊 𝟐

This paragraph discusses the performance of the proposed algorithm evaluating the influence of the parameters Δ, 𝑀, 𝐾 1 and K 2 . The period 𝑇 𝑝 of the oscillation around the power 𝑃 𝑝 is equal to 𝑡 4 -𝑡 0 . Then it can be expressed as:

𝑇 𝑝 = 2𝐾 2 ∆/𝑃 𝑝 𝐾 2 2 -𝐾 1 2 ( 𝑑𝑃 𝑝 𝑑𝐺 ) 2 + 2(𝑀 -𝐾 2 )∆/𝑃 𝑝 (𝑀 -𝐾 2 ) 2 -𝐾 1 2 ( 𝑑𝑃 𝑝 𝑑𝐺 ) 2 (2.35)
When the algorithm attains 𝑃 𝑚𝑝 , we have

𝑑𝑃 𝑝 𝑑𝐺 ≅ 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 ≅ 0.
We can also remark that in the interval [𝑡 0 , 𝑡 4 ], we have:

𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (1,0) + 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (1,-1) + 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (-1,-1) + 𝑑𝑃 𝑝 𝑟𝑒𝑓 𝑑𝐺 | (-1,0) = 𝐾 2 𝐾 1 + 𝐾 2 -𝑀 𝐾 1 + 𝑀 -𝐾 2 𝐾 1 - 𝐾 2 𝐾 1 = 0 (2.36)
Then, when the algorithm attains the maximum, we have:

𝑇 𝑝 = 𝑇 𝑚𝑝 = 2∆𝑀 𝐾 2 (𝑀 -𝐾 2 )𝑃 𝑚𝑝 (2.37)
The associated frequency is given by:

𝑓 𝑚𝑝 = 𝐾 2 (𝑀 -𝐾 2 )𝑃 𝑚𝑝 2∆𝑀 (2.38)
where ∆ and 𝐾 2 are fixed, considering the constraints As shown in figure 2.7, the amplitude of the oscillation in the power reference is given by:

2𝐾 2 < 𝑀 < ∞, 𝑓 𝑚𝑝 ∈ ] 𝐾 2 𝑃 𝑚𝑝
∆𝑃 𝑃𝑟𝑒𝑓 = 𝑃 𝑃𝑟𝑒𝑓 (𝑡 1 ) -𝑃 𝑃𝑟𝑒𝑓 (𝑡 3 ) = -(𝐾 2 -𝑀)𝑃 𝑝 (𝑡 2 -𝑡 1 ) -(𝐾 2 -𝑀)𝑃 𝑝 (𝑡 3 -𝑡 2 ) = (𝑀 -𝐾 2 )𝑃 𝑝 (𝑡 3 -𝑡 1 ) = 2(𝑀 -𝐾 2 ) 2 ∆ (𝑀 -𝐾 2 ) 2 -𝐾 1 2 ( 𝑑𝑃 𝑝 𝑑𝐺 ) 2 (2.39)
At the maximum, we recover that 𝑑𝑃 𝑝 𝑑𝐺 ≅ 0 and as expected ∆𝑃 = 2∆. The parameter ∆ is the only parameter involved in the expression of power fluctuations amplitude. Now, the conductance is constrained to lie in a band of width ∆𝐺 = 𝐺(𝑡 2 ) -𝐺(𝑡 4 ) given by:

∆𝐺 = 𝐾 1 ∆𝑀 [𝑀 -𝐾 2 -𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] [𝐾 2 +𝐾 1 𝑑𝑃 𝑝 𝑑𝐺 ] (2.40)
At the maximum point, because 𝑑𝑃 𝑝 𝑑𝐺 ≅ 0, we have:

∆𝐺 = 𝐾 1 ∆𝑀 (𝑀 -𝐾 2 )𝐾 2 (2.41) considering the constraints 2𝐾 2 < 𝑀 < ∞, ∆𝐺 ∈ ] 𝐾 1 ∆ 𝐾 2 , 2 𝐾 1 ∆ 𝐾 2 [.
We can conclude that the value of ∆ controls the amplitude of the power fluctuations around 𝑃 𝑚𝑝 and that we have three parameters, namely, 𝑀, 𝐾 1 and K 2 , for controlling the frequency and the conductance fluctuations around the maximum. These constants have to satisfy some constraints recalled here

𝑀 > 2𝐾 2 𝑎𝑛𝑑 𝐾 2 > | 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥 𝐾 1 (2.42)
Analyzing the last inequality, we can remark that the smaller 𝐾 1 is , more insensitive the algorithm is to

| 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥
, but the weaker the action on 𝐺(𝑡) is. This action has to be sufficient to guarantee a good behavior of the algorithm.

Behavior working with irradiance variations

When irradiance is constant, the behavior of the power of a PV module is represented by a single curve corresponding to the characteristics of the PV module, which is described here as a function of the conductance (P-G). When a sudden change in irradiance occurs, we can consider two cases:

-Case a: it corresponds to an increasing of irradiance. Before the change, 𝑃 𝑃𝑟𝑒𝑓 (𝑡) is at the admissible maximum power, for example 𝑃 𝑚𝑝2 (see figure 2 In that case, we have 𝜀(𝑡 𝑚𝑝2 + ) = 𝑃 𝑝 𝑟𝑒𝑓 (𝑡 𝑚𝑝2 + ) -𝑃 𝑝 (𝑡 𝑚𝑝2 + ) < 0. For 𝑡 > 𝑡 𝑚𝑝2 + 𝛿, 𝛿 being positive, we have: Two cases must be considered on the new power characteristic curve:

𝑑𝑃 𝑃𝑟𝑒𝑓 (𝑡) 𝑑𝑡 = 𝐾 2 𝑃 𝑝 (𝑡) > 0 and 𝑑𝐺(𝑡) 𝑑𝑡 = -𝐾 1 𝑃 𝑝 (𝑡) < 0 (2.43)
 If 𝑑𝑃 𝑃 (𝑡 𝑚𝑝2 + )
𝑑𝐺 >0, 𝑃 𝑃𝑟𝑒𝑓 (𝑡) is increasing and 𝑃 𝑝 (𝑡) is decreasing until they will be equal. From the analysis developed in paragraphs 2.

ASSESSMENT OF THE ALGORITHM

In order to show how the algorithm practically works and to illustrate the theory presented in the previous paragraphs, we consider three characteristics defined in the plane f-u as follow:

𝑓 1 = 800 -20(𝑢 -2) 2 𝑓 2 = 200 -10(𝑢 -6) 2 𝑓 3 = 600 -25(𝑢 -3) 2
The derivatives for each characteristic are given by:

𝑓 1 ′ = -40(𝑢 -2) 𝑓 2 ′ = -20(𝑢 -6) 𝑓 3 ′ = -50(𝑢 -3)
The maximal value of derivatives can be determined from the above expressions. To The behavior of the variable u shows the smooth convergence to the maximum points. A zoom on the trajectory when the algorithm is growing up to the first maximum is represented in Figure 2.13. The reference is behaving exactly as the theory predicts. The original characteristic is represented in black and the reference is moving around it. To verify the analysis of paragraph 2.3.5, a zoom of the trajectories of 𝑓 𝑟𝑒𝑓 and 𝑢 at maximum 800 is represented below. With the selected constants, we have:

𝑇 𝑚𝑝 = 2∆𝑀 𝐾 2 (𝑀 -𝐾 2 )𝑃 𝑚𝑝 = 1 480 = 2.1 × 10 -3 𝑠 ∆𝐺 = 𝐾 1 ∆𝑀 (𝑀 -𝐾 2 )𝐾 2 = 1 12 = 8.3 × 10 -2 𝑆
The above theoretical values coincide exactly with the ones extracted from the curves of figure 2.14, this validating the developed theory.

SIMULATION RESULTS

In order to verify the theoretical prediction about the behavior of the MPPT algorithm, a set of simulation has been implemented in PSIM using the model of the panel available in that software [82].

Steady-state of the PV module variables

The figure 2.17 shows the steady-state waveforms when an average power of 19.36 W is supplied by the PV module. The operation at the maximum power point is confirmed because of the double frequency of the power function. The efficiency of the MPPT is computed as 99.8% at this condition. The oscillatory components of the PV module variables have amplitudes of 100 mA (3.4%), 170 mV (4%) and 0.062 W (0.5%).

Steady-state and motion of the MPPT variables

Start-up of the algorithm

A simulation of the start-up of the algorithm has been developed in order to observe the transient behavior from the starting point (zero current) to an equilibrium point corresponding to the maximum power. As observed in figure 2.23, the steady-state is reached in around 300 ms. Note that the power quickly arrives to values in the neighborhood of the maximum power point ensuring an appropriate behavior for the algorithm.

It is important to note that during the start-up, both power reference and measured power show an important proximity, induced by the sliding-mode control. Although it is not possible to identify the same hysteretic behavior for the maximum power point, the power of the panel remains oscillating around the power reference while the maximum power point is reached.

Sliding motion of the MPPT algorithm

A series of positive and negative step-type stimuli has been defined to give the irradiance to the Solar Module in PSIM. Each step has introduced an irradiance jump of 400 W/m 2 starting from an irradiance of 1000 W/m 2 and returning to the same value after two negative and to positive changes. Each irradiance value remains during 2 s.

Figure 2.24 shows the power signals obtained from the above test where it is possible to observe that the power of the panel takes values near the average value of the power reference. Moreover, it is possible to observe that the efficiency of the algorithm with high powers is higher because of the reduction of the amplitude of the oscillatory component. The system has a stable behavior and a smooth transition in the changes of equilibrium points. Figure 2.24 also shows a zoom of both increasing and decreasing transitions to give an approach of the waveform during the transient response. As it can be noted in both tests, the algorithm quickly reaches the maximum power point, also exhibiting a smooth transient response when step disturbances are applied to the power. Further, it is worth to note that the system shows a similar behavior and in general, a good performance because the maximum power is almost reached instantaneously after a power disturbance. Then, although the power reference increases or decreases slower than the power of the module, it moves to the sliding surface and clearly exhibits a sliding motion at the maximum power points.

All of simulated results confirm the theoretical predictions and the expected efficiency of the proposed MPPT method. Now, the last important step in this chapter consists in demonstrating that this method can be physically implemented using simple electronics circuits and that it works in a real application.

EXPERIMENTAL RESULTS

In this paragraph, we report the experimental measurements validating the theoretical predictions and simulation results.

Prototype and experimental set-up

The MPPT control has been implemented using a microcontroller dsPIC30F4011 and some analogic electronic circuits. Both, current and voltage measurements have been conditioned with second order filters which have a cut-off frequency of 100 Hz and a damping factor of 0. Two solar panels BP 585 from BP SOLAR have been used in the experimental test.

The characteristics and parameters are listed in figure 2.29. MPPT efficiency is obtained. The oscillating components of the PV module variables have amplitudes of 200 mA (5%), 1.6 V (6.6%) and 2.6 W (2.8%) respectively.

Steady-state of the PV module variables

Steady-state and motion of the MPPT variables

As shown in figure 2.31, interesting experimental results can be obtained by a repetitive start-up of the converter without changes in the PV module characteristics.

Because the maximum point is the same, the curve in the plane P-G shows the trajectory of the system towards the maximum power point. As expected the same limit cycle is obtained for the power reference when the system attains the maximum. It is possible to observe that the power of the panel is constrained more or less in the middle point between the maximum and minimum power reference. Further, it is possible to observe in the trace of the extracted power (𝑝 𝑝 ) the behavior representing the motion around the maximum power point.

Detail of the sliding motion towards the maximum

When the parameter 𝐾 2 is reduced close to the limit of stability, a quick variation is induced in the power reference. This condition increases the frequency of the reference and slows the motion towards the maximum as illustrated in figure 2.33. The corresponding behavior described in the plane P-G is shown in figure 2.34, where it can be observed that the motion of the power reference is constrained between two envelopes. It is also possible to observe the motion of the power inside the envelopes being very close to the average value of the power reference. Making an important zoom in both variables (P and G) at the steady-state, the result in figure 2.35 is obtained. In this case, the motion of the power reference induces a very small variation of the real power (𝑝 𝑝 ) ensuring the maximum power point with reduced oscillations increasing the performance of the algorithm. However, as shown in figure 2.33, this improved performance can be obtained with a slower transient response.

Start-up of the algorithm

The start-up of the algorithm has been captured in order to evaluate the transient behavior from the open circuit condition to the maximum power point. As it can be observed in figure 2.36, the steady-state is reached in 300 ms approximately. Although the algorithm is able to have a faster start-up, the presence of the filters conditioning the current and voltage has an impact on the transient response of the measured power, introducing a deviation between the real power and the measured power. However, these filters reduce the presence of noise and eliminate the high frequency in the measurement of the inductor current.

Sliding motion of the MPPT algorithm

A programed sequence of I-V characteristics has been introduced in the PV module emulator E4360A in order to have stepped increments and decrements with a holding time each one of 2 s. Figure 2.37 shows the power signals for the above test in the voltage levels of the control circuit. A power of 100 W corresponds to a voltage level of 1 V and a power level of 50 W corresponds to a voltage level of 500 mV. As it can be shown, the system shows a stable behavior and a smooth transition in the changes of equilibrium points. The power reference is constrained into the hysteresis band.

However, the hysteresis is defined for the power error and not for the power reference leading to an extracted power of the panel in a neighborhood of the maximum power with a small variation. Figure 2.37 also shows a zoom in transitions to clearly show the behavior of both power reference and the real power. An oscilloscope capture with the corresponding results is depicted in figure 2.39. As it can be observed in both tests, the algorithm has a significant rapid response.

Furthermore, it is possible to observe that the real power reaches the maximum value before that the power reference reaches the steady-state increasing the general performance and the harvesting of the power.

General approach of the functionality and performance

The general functionality of the algorithm has been experimentally evaluated using the overall microinverter system connected to the grid and fed by a solar panel. As it can be observed in figure 2.40, in this test the P-V characteristic curve of the panel has been captured using the curve tracer , and the more representative waveforms have been captured. The maximum power measured with the tracer instrument is 82.4 W while the maximum power measured by the oscilloscope is 82.24 W. The average value of the power measured by the oscilloscope is 81.82 W and hence the MPPT efficiency is around 99.45%. In the oscilloscope capture, it is possible to observe the power waveform which exhibits the expected double frequency phenomenon at the maximum power point. The power reference and the voltage of the DC bus (400 V) are also measured. For this operational condition, the frequency of the power reference is 20 Hz. These results are only presented to complete this chapter. An accurate description of the overall system will be presented in the chapter 5.

CONCLUSIONS

In this chapter a Sliding-Mode Extremum Seeking Control MPPT algorithm has been developed. A new proposal has been inspired from the self-optimized function based on sliding-mode proposed by Professor Vadim Utkin more than thirty years ago and revisited recently in the photovoltaic field [77]. Our algorithm shows several advantages to operate with high-gain converters because it does not work using the duty cycle as output signal since that approach introduces several limitations in the domain of the objective optimization function (P-D characteristic). By contrast, the conductance is used as output variable leading to a good compatibility with the proposed control law for the DC-DC conversion stage of the microinverter.

The basic idea is to generate a fictitious power reference compared with the instantaneous power measured at the output of the PV module. The resulted error defines the states of non-linear functions increasing or decreasing the output signal (conductance) and the internal reference (power reference). The dynamics of the algorithm correspond to a variable structure system with four structures since four motions are identified in the P-G plane. The attractivity of the sliding surface and also the trajectory towards the maximum point has been modeled, studied and validated.

Only four tuning parameters are required in the algorithm: 𝐾 1 , 𝐾 2 , 𝑀 and ∆, which only depends on the maximum derivative of the power with respect to the conductance. A particular way to evaluate the maximum derivative of a specific PV module has been also proposed using an asymptotic modelling of the PV module characteristics. As a consequence, the algorithm can be easily tuned ensuring a proper operation in a wide range of power.

The high performances of the algorithm have been verified using simulation and experimental results. The algorithm supports sudden changes on the irradiance and temperature levels demonstrating robustness and reliability. These results motivate the study of the functionalities of the algorithm in the context of other applications. The efficiency of the power extraction was evaluated around 99.6% which is comparable with the best reported performances in ESC MPPT algorithms which normally are above 99% [60]- [70]. Further, the trade-off between the performance and the robustness of the algorithm is very favorable in comparison with other methods that use derivatives or logical operator to find the maximum power point.

Chapter 3

DC-DC stage: Quadratic Boost Converter

INTRODUCTION

A microinverter or module integrated converter (AC module) generally uses a transformer-based DC-DC converter with boost capability in order to avoid a bulky transformer operating at the grid frequency. Several topologies derived of the fly-back, push-pull and the full-bridge converter can be found in scientific and industrial literature. The use of this kind of converters can be justified by the galvanic isolation they provide reducing the common-mode noise interference and the grounding problems

[1]- [12]. However, it is well know that the use of transformers in switching circuits introduces an important complexity beside an increment in the size, weight and cost of the circuits. Therefore, taking into account that the low power levels driven by a microinverter are below 250 W, a transformer-less topology can be preferred if a proper trade-off is achieved between efficiency and power density. Moreover, a transformerless solution can be more attractive with a control system taking advantage of its key characteristics [13]. Hence, it is possible to evidence that this kind of converter becomes an important solution in the field of low power renewable energy applications [14]- [17].

Not all transformer-less converter topologies can be proposed to cover the requirements of the application because of the need of a high DC gain to boost the low voltage given by one or two solar panels (15 -30 V) to a high voltage level around 400

V. Therefore, only a limited number of existing converters or some modifications of them can meet the main requirement of high DC gain without using a power transformer. These topologies are treated in the literature as high gain transformer-less topologies [18]- [20]. Among them, it is possible to identify topologies with coupled inductors [21]- [29], switched capacitors [30]- [31], multiplier cells [32]- [33], additional resonant circuits [34], or other alternative ways which improve the efficiency at the expense of some additional complexity [35]- [37]. Further, recently studied topologies with floating output increase the possibilities to attain high gain with a reduced circuit complexity [38]- [39]. All of these topologies are not considered in this work because the study is concentrated in simpler topologies operating in hard switching, and its aim is searching for a competitive solution through the use of an adequate control strategy.

Among the classic DC-DC topologies used to boost a voltage level to another one several times higher, the boost converter is the most used because of its simplicity, high efficiency and the presence of the input inductor which guarantees the compatibility with the primary source and improves the global performance. However, many authors constrain the use of the boost converter to gains lower than 10 to prevent high values of the duty cycle which inhibit the proper operation of the regulation systems, increase the possibility to have irreversible damages and show risk of saturation in the modulator [40]- [41]. Therefore, the boost converter, in spite of being the basis of the converter which will be treated along the work, is not here considered because it should operate with gains always higher than 10.

The possibility to obtain high conversion ratios has been introduced over 40 Hitherto, from the results reported in [73], it is worth to point out the fact that the four quadratic boost converter topologies offer the possibility to work in a more safe area of the converter duty cycle without the need of a transformer or a complex additional circuit. Further, in the four topologies it is also possible to implement an input current control loop because all have an inductor connected to the input port. Additionally, the three configurations with only one switch (𝐶 2 𝑆 2 -𝑄𝐵, 𝑆 4 -𝑄𝐵 and 𝑅𝐶𝑆 2 -𝑄𝐵) require only a control system and the fact of replacing a MOSFET for a diode can be considered an advantage because only one driver circuit is needed. Assuming that this work is mainly devoted to control issues, we decided to study in deep the (𝐶 2 𝑆 2 -𝑄𝐵) converter because our results can be posteriorly applied to the other topologies by exploring singular differences and by highlighting particular advantages or drawbacks.

Despite of its high gain capability and the absence of a power transformer, the modulation strategy used to control the quadratic boost converters plays a determinant role in its final reliability and performance. On the one hand, the use of Pulse Width Modulation (PWM) when the converter works with higher gains increases the risk of modulator saturation and hence increases the failure possibility. On the other hand, the use of a hysteretic modulator introduces a self-oscillating behavior with variable frequency which is constrained by the parameters of the modulator and some parameters of the converter [74]- [75]. Moreover, robust and reliable nonlinear control techniques such as sliding mode (SM) control can be easily implemented using hysteretic comparators , which has captured the interest of many authors to apply this kind of control in DC-DC power converter applications, including higher complexity structures such as the fourth order converters Cûk and SEPIC [76]- [78]. This control is highly suitable to control DC-DC converters given its variable structure nature which ensures robustness, reliability and simplicity [79]- [80].

Some authors have tried to generalize the approach of the use of the sliding-mode control applied in DC-DC power converters [81]- [85]. However, more and more works in different applications and methods or combinations of them permanently complement the state-of-the art of this research field. Despite of the large amount of works, there is always a challenge, for example, the use of a new converter or the study of an operational mode in a converter, the employ of other variables or sliding surfaces, the utilization of new implementation techniques and others. This is the case of applications in the context of the star-up in the converters [86]- [87], digital or hybrid (analogicdigital) control [88]- [89], interleaving or paralleled operation of converters [90]- [91], or the enforcing of desired behaviors [45]. In this work, the possibility for a converter to have a pure resistive behavior at the input port whereas its output port shows a power source characteristic is highly interesting. Ideally, this application can be classified into the Power Output Power Input (POPI) networks described by Singer et al. [92] and more specifically into the Loss Free Resistor concept (LFR) [93]- [96]. With the synthesis of power processing stages using sliding mode control, it has been possible to define a Sliding Mode LFR (SM-LFR) [START_REF] Cid-Pastor | Synthesis of loss-freeresistors based on sliding-mode control and its applications in power processing[END_REF]. Using this kind of approach, it is possible to cover the requirements of important applications such as Power Factor Correction (PFC) in switching rectifiers [98]- [99], driver of HB-LED lamps [100]- [102], or in our case, the impedance matching with photovoltaic modules [103]- [104]. In the case of the DC-DC stage of the microinverter, the sliding mode control allows attaining the SM-LFR behaviour with a relative simple electronic circuit implementation.

In the remainder of the chapter, the theoretical foundations of the steady-state operation of the quadratic boost converter are given for the continuous conduction mode (CCM) together with the analysis of three possible discontinuous conduction modes (DCM) which have been identified from this study. Then, studying the dynamics of the converter by means of the sliding mode control approach lead to the converter stabilization through a sliding surface defined for the input inductor current operating always in CCM. The stability of the converter operating with this control law is studied by means of a Lyapunov candidate function. After that, this control law is modified to impose a power source characteristic at the converter output port while the input port shows a pure resistive behavior. This subject is tackled treating the controlled converter as a SM-LFR, using the input conductance as the control variable. The effect of the proposed control on the converter efficiency is analyzed and discussed. The theoretical predictions of the analysis are first verified by means of MATLAB and PSIM-based simulations. Subsequently, experimental results validating the entire study are also given.

CONVERTER MODEL

Converter structures and conduction modes

As shown in figure 3.2, the quadratic boost converter has five possible structures Then, the quadratic boost converter can be modeled as a variable structure system which commutes between two, three or four structures (states) during a switching period (T s = 1/f s ) originating one of its different conduction modes. The five possible conduction modes and the possible transitions between them are depicted in figure 3.3a. If the converter works in on-state, this changes to off-state depending of the instantaneous value of the input inductor current. Further, being in the off-state, the converter can change to either on-state or DCL2-state. Being in DCL2-state, the converter can only change to the on-state provided the input inductor is externally forced to remain in continuous conduction mode. This is mainly due to the need of ensure the validity of the control strategy and the existence of the hysteresis band.

Modeling of the converter dynamics

From the above analysis, four different conduction modes are defined: CCM, DCL1, DCL2 and DCL12 modes. To model their dynamic behavior, the set of differential equations corresponding to each circuit in figure 3.2 are summarized in table 3.1. From the equations in table 3.1, a compact representation can be obtained using the following multi-linear expression:

𝑑𝑖 𝐿1 𝑑𝑡 = 𝑣 𝑖𝑛 𝐿 1 (1 -𝑢 1 ) - 𝑣 𝐶1 𝐿 1 (1 -𝑢 -𝑢 1 ) 𝑑𝑖 𝐿2 𝑑𝑡 = 𝑣 𝐶1 𝐿 2 (1 -𝑢 2 ) - 𝑣 𝐶2 𝐿 2 (1 -𝑢 -𝑢 2 ) 𝑑𝑣 𝐶1 𝑑𝑡 = - 𝑖 𝐿2 𝐶 1 (1 -𝑢 2 ) + 𝑖 𝐿1 𝐶 1 (1 -𝑢 -𝑢 2 ) 𝑑𝑣 𝐶2 𝑑𝑡 = 𝑖 𝐿2 𝐶 2 (1 -𝑢 -𝑢 2 ) - 𝑣 𝐶2 𝑅𝐶 2 (3.1)
where 𝑢 is the control signal of the controlled switch, 𝑢 1 represents the existence of the discontinuous conduction mode DCL1 and 𝑢 2 represents the existence of the discontinuous conduction mode DCL2. The variable 𝑢 1 takes the value 1 when the converter operates in the mode DCL1, and the variable 𝑢 2 takes the value 1 when the converter operates in the mode DCL2. In the mode DCL12, both 𝑢 1 and 𝑢 2 take the value 1. This model can be adopted to work with any kind of control irrespective of the employed modulator. Nevertheless, particular assumptions can be expected in order to cope with the analysis of discontinuous conduction modes with variable switching frequency.

Steady state of the converter

The steady state of the DC-DC converters is commonly analyzed using the current and voltage waveforms obtaining their mean values and the amplitude of their ripples.

Working with a constant or variable frequency modulation methods does not affect the analysis of steady state since an equilibrium point is characterized by a constant switching frequency in both cases. Thus, from the steady state analysis can be deduced represents the mean values of 𝑢, 1 -𝑢 -𝑢 1 and 1 -𝑢 -𝑢 2 .

DC gain and average values of the converter variables

A. Converter operating in CCM

From the expressions listed in table 3.1, the balances are given by:

〈𝑣 𝐿1 〉 = 〈𝑣 𝑖 〉𝑈𝑇 𝑠 + (〈𝑣 𝑖 〉 -〈𝑣 𝐶1 〉)(1 -𝑈)𝑇 𝑠 𝑇 𝑠 = 0 (3.2) 〈𝑣 𝐿2 〉 = 〈𝑣 𝐶1 〉𝑈𝑇 𝑠 + (〈𝑣 𝐶1 〉 -〈𝑣 𝐶2 〉)(1 -𝑈)𝑇 𝑠 𝑇 𝑠 = 0 (3.3)
The charge balance in 𝐶 1 and 𝐶 2 results in:

〈𝑖 𝐶1 〉 = -〈𝑖 𝐿2 〉𝑈𝑇 𝑠 + (-〈𝑖 𝐿2 〉 + 〈𝑖 𝐿1 〉)(1 -𝑈)𝑇 𝑠 𝑇 𝑠 = 0 (3.4) 〈𝑖 𝐶2 〉 = - 〈𝑣 𝐶2 〉 𝑅 𝑈𝑇 𝑠 + (〈𝑖 𝐿2 〉 - 〈𝑣 𝐶2 〉 𝑅 ) (1 -𝑈)𝑇 𝑠 𝑇 𝑠 = 0 (3.5)
Denoting 𝑉 𝑖 = 〈𝑣 𝑖 〉, the mean values of the converter variables are given by: 4 (3.6) 3 (3.7)

〈𝑖 𝐿1 〉 = 𝐼 𝐿1 = 𝑉 𝑖 𝑅(1 -𝑈)
〈𝑖 𝐿2 〉 = 𝐼 𝐿2 = 𝑉 𝑖 𝑅(1 -𝑈)
〈𝑣 𝐶1 〉 = 𝑉 𝐶1 = 𝑉 𝑖 (1 -𝑈) (3.8)
〈𝑣 𝐶2 〉 = 𝑉 𝐶2 = 𝑉 𝑖 (1 -𝑈) 2 (3.9)

from which it can be deduced the ideal static gain of the converter: 2 (3.10)

𝑀 𝐶𝐶𝑀 (𝑈) = 𝑉 𝐶2 𝑉 𝑖 = 1 (1 -𝑈)

B. Converter operating in DCL1 mode

From the expressions listed in table 3.1, the balances in 𝐿 1 and 𝐿 2 can be characterized by:

〈𝑣 𝐿1 〉 = 〈𝑣 𝑖 〉𝑈𝑇 𝑠 + (〈𝑣 𝑖 〉 -〈𝑣 𝐶1 〉)𝑈 1 𝑇 𝑠 𝑇 𝑠 = 0 (3.11) 〈𝑣 𝐿2 〉 = 〈𝑣 𝐶1 〉𝑈𝑇 𝑠 + (〈𝑣 𝐶1 〉 -〈𝑣 𝐶2 〉)(1 -𝑈)𝑇 𝑠 𝑇 𝑠 = 0 (3.12)
The charge balance in 𝐶 2 is given by:

〈𝑖 𝐶2 〉 = - 〈𝑣 𝐶2 〉 𝑅 𝑈𝑇 𝑠 + (〈𝑖 𝐿2 〉 - 〈𝑣 𝐶2 〉 𝑅 ) (1 -𝑈)𝑇 𝑠 𝑇 𝑠 = 0 (3.13)
By solving (3.11), (3.12) and (3.13), the steady-state expressions for 𝑣 𝐶1 , 𝑣 𝐶2 and 𝑖 𝐿2 are obtained.

〈𝑣 𝐶1 〉 = 𝑉 𝐶1 = 〈𝑣 𝑖 〉(𝑈 + 𝑈 1 ) 𝑈 1 (3.14) 〈𝑣 𝐶2 〉 = 𝑉 𝐶2 = 〈𝑣 𝑖 〉(𝑈 + 𝑈 1 ) 𝑈 1 (1 -𝑈) (3.15) 〈𝑖 𝐿2 〉 = 𝐼 𝐿2 = 〈𝑣 𝐶2 〉 𝑅(1 -𝑈) (3.16)
The average value of 𝑖 𝐿1 can be obtained from a geometrical analysis of the triangular area described by the current waveform shown in figure 3.5. The height of the triangle is defined by the maximum value of 𝑖 𝐿1 , which is denoted as 𝐼 𝐿1 𝑚𝑎𝑥 whereas the base is defined by the time interval (𝑈 + 𝑈 1 )𝑇 𝑠 . The value of 𝐼 𝐿1 𝑚𝑎𝑥 can be obtained using the slope of the inductor current in the onstate as follows:

𝐼 𝐿1 𝑚𝑎𝑥 = ( 〈𝑣 𝑖 〉 𝐿 1 ) 𝑈𝑇 𝑠 (3.17)
Then, we have:

〈𝑖 𝐿1 〉 = 1 𝑇 𝑠 ( 1 2 𝐼 𝐿1 𝑚𝑎𝑥 (𝑈 + 𝑈 1 )𝑇 𝑠 ) = 〈𝑣 𝑖 〉𝑈(𝑈 + 𝑈 1 ) 2𝐿 1 𝑓 𝑠 〈𝑖 𝐿1 〉 = 〈𝑣 𝑖 〉𝑈(𝑈 + 𝑈 1 ) 2𝐿 1 𝑓 𝑠 (3.18)
Note that in DCM operation it is not possible to use the low ripple approximation because of the presence of pulsating currents. Hence, it is necessary to analyze the charge balance with the charge given or extracted by the nearby elements. For instance, while the inductor 𝐿 2 is permanently connected to the capacitor 𝐶 1 , the inductor 𝐿 1 is connected only during the off-state. Therefore, the charge balance in 𝐶 1 is given by

〈𝑖 𝐶1 〉 = -〈𝑖 𝐿2 〉𝑇 𝑠 + 𝑄 𝐿1 𝑂𝐹𝐹 𝑇 𝑠 = 0 (3.19)
where 𝑄 𝐿1 𝑂𝐹𝐹 represents the charge produced and given by inductor 𝐿 1 to capacitor 𝐶 1 across diode 𝐷 3 . Its value can be determined as the area below the current waveform in figure 3.5.

𝑄 𝐿1 𝑂𝐹𝐹 = 〈𝑣 𝑖 〉𝑈𝑈 1 𝑇 𝑠 2𝐿 1 𝑓 𝑠 = 𝑈 1 𝑇 𝑠 2 𝐼 𝐿1 𝑚𝑎𝑥 (3.20)
Thus, by replacing (3.20) in (3.19), another steady-state expression for 𝑖 𝐿2 is obtained.

〈𝑖 𝐿2 〉 = 〈𝑣 𝑖 〉𝑈𝑈 1 2𝐿 1 𝑓 𝑠 (3.21)
By equalizing the expressions (3.16) and (3.21), it is possible to obtain 𝐾 1 , which is a standard parameter to analyze the discontinuous conduction mode [40].

𝐾 1 = 2𝐿 1 𝑅𝑇 𝑠 = 𝑈(1 -𝑈) 2 𝑈 1 2 𝑈 + 𝑈 1 (3.22)
After that, extracting 〈𝑣 𝑖 〉/〈𝑣 𝐶2 〉 from (3.15) and replacing in (3.22), the value of 𝑈 1 is given by:

𝑈 1 = 𝐾 1 ( 〈𝑣 𝐶2 〉 〈𝑣 𝑖𝑛 〉 ) ( 1 𝑈(1 -𝑈) ) = 𝐾 1 2𝑈(1 -𝑈) 2 (1 + √1 + 4𝑈 2 (1 -𝑈) 2 𝐾 1 ) (3.23)
From the aforementioned expressions, the mean values of the converter variables are given by:

〈𝑖 𝐿1 〉 = 𝐼 𝐿1 = 𝑉 𝑖 4𝑅(1 -𝑈) 2 (1 + √1 + 4𝑈 2 (1 -𝑈) 2 𝐾 1 ) 2 (3.24) 〈𝑖 𝐿2 〉 = 𝐼 𝐿2 = 𝑉 𝑖 2𝑅(1 -𝑈) 2 (1 + √1 + 4𝑈 2 (1 -𝑈) 2 𝐾 1 ) (3.25)
〈𝑣 𝐶1 〉 = 𝑉 𝐶1 = 𝑉 𝑖 2 (1 + √1 + 4𝑈 2 (1 -𝑈) 2 𝐾 1 ) (3.26)
〈𝑣 𝐶2 〉 = 𝑉 𝐶2 = 𝑉 𝑖 2(1 -𝑈) (1 + √1 + 4𝑈 2 (1 -𝑈) 2 𝐾 1 ) (3.27)
The voltage conversion ratio of the converter operating in DCL1 mode is

𝑀 𝐷𝐶𝑀1 (𝑈) = 𝑉 𝐶2 𝑉 𝑖 = 1 + √1 + 4𝑈 2 (1 -𝑈) 2 𝐾 1 2(1 -𝑈) (3.28)

C. Converter operating in DCL2 mode

From the expressions listed in table 3

〈𝑣 𝐶2 〉 = 〈𝑣 𝑖 〉(𝑈 + 𝑈 2 ) 𝑈 2 (1 -𝑈) (3.32)
There are two charges involved in the charge balance of the capacitor 𝐶 1 . A positive charge is given by the inductor 𝐿 1 during the interval (1 -𝑈)𝑇 𝑠 whereas a negative charge 𝑄 𝐿2 is extracted for the inductor 𝐿 2 during the interval (𝑈 + 𝑈 2 )𝑇 𝑠 . Then, we have:

〈𝑖 𝐶1 〉 = 〈𝑖 𝐿1 〉(1 -𝑈)𝑇 𝑠 -𝑄 𝐿2 𝑇 𝑠 = 0 (3.33)
The value of 𝑄 𝐿2 is obtained computing the area below the current waveform in figure 3.6 during the interval (𝑈 + 𝑈 2 )𝑇 𝑠 . The value of the charge 𝑄 𝐿2 divided by the switching period is the same mean value of the inductor current 𝑖 𝐿2 . Hence, the following relation is obtained:

〈𝑖 𝐿1 〉 = 〈𝑖 𝐿2 〉 1 (1 -𝑈) (3.34)
The value of 𝑄 𝐿2 is obtained as the area of the triangle below the current waveform of the inductor 𝐿 2 which is shown in figure 3.6b. The height of the triangle is defined by the maximum value of 𝑖 𝐿2 , denoted as 𝐼 𝐿2 𝑚𝑎𝑥 , whereas the basis is the interval (𝑈 + 𝑈 2 )𝑇 𝑠 . The value of 𝐼 𝐿2 𝑚𝑎𝑥 is derived using the slope of the inductor current in the on state:

𝐼 𝐿2 𝑚𝑎𝑥 = ( 〈𝑣 𝐶1 〉 𝐿 2 ) 𝑈𝑇 𝑠 = 1 𝐿 2 ( 〈𝑣 𝑖𝑛 〉 1 -𝑈 ) 𝑈𝑇 𝑠 (3.35)
and therefore , we obtain Hence, it is possible to derive equation (3.37) since the mean value of the output current is the same mean value of the current through diode 𝐷 1 (𝑄 𝐿2𝑂𝐹𝐹 /𝑇 𝑠 ).

𝑄 𝐿2 = 〈𝑣 𝑖 〉 2𝐿 2 𝑓 𝑠 𝑈(𝑈 + 𝑈 2 ) (1 -𝑈) 𝑇 𝑠 (3.36)
〈𝑣 𝐶2 〉 𝑅 = 〈𝑖 𝐷1 〉 = 〈𝑣 𝑖 〉(𝑈 + 𝑈 2 ) 𝑅𝑈 2 (1 -𝑈) (3.37)
The mean value of the diode current is given by:

〈𝑖 𝐷1 〉 = 1 𝑇 𝑠 ( 1 2 𝐼 𝐿2 𝑚𝑎𝑥 𝑈 2 𝑇 𝑠 ) = 〈𝑣 𝑖 〉𝑈𝑈 2 2𝐿 2 𝑓 𝑠 (1 -𝑈) (3.38)
By equalizing expressions (3.37) and (3.38), it is obtained that:

𝐾 2 = 2𝐿 2 𝑅𝑇 𝑠 = 𝑈𝑈 2 2 (𝑈 + 𝑈 2 ) (3.39)
By solving (3.32) for 〈v C2 〉 /〈v in 〉 and replacing in (3.39), the value of 𝐷 1 is deduced.

𝑈 2 = 𝐾 2 ( 〈𝑣 𝐶2 〉 〈𝑣 𝑖 〉 ) ( 1 -𝑈 𝑈 ) = 𝐾 2 𝑈 (1 + √1 + 4𝑈 2 𝐾 2 ) (3.40)
From the abovementioned expressions, the mean values of the converter variables are given by:

〈𝑖 𝐿1 〉 = 𝐼 𝐿1 = 𝑉 𝑖 4𝑅(1 -𝑈) 2 (1 + √1 + 4𝑈 2 𝐾 2 ) 2 (3.41) 〈𝑖 𝐿2 〉 = 𝐼 𝐿2 = 𝑉 𝑖 4𝑅(1 -𝑈) (1 + √1 + 4𝑈 2 𝐾 2 ) 2 (3.42) 〈𝑣 𝐶1 〉 = 𝑉 𝐶1 = 𝑉 𝑖 (1 -𝑈) (3.43)
〈𝑣 𝐶2 〉 = 𝑉 𝐶2 = 𝑉 𝑖 2(1 -𝑈) (1 + √1 + 4𝑈 2 𝐾 2 ) (3.44)
The voltage conversion ratio of the converter operating in DCL2 mode is:

𝑀 𝐷𝐶𝑀2 (𝑈) = 𝑉 𝐶2 𝑉 𝑖 = 1 + √1 + 4𝑈 2 𝐾 2 2(1 -𝑈) (3.45)

D. Converter operating in DCL12, DCL21 modes

From the expressions listed in table 3.1, the balances in 𝐿 1 and 𝐿 2 are given by:

〈𝑣 𝐿1 〉 = 〈𝑣 𝑖 〉𝑈𝑇 𝑠 + (〈𝑣 𝑖 〉 -〈𝑣 𝐶1 〉)
〈𝑣 𝐶2 〉 = 〈𝑣 𝐶1 〉(𝑈 + 𝑈 2 ) 𝑈 2 (3.49)
The charge balances in both capacitors 𝐶 1 and 𝐶 2 can be obtained from the current waveforms shown in figure 3.7.

〈𝑣 𝐶1 〉 〈𝑣 𝑖 〉 = 𝐿 2 𝐿 1 ( 𝑈 1 𝑈 + 𝑈 2 ) (3.54)
By defining the constants 𝐾 1 and 𝐾 2 with the expressions:

𝐾 1 = 2𝐿 1 𝑅𝑇 𝑠 (3.55) 𝐾 2 = 2𝐿 2 𝑅𝑇 𝑠 (3.56)
Then, from (3.54), (3.55) and (3.56), we have:

〈𝑣 𝐶1 〉 〈𝑣 𝑖 〉 = 𝐾 2 𝐾 1 ( 𝑈 1 𝑈 + 𝑈 2 ) (3.57)
Then, by replacing (3.56) in (3.51), we have:

𝐾 2 = 𝑈𝑈 2 2 𝑈 + 𝑈 2 = 〈𝑣 𝐶1 〉 〈𝑣 𝐶2 〉 𝑈𝑈 2 (3.58)
By solving (3.58) for 𝑈 2 and replacing in (3.49), the following relation is obtained:

𝑀 𝑎 (𝑈) = 〈𝑣 𝐶2 〉 〈𝑣 𝐶1 〉 = 1 + √1 + 4𝑈 2 𝐾 2 2 (3.59)
On the other hand, by replacing (3.58) in (3.57), it is derived that:

〈𝑣 𝐶1 〉 〈𝑣 𝑖 〉 = ( 1 𝑀 𝑎 (𝑈) ) 𝑈 1 𝑈𝑈 2 𝐾 1 (𝑈 + 𝑈 2 ) (3.60)
Then, solving (3.49) for 〈𝑣 𝐶1 〉/〈𝑣 𝐶2 〉 and (3.48) for 𝑈 1 , and replacing in (3.60), we deduce:

〈𝑣 𝐶1 〉 〈𝑣 𝑖 〉 = 𝑈 2 𝐾 1 ( 1 𝑀 𝑎 (𝑈) ) 2 ( 〈𝑣 𝐶1 〉 〈𝑣 𝑖 〉 -1)
From which, it is obtained the voltage conversion ratio of the converter operating in DCL12 mode as:

〈𝑣 𝐶2 〉 〈𝑣 𝑖 〉 = 𝑉 𝐶2 𝑉 𝑖 = 𝑀 𝑎 (𝑈) + √[𝑀 𝑎 (𝑈)] 2 + 4𝑈 2 𝐾 1 2 = 1 2 (( 〈𝑣 𝐶2 〉 〈𝑣 𝐶1 〉 ) + √ ( 〈𝑣 𝐶2 〉 〈𝑣 𝐶1 〉 ) 2 + 4𝑈 2 𝐾 1 ) 𝑀 𝐷𝐶𝑀12 (𝑈) = 𝑉 𝐶2 𝑉 𝑖 = 1 4 ( 1 + √1 + 4𝑈 2 𝐾 2 + √ (1 + √1 + 4𝑈 2 𝐾 2 ) 2 + 16𝑈 2 𝐾 1 ) (3.61)
From the aforementioned expressions, the mean values of the converter variables are given by:

〈𝑣 𝐶1 〉 = 𝑉 𝐶1 = 𝑉 𝑖 𝑀 𝐷𝐶𝑀12 (𝑈) 𝑀 𝑎 (𝑈) (3.62) 〈𝑣 𝐶2 〉 = 𝑉 𝐶2 = 𝑉 𝑖 [𝑀 𝐷𝐶𝑀12 (𝑈)] (3.63)
From the input to output power balance, it is obtained that:

〈𝑖 𝐿1 〉 = 𝐼 𝐿1 = 𝑉 𝑖 [𝑀 𝐷𝐶𝑀12 (𝑈)] 2 𝑅 (3.64)
Since〈𝑖 𝐿2 〉 = 𝐼 𝐿2 𝑚𝑎𝑥 (𝑈 + 𝑈 2 )/2, using (3.53) results in the following expression after some mathematical manipulation:

〈𝑖 𝐿2 〉 = 𝐼 𝐿2 = 𝑉 𝑖 𝑀 𝐷𝐶𝑀12 (𝑈)𝑀 𝑎 (𝑈) 𝑅 (3.65)

Boundaries of the conduction modes at constant frequency

The boundaries between the different discontinuous conduction modes are obtained considering the fact that an inductor current reaches the value of zero exactly at the end of the switching period. Then, it is possible to assert that the converter operates in a critical or boundary conduction mode. As expected, one or both parameters 𝑈 1 or 𝑈 2 are exactly equal to 1 -𝑈 allowing the use of this definition to obtain the mathematical conditions. The procedure consists in obtaining a critical value of 𝐾 1 or 𝐾 2 , wherein it is possible to distinguish the operational modes. The values will be denoted 𝐾 1 𝑐𝑟𝑖𝑡 or 𝐾 2 𝑐𝑟𝑖𝑡 respectively. Four different boundaries can be identified:

a) The converter operates in CCM and the current of inductor L1 reaches the critical conduction mode, which is the way to enter in DCL1 mode.

By replacing 𝑈 1 = 1 -𝑈 in (3.22), the following expression is obtained:

𝐾 1 𝑐𝑟𝑖𝑡 = 𝑈(1 -𝑈) 2 (1 -𝑈) 2 𝑈 + (1 -𝑈) = 𝑈(1 -𝑈) 4 (3.66)
Then, it is possible to assert that when the converter operates in CCM, the boundary to enter into the DCL1 mode is defined by:

2𝐿 1 𝑅𝑇 𝑠 < 𝑈(1 -𝑈) 4 (3.67)
b) The converter operates in CCM and the current of inductor 𝐿 2 reaches the critical conduction mode, which is the way to enter into the DCL2 mode.

By replacing 𝑈 2 = 1 -𝑈 in (3.39), it is obtained that:

𝐾 2 𝑐𝑟𝑖𝑡 = 𝑈(1 -𝑈) 2 𝑈 + (1 -𝑈) = 𝑈(1 -𝑈) 2 (3.68)
Then, it is possible to assert that when the converter operates in CCM, the boundary to attain the DCL2 mode is defined by:

2𝐿 2 𝑅𝑇 𝑠 < 𝑈(1 -𝑈) 2 (3.69)
c) The converter operates in the DCL1 mode and the current of inductor 𝐿 2 reaches the critical conduction mode, which is the way to enter into the DCL12 mode.

By replacing 𝑈 2 = 1 -𝑈 in (3.58), the expression (3.68) is anew obtained.

Then, it is possible to assert that, being in the mode DCL1, the boundary to attain the DCL12 mode is also defined by expression (3.69)

d) The converter operates in the DCL2 mode and the current in inductor 𝐿 1 reaches the critical conduction mode, which is the way to enter into the DCL21 mode.

By replacing 𝑈 1 = 1 -𝑈 in (3.48) and (3.57), and identifying the two expressions, we have:

𝐾 1 = 𝐾 2 (1 -𝑈) 2 𝑈 + 𝑈 2 (3.70)
On the other hand, by using relation (3.59), expression (3.58) becomes:

𝐾 2 = 𝑈𝑈 2 𝑀 𝑎 (𝑈) ⇒ 𝑈 2 = 𝑀 𝑎 (𝑈)𝐾 2 𝑈 (3.71)
Therefore (3.70) leads to:

𝐾 1 𝑐𝑟𝑖𝑡 = 2𝐾 2 𝑈(1 -𝑈) 2 2𝑈 2 + 𝐾 2 (1 + √1 + 4𝑈 2 𝐾 2 ) (3.72)
Then, it is possible to conclude that when the converter operates in the DCL2 mode, the boundary to attain the DCL12 mode is defined by: respectively. It is interesting to observe that the limits of the regions are defined by the functions 𝐾 1 (𝑈) and 𝐾 2 (𝑈). Additionally, it is possible to observe that for a constant value of 𝐾 1 (for example 0.06 in the figure 3.8c) or 𝐾 2 (for example 0.08 in the figure 3.8d), the modes DCL12 and DCL21 can only be attained for a delimited region of the duty cycle. Then, for these operating modes, it results that 𝐾 1 is a function of 𝑈 and 𝐾 2 , and 𝐾 2 is a function of 𝑈 and 𝐾 1 .

2𝐿 1 𝑅𝑇 𝑠 < 2𝐾 2 𝑈(1 -𝑈) 2 2𝑈 2 + 𝐾 2 (1 + √1 + 4𝑈 2 𝐾 2 ) ( 3

Conduction modes and DC gain in a specific

SLIDING-MODE CONTROL

The use of the sliding-mode control in DC-DC power converters consist in imposing a desired behavior through a sliding surface by switching between the different structures of the converter. As was studied above, a DC-DC converter such as the quadratic boost converter can have two, three, four or five different structures and switches between them to operate in a desired conduction mode. For example, when the converter operates in CCM, it switches between two structures: the on-state and the off-state. In order to reach the sliding surface and remain there, the converter must change its controlled switch from the on-state to the off-state or vice versa depending on the evolution of the variables involved in the sliding surface. In general, two conditions have to be fulfilled: the reachability of the sliding surface and the permanence or invariance therein after it is reached.

To ensure the reachability of the sliding surface it is necessary that the trajectory generated directly lead the converter dynamics to the sliding surface. On the other hand, to ensure the invariance when the sliding motion is attained, it is necessary that switching between the different structures maintains the system in the sliding surface.

Ideally, this switching takes place at an infinite switching frequency. However, in a real converter, a higher but not infinite frequency is only possible due to the limits of its semiconductor components, which is undesirable in terms of efficiency. Therefore, to impose a finite switching frequency, a hysteretic modulator can be used constraining the sliding surface within a hysteresis band guarantying a uniform switching. Figure 3.11 gives a graphic representation for two of the converter variables (𝑉 𝐶2 and 𝑖 𝐿1 ). It is easily to identify the evolution of the system towards the desired equilibrium point with coordinates (𝑖 𝑖𝑛 ̅̅̅, 𝑣 𝑜𝑢𝑡 ̅̅̅̅̅̅). In the illustrated case, the converter operates in CCM having two possible structures, structure 1 and structure 2. From the origin, the structure 1 allows reaching the sliding surface 𝑆(𝑥) = 𝑖 𝐿1 -𝑖 𝑖𝑛 ̅̅̅ and switching between structure 1 and structure 2 allows the converter dynamics remaining in it. Further, a hysteresis band has been added showing the way to have a sliding surface as the average value of the variable 𝑖 𝐿1 . It is worth to note that the system switches between the two structures within the limits of the hysteresis band whereby the invariance is ensured. However, it is also possible to observe in figure 3.11 that the system reaches the sliding surface long time before reaching the equilibrium point and later it attains the equilibrium point through the sliding surface. This remainder dynamic is known as the ideal sliding dynamic. The control signal u takes the values 0 or 1 in order to change the structure of the system.

To analyze the converter dynamics operating under a sliding motion, a well-known method can be applied to represent the nonlinear control variable u as the fictitious variable 𝑢 𝑒𝑞 in such a way that the discontinuous nature of the control signal disappears but its influence in the dynamic of the converter model is preserved. This method is known in the literature as the equivalent control method introduced by V. Utkin in [START_REF] Utkin | Sliding mode control in electromechanical systems[END_REF]. This method has been applied in many related works and will be the analysis tool in this work.

Hysteresis modulation-based SM controller

For a desired sliding surface 𝑆(𝑥), the ideal expected behavior in sliding motion is based on 𝑆(𝑥) = 0. In a variable structure system with two possible configurations such as a DC-DC converter operating in CCM, a control law based on sliding mode can be given by:

𝑢 = { 1, 𝑤ℎ𝑒𝑛 𝑆(𝑥) < 0 0, 𝑤ℎ𝑒𝑛 𝑆(𝑥) > 0 (3.74)
where it is mandatory that the control action 𝑢 = 1 ensures the increment of 𝑆(𝑥)

when its value is negative and the control action 𝑢 = 0 ensures the decrement of 𝑆(𝑥) when its value is positive. As expected, the possible switching frequency of the systems will be undesirably higher. Then, a proper solution consists in using a hysteresis modulator to modify the control law (3.74) in the following way:

𝑢 = { 1, 𝑤ℎ𝑒𝑛 𝑆(𝑥) < -𝛿 0,
𝑤ℎ𝑒𝑛 𝑆(𝑥) > 𝛿 which allows enforcing the converter to operate with a finite switching frequency. The value of 𝛿 must be sufficiently high to avoid spurious switching produced by interference of noise and sufficiently low to ensure the validity of the approximation.

Then, with the use of the hysteresis comparator, the expected sliding motion 𝑆(𝑥) = 0 will correspond to the average value computed for a switching period. Then, the sliding mode control theory allows analyzing the average behavior of the converter variables when a hysteresis based modulator is used [74], or, in other words, a sliding mode control law can be easily implemented by means of a hysteresis modulator.

Indirect stabilization via input inductor current

As was explained above, a sliding surface can be obtained from the linear combination of the converter variables and/or some reference values. Then, the simpler sliding surface is obtained using only one converter variable and a reference variable which can be constant. Hence, when the output voltage is controlled by using a surface which includes this same output voltage, the converter is said to be stabilized in a direct way.

However, in some converters, such as boost converters, this kind of sliding surfaces leads to an unstable equilibrium point. Therefore, a surface including other variables must be used in order to obtain a stable behavior in the output voltage. In these cases, the proposed surface leads to an indirect control of the output variable [105]- [START_REF] Sira-Ramirez | Control Design Techniques in Power Electronic Devices[END_REF]. A simple indirect control law is obtained by only using the input inductor current which allows immediately attaining the sliding surfaces for a converter affected by disturbances and ensures stability of the equilibrium point. Figure 3.12 illustrates the proposed control applied to the quadratic boost converter.

The sliding surface is

𝑆(𝑥) = 𝑖 𝐿1 -𝑖 𝐸 (3.75)
where 𝑖 𝐸 is a current reference. The value of 𝑖 𝐸 can be a constant value or a slow signal

given by an outer loop in order to achieve a second control objective such as impedance matching [103] or output voltage regulation [50]. In this case, we have that in sliding mode 𝑆(𝑥) = 0 and then 𝑖 𝐿1 = 𝑖 𝐸 (𝑡). The switching function u corresponds to the expression (3.74).

Equivalent control analysis

By considering CCM operation, the model of the quadratic boost converter in (3.1)

becomes

𝑑𝑖 𝐿1 𝑑𝑡 = 𝑣 𝑖𝑛 𝐿 1 - 𝑣 𝐶1 𝐿 1 (1 -𝑢) 𝑑𝑖 𝐿2 𝑑𝑡 = 𝑣 𝐶1 𝐿 2 - 𝑣 𝐶2 𝐿 2 (1 -𝑢) 𝑑𝑣 𝐶1 𝑑𝑡 = - 𝑖 𝐿2 𝐶 1 + 𝑖 𝐿1 𝐶 1 (1 -𝑢) 𝑑𝑣 𝐶2 𝑑𝑡 = 𝑖 𝐿2 𝐶 2 (1 -𝑢) - 𝑣 𝐶2 𝑅𝐶 2 - 𝑖 0 𝐶 2 (3.76)
A current source 𝑖 0 is introduced in order to model disturbances in the output load.

A. Equivalent control

Applying the invariance conditions (𝑆(𝑥) = 0 and 𝑆 (𝑥) = 0) in (3.75) The magnitudes of the applied excitation to obtain the waveforms in figure 3.14 have been computed ensuring the steady-state of the output voltage in 350, 400 and 450 V.

𝐴(𝑠) = 𝑚𝐶

As can be observed, the linearized model represents with a good precision the dynamical behavior of the converter in this set of magnitudes for the considered disturbances.

Stability analysis using a Lyapunov candidate function

When 𝐼 𝐸 (𝑡) is constant, the ideal sliding dynamics in (3.79)

If ( i L2 ̅̅̅̅ v C2 ̅̅̅̅̅ v C1 v C1 ̅̅̅̅̅ ) i L2 ̅̅̅̅ v C2 ̅̅̅̅̅ - i L2 ̅̅̅̅ 2 4v C1 2 = i L2 ̅̅̅̅ 2 v C1 ( 1 v C1 ̅̅̅̅̅ - 1 4v C1
) is positive, then Q ≥ 0. This is true if 4v C1 > v C1 ̅̅̅̅, which is the same that:

𝐯 𝐂𝟏 > 𝟏 𝟒 𝐯 𝐂𝟏 ̅̅̅̅ (3.97)
In conclusion, if the conditions i L2 < 2i L2 ̅̅̅̅ and v C1 >

1 4 v C1 ̅̅̅̅ are satisfied, then V ̇(i L2 , v C1 , v C2 ) ≤ 0. Moreover, V ̇(i L2 , v C1 , v C2 ) = 0 for v C1 = v C1 ̅̅̅̅ and v C2 = v C2 ̅̅̅̅,
which, considering the above conditions, implies i L2 = i L2 ̅̅̅̅, i L1 being equal to 𝐼 𝐸 .

Then, V ̇(i L2 , v C1 , v C2 ) = 0, only for the equilibrium point. Therefore, the system is asymptotically stable by the LaSalle's invariance principle [110].

Converter operating as a Loss-Free Resistor

The DC transformer, the DC gyrator and the Loss Free Resistor (LFR) are the three canonical elements to model a DC processing system. These elements can be classified into the two-port circuits denominated as POPI (Power Output = Power Input) which has been introduced by Singer in [92]. As shown in figure 3.15, the DC-DC stage must be a type of POPI circuit which extract the maximum power to process, and make it available to supply the DC-AC stage which has been represented as a constant current load. So, the converter used in the DC-DC stage must show a resistive behavior at the input port whereas it shows a power source behavior at the output port. To obtain such a behavior, a DC-DC power converter such as the quadratic boost one can be enforced to be a LFR if a proportional relation exists between the input voltage and the input current. Then, the control of the input port can be designed using either the input resistance or the input conductance as a control variable. A sliding surface which involves these two variables and a proportional constant (R or G), can be proposed to obtain a SM-LFR property. or the load 𝐼 𝑜 (𝑠). To regulate the output voltage it is possible to introduce an external control loop. As shown in figure 3.20, it is possible to consider that the current load connected to the converter is defined as a function of the output voltage. In this approach the energy of the output capacitor is maintained constant. The output power of the converter is defined by the output voltage and the output current which are internal variables of the control system. This approach requires a controller operating over the load (DC-AC stage) defining the instantaneous current demanded to the DC-DC stage so that the output voltage can be stabilized and regulated. As it can be shown in figure 3.20, the regulation of the output voltage around 400 V is accomplished. Further, positive and negative disturbances are applied on the input voltage at 0.1 and 0.15 s in order to obtain a transient deviation of 5% from the steadystate value. Moreover, positive and negative disturbances are applied on the conductance at 0.2 and 0.25 s respectively to have a similar transient deviation of 5%. A simple controller system allows efficiently controlling the quadratic boost converter. It is worth to note, that this illustrative approach considers a constant load, which is indeed the case of a power converter connected to a grid. Hence, this control problem will be explained in more details in the chapters devoted to the DC-AC conversion stage.

Analysis with constant current load

EFFICIENCY ASPECTS

A complete study of the efficiency of a quadratic boost converter operating in conditions similar to the one required in our application has been reported in [59]. In that work the quadratic boost converter is controlled using a two-loop controller to regulate the output voltage at 400 VDC. An inner loop corresponds to the sliding mode indirect stabilization described in subsection 3.3.2. To evaluate the efficiency, it is considered that the converter must operate in a range of input voltage and output power.

This section compiles the main elements of that work and proposes other experimental results with a different set of parameter in order to complete and enhance the understanding about the effects of a sliding mode strategy on the converter efficiency.

As analyzed in the above sections, the use of a hysteretic comparator to enforce a sliding surface leads to a non-constant frequency operation depending on the input voltage (𝑉 𝑖 ) and the output load (𝑃 𝑜 ). However, it is worth nothing that the frequency is constant for each point (𝑉 𝑖 , 𝑃 𝑜 ) in the overall operational range of the converter.

Therefore, the frequency variations affect the efficiency because the switching losses of the semiconductor devices are proportional to the switching frequency. In addition, this fact constitutes a significant difference with respect to a PWM -based control which deserves a special attention.

Conduction losses

In order to analyze the effect of the conduction losses on the converter gain and the efficiency of the converter, the ideal circuit configurations depicted in figure 3. This model is adapted for evaluating theoretically the converter efficiency or the converter dynamics. However, introducing the parasitic elements in differential equations can increase considerably the required computation. Hence, the complete model has been only used in the steady-state analysis of the converter to evaluate the efficiency within its operational range.

𝐿 1 𝑑𝑖 𝐿1 𝑑𝑡 = 𝑣 𝑖 -𝑣 𝐶1 (1 -𝑢) -𝑖 𝐿1 [𝑅 𝐿1 + (𝑅 𝐷2 + 𝑅 𝑀 )𝑢 + (𝑅 𝐷3 + 𝑅 𝐶1 )(1 -𝑢)] -𝑖 𝐿2 [𝑅 𝑀 𝑢 + 𝑅 𝐶1 (1 -𝑢)] -𝑉 𝐷2 𝑢 -𝑉 𝐷3 (1 -𝑢) 𝐿 2 𝑑𝑖 𝐿2 𝑑𝑡 = 𝑣 𝐶1 -𝑣 𝐶2 𝐾 𝑜 (1 -𝑢) -𝑖 𝐿1 [𝑅 𝑀 𝑢 -𝑅 𝐶1 (1 -𝑢)] -𝑖 𝐿2 [𝑅 𝐿2 + 𝑅 𝐶1 + 𝑅 𝑀 𝑢 + (𝑅 𝐷1 + 𝑅 𝐶2 𝐾 𝑜𝑢𝑡 )(1 -𝑢)] -𝑣 𝐷1 (1 -𝑢) 𝐶 1 𝑑𝑣 𝐶1 𝑑𝑡 = -𝑖 𝐿2 + 𝑖 𝐿1 (1 -𝑢) 𝐶 2 𝑑𝑣 𝐶2 𝑑𝑡 = 𝑖 𝐿2 (1 -𝑢) - 𝑣 𝐶2 𝑅 (3.118)
where 𝐾 𝑜 = 𝑅/(𝑅 + 𝑅 𝐶2 ). Evaluating (3.118) at an equilibrium point (derivatives equal to zero and 𝑢 = 𝑈), it is possible to derive the following linear system

                                                  0 0 ' ' 1 0 ' 0 0 0 1 ' ' 1 ' ' 0 ' ' ' 1 3 2 2 1 2 1 1 2 1 1 1 U V U V U V V V V I I RU U U K U R U R R R U R U R U U R U R U R U R R D D D i C C L L o e M C L C M C M b a L where 𝑈 ′ = 1 -𝑈, 2 D M a R R R   , 1 3 C D b R R R  
, and

2 1 C o D e R K R R  
. Hence, solving this linear system leads to the mean value of the converter variables. Further, the effect of the voltage drops in resistive elements can be observed evaluating the converter gain. On the other hand, note that this analysis covers the effect of the voltage drops on the mean values of the variables but it does not include the mean values of the converter variables. Moreover, it is possible to compute the DC conduction losses in the converter using the formulas given in table 3.2 ( S I represents the current of the controlled switch (MOSFET in the case of the table) and D I represents the diode currents).

     2 Capacitors C C C R I P RMS   2 MOSFET DSon S S R I P RMS   2 Diodes D D D D D V I R I P RMS     2
The RMS value of the currents is slightly different of the average value in normal load conditions but is considerably different in lower load conditions because the amplitude of the current ripple begins to be comparable with the average. An accurate estimation of the conduction power losses must compute both components (AC and DC) for the case of triangular or trapezoidal waveforms.

Switching losses

Quantifying the switching losses is a really complex problem even for the more basic converters. For a power converter operating with hard switching at a switching frequency s f , some approximations use the analysis of the switching transitions (rise and fall) by computing the energy required to change from one state to another. Figure 3.23 shows these transitions and the involved variables considering diodes with no reverse recovery (SiC diodes).

Element Expression

ON transition

s C r L L L ON f V t I I I P              2 2 2 1 2 1 2 1  OFF transition s C f L L L OFF f V t I I I P              2 2 2 1 2 1 2 1  Gate charge s gate gate GATE f C V P    2 2 1

Estimation of the converter efficiency

In order to estimate the efficiency of the converter, a MATLAB algorithm has been performed following the diagram in figure 3.24. The algorithm uses the model in (3.116) and the expressions to compute both conduction and switching losses (Tables 3.2 and 3.3). In step1, the ideal duty cycle is computed using only the input and output voltage (desired value). After that, using the loops with step 2, the algorithm computes the required duty cycle to obtain the desired output voltage with the voltage drops. At this point we have the average values of the converter variables. In steps 3 and 5 the conduction losses are evaluated. In step 4, the switching losses are computed involving the switching frequency which is calculated for each operational point using (3.118) [59]. With the output power and the input power which is computed in step 6, it is possible to calculate the converter efficiency. This procedure is repeated for a complete set of operational conditions and the resulting data is represented graphically in figure 3.25 for the set of parameters listed in tables 3.4.

𝑓 𝑠 = 1 𝑇 𝑠 = 𝑉 𝑖 𝑈 2𝐿 1 Δ = 𝑉 𝑖 2𝐿 1 Δ (1 -√ 𝑉 𝑖 𝑉 𝐶2 ) (3.118)

EXPERIMENTAL RESULTS

Several theoretical expressions have been derived to explain the static and dynamic characteristics of the quadratic boost converter as a competitive solution to cover the requirements of the DC-DC stage in a dual stage microinverter. Each of the different topics has been illustrated and verified through simulation results using specialized computational tools such as PSIM and MATLAB. The interest of this section is to highlight some topics directly related with the final application examining the behaviors induced by the use of sliding mode control.

Experimental set-up and converter prototype

A 100 W prototype of the quadratic boost converter has been implemented to work with an output voltage of 400 V. The input voltage is constrained between 20 and 30 V.

The parameters of the converter are listed in table 3.4. The values of the passive elements have been selected to have low ripples on the inductance currents and the capacitor voltages [40], also considering the estimated limits of the frequency in all operational conditions. The conditions to ensure the continuous conduction mode in both inductors have been exceeded in order to provide an acceptable range to support transitory variation in variables. As will be observed in the results of the experimental validation, the design of the converter perfectly covers the stability conditions. The control circuit has been implemented using simple analog electronic circuits. The current reference is generated using an analog multiplier and simple conditioning circuits for the signal G and the input voltage 𝑣 𝑝 . The hysteresis comparator is implemented using two comparators and a flip-flop circuit obtaining the limits of the hysteresis band from op-amp-based circuits. The driver of the power MOSFET is implemented using a totem-pole configuration. The schematic diagram of the circuit is depicted in figure 3.26.

Efficiency measurements

To test the efficiency of the converter, a control loop has been introduced in order to maintain regulated the output voltage as shown figure 3.31. As observed in figure 3.32, an efficiency surface is obtained. As expected from the simulated results with the prediction algorithm, a wide region of the operational range have similar efficiencies. A maximum efficiency of 93.1% is obtained at high loads (100 W) and high input voltages (30 V). The average efficiency is about 89 % which is more or less the efficiency at mid power levels. A minimum efficiency of 82% is identified at lower powers (20 W) and lower voltages (20 W). By using a lower input voltage such as 15 V, lower efficiencies around 75% are obtained.

CONCLUSIONS

A deep study of the conventional quadratic boost converter has been provided in this chapter. A comprehensive description of the converter as a variable structure system with all possible discontinuous conduction modes and the transitions between them has been developed. A theoretical analysis allows the derivation of the converter gains in each case.

Being central in this work, the case of the control of the quadratic boost converter operating in continuous conduction mode using a sliding mode indirect stabilization strategy has been particularly investigated. After linearizing the reduced model obtained by the equivalent control method, the main transfer functions of interest have been obtained and the resulting model has been validated. The associated stability analysis has been also developed in the context of the Lyapunov stability theory. The sliding mode indirect stabilization strategy has been particularly adopted for enforcing the converter to operate as a Loss Free Resistor, this playing a central role in the microinverter application here reported.

The chapter ends by an analysis of the converter efficiency controlled by sliding modes validating the approach. Through the chapter, the theoretical developments have been validated by simulations with PSIM and MATLAB, and by experimental implementations.

Chapter 4

DC-AC stage:

Full-Bridge Power Source Inverter

INTRODUCTION

As introduced in chapter 1, many converter topologies can be used to accomplish the grid-connection objective [1]- [6]. The selection of an appropriate topology is determinant for efficiency, dynamic performance, reliability and power density.

Nowadays, the use of a low frequency transformer to give isolation and to boost the voltage at the AC side is less and less considered because of the size and cost of the transformer [7]. This fact has put in evidence the advantages of topologies with high frequency transformers based on the fly-back converter [8]- [13], or the push-pull converter [14]- [15]. However, at low power levels, it is not necessary to have an isolation, which motivates the use of transformer-less topologies [16]- [21] without ignoring the advantages to use an isolation transformer. Among these, the multilevel inverters and the cascade connection of inverters have been recently proposed in singlephase applications improving power quality aspects [22]- [28]. Also, it is worth to note that many of the existing topologies use one or more full-bridge inverters or a slight variation, this justifying our interest for this kind of structure.

On the other hand, it is possible to classify the grid-connected inverters in topologies involving DC-Link, pseudo DC-Link or no DC-Link as it is described and studied in [29]. These topologies can be distinguished not only from the configuration of the converter but also from the control strategy. Since the control applied to the DC-DC stage of the microinverter enforces a power source with an unregulated DC voltage to feed the DC-AC stage, the full bridge inverter is controlled by a sliding mode control strategy which in addition regulates the voltage of the DC link.

The full bridge converter connected to the grid can be used as either a power conditioner or a grid-tied inverter. Having in mind that in both cases the grid voltage is not a controlled variable but it is constant, then, the energy flow is fixed by the current delivered by the converter. The connection between the converter and the grid is done using a passive array such as L, CL, LC, LCL networks and hence the control objective consists on enforcing a desired value for the current of this coupling element [30]. In power conditioning, the inverter contributes totally or partially to both distortion and reactive power and has to prevent that the corresponding components circulate through the electrical networks. The control must generate a corresponding current reference to compensate the harmonics, the reactive power and the power losses. In this application, the DC side is only connected to a capacitive storage element (not to a load or a source) whose voltage is regulated to ensure power balance in the converter [31]- [33]. On the other hand, the grid-tied inverter converts a certain amount of power available at the DC side injecting it in the grid [34]- [36]. As a grid-connected inverter, the full-bridge shows a buck behaviour and hence it must be ensured that the voltage of the DC link be always higher that the peak value of the grid voltage.

The control of a grid-connected inverter requires the generation of a proper current reference and a way to robustly track it. To generate this reference it is necessary to measure the frequency and phase of the grid voltage which can be extracted either directly using the grid voltage signal or indirectly using a synchronization system.

Among these two possibilities, the more relevant difference is that in the direct way approach, the harmonic content of the voltage is transferred to the current increasing the levels of its Total Harmonic Distortion (THD). In the indirect case, a Phase Looked Loop (PLL) is used in order to generate a synchronized sinusoidal signal with normalized amplitude and a very low harmonic content [37]- [39]. The amplitude of the current can be given by a control loop ensuring the power balance of the overall system [40]- [42]. Using a dual stage topology, the control of the inverter can regulate the voltage of the DC link by modifying the power injected to the grid through the modification of the output current.

In order to obtain the tracking of the current reference, many control methods have been applied. The simplest one consists in using an analog comparator operating at constant frequency [43]. Other method which implies a lower complexity is the hysteresis control which can be applied using either a constant hysteresis band or an adaptive hysteresis band or other variants [44]- [47]. Like in other reference tracking applications, the predictive control is also an interesting option when digital implementation is possible [48]- [49]. Beside of the study of these kinds of control, the sliding mode control has been also used in grid-connected applications due to its wellknown robustness, stability and simple implementation [50]- [53]. Sliding mode control strategies for grid-tied inverters has been also implemented using digital devices retaining its main advantages [54].

This chapter introduces the full bridge inverter working as a Power Source Inverter (PSI) using sliding-mode to ensure a flawless tracking of the sinusoidal current reference. A first order sliding surface is employed to control the output current of the inverter which is studied using the equivalent control method [55]. The resulting sliding motion is used to derive a linear model between the DC link voltage and the amplitude of the output current. This model is used to design a two loop controller with a current controller as the inner loop and a voltage regulator in the outer one. The voltage regulator defines the instantaneous amplitude of the output current while the sinusoidal waveform synchronized with the grid is given by a PLL. The relation between the Total Harmonic Distortion (THD) and the Power Factor (PF) with the voltage of the DC link is modelled and analyzed through the harmonic balance method. A linear controller is proposed to have a simple but effective reduction of the oscillations in the output of the outer control loop reducing the harmonics. A feed-forward strategy is also proposed to predict the ripple and compensate it. The theoretical analysis is verified along the chapter by means of simulation, which is complemented with experimental results.

CONVERTER MODEL

The connection of a full bridge converter to the grid through a series inductor is a well-known topology which can be used as a rectifier, inverter or shunt active power filter, among other applications. To control these structures it is commonly employed a current tracking loop combined with other algorithms to meet the control objectives. As shown in figure 4.1, the full bridge converter operating as an inverter works with an energy source always available at the DC side. Depending on the nature of this source, the inverters have been classified as Voltage Source Inverters (VSI) or Current Source Inverters (CSI) [56]. However, considering that the microinverter structure is a doublestage topology and also that the first stage behaves as a power source, the inverter will be treated as a Power Source Inverter (PSI). This characteristic follows from the operation of the first stage as a SM-LFR ensuring the availability of a certain amount of power and having the H-bridge as load.

Bipolar commutation

Using bipolar or two-level commutation, the switches of each leg operate complementarily, in such a way that, the high-side switch is off when the low-side switch is on and the high-side switch is on when the low-side switch is off. Further, each leg operates complementarily with respect to the other one. Due to this, the inverter allows tracking a current reference with relative simplicity. Further, it is worth to mention that some studies reveal that in photovoltaic systems the ground current is less important when the inverter uses bipolar commutation [57]. However, the switching losses increase because both legs commute at high-frequency and the input current has a higher harmonic content. In order to obtain the converter model, the diagrams of the two possible structures of the circuit and its corresponding equations are shown in table 4.1.

Circuit configuration and power flow Expressions

Unipolar commutation

The unipolar operation is obtained forcing a leg to switch at high frequency while the other one switches at grid frequency. Also, it is possible to switch the high side switches of both legs at high frequency while the low side switches commute at grid frequency.

Using this method, the switching losses are reduced because only two switches operate at high frequency and also because the switching frequency is lower than in the bipolar case for a fixed hysteresis band. Four circuit structures are possible, two for each halfcycle. Diagrams of the two possible structures of the circuit operating in the positive half-cycle and its corresponding equations are shown in table 4.2.

Circuit configuration and power flow Expressions

Circuit configuration and power flow Expressions

𝐶 𝑑𝑣 𝐶 𝑑𝑡 = 𝑃 𝑣 𝑐 -𝑖 𝐿 𝑤 + 𝑖 𝐿 2 (1 -sign(𝑣 𝑔 )) (4.8)

FUNDAMENTALS OF GRID-CONNECTED GENERATION

Since the lower harmonic distortion of the grid voltage is normally below the 3%, the grid is ideally considered as a pure sine-wave AC voltage source of the form:

𝑣 𝑔 = 𝑉 𝑚𝑎𝑥 sin 𝜔𝑡 (4.9)
where 𝜔 = 2𝜋𝑓, and f is the fundamental frequency. A power generator can inject real power to the grid at unitary power factor when its current is defined as:

𝑖 𝑔 = 𝐼 𝑚𝑎𝑥 sin(𝜔𝑡 + 𝜑) (4.10)
being 𝐼 𝑚𝑎𝑥 a constant value which determines the amount of power injected into the grid and the displacement angle 𝜑 is almost zero . Then, the real power is given by:

𝑃 = 𝑃 𝑔 = 𝐼 𝑚𝑎𝑥 𝑉 𝑚𝑎𝑥 2 = 𝐼 𝑔 • 𝑉 𝑔 (4.11)
, where 𝐼 𝑔 and 𝑉 𝑔 are the RMS values of the grid voltage and current respectively.

However, depending on the control performance, the current injected may be more or less distorted or may be more or less out of phase with respect to the main voltage. In these cases, the above definition of the grid current can be generalized as:

𝑖 𝑔 = 𝐼 1 sin(𝜔𝑡 + 𝜑) + ∑ 𝐼 𝑛 sin(𝑛𝜔𝑡 + 𝜑 𝑛 ) 𝑛 𝑘=2 (4.12)
The current THD-R (relative to the RMS value) is computed as follows:

𝑇𝐻𝐷 -𝑅 = 𝐼 ℎ 𝐼 𝑔 = √ 1 -( 𝐼 1 𝐼 𝑔 ) 2 (4.13)
, where 𝐼 ℎ is the RMS value of the harmonic content in the current 𝐼 𝑔 and the power factor (PF) is defined by:

𝑃𝐹 = 𝑃 𝑆 = 𝑉 𝑔 • 𝐼 1 cos(𝜑) 𝑉 𝑔 • 𝐼 𝑔 (4.14)
Therefore, the DC-AC converter must convert the power available in the DC link in order to inject it into the grid. This has to be done ensuring a high power quality level in the grid connection, or in other words a unitary PF along with a lower current THD.

These two problems are the challenge for the control of a grid-connected inverter.

Due to the adopted control strategy, the amplitude of the current injected to the grid is not constant introducing some differences with respect to the expected ideal behavior.

Therefore the current of the inverter will be forced to follow a current reference given by 𝑖 𝑔 𝑟𝑒𝑓 = 𝑖 𝑚𝑎𝑥 sin 𝜔𝑡.

SLIDING MODE CURRENT CONTROL

Both types of commutation for the full-bridge inverter allow obtaining the tracking of a current reference to inject power into the grid. The use of the sliding mode to control this kind of system consists in defining a sliding surface using a sinusoidal current reference for the output current of the inverter. A fundamental condition ensuring that this strategy is efficient is that the dynamic behavior of the current reference is significantly slower than the current dynamics of the inductive array that couples the inverter to the grid. This fact imposes a switching frequency for the converter much higher than the frequency of the grid. Under these conditions, the output current of the inverter is constrained in a neighborhood of the current reference, obtaining the expected power injection. To prevent the problems associated with a very high switching frequency, a hysteresis band is introduced limiting the current variation around the current reference in the way illustrated in figure 4.2. Our full bridge inverter is coupled to the grid through a single inductor and decoupled to the DC link through a simple capacitor, so that it can be described by a second order PSI. Applying the equivalent control method to analyze the sliding motion [58], a reduced model of the converter dynamics is obtained. This model describes the dynamics of the system for a current reference imposed by the sliding surface.

However, if some harmonics are introduced in the reference by the control, they will also appear in the current of the inverter and therefore it will be necessary to consider the amplitude of the reference as a time-varying signal 𝑖 𝑚𝑎𝑥 .

Bipolar commutation

Defining the first order sliding surface:

𝑆(𝑥) = 𝑖 𝑔 𝑟𝑒𝑓 -𝑖 𝐿 = 𝑖 𝑚𝑎𝑥 sin 𝜔𝑡 -𝑖 𝐿 (4.15)
the current loop enforces that 𝑖 𝐿 = 𝑖 𝑔 𝑟𝑒𝑓 , and hence 𝑖 𝑔 = 𝑖 𝑚𝑎𝑥 sin 𝜔𝑡. The sliding motion is achieved using the control law:

𝑢 = { 1 𝑤ℎ𝑒𝑛 𝑆(𝑥) > 0 -1 𝑤ℎ𝑒𝑛 𝑆(𝑥) < 0 (4.16)
Applying the invariance conditions (𝑆(𝑥) = 0, 𝑆 (𝑥)| 𝑢=𝑢 𝑒𝑞 = 0) leads to:

𝑑𝑆(𝑥) 𝑑𝑡 = 𝑖 𝑔 𝑟𝑒𝑓 𝑑𝑡 - 𝑑𝑖 𝐿 𝑑𝑡 = 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 sin 𝜔𝑡 + 𝜔𝑖 𝑚𝑎𝑥 cos 𝜔𝑡 + 𝑣 𝑔 𝐿 - 𝑣 𝐶 𝑢 𝐿 = 0 (4.17)
The equivalent control is deduced as: When 𝑖 𝑚𝑎𝑥 is constant and equal to 𝐼 𝑚𝑎𝑥 , the reachability of the sliding surface can be obtained analyzing the condition 𝑆(𝑥) 𝑆 (𝑥) < 0, which leads to:

𝑢 𝑒𝑞 =
𝑣 𝐶 (𝑡) > √ 𝐿 2 𝜔 2 𝐼 𝑚𝑎𝑥 2 + 𝑉 𝑚𝑎𝑥 2 sin (𝜔𝑡 + atan [ 𝜔𝐿𝐼 𝑚𝑎𝑥 𝑉 𝑚𝑎𝑥 ]) (4.20)

Unipolar commutation

Defining the previous first order sliding surface 𝑆(𝑥) = 𝑖 𝑔 𝑟𝑒𝑓 -𝑖 𝐿 , the equality 𝑖 𝐿 = 𝑖 𝑔 𝑟𝑒𝑓 is enforced. The sliding motion is achieved using the same control law (4.16). Now, the invariance conditions (𝑆(𝑥) = 0, 𝑆 (𝑥)| 𝑢=𝑢 𝑒𝑞 = 0), lead to

The equivalent control can be deduced and is given by:

𝑢 𝑒𝑞 = 1 𝑣 𝐶 (𝐿 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 sin 𝜔𝑡 + 𝜔𝐿𝑖 𝑚𝑎𝑥 cos 𝜔𝑡 + 𝑉 𝑚𝑎𝑥 sin 𝜔𝑡 + 𝑣 𝐶 2 (1 -sign(𝑣 𝑔 ))) (4.22)
This equivalent control is then constrained to be in the interval -1 < 𝑢 𝑒𝑞 < 1.

Replacing the equivalent control (4. The bipolar operation of the inverter and the corresponding model will be retained in the following in order to simplify the experimental validation of the control design.

CONTROL OF THE DC-LINK

An interesting way to stabilize and regulate the DC voltage consists in considering the current control as the inner loop of a cascade type controller whose voltage outer controller modifies the current amplitude of the reference of the inner loop (see figure 4.4). In fact, the outer loop ensures that the average value of the voltage in the DC link is permanently regulated. Thus, the power balance on the overall microinverter is guaranteed because the voltage in the storage energy element of the middle-point (DClink) is kept around a constant value. Now, a dynamic model having the voltage of the DC link as function of the amplitude of the output current is necessary for designing appropriately the outer loop controller 𝐶(𝑠).

Model of the DC-Link voltage

Note that both types of commutation (unipolar, bipolar) lead to the same resulting sliding dynamics.

Recalling that sin 𝜔𝑡 cos 𝜔𝑡 = sin 2𝜔𝑡 /2 and sin 2

As it can be noted in (4.30), the voltage of the DC link is affected by the amplitude of the output current and also by the input power. Because the amplitude of the current will be considered as the manipulated variable, the power will be considered as a disturbance. By applying a simple feedback strategy, the block diagram depicted in figure 4.5 represents the closed loop system. Note that the transfer function from 𝐼 𝑚𝑎𝑥 (𝑠) to 𝑉 𝐶 (𝑠) exhibits a negative sign, and it is, in fact, a simple integrator with a stable zero. Note also that integration characterizes the transfer function from power to voltage variation. If we consider that the power varies by steps, the loop will be affected by disturbances of order 2 (ramps). The controller design has to be done taking into account all these specificities.

Synthesis of a linear controller

From the above discussion, since the system naturally possesses one integration, the controller must add another one. Moreover, the controller must attenuate the effect of the DC voltage ripple in the loop signals. For this reason, a first order filter is introduced in the controller; its bandwidth will be tuned for filtering the ripple whose frequency is 2𝜔. Considering the previous analysis the selected controller will be fixed as:

𝐶(𝑠) = - 𝐾 𝑐 (𝑇 𝑐 𝑠 + 1)
𝑠(𝑇 𝑓 𝑠 + 1) , 𝐾 𝐶 , 𝑇 𝐶 , 𝑇 𝑓 > 0 (

, where 𝑇 𝑓 is fixed to filter the ripple of the error signal and the sign minus compensates the sign of the converter transfer function. In addition, an analysis of the zero of the converter transfer function shows that, for the frequencies involved, it can be neglected, this leading to a simplified transfer function 𝐺(𝑠) given by:

𝐺(𝑠) = 𝑉 𝐶 (𝑠) 𝐼 𝑚𝑎𝑥 (𝑠) = - 𝑉 𝑚𝑎𝑥 2𝐶𝑉 𝑟𝑒𝑓 𝑠 = - 𝐾 𝑚𝑎𝑥 𝑠 (4.32)
For example, the value of 𝐿 considered in section 4.9 (experimental results) is 10 mH, Writing 𝑇 𝑐 = 𝑎𝑇 𝑓 with 𝑎 > 0, the discriminant ∆ will be positive if:

𝑉 𝑚𝑎𝑥 = 220√2 𝑉
𝑎 > 3 + 2√2 > 1 (4.37)
And the roots can be expressed in terms of 𝑇 𝑓 and 𝑎 as:

𝜔 1,2 = 1 2𝑎𝑇 𝑓 ((𝑎 -1) ± √ 𝑎 2 -6𝑎 + 1) (4.38)
It is easy to verify that the two roots are positive for the values of 𝑎 satisfying (4.37).

For the two frequencies 𝜔 1,2 , the phase of the controlled open loop transfer function is equal to -135°. If the gain for 𝜔 1,2 is equal to 1, then the phase margin will be equal to 45°. This is true if the gain of the controller 𝐾 𝑐 is selected as: Finally, the design method can be summarized by the following step by step procedure:

𝐾 𝑐 1,2 =
-Select 𝑇 𝑓 = 1/2𝜋𝑓 𝑓 to reject the ripple in the measurement of the DC voltage.

Values of 𝑓 𝑓 around 50Hz show a good trade-off between ripple-rejection and time response performance.

-Select the value of 𝑎 satisfying the condition (4.37). The value of 𝑎 affects considerably the time response only in a range near to the constraint (4.37).

-Compute the value of 𝑇 𝑐 , 𝜔 1 and consequently the controller gain 𝐾 𝐶 = 𝐾 1 . To design the controller using the aforementioned method, the set of parameters for the power converter and the power sources are listed in Table 4.4, which will be used afterwards to illustrate the theory by simulations and experimental results.

Simulation results

In order to verify the theoretical approach, several simulations have been implemented

in MATLAB for the converter model and PSIM for the converter circuit. In the PSIM model, the power source is implemented using a voltage controlled source whose value is defined by the output current for a constant value of power.

The ripple in the voltage of the DC bus is due to the AC nature of the output power but also to the harmonic content of the grid voltage. However, the rejection of the component oscillating at double frequency of the grid by a low pass filter ensures that the other components must be mainly attenuated. Hence, the value of the parameter 𝑇 𝑓 THD values lower than 5% are obtained for both cases.

POWER QUALITY ANALYSIS

Modeling of the voltage ripple of the DC-Link

In order to have a more accurate representation of the system variables and quantify the parametric dependence between them, it is possible to study the harmonic balance of the power between the DC side and the grid side. The AC side is defined by the expressions:

𝑣 𝑔 (𝑡) = 𝑉 𝑚𝑎𝑥 𝑠𝑖𝑛 𝜔𝑡 (4.41)

𝑖 𝑔 (𝑡) = 𝑖 𝑚𝑎𝑥 (𝑡) 𝑠𝑖𝑛 𝜔𝑡 (4.42)

Whereas the voltage of the DC side can be defined as:

𝑣 𝐶 (𝑡) = 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛(2𝜔𝑡 + 𝛾) (4.43)
Note that only an oscillatory component at the double frequency of the grid has been introduced and also note that this component has a phase displacement 𝛾 with respect to the grid voltage, which is considered as the phase reference. This assumption allows working with a moderate mathematical complexity in the analysis. As a consequence of the voltage regulation loop, the oscillatory component of the DC voltage appears in the current amplitude 𝑖 𝑚𝑎𝑥 (𝑡) with an average value of 𝐼 𝑚𝑎𝑥 and a harmonic of amplitude 𝐼 2 and phase 𝜙 which will depend on the frequency response of the loop compensator.

Then, 𝑖 𝑚𝑎𝑥 (𝑡) is expressed as:

𝑖 𝑚𝑎𝑥 (𝑡) = 𝐼 𝑚𝑎𝑥 + 𝐼 2 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) (4.44)
The sliding surface is defined by 𝑆(𝑥) = 𝑖 𝑔 -𝑖 𝐿 . At the surface, the invariance condition is fulfilled if 𝑆(𝑥) = 0 and 𝑆 (𝑥) = 0. The derivative of the current reference is given by:

𝑑𝑖

Total Harmonic Distortion of the output current

The presence of the component 𝐼 2 is a consequence of the ripple in the voltage of the DC bus and its propagation through the outer control loop. To quantify this aspect, the frequency response of the controller must be analyzed. We have: The validation of the above expression is graphically presented in figure 4.13. A reference of 400 V is used to the voltage regulation loop. The inverter is connected to a 220 V/ 50 Hz grid injecting a power of 100 W. The transfer function of the voltage controller is defined as follows:

𝐶(𝑠) = 𝐼 𝑚𝑎𝑥 (
𝐶(𝑠) = -0.1(0.06𝑠 + 1) 𝑠(0.005𝑠 + 1)

(4.75)
The 100 Hz component of the voltage ripple is attenuated for the controller with a gain of -54.7 dB. Further, a phase lead of 74° is observed between the AC components of the voltage and the current reference.

Considering that the PLL gives a perfect sinusoidal waveform with normalized amplitude, the harmonic content of the current can be only introduced through the amplitude signal 𝑖 𝑚𝑎𝑥 . The current 𝑖 𝑔 in expression (4.57) can be rewritten as: where 𝑖 𝑔1 = 𝐼 𝑔1 𝑠𝑖𝑛(𝜔𝑡 + 𝜑) and 𝑖 𝑔3 = 𝐼 𝑔3 cos(3𝜔𝑡 + 𝜙). Then, we have: However, the ripple content of the current has a non-negligible effect in the THD and hence must be also considered. Although the frequency varies in the ripple waveform, because of its triangular shape, the RMS value of the ripple can be approximated as 𝑖 𝑟𝑖𝑝 = Δ/√3. Then, it is possible to obtain the RMS values of the current components and then to compute the total harmonic distortion given by: 𝑇𝐻𝐷 -𝐹 ≅ √ 𝑅𝑀𝑆(𝑖 𝑔3 ) Note that a reduced current component 𝐼 2 directly increases the power factor; also, observe the effect of the angle 𝜙 in the power factor. As an illustrative example, the DPF and the PF have been computed for the same parameters used in the simulated results in subsection 4.5.2. For the first case (THD=7.98%) which corresponds to the worst THD, the values of DPF and PF are 0.997 and 0.994 respectively. For the second case (THD=4.45%) which corresponds to the best THD, the values of DPF and PF are 1.000 and 0.999 respectively.

𝑖 𝑔 = 1 2 √ 4𝐼 𝑚𝑎𝑥 2 -
𝑖 𝑔 =

Effect of a DC component in the current reference

The existence of a DC component in the current reference is a spurious phenomenon which can affect the general behavior of the grid-connected inverter either because the output current will exhibit a DC component or because this component can generate other effects [58]. The following analysis allows quantifying this effect.

If the current reference is given by:

𝑖 𝑔 (𝑡) = 𝑖 𝑚𝑎𝑥 (𝑡)[𝑎 + 𝑠𝑖𝑛(𝜔𝑡)]
, where 𝑎 is the DC component, then, we will have:

𝑖 𝑔 (𝑡) = [𝐼 𝑚𝑎𝑥 + 𝐼 2 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙)][𝑎 + 𝑠𝑖𝑛(𝜔𝑡)] (4.83)

RIPPLE COMPENSATION USING A FEED-FORWARD LOOP

Despite the presence of a low pass filter in the control loop, a small double frequency component is always perturbing current 𝑖 𝑚𝑎𝑥 . Hence, a complementary compensation can be added in order to obtain a more important reduction of the spurious component without increasing the order of the control system. This section introduces a feedforward approach which predicts the amplitude of the voltage ripple and reproduces it using the PLL system to generate a sinusoidal component with a frequency two times higher than the frequency of the grid. Thus, the main component of the ripple can be effectively attenuated before entering into the control loop, so that the filter can remove the remaining component more easily. The amplitude of the voltage ripple is predicted from (4.68) reproduced below:

𝑉 2 = √ (𝜔𝐿𝐼 𝑚𝑎𝑥 2 ) 2 + (𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 ) 2
4𝐶𝑉 𝑑𝑐 𝜔 Figure 4.16 shows the ripple compensation scheme. The PLL is one of the more important components of the system because it is used to generate both the sinusoidal grid frequency and the double grid frequency components.

For the mathematical operations required to compute the amplitude of the ripple, a digital implementation is preferred for practical reasons, taking into account that a complete digital implementation of the overall control scheme is possible. In this case, the current reference 𝑖 𝑟𝑒𝑓 is generated and given to an analog hysteresis comparator to obtain the desired tracking with a good frequency response. Thus, the proposed system requires the measurement of the DC bus voltage, the current in the coupling inductor and the voltage of the grid. Depending on the technique used in the PLL, this measure can be either detect the zero crossing or obtain an instantaneous measurement of the grid voltage signal. In a first approach, we propose the use of a zero crossing detection circuit which exploits a discrete square wave synchronized with the grid frequency. This type of PLL offers the possibility to have another square wave output with a frequency multiple of the grid frequency. This signal can be used to define a sampling frequency with the same number of samples per period. With this system, the mathematical computations required for the sine-wave generation are reduced since the signal waveform can be digitally stored in an electronic memory. In that case, it is possible during the released computational time to execute the required calculations to predict the ripple amplitude. The proposed control has been implemented in the PSIM simulations obtaining the results shown in Figure 4.18. As it can be noted, a reduction of 0.6% is obtained with the ripple compensation.

INPUT CURRENT OF THE INVERTER

In the microinverter structure, the inverter shown in Figure 4.19 is fed by a DC-DC converter connected to A PV module. The converter has an output capacitor which is also the coupling element of the inverter. Hence, the interaction between both converters can be studied determining the behavior of the input current 𝑖 𝑖𝑛 . In order to simplify the study, we apply the change of coordinates 𝑣 𝐶 ′ = 𝑉 2 𝑠𝑖𝑛(2𝜔𝑡′)

and then: After the substitutions𝑥 = 2𝜔𝑡′ and 𝑦 = 𝑉 𝑑𝑐 + 𝑉 2 sin 𝑥, we have:

𝑖 𝑑𝑐 ′ (𝑡) = 𝑃 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′ (
𝑎 1 = 𝑃 2𝜋𝑉 2 ln(𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′)| - 𝜋 2 𝜋 2 (4.94)
From which it is obtained that 𝑎 1 = 0. On the other hand:

𝑏 1 = 2 𝑇 ∫ 𝑃 sin 2𝜔𝑡′ 𝑑𝜔𝑡 ′ 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′ 𝜋 2 - 𝜋 2 = 𝑃 𝜋 ∫ sin 2𝜔𝑡′ 𝑑𝜔𝑡 ′ 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′ 𝜋 2 - 𝜋 2 (4.95)
By applying the substitution 𝑥 = 2𝜔𝑡′, we have:

𝑏 1 = 𝑃 2𝜋𝑉 2 [∫ 𝑑𝑥 - 𝛽 𝛼 𝑉 𝑑𝑐 ∫ 𝑑𝑢 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 𝑢 𝛽 𝛼 ] (4.96)
where 𝛼 = -𝜋 and 𝛽 = 𝜋. Note that the second integral is the same in (4.90), from which it is obtained that: Considering that the switching frequency is much higher than the grid frequency and that the hysteresis band is small when compared with the current amplitude, it is possible to analyze the motion of the current waveform considering a linear evolution during both intervals of the variable u (u=1, u=-1). Hence, the switching period and switching frequency in the bipolar commutation mode can be determined as follows:

𝑏 1 = 𝑃 2𝜋𝑉 2 [ 2𝜔𝑡′ - 1 √ 1 -( 𝑉 2 𝑉 𝑑𝑐 ) 2 [ atan ( tan 2𝜔𝑡′ 2 + 𝑉 2 𝑉 𝑑𝑐 √ 1 -( 𝑉 2 𝑉 𝑑𝑐 ) 2 ) ] ] -𝜋 2 𝜋 2 = 2𝑃 𝑉 2 ( -𝑉 𝑑𝑐 + √ 𝑉 𝑑𝑐 2 -𝑉 2 2 √ 𝑉 𝑑𝑐 2 -𝑉 2
𝑇 𝑠 = ∆𝑡| 𝑢=1 + ∆𝑡| 𝑢=-1 = 2𝛿𝐿 𝑉 𝐶 -𝑉 𝑔 + 2𝛿𝐿 𝑉 𝐶 + 𝑉 𝑔 = 4𝛿𝐿𝑉 𝐶 𝑉 𝐶 2 -𝑉 𝑔 2 𝑓 𝑠 = 𝑉 𝐶 2 -𝑉 𝑔 2 4𝛿𝐿𝑉 𝐶 (4.106)
The switching frequency is constrained within the interval 𝑓 𝑠 𝑚𝑖𝑛 > 𝑓 𝑠 > 𝑓 𝑠 𝑚𝑎𝑥 . The maximum frequency corresponds to the zero crossing of 𝑉 𝑔 and is given by:

𝑓 𝑠 𝑚𝑎𝑥 = 𝑉 𝐶 4𝛿𝐿 (4.107)
while the minimum one corresponds to the maximum value of 𝑉 𝑔 , having the expression:

𝑓 𝑠 𝑚𝑖𝑛 = 𝑉 𝐶 2 -𝑉 𝑚𝑎𝑥 2 4𝛿𝐿𝑉 𝐶 (4.108)
Using the same analysis for unipolar commutation, it is obtained that:

𝑇 𝑠 = ∆𝑡| 𝑢=1 + ∆𝑡| 𝑢=-1 = 2𝛿𝐿 𝑉 𝐶 -𝑉 𝑔 + 2𝛿𝐿 𝑉 𝑔 = 2𝛿𝐿𝑉 𝐶 𝑉 𝑔 (𝑉 𝐶 -𝑉 𝑔 ) 𝑓 𝑠 = 𝑉 𝑔 (𝑉 𝐶 -𝑉 𝑔 ) 2𝛿𝐿𝑉 𝐶 (4.109)
Only the maximum frequency can be determined by equalizing 𝑑𝑓 𝑠 /𝑑𝑉 𝑔 to zero, having the maximum frequency when the voltage 𝑉 𝑔 is equal to 𝑉 𝐶 /2, and:

𝑓 𝑠 𝑚𝑎𝑥 = 𝑉 𝐶 8𝛿𝐿 (4.110)

Efficiency analysis

In order to assess the efficiency of the sliding-mode current controlled inverter, both conduction and switching losses are considered. The conduction losses are produced in the ON resistance of the switches and also in the copper resistance of the inductor. To compute these losses, it is considered that the inductor current which is the same current flowing from inverter to the grid, continuously passes through two switches and the series resistance of the inductor. Then, considering that an amplitude value of the current ripple below 30% does not introduce a considerable deviation of the conduction power losses [59], we have

𝑃 𝑃 𝑐𝑜𝑛 = 𝐼 𝑚𝑎𝑥 2 √2 (2𝑅 𝑀 + 𝑅 𝐿 ) (4.111)
where 𝑅 𝑀 is the on-resistance of the MOSFETs and 𝑅 𝐿 is the equivalent series resistor of the inductor. The coupled capacitor is not considered in this analysis because it is included in chapter 2 as a part of the DC-DC converter. To obtain an estimation of the switching losses, the analysis of the current and voltage transitions in both turn-on and turn-off commutations is proposed. the switching losses in each power switch can be estimated as:

𝐸 𝑃 𝑂𝑁 = 1

Simulated results

From the above analysis it is possible to estimate the efficiency of the inverter by means of a computational tool. In this case a MATLAB-based algorithm has been created in order to have an approximation of the converter efficiency for different operational points of the inverter.

Then, the efficiency is computed for a set of values of grid voltage and output power considering the voltage in the DC bus regulated at a constant value. Figure 4.24 shows the simulated results for bus voltage of 400 VDC using the parameters listed in table 4.6. Intervals 𝑡 𝑜𝑓𝑓 and 𝑡 𝑜𝑛 have been approximated by 𝑡 𝑓 and 𝑡 𝑟 respectively. The control circuit is implemented using a microcontroller dsPIC30F4011 and some classical discrete electronic circuits. The current reference is stored in the microcontroller using a pre-sampled table with 1024 samples per period. The microcontroller uses an external signal given by an external PLL to make the corresponding counter. The current controller is digitally implemented by means of the microcontroller whereas the voltage controller is implemented with analog electronics.

Two integrated circuit IR2120 has been used as MOSFET drivers.

CONCLUSIONS

In this chapter, the concept of Power Source Inverter (PSI) has been introduced in order to theoretically obtain all the parameters which describe the dynamic performance and the power quality of a grid-connected inverter operating with a first order sliding mode control law tracking a pure sine current reference. From the ideal sliding dynamic model a proportional-integral controller with an additional pole has been designed to regulate the voltage of the DC bus attenuating at the same time the oscillatory component at double grid frequency. This design has been verified in simulations and experimental results.

Although the full-bridge inverter is a well-known topology and its behavior has been deeply studied for inverters working with PWM and hysteresis comparators, this chapter has proposed an analysis of the operation of the converter in an ideal sliding motion connected to the grid, this involving many aspects which have not been discussed previously in the literature. Among the most important ones, these aspects are the power factor, the displacement power factor, the total harmonic distortion, the effect of a DC component in the current and in general, the harmonic balance of the converter.

Simulations results have illustrated the theory and have been complemented at the end of the chapter by means of experimental results, which has shown the potentialities of the proposed PSI under sliding mode control.

Chapter 5

Microinverter: A Complete Sliding-Mode Approach

INTRODUCTION

In the introduction of this dissertation, a state-of the art of the module-integrated converters [1]- [4], micro-inverters [5]- [10] or AC modules [11]- [12] has been presented and our proposal concerning an "entirely-sliding-mode-controlled transformer-less single-phase dual-stage grid-connected micro-inverter" introduced [13]. As it can be observed in figure 5.1a, the proposed microinverter is composed of a power and a control blocks. A DC-DC and DC-AC converters are connected in series through a bus of 400 VDC.

In terms of control, as presented in figure 5

DERIVATION OF A GLOBAL MODEL

In chapter 3, the quadratic boost converter [14]- [15] has been studied to cover the requirements of the DC-DC stage of the microinverter. A linear model of the ideal sliding dynamics has been derived for the converter operating as a SM-LFR having a current source as load [16]. In that model, the system has three inputs and one output.

The inputs are the incremental values of conductance (𝐺), input voltage (𝑣 𝑝 ) and output current (𝑖 𝑜 ), and the output is the voltage of the output capacitor which is the same voltage of the DC bus (𝑣 𝑑𝑐 ). To complete that model to give a more realistic approach of the DC-DC converter inserted in the microinverter structure, we propose the insertion of the input capacitor obtaining the input current (𝑖 𝑝 ) as the input variable given by the PV module retaining the conductance (𝐺) as a variable given by the MPPT.

Furthermore, the DC-AC converter is modeled as a load considering the DC component of its input current (𝑖 𝑜 ) as a function of the control loop used to regulate the voltage of the DC bus. Then, we have an averaged model of the internal microinverter dynamics with two input variables and an output variable.

DC-DC converter including the input capacitor

First of all, a coupling capacitor is inserted in the input port of the DC-DC converter changing the model of the quadratic boost converter studied in chapter Recalling from chapter 3, the control strategy defining the SM-LFR behavior can be expressed as follows:

𝑆(𝑥) = 𝑖 𝐿1 -𝑣 𝑝 (𝑡)𝐺(𝑡) (5.2)

Applying the invariance conditions (𝑆(𝑥) = 0 and 𝑆 (𝑥) = 0) in (5.1) and (5.2) leads to the equivalent control: The equilibrium point is then defined by: 𝑖 𝐿1 ̅̅̅̅ = 𝑖 𝑝 ̅ = 𝑉 𝑝 𝐺; 𝑖 𝐿2 ̅̅̅̅ = (𝑉 𝑝 𝐺𝐼 0 ) 1 2 ;

(5.5) 𝑣 𝐶1 ̅̅̅̅ = 𝑉 𝑝 3 2( 𝐺/𝐼 0 ) 1 2 ; 𝑣 𝐶2 ̅̅̅̅ = (𝑉 𝑝 2 𝐺)/𝐼 0 Linearizing (5.4) around the equilibrium point (5.5), the equation system (5.6) is obtained where the superscript (~) stands for the increments of the variables in (5.4) around the corresponding equilibrium values. (5.11)

Modeling of the grid-connected DC-AC converter as a load

In chapter 4, a linear model has been derived for the ideal sliding dynamics of the inverter when it tracks a sinusoidal current reference. In that study, the converter has been treated as a Power Source Inverter (PSI) including the power decoupling capacitor and a voltage controller regulating the voltage of the DC bus. However, the coupling capacitor is also included in the study of the quadratic boost converter. Recalling that:

𝑣 𝑔 = 𝑉 𝑚𝑎𝑥 sin 𝜔𝑡 (5.12)

𝑖 𝑔 = 𝐼 𝑚𝑎𝑥 sin(𝜔𝑡 + 𝜑) (5.13) being the displacement angle 𝜑 near to zero . Then, the real power is given by: 𝑃 𝑔 ≈ 𝐼 𝑚𝑎𝑥 𝑉 𝑚𝑎𝑥 2 (5.14)

Ideally the power at the DC bus is equal to the output power and then, we can write: The loop transfer function is given by: (5.24)

𝑃 𝑑𝑐 =
As it can be noted in (5.24) the coefficients of the characteristic polynomial depends on the equilibrium values of the input and output voltages (𝑚), the converter parameters, the instantaneous conductance given by the MPPT (𝐺) and the voltage controller parameters. The stability of the system in the neighborhood of each equilibrium point can be evaluated. However, the operational range of the PV module variables is required and hence a brief analysis is presented before.

Characterization of the PV module

The operational ranges of the PV module have been obtained from the I-V and P-V characteristics shown in figure 5 𝑁 𝑠 is the number of cells conforming the PV module, A is the diode quality factor, K is the Boltzmann constant (1.3806504 × 10 -23 J/K) and q is the electron charge

Stability of the linear model

In order to assess the stability of the closed loop system, the characteristic polynomial is numerically evaluated using values in table 5.2. The parameters of the PI controller are the same as in chapter 4 (𝐾 𝑐 = 0.7138, 𝑇 𝑐 = 0.06, 𝑇 𝑓 = 0.004). The roots of the characteristic polynomial are evaluated for all the limit conditions (ranges of parameters M, G, and 𝑉 𝑚𝑎𝑥 ) finding that the system is asymptotically stable.

The resulted loci are shown in figure 5.4. As it can be noted in figure 5.4a, the system has five roots. One root is completely real and two pairs are complex conjugate. The real root is far from the origin and hence their influence is negligible. Figure 5.4b shows that a pair of complex conjugate roots are much closer to the origin that the other one and hence the pair of roots depicted in figure 5.4c is dominant. Also, it is possible to observe that the system is always stable for the complete range of operational conditions and the given set of parameters. As it can be observed, the PV module is modeled as the incremental variation of the current controlled source 𝐼 𝑝 (𝑠) which is a function of the incremental variation in the conductance 𝐺(𝑠) and the maximum power 𝑃 𝑚 (𝑠). Then, from this approach, each maximum power point is represented by its associated conductance 𝐺 𝑚 (𝑠). including the high frequency component) which implies a power factor near one (0.99).

Inclusion of the MPPT algorithm

The average power is computed showing that the total amount of power available at the PV module side is injected into the grid. It is worth to note that these simulations have been performed without including parasitic resistive elements.

Figure 5.12 shows the simulated steady-state waveforms of the DC bus. The voltage of the bus is regulated to the desired value of 400 V showing an acceptable voltage ripple.

The input current of the inverter is depicted and its average value is about 0.28 A, which allows injecting a power of about 120 W. These results confirm that the control of the DC-AC stage as in the way studied in chapter 4 is consistent and compatible with the other control parts of the microinverter. It is worth to note that during the start-up, the voltage controller of the DC bus suffers a large deviation which is quickly controlled. The maximum observed variation in this case is around 100 V over the reference value of 400 V.

EXPERIMENTAL RESULTS

The entire microinverter system has been studied in a single experimental set-up in order to validate the theoretical predictions and simulation results.

Prototype and experimental set-up

The microinverter system is obtained by assembling the circuits developed in chapter

Figure 1 .

 1 Figure 1. Schéma du micro-onduleur identifiant les étapes de conversion d'énergie et les différentes fonctionnent de commande

Figure 2 .

 2 Figure 2. Trajectoire non linéaire de la puissance vers le maximum et représentation graphique dans le temps des variables de l'algorithme MPPT dans un régime établi

Figure 3 .

 3 Figure 3. Convertisseur élévateur quadratique

Figure 4 .

 4 Figure 4. Diagramme d'évènements des modes de conduction du convertisseur élévateur quadratique : a) Représentation générale ; b) Mode de conduction continue ; c) Mode de condition discontinu dans l'inductance L1 ; d) Mode de conduction discontinue dans l'inductance L2 ; e) et f) Modes de conduction discontinue dans les deux inductances.

Figure 6 .

 6 Figure 6. Diagramme du circuit de l'onduleur de source de puissance de pont complet raccordée au réseau.

Figure 7 .

 7 Figure 7. Schéma fonctionnel du control du l'onduleur raccordée au réseau

Figure 7 .

 7 Figure 7. Représentation du system de control montrant l'interaction entre l'étape CC-CC et l'étape DC-AC à travers de ses fonctionnes transfert

Figure 9 .

 9 Figure 9. Montage expérimental du micro-onduleur complet

Figure 10 .

 10 Figure 10. Mesures expérimentaux du régime établi et de la réponse dynamique du system complet

Fig. 1 . 1 .

 11 Fig. 1.1. Different types of photovoltaic installation: a) centralized inverters, b) string and multi-string inverters, c) module oriented inverters.

Fig. 1 . 2 .

 12 Fig. 1.2. Photovoltaic system based on "microinverters". [Source: www.emphase.com]

Fig. 1 . 3 .

 13 Fig. 1.3. Technology of interconnection of micro-grid including micro-inverters and smart-phone apps

Fig. 1 . 4 .

 14 Fig. 1.4. General scheme of the sliding-mode controlled microinverter

Fig. 2 . 1 .

 21 Fig. 2.1. Asymptotic approach of the characteristic curves of the PV module: a) I-V; b) P-V.

3 )where

 3 𝐺 2 = 𝐼 𝑠𝑐 2 /𝑃 𝑚𝑝 and 𝐺 1 = (𝐺 0 2 𝑉 𝑜𝑐 2 [1 -√1 -4𝑃 𝑚𝑝 /(𝑉 𝑜𝑐 2 𝐺 0 )] -2𝐺 0 𝑃 𝑚𝑝 ) /2𝑃 𝑚𝑝 . The resulting characteristic is depicted in figure 2.2.

Fig. 2 . 2 .

 22 Fig. 2.2. Asymptotic approach of the characteristic curves of the PV module: a) P-V; b) P-G.

Fig. 2 . 4 .

 24 Fig. 2.4. Ideal switching functions of the proposed sliding-mode MPPT.

Fig. 2 . 5 .

 25 Fig. 2.5. Switching functions of the proposed sliding-mode MPPT.

Figure 2 . 6 .

 26 Figure 2.6. Implemented nonlinearities, waveforms and trajectory on the power characteristic

  Hence, it is possible to control the frequency 𝑓 𝑚𝑝 by an appropriate selection of the free parameters Δ, 𝑀 and K 2 . Now, the amplitude of the fluctuations of the power reference around the maximum point 𝑃 𝑚𝑝 is evaluated. Its determination is illustrated in figure 2.7.

Figure 2 . 7 .

 27 Figure 2.7. Determination of the amplitude of power reference and conductance at the maximum power point.

  .8). Due to this sudden change at 𝑡 𝑚𝑝2 , 𝑃 𝑝 (𝑡) suffers a discontinuity and the panel characteristic suddenly changes. It corresponds to a new characteristic whose powers are higher (see figure 2.8).

Fig. 2 . 8 .

 28 Fig. 2.8. Power point after sudden irradiance changes from a lower to a higher irradiance.

2

 2 cover all cases the maximum value is equal to 50. The constants of the algorithm must verify 𝐾 constants satisfying all the constraints are given by 𝑀 = 100, 𝐾 1 = 0.1, 𝐾 2 = 40 and Δ = 20. To test the algorithm, two commutations among the previous f-u characteristics are made as depicted in the figure 2.10. For the first commutation occurring at 𝑢 = 4, the algorithm jumps from the curve 𝑓 1 (maximum of 800) to the curve 𝑓 2 (maximum of 200). The second commutation occurs at 𝑢 = 8, and the algorithm jumps from the curve 𝑓 2 (maximum of 200) to the curve 𝑓 3 (maximum of 600).

Fig. 2 .

 2 Fig. 2.10. Simulated commutations

Fig. 2 .

 2 Fig. 2.11. a)Evolution of fref and f versus time; b) evolution of the variable u versus time

Fig. 2 . 12 .

 212 Fig. 2.12. Trajectory in the plane f ref versus u.

Fig. 2 .

 2 Fig. 2.13. Zoom of the trajectories in the f-u plane.

Fig. 2 . 14 .

 214 Fig. 2.14. Zoom on trajectories of f ref and u versus time for the maximum 800.

  The general block diagram of the implemented model is depicted in figure 2.15 while the PV module parameters are shown in figure 2.16. Note that a block representing the DC-DC converter (quadratic boost converter) and a block representing the solar panel has been included to evaluate the relation between the MPPT variables and the system variables.

Fig. 2 .

 2 Fig. 2.15. Block diagram of the power stage connected to the PV module and the MPPT (LPF: Low Pass Filter; SM-LFR: Sliding-Mode Loss-Free-Resistor)

Fig. 2 .

 2 Fig. 2.16. Graphic interface defining the parameters of the Solar Module in PSIM

Figure 2 .

 2 Figure 2.18 shows the trajectory of the system in the P-G plane towards the maximum power point. The power reference shows a limit cycle when the system yields the maximum whereas the output power is shown invariant. Moreover, it is possible to observe the motion of the system towards the maximum power point. Complementarily, the figure 2.19 depicts the stationary state of the MPPT variables showing the relation between the time response and the P-G plane representation.

Fig. 2 .

 2 Fig. 2.17. Detail of the simulated oscillatory components of current, voltage and powers.

Fig. 2 .

 2 Fig. 2.18. Steady-state behavior of the proposed MPPT algorithm

Fig. 2 .

 2 Fig. 2.20. Evolution of the power towards the maximum power point after the system's start-up

Fig. 2 .

 2 Fig. 2.22. Zoom of the simulated steady state of power and conductance in the P-G plane.

Fig. 2 . 23 .

 223 Fig. 2.23. Response of voltage, current and power during the system's start-up (K 1 = 0.2, K 2 = 10 and M = 40)

Fig. 2 . 2 . 5 . 6

 2256 Fig. 2.24. Simulated time response of the algorithm to irradiance changes

Fig. 2 .

 2 Fig. 2.25. Simulated results using sudden changes of 9 W in the maximum power P-V characteristic

7 .

 7 The filters are used to mitigate the high frequency ripple component in the case of the current and the peaks induced in the voltage as effect of the commutation of the DC-DC converter. A constant sampling frequency of 50 kHz has been established in the microcontroller. The serial digital-to-analog (DAC) converter MCP4812 has been used in order to obtain the continuous signal 𝐺 which used in the control of the DC-DC converter and the signal 𝑃 𝑝 𝑟𝑒𝑓 (VOUTB) which is observed for analysis. Two digital pins of the micro-controller are used to reproduce the corresponding signals of the non-linear functions 𝑢 and 𝑣 for the experiments.Figure2.27 illustrates the schematic diagram of the MPPT circuit.

Fig. 2 .

 2 Fig. 2.27. Schematic electronic circuit for the MPPT algorithm

Figure 2 .

 2 Figure 2.30 shows the steady-state waveforms when an average power of 91.2 W is extracted from the PV emulator and injected in a voltage load of 400 V through a quadratic boost converter. The PV emulator has been configured with an Isc of 4 A and a Voc of 25 V. The maximum derivative for this module using the asymptotic model of the PV model and equations (4.3) and (4.4) is | dP p dG | max = 531. Then, the parameters of the algorithm are defined as K 1 = 0.015, K 2 = 20, M =500, Δ = 5. The double frequency of the power waveform in comparison with the frequency of the voltage andcurrent waveforms confirms that the system works at the maximum power point (considering a single maximum operational condition). Further, the oscilloscope capture shows the measured power computed from the product of the filtered and conditioned current and voltage, which is compared with the power reference.The static performance of the MPPT algorithm operating in the steady state (one cycle) can be obtained using the expression 𝜼 = 𝑷 𝒑 ̅̅̅̅ /𝑷 𝒑 𝒎𝒂𝒙 , when 𝑃 𝑝 ̅ is the average value of the power and 𝑃 𝑝 𝑚𝑎𝑥 is the maximum power. Using this definition, a 99.2%

Fig. 2 .

 2 Fig. 2.30. Detail of the experimental oscillating components of the current, voltage and powers.

Fig. 2 .

 2 Fig. 2.31. Steady-state behavior of the proposed MPPT algorithm

Fig. 2 .

 2 Fig. 2.32. Steady-state behavior of the proposed MPPT algorithm

Fig. 2 .

 2 Fig. 2.33. Response of the power towards the maximum power point after the system's start-up

Fig. 2 .

 2 Fig. 2.34. Measured trajectory of the power towards the maximum power point in the P-G plane showing the variable structure behavior while approaching to the maximum power point

Fig. 2 .

 2 Fig. 2.35. Zoom of the experimental steady -state behavior of powers and conductance in the P-G plane.

Fig. 2 .

 2 Fig. 2.36. Time response of the voltage, current and power of the system during a start-up

Fig. 2 .

 2 Fig. 2.37. Time response of the algorithm to irradiance changes

Fig. 2 .

 2 Fig. 2.38. Experimental results for 30 W sudden changes in the P-V characteristic curve.

Fig. 2 .

 2 Fig. 2.40. General validation of the performance of the proposed MPPT algorithm.

  photovoltaic conversion[72].Considering these two topologies, namely, the Rotated Cell Single Switch (RC𝑆 2 -𝑄𝐵) and the Software Synthesized Single Switch (𝑆 4 -𝑄𝐵), and adding the Cascaded Connected Double Switch (𝐶 2 𝐷𝑆 -𝑄𝐵) and the Cascaded Connected Single Switch (𝐶 2 𝑆 2 -𝑄𝐵), we report a comparative study of efficiency and performance based on simulation results which is also complemented with experimental results[73]. Figure3.1 shows the aforementioned four quadratic boost converter topologies.

Fig. 3 . 1 .

 31 Fig. 3.1. Circuit diagram of the quadratic boost converter topologies: a) 𝐶 2 𝐷𝑆 -𝑄𝐵; b) 𝐶 2 𝑆 2 -𝑄𝐵: c) RC𝑆 2 -𝑄𝐵; and, d) 𝑆 4 -𝑄𝐵.

(

  states) giving origin to a continuous conduction mode and three different discontinuous conduction modes. Two of these structures are attained when only one inductor current either 𝐿 1 or 𝐿 2 reaches zero before a switching period is finished, thus originating the DCL1 state (Discontinuous Conduction on inductor 𝐿 1 ) or the DCL2 state (Discontinuous Conduction on inductor 𝐿 2 ) respectively. The third structure is found when the current of both inductors reach the value zero before a switching period is finished (DCL12 state).

Fig. 3 . 2 .

 32 Fig. 3.2. Circuit diagrams of quadratic boost converter states: a) on-state; b) off-state: c) DCL1-state; d) DCL2-state; and, e) DCL12-state.

Fig. 3 . 3 .

 33 Fig. 3.3. Event diagram of the quadratic boost converter conduction modes: a) general representations of states and transitions, b) CCM mode, c) DCL1 mode, d) DCL2 mode, e) DCL12 mode, f) DCL21 mode.

Fig. 3 . 4 .

 34 Fig. 3.4. Event diagram of the quadratic boost converter using a hysteresis comparator.

  the gain of the converter as a function of either the duty cycle (CCM) or the duty cycle and the output load (DCL1, DCL2 and DCL12 modes). It is also possible to obtain the static relations between the average values of either the capacitor voltages and the input voltage or the inductor currents and the input and load currents. Further, a more complex relation is obtained when the converter enters in the mode DCL2 using a hysteresis modulator because of the interdependence among the switching frequency, the voltage gain and the steady state average value of the voltages. All of them are now discussed in detail. The mean values of the converter variables in steady-state are represented by the notation 〈•〉. The control variables are substituted by the variables 𝑈, 𝑈 1 and 𝑈 2 which

Fig. 3 . 5 .

 35 Fig. 3.5. Current waveform of the inductor L1 when the converter operates in DCL1 mode.

Fig. 3 . 6 .

 36 Fig. 3.6. Current waveforms when the converter operates in DCL2 mode: a) diode D1, b) inductor L2

Fig. 3 . 7 .

 37 Fig. 3.7. Current waveforms when the converter operates in DCL12 mode: a) inductor L1, b) diode D1, c) inductor L2.

. 73 )Figure 3 .

 733 Figure 3.8 shows the domains of the different conduction modes where it is possible to identify the regions for CCM, DCL1, DCL2 and DCL12 modes as a function of the duty cycle U and the DCM constants 𝐾 1 and 𝐾 2 . As it can be observed, the discontinuous regions are concentrated around the low cycles and low values of 𝐾 1 and 𝐾 2 , which correspond to low power levels. We can note that the converter cannot work in the region of high duty cycles because the existence of the discontinuous conduction modes requires always to satisfy the conditions 𝑈 1 > 1 -𝑈 or 𝑈 2 > 1 -𝑈 or both.

Fig. 3 . 8 .Figures 3 .

 383 Fig. 3.8. Graphical representation of the critical values 𝐾 1 𝑐𝑟𝑖𝑡 and 𝐾 2 𝑐𝑟𝑖𝑡 as functions of the duty cycle U: a) CCM and DCL1 mode (𝐾 1 vs. U), b) CCM and DCL2 mode (𝐾 2 vs. U), c) CCM, DCL1 mode and DCL12 mode (𝐾 2 vs. U, 𝐾 1 ); d) CCM, DCL2 mode and DCL21 mode (𝐾 1 vs. U, 𝐾 2 ).

  figure 3.9, for a set of converter parameters (𝐿 1 , 𝐿 2 , 𝐶 1 , 𝐶 2 and 𝑓 𝑠 ) listed in the figure caption, the gain of the converter is represented for the different conduction modes as a function of the duty cycle and output load R showing their influence.

Fig. 3 . 9 .

 39 Fig. 3.9. Ideal converter gain vs. duty cycle (Parameters: 𝐿 1 = 120 𝜇𝐻, 𝐿 2 = 820 𝜇𝐻, 𝐶 1 = 𝐶 2 = 10 𝜇𝐹)

Fig. 3 .

 3 Fig. 3.11. Graphical representation of the reachability and invariance properties of the sliding mode control

Fig. 3 .

 3 Fig. 3.12. Block diagram of the indirect stabilization of the quadratic boost converter

  𝐴(𝑠)𝑉 𝐶2 (𝑠) = 𝐵(𝑠)𝐼 𝐸 (𝑠) + 𝐶(𝑠)𝑉 𝑖 (𝑠) + 𝐷(𝑠)𝐼 0 (𝑠) (3.82) This function can be represented by means of the block diagram in figure 3.13.

Fig. 3 . 13 .

 313 Fig. 3.13. Block diagram of the linearized model

Fig. 3 . 14 .

 314 Fig. 3.14. Time response of the switching model compared with the linearized model when the converter operates with an input current controller (Parameters: L1=120 µH, L2=4.7 mH, C1=C2=11 µF, R=1600 Ω): a) output voltage V C2 (s), b) input current compared with the input current reference, c) input waveforms 𝐼 𝐸 (s), 𝑉 𝑖 (s) and 𝐼 0 (s).

Figure 3 .

 3 Figure 3.14 shows the transient response produced in the output voltage when a sequence of positive or a negative excitation is applied firstly in the current reference 𝐼 𝐸 (𝑠), subsequently in the input voltage 𝑉 𝑖 (𝑠) and finally in the output current 𝐼 𝑜 (𝑠).

Fig. 3 . 15 .

 315 Fig. 3.15. Representation of a Loss-Free-Resistor (LFR) as a two-port conversion system.

Figure 3 .

 3 16 shows the diagram of the corresponding control system.

Fig. 3 . 16 .

 316 Fig. 3.16. Block diagram of the quadratic boost converter operating as a SM-LFR

  𝑉 𝐶2 (𝑠) = 𝐺 𝑔 (𝑠)𝐺(𝑠) + 𝐺 𝑣𝑖 (𝑠)𝑉 𝑖 (𝑠) + 𝐺 𝑖𝑜 (𝑠)𝐼 0 (𝑠) (3.105) This function can be represented by means of the block diagram in figure 3.17

  .

Fig. 3 . 17 .

 317 Fig. 3.17. Block diagram of the linearized model when the converter operates as a SM-LFR

Fig. 3 .

 3 Fig. 3.18. Time response of the switching model compared with the linearized model when the converter operates with an input current controller (Parameters: L1=120 µH, L2=4.7 mH, C1=C2=11 µF, R=1600 Ω): a) output voltage V C2 (s), b) input current compared with the input current reference, c) input waveforms𝐺(s), 𝑉 𝑖 (s) and 𝐼 0 (s).

By linearizing ( 3 .

 3 101) around the equilibrium point (3.103), the equation system (3.113) is obtained where the superscript (~) stands for the increments of the variables in (3.101) around the corresponding equilibrium values.

Fig. 3 .

 3 Fig. 3.19. Time response of the switching model compared with the linearized model when the converter operates with an input current controller (Parameters: L1=120 µH, L2=4.7 mH, C1=C2=11 µF): a) output voltage V C2 (s), b) input current compared with the input current reference, c) input waveforms𝐺(s), 𝑉 𝑖 (s) and 𝐼 0 (s).

Fig. 3 .

 3 Fig. 3.20. Block diagram of the controlled system stabilizing the SM-LFR operation when converter copes with a constant current source load.

Fig. 3 .

 3 Fig. 3.20. Comparison of the switching model with the linearized model when the converter operates as a SM-LFR with output voltage regulation via output current (Parameters: L1=120 µH, L2=4.7 mH, C1=C2=11 µF, Kp=0.01 and Ki=1): a) output voltage v C2 (t); b) input current i L1 (t) and current reference i E (𝑡); c) current on inductor L2 i 𝐿2 (𝑡); and d) input disturbance waveforms 𝐺(s) and 𝑉 𝑖 (s).

  2 can be complemented by introducing the parasitic resistive components in inductors, capacitors and controlled switches. Further, a complete model of diodes can be included using a series resistor and a DC source which allows representing the effect of the current in the forward voltage. The circuit diagrams corresponding to the improved model of the converter operating in CCM are shown if figure 3.21.

Fig. 3 .

 3 Fig. 3.21. Circuit diagram of the ON-state of the non-ideal quadratic boost converter

Figure 3 .

 3 22 shows a comparison of output voltage curves obtained with the ideal model, two different values of resistive load and two different values of output voltage. As it can be observed, the converter must operate in a range of duty cycle from 0.72 to 0.83 approximately to obtain an output voltage of 400 V for input voltages between 15 V and 30 V, and output loads between 16 W and 100 W. However, it is worth mentioning that the switching losses have not been already considered and hence some deviations can be observed.

Fig. 3 .

 3 Fig. 3.22. Output voltage vs. duty cycle for different input voltages and output loads

Fig. 3 .

 3 Fig. 3.23. Circuit diagram of the on-state of the non-ideal quadratic boost converter

Fig. 3 .

 3 Fig. 3.24. Algorithm to estimate the efficiency of the quadratic boost converter.

Fig. 3 .

 3 Fig. 3.25. Results of the efficiency prediction algorithm: a) Efficiency surface; b) Switching frequency surface; c) Efficiency as a function of the output load; d) Frequency as a function of the input voltage.

Fig. 3 .

 3 Fig. 3.26. Schematic circuit of the control board of the DC-DC converter

Fig. 3 .

 3 Fig. 3.27. Experimental set-up and converter prototype

Fig. 3 .

 3 Fig. 3.30. Oscilloscope waveforms of the output voltage when step type changes are introduced in the input voltage: a) 400 V -340 V -400 V, b) 400 V -470 V -400 V.

Fig. 3 .

 3 Fig. 3.31. Block diagram of the converter control used in the efficiency measurements

Fig. 3 .

 3 Fig. 3.32. Efficiency surface as a function of the input voltage and the output power (Parameters: 𝐿 1 = 120 𝜇𝐻, 𝐿 2 = 4.7 𝑚𝐻, 𝐶 1 = 𝐶 2 = 10 𝜇𝐹, 𝑅 = 10 𝑘Ω, 𝑉 𝑜 = 400 𝑉)

Fig. 4 .

 4 Fig. 4.1. H-bridge grid-connected PSI.

  𝑺 𝟒 are ON 𝑺 𝟐 , 𝑺 𝟑 are OFF 𝑣 𝐿 = 𝑣 𝐶 -𝑣 𝑔 𝑖 𝐶 = 𝑖 𝑑𝑐 -𝑖 𝑖𝑛 𝑖 𝑖𝑛 = 𝑖 𝐿 b) 𝑺 𝟏 , 𝑺 𝟒 are OFF 𝑺 𝟐 , 𝑺 𝟑 are ON 𝑣 𝐿 = -𝑣 𝐶 -𝑣 𝑔 𝑖 𝐶 = 𝑖 𝑑𝑐 -𝑖 𝑖𝑛 𝑖 𝑖𝑛 = -𝑖 𝐿 By defining the control variable 𝑢 such that 𝑢 = 1 when 𝑆 1 and 𝑆 4 are on, and 𝑢 = -1 when 𝑆 2 and 𝑆 3 are on, the following equations are obtained:

4 )

 4 𝑺 𝟒 are ON 𝑺 𝟐 , 𝑺 𝟑 are OFF 𝑣 𝐿 = 𝑣 𝐶 -𝑣 𝑔 𝑖 𝐶 = 𝑖 𝑑𝑐 -𝑖 𝑖𝑛 𝑖 𝑖𝑛 = 𝑖 𝐿 b) 𝑺 𝟐 , 𝑺 𝟒 are ON 𝑺 𝟏 , 𝑺 𝟑 are OFF 𝑣 𝐿 = -𝑣 𝑔 𝑖 𝐶 = 𝑖 𝑑𝑐 By defining the control variable 𝑤 such that 𝑤 = 1 when 𝑆 1 and 𝑆 4 are on, and 𝑤 = -1 when 𝑆 2 and 𝑆 3 are on, the following equations are obtained for the positive half-cycle of the grid voltage: Operation in the negative half-cycle is similar but the inductor current flows in the opposite direction. Diagrams of the two possible structures of the circuit operating in the positive half-cycle and its corresponding equations are shown in table 4.3. The following equations are obtained for the negative half-cycle of the grid voltage: 𝐿 𝑑𝑖 𝐿 𝑑𝑡 = -𝑣 𝑔 -𝑣 𝐶 (1 -𝑤) (4.5) 𝐶 𝑑𝑣 𝐶 𝑑𝑡 = 𝑖 𝑑𝑐 -𝑖 𝐿 𝑤 = 𝑃 𝑣 𝑐 + 𝑖 𝐿 (1 -𝑤) (4.6)

  𝑺 𝟒 are OFF 𝑺 𝟐 , 𝑺 𝟑 are ON 𝑣 𝐿 = -𝑣 𝐶 -𝑣 𝑔 𝑖 𝐶 = 𝑖 𝑑𝑐 -𝑖 𝑖𝑛 𝑖 𝑖𝑛 = -𝑖 𝐿 b) 𝑺 𝟐 , 𝑺 𝟒 are OFF 𝑺 𝟏 , 𝑺 𝟑 are ON 𝑣 𝐿 = -𝑣 𝑔 𝑖 𝐶 = 𝑖 𝑑𝑐 Then, it is possible to write equations (4.3), (4.4), (4.5) and (4.6) in a more compact form given below: 𝐿 𝑑𝑖 𝐿 𝑑𝑡 = -𝑣 𝑔 + 𝑣 𝐶 𝑤 -𝑣 𝐶 2 (1 -sign(𝑣 𝑔 )) (4.7)

Fig. 4 .

 4 Fig. 4.2. H-bridge grid-connected PSI.

Figure 4 .

 4 Figure 4.3 shows the use of a current control loop in the inverter, obtaining the reference calculated from the value of the available power in the DC bus. The sine waveform synchronized with the grid is obtained from a PLL. However, the stability of the voltage in the DC bus is not guaranteed and hence it is necessary to complete the control strategy using an outer loop.

Fig. 4 . 3 .

 43 Fig. 4.3. Block diagram representing the current loop control of the inverter.

Fig. 4 . 4 .

 44 Fig. 4.4. Block diagram of the two-loop controller for the grid-connected inverter.

Fig. 4 . 5 .

 45 Fig. 4.5. Block diagram of the closed-loop control system.

Fig. 4 . 6 .

 46 Fig. 4.6. Bode plot showing the two possible frequencies to achieve a phase margin of 45° using the proposed controller

Figure 4 .

 4 Figure 4.7 shows the transient responses of the closed-loop system using the results in Table 4.5. The responses have been generated considering a step change in the voltage reference of 20 V. It is worth pointing out that only a slight difference can be noted in the overshoot and settling time. Also, increasing the value of 𝑎 does not lead to significant changes in the form of transient response.

Fig. 4 . 7 .

 47 Fig. 4.7. Responses of the closed-loop system to a step change in the reference of 50V, having controllers for the values of 𝑎 in table 4.5: a) 𝐾 𝑐 = 𝐾 𝑐 1 ; and, b) 𝐾 𝑐 = 𝐾 𝑐 2

Fig. 4 . 8 .

 48 Fig. 4.8. Disturbance rejection of the closed-loop system using the values of 𝑎 in table 4.5: a) responses for 𝐾 𝑐 = 𝐾 𝑐 1 ; and, b) responses for 𝐾 𝑐 = 𝐾 𝑐 2

Figure 4 .Fig. 4 . 9 .

 449 Figure 4.9 shows the transient response of the system when a step is applied to the reference value. When the equilibrium point of 400 V for the DC bus voltage is attained, the reference value changes to 450 V at 0.6 s. The reference value returns to 400 V abruptly after one second. It is possible to observe that the controlled system has the desired regulation with zero error and shows a good performance in the time response.

Figure 4 .

 4 Figure 4.10 shows the transient responses of the system when a sudden additive disturbance is applied to the input power. After reaching the equilibrium point of 400 V for an input power of 50 W, the input power changes to 100 W at 0.2 s. The power returns to 50 W after 0.5 seconds. It is possible to observe that the controlled system rejects the disturbance showing a maximum voltage deviation of 15% controlled within 0.2 s.

Fig. 4 .

 4 Fig. 4.10. Simulation results of disturbance rejection for sudden changes in the input power

Figure 4 .

 4 11 shows the waveform of the inductor current for the two limits of power in the operational range: 20 W and 100W.

Fig. 4 .

 4 Fig. 4.11. Current waveforms for two operational points of the grid connected inverter: a) 20 W; b) 100 W.

(4. 65 )

 65 times lower. Then, the desired maximum amplitude of the voltage ripple can be imposed by means of the passive elements. The inductor constrains the derivative of the output current and hence is designed for a maximum current value and a hysteresis size, so that, the parameter which defines the voltage ripple is the DC bus capacitor.

Fig. 4 .

 4 Fig. 4.12. Validation of the predicted parameters of the ripple in the voltage of the DC bus a) case 1 (L = 100 mH, C = 27 µF); b) case 2 (L = 10mH, C = 47 µF).

Fig. 4 .

 4 Fig. 4.13. Validation of the predicted parameters of the current amplitude 𝑖 𝑚𝑎𝑥 (L = 10mH, C = 47 µF).

(4. 85 )Fig. 4 .

 854 Fig. 4.15. Simulated voltage ripple on the DC bus: a) without DC component; b) with DC component.

Figure 4 .

 4 Figure 4.15 shows some simulated results when a DC component of 0.2A is introduced in the current reference. As it can be observed a component at the grid frequency appears in the voltage of the DC bus also increasing the effective amplitude of the ripple.

Fig. 4 .

 4 Fig. 4.16. Block diagram of the two-loop controller for the grid-connected inverter including the feed-forward ripple compensation based on the double-frequency PLL.

Figure 4 .

 4 Figure 4.17 illustrates the proposed sine-wave generation using a low sampling frequency to obtain a better graphical representation. It is possible to identify both sampled sine-waves using the same switching frequency: the high frequency signal (HF) generated by means of the discrete PLL and the zero crossing signal. Although this solution is not efficient with distorted voltage signals, it is important to note that the microinverter will work connected to a system which must satisfy the international standards which impose a THD less than 5% in the grid voltage. The sampling frequency used in the simulations is 2 kHz and it can be increased until 40 or 50 kHz.

Fig. 4 .

 4 Fig. 4.17. Signals involved in the proposed PLL system.

Fig. 4 .

 4 Fig. 4.18. Influence of the voltage ripple in the current harmonic distortion (L = 10 mH, C = 27 µF): a) case 1 (L = 100 mH, C = 27 µF); b) case 2 (L = 10mH, C = 47 µF).

Fig. 4 .

 4 Fig. 4.19. Currents in the H-bridge grid-connected PSI.

(4. 101 )Fig. 4 . 1 Figure 4 .Fig. 4 .

 1014144 Fig. 4.20. a) Current waveforms of the inverter; b) average and fundamental current components.

Fig. 4 .

 4 Fig. 4.22. Graphical analysis of the turn-off and turn-on commutations in the power MOSFETs.

Figure 4 .

 4 22 shows the approximation of these transitions without considering the ripple values in the current or the effect of the body diodes of the MOSFET. This assumption allows concentrating the complexity of the analysis problem in the frequency variation introduced by the hysteresis-based implementation of the sliding mode control. Now,

Figure 4 .

 4 Figure 4.23 shows the grid voltage and current waveforms as references to analyze the frequency and switching power losses waveforms. The average values of the frequency and the power switching losses are identified to be subsequently introduced in the expressions of the losses.

Fig. 4 .

 4 Fig. 4.23. Waveforms to analyze the power switching losses: a) grid voltage, b) average grid current, c) instantaneous switching frequency, and d) instantaneous and average power switching losses.

Fig. 4 .

 4 Fig. 4.24. Inverter efficiency as a function of the grid voltage and output power considering a regulated DC-bus.

Fig. 4 . 26 .

 426 Fig. 4.26. Experimental set-up and inverter prototype.

Figure 4 .

 4 Fig. 4.27. Oscilloscope captures: a) grid voltage, output current and control signals (gate signals); b) grid voltage, output current and synchronization signal (90° of phase displacement).

4. 10 . 3

 103 figure 4.28b, the change of the power involves the 100 W and 20 W power levels.

Fig. 4 . 4 . 10 . 4

 44104 Fig. 4.28. Experimental results showing the transient response of the DC voltage loop: a) power changes from 50 W to 80 W and vice versa; b) power changes from 20 W to 100 W and vice versa.

Fig. 4 .

 4 Fig. 4.29. Experimental results showing the filter effect a) T f = 0.0005; b) T f = 0.005.

  .1b, three building blocks constitute the microinverter: the MPPT algorithm, the DC-DC converter and the DC-AC converter decomposing the microinverter and following respectively the contents of chapters 2, 3 and 4 respectively. The PV module and the DC-DC converter interact through the voltage (𝑣 𝑝 ) which is the voltage of an input capacitor connected in parallel with the PV module absorbing the high frequency component of the input current of the converter and smoothing the rapid changes of the PV voltage. The MPPT algorithm uses the measurement of the input voltage (𝑣 𝑝 ) and the input current (𝑖 𝑝 ) to compute the input power (𝑝 𝑝 ) and to set the conductance reference (𝐺). The value of the conductance is then used in the control of the DC-DC converter to compute the corresponding current reference with also the measurement of the voltage (𝑣 𝑝 ). The output of the DC-DC stage interacts with its load (DC-AC stage) through the output current (𝑖 𝑜 ). The control of the DC-AC stage regulates the DC bus voltage and fixes the amplitude of the grid current reference (𝑖 𝑔 𝑟𝑒𝑓 ) defining the power output. The DC-AC converter is connected with the grid through the AC output current (𝑖 𝑔 ). The measurement of the grid voltage (𝑣 𝑔 ) is only used to synchronize the current reference.

Fig. 5 . 1 .

 51 Fig. 5.1. General scheme of the sliding-mode controlled microinverter: a) Differentiation of the power stage and the control stages; b) differentiation of the studied pieces

5. 2 . 3 .

 23 Derivation of a model for the chain DC-DC-AC Considering the voltage regulation loop developed in Chapter 4, the current 𝐼 𝑚𝑎𝑥 (𝑠) is defined by a compensator with the form: 𝐶 𝑑𝑐 (𝑠) = -𝐾 𝑐 (𝑇 𝑐 𝑠 + 1) 𝑠(𝑇 𝑓 𝑠 + 1) (5.20) Then, the block diagram depicted in figure 5.2 can be deduced.

Fig. 5 . 2 .

 52 Fig. 5.2. Block diagram of the linear approach coupling the DC-DC stage and the DC-AC stage

(- 1 .

 1 602 × 10 -19 C). The values of the parameters are listed in table 5.1. It is possible to note in figure 5.3 that the voltage of the PV module varies in a reduced interval (20.6 to 22.2 V) working with a wide range of powers (24.81 to 103.5 W). The conductance at the maximum power point of the PV module takes values between 0.06 and 0.21 S for currents between 1.203 and 4.602 A. Thus, a change in the power of the PV module results from a simultaneous change of conductance 𝐺(𝑠) and input current 𝐼 𝑝 (𝑠).

Fig. 5 . 3 .

 53 Fig. 5.3. Characteristics of the PV module: a) I-V; b) P-V.

Fig. 5 . 4 .

 54 Fig. 5.4. Roots of the closed-loop system in the complex plane for different equilibrium points.

Figure 5 .

 5 Figure 5.2 shows a block diagram which represents the average model of the chain DC-DC-AC of the microinverter. However, that model does not include the MPPT dynamics or the PV module equation. Considering the analysis of chapter 2, the MPPT can be represented as a stable system which tracks the maximum power 𝑝 𝑚 (𝑡) and fix the values of 𝑖 𝑝 (𝑡) and 𝐺(𝑡) in the way depicted in figure 5.5 (variables in frequency domain).

Fig. 5 . 5 .

 55 Fig. 5.5. Block diagram of the linear approach including the MPPT and the PV module

  After some manipulations, it is possible to transform the block diagram of figure 5.5 into the one in figure 5.6. From the analysis of paragraph 5.2.3, the stability of the transfer function from 𝑃 𝑚 (𝑠) to 𝐼 𝑚𝑎𝑥 (𝑠) is ensured if polynomials 𝐴(𝑠) and 1 + 𝐾 𝑔 𝐶 𝑑𝑐 (𝑠)𝐺 𝑖𝑜 (𝑠) are Hurwitz.

Fig. 5 . 6 .

 56 Fig. 5.6. Block diagram of the transfer from P m (s) to I max (s)

Fig. 5 . 9 .

 59 Fig. 5.9. Simulation including the MPPT as a stable second order system

Fig. 5 .

 5 Fig. 5.10. Simulation including the MPPT as a stable second order system

Fig. 5 . 11 .

 511 Fig. 5.11. Detail of the steady state of the current and voltage waveforms at the grid side.

Figure 5 .

 5 Figure 5.11 shows the simulated steady-state waveforms of the microinverter at the grid side working with the irradiance value of 600 W/m 2 . As it can be observed, the grid

Fig. 5 .

 5 Fig. 5.12. Detail of the steady state of the current and voltage waveforms at the DC bus.

Figure 5 .

 5 Figure 5.13 shows the simulated steady-state waveforms of the microinverter at the PV module side including also the more relevant MPPT signals. It is possible to observethat the extracted power is around the maximum available power and also that the power presents the characteristic double frequency with respect to the current and voltage waveforms. The power reference of the algorithm is depicted to show that its average value corresponds to the maximum available power and is predicted by the existing sliding motion. The conductance waveform is also depicted showing its coherence with the power reference signal and also confirming the average values used to validate the linear model. In general, the steady-state behavior of the signals is the expected one and validates the application of the MPPT algorithm and the DC-DC converter control laws when the overall microinverter is considered.

Fig. 5 .

 5 Fig. 5.14. Simulated results using sudden changes of 80 W in the maximum power P-V characteristic.

2, 3 and 4 .

 4 The set-up for measurements, photovoltaic modules, microinverter prototype and control circuit are shown in figure 5.16. The experimental set-up is composed of a solar panel emulator E4360A from Agilent, an oscilloscope MSO3014 from Tektronix, current probes TCP202 and TCPA300 from Tektronix, high voltage differential probe P5210 from Tektronix, power supply Multimeteix XA3033, multimeter HP34401A from Hewlett Packard, electronic load IT8512B from ITEC, I-V Curve Tracer MP-160 from EKO and pyranometer SP LITE from KIPP & ZONEN.

Fig. 5 .

 5 Fig. 5.16. Microinverter prototype and experimental set-up

Fig. 5 .A1. 1 . 1 )

 511 Fig. 5.20. Experimental results for sudden changes of 55 W in the maximum power of the PV module

Table 1 .

 1

	Characteristics	Input	MPPT	Maximum	Weighted	Nominal	Power	Current
	Manufacturer	voltage	Efficiency	Efficiency	Efficiency	Power	Factor	THD
	Aurora [79]	25-60 V	*NS	95.5%	95%	300 W	>0.95	*NS
	Direct-grid [80]	53-75 V	99%	*NS	92%	460 W	>0.99	*NS
	Emphase [81]	27-39 V	99.4%	96.5%	96.5%	215 W	>0.95	*NS
	Enecsys [82]	27-54 V	*NS	94%	91%	200 W	>0.95	<5%
	iEnergy [83]	25-58 V	99.3%	94%	93%	240 W	>0.95	<3%
	Involar [84]	24-40 V	*NS	95%	94%	250 W	>0.99	<3%
	SolarBridge [85]	28-36 V	99.6%	95.5%	94.5%	240 W	>0.99	<5%
	SolarEuropa [86]	24-50 V	*NS	95%	94%	250 W	>0.99	<3%
	Sparq [87]	32-50 V	*NS	95%	93.4%	190 W	>0.99	<2%

1. Summary of commercial microinverter characteristics

Table 1 .

 1 2. Summary of commercial microinverter characteristics

	Characteristics Manufacturer	Warranty Years	Nominal Power	Maximum Efficiency	Weight [kg]	Size W [cm] H [cm] D [cm]	Power Density
	Aurora [79]	*NS	300 W	95.5%	1.50	21.00	16.00	5.00	0.18 W/cm
	Direct-grid [80]	20	460 W	*NS	2.17	22.53	22.76	3.05	0.29 W/cm
	Emphase [81]	25	215 W	96.5%	1.60	16.30	17.30	2.50	0.30 W/cm
	Enecsys [82]	20	200 W	94%	1.80	26.20	16.00	3.50	0.14 W/cm
	iEnergy [83]	25	240 W	94%	1.60	23.20	21.10	4.31	0.11 W/cm
	Involar [84]	15	250 W	95%	2.90	23.50	14.40	5.30	0.14 W/cm
	SolarBridge [85]	25	240 W	95.5%	2.41	20.32	16.51	3.17	0.22 W/cm
	SolarEuropa [86]	15	250 W	95%	2.40	23.00	13.80	3.50	0.22 W/cm
	Sparq [87]	25	190 W	95%	1.45	19.05	12.70	3.04	0.26 W/cm
	SunSine [88]	20	200 W	*NS	*NS	*NS	*NS	*NS	*NS
	ReneSola [89]	25	250 W	96.3%	2.00	23.00	13.80	3.50	0.22 W/cm
	APS [90]	25	250 W	95.5%	1.50	16.00	15.00	2.90	0.36 W/cm
	Siemens [91]	25	215 W	96.3%	1.60	17.30	16.40	2.50	0.30 W/cm
	ABB [92]	10	250 W	96.5%	1.65	26.60	24.60	3.50	0.11 W/cm
	AE Conversion [93]	*NS	250 W	93.5%	2.50	31.4	26.7	6.65	0.04 W/cm

*NS: No Specified

2.3.3 Equivalent control If

  we suppose that the algorithm has reached the sliding surface, we have 𝜀(𝑡) = 0 and 𝑃 𝑝 (𝑡) = 𝑃 𝑃𝑟𝑒𝑓 (𝑡). The equivalent controls are defined equating the derivative of 𝜀 to zero. Then, denoting equivalent controls by 𝑢 𝑒𝑞 and 𝑣 𝑒𝑞 , they must satisfy:

	𝑑𝜀 𝑑𝑡	= 𝑀𝑃 𝑝 𝑣 𝑒𝑞 + 𝐾 2 𝑃 𝑝 -	𝑑𝑃 𝑝 𝑑𝐺	𝐾 1 𝑃 𝑝 𝑢 𝑒𝑞 = 0 with -1 < 𝑣 𝑒𝑞 < 0 and -1 < 𝑢 𝑒𝑞 < 1	(2.21)
	If 𝑃 𝑃𝑟𝑒𝑓 (𝑡) changes and if the reachability conditions are satisfied, 𝑃 𝑝 (𝑡) tracks the
	reference 𝑃 𝑃𝑟𝑒𝑓 (𝑡). Now it is necessary to determine under what conditions, the
	𝐾 1 | 𝑑𝑃 𝑝 𝑑𝐺 algorithm increases 𝑃 𝑃𝑟𝑒𝑓 (𝑡) until it reaches 𝑃 𝑚𝑝 . | 𝑚𝑎𝑥 ] 𝑃 𝑝 (0 + ) > 0
	and 𝐶 = min{𝐶 1 , 𝐶 2 }. From (2.16) and (2.18), we can deduce, remarking that |𝜀| =
				1
	√2𝐶[𝑽(𝜀)]	2 :
	𝑽 ̇(𝜀) + √2𝐶[𝑽(𝜀)] 1 2 ≤ 0	∀ 𝜀
	From theorem 4.2 in [81] (p. 760), we can conclude that the error 𝜀(𝑡) is finite-time
	stable, meaning that 𝜀(𝑡) attains the sliding-surface in a finite time 𝑇 𝜀 satisfying (see
	[80]):
	𝑇 𝜀 ≤	|𝜀(0 + )| 𝐶
	Another remarkable point is that the previous conditions only depend on the
	knowledge of | 𝑑𝑃 𝑝 𝑑𝐺	| 𝑚𝑎𝑥	. From the panel characteristic curve and in particular from the
	approximation developed in paragraph 2, it is possible to easily evaluate | 𝑑𝑃 𝑝 𝑑𝐺	| 𝑚𝑎𝑥	. In the
	sequel, we consider that the different constants in the algorithm are selected according
	to the above reachability conditions.

Table 2 .

 2

1. Modeling of the four MPPT structures in the plane P-G Differential equations describing the dynamic behavior of 𝑃 𝑝 𝑟𝑒𝑓 and G

Table 2 . 2

 22

. Modeling of the MPPT power error Differential equations describing the dynamic behavior of 𝜀 𝑑𝜀

  this implies 𝑃 𝑝 𝑟𝑒𝑓 (𝑡 4 ) > 𝑃 𝑝 𝑟𝑒𝑓 (𝑡 0 ). Since the derivative of 𝑓 is negative, 𝐾 2 and 𝑀 -𝐾 2 ≠ 𝐾 1

	𝑑𝑃 𝑝 𝑑𝐺	| 𝑚𝑎𝑥	is needed, but as pointed out before, | 𝑑𝑃 𝑝 𝑑𝐺	| 𝑚𝑎𝑥
			𝑑𝑃 𝑝

𝑑𝐺

from reachability conditions, it can be concluded that 𝑓(𝐾 2 ) -𝑓(𝑀 -𝐾 2 ) > 0 if and only if 𝑀 -𝐾 2 > 𝐾 2 , which can also be written as 𝑀 > 2𝐾 2 . Adding this last condition to the reachability ones, we obtain all the constraints the parameters 𝑀, 𝐾 1 𝑎𝑛𝑑 𝐾 2 must satisfy. Note the important fact that only the knowledge of |

  𝑃 𝑃𝑟𝑒𝑓 (𝑡) and 𝑃 𝑝 (𝑡) are increasing. But from reachability conditions 𝑃 𝑃𝑟𝑒𝑓 (𝑡) is growing up faster than 𝑃 𝑃 (𝑡), at a moment they will be equal and then we can deduce that 𝑃 𝑝 𝑟𝑒𝑓 (𝑡) will move toward the new maximum from the analysis developed in paragraph 2.3.1 to 2.3.5. 𝑃 𝑃𝑟𝑒𝑓 (𝑡) is decreasing and 𝑃 𝑝 (𝑡) is increasing until they will be equal. From the analysis developed in paragraph 2.3.1 to 2.3.5, 𝑃 𝑝 𝑟𝑒𝑓 (𝑡) will move toward the new maximum.

	 If <0, we have: 𝑑𝑃 𝑃 (𝑡 𝑚𝑝2 + ) 𝑑𝐺 𝑑𝑃 𝑃𝑟𝑒𝑓 (𝑡) 𝑑𝐺 = -𝐾 2 𝐾 1 <  If 𝑑𝑃 𝑃 (𝑡 𝑚𝑝2 + ) 𝑑𝐺 >0,  If 𝑑𝑃 𝑃 (𝑡 𝑚𝑝2 + ) 𝑑𝐺 <0, 𝑃 𝑃𝑟𝑒𝑓 (𝑡) and 𝑃 𝑝 (𝑡) are decreasing. But from reachability 𝑑𝑃 𝑝 (𝑡) 𝑑𝐺 ⇒ 𝑑𝑃 𝑃𝑟𝑒𝑓 (𝑡) 𝑑𝑡 > 𝑑𝑃 𝑝 (𝑡) 𝑑𝑡 (2.44) conditions we have:
	
	𝑑𝑃 𝑃𝑟𝑒𝑓 (𝑡) 𝑑𝐺	=	𝐾 2 -𝑀 𝐾 1	<	𝑑𝑃 𝑝 (𝑡) 𝑑𝐺	⇒	𝑑𝑃 𝑃𝑟𝑒𝑓 (𝑡) 𝑑𝑡	<	𝑑𝑡 𝑑𝑃 𝑝 (𝑡)	(2.46)
	𝑃 𝑑𝑃 𝑝 𝑑𝐺	|

3.1 to 2.3.5, 𝑃 𝑝 𝑟𝑒𝑓 (𝑡) will move towards the new maximum. -Case b: it corresponds to a decreasing of irradiance. With the same notations as before, we have 𝜀(𝑡 𝑚𝑝2 + ) = 𝑃 𝑝 𝑟𝑒𝑓 (𝑡 𝑚𝑝2 + ) -𝑃 𝑝 (𝑡 𝑚𝑝2 + ) > 0 (see figure 2.9). For 𝑡 > 𝑡 𝑚𝑝2 + 𝛿, 𝛿 positive, we have: 𝑑𝑃 𝑃𝑟𝑒𝑓 (𝑡) 𝑑𝑡 = (𝐾 2 -𝑀)𝑃 𝑝 (𝑡) < 0 and 𝑑𝐺(𝑡) 𝑑𝑡 = 𝐾 1 𝑃 𝑝 (𝑡) > 0 (2.45) Fig. 2.9. Power point after sudden irradiance changes from a higher to a lower irradiance Two cases must be considered on the new power characteristic curve: 𝑃𝑟𝑒𝑓 (𝑡) is falling down faster than 𝑃 𝑃 (𝑡), at a moment they will be equal and then we can deduce that 𝑃 𝑝 𝑟𝑒𝑓 (𝑡) will move toward the new maximum from the analysis developed in paragraph 2.3.1 to 2.3.5. The analysis of the algorithm is now complete. Even if sudden changes of irradiance occur, the proposed algorithm will allow extracting the maximum power of the PV module, provided | 𝑑𝑃 𝑝 𝑑𝐺 | 𝑚𝑎𝑥 is evaluated appropriately. Recall that the analysis developed in paragraph 2.2 can be used to evaluate the value of | 𝑚𝑎𝑥 and to select the constants Δ, 𝑀, 𝐾 1 and 𝐾 2 .

Table 3 .

 3 1. Set of differential equations of the converter states

			ON-State	OFF-State	DCL1-State	DCL2-State	DCL12-State
	𝑣 𝐿1 = 𝐿 1	𝑑𝑖 𝐿1 𝑑𝑡	𝑣 𝑖𝑛	𝑣 𝑖𝑛 -𝑣 𝐶1	0		𝑣 𝑖𝑛 -𝑣 𝐶1		0
	𝑣 𝐿2 = 𝐿 2	𝑑𝑖 𝐿2 𝑑𝑡	𝑣 𝐶1	𝑣 𝐶1 -𝑣 𝐶2	𝑣 𝐶1 -𝑣 𝐶2		0		0
	𝑖 𝐶1 = 𝐶 1	𝑑𝑣 𝐶1 𝑑𝑡	-𝑖 𝐿2	𝑖 𝐿1 -𝑖 𝐿2	-𝑖 𝐿2	𝑖 𝐿1		0
	𝑖 𝐶2 = 𝐶 2	𝑑𝑣 𝐶2 𝑑𝑡	-	𝑣 𝐶2 𝑅	𝑖 𝐿2 -	𝑣 𝐶2 𝑅	𝑖 𝐿2 -	𝑣 𝐶2 𝑅	-	𝑣 𝐶2 𝑅	-	𝑣 𝐶2 𝑅

  .1, the volts-second balances are given by By solving (3.29) and (3.30), the steady-state expressions for 𝑣 𝐶1 and 𝑣 𝐶2 are obtained.

	〈𝑣 𝐿1 〉 =	〈𝑣 𝑖 〉𝑈𝑇 𝑠 + (〈𝑣 𝑖 〉 -〈𝑣 𝐶1 〉)(1 -𝑈)𝑇 𝑠 𝑇 𝑠	= 0	(3.29)
	〈𝑣 𝐿2 〉 =	〈𝑣 𝐶1 〉𝑈𝑇 𝑠 + (〈𝑣 𝐶1 〉 -〈𝑣 𝐶2 〉)𝑈 2 𝑇 𝑠 𝑇 𝑠	= 0	(3.30)
	〈𝑣 𝐶1 〉 =	〈𝑣 𝑖 〉 1 -𝑈			(3.31)

  By solving(3.46) and(3.47), the steady-state expressions for 𝑣 𝐶1 and 𝑣 𝐶2 are obtained.

	〈𝑣 𝐿2 〉 =	〈𝑣 𝐶1 〉𝑈𝑇 𝑠 + (〈𝑣 𝐶1 〉 -〈𝑣 𝐶2 〉)𝑈 2 𝑇 𝑠 𝑇 𝑠	= 0	(3.47)
	〈𝑣 𝐶1 〉 =	〈𝑣 𝑖 〉(𝑈 + 𝑈 1 ) 𝑈 1			(3.48)
		𝑇 𝑠	𝑈 1 𝑇 𝑠	= 0	(3.46)

  can be rewritten as:

	𝑽 ̇(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 ) Expressing (3.95) in matrix form, namely,V ̇= x′Qx, results in
	𝐿 2 Developing each term of the previous expression leads to: 𝑑𝑖 𝐿2 𝑑𝑡 = 𝑣 𝐶1 -𝑣 𝐶1 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ 2 𝑣 𝐶2 = 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ (𝑖 𝐿2 -𝑖 𝐿2 𝑑𝑖 𝐿2 𝑽 ̇(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 ) ≤ ̅̅̅̅)𝐿 2 𝑑𝑡 + [( 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ + 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ) (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) -1 2 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 2 (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2 ] 𝐶 1 + 𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅ (𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅)𝑅𝐶 2 𝑑𝑡 𝑄 𝑑𝑣 𝐶2 -[(𝑖 𝐿2 -𝑖 𝐿2 ̅̅̅̅) (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) (𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅)] ⏟ 𝑥 ′ [ 0 0 0 0 𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅̅ 𝑣 𝐶1 𝑣 𝐶1 ̅̅̅̅̅ 𝑖 𝐿2 ̅̅̅̅ 2𝑣 𝐶1 0 𝑖 𝐿2 ̅̅̅̅ 2𝑣 𝐶1 𝑖 𝐿2 𝑣 𝐶2 ̅̅̅̅̅ ] 𝑥 𝑑𝑣 𝐶1 𝑑𝑡 ⏟ 𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅ ⏟ ̅̅̅̅ [ 𝑖 𝐿2 -𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅ ]	(3.92) (3.96)
	𝐶 1 𝑽 ̇(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 ) 𝑑𝑣 𝐶1 𝑑𝑡 = -𝑖 𝐿2 +	𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ 𝑣 𝐶1	(3.90)
	𝑅𝐶 2	𝑑𝑣 𝐶2 𝑑𝑡	= -𝑣 𝐶2 + = 1 2 𝑣 𝐶2 𝑣 𝐶1 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅ 𝑖 𝐿2 𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ 𝑖 𝐿2 𝑣 𝐶1 2 (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2 -	1 2	𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 3	(𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2
	-After some simple manipulations, (3.90) can be expressed as follows: 𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 ( 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ + 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ) (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2 -𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 (𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅) 2 ̅̅̅̅ -𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅)(𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅)
	𝐿 2 𝐶 1 But, 𝑖 𝐿2 𝑑𝑖 𝐿2 𝑑𝑡 𝑑𝑣 𝐶1 𝑑𝑡 ̅̅̅̅ > 0, 𝑣 𝐶1 = (1 + 𝑣 𝐶1 ̅̅̅̅ 𝑣 𝐶1 = -(𝑖 𝐿2 -𝑖 𝐿2 ) (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) -̅̅̅̅) -𝑖 𝐿2 ̅̅̅̅ > 0, 𝑣 𝐶2 ̅̅̅̅ > 0 because of the nature of the power circuit, then, 𝑣 𝐶1 ̅̅̅̅ 2 𝑣 𝐶2 (𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅) ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ 𝑣 𝐶1 (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) -i L2 ̅̅̅̅ v C1 ̅̅̅̅̅ v C2 ̅̅̅̅̅ v C1 3 (v C1 -v C1 ̅̅̅̅) 2 < 0. Therefore,
	𝑅𝐶 2 𝑽 ̇(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 ) 𝑑𝑣 𝐶2 𝑑𝑡 = -(𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅) +	𝑣 𝐶1 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅ 𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1	(𝑖 𝐿2 -𝑖 𝐿2 ̅̅̅̅) -	𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1	̅̅̅̅) (𝑣 𝐶1 -𝑣 𝐶1
	Consider now the Lyapunov candidate function defined by: ≤ 1 2 𝑣 𝐶2 ̅̅̅̅ 𝑖 𝐿2 𝑣 𝐶1 2 (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2 -𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1 ( 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ + 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ) (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2	(3.93)
	𝑽(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 )	-	𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅	(𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅) 2 -	𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1	(𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅)(𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅)
	If	1 2	= ̅̅̅̅, which is the same that: 1 2 [𝐿 2 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ (𝑖 𝐿2 -𝑖 𝐿2 ̅̅̅̅) 2 + 𝐶 1 ( 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 ̅̅̅̅ i L2 < i L2	+	𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1	) (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2	(3.91)
	𝐢 𝐋𝟐 < 𝟐𝐢 𝐋𝟐 ̅̅̅̅	+	𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅	𝑅𝐶 2 (𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅) 2 ]	(3.94)
	Note that 𝑉(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 ) > 0, 𝑉(𝑖 𝐿2 ̅̅̅̅, 𝑣 𝐶1 ̅̅̅̅, 𝑣 𝐶2 then: ̅̅̅̅) = 0 , and that this function is well 𝑽 ̇(𝑖 𝐿2 , 𝑣 𝐶1 , 𝑣 𝐶2 ) defined because 𝑣 𝐶1 > 0 (see condition in (3.78)). The time derivative of V is given by: ≤ -𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅ 𝑣 𝐶1 𝑣 𝐶1 ̅̅̅̅ (𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅) 2 -𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶2 ̅̅̅̅ (𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅) 2 (3.95)
						-	𝑖 𝐿2 ̅̅̅̅ 𝑣 𝐶1	(𝑣 𝐶1 -𝑣 𝐶1 ̅̅̅̅)(𝑣 𝐶2 -𝑣 𝐶2 ̅̅̅̅)

  As in the case of resistive load, it is possible to obtain the transfer function from the incremental conductance to the incremental output voltage 𝐺 𝑔 (𝑠) = 𝑉 𝐶2 (𝑠)/𝐺(𝑠) = function from either the input voltage variation or the output current perturbation to the incremental output voltage. The resulted transfer functions are:

	𝐵(𝑠)/𝐴(𝑠):
	𝐺 𝑔 (𝑠) =	-𝐼 0 𝑚𝐿 1 𝐶 1 𝑠 3 + 𝑚𝐶 1 𝐶 2 𝑠 3 + 𝐺𝐶 2 𝑉 𝑖 𝑚𝐿 1 𝐶 1 𝐿 2 𝑚 𝑠 2 + 𝑚𝐿 2 𝑠 2 -1 (2𝑚 2 𝐶 2 + 𝐶 1 )𝑠 + 𝐼 0 𝑚 (1 + 𝑚 2 𝐿 1 𝐿 2 ) 𝑠 + 𝑚 3 𝐿 2 2𝑉 𝑖 𝑚𝐿 2 2𝐺	(3.114)
	and the transfer 𝐺 𝑣𝑖 (𝑠) = -𝐼 0 𝑚𝐺𝐿 1 𝐶 1 𝑉 𝑖 𝑚𝐶 1 𝐶 2 𝑠 3 + 𝑠 3 + 𝑚𝐶 1 [ 𝐺𝐶 2 𝐼 0 𝑉 𝑖 + 𝑚 𝑠 2 + 𝐺𝐿 1 𝐿 2 𝑚𝐿 2 ] 𝑠 2 -[ 1 (2𝑚 2 𝐶 2 + 𝐶 1 )𝑠 + 𝐼 0 𝑚𝐿 1 𝐺 𝑉 𝑖 𝐿 2 + 𝑚𝐶 1 𝐿 2	+ 𝑚 3 𝐿 2 𝐼 0 𝐺 𝑉 𝑖 𝑚 ] 𝑠 + 2𝐺	𝐼 0 𝑚 𝑉 𝑖 𝐿 2	(3.115)
	𝐺 𝑖𝑜 (𝑠) =	𝑚𝐶 1 𝐶 2 𝑠 3 +	-𝑚𝐶 1 𝑠 2 -𝐺𝐶 2 𝑚 𝑠 2 + 1 𝑚𝐿 2 𝑚 𝐺 (2𝑚 2 𝐶 2 + 𝐶 1 )𝑠 + 2𝑚 𝑠 -𝐿 2	𝑚 3 𝐿 2 2𝐺	(3.116)
	From the Routh Hurwitz test, this system is stable satisfying the condition:
	𝒎 > √	𝑪 𝟏 𝟐𝑪 𝟐								(3.117)
	𝑑𝑖 𝐿2 d𝑡	=		2 𝐿 2	𝑣 𝐶1 ̃-1 𝑚𝐿 2	𝑣 𝐶2 ̃-𝑚 𝐿 2	𝑣 𝑖 ̃+ 𝑚𝐺𝐿 1 𝐿 2	𝑑𝑣 𝑖 d 𝑡	+	𝐿 2 𝑚𝐿 1 𝑉 𝑖	𝑑𝐺 d𝑡
	𝑑𝑣 𝐶1 d𝑡	=	-1 𝐶 1	𝑖 𝐿2 ̃-𝐺 𝑚 2 𝐶 1	𝑣 𝐶1 ̃+ 2𝐺 𝑚𝐶 1	𝑣 𝑖 ̃+ 𝑉 𝑖 𝑚𝐶 1	𝐺 ̃-𝑉 𝑖 𝐺𝐿 1 𝑚𝐶 1	𝑑𝐺 d𝑡	-	𝐺 2 𝐿 1 𝑚𝐶 1	𝑑𝑣 𝑖 d 𝑡	(3.113)
	𝑑𝑣 𝐶2 d𝑡	=	1 𝑚𝐶 2	𝑖 𝐿2 ̃-𝐼 0 𝐶 2 𝑚𝑉 𝑖	𝑣 𝐶1 ̃+ 𝐼 0 𝐶 2 𝑉 𝑖	𝑣 𝑖 ̃-𝐼 0 𝐺𝐿 1 𝑉 𝑖 𝐶 2	𝑑𝑣 𝑖 d 𝑡	-	𝐼 0 𝐿 1 𝐶 2	𝑑𝐺 d𝑡	-	1 𝐶 2	𝑖 0 ̃

. As it is shown in figure

3

.19, a simulation test in PSIM validating the model by comparing the linearized model with the switching model when different input excitations are applied in either the input variable 𝐺(𝑠) or the disturbance variable 𝑉 𝑖

(𝑠)

Table 3 .

 3 2. Expressions to compute the power conduction losses

	Element			Expressions
	Inductors
	P	P	P	P	I	R
	L	core	Copper	core	L	L
					RMS

Table 3 .

 3

3. Expressions to compute the power switching losses

Table 3 .

 3 4. Details of the power semiconductor used in the converter prototype

	Element	Parameter	Symbol Value
	Inductor L1	Inductance	𝐿 1	120 µH
		Equivalent Series Resistance (ESR)	𝑅 𝐿1	28 mΩ
	Inductor L2	Inductance	𝐿 2	4.7 mH
		Equivalent Series Resistance (ESR)	𝑅 𝐿1	1.14 Ω
	Capacitors C1 and C2	Capacitance	𝐶 1,2	10 µF
	(C4ATHBW4900A3LJ)	Equivalent Series Resistance (ESR)	𝑅 𝐶1,𝐶2	1.9 mΩ
	MOSFET	Drain -Source ON-resistance	𝑅 𝑀	35 mΩ
	(APT94N60L2C3)	Rise time	𝑡 𝑟	27 ns
		Fall time	𝑡 𝑓	8 ns
		Gate voltage	𝑉 𝑔𝑎𝑡𝑒	12 V
		Input capacitance	𝐶 𝑔𝑎𝑡𝑒	13.6 nF
	Diodes	Static forward voltage	𝑅 𝐷𝑥	0.75 V
	(C3D06060A)	Series resistance	𝑉 𝐷𝑥	110 mΩ

Table 4 .

 4

1. Converter circuit structures using bipolar commutation.

Table 4 . 2

 42

. Converter circuit structures for the positive half cycle using unipolar commutation.

Table 4 .

 4

3. Converter circuit structures for the negative half cycle using unipolar commutation.

  term on the right side is nonlinear. Denoting by * ̂ the linearized variables, the linearized model around 𝐼 𝑚𝑎𝑥 , 𝑉 𝑚𝑎𝑥 , 𝑉 𝐶 , 𝑃 ̅ is obtained:

	1 2	𝐶 ( 𝑣 𝑐	2 (𝑡 + 𝑇) -𝑣 𝑐 𝑇	2 (𝑡)	) ≅ 𝐶	𝑑𝑣 𝐶 𝑑𝑡
	and the model is given by:
	𝐶	𝑑𝑣 𝐶 𝑑𝑡	≅ 𝑃 -	𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2	-	𝐿𝑖 𝑚𝑎𝑥 2	𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡	(4.28)
	𝜔𝑡 = (4.29) (4.25) Taking the Laplace transform of the previous expression for zero initial conditions, we (1 -cos 2𝜔𝑡)/2, expression (4.19) can be equivalently written as: 𝐶𝑣 𝐶 𝑑𝑣 𝐶 𝑑𝑡 = 𝑃 -𝐿𝑖 𝑚𝑎𝑥 2 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 + 𝐿𝑖 𝑚𝑎𝑥 2 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 cos 2𝜔𝑡 -𝐿𝜔𝑖 𝑚𝑎𝑥 2 2 sin 2𝜔𝑡 -𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 2 + where the last 𝐶 𝑑𝑣 ̂𝐶 𝑑𝑡 = 1 𝑉 𝐶 𝑃 ̂-𝑉 𝑚𝑎𝑥 2𝑉 𝐶 𝑖m 𝑎𝑥 -𝐿𝐼 𝑚𝑎𝑥 2𝑉 𝐶 𝑑𝑖m 𝑎𝑥 𝑑𝑡 𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 cos 2𝜔𝑡 2
	have:
	Now, if 𝑖 𝑑𝑖 𝑚𝑎𝑥 𝑉 𝐶 (𝑠) = -𝐿𝐼 𝑚𝑎𝑥 𝑠 + 𝑉 𝑚𝑎𝑥 2𝐶𝑉 𝑟𝑒𝑓 𝑠 𝐼 𝑚𝑎𝑥 (𝑠) + 1 𝐶𝑉 𝑟𝑒𝑓 𝑠 𝑃(𝑠)
	1 𝑇	∫ 𝐶𝑣 𝐶 𝑡+𝑇 𝑡	𝑑𝑣 𝐶 𝑑𝑢	𝑑𝑢 = 𝑃 -	𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 2	-	𝐿𝑖 𝑚𝑎𝑥 2	𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡
	It is obtained that:
	1 2	𝐶 [ 𝑣 𝑐	2 (𝑡 + 𝑇) -𝑣 𝑐 𝑇	2 (𝑡)	] = 𝑃 -	𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 2	-	𝐿𝑖 𝑚𝑎𝑥 2	𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡	(4.26)
	which leads to the equilibrium point where 𝑣 𝑐 (𝑡 + 𝑇) ≅ 𝑣 𝑐 (𝑡):
	𝑃 ̅ =	𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2			(4.27)
	Since 𝑇 =	𝜋 𝜔	is small, the following approximation can be considered:

𝑚𝑎𝑥 (𝑡) is not a constant but a slow variable over an interval of time of length 𝑇 = 𝜋/ 𝜔, we can suppose that 𝑖 𝑚𝑎𝑥 (𝑡), 𝑑𝑡 (𝑡) and 𝑃 do not vary significantly. Then, integrating (4.25) from 𝑡 to 𝑡 + 𝑇 leads to:

  and 𝐼 𝑚𝑎𝑥 ≤ 0.65 𝐴 (𝑃 ≤ 200 𝑊), leading to a zero in 𝑠 = 𝑇 𝑓 being fixed, the free parameters are 𝐾 𝑐 and 𝑇 𝑐 . They will be determined in order to have a phase margin of 45°. The first step consists in identifying the frequencies corresponding to a phase of -135° for the controlled open loop transfer function. We have:1+𝑇 𝑐 𝑇 𝑓 𝜔 2 = 1, which implies that 𝑇 𝑐 𝑇 𝑓 𝜔 2 -𝜔(𝑇 𝑐 -𝑇 𝑓 ) + 1 = 0. Since the

	310.2 0.65 * 10 10 3 𝑅𝑎𝑑/𝑠. From the above analysis, the controlled open loop transfer function is: 𝐿(𝑠) = 𝐶(𝑠)𝐺(𝑠) = 𝐾 𝑚𝑎𝑥 𝐾 𝑐 (𝑇 𝑐 𝑠 + 1) 𝑠 2 (𝑇 𝑓 𝑠 + 1) (4.33) × (4.34) or equivalently, atan(𝑇 𝑐 𝜔) -atan(𝑇 𝑓 𝜔) = 45°. Using the identity atan(𝑥) -atan(𝑦) = atan ( 𝑥+𝑦 1-𝑥𝑦 ), then, the previous equation is expressed as: atan(𝑇 𝑐 𝜔) -atan(𝑇 𝑓 𝜔) = atan ( 𝜔(𝑇 𝑐 -𝑇 𝑓 ) 1 + 𝑇 𝑐 𝑇 𝑓 𝜔 2 ) = 45° (4.35) Then, 𝜔(𝑇 𝑐 -𝑇 𝑓 ) discriminant ∆= 𝑇 𝑓 2 -6𝑇 𝑐 𝑇 𝑓 + 𝑇 𝑐 2 > 0, there exist two real roots given by: , where Arg[𝐿(𝑗𝜔)] = -180°-atan(𝑇 𝑓 𝜔) + atan(𝑇 𝑐 𝜔) = -135° 𝜔 1,2 = 1 2𝑇 𝑓 𝑇 𝑐 (-𝑇 𝑓 + 𝑇 𝑐 ± √𝑇 𝑓 2 -6𝑇 𝑐 𝑇 𝑓 + 𝑇 𝑐 2 ) (4.36)

Table 4 .

 4 For the set of parameters in table 4.4, the controller parameters are listed in table4.5 for different values of 𝑎. The values of 𝑎 are generated by a MATLAB algorithm which considers 𝑎 𝑘 = 1 + 0.05𝑎 𝑘-1 for values of 𝑘 between 1 and 6, having 𝑎 0 = 5.828

	4. Parameters of the converter and power sources
	Parameter	Symbol	Value
	Inductance	𝐿	10 𝑚𝐻
	DC bus Capacitor	𝐶	22 𝜇𝐹
	DC bus Voltage	𝑉 𝐶	400 𝑉
	Grid RMS Voltage	𝑉 𝑔	220 𝑉
	Grid Frequency	𝑓 𝑔	50 𝐻𝑧
	Power Range	𝑃	20 -100 𝑊
	satisfying (4.37).

Table 4 .

 4 5. Controller parameters for 𝑇 𝑓 = 0.004

	Parameter	𝑎 = 5.828 𝑎 = 6.119 𝑎 = 6.731 𝑎 = 7.741 𝑎 = 9.289	𝑎 = 11.614
	𝜔 1 (𝑅𝑎𝑑/𝑠)	103.55	77.68	61.22	47.40	35.95	26.67
	𝜔 2 (𝑅𝑎𝑑/𝑠)	103.55	131.46	151.63	170.30	187.13	201.80
	𝑇 𝑐 (𝑠)	0.0233	0.0245	0.0269	0.0310	0.0372	0.0464
	𝐾 𝑐 1 (Ω -1 /𝑠)	0.5711	0.3781	0.2573	0.1656	0.1006	0.0578
	𝐾 𝑐 2 (Ω -1 /𝑠)	0.5711	0.7450	0.8224	0.8405	0.8005	0.7138

  in the controller transfer function must be selected to reject the double frequency component without enforcing a slow transient response. Considering a grid frequency of 50 Hz, then the low pass filter must reject a 100 Hz (628 Rad/s) component. If a value of 5 ms is selected for 𝑇 𝑓 , then, an attenuation of more or less -10 dB is ensured at the double grid frequency. So, from the analysis in the above section, a value of 𝑎 = 12 is selected, with a value of 0.06 s for 𝑇 𝑐 . A value of 0.4477 for 𝐾 𝑐 is then computed for a frequency 𝜔 2 = 162.86 Rad/s. The resulted controller is

	𝐶(𝑠) = -	0.4477(0.06𝑠 + 1) 𝑠(0.005𝑠 + 1)	(4.40)

  Replacing in the equation describing the motion of 𝑣 𝐶 , we obtain:

	𝐶𝑣 𝐶 The mean value of the power is obtained as: 𝑑𝑣 𝐶 𝑑𝑡 = 𝑃 -𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2 + 𝑉 𝑚𝑎𝑥 𝐼 2 4 sin 𝜙 -𝑉 𝑚𝑎𝑥 𝐼 2 2 Then, we have: 𝑑𝑆 𝑑𝑡 = 𝑑𝑖 𝑔 𝑑𝑡 -𝑑𝑖 𝐿 𝑑𝑡 = 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 𝑠𝑖𝑛 𝜔𝑡 + 𝜔𝑖 𝑚𝑎𝑥 cos 𝜔𝑡 + 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) 𝑣 𝑔 𝐿 -𝑢 𝑣 𝐶 𝐿 + 𝐿𝐼 2 2 𝜔 8 sin(2𝜔𝑡 + 2𝜙) -𝐿𝐼 𝑚𝑎𝑥 𝐼 2 𝜔 cos(2𝜔𝑡 + 𝜙) -𝐿𝜔𝐼 𝑚𝑎𝑥 2 2 𝑠𝑖𝑛 2𝜔𝑡 -𝐿𝐼 2 2 𝜔 4 𝑠𝑖𝑛 2𝜔𝑡 + 𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2 cos 2𝜔𝑡 |𝑝 𝑔 ̅̅̅| = 𝑉 𝑚𝑎𝑥 2 ( √ 𝐼 𝑚𝑎𝑥 2 -𝐼 𝑚𝑎𝑥 𝐼 2 sin 𝜙 + 𝐼 2 2 4 ) cos [atan ( 𝐼 2 cos 𝜙 2𝐼 𝑚𝑎𝑥 -𝐼 2 sin 𝜙	)]	(4.46) (4.61)
	Since Since cos [atan ( 𝑑𝑆 𝑑𝑡 | 𝑢=𝑢 𝑒𝑞 = 0 , it can be derived: + 𝑉 𝑚𝑎𝑥 𝐼 2 4 sin(4𝜔𝑡 + 𝜙) + 𝐿𝐼 𝑚𝑎𝑥 𝐼 2 𝜔 cos(4𝜔𝑡 + 𝜙) -𝐿𝐼 2 2 𝜔 2 sin(4𝜔𝑡 + 2𝜙) + 𝐿𝐼 2 2 𝜔 4 sin(6𝜔𝑡 + 2𝜙) 𝐼 2 cos 𝜙 2𝐼 𝑚𝑎𝑥 -𝐼 2 sin 𝜙 )] = 1 √1+𝑥 2 , we obtain:
	𝑢 𝑒𝑞 = 𝑝 𝑔 ̅̅̅ = 𝑉 𝑚𝑎𝑥 1 𝑣 𝐶 (𝐿 2 (𝐼 𝑚𝑎𝑥 -𝑑𝑖 𝑚𝑎𝑥 𝑠𝑖𝑛 𝜔𝑡 + 𝐿𝜔𝑖 𝑚𝑎𝑥 𝑐𝑜𝑠 𝜔𝑡 + 𝑉 𝑚𝑎𝑥 𝑠𝑖𝑛 𝜔𝑡) + 𝐿𝜔𝐼 2 2 8 sin(6𝜔𝑡 + 2𝜙) -𝐿𝐼 2 4 𝑑𝐼 2 𝑑𝑡 + 𝐿𝐼 2 4 𝑑𝐼 𝑚𝑎𝑥 𝑑𝑡 sin 𝜙 𝐼 2 𝑠𝑖𝑛 𝜙) 2 𝑑𝑡 + 𝐿𝐼 𝑚𝑎𝑥 4 𝑑𝐼 2 𝑑𝑡 sin 𝜙 + 𝐿𝐼 𝑚𝑎𝑥 𝐼 2 4 𝑑𝜙 𝑑𝑡 𝐿𝐼 𝑚𝑎𝑥 𝑑𝐼 𝑚𝑎𝑥 cos 𝜙 -2 𝑑𝑡 + 𝐿𝐼 𝑚𝑎𝑥 2 𝑑𝐼 𝑚𝑎𝑥 𝑑𝑡 𝑐𝑜𝑠 2𝜔𝑡 -𝐿𝐼 𝑚𝑎𝑥 𝐼 2 2 𝑑𝑡 cos(2𝜔𝑡 + 𝜙) 𝑑𝜙 , which confirms (4.55) for the mean power on the DC side. From (4.43), we have: (4.47) (4.62) (4.52)
	𝐶 𝑑𝑣 𝐶 𝑑𝑣 𝐶 𝑑𝑡 𝑑𝑡 = = 𝑑𝑉 𝑑𝑐 𝑃 𝑣 𝐶 -𝑑𝑡 + 2𝜔𝑉 2 cos(2𝜔𝑡 + 𝛾) + 𝑉 2 cos(2𝜔𝑡 + 𝛾) 1 𝑣 𝐶 𝑖 𝑚𝑎𝑥 𝑠𝑖𝑛 𝜔𝑡 (𝐿 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 𝑠𝑖𝑛 𝜔𝑡 + 𝐿𝜔𝑖 𝑚𝑎𝑥 𝑐𝑜𝑠 𝜔𝑡 + 𝑉 𝑚𝑎𝑥 𝑠𝑖𝑛 𝜔𝑡) -𝐿𝐼 𝑚𝑎𝑥 2 𝑑𝐼 2 𝑑𝑡 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) -𝐿𝐼 2 2 𝑑𝐼 𝑚𝑎𝑥 𝑑𝑡 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) + 𝐿𝐼 2 8 𝑑𝜙 𝑑𝑡 sin(2𝜔𝑡 + 2𝜙) + 𝐿𝐼 2 4 𝑑𝑡 cos 2𝜔𝑡 𝑑𝐼 2 2 𝑑𝛾 𝑑𝑡 + 𝑑𝑉 2 𝑑𝑡 sin(2𝜔𝑡 + 𝛾)	(4.48) (4.63)
	Then: and then:	-	𝐿𝐼 2 8	𝑑𝐼 2 𝑑𝑡	cos(2𝜔𝑡 + 2𝜙) -	𝐿𝐼 2 4	2	𝑑𝑡 𝑑𝜙	sin(4𝜔𝑡 + 2𝜙)
	𝐶𝑣 𝐶 𝐶𝑣 𝐶	𝑑𝑣 𝐶 𝑑𝑡 𝑑𝑣 𝐶 𝑑𝑡	= 𝑃 -𝐿𝑖 𝑚𝑎𝑥 -𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 𝑠𝑖𝑛 2 𝜔𝑡 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 𝑠𝑖𝑛 2 𝜔𝑡 -𝐿𝜔𝑖 𝑚𝑎𝑥 + 𝐿𝐼 2 4 𝑑𝐼 2 𝑑𝑡 cos(4𝜔𝑡 + 2𝜙) + 𝐿𝐼 𝑚𝑎𝑥 𝐼 2 2 𝑠𝑖𝑛 𝜔𝑡 𝑐𝑜𝑠 𝜔𝑡 4 𝑑𝜙 𝑑𝑡 cos(4𝜔𝑡 + 𝜙) = 𝐶 [𝑉 𝑑𝑐 𝑑𝑉 𝑑𝑐 𝑑𝑡 + 𝑉 2 𝑑𝑉 𝑑𝑐 sin(2𝜔𝑡 + 𝛾) + 2𝑉 𝑑𝑐 𝑉 2 𝜔 𝑐𝑜𝑠(2𝜔𝑡 + 𝛾) 𝑑𝑡 + 𝐿𝐼 𝑚𝑎𝑥 4 𝑑𝐼 2 𝑑𝑡 sin(4𝜔𝑡 + 𝜙) + 𝐿𝐼 2 4 𝑑𝐼 𝑚𝑎𝑥 𝑑𝑡 sin(4𝜔𝑡 + 𝜙) + 𝑉 𝑑𝑐 𝑉 2 𝑑𝛾 𝑑𝑡 cos(2𝜔𝑡 + 𝛾) + 𝑉 𝑑𝑐 𝑑𝑉 2 𝑑𝑡 𝑠𝑖𝑛(2𝜔𝑡 + 𝛾)	(4.49)
	𝐶𝑣 𝐶 At the equilibrium point (derivatives equal to zero), we have: 𝑑𝑣 𝐶 𝑑𝑡 = 𝑃 -𝐿𝑖 𝑚𝑎𝑥 2 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 (1 -𝑐𝑜𝑠 2𝜔𝑡) -𝐿𝜔𝑖 𝑚𝑎𝑥 2 2 𝑠𝑖𝑛2𝜔𝑡 -𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 2 (1 -𝑐𝑜𝑠 2𝜔𝑡) -𝑉 2 𝑑𝑡 cos(4𝜔𝑡 + 2𝛾)] 𝑑𝑉 2 + 𝐿𝐼 2 2 8 𝑑𝜙 𝑑𝑡 sin(6𝜔𝑡 + 2𝜙) -𝐿𝐼 2 8 𝑑𝐼 2 𝑑𝑡 cos(6𝜔𝑡 + 2𝜙) + 𝑉 2 2 𝜔 𝑠𝑖𝑛(4𝜔𝑡 + 2𝛾) + 2𝑉 2 2 𝑑𝛾 𝑑𝑡 sin(4𝜔𝑡 + 2𝛾) +	2 𝑉 2	𝑑𝑡 𝑑𝑉 2	(4.50) (4.64)
	And:
	𝑚𝑎𝑥 𝑑𝑡 𝐶𝑣 𝐶 𝑑𝑣 𝐶 = 𝑑𝑡 𝑃 = 𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 𝑑𝐼 𝑚𝑎𝑥 𝑑𝑡 + + 2𝐼 2 𝜔 cos(2𝜔𝑡 + 𝜙) + 𝐼 2 𝑑𝐼 2 𝑑𝑡 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) = 𝑃 -𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 2 -𝐿𝑖 𝑚𝑎𝑥 2 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 -𝐿𝜔𝑖 𝑚𝑎𝑥 𝑑𝜙 𝑑𝑡 cos(2𝜔𝑡 + 𝜙) 2 2 sin 2𝜔𝑡 + 𝐿𝑖 𝑚𝑎𝑥 2 𝑑𝑖 𝑚𝑎𝑥 𝑑𝑡 cos 2𝜔𝑡 + 2 -𝑉 𝑚𝑎𝑥 𝐼 2 4 sin 𝜙 + 𝑉 𝑚𝑎𝑥 𝐼 2 2 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) -𝐿𝐼 2 By equalizing (4.52) and (4.64), evaluated at the equilibrium point (derivatives equal (4.45) (4.51) 2 𝜔 8 sin(2𝜔𝑡 + 2𝜙) to zero) and keeping only the double frequency components, we have: 𝑉 𝑚𝑎𝑥 𝑖 𝑚𝑎𝑥 2 cos 2𝜔𝑡 Considering (4.42) and (4.44), (4.51) becomes after some tedious calculations: + 𝐿𝐼 𝑚𝑎𝑥 𝐼 2 𝜔 cos(2𝜔𝑡 + 𝜙) + 𝐿𝜔𝐼 𝑚𝑎𝑥 2 2 𝑠𝑖𝑛 2𝜔𝑡 + 𝐿𝐼 2 2 𝜔 𝑠𝑖𝑛 2𝜔𝑡 4 -𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2 cos 2𝜔𝑡 -2𝐶𝑉 𝑑𝑐 𝑉 2 𝜔 𝑐𝑜𝑠(2𝜔𝑡 + 𝛾) 𝑉 𝑚𝑎𝑥 𝐼 2 4 sin(4𝜔𝑡 + 𝜙) -𝐿𝐼 𝑚𝑎𝑥 𝐼 2 𝜔 cos(4𝜔𝑡 + 𝜙) + 𝐿𝐼 2 (4.53) = -𝑉 𝑚𝑎𝑥 𝐼 2 2 𝑠𝑖𝑛(2𝜔𝑡 + 𝜙) + 𝐿𝐼 2 2 𝜔 sin(2𝜔𝑡 + 2𝜙) 8 2 𝜔 2 sin(4𝜔𝑡 + 2𝜙) -𝐿𝐼 2 2 𝜔 4 sin(6𝜔𝑡 + 2𝜙) -2 -𝐿𝐼 𝑚𝑎𝑥 𝐼 2 𝜔 cos(2𝜔𝑡 + 𝜙) -𝐿𝜔𝐼 𝑚𝑎𝑥 2 2 𝑠𝑖𝑛 2𝜔𝑡 -𝐿𝐼 2 2 𝜔 𝑠𝑖𝑛 2𝜔𝑡 4 𝐿𝜔𝐼 2 8 sin(6𝜔𝑡 + 2𝜙) + 𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2 cos 2𝜔𝑡

  The value of 𝐼 𝑝 2 can be deduced from the Fourier series expansion, having that 𝐼 𝑝 2 =

	𝐼 𝑝 𝑑𝑐 =	𝑃 2𝜋	𝜋 -𝜋 ∫	𝑑𝜔𝑡 ′ 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′
	𝐼 𝑝 𝑑𝑐 =	𝑃 2𝑉 𝑑𝑐 𝜋	𝜋 -𝜋 ∫	1 +	𝑑𝜔𝑡 ′ 𝑉 2 𝑉 𝑑𝑐 𝑠𝑖𝑛 2𝜔𝑡′	(4.90)
	Applying the change of variable 𝑡 = tan	𝜔𝑡 2	, the integral can be evaluated and leads to:
	𝐼 𝑝 𝑑𝑐 =	√ 𝑉 𝑑𝑐	𝑃 2 -𝑉 2	2	(4.91)
	√ 𝑎 1	2 + 𝑏 1	2 , where 𝑎 1 and 𝑏 1 are the corresponding Fourier coefficients of the
	fundamental frequency, that is:
	𝑖 𝑑𝑐 ′ (𝑡) = 𝐼 𝑝 𝑑𝑐 + 𝑖 𝑝 2 = 𝐼 𝑝 𝑑𝑐 + 𝑎 1 cos 2𝜔𝑡′ + 𝑏 1 sin 2𝜔𝑡′	(4.92)
	with
	𝑎 1 =	2 𝑇	∫ -𝜋 2 𝜋 2	𝑃 cos 2𝜔𝑡′ 𝑑𝜔𝑡 ′ 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′	=	𝑃 𝜋	∫ -𝜋 𝜋	cos 2𝜔𝑡′ 𝑑𝜔𝑡 ′ 𝑉 𝑑𝑐 + 𝑉 2 𝑠𝑖𝑛 2𝜔𝑡′	(4.93)
										4.88)
	Considering only the DC and a double frequency component of the input
	current 𝑖 𝑑𝑐 ′ (𝑡), we have:
	𝑖 𝑑𝑐 ′ (𝑡) = 𝐼 𝑝 𝑑𝑐 + 𝑖 𝑝 2 = 𝐼 𝑝 𝑑𝑐 + 𝐼 𝑝 2 𝑠𝑖𝑛(2𝜔𝑡 ′ + 𝛽)	(4.89)
	The magnitude 𝐼 𝑝 𝑑𝑐 can be deduced as follows:

  𝑓 𝑠 has been taken of (4.108). Then, the power switching losses can be computed as the average of the function 𝑃 𝑃 𝑠𝑤 (𝑡) during a half period of the grid (𝑇 = 𝜋):

	where 𝑃 𝑃 𝑠𝑤 ̅̅̅̅̅̅ =	1 𝑇	𝑇 ∫ 𝑃 𝑃 𝑠𝑤 (𝑡) 0 = 𝐼 𝑚𝑎𝑥 𝑉 𝐶 [𝑡 𝑜𝑛 + 𝑡 𝑜𝑓𝑓 ] 𝑑𝑡 32𝜋𝛿𝐿𝑉 𝐶	0 ∫ [𝑉 𝐶 𝜋	2 sin 𝜔𝑡 -𝑉 𝑚𝑎𝑥	2 sin 3 𝜔𝑡]	𝑑𝜔𝑡	(4.115)
	After solving some simple calculations, the power switching losses are computed as
	follows:
	𝑃 𝑃 𝑠𝑤 ̅̅̅̅̅̅ =	𝐼 𝑚𝑎𝑥 [𝑡 𝑜𝑛 + 𝑡 𝑜𝑓𝑓 ] 16𝛿𝐿𝜋	(𝑉 𝐶	2 -	2 3	𝑉 𝑚𝑎𝑥	2 )	(4.116)
		8	𝐼 𝐿 𝑉 𝐶 𝑡 𝑜𝑛				(4.112)
	𝐸 𝑃 𝑂𝐹𝐹 =	1 8	𝐼 𝐿 𝑉 𝐶 𝑡 𝑜𝑓𝑓				(4.113)
	However, the time varying nature of the current waveform leads to the expression:
	𝑃 𝑃 𝑠𝑤 (𝑡) =	𝑉 𝐶 𝑓 𝑠 𝐼 𝑚𝑎𝑥 (𝑡 𝑜𝑛 + 𝑡 𝑜𝑓𝑓 ) 8 = 32𝜋𝛿𝐿 𝐼 𝑚𝑎𝑥 (𝑡 𝑜𝑛 + 𝑡 𝑜𝑓𝑓 ) sin 𝜔𝑡 [𝑉 𝐶 2 sin 𝜔𝑡 -𝑉 𝑚𝑎𝑥	2 sin 3 𝜔𝑡]	(4.114)

Table 4 .

 4 6. Passive elements and semiconductor used in the converter prototype

	Element	Parameter	Symbol Value
	Inductor L	Inductance	𝐿	10 mH
		Equivalent Series Resistance (ESR)	𝑅 𝐿	28 mΩ
	Capacitor C	Capacitance	𝐶	27 µF
	(C4ATHBW4900A3LJ)	Equivalent Series Resistance (ESR)	𝑅 𝐶	1.9 mΩ
	MOSFETs	Drain -Source ON-resistance	𝑅 𝑀	35 mΩ
	(APT94N60L2C3)	Rise time	𝑡 𝑟	27 ns
		Fall time	𝑡 𝑓	8 ns

  By introducing both 𝑆(𝑥) = 0 and the expression of the equivalent control in (5.1), the equation system (5.4) will represent the resulting ideal sliding dynamics of the converter.

	1 -𝑢 𝑒𝑞 =	𝑣 𝑝 𝑣 𝐶1	-	𝐿 1 𝑣 𝐶1	(𝑣 𝑝	𝑑𝐺 𝑑𝑡	+ 𝐺	𝑑𝑣 𝑝 𝑑𝑡	)	(5.3)
	𝑑𝑣 𝑝 𝑑𝑡	=	𝑖 𝑝 𝐶 𝑝	-	𝑣 𝑝 𝐺 𝐶 𝑝
	𝑑𝑖 𝐿2 𝑑𝑡 𝑑𝑣 𝐶1 𝑑𝑡	= = -𝑣 𝐶1 𝐿 2 𝑖 𝐿2 -𝐶 1	𝑣 𝐶2 𝐿 2 + 𝑖 𝐿1 [ 𝑣 𝐶1 𝑣 𝑝 𝐶 1 [ 𝑣 𝑝 -𝑣 𝐶1 -𝐿 1 𝑣 𝐶1 𝐿 1 (𝑣 𝑝 𝑣 𝐶1 (𝑣 𝑖 𝑑𝐺 𝑑𝑡 𝑑𝐺 + 𝐺 𝑑𝑡 + 𝐺 𝑑𝑣 𝑝 𝑑𝑡 𝑑𝑣 𝑝 )] 𝑑𝑡 )]	(5.4)
	𝑑𝑣 𝐶2 𝑑𝑡	=	𝑖 𝐿2 𝐶 2	[	𝑣 𝑝 𝑣 𝐶1	-	𝐿 1 𝑣 𝐶1	(𝑣 𝑝	𝑑𝐺 𝑑𝑡	+ 𝐺	𝑑𝑣 𝑝 𝑑𝑡	)] -	𝑖 0 𝐶 2

  where 𝑚 = 𝑣 𝐶1 ̅̅̅̅/𝑉 𝑖 = 𝑣 𝐶2 ̅̅̅̅/𝑣 𝐶1 ̅̅̅̅ and 𝐺 = 𝑚 2 𝐺/𝐼 𝑜 . Taking the Laplace transform of (5.6), the following multiple -input single-output transfer equation can be obtained:𝐵(𝑠) = -𝑚𝐼 𝑜 𝐿 1 𝐶 1 𝐶 𝑝 𝑠 4 + 𝑚𝐶 1 𝐶 𝑝 𝑉 𝑝 𝐿 1

	𝑑𝑣 𝑝 d𝑡	=	1 𝐶 𝑝	𝑖 𝑝 ̃-𝐺 𝐶 𝑝	𝑣 𝑝 ̃+ 𝑉 𝑝 𝐶 𝑝	𝐺 ̃
	𝑑𝑖 𝐿2 d𝑡		=	2 𝐿 2	𝑣 𝐶1 ̃-1 𝑚𝐿 2	𝑣 𝐶2 ̃-𝑚 𝐿 2	𝑣 𝑝 ̃+ 𝑚𝐺𝐿 1 𝐿 2	𝑑𝑣 𝑝 d𝑡	+	𝐿 2 𝑚𝐿 1 𝑉 𝑝	𝑑𝐺 d𝑡
	𝑑𝑣 𝐶1 d𝑡	=	-1 𝐶 1	𝑖 𝐿2 ̃-𝐺 𝑚 2 𝐶 1	𝑣 𝐶1 ̃+ 2𝐺 𝑚𝐶 1	𝑣 𝑝 ̃+ 𝑉 𝑝 𝑚𝐶 1	𝐺 ̃-𝑉 𝑝 𝐺𝐿 1 𝑚𝐶 1	𝑑𝐺 d𝑡	-	𝑚𝐶 1 𝐺 2 𝐿 1	𝑑𝑣 𝑝 d𝑡	(5.6)
	𝑑𝑣 𝐶2 d𝑡	=	1 𝑚𝐶 2	𝑖 𝐿2 ̃-𝐼 0 𝑚𝐶 2 𝑉 𝑝	𝑣 𝐶1 ̃+ 𝐼 0 𝐶 2 𝑉 𝑝	𝑣 𝑝 ̃-𝐺𝐿 1 𝐼 0 𝐶 2 𝑉 𝑝	𝑑𝑣 𝑝 d𝑡	-	𝐿 1 𝐼 0 𝐶 2	𝑑𝐺 d𝑡	-	1 𝐶 2	𝑖 0 ̃
	𝐴(𝑠)𝑉 𝐶2 (𝑠) = 𝐵(𝑠)𝐺(𝑠) + 𝐶(𝑠)𝐼 𝑝 (𝑠) + 𝐷(𝑠)𝐼 0 (𝑠)	(5.7)
	where:
	𝐴(𝑠) = 𝑚𝐶 1 𝐶 2 𝐶 𝑝 𝑠 4 + ( 𝐺𝐶 2 𝐶 𝑝 𝑚 + [ 2𝐺𝐶 𝑝 𝐿 2 𝑚 3 + 𝐺 𝐿 2 𝑚 + 𝑚𝐶 1 𝐶 2 𝐺) 𝑠 3 + [ 𝐿 2 𝑚 𝐶 𝑝 𝐿 2 𝑚 3 (2𝑚 2 𝐶 2 + 𝐶 1 ) + (2𝑚 2 𝐶 2 + 𝐶 1 )] 𝑠 + 2𝐺 2	𝐺 2 𝐶 2 𝑚	]	(5.8)
									+ [	2𝑉 𝑝 𝐶 𝑝 𝐿 2 𝑚	-	𝐿 2 𝐼 𝑜 𝐺𝑚𝐿 1 𝐿 2	𝑠 3 -[ + 𝑚𝐶 1 𝑉 𝑝 𝐼 𝑜 𝐶 𝑝 𝑚 𝐿 2 ] 𝑠 -(1 + 𝑚 2 𝐿 1 𝐿 2 𝐿 2 𝑚 2𝑉 𝑝 𝐺	) + 𝑚𝐶 1 𝐼 𝑜 ] 𝑠 2	(5.9)
	𝐶(𝑠) = -	𝐺 2 𝐶 1 𝐿 1 𝑚 𝐿 2 𝑚 𝑠 3 + 𝑚𝐶 1 ( 𝑚 2 + 𝐺 + 4𝐺	𝐺𝐿 1 𝐿 2	) 𝑠 2 -( 𝑚 2 + 𝐺 2	𝐶 1 𝑚 𝐿 2	+	2𝐺 2 𝐿 1 𝑚𝐿 2	) 𝑠	(5.10)
	𝐷(𝑠) = -𝑚𝐶 1 𝐶 𝑝 𝑠 3 -(	𝐺𝐶 𝑝 𝑚	+ 𝐶 1 𝐺𝑚) 𝑠 2 -(	2𝑚𝐶 𝑝 𝐿 2	+	𝐺 2 𝑚	) 𝑠 -	2𝑚𝐺 𝐿 2

  𝑃 𝑔 ≈ 𝑣 𝐶2 ̅̅̅̅𝐼 𝑑𝑐 = 𝑉 𝑑𝑐 𝐼 𝑜 ≈ 𝐼 𝑚𝑎𝑥 𝑉 𝑚𝑎𝑥 2 (5.15)where 𝑉 𝑑𝑐 is the reference of the regulated voltage of the DC bus and 𝐼 𝑑𝑐 is the mean value of the output current of the DC-DC converter (which is the input current of the DC-AC converter). Then, the current 𝐼 𝑜 can be approximated by: Then, the relation between incremental variations of the current 𝐼 𝑜 and the amplitude of the output current is given by: This last expression can be approximated as in(5.18) supposing that the amplitude of the oscillatory component of the voltage of the DC bus is considerably small in comparison with its average value. Further, the factor

			𝑉 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 2𝑉 𝑑𝑐 2	is a few times smaller
	than one which further reduces the effect of that component.
	𝑖 𝑜 ̃≈ 𝑉 𝑚𝑎𝑥 2𝑉 𝑑𝑐	𝑖 𝑚𝑎𝑥 ̃≈ 𝐾 𝑔 𝑖 𝑚𝑎𝑥 ̃	(5.18)
	Taking the Laplace transform, it is obtained that:
	𝐼 𝑜 (𝑠) = 𝐾 𝑔 𝐼 𝑚𝑎𝑥 (𝑠)	(5.19)
	𝐼 𝑜 ≈	𝐼 𝑚𝑎𝑥 𝑉 𝑚𝑎𝑥 2𝑉 𝑑𝑐	(5.16)
	𝑖 𝑜 ̃≈ 𝑉 𝑚𝑎𝑥 2𝑉 𝑑𝑐	𝑖 𝑚𝑎𝑥 ̃-𝐼 𝑚𝑎𝑥 𝑉 𝑚𝑎𝑥 2𝑉 𝑑𝑐 2 𝑣 𝑑𝑐 ̃	(5.17)

  𝑁𝑢𝑚(𝑠) = 𝐾 𝑐 𝐾 𝑔 𝑇 𝑐 𝐶 1 𝐶 𝑝 𝑚𝑠 4 + 𝐾 𝑐 𝐾 𝑔 (𝐶 1 𝐶 𝑝 𝑚 + 𝑇 𝑐 𝐺𝐶 𝑝 𝑚 + 𝑇 𝑐 𝐶 1 𝑚𝐺) 𝑠 3 𝐷𝑒𝑛(𝑠) = 𝑚𝐶 1 𝐶 2 𝐶 𝑝 𝑇 𝑓 𝑠 6 + (𝑚𝐶 1 𝐶 2 𝐶 𝑝 + 𝐺𝐶 2 𝐶 𝑝 𝑇 𝑓 𝑚 + 𝑚𝐶 1 𝐶 2 𝐺𝑇 𝑓 ) 𝑠 5 Then, the characteristic polynomial is 𝑃(𝑠) = 𝐷𝑒𝑛(𝑠) + 𝑁𝑢𝑚(𝑠): 𝑃(𝑠) = 𝑚𝐶 1 𝐶 2 𝐶 𝑝 𝑇 𝑓 𝑠 6 + (𝑚𝐶 1 𝐶 2 𝐶 𝑝 + 𝐺𝐶 2 𝐶 𝑝 𝑇 𝑓 𝑚 + 𝑚𝐶 1 𝐶 2 𝐺𝑇 𝑓 ) 𝑠 5

	𝐿(𝑠) =	𝑁𝑢𝑚(𝑠) 𝐷𝑒𝑛(𝑠)	= 𝐾 𝑔 𝐺 𝑖𝑔 (𝑠) (	𝐷(𝑠) 𝐴(𝑠)	) =	𝐾 𝑔 𝐾 𝑐 (𝑇 𝑐 𝑠 + 1) 𝑠(𝑇 𝑓 𝑠 + 1)	( 𝐷(𝑠) 𝐴(𝑠)	)	(5.21)
	where:
		+ 𝐾 𝑐 𝐾 𝑔 (	𝐺𝐶 𝑝 𝑚	+ 𝐶 1 𝐺𝑚 +	2𝑇 𝑐 𝑚𝐶 𝑝 𝐿 2	+	𝑇 𝑐 𝐺 2 𝑚	) 𝑠 2	(5.22)
		+ 𝐾 𝑐 𝐾 𝑔 (	2𝑚𝐶 𝑝 𝐿 2	+	𝐺 2 𝑚	+	2𝑚𝐺𝑇 𝑐 𝐿 2	) 𝑠 +	2𝐾 𝑐 𝐾 𝑔 𝑚𝐺 𝐿 2
		+ [ 𝐺𝐶 2 𝐶 𝑝 𝑚	+ 𝑚𝐶 1 𝐶 2 𝐺 +	𝐶 𝑝 𝑇 𝑓 𝐿 2 𝑚	(2𝑚 2 𝐶 2 + 𝐶 1 ) +	𝐺 2 𝐶 2 𝑇 𝑓 𝑚	] 𝑠 4
		+ [( 𝐿 2 𝑚 𝐶 𝑝	+	𝐺𝑇 𝑓 𝐿 2 𝑚	) (2𝑚 2 𝐶 2 + 𝐶 1 ) +	𝐺 2 𝐶 2 𝑚	+	𝐿 2 𝑚 3 ] 𝑠 3 2𝐺𝑇 𝑓	(5.23)
		+ [ 𝐿 2 𝑚 3 (𝐶 𝑝 + 𝐺𝑇 𝑓 ) + 2𝐺	𝐺 𝐿 2 𝑚	(2𝑚 2 𝐶 2 + 𝐶 1 ) +	2𝐺 2 𝑇 𝑓 𝐿 2 𝑚 3 ] 𝑠 2 +	2𝐺 2 𝐿 2 𝑚 3 𝑠
		+ [ 𝐺𝐶 2 𝐶 𝑝 𝑚	+ 𝑚𝐶 1 𝐶 2 𝐺 +	𝐶 𝑝 𝑇 𝑓 𝐿 2 𝑚	(2𝑚 2 𝐶 2 + 𝐶 1 ) +	𝐺 2 𝐶 2 𝑇 𝑓 𝑚
		+ 𝐾 𝑐 𝐾 𝑔 𝑇 𝑐 𝐶 1 𝐶 𝑝 𝑚] 𝑠 4
		+ [( 𝐶 𝑝 + 𝐺𝑇 𝑓 𝐿 2 𝑚	) (2𝑚 2 𝐶 2 + 𝐶 1 ) +	𝐺 2 𝐶 2 𝑚	+	2𝐺𝑇 𝑓 𝐿 2 𝑚 3
		+ 𝐾 𝑐 𝐾 𝑔 (𝐶 1 𝐶 𝑝 𝑚 +	𝑇 𝑐 𝐺𝐶 𝑝 𝑚	+ 𝑇 𝑐 𝐶 1 𝑚𝐺)] 𝑠 3
		+ [ 𝐿 2 𝑚 3 (𝐶 𝑝 + 𝐺𝑇 𝑓 ) + 2𝐺	𝐺 𝐿 2 𝑚	(2𝑚 2 𝐶 2 + 𝐶 1 )
		+	2𝐺 2 𝑇 𝑓 𝐿 2 𝑚 3 𝐾 𝑐 𝐾 𝑔 (	𝐺𝐶 𝑝 𝑚	+ 𝐶 1 𝐺𝑚 +	2𝑇 𝑐 𝑀𝐶 𝑝 𝐿 2	+	𝑇 𝑐 𝐺 2 𝑚	)] 𝑠 2
		+ [𝐾 𝑐 𝐾 𝑔 (	2𝑚𝐶 𝑝 𝐿 2		+	𝐺 2 𝑚	+	2𝑚𝐺𝑇 𝑐 𝐿 2	) +	2𝐺 2 𝐿 2 𝑚 3 ] 𝑠 +	2𝐾 𝑐 𝐾 𝑔 𝑚𝐺 𝐿 2

  .3. The curves have been generated by means of the analytical model of the PV module using the Lambert-W function:𝐼 0 is the saturation current, 𝑅 𝑠 is the series resistance, 𝐼 𝑝𝑣 is the photogenerated current and 𝑉 𝑡𝑎 is the thermal voltage. 𝐼 𝑝𝑣 and 𝑉 𝑡𝑎 are given by:

	𝑖 𝑝 = 𝐼 𝑝𝑣 + 𝐼 0 -	𝑉 𝑡𝑎 𝑅 𝑠	𝑊 (	𝑅 𝑠 𝐼 0 𝑉 𝑡𝑎	exp ( 𝑣 𝑝 + 𝑅 𝑠 (𝐼 𝑝𝑣 + 𝐼 0 ) 𝑉 𝑡𝑎	))	(5.25)
	𝐼 𝑝𝑣 = 𝐼 𝑠𝑐 (	𝑆 𝑆 𝑛	+ 𝐶 𝑡 (𝑇 -𝑇 𝑛 ))	(5.26)
	𝑉 𝑡𝑎 = 𝑁 𝑠 𝐴𝐾𝑇/𝑞				(5.27)

Table 5 .

 5 1. Parameters to define the I-V characteristic of the PV module

	Parameter	Symbol	Value
	Short Circuit Current Nominal Light Intensity Light Intensity	𝐼 𝑠𝑐 𝑆 𝑛 𝑆	6 A 1000 W/𝑚 2 1000 W/𝑚 2
	Temperature Coefficient	𝐶 𝑡	0.0039 A/C
	Nominal Temperature	𝑇 𝑛	25 °C
	PV module temperature	𝑇	25 °C
	Saturation Current	𝐼 0	6.87 × 10 -11 A
	Number of cells	𝑁 𝑠	36
	Diode Quality Factor	𝐴	1.2
	Series resistance	𝑅 𝑠	0.0137 Ω

Table 5 .

 5

	2. Parameters of the DC-DC converter
	Parameter	Symbol	Value
	Input inductance	𝐿 1	120 𝜇𝐻
	Intermediate inductance	𝐿 2	4.7 m𝐻
	Input capacitor	𝐶 𝑝	11 𝜇𝐹
	Intermediate capacitor	𝐶 1	11 𝜇𝐹
	DC bus Capacitor	𝐶 2	20 𝜇𝐹
	DC bus Voltage	𝑉 dc	400 𝑉
	Grid RMS Voltage	𝑉 𝑔	220 𝑉 ± 10%

Acknowledgements

Dynamical behavior tests

Changes in the input reference (conductance G)

Two different step changes are applied in order to observe the transient behavior of the converter working with conductance variations. In both cases the converter has a resistance of 4.58 kΩ as output load and is fed by a constant DC source of 20 V. The conductance signal is a square wave that allows 50 V changes in the output voltage. In the first test, which is shown in figure 3.28a, the output voltage changes from 350 to 400 V and vice versa. In the second case shown in figure 3.28b, the output voltage changes from 400 to 450 V and vice-versa. A first order response is observed with a time constant of approximately 150 ms.

Changes in the output current

Two different step changes are applied in order to observe the transient behavior of the converter working with output current variations. In both cases the converter is fed by a constant DC source of 20 V. A square wave signal is used to switch a resistive load connected to the output voltage thought a power MOSFET in such a way that the output resistance changes between 3.14 and 4.58 kΩ. In the first test shown in figure 3.29a, the conductance is adjusted to have the maximum output voltage aligned at 400 VDC. In the second case shown in figure 3.29b, the conductance is adjusted to have the minimum output voltage aligned at 400 VDC. A first order response is also observed in both cases with a time constant of approximately 110 ms.

By averaging the expression for a period of , the average power is obtained as:

sin 𝜙 (4.54)

Then, since P = P ̅ , then:

On the other hand, from (4.42) and (4.44) after some simple manipulations, we have:

where:

The power on the AC side is computed as 𝑝 𝑔 (𝑡) = 𝑣 𝑔 (𝑡)𝑖 𝑔 (𝑡): , from which, after some tedious calculations, we have: Considering the ideal case with I 2 = 0, and ϕ = 0, then, we have:

and that:

Therefore, the parameters which define the DC link voltage ripple have been determined. Figure 4.12 shows simulation results of the ripple voltage obtained in PSIM for different sets of parameters validating the above theoretical predictions.

In both cases of figure 4.12 a regulated DC bus voltage of 800 V and an input power of 1 kW have been used. The levels of voltage and power have been selected in order to better illustrate the studied behaviors in the voltage ripple. A grid source of 220 V / 50

Hz is used. Differences between the amplitude and phase displacement of the voltage ripple are observed for two set of values of the passive parameters (L and C). The differences between the simulated values and the theoretically computed values of the amplitude and phase displacement of the voltage ripple are lower than 0.5%. It is worth to note that the phase displacement appears for higher values of inductance; however, for an equal value of power the effect is negligible when the inductance has a value ten It is worth to note that the effect of the ripple component of the current is considerable.

Considering an ideal sliding motion in the current (Δ = 0), the THD reduces to 7.01% and 2.17% for the cases in figure 4.13 respectively.

Power factor and displacement power factor

The power factor is a key aspect of the grid connected systems. Because of the presence of some harmonic content, it is possible to differentiate the effect of the current phase displacement using the Displacement Power Factor (DPF) or to have combined the effect of the current displacement and the harmonic content in the Power Factor (PF). From (4.58), we have:

Hence, the displacement power factor is obtained as: To be complete, it would be interesting to obtain a model for the MPPT block of figure 5.6. We will see in the following section that it can be approximated by a unitary gain linear model of order one or two, that is:

(5.28)

SIMULATION RESULTS

In order to verify the theoretical predictions done in this work and summarized in this chapter, a simulation of the overall system has been performed in PSIM verifying the derived linear models, the stability and the performance of the control system.

Validation of the model of the chain DC-DC-AC

In order to test the validity of the derived linear model, the block diagram depicted in In order to complete the above described simulation, the MPPT block in figure 5.6 has been represented by the following second order numerical transfer function:

(5.29)

A change in the irradiance value of the model has been applied in order to evaluate the dynamic response of the system. Note that this test involves a dynamical motion in the main variables of the microinverter in the PV module side, the DC bus and the grid side.

Dynamic response in front of changes in the P-V characteristic

A simulation is performed in order to assess the dynamic response of the microinverter system working with changes in irradiance of the Solar Module from 600 W/m 2 to 200 W/m 2 and vice versa holding each value during 0.5 s. Note that these changes are equivalent to positive and negative step disturbances of 80 W. The simulated results

shown in figure 5.14 demonstrate that the system remains stable showing robustness at PV side, DC bus, grid side and MPPT algorithm.

Start-up of the microinverter system

The start-up of the microinverter has been simulated to show the transient behavior from the start (zero initial conditions) to an equilibrium point at a maximum power of 120 W. As it can be observed in figure 5.15, the steady-state is reached in around 100 ms. Note that the current of the PV module evolves quickly to the short circuit value and in consequence the power attains values in the neighborhood of the maximum power point ensuring a good performance.

Two solar panels BP 585 from BP SOLAR has been used in the experimental test. The characteristics and parameters are listed in figure 5.17.

Start-up of the microinverter system

The start-up of the microinverter has been captured in figure 5

CONCLUSIONS

In this chapter, the study of an entirely-sliding-mode-controlled transformer-less single-phase dual-stage grid-connected microinverter has been presented. The coupling of the building blocks has been done studying the cascade connection of the DC-DC and DC-AC conversions with the MPPT algorithm. A global linear model has been derived

for the cascade connection of the DC-DC and the DC-AC stages and its stability has been numerically evaluated showing that the chain DC-DC-AC is asymptotically stable in the overall range of the operational conditions. The operation of the overall system has been tested using simulation and experimental results confirming the initial hypothesis of the compatibility of the proposed independent control laws and the feasibility of a microinverter satisfying all the desired control objectives.

Conclusions and perspectives

This dissertation has summarized the theoretical analysis, discussions, simulation and experimental results of our study about the control problems associated with a low power photovoltaic generator. Using a dual-stage grid-connected topology of module oriented converter or microinverter as it is also called, it has been possible to study the problem of the maximum power point tracking, the problem of input impedance matching of a high-gain DC-DC converter, and the problem of the grid-connection of a DC-AC converter, all from the perspective of the sliding mode control theory. We have proposed and analyzed in detail a new MPPT algorithm which is performant and is compatible with the control of the DC-DC stage through the input conductance. In the DC-DC stage, we have intensively studied the quadratic boost converter and its operational modes using the Loss-Free-Resistor concept. The DC-AC converter has been controlled to ensure the stability of the power flow in the microinverter regulating the voltage in the DC bus and injecting real power to the grid with a high power quality level. The control of both converters has been obtained using current control loops based on simple hysteresis comparators which ensure the sliding motion giving simplicity and robustness to the system. Finally, the three constitutive elements (MPPT, DC-DC and DC-AC) have been studied together deriving a global linear mode of the microinverter which allows understanding the dynamic of the entire system. Therefore, it has been demonstrated that the proposed control laws are feasible and reliable, compatible with a real application with microinverters.

The development of this thesis has also generated research questions related with the aspects listed below and which concentrate our future efforts:

 Complement the study and comparison of the quadratic boost converter topologies and operational modes concluding about the best obtained performance.

 Optimize the overall system applying appropriate assembling techniques and layout optimization to minimize the size of the prototype close to the commercial products.

 Integrate the implementation of the entire control system into a single digital device studying the effect of discretization in the performance and stability of the system.

 Extend the results of the MPPT algorithm to multivariable systems and give its properties to work in mismatched conditions. To design the values of the passive components, it is necessary to define the power and voltage levels of the converter. Considering that the output voltage of the DC-DC converter is the same DC bus voltage, the value of the capacitor 𝐶 2 will be obtained from the analysis of the DC-AC stage assuming the voltage at this point as constant.

The figure A1.2 shows the waveforms of the current on inductors 𝐿 1 and 𝐿 2 , and also the voltage on the capacitor 𝐶 1 .

Using the above analysis, the value of the inductor 𝐿 1 can be selected using the switching frequency, the input and output voltages, and the width of the hysteresis band 2∆ 𝑖 𝐿1 . Then, the expression (3.119) can be solved for determining 𝐿 1 and leads to:

and 𝑉 𝐶2 the regulated voltage of the DC bus.

A1.1. Inductor 𝑳 𝟐

Using the same criteria for the inductor 𝐿 1 , the continuous conduction mode is ensured by:

It is important to mention that the dynamic behavior of the inductor current 𝑖 𝐿2 is in fact a result of the ideal sliding dynamic induced through the inductor current 𝑖 𝐿1 .

Besides, the transient response on the inductor current 𝑖 𝐿2 can exhibit deviations with respect to the equilibrium point during changes of the reference or disturbances.

Therefore selection of ℎ 𝐿2 is a key design aspect. The value of the inductor can be selected by satisfying:

The design of the capacitor 𝐶 1 must be done to satisfy the stability condition (3.117).

That expression can be rewritten as:

Further, the value of 𝐶 1 must ensure a low ripple in order to satisfy the approximations introduced by the use of the equivalent control for analyzing the sliding motion. Using definition of 𝑄 𝐶1 in figure A1.2 after simple mathematical manipulations, the following expression is obtained:

Appendix 2:

DC-AC converter design criteria

This appendix presents the main criteria to select the value of the inductor and the capacitor of the full bridge converter shown in figure A2.1 using bipolar commutation.

As it can be observed, the inductor is the main element of the grid connection and the capacitor is the coupling element which connects the DC-AC stage to its power source (DC-DC converter operating as SM-LFR). After defining the value of the inductor, it is then possible to determine a value for the capacitor considering a maximum allowable voltage ripple at the DC bus. For this, the expression (4.86) is solved for 𝐶 as follows:

being 𝑉 𝑑𝑐 the regulated value of the voltage 𝑉 𝐶2 of the quadratic boost converter. It is important to recall that at this point the condition (3.117) must also be verified.

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