Mohsen Rezayati

Farzad Tahami

Jean-Luc Schanen

Prof Yves Lembeye

Dr Mohammadreza Zolghadri

Prof Shahriar

Nadine Michel

Laurent Gerbaud

David Frey

Mahmoud Fotouhi

Yvan Avenas

Morteza

Reza Soleiman

Mahdi, Baoling Alexis Ali

Adrien Yousra

Bakr, Blazej Damien Andre

Nisith Nicolas Lucas

Nasreddin

Keywords: State Plane Analysis and Optimization of CLLC Resonant Converter for Bidirectional EV Charger Bidirectional EV charger, Magnetic components design, Multiple operating points, Optimization, Power loss modeling, Resonant converter, Sequential quadratic programming, Soft switching, State-plane analysis State Plane Analysis and Optimization of CLLC Resonant Converter for Bidirectional EV Charger

Titre: Analyse par le plan de phase et optimisation d'un convertisseur résonant CLLC pour chargeur de véhicule bidirectionnel Mots clés: Chargeur EV bidirectionnel; Conception de composants magnétiques; Plusieurs points de fonctionnement; Optimisation; Modélisation des pertes; Convertisseur résonant; Programmation quadratique séquentielle; Commutation douce; Analyse par le plan de phase.

Résumé: Le convertisseur résonant CLLC est un candidat prometteur pour les applications de transfert d'énergie bidirectionnelle à haute efficacité telles que le chargeur embarqué véhicule-réseau (V2G). Usuellement, il est étudié via l'approximation du premier harmonique (FHA), qui a une faible précision. Dans cette thèse, une méthode d'analyse de circuit basée sur un changement de variables est présentée qui décrit les équations d'état au moyen de deux jeux d'équations découplées, puis les résultats sont reconstruits dans le référentiel d'origine. La méthode proposée est utilisée pour analyser en profondeur le convertisseur résonant CLLC, ainsi que dans son processus d'optimisation. La charge d'une batterie lithium-ion comporte une large gamme de tensions et de courants, par conséquent, le chargeur doit être conçu et optimisé pour faire face à de multiples points de fonctionnement. Dans cette thèse, une méthodologie d'optimisation prenant en compte un ensemble multiple de points de fonctionnement est présentée. L'algorithme de programmation quadratique séquentielle (SQP) est utilisé lors de l'optimisation, ce qui impose à l'ensemble des paramètres et des modèles d'être dérivables. On raisonne donc dans un espace imaginaire, puisque tous les paramètres ne sont pas physiquement implémentables. Une fois l'optimum obtenu, le retour au monde réel (discrétisé et physiquement implémentable) est proposé, en sélectionnant les valeurs discrètes inférieurs et supérieures. On obtient ainsi 24 convertisseurs physiquement réalisables, au sein duquel on sélectionne le meilleur. Pour vérifier cette méthodologie, deux prototypes expérimentaux sont mis en oeuvre: un convertisseur résonnant CLLC optimal de 3,3 kW et un convertisseur résonant CLLC de plus petite taille. Il est montré que le convertisseur optimal répond aux contraintes des points de fonctionnement multiples avec une puissance de 3,3kW et un rendement moyen de 98%, alors qu'il n'est pas possible pour le plus petit convertisseur d'atteindre la puissance de 3,3kW et de répondre à toutes les contraintes.

v
proposed method is then used to thoroughly analyze the CLLC resonant converter operating in the CCM and discontinuous conduction mode (DCM). The voltage and current stresses of components, zero voltage switching condition, output voltage gain, output characteristic diagram, and mode boundary of CCM/DCM are then obtained. The accuracy of the proposed approach is verified by the simulation on a 3.3 kW bidirectional CLLC resonant converter with GaN transistors as the primary and secondary side switches. The results confirm the accuracy of the proposed state-plane analysis of the CLLC resonant converter operating in either direction of power transfer.

By completing the exact analysis of the CLLC resonant converter, the second aim of this dissertation is relative to the optimization of the CLLC resonant converter. The CADES Software is used as the platform of optimization and the state plane analysis is implemented on it using C++. Furthermore, the accuracy of the implemented method is verified by Maxwell simulation to be able to be used in the optimization process and check the value of magnetic components, saturation of magnetic material, etc.

The common optimization of a resonant converter has been based on the single operating point. The selection of a single operating point leads the designed converter not to meet all the constraints when the converter operates in a different operating point. Based on the charging process of a Lithium-Ion battery, its operating region contains a wide range of voltages and currents. Therefore, there exist multiple operating points for a bidirectional EV charger. In this dissertation, a new algorithm for multiple operating points optimization is presented. The Sequential Quadratic Programming (SQP) algorithm is adopted to find the optimum results, where the second derivative of the objective functions must be available. However, there are some discrete parameters for optimizing a bidirectional EV charger, such as core sizes, number of turns, and number of turns in each layer. Therefore, in the proposed method, all the discrete parameters are considered as continuous parameters for the availability of the derivatives. This set of parameters is entitled to the imaginary world due to the fact that these parameters are not physically implementable. The proposed method optimizes the CLLC resonant converter in the imaginary world and with some steps tries to find the optimum solution in the real world (i.e. the physically implementable parameters). In each step, the discrete parameters will be fixed by putting a floor under them and setting a ceiling on them. At the end of the steps, the optimum charger is selected between 24 samples. Finally, the two aims of this dissertation, i.e. analysis and optimization, are verified by the experimental tests and comparison with the simulation results and FHA-based results. The vi accuracy of the proposed state-plane analysis is verified by the experiments on a 3.3 kW bidirectional CLLC resonant converter with GaN transistors as the primary and secondary side switches. The results confirm the accuracy of the proposed state-plane analysis of the CLLC resonant converter operating in either direction of power transfer. The optimization algorithm is also verified using two experimental setups, i.e. the optimum converter and a smaller in-size converter.

vii Résumé À mesure que de plus en plus d'énergies renouvelables entrent en scène, le changement en cours sera radical pour le réseau électrique. Le secteur de l'énergie doit trouver de nouvelles façons d'équilibrer la production et la consommation d'énergie. Les batteries de véhicules électriques (EV) sont de loin la forme de stockage d'énergie la plus rentable car elles ne nécessitent aucun investissement supplémentaire en matériel. Un chargeur EV bidirectionnel est une nouvelle technologie qui permet de renvoyer l'énergie vers le réseau électrique et les charges auxiliaires depuis la batterie. Les exigences électriques du chargeur EV sont examinées en détail pour trouver un convertisseur d'électronique de puissance approprié pour répondre à toutes les exigences.

Le convertisseur résonant CLLC est un candidat prometteur pour les applications de transfert de puissance bidirectionnelle à haut rendement telles que les chargeurs embarqués véhiculeréseau (V2G) et les convertisseurs CC-CC de véhicules hybrides. Les principes fondamentaux du chargeur EV bidirectionnel basé sur le convertisseur résonnant CLLC et ses principaux composants, y compris le transformateur, les inductances résonantes, les condensateurs résonants, la partie onduleur et redresseur, etc. sont étudiés en profondeur, où les avantages et les inconvénients du convertisseur résonant CLLC et les candidats possibles sont mis en évidence.

Une méthodologie de conception globale comprenant la conception des transformateurs, la conception des inductances, la conception des condensateurs, les pertes des semi-conducteurs, les pertes des diodes, les pertes du noyau magnétique et les pertes des condensateurs est étudiée pour définir un problème d'optimisation correspondant à la fonction objectif et aux contraintes.

Une méthode basée sur l'interpolation est proposée pour les pertes de commutation du transistor GaN avec les différents niveaux de tension et de courant, les circuits de commande de grille, la température de jonction, les condensateurs parasites, etc. Un modèle de perte de conduction est également présenté sur la base de la fiche technique des transistors GaN. La fonction objectif peut être considérée comme maximisant l'efficacité et la densité de puissance, ou minimisant la masse, le volume et les pertes. Les contraintes sont également mises en évidence, qui sont la région de fonctionnement du chargeur de batterie, la température maximale, le courant RMS maximal des éléments résonants, la tension maximale, la condition de commutation douce et le fonctionnement en mode de conduction continue (CCM). Dans de nombreux travaux, la plupart de ces contraintes et l'analyse du convertisseur résonant CLLC sont calculées par la méthode viii d'approximation du premier harmonique (FHA), qui a une faible précision. Par conséquent, l'analyse de CLLC reste difficile en raison de sa nature multi-résonnante complexe et de plusieurs éléments de stockage.

Dans cette thèse, une méthode d'analyse de circuit basée sur le changement de variables est présentée qui représente les équations de l'espace d'état en deux ensembles découplés d'équations. Les analyses sont effectuées dans deux systèmes de coordonnées de plan d'état, puis les résultats sont reprojetés sur la région d'origine. Les deux jeux d'équations découplés sont similaires au celles de deux convertisseurs à résonance série. La méthode proposée est ensuite utilisée pour analyser en profondeur le convertisseur résonant CLLC fonctionnant en mode de conduction continue (CCM) et discontinue (DCM). Les contraintes de tension et de courant des composants, la condition de commutation de tension nulle, le gain de tension de sortie, le diagramme de caractéristique de sortie et la limite de mode de CCM/DCM sont ensuite obtenus. La précision de l'approche proposée est vérifiée par la simulation sur un convertisseur résonnant CLLC bidirectionnel de 3,3 kW avec des transistors GaN comme commutateurs côté primaire et secondaire. Les résultats confirment la précision de l'analyse du plan d'état proposée du convertisseur résonnant CLLC fonctionnant dans les deux sens de transfert de puissance.

En complétant l'analyse exacte du convertisseur résonnant CLLC, le deuxième objectif de cette thèse est relatif à l'optimisation du convertisseur résonnant CLLC. Le logiciel CADES est utilisé comme plate-forme d'optimisation et l'analyse du plan déphase y est implémentée à l'aide de C++. De plus, la précision de la méthode mise en oeuvre est vérifiée par la simulation éléments finis Maxwell pour pouvoir être utilisée dans le processus d'optimisation et vérifier la valeur des composants magnétiques, la saturation du matériau magnétique, etc. L'optimisation d'un convertisseur résonnant a été basée sur le point de fonctionnement unique. La sélection d'un point de fonctionnement unique conduit le convertisseur conçu à ne pas respecter toutes les contraintes lorsque le convertisseur fonctionne pour un point de fonctionnement différent. Basé sur le processus de charge d'une batterie lithium-ion, sa région de fonctionnement contient une large gamme de tensions et de courants. Par conséquent, il existe plusieurs points de fonctionnement pour un chargeur de VE bidirectionnel. Dans cette thèse, un nouvel algorithme d'optimisation de points de fonctionnement multiples est présenté. L'algorithme de programmation quadratique séquentielle (SQP) est adopté pour trouver les résultats optimaux, où la dérivée seconde des fonctions objectives doit être disponible.

Cependant, il existe certains paramètres discrets pour optimiser un chargeur EV bidirectionnel, tels que la taille des noyaux magnétiques, le nombre de tours et le nombre de tours dans chaque ix couche. Par conséquent, dans la méthode proposée, tous les paramètres discrets sont considérés comme des paramètres continus pour calculer des dérivées. Ceci correspond à raisonner dans un monde "imaginaire"t du fait que ces paramètres ne sont pas physiquement implémentables.

La méthode proposée optimise le convertisseur résonnant CLLC dans ce monde imaginaire et avec quelques étapes tente de trouver la solution optimale dans le monde réel (c'est-à-dire les paramètres physiquement implémentables). À chaque étape, les paramètres discrets seront continus en mettant un plancher sous eux et en leur fixant un plafond. A la fin des étapes, le chargeur optimal entre 24 possibilisées discrètes est sélectionné.

Enfin, les deux objectifs de cette thèse, à savoir l'analyse et l'optimisation, sont vérifiés par les tests expérimentaux et la comparaison avec les résultats de simulation et les résultats basés sur la FHA. La précision de l'analyse du plan d'état proposée est vérifiée par les expériences sur un convertisseur résonant CLLC bidirectionnel de 3,3 kW avec des transistors GaN comme interrupteurs au primaire et au secondaire. Les résultats confirment la précision de l'analyse du plan d'état proposée du convertisseur résonnant CLLC fonctionnant dans les deux sens de transfert de puissance. L'algorithme d'optimisation est également vérifié à l'aide de deux configurations expérimentales, à savoir le convertisseur optimal et un convertisseur de plus petite taille. The angle of between zero crossing of iL1 and iL2 δ Skin depth of a wire δN

List of Tables

The defined angle in Fig. 5 The defined angle in Fig. Fossil fuels have been the main source of energy supply in the world [START_REF] Balat | Status of Fossil Energy Resources: A Global Perspective[END_REF]. With the development in the transportation industries, as well as the increase in energy consumption, the consumption of fossil fuels has increased even more [START_REF] Neves | The substitution of fossil fuels in the US transportation energy mix: Are emissions decoupling from economic growth?[END_REF]. It causes more environmental pollution and a reduction in fossil fuel resources. CO2 emissions from fossil fuels have been about 35 billion tons [START_REF] Friedlingstein | Global Carbon Project[END_REF] and it is increasing rapidly as shown in Fig. 1-1. The transportation industry can also increase CO2 emissions if conventional vehicles are supplied with fossil fuels. These reasons and the limited resources of fossil fuels and growing oil prices have led the international community to use clean and renewable energies as the main energy resource [START_REF] Marinescu | Toward a practical solution for residential RES based EV charging system[END_REF]. Electric vehicles (EVs) are a new solution due to their contribution towards reducing greenhouse gas emissions.

Fig. 1-1: Annual CO2 emission from fossil fuels and industry.

Source: Global Carbon Project [START_REF] Friedlingstein | Global Carbon Project[END_REF] Fig. 12: EV sales in the world from 2012 to 2021 [START_REF] Irle | Global EV Sales for 2022 H1[END_REF] The above figure manifests that global EV sales have exponentially increased since 2012, and the US Energy Information Administration (EIA) expects to grow them to 2.21 billion vehicles by 2050 [START_REF] Dwyer | EIA projects global conventional vehicle fleet will peak in 2038[END_REF].

In general, the advantages of electric vehicles in comparison with internal combustion vehicles can be presented as follows [START_REF] Emadi | Advanced electric drive vehicles[END_REF]: Despite the many advantages of electric vehicles, one of their most important drawbacks is their short driving distance, which can be solved by using a larger capacity battery, improving the technologies of the batteries, using hybrid configuration, etc. [START_REF] Sawsan | Review on State-of-the-Art Unidirectional Non-Isolated Power Factor Correction Converters for Short-/Long-Distance Electric Vehicles[END_REF].

•
The second disadvantage of electric vehicles is the limited number of charging stations in the world. Until 2021, there are only 569,000 fast public chargers in the world [START_REF]Global EV Outlook 2022: Securing supplies for an electric future[END_REF]. However, it is expected to reach 1.3 million public charger stations by 2025, and also 2.9 million stations by 2030 [START_REF]The state of EV charging infrastructure in Europe by 2030[END_REF].

The third disadvantage of electric vehicles is the diversification of the existing EV charging standards which makes the designers perplexing. As an example, Table 1-1 shows the current and voltage levels of IEC and SAE standards for DC and AC charging [START_REF] Syed Muhammad | Review of Electric Vehicle Technologies, Charging Methods, Standards and Optimization Techniques[END_REF]: The sales of different electric vehicles have increased from less than 210 thousand in 2014 to more than 6.7 million in 2021 [START_REF] Irle | Global EV Sales for 2022 H1[END_REF]. The electric vehicle consists of several parts, including the electric motor, battery pack, onboard charger, power electronics controller, battery management system (BMS), and thermal cooling system. . For EV owners, the most reliable place to charge batteries is their home [START_REF] Chu | Vehicle Electrification: Status and Issues[END_REF]. On-board chargers help EV owners to charge their vehicles wherever there is an appropriate electric outlet [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF].

High temperature applications

High voltage operation

High frequency switching

Fig. 1-5: General comparison between Si, SiC, and GaN semiconductors [START_REF] Bigorra | Innovative electrical and electronic architecture for vehicles with dual voltage power networks. In-vehicle application[END_REF] This advancement in the sale of electric vehicles has been made possible by the advent of wide band gap (WBG) semiconductors [START_REF] Matallana | Power module electronics in HEV/EV applications: New trends in wide-bandgap semiconductor technologies and design aspects[END_REF], which provide high-frequency and high-efficiency capabilities. GaN and SiC semiconductors are common WBG devices. The landscape for the use of SiC and GaN is estimated that the sales of these devices will increase from $400 million sales to over $3000 million sales from 2018 to 2025 [START_REF] Voora | Global Market Insights report Inc. report[END_REF]. As shown in Fig. 12345, GaN devices are suitable for high frequency and high voltage applications [START_REF] Bigorra | Innovative electrical and electronic architecture for vehicles with dual voltage power networks. In-vehicle application[END_REF], where research on GaN semiconductors began in 2000 [START_REF] Grigory | 7.5 kW/mm2 current switch using AlGaN/GaN metal-oxide-semiconductor heterostructure field effect transistors on SiC substrates[END_REF]. In this dissertation, GaN transistors are selected as the main switches.

The future of electric vehicle chargers is a bidirectional charger [START_REF] Tran | The state of the art of battery charging infrastructure for electrical vehicles: Topologies, power control strategies, and future trend[END_REF], which is an advanced EV charger and can charge or discharge the main battery if needed. To make it clear, in a unidirectional EV charger, energy flows from the grid into the battery packs but in a bidirectional EV charger, it can also be pushed back into the grid or other electricity-consuming equipment. Bidirectional power transferring for an EV charger can be used as vehicle-to-grid (V2G), vehicle-to-home (V2H), and vehicle-to-load (V2L) applications as shown in Fig. 1-6 [START_REF] Kwon | An Electrolytic Capacitorless Bidirectional EV Charger for V2G and V2H Applications[END_REF] and [START_REF] Wang | Design, and Control of a SiC Isolated Bidirectional Power Converter for V2L Applications to both DC and AC Load[END_REF]. applications For V2G applications, the stored energy in the batteries can be sent into the grid. For this purpose, it is mandatory to invert the DC voltage of the batteries into AC voltage with grid synchronization (matching the frequency and voltage) [START_REF] Vaishnav | Single-stage isolated bi-directional converter topology using high frequency AC link for charging and V2G applications of PHEV[END_REF]. It means the structure of a bidirectional charger is more sophisticated than a unidirectional charger. In the V2G applications, the battery can store energy during low profile and push it back to the grid in high demand. It tends to not only reduce the peak demand of power plants and avoid blackouts but also can earn money by charging the batteries at low-price hours and selling the stored electricity at high-price hours [START_REF] Honarmand | Optimal scheduling of electric vehicles in an intelligent parking lot considering vehicle-to-grid concept and battery condition[END_REF]. Furthermore, the surplus energy produced by renewable energies (such as solar and wind) can be stored in EVs and pushed back into the grid at required times. For example, CHAdeMO is an off-board charger designed by some major companies in Japan [START_REF] Das | Electric vehicles standards, charging infrastructure, and impact on grid integration: A technological review[END_REF], which supports V2G applications.

In V2H applications, the EV plays the role of uninterruptable power supply (UPS) for domestic applications [START_REF] Monteiro | Operation Modes for the Electric Vehicle in Smart Grids and Smart Homes: Present and Proposed Modes[END_REF] and can directly supply the required power of a home without any grid connection and synchronization, which makes its installation easy. The V2H application is useful for intelligent buildings with rooftop solar cells [START_REF] Nazari | Electric vehicles for smart buildings: a survey on applications, energy management methods, and battery degradation[END_REF].

V2L applications can supply the appliances inside the vehicle such as laptops, lighting, telecommunication, etc. with the main battery packs of an EV [START_REF] Bai | Advanced Components for Electric Vehicle System: Vehicle-to-Grid Metering, Vehicleto-Vehicle Charging, and High-Power Charge Circuit Interrupting Devices[END_REF]. In emergency conditions, the acid battery can be excited and the empty battery of other EVs can be charged shortly with an initial charge of the bidirectional V2L charger.

Generally, on-board chargers have high power density, high efficiency, and low weight.

Also, they increase the flexibility of driving because they can be plugged in anywhere there is an electric outlet. However, due to the limited space in a vehicle, on-board chargers have a limited power level to achieve and their nominal power usually is lower than 3.5 kW [START_REF] Haghbin | Grid-Connected Integrated Battery Chargers in Vehicle Applications: Review and New Solution[END_REF].

Nowadays, the power rating of on-board chargers has been increased significantly with the progress of semiconductor devices (e.g. GaN transistors) and introducing the new high powerdensity power electronics converters (e.g. CLLC resonant converter). Considering these features, on-board charging can be the main type of EV charging by offering fast charging, high efficiency, high power density, bidirectional power transferring, reliability, and low manufacturing cost capabilities to the EV market. As a result, an optimization algorithm can design a bidirectional EV charger optimally while all requirements are also met.

Problem Statement and Challenges

Owing to surpassing the EV market, the designed EV charger should offer competitive efficiency, power density, bidirectionally, and charging time. Although on-board chargers have been studied in recent publications, there are several challenges moving toward high efficiency and high power density bidirectional EV chargers. These challenges are related to various parts associated with the on-board chargers such as the selected converter, analysis method, optimization algorithm, implementation of magnetic components, control, and appropriate components selection.

According to the policies of some European Unions and the U.S., the production of petroleum vehicles will stop in the next few years and the world will turn to electric vehicles [START_REF] Plumer | 6 Automakers and 30 Countries Say They'll Phase Out Gasoline Car Sales[END_REF]. It means the widespread use of EVs will emerge in the world, and obviously, it leads to EV chargers being used every day and on a large scale. Therefore, a charger should be designed correctly to meet the following objectives.

• Operating in a higher efficiency: by increasing the total efficiency or reducing the losses associated with the charger, even by lower than a percent, global energy consumption will be decreased and more fossil fuel resources will be saved, due to the widespread use of the EV chargers. Therefore, designing a charger with higher efficiency is preferable.

• Preparing required voltage ranges and current ranges: the voltage and current of the battery packs vary according to the state-of-the-charge (SOC) of the batteries. Therefore, the charger should be designed correctly to prepare the required voltages and currents.

• Having lower volume and mass of product: a crucial step towards achieving a bestselling product is to reduce the volume and/or mass of that product. A designer can perform an analysis to design an impacted charger. Therefore, geometrical considerations are also mandatory in the designing process.

• Ensuring ZVS and ZCS in the feasible operating regions: there is a need for zero voltage switching and zero current switching capability for the designed charger to reduce switching losses. They can provide a higher efficiency charger with a lower operating temperature, which can also increase the lifespan of the charger.

• Lower conducted and radiated emissions: conducted emission is unintended energy propagated through the cables or harness to the components. Radiated emission is also unintended energy, which propagates through the electric and/or electromagnetic fields without any harness. These emissions are generated by a component and affect other components. Some standard levels are defined to clarify the allowable noise levels. A designer can decrease these emissions by adding LC filters, common mode chokes, ground-plane in the printed circuit board (PCB), shielding, etc.

• Establishing power factor correction: for the grid-connected converters, the power factor close to unity can reduce losses in the distribution system. In this dissertation, a bidirectional battery charger will be designed, which supplies from the grid. So, this part should be considered in the design process.

• Reducing costs: the cost of a product has a direct effect on the rapid market progress and success of that product to sell like hotcakes. Design considerations can play a crucial role in reducing costs.

• Increasing reliability and accessibility: precise peak and RMS values analysis related to the voltage across and current passed through a component, real conditions (such as ambient temperature, nonlinear behavior of a component, magnetic cores saturation), applied fields (such as magnetic field and solar irradiance), physical stresses on the components, humidity, etc. have to be considered for reliability and accessibility assessment [START_REF] Gurpinar | Reliability-Driven Assessment of GaN HEMTs and Si IGBTs in 3L-ANPC PV Inverters[END_REF]. A designer should adopt comprehensive analysis and proper component selection to meet the reliability criteria.

• etc.

The role of analysis in design and optimization has received increased attention across several pieces of research in recent years [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF], [START_REF] Choi | Design of a 3.3 kW 100 kHz EV Charger Based on Flyback Converter With Active Snubber[END_REF], [START_REF] Bosshard | Multi-Objective Optimization of 50 kW/85 kHz IPT System for Public Transport[END_REF], [START_REF] Ali | Multiobjective Optimized Smart Charge Controller for Electric Vehicle Applications[END_REF], [START_REF] Dube | Analysis and Design of an Integrated Bidirectional Three-Phase AC-DC Resonant Converter[END_REF], [START_REF] Ramezani | Optimal Design of Fully Integrated Magnetic Structure for Wireless Charging of Electric Vehicles[END_REF], [START_REF] Singh | Design of GaN based 72V,3.3KW LLC Resonant Converter for on-board EV Charger[END_REF], [START_REF] Qi | 650V GaN Based 3.3kW Bi-Directional DC-DC Converter for High Efficiency Battery Charger with Wide Battery Voltage Range[END_REF], [START_REF] Zou | Modeling and Optimization of an Integrated Transformer for Electric Vehicle On-Board Charger Applications[END_REF], [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF],

and [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF]. Analysis methods using such an assumption reduce the accuracy of the designed and optimized converter. Furthermore, an analysis method without a closed form requires some iterations to find the parameters and increases run time and complexity. Therefore, the following considerations should be attended to for an analysis method:

• Precise • Closed-form • Fast • Comprehensive
An optimized battery charger with the minimum volume/mass, which operates in multiple operating points and satisfies several constraints is a fundamental property of a world-class EV charger. High efficiency and high power density battery charger are necessary to achieve them.

A set of inputs is found to find the minimum/maximum of an objective function while the constraints have been met. For the optimization process, the following characteristics are mandatory:

• Accuracy and convergence: it is obvious that the accuracy of an optimization method plays a vital role in the selection of that method. Stochastic and recursive algorithms suffer from a lack of well-accuracy convergence. Deterministic algorithms can improve the accuracy of the optimization. However, it lacks regarding for the absolute extremum value convergence and may converge into relative or local extremum values.

Initialization with different starting points may resolve this issue as shown in Fig. 1234567.

Global maximum

Local maximum

Starting point 1

Starting point 2 • Multi operating points capability: there are multiple operating points for an EV battery charger since it operates in different voltage levels and currents. Previous studies have mainly concentrated on a single operating point (e.g. [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF] and [START_REF] Choi | Design of a 3.3 kW 100 kHz EV Charger Based on Flyback Converter With Active Snubber[END_REF]). Furthermore, some EV chargers are designed for battery charging applications regarding the wide output voltage and different current levels (e.g. [START_REF] Bosshard | Multi-Objective Optimization of 50 kW/85 kHz IPT System for Public Transport[END_REF] and [START_REF] Ramezani | Optimal Design of Fully Integrated Magnetic Structure for Wireless Charging of Electric Vehicles[END_REF]). However, the designed converters suffer from the lack of optimum design for the chargers with multiple operating points. So, it makes adequate performance difficult to obtain.

Therefore, the main challenges of this dissertation are to obtain a new exact analyzing method, to present a design methodology for an EV battery charger design, and to attain an optimized battery charger with the maximum power density regarding multiple operating points.

It should be mentioned that the connection of battery chargers to the grid can produce detrimental harmonic effects on it and increase total harmonic distortion and noise emission.

This drawback can be mitigated by using differential and common mode chokes as an input filter, reducing the rising time and falling time of sharp waveforms, and using an active frontend rectifier for the rectifying part. In this dissertation, only DC to DC converter will be analyzed, because designing an EMI filter, switching time control, etc. are beyond the scope of this dissertation.

The DAB, LLC, and CLLC converters are commonly used as the DC-to-DC stages of EV chargers. Therefore, the proper selection of the main DC-to-DC converter is the first challenge of an EV charger. Besides, the current on-board chargers, specially DAB-based, LLC-based, and CLLC-based chargers, have been analyzed based on the conventional analyzing method, i.e. the first harmonic approximation (FHA) method [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF], [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], [START_REF] Malan | Modeling and Control of a Resonant Dual Active Bridge With a Tuned CLLC Network[END_REF], [START_REF] Li | Bi-directional on-board charger architecture and control for achieving ultra-high efficiency with wide battery voltage range[END_REF], and [START_REF] Kim | High-Efficiency Isolated Bidirectional AC-DC Converter for a DC Distribution System[END_REF].

Furthermore, the efficiency of the designed charger based on the conventional method deteriorates when the load is far away from the nominal point. This deviation also occurs for power density and other features of the EV charger. Moreover, there is no precise method for the analysis of high-order resonant converters in order to be used in this dissertation. In addition to approximate methods, the parameters sweeping method has been adopted in several publications (e.g. [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF], [START_REF] Li | High-Frequency Transformer Design with High-Voltage Insulation for Modular Power Conversion from Medium-Voltage AC to 400-V DC[END_REF], [START_REF] Sun | Design and Multi-Objective Optimization of an Auxiliary Wireless Power Transfer Converter in Medium-Voltage Modular Conversion Systems[END_REF]) and its accuracy is challenging.

Generally, the resonant tank of a CLLC resonant converter has a transformer with specific magnetizing inductance, two resonant inductors, and two resonant capacitors [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF]. This converter realizes soft switching in a wide operating region. Most previous works have designed CLLC resonant converters on a single operating point. However, owing to the battery charging process, the charger operates in constant current (CC) and constant voltage (CV) modes.

Therefore, the on-board converter should be designed in corresponding with the whole of the operating region. It is needed to provide a global design methodology for the converter, and especially for magnetic design of the resonant tank.

The new semiconductor devices, like GaN transistors, introduce high power-density capability for power electronics converters [START_REF] He | Design of a 1-MHz high-efficiency high-powerdensity bidirectional GaN-based CLLC converter for electric vehicles[END_REF]. Understanding the GaN power loss modeling, whether switching loss or conduction loss, is one of the most important parts of the optimum design of GaN-based EV chargers. Despite a considerable number of analytical loss measurements, there is no simple and accurate analytical model appropriate for design engineers. Most of the works carried out on the switching loss analysis use an integral form which leads to a complicated and time-consuming analyzing model. Furthermore, for one device, the switching losses are provided at one operating point associated with the datasheet's value, and for a specific gate-driver circuit [START_REF] Rezayati | Interpolation-Based Switching and Conduction Losses Model of GaN Transistors[END_REF]. Therefore, a model is needed to predict the losses for other operating points and different parameters.

Designing a high efficiency and high power density on-board charger is another challenge for an on-board charger in the EV market. Due to the limited space inside an EV, it is needed the power density to be considered in the optimization process. It means a mathematical-based formula of power density shall be obtained and put into the optimization process. Conventional algorithms have not considered power density as a function of electric parameters to optimize in it the whole of the system. The conventional methods have used parameter sweeping or iteration-based algorithms. Also, in several studies, the trial-and-error method has been considered to find the optimum result. Therefore, there is not any guarantee that the obtained results are the best possible result. Besides, a single operating point at a fixed switching frequency has been considered in literature but the charger should be optimized in the whole operating region of the charger.

Research Scopes and Objectives

According to the previous discussions, the research objectives are:

1.3.1. Analysis
Firstly, among the commonly used converters, such as DAB, LLC, and CLLC converters, the LLC is not suitable for bidirectional power transfer. Between the DAB and CLLC converters, the CLLC converters are slightly better than the DAB converters in terms of efficiency, power density, soft switching region, current stresses, size, and cost, for bidirectional EV charger applications [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF]. Therefore, the CLLC resonant converter is selected in this dissertation.

Despite the advantages mentioned for the CLLC converter, it is difficult to achieve its inherent features and characteristics due to the complex structure of the resonant tank and its different operating modes, which make the design and control of the CLLC converter difficult.

Also, as stated in the literature (and shown by the simulation and experimental results in this dissertation), if the first harmonic approximation (FHA) method is used for obtaining the CLLC converter parameters [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF] and [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF]: such as the voltage gain ratio, switching stresses, maximum voltage of capacitors, and maximum current of inductors, there is a crucial difference between the FHA-based results and actual results, which encounters a great obstacle in design and optimization [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF]. Therefore, the first objective of this dissertation is to provide a detailed and exact analysis of the CLLC resonant converter based on the state-plane analysis method, which can be used to achieve all circuit parameters in different operating modes, loads, and frequencies.

Another analyzing part that has not been done yet for the CLLC resonant converter is to check and provide a precise relationship for the ZVS condition of the main primary switches.

In the previous papers and dissertations about CLLC resonant converter, Eq. (1.1) was obtained using the first harmonic approximation method to check the ZVS condition of this converter [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF], and [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF]:

16 dt s ,max t Lm Css f  (1.1)
where tdt is dead-time, CSS is the parasitic capacitor of a GaN transistor, Lm is the magnetizing inductance, and fs,max is the maximum switching frequency. According to Eq. (1.1), it is obvious that this equation is not dependent on the load current and other circuit parameters which leads to deteriorate in accuracy, and causes the converter operates in a non-soft-switching region. This inaccuracy will be shown in chapter 3 by simulation and in chapter 5 by experimental results. Also, the FHA method cannot be used in discontinuous conduction mode [START_REF] Safaee | Time-Domain Analysis of Voltage-Driven Series-Series Compensated Inductive Power Transfer Topology[END_REF], and its accuracy deteriorates when the switching frequency deviates from the resonant frequency.

In [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF], an attempt has been done to provide a detailed analysis of this converter. In this reference, the differential equations corresponding to the state variables were solved and the relationships between the current of inductors and the voltage of capacitors were obtained in the time domain. But for the ZVS condition checking, the same approximate method (i.e. FHA) was used. Whether the proposed methods in [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF] and [START_REF] Liu | A Modified Gain Model and the Corresponding Design Method for an LLC Resonant Converter[END_REF] are remarkably accurate or not, possessing an exact closed-form method, with the ability of transient and steady-state analysis, and easy to obtain the output characteristics, gain voltage ratio, and switching stresses is mandatory. The state-plane analysis method can be the solution.

By reviewing the literature, the state-plane analysis of this converter has not been presented so far. Therefore, one of the objectives of this dissertation is to provide a precise analysis of the CLLC resonant converter in the field of state-space. The desired outcomes are:

• The voltage gain ratio; It should be mentioned that for completing the discussion, the dynamic analysis for transient behavior and startup conditions of this converter can also be examined, but it is beyond the scope of this dissertation.

Furthermore, there is another important aspect of exact analysis. Increasing the operating frequency of a power electronics converter is a promising road map for the power electronics industry, which leads to obtaining a high power density converter. However, operating in high switching frequencies make the influence of parasitic components in the converter performance essential. In this case, the inductance of tracks of a printed circuit board (PCB), parasitic junction capacitors of transistors and diodes, and the leakage inductance of transformers participate actively in resonance. These parasitic elements convert a low-order resonant circuit to a high-order circuit, and its exact analysis will not be possible with the previous methods.

For example, if a series resonant converter operates at a high frequency, this converter may convert to LLC, LLLC, or another high-order resonant converter, which requires a precise analysis method. Therefore, the analysis method performed in this dissertation can also be used not only for high-order resonant converters but also for low-order resonant converters while operating in high-frequency applications.

Optimization

Compared with unidirectional LLC resonant converters and DAB converters, the bidirectional CLLC resonant converter can prepare higher efficiency in two opposite power directions because it can ensure ZVS condition for primary side switches and ZCS for secondary side rectifiers over a wide load range, which is useful for high variation voltage and current applications [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF]. Furthermore, it has a low turn-off current for the primary switches and a low di/dt current for the secondary rectifiers. But these benefits are met when a proper resonant tank is designed based on an optimization strategy. For example, a high value of magnetizing inductance can decrease the circulating current by increasing the impedance of the parallel path, which reduces circulating loss and increases efficiency. However, according to Eq. (1.1) and by increasing Lm, the required dead-time for ZVS operating will be increased.

High dead-time not only reduces the amount of power transmitted to the load and reduces efficiency, but also can impose a restriction on increasing the switching frequency, which leads to the reduction of switching losses and also reduction of power density by a massive selection of passive components. As a result, there is a tradeoff for magnetizing inductance selection and an optimization strategy can solve this requirement. Some tradeoffs may need to be used for other components to obtain a high efficiency and high power density bidirectional CLLC resonant converter.

There is another important aspect of optimization due to the application of the selected resonant converter. The battery charging process includes a wide range of input and output voltages, as well as a whole range of currents from full-load to near-no-load. Therefore, it is necessary to design an optimum charger for the entire voltage and current ranges of the battery, using accurate analysis. A lot of references in the literature considered only one operating point, as a rated point, to optimize the CLLC resonant converter. Also, bidirectional operating must be considered in the optimization process.

It is not possible to increase the volume and mass of an on-board charger whatever you want due to the limited space inside an electric vehicle. Consequently, power density is a crucial rule in the design of an on-board charger. Furthermore, electric vehicles will emerge in widespread use in the world, and obviously, operating at high efficiency is imperative. Therefore, efficiency can be selected as an objective function or constraint to fulfill the aforementioned requirement.

Besides, the first harmonic approximation method was performed in the previous works (e.g. [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF]), which decreases precision. The proposed method based on state-plane analysis can improve the accuracy of the optimization result. In summary, the objectives of the optimization part are as follows:

• There are some tradeoffs in the component selection, which can be solved by optimization; • A wide range of voltages and currents;

• Multiple operating points based on the charging process;

• A bidirectional capability that may change the objective functions;

• Limited space inside an EV; • High efficiency operating constraint;

• Using an exact method in the optimization part for obtaining the objective functions and constraints.

Thesis Outline

This dissertation contains 6 chapters, which focus on the analysis and optimization of a bidirectional full-bridge CLLC resonant converter.

Chapter 1 gives an introduction to the research motivation, types of EV chargers, and the selected power electronics converter.

Chapter 2 provides a literature review of types of electric vehicles, types of EV chargers, the importance of bidirectional transferring, and a general review of CLLC resonant converter in terms of analysis and optimization.

Chapter 3 gives the first harmonic approximation method for the CLLC resonant converter.

The obtained results are compared with PSIM simulation.

Chapter 4 presents the fundamental concepts of the global design, and optimization for using bidirectional EV charging, an optimization algorithm is adopted and the objective function is expressed. Corresponding constraints in EV applications are also investigated. Then, the previous methods for the analysis of the CLLC resonant converter are evaluated and it is shown that there are some design challenges in using previous methods.

Chapter [START_REF] Chrisafis | France to ban sales of petrol and diesel cars by 2040[END_REF] proposes a new approach for the exact analysis and design of a full-bridge resonant converter, as the state-plane analysis. Based on this method, the switching stresses of the components, output characteristic, control plane diagram, and soft switching condition are obtained in continuous conduction mode (CCM) and discontinuous conduction mode (DCM).

Furthermore, the CCM and DCM boundaries are expressed. By simulation, the proposed stateplane analysis for CLLC resonant converter is verified.

Chapter 6 provides an algorithm for the optimization of multi operating points problems in bidirectional EV charging applications. At the first of this chapter, the input parameters, output parameters, and constraints are expressed. Then state-plane analysis codes are implemented into CADES software based on C++ integrated design environment. The implemented codes are verified using Maxwell simulation. In CADES software, a single operating point optimization is done and it is shown that the designed converter based on the single operating point cannot meet all the constraints for the other points. Therefore, the multi operating points optimization algorithm is proposed which can find an optimum converter through 9 steps, when all the constraints are met.

Chapter 7 is dedicated to the conclusions and future work.

Literature Review

In this chapter, a review of previous works is presented, and transportation electrifications background, types of battery chargers of these vehicles, vehicle-to-grid (V2G) power transfer, and a detailed review of resonant converters, especially the CLLC resonant converter are described.

One of the aspects of a charger is that the charger is installed on the vehicle, i.e. the on-board charger, or a charger is installed outside the vehicle and at the charging station, i.e., the offboard (stand-alone) charger, as shown in Fig. 2-1.

Located charger

Charging station on-board charger off-board charger Battery pack

Fig. 2-1: Basic concept of an Off-board charger (left side) and On-board charger (right side).

On-board chargers have high power density, high efficiency, and low weight. Also, they increase the flexibility of driving because they can be plugged in anywhere there is an electric outlet. However, due to the limited space in a vehicle, on-board chargers have a limited power level to achieve and their nominal power usually is lower than 3.5 kW [START_REF] Haghbin | Grid-Connected Integrated Battery Chargers in Vehicle Applications: Review and New Solution[END_REF]. Off-board chargers can be used as a charging station with fast charging technology and high power transfer. But, due to the flexibility and household applications of on-board chargers, in this dissertation, the on-board charger will be discussed.

Also, depending on whether the power transfer to the charger is done directly through the wire or it is transferred wirelessly through a magnetic coupling, a charger can be classified as a wired or wireless charger, respectively, as shown in Fig. 2-2.

Fig. 2-3: Conceptual diagram of a bidirectional charger for V2G applications

Using a bidirectional charger, energy can be stored in the vehicle during low load hours and returned to the grid during peak hours, where the vehicle helps the power grid as a distributed generation (DG). Also, if solar energy is used to charge the battery, the amount of excess produced energy can be returned to the grid. Given the importance of this issue in the future of the power grid, this dissertation will use a bidirectional charger.

In terms of classification according to the standard voltage level, three levels are defined: Level 1 has an input voltage of 120V, a current of 20A, and a power of 1.9 kW and is installed on the vehicle. Level 2 includes voltages from 208V to 450V, a current of 3.3A to 14.4A and is installed on the vehicle. Level 2 chargers have the advantage that they can be powered from both private homes and public places as they require AC 240V. Level 3 also requires highvoltage DC power, which is used for fast charging in less than 1 hour [START_REF] Emadi | Advanced electric drive vehicles[END_REF]. Since level 2 can be charged via household electricity, this standard level will be used in this dissertation.

In terms of a multi-stage converter, it can also be considered that first, a converter raises the input voltage to the desired level and second, another converter performs the task of controlling the battery voltage. For grid-connected applications, usually, there is a boost converter in the input part as the first stage to use as power factor correction, and the other stage controls the required power. So, the selected charger for this dissertation can be considered as a two-stage charger. The reason for using a boost converter in the first stage is the simple structure, continuous input current, and low total harmonic distortion (THD) [START_REF] Oh | A High-Efficient Nonisolated Single-Stage On-Board Battery Charger for Electric Vehicles[END_REF].

By determining the characteristics of the required charger as a charger with voltage level 2, bidirectional, on-board, power transmission with wire, and two stages, the most important feature of the charger is its converter structure, which will be evaluated in the next section. where the grid AC voltage rectifies via diode rectification for the unidirectional charger or via synchronous rectification for the bidirectional charger. Then the rectified voltage is applied to a boost converter for power factor correction and also increasing the voltage. It should be mentioned that there exist some structures which integrate the rectifier part with the PFC part [START_REF] Prasanna | Novel bidirectional single-phase single-stage isolated AC-DC converter with PFC for charging of electric vehicles[END_REF]. Finally, a power electronics converter charges the battery packs to the required voltage and current. The main goal of this dissertation is to analyze and optimize the main DC-to-DC converter for battery charging, therefore the rectifier part and PFC part are ignored in our analysis.

According to the mission of this dissertation in designing an optimal bidirectional EV charger, it is necessary to provide conditions such as high efficiency, high power density, high switching frequency, bidirectional power transferring, and soft switching for a wide range of loads. For these reasons and especially for high-efficiency power transferring, the solution is to use resonant converters [START_REF] Russi | Coupled-Filter-Inductor Soft-Switching Techniques: Principles and Topologies[END_REF]. For high-power DC-DC applications, resonant converters have become a desirable and widely used selection [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF] and [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], where the resonant converter can be designed as highly impacted and high-efficiency applications, without any snubber [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF]. The well-designed resonant converters can operate in soft switching conditions for wide operating areas. Therefore, a resonant converter will be used in this dissertation.

The general structure of a battery charger based on a resonant converter can be shown in Fig. 2345 where the inverter part can be a full-bridge or half-bridge and the rectifier part can be a diode rectifier or synchronous rectification. In the next part, a review of resonant converters and their utilization in bidirectional power transfer will be presented.

Resonant converters are classified based on the number of inductors and capacitors and also their positions, usually such as the series resonant converter (SRC), parallel resonant converter (PRC), dual active bridge (DAB), LLC, LCC, CLLC, CLLLC, etc.

The schematic diagram of the series resonant converter (SRC) is shown in Fig. 23456. SRC operates in ZVS for above the resonant frequency and ZCS for below the resonant frequency.

The gain is unity at the resonant frequency. The circulating current and turn OFF current are low at frequencies near the resonant frequency. However, for light load and high input voltage conditions, the voltage regulation is hard to fulfill and also the circulating current is increased.

Also, the series resonant converter is not a good candidate for grid-connected and front-end applications [START_REF] Huang | Classification and selection methodology for multi-element resonant converters[END_REF]. Furthermore, the efficiency of SRC is decreased when the switching frequency deviates from the resonant frequency.

V g Q 1 Q 3 Q 2 Q 4 C L n:1 C ss Q 5 Q 7 Q 6 Q 8 R C F

Fig. 2-6: Schematic diagram of a full bridge series resonant converter (SRC).

The parallel resonant converter (PRC), as shown in Fig. 234567, plays a role of a step-up converter and can obtain ZVS and ZCS depending on the required current and voltage. But this converter suffers from a high circulating current and high turn OFF current. Also, this converter is not suitable for grid-connected and front-end applications [START_REF] Huang | Classification and selection methodology for multi-element resonant converters[END_REF].

V g Q 1 Q 3 Q 2 Q 4 C L L F n:1 C ss Q 5 Q 7 Q 6 Q 8 R C F

Fig. 2-7: Schematic diagram of a full bridge parallel resonant converter (PRC).

The LCC resonant converter is composed of SRC and PRC converters. Fig. 2345678shows the schematic diagram of the LLC resonant converter. This converter has a lower circulating current in comparison with PRC, and lower load sensitivity in comparison with SRC. But conduction loss and switching loss are raised in high input voltage and it is not suitable for EV charging.

V g Q 1 Q 3 Q 2 Q 4 C 1 L C 2 n:1 C ss Q 5 Q 7 Q 6 Q 8 R C F L F Fig. 2-8: Schematic diagram of a full bridge LCC resonant converter.
Fig. 23456789shows the schematic diagram of the dual active bridge (DAB) converter, which is popular for various applications, such as EV battery chargers, DC distribution systems, etc. It has a wide soft switching region and works in ZVS for both the primary and secondary sides [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF]. Galvanic isolation, high switching frequency, high power density, and bidirectional power transferring make the DAB converter a proper candidate for wired and wireless EV chargers [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF]. By now, the DAB converter can be selected as the main DC-to-DC converter of this dissertation, but it is not the final selection.

V g Q 1 Q 3 Q 2 Q 4 L n:1 C ss Q 5 Q 7 Q 6 Q 8 R C F

Fig. 2-9: Schematic diagram of a dual-active bridge converter (DAB).

The full-bridge (see Fig. 2-10) and half-bridge LLC resonant converters are broadly used in many applications, such as EV chargers, LED drivers, renewable energy systems, etc. [], due to the advantage of wide voltage gain, wide soft switching region, galvanic isolation, high efficiency, high switching frequency. Furthermore, the main primary switches operate with ZVS and the output rectifier part operates in ZCS [START_REF] Huang | Classification and selection methodology for multi-element resonant converters[END_REF], which is caused lower switching loss and circulating current in comparison with SRC, PRC, and LCC, over wide load ranges. Also, the LLC resonant converter can be designed with high power density by integrating Lm into the transformer as its magnetizing inductance and L1 as its leakage inductance. Therefore, the overall volume of magnetic components will be reduced. According to the aforementioned advantages, the LLC resonant converter can be also selected as the main DC-to-DC converter of this dissertation.

V g Q 1 Q 3 Q 2 Q 4 C 1 L 1 L m n:1 C ss Q 5 Q 7 Q 6 Q 8 R C F Fig. 2-10: Schematic diagram of a full bridge LLC resonant converter.
By increasing the number of resonant elements, some benefits such as short circuit protection and bidirectional operating can be obtained [START_REF] Fu | Novel Multi-Element Resonant Converters for Front-end DC/DC Converters[END_REF]. It means a better topology selection is always challenging [START_REF] Huang | Classification and selection methodology for multi-element resonant converters[END_REF].

According to the previous discussion about different topologies, the dual-active bridge converter and LLC resonant converter are good candidates for the main DC to DC converter selection of bidirectional EV charger. If we use LLC resonant converter, in vehicle-to-grid condition, the magnetizing inductance (Lm) is clamped by the battery voltage because it is in parallel with a DC voltage source as shown in Fig. 2-11. Therefore, the LLC resonant converter operates similar to a series resonant converter (SRC) [START_REF] Li | Unified Modeling, Analysis, and Design of Isolated Bidirectional CLLC Resonant DC-DC Converters[END_REF] and the advantage of the LLC resonant converter is dissipated, e.g. efficiency will be decreased when the switching frequency deviates from the main resonant frequency [START_REF] Shyu | Using automatic frequency shifting techniques for LLC-SRC output voltage regulation[END_REF]. As a result, the traditional LLC resonant converter is unidirectional and cannot be used effectively in V2G applications. For bidirectional LLC resonant converter [START_REF] Li | A Bidirectional Synchronous/Asynchronous Rectifier Control for Wide Battery Voltage Range in SiC Bidirectional LLC Chargers[END_REF], the functional behavior of the converter, e.g. voltage gain ratio and switching stresses, are different in the forward mode and reverse mode. So, it causes different optimum components for each direction. Also, the voltage gain of bidirectional LLC resonant converters in reverse mode is similar to series resonant converters [START_REF] Li | A Bidirectional Synchronous/Asynchronous Rectifier Control for Wide Battery Voltage Range in SiC Bidirectional LLC Chargers[END_REF] and loses its benefit. Therefore, the LLC resonant converter cannot be selected for the main DC-to-DC converter of this dissertation.

Vin Another candidate is the DAB converter. In [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], it is shown that the efficiency and power density of the DAB converter is lower than the capacitor-inductor-inductor-capacitor (CLLC) resonant converter. Therefore, the CLLC resonant converter is superior to the conventional dual active bridge converter in terms of efficiency and power density [START_REF] Ryu | Effective Test Bed of 380-V DC Distribution System Using Isolated Power Converters[END_REF], [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], and [START_REF] Xue | Dual Active Bridge-Based Battery Charger for Plug-in Hybrid Electric Vehicle With Charging Current Containing Low Frequency Ripple[END_REF] and to the LLC resonant converter in terms of bidirectional capability [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF].

C 1 L 1 L m V battery LLC resonant tank
The circuit diagram of the bidirectional full-bridge CLLC resonant converter is shown in Fig. 2-12. The primary active full bridge is composed of four GaN transistors. The switching network of the secondary side is also formed in a similar manner as the synchronous rectifier [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF]. The CLLC resonant converter has a high-order resonant tank, which is comprised of capacitors C1 and C2, the transformer magnetizing inductance Lm, and the primary and secondary leakage inductances L1 and L2. Depending on the desired power flow direction, the bridge of one side acts as an inverter and that of the other side acts as a synchronous rectifier.

V g Q 1 Q 3 Q 2 Q 4 C 1 L 1 C 2 L 2 L m n:1 C ss V O + _ Q 5 Q 7 Q 6 Q 8 Fig. 2-12: Schematic diagram of a bidirectional full-bridge CLLC resonant converter.
The CLLC resonant converter is the bidirectional version of the LLC resonant converter [START_REF] Kim | Enhancing Efficiency in Bidirectional Resonant DC-DC Converter[END_REF]. The structure of a CLLC resonant converter has three inductors, therefore this converter can be named a CLLLC resonant converter. But in this dissertation, it is stated as CLLC resonant converter [START_REF] Zahid | Design, Modeling and Control of Bidirectional Resonant Converter for Vehicleto-Grid (V2G) Applications[END_REF]. The reason is that there is an inductive cutset in the resonant tank, and by having the current of two inductors, the current of the other inductor can be calculated independently. It means only two inductors have an effect on state space equations and therefore CLLC notation is selected.

For grid-to-vehicle (G2V) applications, the primary side active bridge acts as an inverter to apply the inverted voltage to the resonant tank, and the secondary side active bridge acts as a rectifier using synchronous rectification or antiparallel diodes of the transistors when these transistors are OFF [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF]. Finally, this voltage is refined by a filter which is formed by only one capacitor. For vehicle-to-grid (V2G) applications, the secondary side active bridge inverts the voltage of batteries and applies it to the resonant tank, and the primary side active bridge rectifies the derived AC voltage of the resonant tank.

According to the schematic diagram of the CLLC resonant converter, this converter comprises two LLC resonant converters on each side [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], which ensure ZVS for the main primary switches and ZCS for the secondary rectifying switches. The advantage of the CLLC resonant converter can be expressed as:

• Bidirectional power transferring capability;

• Wide voltage gain ratio;

• Wide soft switching region under wide load ranges (from no load to full load);

• Ensuring ZVS in G2V mode and ZCS in V2G mode for the primary switches;

• Ensuring ZCS in G2V mode and ZVS in V2G mode for the secondary switches;

• Higher efficiency in comparison to DAB and LLC;

• Lower turn OFF current in comparison to other resonant converters;

• Frequency modulation control method;

• Galvanic isolation;

• Short circuit protection capability.

A common way to charge lithium batteries is to first charge the battery with a constant current equal to the maximum tolerable current of the battery. This area is called the constant current (CC) area. The battery voltage then increases to reach its nominal value. After that, the battery voltage will remain constant and the battery will be charged at a constant voltage; until the battery current is close to zero. This area is called the constant voltage (CV) area [START_REF] Lin | A Li-Ion Battery Charger With Smooth Control Circuit and Built-In Resistance Compensator for Achieving Stable and Fast Charging[END_REF]. This method of charging for a battery cell is shown in Fig. 2345678910111213. According to the above figure, it is clear that the process of charging the battery, firstly, includes a wide range of voltages, and secondly, this process will continue from full load to near no load. Therefore, one of the most important criteria for choosing a charger is that the charger operates at the optimum point and maintains efficiency of the charger throughout the battery charging process in a wide range of current and voltage variations. The selected CLLC resonant converter must be designed in different operating points correctly, which is one of the goals of this dissertation.

Half-bridge and full-bridge configurations are the most conventional topologies used in bidirectional converters [START_REF] Fahem | On-board Bidirectional Battery Chargers Topologies for Plug-in Hybrid Electric Vehicles[END_REF]. A full-bridge and half-bridge single-phase DC-to-DC converters were presented in [START_REF] Pinto | On-Board Reconfigurable Battery Charger for Electric Vehicles with Traction-to-Auxiliary Mode[END_REF] to provide G2V, V2G, and traction-to-auxiliary (T2A) capabilities. A bidirectional AC-to-DC converter was discussed in [START_REF] Kim | Multi-Functional On-Board Battery Charger for Plug-in Electric Vehicles[END_REF] for operating in V2G applications. In [START_REF] Zheng | A Hybrid Cascaded Multilevel Converter for Battery Energy Management Applied in Electric Vehicles[END_REF], a V2G multilevel converter was proposed for battery energy management, where each battery cell can be controlled to be connected to the circuit or to be bypassed by a half-bridge converter. About the CLLC resonant converter, this converter can be implemented as a halfbridge or full-bridge. In [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], four 1kW bidirectional converters, i.e. half-bridge and full-bridge DAB converters and half-bridge and full-bridge CLLC converters were compared, and it was shown that for bidirectional EV charging applications, the CLLC converters are slightly better than the DAB converters, in terms of the efficiency, power density, size, and cost. Future chargers will move towards being bidirectional and the CLLC resonant converter is a proper candidate.

In summary, the advantage of a bidirectional charger is:

• Begin the consistent trends of the future;

• Can earn money by selling electricity;

• More using renewable energies capabilities;

• Reduce the peak demand for power plants;

• Use as an uninterruptable power supply and avoid blackouts;

• Associate with intelligent building systems;

• Supply appliances inside the vehicle;

• EVs play a role of distributed generation systems;

• Useful for emergency modes.

However, it has the following disadvantage:

• High cost;

• Complicated production and control;

• Use synchronization with the grid for V2G applications.

Based on the aforementioned discussion, a two-stage bidirectional on-board charger is selected, when the full-bridge CLLC resonant converter is selected as the main DC-to-DC converter.

The bidirectional capability of the CLLC resonant converter was first proposed in [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF]. The CLLC resonant converter is a promising converter for bidirectional power conversion in applications such as bidirectional EV chargers and hybrid vehicle DC-DC converters [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF], [START_REF] Ryu | Effective Test Bed of 380-V DC Distribution System Using Isolated Power Converters[END_REF],

[76], [START_REF] Ryu | Test bed implementation of 380V DC distribution system using isolated bidirectional power converters[END_REF], [START_REF] Min | Bidirectional Resonant CLLC Charger for Wide Battery Voltage Range: Asymmetric Parameters Methodology[END_REF], [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF], [START_REF] Huang | Implementation of Bidirectional Resonant DC Transformer in Hybrid AC/DC Micro-Grid[END_REF], [START_REF] Zou | Bi-Directional CLLC Converter With Synchronous Rectification for Plug-In Electric Vehicles[END_REF], and [START_REF] Endres | 6 kW Bidirectional, Insulated Onboard Charger with Normally-Off GaN Gate Injection Transistors[END_REF]. This converter is bidirectional in nature [START_REF] Ryu | Effective Test Bed of 380-V DC Distribution System Using Isolated Power Converters[END_REF], where the power can be transferred as the forward mode (i.e. G2V) and the reverse mode (e.g. V2G)

at high efficiency, independently. However, a symmetrical structure of a CLLC resonant converter allows the same operation in two different power flow directions [START_REF] Ryu | Effective Test Bed of 380-V DC Distribution System Using Isolated Power Converters[END_REF], and makes its design easy. The symmetrical structure of the CLLC resonant converter was first proposed in [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], which was analyzed the voltage gain ratio in terms of switching frequency variation. The symmetrical structure of CLLC resonant converter makes its analysis, design, optimization, and control easy [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], [START_REF] Malan | Modeling and Control of a Resonant Dual Active Bridge With a Tuned CLLC Network[END_REF], [START_REF] Huang | Comprehensive Coordinated Frequency Control of Symmetrical CLLC-DC Transformer in Hybrid AC/DC Microgrids[END_REF], [START_REF] Huang | Robust Circuit Parameters Design for the CLLC-Type DC Transformer in the Hybrid AC-DC Microgrid[END_REF], [START_REF] He | Design of 1 kW Bidirectional Half-bridge CLLC Converter for Electric Vehicle Charging Systems[END_REF], and [START_REF] Huang | Simplified Resonant Parameter Design of the Asymmetrical CLLC-Type DC Transformer in the Renewable Energy System via Semi-Artificial Intelligent Optimal Scheme[END_REF]. A detailed design procedure of the bidirectional CLLC resonant converter for a battery charging application is introduced in [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF] to simply the CLLLC converter to CLLC converter. In [START_REF] Min | Bidirectional Resonant CLLC Charger for Wide Battery Voltage Range: Asymmetric Parameters Methodology[END_REF], a design methodology for an asymmetric parameter design is proposed, but the corresponding relationship between the circuit parameters of charging and discharging modes has not been analyzed fairly. The voltage gain ratio of a symmetrical half-bridge CLLC resonant converter in terms of charging mode and discharging mode is expressed in [START_REF] Zou | Bi-Directional CLLC Converter With Synchronous Rectification for Plug-In Electric Vehicles[END_REF]. A design methodology for maximum efficiency tracking is proposed in [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF], and the circulating current and efficiency are compared for LLC and CLLC resonant converters, where was shown the conditions are better for CLLC in comparison with LLC, but without any experimental results.

The first harmonic approximation (FHA) is a simple analysis method, which is widely used in the analysis and design of CLLC resonant converters [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF], [START_REF] Zou | Bi-Directional CLLC Converter With Synchronous Rectification for Plug-In Electric Vehicles[END_REF], and [START_REF] Zhang | Integrated Half-Bridge CLLC Bidirectional Converter for Energy Storage Systems[END_REF]. However, the accuracy of this method deteriorates when the voltage and current of load (i.e. the battery) are widely changed or the converter operates in discontinuous conduction mode (DCM) [START_REF] Safaee | Time-Domain Analysis of Voltage-Driven Series-Series Compensated Inductive Power Transfer Topology[END_REF].

Time domain analysis (TDA) or state space analysis (SSA) is a general steady-state approach for the exact analysis of resonant converters, which was performed in [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF] for state analysis of CLLC resonant converter. In this method, the state space equations of each subinterval are solved and the time domain expressions of state variables are obtained over one switching period. TDA is particularly useful for simulation purposes but it is too complicated to be used for the design of a resonant converter with several energy storage elements [START_REF] Wei | Analysis and Design of the LLC Resonant Converter With Variable Inductor Control Based on Time-Domain Analysis[END_REF]. In [START_REF] Ditze | Steady-state analysis of the bidirectional CLLLC resonant converter in time domain[END_REF], it was shown that the CLLC resonant converter has multiple resonant frequencies based on C1, and L1, as well as C2 and L2, to which Lm can also be added. It is also mentioned that for frequencies above the resonant frequency of C1 and L1, the converter operates in ZVS, which is lost for those below this resonant frequency. However, this statement is generally incorrect, and the adequate proof is mandatory. In [START_REF] Lai | A high-efficiency 3.3-kW bidirectional on-board charger[END_REF], a CLLC resonant converter with a power of 3.3 kW was built for a bidirectional EV charger. In [START_REF] Radimov | Bi-directional CLLC converter front-end for off-grid battery inverters[END_REF], a CLLC resonant converter operates as an inverter to transfer power from the batteries to the grid, while the first harmonic approximation method was considered for analysis. A relationship for leakage inductance calculation of the main transformer according to its dimensions was presented in [START_REF] Liu | Bidirectional CLLC resonant DC-DC converter with integrated magnetic for OBCM application[END_REF] for a bidirectional on-board charger. An effective test bed was proposed in [START_REF] Ryu | Effective Test Bed of 380-V DC Distribution System Using Isolated Power Converters[END_REF] for a bidirectional DC distribution system based on a 5 kW CLLC converter, 3 kW DAB converter, and 3 kW LLC converter, for the main DC-bus voltage control, battery voltage interface, and the renewable energy simulator, respectively.

A graphical and geometrical method for exact analyzing of the resonant converter is the state-plane analysis where the current or voltage of the resonant tank elements is calculated as a function of the other variables and plotted in a 2D or 3D diagram [START_REF] Oruganti | Resonant Power Processors, Part I---State Plane Analysis[END_REF] and [START_REF] Chen | Generalized Optimal Trajectory Control for Closed Loop Control of Series-Parallel Resonant Converter[END_REF]. The exact circuit behavior and design criteria in CCM and DCM operation can be obtained using this method [START_REF] Anand A G | An Unidirectional Single Stage Single Phase Soft-Switched Resonant High Frequency Link Inverter[END_REF] and [START_REF] Gücin | A Series Resonant Converter-Based Multichannel LED Driver With Inherent Current Balancing and Dimming Capability[END_REF]. The analytical equations extracted by state-plane analysis can be used directly in the optimization process to increase accuracy. State-plane analysis has been applied

to the converters with two or three resonant elements [START_REF] Anand A G | An Unidirectional Single Stage Single Phase Soft-Switched Resonant High Frequency Link Inverter[END_REF], [START_REF] Xu | Multimode Constant Power Control Strategy for LCC Resonant Capacitor Charging Power Supply Based on State Plane Analysis[END_REF], [START_REF] Zurbriggen | Near-Time-Optimal Dynamics in PWM DC-DC Converters: Dual-Loop Geometric Control[END_REF], and [START_REF] Barrero | A modified state-plane control of a bi-directional Series Resonant Converter for an EDLC Energy Storage System in Hybrid Electric Vehicles[END_REF], but the CLLC resonant converter has five resonance elements (two series capacitor-inductor and one transformer) and state-plane analysis is hard to accomplish. Therefore, most of the research has been focused on FHA and TDA methods [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF], [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF], [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF], [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], [START_REF] Malan | Modeling and Control of a Resonant Dual Active Bridge With a Tuned CLLC Network[END_REF], [START_REF] Zou | Bi-Directional CLLC Converter With Synchronous Rectification for Plug-In Electric Vehicles[END_REF], [START_REF] Ditze | Steady-state analysis of the bidirectional CLLLC resonant converter in time domain[END_REF], [START_REF] Li | Bi-directional on-board charger architecture and control for achieving ultra-high efficiency with wide battery voltage range[END_REF],

and [START_REF] Kim | High-Efficiency Isolated Bidirectional AC-DC Converter for a DC Distribution System[END_REF]. Table 2-1 demonstrates a comparison between the analyzing methods.

Stateplane

Owing to the electrical constraints, it is mandatory to find a closed-form of them to utilize in the design and optimization process. The first harmonic approximation (FHA) is a simple analysis method, which only considers the fundamental components of applied voltage and current to the resonant tank [START_REF] Steigerwald | A comparison of half-bridge resonant converter topologies[END_REF]. The accuracy of the FHA method for the CLLC resonant converter is deteriorative [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF] and it is needed to present a new exact method.

In [START_REF] Malan | Frequency modulation of a series resonant dual active bridge to minimize the circulating reactive currents in the high frequency link[END_REF], another type of CLLC resonant converter was proposed, where mutual inductors were used instead of typical inductors and a transformer. Triple phase shift modulation was implemented to reduce circulating current and increase power factor. A dual three-level neutral point clamped half-bridge was used in place of the inverter part of the CLLC resonant converter for applying in high voltage applications [START_REF] Cai | Design of a high-frequency isolated DTHB CLLC bidirectional resonant DC-DC converter[END_REF]. But in this reference, the magnetizing inductance was considered as infinity, or in other words, the transformer was imagined as an ideal transformer, which makes the behavior of the proposed CLLC converter different from conventional CLLC resonant converters.

Laplace analysis was employed in [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF] to find maximum efficiency tracking, where switching near the main resonant frequency can fulfill this goal by decreasing the circulating current, turn-OFF current of the main switches, and di/dt of the secondary side diodes. The variable DC link voltage control method was considered in [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF] to operate at the resonant frequency and obtain maximum efficiency using the sinusoidal approximation method. However, it may cause a low-frequency oscillation in the output voltage. A two-stage AC to DC converter based on a boost converter and CLLC resonant converter was presented in [START_REF] Lu | Variable DC Link Control Loop Design for an Integrated Two-Stage AC/DC Converter[END_REF],

where the boost converter can control the DC link voltage while preparing power factor correction, and the CLLC resonant converter can transfer the required power in a high switching frequency and galvanic insulation. In [START_REF] Zhang | Integrated Half-Bridge CLLC Bidirectional Converter for Energy Storage Systems[END_REF], a two-stage DC-to-DC converter was proposed, where the first stage is a buck/boost converter for decreasing/increasing the output voltage, and the second stage is the CLLC resonant converter for controlling the voltage gain ratio and preparing soft switching conditions. However, the first harmonic approximation was used in this reference, which reduces accuracy.

The influence of the junction capacitance of the secondary side diodes on output power and switching stresses was evaluated in [START_REF] Ditze | Influence of the junction capacitance of the secondary rectifier diodes on output characteristics in multi-resonant converters[END_REF]. A 6-kW on-board charger with a variable turn ratio was implemented in [START_REF] Endres | 6 kW Bidirectional, Insulated Onboard Charger with Normally-Off GaN Gate Injection Transistors[END_REF] based on 600 V GaN transistors due to their low output capacitance and reverse current operating capability.

In [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], full-bridge and half-bridge CLLC resonant converters as well as full-bridge and halfbridge DAB converters were compared for EV charger applications. Among them, the halfbridge CLLC resonant converter has the highest power density, the highest efficiency, and the lowest cost. Also, the full-bridge CLLC resonant converter has the lowest peak current. As a result, CLLC resonant converters are better than DAB converters for bidirectional EV charger applications.

The low accuracy of the first harmonic approximation method for obtaining the voltage gain ratio and operating modes of the CLLC resonant converter was investigated in [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF], which causes an obstacle in the design and control of the CLLC resonant converter. So, an accurate operating mode analysis in the time domain was proposed. However, the ZVS condition is based on the first harmonic approximation method and suffers from low accuracy [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF]. Some research has been done to optimize the CLLC resonant converter. In [START_REF] Sun | Design and Multi-Objective Optimization of an Auxiliary Wireless Power Transfer Converter in Medium-Voltage Modular Conversion Systems[END_REF], with the first harmonic approximation method, the important parameters of the resonant tank were obtained, where the objective functions are maximizing efficiency, minimizing the coupling capacitance of magnetic parts, and adequate insulating for the coil. The optimization was done by parameter sweeping. It should be mentioned that in this reference, the inductance of the coil was obtained numerically with MATLAB, which is hard to implement. In [START_REF] Wang | A Time-Domain Optimization Design Methodology for CLLC Resonant Converter[END_REF], with timedomain analysis, the efficiency was maximized with the constraints of ZVS boundary, maximum gain voltage, minimum gain voltage, and operating mode boundary, to find the value of the resonant tank's elements. A design methodology of high-frequency transformer design based on the Pareto optimization was established in [START_REF] Zhao | Design and Optimization of the High Frequency Transformer for 100kW CLLC Converter[END_REF] for one operating point to maximize efficiency and power density. In [START_REF] Li | High-Frequency Transformer Design with High-Voltage Insulation for Modular Power Conversion from Medium-Voltage AC to 400-V DC[END_REF], a high-frequency transformer was optimized by sweeping the parameters to find minimum losses at low volume. A planar transformer was optimized in [START_REF] Zhang | Investigation of Design Methodology of Planar Transformers for EV On Board Chargers[END_REF] by selecting the combinations of some discrete parameters and comparing all the local optimal designs at one operating point. A modular high-frequency transformer was designed in [START_REF] Zhao | High-Frequency Transformer Design for Modular Power Conversion From Medium-Voltage AC to 400 VDC[END_REF] for a 400 V CLLC resonant converter, while the maximum efficiency reaches 98%. A symmetrical PCB transformer structure was presented in [START_REF] Li | A symmetrical resonant converter and PCB transformer structure for common mode noise reduction[END_REF], where it is shown that operating in high switching frequencies can actualize PCB-winding transformers. But it causes a higher level of common mode noise, which was solved by splitting the cores. The optimization strategy for an LLC resonant converter was proposed in [START_REF] Zhang | Gapped Transformer Design Methodology and Implementation for LLC Resonant Converters[END_REF] in terms of the relative permeability and air gap to minimize losses.

A symmetrical CLLC resonant converter was optimized for a constant power load and the fixed output voltage, with the particle swarm optimization (PSO) method [START_REF] Lin | Design of Symmetrical CLLC-Resonant DC Transformer Considering Voltage Transfer Ratio and Cascaded System Stability[END_REF] and iteration.

The same method was used in [START_REF] Lin | Parameter Design for Symmetrical CLLC-Type DC Transformer Considering Cascaded System Stability and Power Efficiency[END_REF] to maximize power efficiency. By sweeping some variables, a CLLC resonant converter was optimized in [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF] to minimize losses and minimum volume in one operating point. In [START_REF] Ge | A Parameter Design Method of CLLC Resonant Converter Based On Time-Domain Model[END_REF], a parameter design methodology based on the ratio of magnetizing inductance to leakage inductance was proposed to increase efficiency. A two-stage optimization method was presented in [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF] to minimize the power losses of a symmetrical CLLC resonant converter, where the first stage found the optimum value of the resonant tank by minimizing total power losses, and in the second stage, the thickness of the air gap and distance between the primary winding and the secondary winding. In [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF], the total losses were optimized using the genetic algorithm (GA) and particle swarm optimization (PSO) with the constraints of minimum window area and maximum allowable flux density.

Table 2-2: Previous methods and parameters done for CLLC resonant converter

Parameters→ Voltage gain ratio

Switching stresses ZVS condition Optimization Methods↓

First Harmonic Approximation [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF], [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], [START_REF] Malan | Modeling and Control of a Resonant Dual Active Bridge With a Tuned CLLC Network[END_REF], [START_REF] Kim | High-Efficiency Isolated Bidirectional AC-DC Converter for a DC Distribution System[END_REF], [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF] [46], [START_REF] Malan | Modeling and Control of a Resonant Dual Active Bridge With a Tuned CLLC Network[END_REF], [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF] [18], [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF], [START_REF] Chen | Snubberless Bidirectional DC-DC Converter With New CLLC Resonant Tank Featuring Minimized Switching Loss[END_REF], [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], [START_REF] Kim | High-Efficiency Isolated Bidirectional AC-DC Converter for a DC Distribution System[END_REF], [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF] [18], [START_REF] Zahid | Design of Bidirectional DC-DC Resonant Converter for Vehicle-to-Grid (V2G) Applications[END_REF], [START_REF] Malan | Modeling and Control of a Resonant Dual Active Bridge With a Tuned CLLC Network[END_REF], [START_REF] Li | Bi-directional on-board charger architecture and control for achieving ultra-high efficiency with wide battery voltage range[END_REF], [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF] Time-domain Analysis [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF], [START_REF] Ditze | Steady-state analysis of the bidirectional CLLLC resonant converter in time domain[END_REF] [38], [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF] [38] [START_REF] Dube | Analysis and Design of an Integrated Bidirectional Three-Phase AC-DC Resonant Converter[END_REF], [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF] State-plane Analysis Not done Not done Not done Not done Nevertheless, none of the aforementioned studies provides any general optimization method in terms of bidirectional power transferring, using an exact method for optimization, a wide range of voltages and currents, power density criteria, effects of core sizes on the power losses, ZVS conditions, constraints of an on-board charger, etc, as shown in Table 2-2. These requirements will be considered in this dissertation.

First Harmonic Approximation Model of CLLC Resonant Converter

Introduction

One of the most widely used methods in modeling resonant converters is the first harmonic approximation (FHA) method or the sinusoidal approximation method, which only considers the fundamental components of applied voltage and current to the resonant tank [START_REF] Steigerwald | A comparison of half-bridge resonant converter topologies[END_REF]. Due to its simplicity, the FHA method is widely used in the analysis and design of CLLC resonant converters [START_REF] Jung | Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems[END_REF], [START_REF] Liu | A CLLC resonant converter based bidirectional EV charger with maximum efficiency tracking[END_REF], [START_REF] Zou | Bi-Directional CLLC Converter With Synchronous Rectification for Plug-In Electric Vehicles[END_REF], and [START_REF] Zhang | Integrated Half-Bridge CLLC Bidirectional Converter for Energy Storage Systems[END_REF]. In this part, the FHA method is utilized for obtaining electrical constraints, where the presented method in [START_REF] Erickson | Fundamentals of power electronics, Principles of resonant power conversion[END_REF] is extracted to apply to the FHA model of the CLLC resonant converter.

FHA Method

According to this method, the primary and secondary sides of the resonant tank are considered as a square voltage source as shown in Fig. 3-1.

For this square voltage source (see Fig. 3-2), only its fundamental harmonic (switching frequency) is comprised. For the primary voltage, the following condition is considered:

𝑣 1 (𝑡) ≈ 4 𝜋 𝑉 𝑖𝑛 sin 𝜔 𝑠 𝑡 (3.1)
Based on the applied sinusoidal voltage, the primary current is also sinusoidal because it is applied to a linear LC circuit. This current is determined as (3.2).

𝑖 1 (𝑡) ≈ 𝐼 1 sin(𝜔 𝑠 𝑡 -𝜑 1 ) (3.2)
where I1 is the maximum value of the primary current, and φ1 is the phase shift between the voltage and current. The input current is the average of the absolute value of I1, so we have:

V in Q 1 Q 3 Q 2 Q 4 v 1 + _ Primary side (inverter) V in V in -V in V out + _ v 2 + _ Secondary side (rectifier) V out V out -V out i 1 i 2 i in i out
〈|𝑖 𝑖𝑛 (𝑡)|〉| 𝑇 𝑠 = 2 𝑇 𝑠 ∫ 𝑖 𝑖𝑛 (𝑡)𝑑𝑡 𝑇 𝑠 2 ⁄ 0 (3.3) ≈ 2 𝑇 𝑠 ∫ 𝐼 1 sin(𝜔 𝑠 𝑡 -𝜑 1 ) 𝑑𝑡 𝑇 𝑠 2 ⁄ 0 = 2 𝜋 𝐼 1 cos 𝜑 1
So, the approximated model of the inverter part can be modeled as a dependent voltage source, as shown in Fig. 3-3.

V in Q 1 Q 3 Q 2 Q 4 FHA + _ 2 π I 1 cos(φ 1 )
4Vin π sin(2πf s t)

I 1 sin(2πf s t-φ 1 )

Fig. 3-3: FHA model of the full-bridge inverter

Now, let's consider the secondary side. The first harmonic of the secondary side rectifier is similar to the primary. So we have:

𝑣 2 (𝑡) ≈ 4 𝜋 𝑉 𝑜𝑢𝑡 sin(𝜔 𝑠 𝑡 -𝜑 2 ) (3.4)
where φ2 is the phase shift between the primary voltage and secondary voltage.

Owing to the linear resonant tank, the secondary current is also sinusoidal with the same phase shift:

𝑖 2 (𝑡) ≈ 𝐼 2 sin(𝜔 𝑠 𝑡 -𝜑 2 ) (3.5)
where I2 is the maximum value of the secondary side current.

It should be mentioned that i2 and v2 have the same phase (φ2) because of the zero crossing of the rectifier part.

The load current is equal to the average of the absolute value of i2, i.e.:

𝐼 = 〈|𝑖 2 (𝑡)|〉| 𝑇 𝑠 = 2 𝑇 𝑠 ∫ 𝑖 2 (𝑡)𝑑𝑡 𝑇 𝑠 2 ⁄ 0 (3.6)
Hence we have:

𝐼 ≈ 2 𝑇 𝑠 ∫ 𝐼 2 sin(𝜔 𝑠 𝑡 -𝜑 2 ) 𝑑𝑡 𝑇 𝑠 2 ⁄ 0 = 2 𝜋 𝐼 2 (3.7)
The effective resistance of the rectifier input port is determined as follows:

𝑅 𝑒 = 𝑣 2 (𝑡) 𝑖 2 (𝑡) = 8 𝜋 2 𝑉 𝑜𝑢𝑡 𝐼 = 8 𝜋 2 𝑅 𝐿𝑜𝑎𝑑 (3.8)
Therefore, the equivalent circuit model of the rectifier part is shown in Fig. 34.

FHA R e = R 2 π I 2

I 2 sin(2πf s t-φ 2 )
8. π 2

Fig. 3-4: FHA model of the full-bridge rectifier

Now it is time to get the FHA model for the CLLC resonant converter. This modeling is depicted in the next part.

FHA Model for the CLLC Resonant Converter

Let's consider the schematic diagram of the full-bridge CLLC resonant converter, which was shown in Fig. 23456789101112. By substituting the inverter and rectifier part, the first harmonic approximation model of the CLLC resonant converter can be drawn as Fig. 345.

C 1 L 1 C 2 L 2 L m n:1 V O + _ 2 π I 1 cos(ϕ 1 ) 4V in π sin(2πf s t) V in + _ R e = R 8. π 2 R 2 π I 2 I 2 I 1

Fig. 3-5: FHA model of the full-bridge CLLC resonant converter

By solving the above figure, the electrical parameters can be obtained as:

𝐼 1 = 4 𝜋 𝑉 𝑖𝑛 × 𝑛 2 √𝑎 2 + 𝑏 2 × √ 𝑅 𝑒 2 × (2𝜋𝑓 𝑠 ) 4 + (𝐿 2 × (2𝜋𝑓 𝑠 ) 3 + 𝐿 𝑚 × 𝑛 2 × (2𝜋𝑓 𝑠 ) 3 - 2𝜋𝑓 𝑠 𝐶 2 ) 2 (3.9) 𝐼 2 = 4 𝜋 𝑉 𝑖𝑛 × 𝐿 𝑚 × 𝑛 3 × (2𝜋𝑓 𝑠 ) 3 √𝑎 2 + 𝑏 2 (3.10) 𝑉 𝐶1 = 𝐼 1 2 × 𝐶 1 × 𝜋 × 𝑓 𝑠 (3.11) 𝑉 𝐶2 = 𝐼 2 2 × 𝐶 2 × 𝜋 × 𝑓 𝑠 (3.12) 𝑉 𝑂 = 2 𝜋 𝐼 2 (3.13)
where

𝑎 = -(𝐿 1 𝐿 2 𝑛 2 + 𝐿 1 𝐿 𝑚 𝑛 4 + 𝐿 2 𝐿 𝑚 𝑛 2 ) × (2𝜋𝑓 𝑠 ) 4 + ( 𝐿 1 𝑛 2 𝐶 2 + 𝐿 2 𝑛 2 𝐶 2 + 𝐿 𝑚 𝑛 2 𝐶 2 + 𝐿 𝑚 𝑛 4 𝐶 1 ) × (2𝜋𝑓 𝑠 ) 2 - 𝑛 2 𝐶 1 𝐶 2 (3.14) 𝑏 = 𝑅 𝑒 × (𝐿 1 𝑛 2 (2𝜋𝑓 𝑠 ) 3 + 𝐿 𝑚 𝑛 2 (2𝜋𝑓 𝑠 ) 3 - 2𝜋𝑛 2 𝑓 𝑠 𝐶 1 ) (3.15)
and

𝑅 𝑒 = 8 × 𝑅 𝜋 2 (3.16)
Now, it is needed to evaluate the FHA model for the CLLC resonant converter.

Evaluation of Electrical Constraints

So far, the bidirectional CLLC resonant converter was selected for the main power converter of an onboard charger, and a model was presented to find an optimum result subject to the constraints. Therefore, it is needed to examine the accuracy of the models. In the beginning, the FHA is used as a modeling framework for the evaluation of electrical constraints. Now, the electrical parameters of the CLLC resonant converter are examined.

Current Evaluation based on FHA

The primary and secondary inductor currents are illustrated in Fig. 3456: According to the above figure, it is manifest that the FHA-based waveforms are not exactly matched to the simulation-based results. Also, the maximum iL1 of time-domain simulation is higher than the FHA model, however, this condition is opposed to iL2. For a favorable comparison, the maximum values and RMS values are also compared in Table 3-2. 3-3 for a better evaluation of electrical constraints. The above table describes that the error for the maximum value reaches 11.68% and it is 12.39% for the RMS value. Therefore, the designed or optimized converter may deviate from the desired converter. It should be mentioned that these errors were obtained in one specific operating point, and there is a possibility for the error to be increased in different operating points.

ZVS Evaluation based on FHA

In literature, a ZVS condition formula for CLLC resonant converters is based on the FHA method and is derived from the LLC resonant converter, which was presented in (1.1).

Using the listed values of The above figure shows that the ZVS is also satisfied for the value above than the value shown in (3.17). This simulation proves that (3.17) is not the boundary condition for ZVS.

Another examination is that (3.17) is independent of the load and other parameters. Therefore, the FHA method is not a good method for ensuring ZVS operation.

Summary

Table 3-4 shows the results obtained from the FHA model and time-domain simulation. Two scenarios have been considered to evaluate the precision of the FHA-based results.

Global design methodology

Introduction

The half-bridge and full-bridge CLLC resonant converters (and other resonant converters)

have been designed as a conventional method by parameter sweeping to select a proper value for resonant tank elements. Furthermore, in this method, it is supposed that the converter operates in one operating point. Therefore, using conventional methods for the optimization of bidirectional CLLC resonant converters has the following drawbacks:

1. The CLLC resonant converter is a high-order resonant converter and selecting appropriate parameters for parameter sweeping is difficult. Therefore, it needs comprehensive knowledge of circuit behaviors as well as multiple tries and errors to meet desired requirements.

2. Conventional methods have employed approximated methods for optimization, which leads to inaccuracy in the solution.

3. The operating region of an on-board charger contains a wide range of voltages and currents. So, the designed converter should be optimized in corresponding with the whole operating region. But, a single operating point at a fixed switching frequency has been considered in conventional optimization methods.

4. Due to the limited space inside an EV, it is needed the power density to be considered in the optimization process. It means a mathematical-based formula of power density shall be obtained and put into the optimization process. Conventional algorithms have not considered power density as a function of electric parameters to optimize the whole of the system in one step.

5. The conventional methods have used parameter sweeping or iteration-based algorithms.

Therefore, there is not any guarantee that the obtained results are the best possible result.

This dissertation contains an optimization method based on an exact, deterministic, singlestage, and multi-operating point methodology to optimize power density.

Optimization Problem Definition

In an optimization problem, the goal is to find the minimum or maximum value of an objective function by changing the inputs. The operating region can be either constrained or not. In the scope of this dissertation, there are some constraints based on the voltages, currents, temperatures, space, power density, etc. Therefore, the optimization problem of this dissertation is a constrained optimization problem, which is presented as (4.1) [START_REF] Delhommais | Méthode de pré-design par optimisation en électronique de puissance[END_REF]:

12 min max () ( ) 0 1,2,..., ( ) 0 1,2,..., [ , ,..., ] 1,2,..., i i T n i i i Maximize y x Subject to g x i p h x i m x x x x x x x i n = == =   = (4.1)
where y is the objective function, x is the vector of input variables, g is the inequality constraints function with the number of p functions, h is the equality constraints function with the number of m functions, and n is the number of input variables.

Fig. 4-1: The operating region and constraint of a single-variable objective function

In this dissertation, it is assumed that at least the first derivative of the objective function and constraint functions is available. Some optimization algorithms have been presented to solve (4.1) and find the global optimum point. In the following part, a review of the different optimization algorithms is expressed.

Optimization Algorithms

An optimization algorithm is a method that seeks the inputs to find the minimum or maximum of an objective function while constraints have been met.

Stochastic Algorithms

In stochastic algorithms, random algorithms are used as heuristic searching, where may or may not find the optimum result. These algorithms are inspired by stochastic, environmental, biological, or physical processes. In the genetic algorithm as an example, a population of solutions is evolved toward a better solution with a population generation. The fitness of each generation is evaluated and the more fit generations are selected and the others are recombined or mutated into a new generation. Finally, the genetic algorithm is finished when either the number of generations reaches its maximum value or a fitness level is lower than the required level.

Stochastic algorithms such as Simulated Annealing (SA) [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF], Bee Colony (BC) [START_REF] Teodorovic | Bee colony optimization: principles and applications[END_REF],

Stochastic Hill Climbing (SHC) [START_REF] Rohan | Development of intelligent drone battery charging system based on wireless power transmission using hill climbing algorithm[END_REF], Genetic Algorithm (GA) [START_REF] Weile | Genetic algorithm optimization applied to electromagnetics: A review[END_REF], Particle Swarm Optimization (PSO) [START_REF] Poli | Particle swarm optimization[END_REF], etc. have been used in some applications. For example, the GA and PSO algorithms were used for two-stage optimization for a hybrid AC-DC micro-grid application [START_REF] Zhao | AI Algorithm-Based Two-Stage Optimal Design Methodology of High-Efficiency CLLC Resonant Converters for the Hybrid AC-DC Microgrid Applications[END_REF], however, it may cause to get stuck in a local optimum point instead of the absolute optimum point.

For the stochastic algorithms, it is obvious that any single run of the algorithm will be different due to different starting points, the randomness of the algorithm, the selection of the number of iterations, and the selected fitness level. As a result, the stochastic algorithm may have a high convergence speed and be highly sensitive to the initial parameters. Also, it may not converge into a result, and there is no guarantee that the obtained result is the best possible result.

Deterministic Algorithms

A deterministic algorithm can provide the absolute (global) extremum or at least a local extremum can be guaranteed using the deterministic algorithm [START_REF] Barhen | TRUST: A deterministic algorithm for global optimization[END_REF], where there exists a theoretical guarantee within a predefined tolerance for obtaining the results.

Linear programming (LP), which contains linear equations, is able to efficiently solve an optimization problem involving even millions of variables to find the local or global extremum point [START_REF] Bienstock | Solving LP relaxations of large-scale precedence constrained problems[END_REF]. Non-linear programming (NLP) is a deterministic optimization method for solving non-linear equations for global optimization [START_REF] Sun | Optimization theory and methods: nonlinear programming[END_REF]. The branch-and-bound [START_REF] Leyffer | Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming[END_REF], DIRECT [START_REF] Gablonsky | Modifications of the DIRECT algorithm[END_REF], outer-approximation [START_REF] Fletcher | Solving mixed integer nonlinear programs by outer approximation[END_REF], and extended cutting plane [START_REF] Westerlund | An Extended Cutting Plane Method for Solving Convex MINLP Problems[END_REF] algorithms are other types of deterministic algorithms. In the branch-and-bound algorithm as an example, the search space of optimization is the whole range of all variables, and each range is bounded to find an optimum value. However, it can find the optimum solution when each range can be solved. Also, it may cause the optimization process to get stuck in a local extremum. The Sequential Quadratic Programming (SQP) algorithm is a gradient-based optimization technique, which is an appropriate method for non-linear programming optimization in a constrained space. The SQP has been applied for different optimization applications. In Table 4-1, a comparison of different optimization algorithms is shown:

Table 4-1:
The main properties of some optimization algorithms [START_REF] Delhommais | Méthode de pré-design par optimisation en électronique de puissance[END_REF] Type In this dissertation, the SQP method will be utilized as the optimization algorithm, which is discussed in the next part.

Sequential Quadratic Programming (SQP)

An SQP problem can be iteratively solved by linearization of Eq. (4.1) using Taylor's approximation to find the searching direction at each iteration. The Jacobian matrix is calculated to find the right direction of the new system variable as (4.2).

ℒ(𝑥, 𝜆, 𝜎) = 𝑦(𝑥) -𝜆𝑔(𝑥) -𝜎ℎ(𝑥)

(4.2)
where 𝜆 and 𝜎 are Lagrange multipliers. According to the above equation, the first and second derivatives of the objective function must be available. However, there exist some discrete parameters in the power electronics circuits, which must be derivable. This modeling is presented in the next part.

Continuous Modeling of Discrete Parameters

The discrete parameters of the CLLC-based EV charger can be classified into two parameters, i.e. standard core sizes and the number of turns. The parameter X is defined as a continuous variable to make the core sizes continuous using interpolation with Matlab. The interpolation of some variables is shown in Fig. 4-3. The continuous model of each variable is listed in Using parameter X, all the variables corresponding to the core sizes will be continuous, if the parameter X is selected as a free positive number.

The number of turns

Another discrete parameter is the number of turns and also the number of turns in each layer.

No interpolation is needed and these values can be directly considered as continuous parameters.

Table 4-3: Continuous equations for the core sizes of conventional and planar E cores

Design Methodology and Models

Whether stochastic or deterministic algorithms are selected, it is mandatory to obtain the objective function precisely due to the fact that the objective function is calculated repeatedly and a small error in the models can cause a high error in the final result. In this part, the total losses and design methodology will be presented.

Semiconductor Losses

The calculation of semiconductor losses depends on the selected technology of switching and circuit behavior. Gallium Nitride (GaN) transistors and Schottky diodes are used in this dissertation as the semiconductor parts.

Proposed Switching Losses Calculation for GaN transistors

Understanding the GaN power loss modeling, whether switching loss or conduction loss, is one of the most important parts of the optimum design of GaN-based EV chargers. The switching losses, involving turn-on and turn-off situations, are commonly calculated by multiplying the switching frequency by the energy losses which are provided in the datasheet of a GaN transistor. It should be mentioned that the losses are usually given at a specific operating point [START_REF] Babaei | Multi-input high step-up inverter with soft-switching capability, applicable in photovoltaic systems[END_REF]. Even these data are not always given within the datasheet of some GaN transistors. So, it is necessary to propose a method for switching loss modeling.

In [START_REF] Huang | Analytical loss model of high voltage GaN HEMT in cascode configuration[END_REF], authors presented an analytical loss model in the integral form for a high-voltage

GaN transistor in cascode configuration considering parasitic inductance and nonlinearity of capacitors. In [START_REF] Chen | A complete switching analytical model of lowvoltage eGaN HEMTs and its application in loss analysis[END_REF], the loss analysis of a low-voltage GaN transistor was calculated according to the switching processes modeling for a synchronous buck converter and the qualitative analysis for charging the gate resistances, the drain-source capacitances, etc. was presented. In [START_REF] Xin | Analytical switching loss model for GaN-based control switch and synchronous rectifier in lowvoltage buck converters[END_REF], an analytical switching loss model was presented for a GaN-based buck converter; the proposed equations of power losses are based on buck converter parameters and cannot be applied to other topologies. In [START_REF] Wang | An improved switching loss model for a 650V enhancement-mode GaN transistor[END_REF], a switching loss model by the interpolation of the input capacitance and transconductance was presented for a 650V GaN transistor. A datasheet-based analytical switching loss model was presented in [START_REF] Li | Analytical Switching Loss Model of Cascode GaN HEMTs Based Totem-Pole PFC Converters Considering Stray Inductances[END_REF] for totem-pole PFC converters. In [START_REF] Korada | Loss characterization and analysis of high voltage e-mode GaN HEMT in soft-switching application[END_REF],

the LTSPICE simulation was carried out on the preliminary calculation of the EON and EOFF.

In [START_REF] Huang | Evaluation and applications of 600V/650V enhancement-mode GaN devices[END_REF], the impact of GaN transistor packaging on switching loss was illustrated. In [START_REF] Zhang | An improved analytical model of GaN HEMT in cascode configuration during turn-on transition[END_REF] with an analytical method, all the parasitic inductors and capacitors were considered for a cascode configuration and only the turn-ON transition was analyzed. In [START_REF] Dadanema | Analytical losses model for SiC semiconductors dedicated to optimization operations[END_REF] authors used the analysis of the switching waveforms to obtain loss prediction of SiC and GaN devices.

However, despite a considerable number of analytical loss measurements, there is no simple and accurate analytical model appropriate for design engineers. Most of the works carried out on the switching loss analysis use an integral form which leads to a complicated and timeconsuming analyzing model. Furthermore, for one device, the switching losses are provided at one operating point associated with the datasheet's value. Therefore, a model is needed to predict the losses for other operating points. The value of LLoad is considered to prepare the required current for the double pulse tester of the selected GaN device [START_REF] Yao | A multifunctional double pulse tester for cascode GaN devices[END_REF]. For this circuit, one specific condition is considered as the base value of GaN parameters. Table 4- The proposed method must possess high accuracy for any type of circuit parameters, e.g. different voltage, current, gate resistor, GaN transistor, etc. The criteria for general loss modeling analysis are obtained by comparing them with the base values. For this purpose, the LTSPICE simulation is performed and then, an algebraic equation is presented by the interpolation.

There are four kinds of parameters for a circuit based on its operating point and components:

• Electrical operating point: VDS, ID,

• Thermal operating point: Tj

• Circuit operating point: Rgate, Vgate, Lp, etc.

• Device characteristics: Ciss, Coss, Crss.

Owing to the GaN circuit parameters dependencies, the proposed method is divided into two types of energy loss equations. The first type is a function with two variables and the second type is a function with one variable.

a) Type I (Two Variables Function):

It is obvious that in the switching power devices, there is a close relationship between the drain current, ID, and junction temperature, Tj.

Another alternative explanation is the clear association between the temperature and current, which are not independent. Current generates losses and increases the temperature indeed. Therefore, a combined formulation is used. where coefficients P1, P2, … , P7 are defined in ( ) ( )

32 1 2 3 4 base EX Eb X P X P X P X P E = = + + + (4.6)
where coefficients P1, P2, P3, and P4 are also defined in the negative gate-to-source voltage has a low effect on switching losses. By reducing the positive gate-to-source voltage, the switching losses will be increased which may be affected by increasing the ON state resistance (Fig. 456). Furthermore, the gate inductance has a nonlinear relationship with switching losses (Fig. 456). The switching loss will be increased by increasing Ciss, Coss, and Crss (Fig. 4-6 g, h, andi). Other parameters, such as drain and source parasitic inductors, and the DC line inductance, have not been shown in Fig. 456.

As a result, Eq. (4.8) is used to calculate the switching losses of GaN transistors, when ID<31A, VDS<650V, and Tj<150°C.

It should be mentioned that the proposed method can be used for different GaN transistors because the switching loss relationship is also based on the GaN transistor's parameters (e.g. Ciss, Coss, Crss, Lpara, and Lex). In this work, the selected technology for switching is predetermined and the GS66508B GaN transistor has been selected. So, the aforementioned parameters are fixed and the switching loss formula can be simplified as (4.9). where ID is its current at the switching instance, and Tj is the junction temperature. For G2V applications, VDS is 400V for the primary switches (inverter) and it is equal to Vout for the secondary switches (rectifier). For V2G applications, it depends on the discharging profile for the secondary switches (inverter), and it is equal to Vout for the primary switches (rectifier). ID is related to the charging/discharging current (defined operating points).

𝑃

Proposed Conduction Losses Calculation for GaN transistors

The conduction losses are usually calculated by multiplying the ON resistance by the RMS current to the square [START_REF] Mirjafari | Survey of modelling techniques used in optimisation of power electronic components[END_REF]. In [START_REF] Wang | Characterization and modelling of frequencydependent ON-resistance for GaN devices at high-frequencies[END_REF], the authors presented the modeling of frequency-based ON resistance for GaN transistors, especially at frequencies above MHz and an electrical circuit was considered for the ON resistance calculation. Impacts of soft-switching and hard-switching on dynamic ON-resistance of GaN transistors were analyzed in [START_REF] Li | Dynamic on-state resistance test and evaluation of GaN power devices under hard-and soft-switching conditions by double and multiple pulses[END_REF]. However, experiments and simulations have indicated temperature-dependency in the ON state resistance of GaN transistors. This dependency demonstrates a need to model the relationship between the ON state resistance and temperature.

An easy and exact modeling method is presented for obtaining the conduction power loss which is directly relevant to the ON resistance of the GaN transistor. For this purpose, the values of the ON resistance at two different temperatures can be found on the GaN transistor's datasheet, T1 and T2, typically 25°C and 150°C. Table 4- Some simulations for the measurement of ON resistance in different temperatures have been carried out which consistently show that there is a direct relationship between the temperature and ON resistance. Therefore, an algebraic equation can be fitted as (4.12).

( )

2 1 2 1 1 2 2 1 2 1 ON jj R R T R T R R T T T T T T -- =+ -- (4.12)
where the values of ON resistance and related temperatures for some GaN transistors have been presented in Finally, the conduction power loss can be calculated as (4.13).

.

ON RMS cond

P R I = (4.13)
where IRMS is the root mean square current of the GaN transistor and depends on the type of the converter and the load current.

In summary for obtaining conduction losses, only the ON state resistance is needed, which is temperature-dependent. So, using the model presented in (4.12), the conduction losses are computed as (4.13).

Diode Losses

Transistors can be applied as synchronous rectification to increase the total efficiency. However, a controller and external components are needed to detect the exact transition time.

In this dissertation, an extra diode is parallel to each GaN transistor for having an alternative solution. In this case, the gate driver of GaN transistors is OFF and the current will pass through the corresponding diodes. Therefore, it is needed to obtain the diode losses, which are usually calculated using multiplying the diode voltage drop by the DC current as (4.14). Although the voltage drop is a function of the current and junction temperature, in this dissertation, only the typical value in its datasheet is considered. For more accuracy, considering V-I characteristics of the diode is preferred.

Core Losses in Magnetic Components

The core losses are the sum of eddy-current loss and hysteresis loss, and anomalous loss, which play an important role in the design and optimization of power electronics converters.

There are some methods to evaluate the core losses. The simplest method is the Steinmetz equation captured using an empirical approach and is precise for pure sinusoidal waveforms.

The Steinmetz equation is expressed as (4.15). where Pcore is the core losses, Vcore is the core's volume, B is the maximum flux density of the core, and K, α, and β are the coefficients related to the core's material and geometry, and usually mentioned in the core's datasheet. However, the waveforms are not pure sinusoidal in the power electronics converter, and the accuracy of the Steinmetz equation may deteriorate for non-sinusoidal flux waveforms. It should be mentioned that due to the non-linear behavior of the core loss equation, the superposition cannot be applied.

In [START_REF] Yan | Optimized Design of Integrated Planar Matrix Transformer for LLC Converter in Consumer Electronics[END_REF], a temperature-dependent form of the Steinmetz equation was considered as (4.16).

( )

2 0 1 2 core core C C T C T P K V f B    - + = (4.16)
But there is not any information about the value of C0, C1, and C2. In [START_REF] Li | Dynamic on-state resistance test and evaluation of GaN power devices under hard-and soft-switching conditions by double and multiple pulses[END_REF], the Steinmetz equation was modified and an equivalent frequency was introduced for non-sinusoidal waveforms:

( )

2 22 0 2 T eq dB t dt dt B f      =  (4.17)
and the core loss is calculated as (4.18).

1 core eq r core f P K V f B  -    = (4.18)
Another modification was the generalized Steinmetz equation as [START_REF] Li | Coupled-inductor design optimization for fast-response low-voltage[END_REF]:

( ) ( ) 0 1 T i core dB t K B t dt T dt P   - =  (4.19)
The Steinmetz equation was extended for rectangular and trapezoidal waveforms in [START_REF] Agheb | Core loss behavior in high frequency high power transformers-II: Arbitrary excitation[END_REF],

and the simpler version of this method was discussed in [START_REF] Zhang | Design and Optimization of a Magnetron DC-DC Isolated Power Supply With High Efficiency[END_REF] as (4.20).

1 1 2 8 core core P K V f B    - -     = (4.20)
An experimental-based methodology for core loss calculation was presented in [START_REF] Kalra | A Power Loss Measurement Technique for Inductive Power Transfer Magnetic Couplers[END_REF], but it is hard to implement into the optimization process.

In this dissertation, the more simple version of the Steinmetz equation i.e. Eq. (4.15) is selected because the CLLC resonant tank plays a role of a filter for the applied DC voltage, and the magnetic component waveforms in each subinterval are a part of a sinusoid. Also, it is difficult to implement integral forms in the optimization process.

Copper Losses in wires

In high-power applications, the copper loss is a considerable part of the total power losses, which is created by a current path through a conductor. For a conductor, the DC resistance is calculated as (4.21).

eff DC eff l R A   = (4.21)
where ρ is the specific resistance and it is 1.68×10 -8 Ω/m for copper, Aeff is the effective cross-section area of the conductor, and leff is the total effective length of the conductor.

However, owing to the frequency of the current path through a wire, adjacent magnetic fields of other wires, and the placements of each turn in a winding, additional losses are caused, which are called the skin effect and proximity effect.

The skin effect is the tendency of AC current to follow near the surface of a conductor and it leads to a decrease in the skin depth by increasing the frequency. As shown in Fig. 4567 where rw is the radius of a wire.

Although this method can be used for skin effect consideration, this behavior is complicated.

Furthermore, the proximity effect has another effect on copper loss. The Dowel equation [START_REF] Kazimierczuk | High-frequency magnetic components[END_REF] can model these effects via obtaining AC resistance RAC instead of DC resistance RDC as [START_REF] Mirjafari | Survey of modelling techniques used in optimisation of power electronic components[END_REF]:

( ) 2 21 sinh 2 sin 2 sinh sin cosh 2 cos 2 3 cosh cos l DC AC N A A A A A A A A A RR  - +-    + -+   = (4.24)
Where Nl is the number of layers and It should be mentioned that the skin effect can be dramatically reduced using Litz wires for conventional magnetic components or using planar transformers and inductors. The proximity effect can be also decreased via the sandwich formation of layers.

Capacitor Losses

The equivalent series resistor (ESR) of a capacitor represents the power loss as: where δ is the loss angle, which depends on the dielectric material and usually is mentioned in the datasheet.

It should be mentioned that multilayer ceramic capacitors (MLCCs) have high efficiency and low ESR, which offer low losses for resonant converters. In this dissertation, the MLCCs are used as the capacitors of the resonant tank. The reluctance ℜ (magnetic resistance) of a portion of the core is defined as:

Inductor Design

0 c c rc l A  =  (4.29)
where lc is the length of the core, Ac is the area of the core, µ0=4π×10 -7 H/m is the permeability of free space, and µr is the magnetic permeability of the magnetic material. The reluctance of an air gap can be obtained as:

0 g g g A  =  (4.30)
where g is the length of the air gap, and Ag is the area of the air gap.

Basically, the reluctance of magnetic materials is much lower than the air gap reluctance and can be neglected. However, a general solution is considered in this dissertation.

The value of inductance can be calculated as:

2 total N L =  (4.31)
where N is the number of turns, and ℜtotal is the equivalent reluctance. The above equation establishes a relationship between the electrical parameter L and physical parameters N and ℜ, which will be used in the optimization process.

However, the physical parameters are important to build the converter. To select the right core for the inductor, it is necessary to fit the number of turns in the selected core, and also to check the effect of the core saturation.

To do this, one core is selected from the existing database of cores. Then, according to the required inductance (obtained in optimization or based on the designer's opinion), the required number of turns is calculated using (4.32).

, total L N L =  (4.32)
where ℜtotal, L is the equivalent reluctance of L.

Then, it needs to be checked if the core is saturated with this number of turns or not. To do this, the maximum number of turns to avoid saturation is calculated as (4.33). where le is the effective length of the core, Bmax is the maximum flux density, and iL,max is the maximum current of the inductor. Now, if the selected number of turns as (4.32) is less than the maximum number of turns as (4.33), the selected core will be suitable, otherwise, a larger core should be selected. Finally, the feasibility of the turns to fit in the core window must be checked. Consequently, the required number of turns, cross-section, and effective length of the core will be calculated. Also, according to the selected core, the volume of the core can be calculated, which will be required for the optimization process with the power density objective function.

Transformer Design

A transformer is usually designed at the minimum switching frequency to have the minimum losses. On the other hand, the size of a transformer is decreased by increasing the switching frequency. Hence, there is a trade-off for switching frequency selection. In EV charger applications, the switching frequency usually changes to regulate the output voltage. Therefore, it is suitable that the range of switching frequencies should not be wide. The transformer turn ratio can help to reduce this range. For the CLLC resonant converter as an example, operating near the resonant frequency leads to the maximum output voltage. Therefore, the transformer turn ratio can be calculated as (4.34) to meet all the voltage ranges:

, max max out in V VM n  = (4.34)
In our design, Vout, max = 400V, Vin = 400V, and Mmax = 1, which leads n = 1. A unity turn ratio also leads to a symmetrical transformer. Now, other aspects of the transformer design are examined. According to Eq. (1.1) for the ZVS condition, the minimum required dead-time is based on the initial current of the primary side inductor, input DC voltage, and the capacitor parallel to the GaN transistors. It should be noted that during the dead time, no power will be applied to the resonant tank (and also load).

But in general, the magnetizing inductor has also a great effect on the minimum dead-time [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF].

To examine this effect, it can be said that the minimum dead-time is directly (but not linearly) related to the amount of the magnetizing inductor. In other words, by increasing the value of Lm, the minimum dead time value increases, as shown in Fig. 45678910. But this will lead to an increase in the circulating current and conduction losses because Lm is located in parallel. As shown in Fig. 4-11, Lm varies from 100µH to 180µH. As shown in the above figure, by increasing Lm, the maximum current of the primary side (iL1, max) decreases and the maximum current of the secondary side (iL2, max) increases. Similarly, the maximum voltage of the primary side capacitor (vC1, max) decreases and the maximum voltage of the secondary side capacitor (vC2, max) increases. Also, to investigate the effect Lm on the output characteristic (i.e. the normalized output current, J, versus normalized output voltage, M), Fig. 4-12 can be analyzed:

Increasing Lm

Fig. 4-12: Effect of magnetizing inductance on the J-M curve

whereby increasing Lm, the output characteristic curves step up, and more current can be obtained at the same voltage. However, this increasing leads to the CLLC resonant converter operates as like as a series resonant converter, and losing its advantages. Therefore, a trade-off exists for Lm calculation. These investigations show that the transformer has a direct effect on the converter behaviors.

Therefore, the following criteria are considered in the optimization process and discussed as follows.

Core Selection

To choose the right core for the transformer, the following criteria should be met:

• The number of turns for the primary side and secondary side must be fitted in the core window;

• The required transformer turn ratio must be prepared;

• The transformer must not work in saturation region for each operating point;

• The core material should be selected based on the maximum frequency.

To do this, a database is created from existing cores and one of them is selected. In this dissertation, ferrite cores with 3C90 or 3F3 will be used based on the maximum frequency.

Number of Turns

According to the above 4 criteria, the number of turns of the windings is calculated. E-type cores are selected with the dimensions shown in Fig. 4-2. For obtaining the number of turns, one of the above cores is selected. Based on the selected core and required magnetizing inductance, the number of turns for the primary winding is calculated as (4.35). where N1 is the number of turns for the primary side and ℜT is the reluctance (magnetic resistance) of the transformer core. It is assumed that Lm is located on the primary side and therefore, N1 was obtained.

Then, core saturation should be checked. The maximum allowable turns for operating in the linear region are: where le is the effective length of the core, Bmax is the maximum flux density, and i1,max is the maximum current of the primary side.

Therefore, the number of turns calculated in (4.35) should be less than the number of turns in (4.36). After passing this step, the number of secondary side turns is calculated using the transformer conversion ratio as (4.37). If necessary, the saturation condition for secondary winding can be also checked.

Finally, it must be calculated whether the number of turns is fitted to the desired core window or not. If this is not possible, a larger core should be selected, and this process continues to meet all the constraints.

Selection of Resonant Capacitor

Power electronics circuit designers can select the resonant frequency, fres, based on their knowledge, the design requirements, or independently through the optimization algorithm. The value of the resonant capacitor, C, is obtained as (4.38). It should be mentioned that for CLLC resonant converter, the aforementioned resonant frequency (fres) is the resonant frequency of series L and C without the magnetizing inductance Lm.

Objective Functions Evaluation

Conventional EV chargers have been facing the challenge of efficiency increment. Using special magnetic cores like planar cores, and high technology switching components like GaN transistors, the existing EV onboard chargers are transferring into a high-efficiency and high power density converter, the so-called optimum onboard charger. The objective function of the optimization process can be examined in different aspects:

• Size: the optimal size of an onboard charger is to decrease the volume and/or weight of the charger to determine a suitable battery capacity in the limited space. Therefore, a small-in-size charger is preferable.

• Efficiency: the efficiency of the charger should remain high at different operating points. Therefore, a high-in-efficiency charger is preferred.

• Losses: power loss in magnetic components and switching components can increase the temperature of the components which reduces the lifespan of the converter and leads to failure. Therefore, a small-in-losses charger is preferable. On the other hand, increasing losses will decrease efficiency. However, low losses have the same meaning as high efficiency and therefore will not be considered in this dissertation as the objective function.

• Power Density: for a power electronics converter, possessing high output power in a low volume is increased the power density. The high power density was obtained by reducing the size of the passive components in [START_REF] Chen | Design of Planar Integrated Passive Module for Zero-Voltage-Switched Asymmetrical Half-Bridge PWM Converter[END_REF], and minimizing losses and volume in [START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF], whereas it was achieved by minimizing the output power divided by volume in [START_REF] Biela | Impact of Power Density Maximization on Efficiency of DC-DC Converter Systems[END_REF], and minimizing the output power divided by mass in [START_REF] Nawawi | Design and Demonstration of High Power Density Inverter for Aircraft Applications[END_REF]. It means the power density is not fairly independent of the size, efficiency, and losses.

• Cost: a low-cost product can make it a leader in the competitive market. So, a smallin-cost charger is preferred. However, it is beyond the scope of this dissertation.

• Reliability: EV's driver dissatisfaction is expressed with charger failure. Proper component selection and derating, good printed circuit board (PCB) designing with minimum inter-connections, decreasing operating temperature, etc. can increase the reliability of an onboard charger. However, reliability is also beyond the scope of this dissertation. So, in this thesis, the size and efficiency of the charger are simultaneously selected as the objective function, which should be minimized and maximized, respectively. Therefore, the objective function can be computed as:

𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = max { 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑆𝑖𝑧𝑒 } 𝑜𝑟 min{ 𝑆𝑖𝑧𝑒 𝐸𝑓𝑓𝑖𝑛𝑖𝑒𝑛𝑐𝑦 } (4.39)
where Size can be either weight or volume, and efficiency is the efficiency of the selected point or the average efficiency of all operating points. We select a weight for the objective function, and volume will be considered as a constraint. Therefore, the objective function is to maximize the new version of power density as (4.40).

𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = max { 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑆𝑖𝑧𝑒 } (4.40)

Constraints Discussion

The solution and input variables must satisfy some conditions called constraints: ( ) , () ()

T n x x x x Maximize f x Subject to g x A h x B = =  (4.41)
where g(x) is the equality constraints, h(x) is the inequality constraints, n is the number of input variables, and A and B are circuit-based vectors. For onboard charger applications, the following constraints must be considered in the optimization process:

Operating Region

Owing to the battery charging process, the voltage of the battery varies between VMIN at the start of CC mode to VMAX at the end of CV mode. Therefore, there exists the following constraint for the output voltage:

𝑉 𝑀𝐼𝑁 ≤ 𝑉 𝑂𝑈𝑇 ≤ 𝑉 𝑀𝐴𝑋 (4.42)
where we select VMIN=250V and VMAX=400V. This voltage constraint is shown in Fig. where we select IMIN=1A and IMAX=12A. This current constraint is shown in Fig. 4-13 (b).

4-13(a).

The operating region of an onboard charger is also limited by the charger power rating. It is assumed that the charger is designed to prepare a power between PMIN to PMAX as shown in Fig.

4-13 (c).

𝑃 𝑀𝐼𝑁 ≤ 𝑃 𝑂𝑈𝑇 ≤ 𝑃 𝑀𝐴𝑋 (4.44)

Consequently, the complete operating region of an onboard charger can be shown in Fig.

4-13 (d) by superimposing all the constraints.

Maximum Temperature

Ambient temperature affects the operating temperature of components. Based on their datasheet, hardware requirements, and reliability reasons, the maximum temperature must not be exceeded. In the following, the temperature constraints of the components are examined.

1-Semiconductors:

The junction temperature (Tj) of semiconductors must be limited to a specific value entitled maximum junction temperature, TjMAX, which is typically 150°C. It should be mentioned that this value is the upper margin of the junction temperature and approaching this value will exponentially decrease the lifespan.

As shown in Fig. 4-14, for a semiconductor device, the junction temperature (TJ) can be obtained as (4.45). where TA is the ambient temperature, RJA is the junction-to-ambient thermal resistance, and PLoss is the total power dissipation of the semiconductor, which is obtained for GaN transistors and diodes in (4.9) and (4.14), respectively. The typical thermal resistance of some semiconductor devices is listed in Table 4-7.

where TJ is the junction temperature.

2-Magnetic components:

The core loss and copper loss generate heat, which leads to either changing the magnetic behavior of the core or damage to the wire insulation and causes a short circuit. Therefore, it is necessary to impose a constraint on the temperature of magnetic components. Eq. (4.45) can be also used for the calculation of the hotspot in a magnetic core. However, it is hard to model the thermal resistance of a magnetic core. Also, using approximate methods lead to low accuracy.

Hence, finite element methods (FEMs) have been developed to determine temperature, which is beyond the scope of this thesis.

On the other hand, the Curie temperature of common magnetic materials such as 3C90 and 3F4 is higher than 200°C [START_REF] Ferroxcube | Datasheet, 3C90 Material Specification[END_REF](see Fig. 4-15), which is too high. It should be mentioned that the full thermal modeling of the transformer is difficult to implement and also it is beyond the scope of this dissertation. Therefore, we replace it with a rule of thumb constraints, i.e. checking the saturation condition for a transformer to avoid approaching the Curie temperature. (right circuit) [START_REF] Ferroxcube | Datasheet, 3F4 Material Specifications[END_REF].

3-Wires:

Copper loss sources are the windings of magnetic components, and PCB tracks. In [START_REF] Ansi/Ipc | IPC -2221[END_REF],

there exists an empirical curve for PCB tracks to obtain the trace width based on the desired temperature rise. However, for a pre-selected trace, temperature calculation can be done like a conductor resistance. For the windings, this condition is the same. Therefore, the power loss and temperature of a conductor are calculated.

A conductor with the cross-sectional area A and the length l is presumed (see Fig. 4567). The DC resistance of this conductor was calculated in (4.24). However, due to the frequency of the current passing through the copper and also the position of the transformer layers relative to each other, two effects can change the amount of DC resistance, skin effect and proximity effect, which were examined in part 4.5.4.

Eq. (4.23) showed that the effective area is decreased, which leads to an increase in the resistance. The core temperature of a resistor can be obtained according to the following conventional equation as (4.47).

, J Loss A th JA T T R P = +  (4.47)
Where PLoss is the power loss obtained in (4.26), and Rth,JA is the core-to-ambient thermal resistance, which is affected by the width of wires, choice of material, soldering pattern, insulation, terminations, and cooling system [START_REF] Vishay | Application Note, Power Dissipation in High Precision Vishay Sfernice Chip Resistors and Arrays (P Thin Film, PRA Arrays, CHP Thick Film)[END_REF].

Therefore, this power loss increases the temperature of the resistor. It is a well-known fact that the conductor resistance varies linearly with the temperature as shown in (4.48).

( ) ( ) ( )

0 1 ref R T R T T  = + - (4.48)
where R(T) is the conductor resistance at the required temperature T, R0 is the conductor resistance at the reference temperature (Tref), usually 20°C, and α is the temperature coefficient for the conductor material, which is equal 0.004041 °C-1 for copper.

In this dissertation, the following constraint is considered for wires:

2 5 A J mm  (4.49)
where J is the current density of a wire.

Maximum RMS Current for resonant capacitors and inductors

The absolute maximum rating current of capacitors must not exceed the specified value in their datasheet. With rising the current, the capacitor temperature may exceed the allowable hot spot temperature. It causes damage to the dielectric material and reduces the capacitor lifespan.

Therefore, it is necessary to ensure the following constraint [START_REF] Kolar | Analytical calculation of the RMS current stress on the DC link capacitor of voltage DC link PWM converter systems[END_REF]:

, , , C RMS C RMS MAX II  (4.50)
The multilayer ceramic capacitors (MLCCs) are selected in this dissertation owing to their low equivalent series resistance (ESR), low cost, high frequency, low size, and high reliability.

TDK capacitors are chosen for the resonant capacitors that the following constraint can be considered for the maximum RMS current based on their datasheets: For inductors, the maximum allowable current density for a wire can be considered equal to J=5 A/mm2. Therefore, based on the RMS current of an inductor, the minimum diameter of the wire should be:

,, 22 where d is the wire diameter.

Maximum Voltage

Applying a voltage beyond the breakdown voltage causes failure and usually, the capacitor becomes a short circuit. TDK MLCCs have a rated voltage from 4V up to 3kV. Owing to the voltage rating of the onboard chargers, a capacitor with a 1000V voltage rating is acceptable.

Therefore, the following constraint is considered for the capacitor voltage rating. Zero voltage switching (ZVS) condition enables the power electronics converters to operate in low loss, fast switching, high efficiency, and high power density EV applications [START_REF] Gautam | A Zero-Voltage Switching Full-Bridge DC--DC Converter With Capacitive Output Filter for Plug-In Hybrid Electric Vehicle Battery Charging[END_REF]. For accomplishing ZVS operation for the main power switches e.g. S2 and S3, the energy stored in the transformer leakage inductance L1 must discharge the parallel capacitors Cs2 and Cs3 until the switch S2 turns OFF. Therefore, the following inequality must be satisfied for achieving the ZVS condition [START_REF] He | Comprehensive Analyses and Comparison of 1 kW Isolated DC-DC Converters for Bidirectional EV Charging Systems[END_REF]:

dead time OSS SW MAX t Lm Cf   (4.54)
The above constraint can be used as the ZVS operating condition based on the FHA method.

However, the exact relationship of the ZVS condition for the CLLC resonant converter was presented in [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF] and we use the equations as mentioned in Chapter 5.

CCM Condition

Although there are some disadvantages to CCM operating such as turn-off switching loss, proper design of this converter can improve this loss. However, DCM has more disadvantages.

For example, a bulky capacitor is needed for the DC link capacitors, DCM has a higher peak current for switches and other components in comparison with CCM, higher conduction losses, etc. Operating in CCM condition is superior in high current applications. Therefore, in this dissertation, the continuous conduction mode (CCM) is considered as a constraint. This constraint can be met if the voltage of the section before the rectifying part (i.e. v'2) is higher than the section after the rectifying part (i.e. Vout) when the diode current has a zero crossing.

Therefore, (4.55) is considered as the CCM operating: 𝐶𝐶𝑀 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛: 𝑉 2 ′ > 𝑉 𝑂𝑈𝑇 (4.55)

Requirements of EV chargers

There are some requirements for EV battery chargers, which are mandatory to meet. These requirements can be classified into 4 considerations: electrical requirements, placement requirements, environmental requirements, and safety requirements. In this dissertation, electrical requirements will be discussed in detail, and a review of the other requirements will be presented briefly.

Electrical Requirements

The electrical requirements of a household EV charger can be selected as follows [START_REF] Emadi | Advanced electric drive vehicles[END_REF]: Furthermore, there is a boost converter to increase the DC voltage to a fixed value of 400V and also play the role of power factor correction (PFC). Therefore, we have a constant voltage of 400V in the input part of the DC-to-DC converter and any variation of AC input voltage does not affect on our analysis. So, the electrical requirements can be rearranged as:

•
• Input DC voltage: 400V It should be mentioned that the aforementioned requirements are related to the basic electrical characteristics of a charger.

Placement Requirements

Due to the fact that the on-board charger is located inside the EV, the size and weight of the charger shall be selected based on the defined place in the EV. Furthermore, the selected socket outlet and the associated plug shall comply with IEC 60309. GEN2.5 is a 3.3kW battery charger provided by Valeo and Siemens Companies that has a weight of 3.5 kg and a size of 31cm × 19.4cm × 6.9cm. For this on-board charger, the power density is 0.98kW/kg or 1.1 kW/dm 3 [START_REF] Datasheet | On Board Charger (OBC)[END_REF] . Furthermore, CAD332DF-400A is a 3.3kW EV on-board charger from OVARTECH Company that its weight is 8.4kg and its dimension is 30.5cm × 12.5cm × 17cm [159].

However, the placement checking of an on-board charger is beyond the scope of this dissertation.

Environmental Requirements

Temperature, humidity, dust, and vibration are the most important parts of the environmental requirements for the components and the charger. However, in this dissertation, only temperature-based requirement is considered:

• -40°C to 105°C ambient temperature;

• The junction temperature of transistors: based on their datasheets (usually 110°C);

• Magnetic cores temperature: based on the core material (usually 110°C);

• PCB track temperature: based on the allowable over-temperature related to the ambient temperature (usually 10~20°C); • Temperature criteria of other components: based on their datasheet, and standards e.g.:

AEC-Q100 for integrated circuits (ICs), AEC-Q101 for discrete semiconductor components, and AEC-Q200 for passive components.

In this dissertation, only the DC-to-DC converter is analyzed. Therefore, the temperature of all components is considered and the effect of ambient temperature on the components is neglected.

Safety Requirements

High voltage transition rate (dv/dt), high power transfer, and high switching frequency can increase the electromagnetic interference (EMI) of a power electronics converter, where the converter can emit noise to the environment or absorb noise from it. For this reason, some standards are authorized for electromagnetic compatibility (EMC) such as CISPR 25 [160]. In this dissertation, it is supposed to there exists an EMI filter and well-designing for PCB tracks to reduce common mode and differential mode noises. Therefore, noise considerations are neglected. Other safety standards can be considered as:

• IEC 61851-21: Conductive charging system of an EV charger;

• ISO 6469: Safety specifications of electric road vehicles;

• UL 2202: EV charging system equipment;

• ECE R100: Electric shock;

• etc.

It should be mentioned that the aforementioned standards are for safety. However, in this dissertation, only the DC-to-DC converter is analyzed. Therefore, safety considerations are also ignored.

Simulation Verification of Switching and Conduction Losses model

In order to verify the proposed model, some GaN transistors are selected as case studies. The characteristics of these transistors are listed in It can be observed that the proposed method has sufficient accuracy and errors are lower than 8%. So, the most obvious finding to emerge from this analysis is that the conduction losses can be modeled by interpolated algebraic equations with high precision.

Maxwell Verifications of the Magnetic Design

Suppose that there exists an E64 transformer core with 8 turns for the primary and 8 turns for the secondary sides. For the simile implementation in Maxwell, a planar technology is adopted with an 8-layer PCB with FR4 and 2 turns in each layer as shown in Fig. 4

Fig. 4-19: Maxwell simulation result for leakage inductance measuring

Furthermore, the magnetizing inductance based on the CADES analysis is calculated as 86.76µH. This value is obtained as 84.3827µH using the Maxwell, which is shown in Fig. 4-20.

As a result, the calculated magnetizing inductance using CADES and simulated result using Maxwell are approximately equal to each other and it verifies the accuracy of the implemented code. The above result also shows that the magnetizing inductance from the primary side (i.e. for the analysis of the CLLC resonant converter.

Winding DC Resistance (mOhm) Magnetizing Inductance (uH)

Proposed State Plane Analysis

Introduction

The CLLC resonant converter has to be carefully designed, to provide soft switching and proper control over a wide operating range. In this chapter, a detailed exact analysis approach based on state-plane analysis is proposed for the bidirectional full-bridge CLLC resonant converter. For this purpose, a new formulation of state-space equations is presented and the high-order system of the CLLC resonant converter is decoupled into two lower-order systems.

Based on the proposed method, the exact analysis of the converter in CCM, the voltage and current stresses of the resonant elements and switching components, the gain ratio, soft switching criteria, and the boundary between CCM/DCM are obtained. The accuracy and feasibility of the proposed method are verified by simulation and the results are compared with the FHA method.

This chapter is a slightly modified version of [START_REF] Rezayati | Generalized State-Plane Analysis of Bidirectional CLLC Resonant Converter[END_REF] published in IEEE Transactions on Power Electronics and has been reproduced here with the permission of the copyright holder.

Operating Modes of the CLLC Resonant Converter

The circuit diagram of the bidirectional full-bridge CLLC resonant converter is shown in Fig. 5-1, where the voltage polarities and current directions for the forward power conversion are denoted in red and for the reverse power conversion are denoted in blue. Also, the switches can be either a GaN transistor or a MOSFET. The GaN transistors can operate in both directions and the MOSFETs can operate in the forward direction and utilize in the reverse mode using their anti-parallel diode. Thanks to the symmetric structure of the CLLC resonant the circuit analysis of the converter is the same for both cases. In this chapter, only the forward power transfer mode will be discussed in detail. The analysis of the converter in the reverse power transfer mode can be performed in the same manner just by exchanging the notations as shown in Table 5-1.

S D S S C SS V OUT = V IN S 1 S 3 S 2 S 4 C 1 L 1 C 2 L 2 L m n:1 Gate1 Gate2 Gate2 Gate1 + - v 1 + - v 2 S 5
Table 5-1: Parameters exchanging for forward and reverse modes for using a symmetrical method for both

Forward mode

Reverse mode

VIN ↔ VOUT C1 ↔ C2 L1 ↔ L2 Lm ↔ Lm" (=Lm/n 2 ) L'2 (=n 2 L2) ↔ L1" (=L1/n 2 ) C'2 (=C2/n 2 ) ↔ C1" (=n 2 C1) V'OUT (=nVOUT) ↔ VIN" (=VIN/n)
The primary active full bridge is composed of four GaN transistors. The switching network of the secondary side is also formed in a similar manner as the synchronous rectifier. The resonant tank is comprised of capacitors C1 and C2, the transformer magnetizing inductance Lm, and the primary and secondary leakage inductances L1 and L2. Depending on the desired power flow direction, the bridge of one side acts as an inverter and that of the other side acts as a synchronous rectifier.

Typical waveforms are shown in Fig. 5 To have a single voltage level, the equivalent circuits are referred to their primary sides by defining:

v 1 v 2 i L1 i L2 +V IN +V OUT -V IN -V OUT t v C1 v C2 i L1max i L2max v C2,max v C1,max -v C2,max
22 2 2 2 2 2 2 ' , ' / , ' , ' OUT OUT L n L C C n V nV v nv = = = = (5.1) v C1 V IN C 1 L 1 C 2 ' L 2 ' L m V' OUT (a) + _ i L1 v C2' + _ i L2' i Lm v C1 V IN C 1 L 1 C 2 ' L 2 ' L m V' OUT (b) + _ i L1 v C2' + _ i L2' i Lm v C1 V IN C 1 L 1 C 2 ' L 2 ' L m V' OUT (c) + _ i L1 v C2' + _ i L2' i Lm v C1 V IN C 1 L 1 C 2 ' L 2 ' L m V' OUT (d) + _ i L1 v C2' + _ i L2' i Lm

Fig. 5-3: Equivalent circuit of CLLC resonant converter in subinterval 1 (a), subinterval 2 (b), subinterval 3 (c), and subinterval 4 (d).

There are four subintervals in a cycle during each one the converter operates in a particular mode: Subinterval 1 (t0≤t<t1): This interval begins with a positive-going zero crossing of the secondary current i'L2. The primary current, iL1, is negative and passes through S1 and S4 in the reverse mode operating of the GaNs which leads to v1 = VIN. Also, i'L2 is positive and the synchronous rectifying GaN transistors S5 and S8 are conducting causing v'2 = V'OUT. From the equivalent circuit shown in Fig. 5-3 (a), the state space equations of the CLLC resonant converter during subinterval 1 can be expressed as follow:

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 1 2 2 1 2 1 2 2 1 1 1 2 1 2 1 1 C L C L m m L m m C C m m L m m C C dv t it dt C dv ' t i ' t dt C ' di t L' L L' L L L v v ' v v ' dt di ' t L L L L L L v v ' v v ' dt = = + + = - - - -     + + = - - - -     (5.2)
where Γ is defined as:

1 2 1 2 '' mm L L L L L L  = + + (5.3)
This interval ends when the gate signal "Gate 1" to the transistors S1 and S4 is turned OFF. are ON and the input voltage with reverse polarity is applied to the circuit, so v1 = -VIN. The secondary side current is negative and GaN rectifiers S6 and S7 are ON, hence v'2 = -V'OUT. This interval is similar to subinterval 1 except that the polarity of the input and output voltages are reversed. Therefore, a similar analysis can be performed. This interval ends when the gate signal "Gate 2" to the transistors S2 and S3 is turned OFF.

Subinterval 4 (t3≤t<t4):

The circuit diagram during this interval is illustrated in Fig. 5-3 (d).

In this interval, the transistors S1 and S4 are turned ON under ZVS conditions due to the negative current of the tank at the beginning of this interval. This interval ends when i'L2 reaches zero which provides the ZCS condition for transistors S5 and S8. The state space equations of this subinterval are similar to (2) with v1 = VIN and v'2 = -V'OUT.

The four sets of state-space equations are coupled and the state-plane trajectory cannot be represented in two-or three-dimensional coordinates. Therefore, a new method is given here for the state-plane analysis of the CLLC resonant converter.

Proposed New State-Plane Analysis of the CLLC Resonant Converter

This section provides new decoupled sets of state-space equations for analysis of the CLLC resonant converter. Let us define the new state space variables iLP, iLN, vCP, and vCN as (5.4).

LP P LL LN N LL P CP C C N CN C C i a i i ' i a i i ' v e v v ' v e v v ' =+ =- =+ =- (5.4)
where aP, aN, eP, and eN are constants that depend on circuit parameters as described in Appendix I: ( ) ( )

( ) ( ) ( ) ( ) 2
2 1 1 2 1 1 1 1 CP LP P P P P LP CP P dv i dt C a L' a Lm L a Lm di v v v ' dt L        = + + + + = - - -   (5.6) ( ) ( ) 2 1 1 2 1 1 1 1 CN LN N N N N LN CN N dv i dt C a L' a Lm L a Lm di v v v ' dt L          = + - + - = - - +   (5.7)
where CP = CN = C'2, and LP and LN are the equivalent inductances for P and N states: To generalize the analysis, the quantities are presented in normalized form using the base quantities defined in

( )( ) ( ) ( ) 1
dm j dt LC a L' a Lm L a Lm R dj m R R M dt L          = + + + + = - - -   (5.10) ( ) ( ) 2 1 0 0 0 1 1 1 CN LN NN N N N ,N LN CN ,N ,N N dm j dt LC a L' a Lm L a Lm R dj m R R M dt L          = + - + - = - - +   (5.11)
where m and j denote the normalized voltage and current, respectively.

Table 5- One can see that the behavior of the CLLC resonant converter can be modeled by two series resonant converters which are shown in Fig. 5-4. The parameters AP, BP, AN, and BN are defined as follows:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 1 m P P P m P P P m N N N m N N N A a L' L Lm L B L a L L A a L' L Lm L B L a L L = + +   = + +   = + -   = -- 
(5.12)

In Fig. 5-4, the polarity of mIN and mOUT depends on the particular subinterval. In subinterval 1, the applied voltages to the resonant tank are +Vin and -Vout. Therefore, the state plane diagrams of this subinterval are two circles for SRC-P and SRC-N as shown in Fig. 5-5.

r 1,P a 1 +b 1 M m CP j LP r 1,N a 2 +b 2 M m CN j LN

Fig. 5-5: State plane diagram of Subinterval 1 for SRC-P (left) and SRC-N (right)

In subinterval 2, the polarity of the applied voltage changes to +Vout by changing the polarity of the secondary side current. Therefore, the state plane diagrams of this subinterval are two circles for SRC-P and SRC-N as shown in Fig. 56. In subinterval 3, the polarity of the inverter part changes and -Vin is applied to the tank.

Therefore, the state plane diagrams of this subinterval have also two circles for SRC-P and SRC-N as shown in Fig. 567.

2 2 0 0 CN N N LN M A B M J + - + r2 ( ) 2 2 0 0 CP P P LP M A B M J + - + ( ) 2 2 0 0 CN N N LN M A B M J - + + + O1 PP A B M + NN A B M + O2 PP A B M - NN A B M - O3 PP A B M -- NN A B M - -  O4 PP A B M -+  NN A B M - + 
From Fig. 56789, It can be observed that for the half switching period, the sum of the arc angles of subintervals 1 and 2, i.e., αx + βx, satisfies the following condition for x∈{P,N}.

0 0 2 ,x sw x x x sw x f ,F f F f    + = = = (5.13)
where ω0,x is the base value of angular frequency and Fx is the normalized switching frequency with x∈{P,N}.

 -  -+      P N -→ +→ P N   2 2 2 1 1 2 1 4 4 r r a cos ar  -  -+       P N +→ -→ P N   2 2 2 2 2 1 2 1 2 1 2 4 1 4 r r r b M sin r b M r -     --   -         same P N   2 2 2 2 1 12 2 4 4 r r b M cos b M r -  --        P N +→ -→
In the following section, component stresses are analyzed using the state-plane diagram. For βP<π/2, the peak current values, JLPP and JLPN, can be calculated from Fig. 5-10 (a) as:

( )

1 LP P P P J P r sin  =+ (5.14) ( ) 1 LP N N N J N r sin  =- (5.15)
The radii r2,P and r2,N of the state-plane trajectory in Fig. 5-10 (a), can be found as follow:

( ) ( ) ( )

2
-+ =- (5.20) 2 2 2 2 12 1 1 4 1 4 N ,N ,N LP N N ,N r r A J N r Ar     -+ =- (5.21)
Using (5.20) and (5.21), the peak current of the primary and secondary inductors, IL1,max and IL2,max , can be expressed as:

( ) On the other hand for π/2<βP<π/2+θ0, the maximum value of the primary side current happens at the switching instant. So, JLPP and JLPN should be calculated from (5.20) and (5.21) in order to find iL1,max. The peak value of the secondary current happens at ω0t=π/2. Therefore, JLPP and JLPN should be calculated from (5.26) and (5.27) in order to find iL2,max.

Voltage Stresses

In a steady state, the peak tank capacitor voltages occur when the inductor currents reach zero since the resonant capacitor and inductor are in series on both sides, hence the capacitor voltage is maximum when the inductor current is zero. In this condition, one can find from (5.4) that:

1 2 2 1 2 2 LP P L L L LN N L L L i a i i ' i ' i a i i ' i ' = + = = - = - (5.28)
which means iLP = -iLN or in the normalized form: Using the law of cosines in Fig. 5-11, one can write:

( ) ( ) ( ) 2
2 2 2 1 2 2 2 XN XN N N N N N m m r A B M A B M sin d sin d sin d        = + + - + + (5.30) ( ) ( )
where d=tan -1 (Ibase,N/Ibase,P). Therefore, mXP and mXN are calculated as:

( ) ( ) 2 2 2 2 2 X P P P P P P m A B M cos d cos d r sin d A B M = - - + - - (5.32)
( ) ( ) 2 2 2 2 1 X N N N N N N m A B M sin d sin d r cos d A B M = - + + - + (5.33)
Consequently, the maximum voltage across C1, vC1max, can be calculated by transforming normalized variables from P and N states to the real value.

( )

1 1 XP XN C ,max base ,P base ,N PN V m V m V ee =  +  + (5.34)
On the other hand, the maximum voltage across C2, vC2max, occurs when iL2 reaches zero. So, the normalized voltage can be expressed as:

( )

2 P P P CP P m P r cos A B M  = - + (5.35) ( ) 2 N N N CP N m N r cos A B M  = - - (5.36)
where mCPP and mCPN are the normalized voltages when iL2=0, in the SRC-P and SRC-N, respectively.

Knowing mCPP and mCPN, VC2,max can be obtained as:

( )

2 1 C ,max P CP base ,P N CP base ,N PN n V e M P V e M N V ee =  -  +
(5.37)

Soft Switching Conditions

One of the most important features of resonant converters is the zero voltage switching capability of the devices. For CLLC resonant converter, in most literature first harmonic approximation is used to find the soft switching conditions. In this paper, the accurate stateplane analysis is used to examine exact soft switching conditions.

As shown in Fig. 5-2 at the end of subinterval 1, S1 and S4 conduct, iL1 is positive and the voltage across drain-source capacitors CSS1 and CSS3 are 0 and +VIN, respectively. During the dead time as shown in Fig. 5-12, all inverter switches are OFF and the primary side inductor current flows through the transistor parasitic output capacitances:

V g S 1 S 3 S 2 S 4 C 1 L 1 C 2 L 2 L m 1:n to the rectifier C ss1 C ss3 C ss2 C ss4 V Css1 + _ V Css3 + _ i L1
Fig. 5-12: Circuit condition between subintervals 1 and 2 for the ZVS analysis. Therefore:

13 IN CSS CSS V V V += (5.38)
As the switches are similar to the capacitor current formula i = C(dv/dt) one can find that:

13 1 3 1 0 22 CSS CSS CSS CSS L II I I i + = → = = - (5.39)
Eq. (5.39) implies that the total energy of the two parasitic capacitors remains unchanged during this subinterval, as a result, no power is transferred to the resonant network.

This mode will continue until the voltage across CSS3 is discharged to zero. After that, the GaN transistors S2 and S3 conduct in the reverse direction, therefore ZVS can be achieved. By assuming linear discharging, the minimum dead time to ensure ZVS is given by:

2 0 2 dt dt tt IN CSS IN SS SS SS dt tt dt I dV C V V C C t dt t I = = - =   = (5.40)
In steady state, the state-plane trajectory is symmetrical. Hence [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF];

( ) ( )

1 1 0 s L L I I  =- (5.41)
By substituting (5.41) into (5.40), the minimum dead-time which ensures ZVS can be calculated as:

0 2 IN SS dt t CV t I =  (5.42)
Obviously, the dead-time depends on the input voltage, the drain-source capacitance of inverter switches, and the initial value of primary current which is a function of circuit parameters and the load. The procedure to drive the initial value of resonant elements has been discussed in [START_REF] Lv | Mode analysis and optimum design of bidirectional CLLC resonant converter for high-frequency isolation of DC distribution systems[END_REF].

On the other hand, the maximum allowable dead time for remaining in ZVS is α+θ0 when the iL1 reaches zero. Therefore, the dead-time must be selected as (5.43) to meet operating at the ZVS condition.

Output Characteristic

The output characteristics describe how the converter output voltage depends on the load current. As can be seen in Fig. 5-2 during the half switching period where the secondary side current iL2 is positive, the secondary resonant capacitor voltage increases from its negative peak -VC2,max to its positive peak +VC2,max. This voltage change corresponds to the net charge transferred to the load, hence: The output short circuit current, JSC, can be calculated using the output characteristics by substituting M=0 into (5.46).

( ) 22
4

S P P P N N N SC P N PN CF J e r cos A e r cos A ee         = + - - + (5.47)
It can be seen that the converter's short-circuit current is inherently limited.

Control Plane Characteristics

One of the most informative curves for the analysis and design of resonant converters is the steady control plane characteristics i.e. voltage conversion ratio, M, vs. normalized switching frequency, F. The load characteristics can be superimposed on the converter output characteristics to determine the output voltage vs. switching frequency. For a resistive load satisfying I=V/R or in the normalized form: In subinterval III, the secondary side current iL2 remains at zero for a while and therefore, the resonant components of the secondary side are decoupled from the resonance. This mode is denoted by D and it is shown in Fig. 5 Therefore, the state-space equation of SRC-D during interval III can be expressed as:

J MQ = ( 5
-17. V in Q 1 Q 3 Q 2 Q 4 C 1 L 1 C 2 L 2 L m n:1 C ss to rectifier
1 1 1 1 1 1 1 C L C L dv i dt C V in v di dt L Lm  =    -  =  +  (5.50)
The solution of this second-order system of differential equations describes a circle as:

2 2 2 1 1 1 1 1 C L D v L Lm i A V g C V g     + - + =           (5.51)
with radius AD that depends on the initial conditions as shown in Appendix II.

In the DCM, iL2 is zero and vC2 remains constant at vC2,max. Hence:

2 1 2 2 2 1 0 LP L L P CP C max C C max C P i i i a for P : v v v v v e =  = - =  = (5.52) 2 1 2 2 2 1 0 LN L L E CN C max C C max C N i i i a for N : v v v v v e =  = + =  = (5.53)
By substituting either (5.52) or (5.53) into (5.51), and using the normalizing base quantities listed in Table 5-2, the state plane trajectory for P-type and N-type forms are obtained as below:

( )

2 2 2 2 1 1 2 2 CP C max P LP D P P C L Lm m m e j A e C L   + - - + =       (5.54) ( ) 2 2 22 11 2 2 CN C max N LN D N N C L Lm m m e j A e CL  + + - + =    (5.55)
where ( )

2 2 0 1 1 0 2 2 2 Lx Cx Cx x D x x x j C L Lm m m e A e e C L  +   + - = +       (5.56)
where mCx0, mCx2, and jLx0 are the initial values for x∈{P,N}.

The state plane trajectory for steady-state operation in DCM is shown in Fig. 5 The angles in Fig. 5-19 can be calculated as the equations shown in Table 5-5.

Current stress in DCM

The maximum value of the primary side current occurs at point A in Fig. 13 where, jLP,max, jLN,max are maximum. So, the peak value of the primary side current is: where jLP,max and jLN,max are at point B in Fig. 13, the secondary side current reaches its maximum value when the current passes π/2 after its zero crossing.

Table 5-5: Some angles for the state-plane diagrams of SRC-P and SRC-N in DCM

SRC-P SRC-N 1 1 2 LP , PP P j sin r  - =-  1 1 1 LN , N N N j sin r   - =-  1 1 1 LP , P P P j sin r  - =+   1 1 2 LN , N N N j sin r   - =+   0 1 2 P P CP , P P A B M m cos r  - ++   =     0 1 1 LN , N N j sin r  - =   ( ) ( ) 2 2 2 1 1 2 1 1 1 2 P P P P P P P r A B M F S cos r A B M F  - =   + - - -     - -   1 2 LN ,X N N j sin r  - =  
Therefore, the peak value of the secondary current is obtained as:

( ) (5.60)

Boundary Condition between CCM and DCM

Right before subinterval 1, when iL2 reaches zero if the voltage V2 in Fig. 1 is lower than VOUT, the four bridge rectifier transistors cannot conduct and iL2 remains at zero. Therefore,

112 11 2 1 1 0 L Lm Lm Lm L L L m di L di i i i v v dt dt L = → = → = → =
(5.61) Writing Kirchhoff's Voltage Law in Fig. 567891011121314151617181920:

C 1 L 1 C 2 ' L 2 ' L m + _ V 1 = ±V IN + - i L1 v C1 + - i' L2 =0 v' C2 = v' C2,max i Lm =i L1 V 2 ' + _ + - + - v L1 v' L2 v Lm + - Fig. 5-20:
The circuit diagram of the resonant tank for the DCM condition.

1 1 1 0 Lm CL V v v v -+ + + = (5.62) 2 2 2 ' ' ' 0 Lm LC v v v V - + + + = (5.63)
By solving (5.62) and (5.63), and knowing that vL2 is equal to zero, vC2 is maximum and V1

= VIN, then V2' becomes: (5.64) where vC1 at this instant can be obtained from (5.4) and (5.35):

2 1 2,max 11 '' mm IN CC mm LL V V v v L L L L = + + ++
( )

2 2,max 2 1 '
P P IN PC CP C C PP r A B M V v vv v ee + - - - == (5.65)
By substituting (5.65) into (5.64), V2' would then become:

2 2
2,max 1 1 '

P P P m P P IN C m PP e r A B M e L V V v L L e e + + - - =+ + (5.66)
where vC2,max was defined in (5.37).

The constraint for CCM operation (Vrectifier>VOUT) in normalized form is defined as: In this condition equations ( 22) through [START_REF] Honarmand | Optimal scheduling of electric vehicles in an intelligent parking lot considering vehicle-to-grid concept and battery condition[END_REF] will be used to obtain iL1,max, and iL2,max. The peak capacitor voltages vC1,max and vC2,max , can be calculated from ( 34) and (37). A detailed simulation verification will be done in the next part.

( ) ( )( ) ( ) 2
J LP P_i L2 J LP P_i L1 SRC-P SRC-N J LP N_i L2 J LP N_i L1 i L2max i L1max r 2P r 2P r 2P θ 0 β P =π O 1 =O 3 =0 A N +B N α N -A N -B N r 1N r 1N r 1N δ N

Simulation Verification

Experimental Verification

The state-plane analysis obtained in the previous part must be checked to physically verify the results obtained from the proposed state-plane analysis. Also, the results must be checked with simulation.

Specification of the experimental set up

An experimental prototype is considered to verify the proposed state plane analysis of the CLLC resonant converter. Table 5-6 shows the main specifications of the prototype. 5-6, has been developed to validate the proposed analysis. The prototype is built using GS66508B GaN transistors on both sides. As the synchronous rectification is not a primary concern of the proposed method of analysis and for the sake of simplicity, during testing the circuit in each direction (forward or reverse power transfer) the transistors of the secondary side are bypassed by adding anti-parallel diodes.

3kW bidirectional EV charger experimental set up

The prototypes have been designed for the G2V application with the input voltage of 400V (which can be prepared by a power supply or by the rectified grid voltage (220V AC) and a boost converter) to prepare the output voltages between 250V to 400V and ensure the maximum power of 3.3kW.

Semiconductors

The GS66508B GaN transistor is employed for the inverting parts as the main switches. Low ON resistance and high frequency operating are the main advantages of this transistor. The absolute maximum voltage rating of this transistor is 650V, which is suitable for 250V~450V applications. A cooling system is ensured for the GaN transistors thanks to an external fan for each GaN transistor for reducing their junction temperature and fixing the thermal resistance.

The electrical characteristics of this transistor are listed in fast turn-OFF capability. A ferrite bead is also added in series to the gate circuit for highfrequency noise reduction. 2. A signal with typically 50% duty cycle is applied to a specific circuit (dead-time circuit) to provide the high-side and low-side signals with configurable dead-time.

119

A single pole double through (SPDT) switch is applied to prepare the capability of working with one of the aforementioned capabilities. In this dissertation, the second capability is employed. A configurable circuit is provided to control the dead-time from ≈0ns to 1000ns based on the designer's decision. It is done using a NAND integrated circuit 74VHC132MTCX, and an RC circuit to provide the expected dead-time using two potentiometers. This circuit is entitled the dead-time circuit and is shown in According to the bottom layer shown in Fig. 5-33-(b), it is manifest that the GaN transistor can dissipate the generated heat to the air around it by the uncoated metal plate. Furthermore, it is possible to add a heatsink to the plate for further temperature reduction. In this dissertation, we added a fan at the bottom of each active bridge in order to fix the thermal resistance and reduce the junction temperature of GaN transistors.

Main Waveforms and Comparisons

Using two active bridge converters, one in the role of an inverter and the other in the role of a rectifier, it is possible to build a CLLC resonant converter by connecting them to the designed resonance tank. Fig. 5-31 shows the parts related to the transformer and two resonant inductors.

It should be noted that their values were obtained from the initial design and their values are not optimal. Also, by performing several tests, the accuracy of the implemented converter was evaluated.

The voltage of resonant capacitors, the current of resonant inductors, and the voltage of switching parts have been measured thanks to the THDPO200 differential voltage probe and TCP0030 current probe on KEYSIGHT DSOX3024T 200MHz 5GSa/s oscilloscope. The observed waveforms will be compared with the results obtained by the state-plane analysis.

The main waveforms of resonant elements, i.e. the inductors currents and capacitors voltages, for different operating points are depicted in Fig. 5-34.

To verify the accuracy of the proposed model, the results coming from the proposed analysis are compared with those of the FHA method, time domain exact simulation by PSIM®, and experiments.

State-plane Analysis Experimental Verifications

In this part, the proposed state-plane analysis, which was presented in chapter 3, is examined by the experimental tests. These verifications are as follows.

Output and Control Plane Characteristics As can be seen, the results from the proposed method are well matched with the simulation and experimental results. It can also be seen that the gain voltage predicted by FHA is substantially higher than the actual value, which means that care should be taken when the FHA method is used for the design and optimization of the CLLC converter. The proposed analysis gives exact values of the voltage and current of the components and can improve the power loss estimation and design optimization [START_REF] Rezayati | Interpolation-Based Switching and Conduction Losses Model of GaN Transistors[END_REF].

v C1 v C2 i L1 i L2 v C1 v C2 i L1 i L2 v C1 v C2 i L1 i L2 v C1 v C2 i L1 i L2 (
In Fig. 5-36, the output characteristic for fsw = 150 kHz and Pout = 0.9~1.5 kW is compared.

The results based on the proposed state-plane analysis are consistent with experimental and simulation results in a wide range of load variation, in contrary to those of the FHA method.

Maximum Current and Voltage of Resonant Elements

As previously explained, depending on βP there are three possible trajectories that determine the peak inductors current accordingly. Fig. 5-37 shows waveforms for the forward power transfer operation at three operating points corresponding to each type of trajectory. As can be seen in the above figure, the dead-time is minimum and the switch operates in the boundary condition at ZVS. If a smaller value is applied to tdt, the ZVS will be lost.

Boundary and DCM Operating

Investigating the constraint of (5.67) shows that a CCM to DCM transition happens at F = 1.3 and Q = 0.0356. Also, Fig. 5-41 compares the secondary side current, iL2, obtained by circuit simulation and the proposed state space equations. According to Table 5-8, with the same power and voltage in the forward power transfer mode, all the values resulted from the proposed analysis are well matched with the experimentally measured values, and the errors are much lower than those obtained from the FHA method. As an example for iL1max, the error of the value obtained from SPA is 2.03% while the error of the value given by FHA is 15.59%. Also, for vC1max the errors are 5.16% and 14.73% for SPA and FHA respectively. Fig. 5-43 compare the obtained error based on the simulation, SPA, and FHA method with the experimental results, where the FHA-based error is too high. The experimental results of the reverse power transfer mode, are also tabulated in Table 5-9.

Comparing the results of the proposed analysis with the experimental and simulation results

shows that the proposed analysis is also effective for the reverse power transfer operation.

Furthermore, Fig. 5-44 shows that the maximum efficiency of the converter operating in forward power transfer is 95.72% while it is 95.62% for the reverse power transfer.

Proposed Multi Operating Points Two-

Stage Optimization Method

Introduction

The state plane analysis was evaluated in the previous chapter, which proposed a new method for the exact analysis of the CLLC resonant converter. On the other hand, the sequential quadratic programming was examined to prepare the exact optimization of the CLLC resonant converter. In this chapter, both of these methods will be utilized in this chapter.

Software Environment

The in the analysis [START_REF] Enciu | Automatic differentiation for electromagnetic models used in optimization[END_REF], [START_REF] Delinchant | An optimizer using the software component paradigm for the optimization of engineering systems[END_REF], and [START_REF] System | Vesta CADES[END_REF], especially in the power electronics applications. In this thesis, the CADES software version 4.1.0 is used as shown in Fig. 6-1: In this dissertation, the state-plane analysis equations have been written by C++ and the optimization algorithm has been implemented by SML. The exact Jacobian matrix of a gradientbased optimization problem is determined thanks to the automatic differentiation Adol-C integrated with CADES [START_REF] Enciu | Automatic differentiation for sensitivity calculation in electromagnetism: Application for optimization of a linear actuator[END_REF]. It leads to obtaining the derivatives more efficiently [START_REF] Delhommais | Méthode de pré-design par optimisation en électronique de puissance[END_REF].

Optimization Parameters

In an optimization problem, the variables are input, output, or constraint, where the type of variables in CADES can be defined as free, fixed, interval, parametric, database, uncertain, or objective. Furthermore, there exists an objective function in the optimization process, which must be maximized/minimized, when all constraints are met. For the selected bidirectional EV charger, the following objective functions and variables are defined. It means that if minimizing the total mass is selected as the objective function, the coefficients of each operating point as well as the probability of operating in G2V don't have any effect on the optimizing total mass for multi operating points optimization.

Secondly, if the power density is selected as the objective function, the coefficients of each operating point play an important role in maximizing the power density due to the fact that the output power depends on the operating point. However, the following condition is obtained:

𝑂𝐹 = 𝑃 ∑
In the above fractions, the numerator is constant because it depends on the operating points which are selected as input parameters. So, (6.4) can be rewritten as (6.5):

= max 𝐾 × 1 𝑀𝑎𝑠𝑠 (6.5) = max 1 𝑀𝑎𝑠𝑠 𝑜𝑟 min 𝑀𝑎𝑠𝑠
The above equation expresses that the coefficients of each operating point, as well as the probability of operating in G2V, don't have any effect on the optimization of the power density for multi operating points optimization when the output power is known. Nevertheless, the optimization results for minimizing the mass or maximizing the power density are shown in Appendix III. These results are shown that the optimization algorithm is converged to an unreal minimum mass (i.e. 9.23g for all magnetic parts) which is not true for a 3.3kW on-board charger.

Therefore, a new formulation of the objective function is considered for involving the efficiency and corresponding coefficients. Eq. (6.6) shows the objective function:

𝑂𝐹 = max 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑇𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 (6.6)
For the bidirectional multi operating points optimization, the objective function can be rewritten as: In this dissertation, we consider 4 operating points (N=M=4 in which points A, B, C, and D) with a probability of 80% for G2V applications (i.e. P=0.8). Furthermore, due to the symmetrical structure of the selected bidirectional CLLC resonant converter, charging and discharging modes have the same functionality (i.e. 𝜂 𝐶,𝑗 = 𝜂 𝐷,𝑗 for 𝑗 ∈ {1,2,3,4}). Therefore, the objective function can be determined as: where 𝜆 𝐴 = 0.033 , 𝜆 𝐵 = 0.767 , 𝜆 𝐶 = 0.126 , and 𝜆 𝐷 = 0.074 for 4 operating points optimization and 𝜆 = 1 for single operating point optimization.

𝑂𝐹 = max
𝑂𝐹 = max ∑
In this dissertation, Eq. (6.9) is named as a new version of the objective function. It should be mentioned that the common power density, i.e. the nominal output power divided by the mass/volume, is not considered as the objective function since the output power is limited to 3.3kW and the objective function will be independent of the operating points and their coefficients, as previously shown in (6.3).

Input Parameters

The input parameters are categorized into three categories, i.e. electrical parameters, geometrical parameters, and optimization parameters.

Electrical input parameters

Table 6-1 shows the list of the electrical input parameters. In this table, some inputs are fixed and others can be changed in an interval. For example, the input voltage is fixed and equal to 400V, and the output voltages are also fixed for 4 operating points. According to the selected GaN transistor, the value of its parasitic capacitor can be selected as a fixed value (its nominal value). For the interval inputs, it is assumed that the value can change in an interval with an initial guess, minimum, and maximum value. For example, the resonant circuit capacitor can be chosen from 20nF to 500nF with a 70nF initial guess, which is based on experience. It is noted that the resonant capacitors in the real world are also a discrete parameter, but it is possible to obtain the corresponding capacitance using series and parallel configuration. The switching frequency can be changed from 100kHz to 800kHz (based on the core material).

Coefficient of operating points and G2V probability

The coefficients of each operating point and also the probability of operating in G2V applications are considered as the optimization process parameters, as listed in Table 6-3. It should be mentioned that the coefficients are adopted from [START_REF] Fang | Optimal Design Methodology for LLC Resonant Converter in Battery Charging Applications Based on Time-Weighted Average Efficiency[END_REF], which were obtained based on the weighted average model of the energy of an EV charger based on the LLC resonant converter. Furthermore, the probability of the EV working in the G2V application is considered to be 80%. It can be possible to propose a new approach to find these coefficients for bidirectional CLLC-based on-board charger applications.

Constraints

Optimization constraints can affect the different parts i.e. the input part, the internal part of the optimization code, and the output part. For the input part as an example, the maximum voltage of resonant capacitors (V_MAX_C) can be selected as a constraint, nevertheless, we selected it as an input parameter. In this part, some important constraints are discussed. Among the optimization constraints, zero voltage switching (ZVS) and operating in continuous conduction mode (CCM) are critical.

ZVS constraint

For ZVS, the dead-time value must be met by the following inequality obtained in the previous chapter (see 5.3.2) for all the operating points:

0 10 2 SS in dt L CV t i    + (6.11)
The above equation can be split into the following inequalities: Eq. (6.12) is defined as a ZVS checker constraint, which should be always greater than zero.

Eq. (6.13) is another ZVS constraint that ensures the primary current at the start of subinterval III to be positive.

If (6.12) and (6.13) are met for all the operating points, it can be said that the ZVS condition for primary inverters is definitely met. However, it is better to put a margin so that the dead time implementation is possible. For example, if the amount of dead time is calculated as 10ps, it will not be practically possible to implement. Therefore, we set a margin for it, which is assumed to be 20ns. Therefore, (6.13) is rewritten as follows: Consequently, (6.12) and (6.14) are utilized for ZVS condition checking.

CCM constraint

Another optimization constraint is that the CLLC resonant converter operates in CCM for all the operating points. The CCM operating has the benefit of higher efficiency and lower switching stresses in comparison with DCM operating. Let's consider the following circuit as shown in Fig. 6-4:

V g Q 1 Q 3 Q 2 Q 4 C 1 L 1 C 2 L 2 L m 1:n D 5 D 7 D 6 D 8 V2 V OUT + _ + _

Fig. 6-4: Displaying the required voltages for CCM checking

The condition for operating in CCM is that when the current on the secondary side becomes zero and the diodes want to change state, the voltage V2 should be higher than the output voltage to turn ON the corresponding diodes, i.e.:

where Bmax is the maximum allowable flux density of the selected core material, Bmax_L1, Bmax_L2, and Bmax_T are the maximum flux density of the magnetic components in a specific operating point and they are determined as:

𝐵𝑚𝑎𝑥_𝐿1 = 𝜇 0 × 𝑁 𝐿1 × 𝐼 1 𝑙𝑔_𝐿1 (6.22) 𝐵𝑚𝑎𝑥_𝐿2 = 𝜇 0 × 𝑁 𝐿2 × 𝐼 2 𝑙𝑔_𝐿2 (6.23) 𝐵𝑚𝑎𝑥_𝑇 = 𝑉 𝐼𝑁 4 × 𝑁 1 × 𝐹𝑠𝑤 × 𝐴𝑒 (6.24)
where I1 and I2 are the maximum current of the primary and secondary sides, respectively.

Horizontally fit in the core Fig. 4-2 shows an EE core with the corresponding dimensions listed in Table 4-2. The number of wires in each layer is limited to:

( ) ( ) _ -n w + n-1 t + 2 S 0 2 EF Fit H - =     (6.25)
where E and F are shown in Fig. 4-2, n is the number of turns per layer, w is the diameter of a wire, t is the space between turns, S is the spacing between the ending turns and the core, and

Fit_H is the variable for checking horizontal fitting of the layers in a core and must be checked for the magnetic core of the inductors and transformer. It should be noted that t and S are typically specified by the safety rules, required insulation, and fabrication tolerance.

Vertically fit in the core

The number of layers in the core window should be also limited to:

( ) _ 2 2 0 Fit V D m w S = - +  (6. 26
)
where D is shown in Fig. 4-2, m is the number of layers, and Fit_V is the variable for checking the vertical fitting of the layers in a core and must be checked for the magnetic core of the inductors and transformer.

Maximum voltage of the resonant capacitors

The maximum voltage of each resonant capacitor must be lower than the absolute maximum rating mentioned in its datasheet. In this work, the selected resonant capacitors are MLLC capacitors with a 1kV voltage rating. Therefore, the following constraints must be also met: 𝑉_𝐶1_𝐶ℎ𝑒𝑐𝑘𝑒𝑟 = 𝑉_𝑀𝐴𝑋_𝐶 -𝑣 𝐶1,𝑀𝐴𝑋 (6.27) 𝑉_𝐶2_𝐶ℎ𝑒𝑐𝑘𝑒𝑟 = 𝑉_𝑀𝐴𝑋_𝐶 -𝑣 𝐶2,𝑀𝐴𝑋 (6.28) where V_MAX_C is equal to 1kV, and vC1,MAX and vC2,MAX are the maximum voltage of C1 and C2 which were obtained in the previous chapter based on the state plane analysis.

Output parameters

The main output parameters are shown in Table 6-4, where the objective function is selected as a new version of power density. Interval -0 250

Single Operating Point Optimization

In this part, a single operating point is discussed. As shown in Fig. 6 • Point B: this point operates at the nominal power, where the voltage increases and reaches nominal voltage, Vrated, at the maximum current (IMAX,Vrated).

• Point C: this point occurs when the voltage reaches its maximum value (IMAX,VMAX). At this point, the operating mode of the battery charger changes to CV mode.

• Point D: this point is just before the battery will be fully-charged while the current is decreasing (IMIN,VMAX).

Generally, battery chargers are optimized at one operating point such as Prated or PMAX. For better comparison and in this section, each operating point is selected as the desired point and optimization will be done on it.

Table 6-5 compares the optimization results of different operating points for selecting each of them in the optimization process and obtaining its effect on the other operating points, where the red numbers mean that the constraint has not been met. According to Table 6-5, by considering the minimum voltage (point A) as the operating point of the optimization, the magnetic core will be saturated at the maximum power (point C), point B, and point D, and also the converter temperature exceeds the margin and reaches 90.27°C at point B. Therefore, all the constraints cannot be met by considering the minimum voltage (point A) as the operating point of optimization. Also, Table 6-5 demonstrates that by considering points B, C, or D as the optimization operating point, the maximum junction temperature, and saturation will be exceeded at the other points and the constraint cannot be met. Furthermore, this table shows that the optimization at point D has the minimum mass and the maximum objective function.

It should be noted that the results shown in Table 6-5 are in the imaginary world and it is not possible to fully trust the accuracy of the obtained results. However, Appendix IV shows that all constraints of single operating point optimization cannot be met in the real world as well.

Therefore, it can be concluded from the data in Table 6-5 and also Appendix IV that the limitation of single operating point optimization for EV charger applications is that all the constraints cannot be met completely, and it leads to cause a problem in the performance of the charger. Hence, it is needed to present a new method to address this problem and supplement the single operating points by considering multi operating points. This method will be discussed in the next part.

The Proposed Multi Operating Points Optimization (Algorithms and Transferring from the Imaginary World to the Real World)

The objective function of this dissertation was expressed in (6.9) with the defined parameters in part 6.3. This optimization method is implemented in the imaginary world with continuous parameters. Indeed, the magnetic cores, number of turns, and number of turns in each layer are defined as any positive number, but they are only integer numbers in the real world. It means that there is not any guarantee that the obtained results are an implementable optimum result.

Therefore, this optimization algorithm is a pre-optimization or a pre-design method until now, and it is mandatory to obtain the value of real components, i.e. in the real world. This section comprehensively assesses the concept of transferring from the imaginary world to the real world as well. The variables which must be discrete in the optimization algorithm are:

• Core sizes,

• Number of turns,

• The number of turns in each layer.

The proposed optimization algorithm can be considered as the flowchart diagram drawn in Fig. 6-6.

Set the input parameters Run the optimization in the imaginary world

The core size is discretized

The turn numbers are discretized

The turn per layer is discretized

Select the best result

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

Imaginary world Real world The optimization procedure is described as follows:

• Step 1: the input parameters are defined, which are classified into three categories: electrical parameters, geometrical parameters, and optimization parameters (see section 4.2.1). In this case, 4 operating points have been considered.

• Step 2: the optimization will be performed to find the preliminary results as the predesign process. In this step, the state-plane analysis is carried out to find the exact value of the objective function and constraints in the imaginary world. Then, an optimum charger is obtained in this step. This step is a pre-optimization process, which provides a preliminary phase for power electronics designing to present a rapid and global glance for the pre-design results in a short time as the domain of the imaginary world. It helps the designer to find an approximate range of solutions.

However, it is possible that not only these results do not accurately converge to the optimum point, but also the obtained result cannot be implemented experimentally due to the fact that some non-integer results are not possible (e.g. number of turns).

Therefore, proposing a method for transferring from the imaginary world to the optimum and implementable real world is necessary, which will be done in the next steps.

• The optimization results will be obtained to make a decision for the next step.

• Step 5: this step more attempts to leave the imaginary world by considering the second discrete value as a fixed input. For scenario I (when the core size is fixed in step 3), we consider the number of turns per layer as the second discrete value, above and below the obtained non-integer result. This selection is due to the fact that the core size is known and it is optimum to completely fit the turns vertically as shown in Fig. 67. • Scenario I: the total turn numbers for L1, L2, and the transformer were obtained in scenario I as 6.37, 6.49, and 8.53 for the lower sizes and lower turns per layer, which will be 6, 6, and 8 if we put a floor under them, and 7, 7, 9 if we set a ceiling on them.

Fig. 6-7: Horizontal and vertical arrangements of the turns

For the lower sizes and higher turns per layer, the total turn numbers were obtained as 6.08, 6.10, and 7.38, which will be 6, 6, and 7 if we put a floor under them, and 7, 7, 8 if we set a ceiling on them. Also, the total turn numbers for L1, L2, and the transformer were obtained 5.32, 5.62, and 6.55 for the higher sizes and lower turns per layer, which will be 5, 5, and 6 if we put a floor under them, and 6, 6, 7 if we set a ceiling on them. Finally, for the higher sizes and higher turns per layer, the total turn numbers were obtained as 5.12, 5.13, and 11.7, which will be 5, 5, and 11 if we put a floor under them, and 6, 6, 12 if we set a ceiling on them.

• Scenario II: the number of turns per layer for L1, L2, and the transformer was obtained in scenario I as 0.92, 0.42, and 0.90 for the lower sizes and lower turn numbers, which will be 1 if we put a floor under them (0 is prohibited), and 2 if we set a ceiling on them. For the lower sizes and higher turn numbers, the number of turns per layer was obtained as 1.11, 1.16, and 0.92, which will be 1 if we put a floor under them, and 2 if we set a ceiling on them. Also, the number of turns per layer for L1, L2, and the transformer was obtained at 0.90, 0.53, and 0.90 for the higher sizes and lower turn numbers, which will be 1 if we put a floor under them, and 2 if we set a ceiling on them. Finally, for the higher sizes and higher turn numbers, the number of turns The above tables indicates that the objective function is about 0.303 %/g for GA and 0.240 %/g for SPA method. The size of the magnetic cores is also less than for GA, but it is not possible to meet all constraints. In the other words, the optimum result with all constraints cannot be obtained by genetic algorithm. In this case, the junction temperature exceeds the defined limit and the magnetic cores operate in the saturation region due to the high frequency and high current operating.

Experimental Verification

The optimum result obtained in the previous part must be checked to physically verify that not only the obtained result can meet all the constraints, but also this converter can operate correctly in different operating points with high efficiency in bidirectional power transfer. Two different samples are also built as case studies to compare the optimization data with other samples in the experimental environment to consider any probabilistic unobserved physical phenomenon. The main objectives of this part are as follows:

• Examining the objective function and constraints of different prototypes in multiple operating points;

• Comparing the proposed optimum result with two different scenarios to validate the proposed algorithm. It should be mentioned that case II consists of a smaller standard size for magnetic components to demonstrate that case I is the optimum converter in size while all constraints are also met.

There is no guarantee for case II to reach the maximum power in steady state conditions. The proposed optimization method can be verified easily, but it is better to consider a lower size magnetic core and evaluate the constraints.

Considering a Lower Size than the Optimum Setup

Let's consider an experimental setup with lower standard sizes for the magnetic cores to provide 3.3kW in the full-load conditions (however, there is no guarantee to reach 3.3kW with these magnetic cores). Fig. 6-12 shows the resonant elements of this setup. The parameters of this set up were listed in In this case, the maximum power is limited to 1.6kW. This experimental setup, which is lower than the optimum result in-size, expresses that it is not possible to reach the required power at F>1, and also the temperature of components exceeds the constraint (i.e. reaches 100°C) for F≈1. Therefore, this experimental setup cannot meet all the requirements and constraints. As a result, the aforementioned results were shown that the temperature of all components is below 85°C and therefore the constraints related to the temperature are met. The electrical behavior of the components has a reasonable match with the results obtained by the state plane analysis shown in Chapter 5. Therefore, the proposed multi operating points optimization method is verified.

Summary

In this chapter, an algorithm for the optimization of multi operating points problems in bidirectional EV charging applications was presented. The input parameters, output parameters, and constraints were expressed to use in an optimization algorithm. A single operating point optimization was implemented in CADES Software, and it was shown that the designed converter based on the single operating point cannot meet all the constraints for the other points.

Therefore, the multi operating points optimization algorithm was proposed which can find an optimum converter through 9 steps, when all the constraints are met. It should be mentioned that a new version of the objective function (i.e. efficiency divided by mass) was considered instead of typical power density (i.e. nominal output power divided by mass) since the maximum output power is fixed as 3.3kW, and cannot be maximized or minimized. A 3.3-kW prototyping was built. Experimental results were used to verify the proposed method. Simulation and experimental results reveal the accuracy of the proposed analysis in the forward and the reverse power transfer modes for different power and voltage levels 7. Conclusions and Future Work

Conclusions

This dissertation focuses on the analysis and optimization of the full-bridge CLLC resonant converter for bidirectional EV charger applications. The outcomes of this dissertation are an exact method for analyzing the CLLC resonant converter and other high-order resonant converters as well as a multi operating points optimization algorithm for use in the EV charger applications and other wide-range applications.

Firstly, the types of EVs were introduced and the all-electric vehicle or battery electric vehicle (BEV) was selected due to the limitations of fossil fuels, increase in charging stations, advancement of lithium-ion battery technology, and price reduction of the batteries. Among the requirements of BEVs (i.e. electrical, placement, environmental, and safety requirements), only electrical requirements were considered. Secondly, the PWM-based converters and resonantbased converters were examined, and the resonant converters were selected due to their high efficiency, high power density, high switching frequency, bidirectional power transferring, and soft switching for a wide range of loads. Thirdly, the bidirectional capability was also considered for the selected converter because of beginning the consistent trends of the future, more using renewable energies capabilities, using as an uninterruptable power supply and avoiding blackouts, supplying appliances inside the vehicle, etc.

Among the different resonant converters, the CLLC resonant converter is a good candidate with excellent power density and efficiency bidirectionally and also a wide soft switching region. Nevertheless, none of the previous studies has provided any exact analyzing approach and general optimization method in terms of bidirectional power transferring and EV charging constraints.

Before presenting the proposed methods, it is needed to define the optimization problem and find an approach for its solving. The optimization algorithms can be classified into two categories: stochastic algorithms and deterministic algorithms, where sequential quadratic As an example, the magnetizing inductance based on the CADES analysis was calculated as 86.76µH, and it was obtained as 84.3827µH using Maxwell simulation, which is shown the appropriate accuracy.

In the optimization process, it is obvious that an EV charger operates at different voltage and current levels based on the behavior of Lithium-Ion batteries. In this dissertation was shown that if only one operating point was considered and optimization was done based on that point, all the constraints could not be met for the other operating points. Therefore, a new approach was proposed for multi operating points optimization. Based on this method, the discrete variables were considered as continuous variables and optimization was done in the imaginary world. Then and each time for one discrete variable, the discrete variable will be continuous by putting a floor under it, and setting a ceiling on it. This process continued to find the continuous version of all variables, and finally, the best solution was considered as the optimum converter.

It was EE55 core for the inductors and EE65 core for the transformer, and resonant capacitors equal to 100nF.

Two experimental setups were built to verify the optimization algorithm. One of them was the optimum 3.3 kW prototype obtained based on the proposed algorithm with XL1=55, XL2=55, and XT=65, and the other one was a prototype with a smaller-in-size converter (i.e. XL1=47, XL2=47, and XT=55). It was attempted to the converters operated at same operating points and the constraints were evaluated.

Although the optimum converter has successfully operated in the defined operating points, the smaller converter has certain limitations in terms of the required power and high-frequency operating. It was not possible for the smaller converter to provide 3.3kW and this converter was limited to a lower power. Furthermore, the optimum converter run at 3.3kW and the maximum temperature of magnetic cores reached 81.4°C. But, for the smaller converter, the temperature of the primary inductor exceeds 100°C and contradicts the temperature constraints because the maximum junction temperature had been considered as 85°C in the optimization process.

Furthermore, this overheating leads to the inductor behavior changes and directly varies the resonant frequency. Consequently, the designed converter has the minimum mass when 3.3kW is provided and it is not possible to achieve the output power of 3.3kW and the defined constraints with a smaller converter.

Future Work

Regarding future work on the analysis and optimization of bidirectional EV chargers based on the CLLC resonant converter, the following improvements are suggested:

1. The selected converter was a full-bridge CLLC resonant converter. Adding a halfbridge corresponding analysis can make the analysis detailed.

2. The common CLLC-based resonant converters are typically operated above the resonant frequency, similar to this dissertation. For a comprehensive analysis, it is better to add analysis below the resonant frequency.

3. In this dissertation, the resonant inductors were added externally to the transformer to provide the required leakage inductance. Including the resonant inductors and magnetizing inductor into one resonant network can result in finding the global maximum power density.

4. In the optimization process, transferring from one step to another step requires preexamining by the designer and it is not done automatically. In the new version of this optimization algorithm, defining an automatic process is preferred.

5. The optimization algorithm can be implemented into Matlab for better data analysis. 6. The selected platform for the experimental setups is the conventional EE transformers. Using new technologies such as planar transformers can provide an integrated transformer with corresponding leakage inductors to reduce mass/volume significantly [START_REF] Zou | Modeling and Optimization of an Integrated Transformer for Electric Vehicle On-Board Charger Applications[END_REF]. The CLLC implementation in an integrated magnetic part can be done in future work. Also, the anti-parallel junction capacitance of the main switches can be considered in the resonant tank as future work. Eq. ( 4) has always two complex poles, so its reverse of Laplace transform in the time domain can be written as the following equation: In the time domain we have: At the end of this region, iL2 will be zero, and iL1=iLm.

                        = - + + + (A.3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
( ) ( ) ( ) 1
In region B: using iL2=0 and iL1=iLm, we have: Eq. ( 16) has also two complex poles, so using the reverse Laplace transform we have: For other resonant elements, the time domain equations in region B are obtained as: (

( ) ( ) ( ) ( ) 1
( )

1
     - = - - + -                         + - - + - +                                - = - - + -         4 1 3 1 11 1 2 r r B B out s s P cos q P sin q V qZ                      + - - - -                           (0.28)
For the sake of simplicity, we define: r q r q r q r q r q r q r q r q r q r q r q r q

r q r q rq a Y Y b Y Y a Y Y b Y Y
a q S S b q S S a q S S b q S S c Z S q Z S d Z S q Z S c Z S q Z S d Z S q Z S c q Z q Z Y q Z Y d q Z Y q Z Y c q Z Y q Z Y d q Z q = + + = -

f Z S V V q Z S V V f Z S V V q Z S V V f q Z Y V V q Z Y V V q Z V f q Z Y V V q Z Y V V q Z V f Z V Y sin S cos q S cos q qY sin q f Z V       = - + - - = - + + - = + + - - = + - - + = - + - = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )

1 1 1 1 3 1 1 1 1 4 1 1 1 1
2 2 ut r r q q out r r q q out r r q q Y sin S cos q S cos q qY sin q The two above tables demonstrate that by considering an operating point as the point of the optimization, all constraints cannot be met for the other operating points. Therefore, a single operating point is not a good candidate for optimizing a bidirectional EV charger for the wide range of voltages and currents. The proposed multi operating point optimization method solved this issue.

f q Z V Y cos S sin Y cos q S sin q f q Z V Y

x 6 . 3 . 4 .

 634 Output parameters ....................................................................................................... 6.4. SINGLE OPERATING POINT OPTIMIZATION .......................................................................... 6.5. THE PROPOSED MULTI OPERATING POINTS OPTIMIZATION (ALGORITHMS AND TRANSFERRING FROM THE IMAGINARY WORLD TO THE REAL WORLD) ........................................ 6.6. NUMERICAL CLARIFICATION OF THE PROPOSED OPTIMIZATION PROCESS ......................... 6.7. COMPARISON BETWEEN THE OPTIMIZATION RESULTS ........................................................ 6.8. EXPERIMENTAL VERIFICATION ........................................................................................

Fig. 1

 1 Fig. 1-1: Annual CO2 emission from fossil fuels and industry. Source: Global Carbon Project [3] ......

- 1 :Fig. 1 - 2 :Fig. 1 - 3 :Fig. 1 - 4 :Fig. 1 - 5 :Fig. 1 - 6 :Fig. 1 - 7 :Fig. 2 - 1 :Fig. 2 - 2 :Fig. 2 - 3 :Fig. 2 - 4 :Fig. 2 - 5 :Fig. 2 - 6 :Fig. 2 - 7 :Fig. 2 - 8 :Fig. 2 - 9 :Fig. 2 - 10 :Fig. 2 - 11 :Fig. 2 - 12 :Fig. 2 - 13 :Fig. 3 - 1 :Fig. 3 - 3 : 34 Fig. 3 - 4 : 35 Fig. 3 - 5 : 35 Fig. 3 - 6 : 37 Fig. 3 - 7 : 39 Fig. 3 - 8 : 40 Fig. 3 - 9 : 41 Fig. 4 - 1 : 44 Fig. 4 - 2 : 47 Fig. 4 - 3 : 49 Fig. 4 - 4 : 52 Fig. 4 - 5 : 54 Fig. 4 - 6 : 56 Fig. 4 - 7 : 62 Fig. 4 - 8 : 64 Fig. 4 - 9 : 65 Fig. 4 - 10 : 67 Fig. 4 - 11 : 67 Fig. 4 - 12 : 68 Fig. 4 - 13 : 72 xviFig. 4 - 73 Fig. 4 - 75 Fig. 4 - 77 Fig. 4 - 17 : 79 Fig. 4 - 83 Fig. 4 - 19 : 83 Fig. 4 - 20 : 84 Fig. 4 - 84 Fig. 4 - 85 Fig. 5 - 1 : 88 Fig. 5 - 2 : 89 Fig. 5 - 3 : 90 Fig. 5 - 4 : 93 Fig. 5 - 5 : 94 Fig. 5 - 6 : 94 Fig. 5 - 7 : 95 Fig. 5 - 8 : 95 Fig. 5 - 9 : 96 Fig. 5 - 10 : 98 Fig. 5 - 11 : 101 Fig. 5 - 12 : 102 Fig. 5 - 104 Fig. 5 - 105 xviiFig. 5 - 106 Fig. 5 - 107 Fig. 5 - 17 : 108 Fig. 5 - 18 : 108 Fig. 5 - 19 : 110 Fig. 5 - 20 : 112 Fig. 5 - 21 : 113 Fig. 5 - 113 Fig. 5 - 23 : 114 Fig. 5 - 115 Fig. 5 - 25 : 116 Fig. 5 - 118 Fig. 5 - 118 Fig. 5 - 28 : 119 Fig. 5 - 29 : 120 Fig. 5 - 30 : 120 Fig. 5 - 31 : 121 Fig. 5 - 32 : 121 Fig. 5 - 33 : 122 Fig. 5 - 34 : 124 Fig. 5 - 124 Fig. 5 - 125 xviiiFig. 5 - 37 : 126 Fig. 5 - 38 : 127 Fig. 5 - 39 : 128 Fig. 5 - 40 : 128 Fig. 5 - 41 : 129 Fig. 5 - 42 : 129 Fig. 5 - 43 : 131 Fig. 5 - 44 : 131 Fig. 6 - 1 : 133 Fig. 6 - 2 : 134 Fig. 6 - 3 : 135 Fig. 6 - 4 : 142 Fig. 6 - 5 : 146 Fig. 6 - 6 : 149 Fig. 6 - 7 : 150 Fig. 6 - 8 : 163 Fig. 6 - 9 : 164 Fig. 6 - 10 : 165 Fig. 6 - 11 : 166 Fig. 6 - 12 : 166 xixFig. 6 - 13 : 167 Fig. 6 - 167 Fig. 6 - 168 Fig. 6 - 168 Fig. 6 - 17 : 169 Fig. 6 - 18 : 170 Fig. 6 - 19 :

 112131415161721222324252627282921021121221331333434353535363737393840394141444247434944524554465647624864496541067411674126841372473475477417794834198342084484485518852895390549355945694579558955996510985111015121025104510551065107517108518108519110520112521113511352311451155251165118511852811952912053012053112153212153312253412451245125537126538127539128540128541129542129543131544131611336213463135641426514666149671506816369164610165611166612166613167616761686168617169618170619 Fig. 1-1: Annual CO2 emission from fossil fuels and industry. Source: Global Carbon Project [3] ......

  -9 for SRC-N diagram δP The defined angle in Fig. 5-9 for SRC-P diagram λC Optimization coefficient for G2V operating λD Optimization coefficient for V2G operating λN The defined angle in Fig. 5-9 for SRC-N diagram λP

  Fig. 1-2 [8].

Fig. 1 -Fig. 1 - 3 :

 113 Fig. 1-3: The main parts of an all-electric vehicle or battery electric vehicle. *Source: www.afdc.energy.gov

Fig. 1 - 4 :

 14 Fig.1-4: Number of charging stations installed in France from 2012 to 2021[START_REF] Zhang | High-Efficiency High-Power-Density CLLC Resonant Converter With Low-Stray-Capacitance and Well-Heat-Dissipated Planar Transformer for EV On-Board Charger[END_REF]

Fig. 1 - 6 :

 16 Fig. 1-6: The conceptual diagram of power flow direction for G2V, V2G, and V2H

Fig. 1 - 7 :

 17 Fig. 1-7: Different operating points to reach the global or local maximum.

Fig. 2 - 2 :

 22 Fig. 2-2: Basic concept of a wired charger (left side) and wireless charger (right side).

Fig. 2 - 4 :

 24 Fig. 2-4: Typical structure of a two-stage charger.

Fig. 2 - 5 :

 25 Fig. 2-5: Basic structure of a full-bridge resonant converter.

Fig. 2 - 11 :

 211 Fig. 2-11: The applied voltages to the resonant tank of an LLC resonant converter.

Fig. 2 - 13 :

 213 Fig. 2-13: Charging process of a lithium-ion cell in constant current (CC) and constant voltage (CV) processes.

Fig. 3 - 1 :Fig. 3 - 2 :

 3132 Fig. 3-1: The before and after voltages of the primary side (top) and secondary side (bottom)

Fig. 3 - 6 :

 36 Fig. 3-6: The current of L1(top) and L2 (bottom) (red curves) and their FHA model (blue curves)

Fig. 3 - 7 :

 37 Fig. 3-7: The voltage of C1(top) and C2 (bottom) (red curves) and their FHA model (blue curves)

Fig. 3 - 9 :

 39 Fig. 3-8: The switching instance for Lm=58.9µH: switch voltage (brown), switch current (green), and gate-to-source voltages (red and blue)

Fig. 4 -

 4 [START_REF] Balat | Status of Fossil Energy Resources: A Global Perspective[END_REF] shows the corresponding functions and the operating region (yellow region) for a one-variable optimization function (n=1).

4. 5 . 1 .

 51 Fig. 4-2: Geometrical diagram of E-shape core

Fig. 4 - 3 :

 43 Fig. 4-3: Continuous models for some variables using interpolation for parameter X (a); parameter C (b); and the volume of the core (c).

45 E 61 F

 4561 8.708×10 -7 × X 4 +6.652×10 -5 × X 3 +0.000437× X 2 +0.3899×X+1.634 C = 4.9110×10 -6 × X 4 -0.0009099×X 3 +0.06433× X 2 -1.512×X+15.99 D = 4.832×10 -6 × X 4 -0.0009239×X 3 +0.06026×X 2 -1.232×X +13..464×10 -6 × X 5 -0.0002725×X 4 +0.0198×X 3 -0.7021×X 2 +12.79×X-81.36D = F/2 E = -0.0001884×X 3 +0.02389×X 2 -0.03055×X +7.= A -E Mass = 0.08002×X 4 -1.03×X 3 +5.869×X2 -10.19×X+5.938 Volume = 0.002014×X 5 -0.3591×X 4 +24.2×X 3 -754.4×X 2 +1.11×10 4 ×X-6.115×104

Fig. 4 -Fig. 4 - 4 :

 444 Fig.4-4illustrates the switching loss test circuit of the GS66508B GaN transistor[START_REF] Gan | GS66508B Bottom-side cooled 650 V E-mode GaN transistor Preliminary Datasheet[END_REF]. It is noted that this circuit can be utilized for other GaN transistors with different parameters.

  Fig.4-4) with the base parameters (shown in Table4-4) is simulated on LTSPICE and the base value of energy loss is obtained as (4.4) by performing the double pulse tester.

Fig. 4 -

 4 [START_REF] Chrisafis | France to ban sales of petrol and diesel cars by 2040[END_REF] shows the switching energy as a function of ID and Tj, where the switching loss dramatically increases by the temperature and current increments.

Fig. 4 - 5 :

 45 Fig. 4-5: Switching losses based on the variation of junction temperature and drain-to-source current for a GaN transistor operating at the base values

Fig. 4 -

 4 [START_REF]Green Roadmap[END_REF]. MATLAB interpolations for the switching losses evince that, the lost energy can be modeled independently as a thirdorder polynomial function with an acceptable root-mean-square error (RMSE). It should be mentioned that the loss modeling of some parameters shown in Fig.4-6 can be modeled as a first or different-order equation, for example, the variation of switching energy with the negative gate driver voltage, VgNeg, is almost constant and can be modeled as a constant value, but all the type II parameters have been considered as a third order equation for the sake of simplicity. So, the switching energy losses can be modeled as the same equation as (4.6) for each type II parameter.

Fig. 4 - 6 :

 46 Fig. 4-6: Switching losses based on considering the type II parameters for a GaN transistor operating at the base value. Variation of drain-to-source voltage from 50V to 650V (a), the gate resistor for OFF state (b), the gate resistor for ON state (c), the gate-to-source voltage for OFF condition (d), the gate-to-source voltage for ON condition (e), the parasitic inductor for a gate pin with considering internal gate inductance (f), Ciss (g), Coss (h), and Crss (i)capacitors effects consideration for switching losses. The red line, blue line, and black line are OFF switching loss, ON switching loss, and total switching loss, respectively.

( 4 . 8 )Fig. 4 -

 484 Fig.4-6 shows the ON switching loss, OFF switching loss, and total switching loss for a GaN transistor tested under the base values. According to Fig.4-6, it is shown that ONswitching energy losses dominate the total energy losses[START_REF] Huang | Analytical loss model of high voltage GaN HEMT in cascode configuration[END_REF] but for this analysis, OFFswitching energy losses have been also considered. Fig. 4-6 (a) depicts that the switching loss extremely depends on the drain-to-source voltage. As shown in Fig. 4-6 (b), by increasing the OFF gate resistor, the turn-off switching loss slightly increases. Fig. 4-6 (c) shows that the turnon switching loss reduces by decreasing the gate resistance [139]. According to Fig. 4-6 (d),

3 + 7 .

 37 𝑆𝑊 | 𝐺𝑆66508𝐵 = 𝑓 𝑠𝑤 × 53.3𝜇 × (0.2413 + 4.617 × 10 -2 × 𝐼 𝐷 + 3.943 × 10 -3 × 𝑇 𝑗 -2.048 × 10 -6 × (𝐼 𝐷 + 0.4755 × 𝑇 𝑗 ) 7370 × 10 -9 × (𝐼 𝐷 + 0.2121 × 𝑇 𝑗 )

  [START_REF]Green Roadmap[END_REF] shows these values for the selected GaN transistors.Two values of ON resistances, R1 and R2, have been assigned to these temperatures as[START_REF] Marinescu | Toward a practical solution for residential RES based EV charging system[END_REF]

Fig. 4 - 7 :

 47 Fig. 4-7: Cross-section area of a circular wire and its skin depth

63

 63

  d is the wire diameter.Finally, the copper loss is calculated by multiplying the resistance by the RMS current to the square:

  typically mentioned in the capacitor datasheet at different frequencies.Furthermore, the following equation can be used for ESR estimation:

Fig. 4 -Fig. 4 - 8 :

 448 Fig.4-8 shows a typical EE-shape magnetic circuit with N turns, and air gap length g:

Fig. 4 -

 4 Fig. 4-9: The flowchart for core selection

Fig. 4 -Fig. 4 -

 44 Fig. 4-10: Minimum dead-time versus the magnetizing inductance

12 [

 12

Fig. 4 - 13 :

 413 Fig. 4-13: Effect of different constraints on J-M curve: (a) voltage; (b) current; (c) power; (d) the complete operating region

Fig. 4 -

 4 Fig. 4-14: Thermal modeling of a semiconductor

Fig. 4 -

 4 Fig.4-15: Effect of temperature on different materials 3C90 (left circuit)[START_REF] Ferroxcube | Datasheet, 3C90 Material Specification[END_REF], and 3F4

53 )Fig. 4 -

 534 Fig. 4-16: The DC bias characteristic of the CGA6N1C0G3A822J230AE capacitor * Source: product.tdk.com

Fig. 4 -

 4 Fig.4-17, there is also a DC link capacitor to stabilize the DC link voltage. However, these parts are neglected in our analysis.

Fig. 4 -

 4 Fig. 4-18: 2-D Maxwell implementation of a transformer with 8 turns

Fig. 4 - 20 :

 420 Fig. 4-20: Maxwell simulation results of magnetizing inductance

  Fig. 4-21: 3-D Maxwell implementation of the planar transformer with 8 turns

Fig. 4 -

 4 Fig. 4-22: 3-D flux density distribution obtained by Maxwell simulation

Fig. 5 - 1 :

 51 Fig. 5-1: Schematic diagram of the full-bridge bidirectional CLLC resonant converter. The red and blue signs are corresponding to the forward and the reverse power conversion, respectively.

Fig. 5 - 2 :

 52 Fig. 5-2: Typical CLLC resonant converter waveforms for the above resonant frequency in the forward power transfer mode

Subinterval 2 ( 1 .

 21 t1≤t<t2): Fig. 5-3 (b) shows the circuit in this subinterval. At the beginning, the gate signals force all the inverter side transistors to be OFF. The transistors S2 and S3 are turned ON during this subinterval with zero voltage switching (ZVS), and conduct in the reverse mode, hence v1 = -VIN. This subinterval ends when i'L2 reaches zero. Consequently, the zero current switching (ZCS) is obtained for rectifying GaN transistors S6 and S7. The state space equations of this subinterval are similar to (5.2) knowing that v1 has the opposite polarity from subinterval Subinterval 3 (t2≤t<t3): As shown in Fig. 5-3 (c), in this subinterval, the switches S2 and S3

5 )

 5 In which K= C2/C1 represents the ratio of resonant capacitors. By substituting (5.4) into (5.2), the new decoupled state space equations denoted by P and N are obtained.

2 :DefinitionFig. 5 -Fig. 5 - 4 :

 2554 Fig. 5-4 (a) and (b) represent the equivalent circuits described by (5.10) and (5.11) [161], which are denoted by the P-type series resonant converter (SRC-P) and the N-type series resonant converter (SRC-N), respectively.

Fig. 5 - 6 :

 56 Fig. 5-6: State plane diagram of Subinterval 2 for SRC-P (left) and SRC-N (right)

Fig. 5 - 7 :Fig. 5 - 8 :Fig. 5 - 9 :

 575859 Fig. 5-7: State plane diagram of Subinterval 3 for SRC-P (left) and SRC-N (right)

5. 3 . 1 .Fig. 5 -Fig. 5 - 10 :

 315510 Fig. 5-10 shows three possible trajectories depending on the value of βP which is the arc angle of subinterval 1. θ0 is the angle corresponding to the time difference between the zero crossing of iL1 and iL2.

23 )

 23 For βP>π/2+θ0, as shown in Fig.5-10 (c) the peak values JLPP and JLPN can be obtained as:

Fig. 5 - 11 :

 511 Fig. 5-11: The jLP-jLN state-plane trajectory for calculating vC1max.

Fig. 5 -

 5 Fig. 5-13: The ZVS operating region as a function of voltage gain and dead-time equivalent

44 )Fig. 5 -Fig. 5 -

 4455 Fig. 5-14 shows the output characteristics of the CLLC resonant converter for different normalized switching frequencies.

. 48 )Fig. 5 - 5 . 4 .

 48554 Fig. 5-15: Constant stresses diagram based on the different resonant elements. (a) jL1 and jL2 are fixed; (b) jL1 and mC1 are fixed; (c) mC1 and mC2 are fixed; (d) jL2 and mC2 are fixed.

Fig. 5 -

 5 Fig. 5-16: Typical CLLC resonant converter operation in the discontinuous conduction mode during a half switching period

Fig. 5 - 17 :

 517 Fig. 5-17: The subinterval of III for DCM operating

Fig. 5 - 18 :

 518 Fig. 5-18: The resonant tank model in subinterval III of DCM operating

  Fig. 5-19: The complete state-plane analysis of the CLLC resonant converter in DCM

1 LP

 1

3 .

 3 Voltage stress in DCMIt can be shown that the peak voltage of the resonant capacitors C1 and C2 in DCM occurs at points C and D respectively. At these points, the voltage across the C1 and C2 can be calculated:

Fig. 5 -

 5 Fig. 5-21 shows the control plane characteristics of the CLLC resonant converter and the DCM boundary. This map is useful for design purposes as the designer can overlay the

Fig. 5 - 21 :

 521 Fig. 5-21: Control plane diagram (M-F) with CCM/DCM boundaries for different values of Q.

Fig. 5 -

 5 Fig. 5-22: State-plane analysis of SRC-P and SRC-N in the boundary condition.

114

 114

Fig. 5 -

 5 Fig. 5-23-(a) illustrates the time domain modeling based on the state plane analysis for the primary current of the CLLC resonant converter. The corresponding state plane analysis, i.e. SRC-P and SRC-N, are also shown in Fig. 5-23-(b) and Fig. 5-23-(c), respectively.

Fig. 5 - 23 :Fig. 5 -

 5235 Fig. 5-23: The time domain waveform obtained by the state plane analysis of iL1 (a), and the state plane analysis of SRC-P (b), and SRC-N (c).

Fig. 5 -Fig. 5 -

 55 Fig. 5-25: The 3.3kW bidirectional EV charger experimental set up

Fig. 5 -

 5 [START_REF] Honarmand | Optimal scheduling of electric vehicles in an intelligent parking lot considering vehicle-to-grid concept and battery condition[END_REF] demonstrates the complete diagram of the gate driver.

Fig. 5 - 1 .

 51 Fig. 5-26: The -3V and +6V circuit diagram using PES1-S5-S9-M-TR to supply gate driver

Fig. 5 - 28 :Fig. 5 - 120 Fig. 5 - 29 :Fig. 5 - 30 :Fig. 5 - 31 :

 5285120529530531 Fig. 5-28: The dead-time circuit using 74VHC132MTCX NAND IC and RC circuits

Fig. 5 -

 5 Fig. 5-33: The top layer (a) and bottom layer (b) of the final PCB of the dual active fullbridge for using as the inverting and rectifying parts

Fig. 5 -

 5 Fig.5-35 compares the control plane characteristics obtained from the state-plane analysis, FHA method, simulation, and experiment results when the switching frequency changes from

Fig. 5 - 34 :

 534 Fig. 5-34: Experimental waveforms of CLLC resonant converter in CCM for (a) F = 1, Pout = 1.2 kW, (b) F = 1.5, Pout = 960 W, (c) F = 2, Pout = 790 W, and (d) F = 3, Pout = 570 W.

Fig. 5 -

 5 Fig. 5-36: M-J diagram of the CLLC resonant converter.

Fig. 5 - 127 Fig. 5 -Fig. 5 -Fig. 5 -

 5127555 Fig. 5-37 : Current and voltage of the resonant elements for the forward power transferring at: (a) π/2<βP at Pout = 700 W, Vout = 254 V, fsw = 250 kHz, (b) π/2<βP<θ0 at Pout = 1 kW, Vout = 307 V, fsw = 170 kHz, and (c) βP>π/2+θ0 at Pout = 1.4 kW, Vout = 360 V, fsw = 120 kHz.

Fig. 5 - 41 :Fig. 5 -

 5415 Fig. 5-41: The secondary side current in the boundary condition by; (a) simulation, and (b) state-plane analysis.

Fig. 5 -

 5 Fig. 5-43: Error comparison of the different parameters based on simulation, SPA, and FHA method in comparison with the experimental results for (a) POUT=1281W and POUT=527W

Fig. 5 -

 5 Fig. 5-44: Efficiency for the forward and reverse power transferring.

  closed-form state-plane analysis-based equations are implemented in a specific optimization platform, named the Component Architecture for Design of Engineering Systems (CADES) software, which was first developed in the Grenoble Génie Electrique Laboratoire (G2ELab). The CADES software was further expanded by the Vesta System Company, and now has provided the capability of problem formulation, fast and exact optimization, and results

Fig. 6 - 1 :

 61 Fig. 6-1: The diagram of CADES Software

1 2 3 Fig. 6 - 2 :

 362 Fig. 6-2: The default environment of the CADES software

6. 3 . 1 . 1 )Fig. 6 - 3 :+ ( 1 -( 6 . 2 )

 31163162 Fig. 6-3: The probability of operating in G2V and V2G in the EV's lifespan

  𝜂 𝐶 and 𝜂 𝐷 are the efficiency of the selected operating point in charging (C) and discharging (D) modes.

Fig. 6 - 5 :

 65 Fig. 6-5: Selection of 4 operating points based on the lithium battery charging process

Fig. 6 - 6 :

 66 Fig. 6-6: The proposed optimization algorithm for multi operating point optimization in EV charging applications

Step 3 :• Step 4 :

 34 this step initially attempts to leave the imaginary world by considering one of the discrete values as a fixed input value, where scenario I is the selection of the core size, scenario II is the selection of total turn numbers, and scenario III is the selection of the number of turns per layer. The obtained non-integer result is discretized by selecting the real integer values above (like ceil function) and below (like the floor function) of the obtained non-integer result. At the end of this step, one of the three discrete parameters is actualized in the real world and only two continuous parameters remain. the optimization runs again and the corresponding constraints are checked to find an optimum result while one of the discrete values is a fixed input parameter.

6. 8 . 1 .

 81 Specifications of the Experimental PrototypeTwo experimental prototypes are considered to verify the proposed optimization algorithm.

Fig. 6 - 8 :

 68 Fig. 6-8: Two experimental prototypes. Case I is the optimum set up and Case II is the set up with a lower mass

  The resonant tank contains two parts: a magnetic part and resonant capacitors. The magnetic parts are made by EE-shape cores with 3C90 material. The value of leakage inductance and magnetizing inductance is measured by the impedance analyzer KEYSIGHT E4990A 20 Hz-120MHz with Keystone 16047E terminal for F ≤ 100MHz (see Fig.6-9).

Fig. 6 - 9 :Fig. 6 -Fig. 6 -

 6966 Fig. 6-9: The magnetic transformers (left figure) and impedance analyzer to measure leakage inductance and magnetizing inductance

Fig. 6 - 12 :

 612 Fig. 6-12: Resonant tank of the experimental set up with a lower size

Fig. 6 -Fig. 6 - 13 :

 6613 Fig. 6-13: The output current (Iout) versus the output voltage (Vout) of the smaller experimental set up

Fig. 6 -

 6 Fig. 6-14: M-F curve of the smaller experimental set up

Fig. 6 -

 6 Fig. 6-16: The primary inductor current (red), the secondary inductor current (green), the primary capacitor voltage (blue), and the secondary capacitor voltage (orange) for the smaller in-size experimental set up

6. 8 . 4 .

 84 Considering the Optimum Setup Now, let's consider the 3.3 kW optimum experimental setup to evaluate it in the different operating points. The parameters of this set up were listed in Table 6-16 case I. The first operating point is considered as F=1. The temperature distribution of the components is shown in Fig. 6-17-(a). Also, the placement of the component is shown in Fig. 6-17-(b). Furthermore, the corresponding waveforms of the resonant elements and semiconductors are shown in Fig. 6-17-(c) and (d), respectively.

Fig. 6 - 17 :

 617 Fig. 6-17: Some important experimental results at F=1: (a) temperature distribution in a thermal diagram, (b) location of the components for better investigation of the thermal diagram, (c) voltage of resonant capacitors and current of resonant inductors, (d) drain-tosource and gate-to-source voltages of GaN transistor, and voltage and current of SiC diode.

Fig. 6 - 18 :Fig. 6 - 19 :

 618619 Fig. 6-18: Some important experimental results at F=2: (a) temperature distribution in a thermal diagram, (b) location of the components for better investigation of the thermal diagram, (c) voltage of resonant capacitors and current of resonant inductors, (d) drain-tosource and gate-to-source voltages of GaN transistor, and voltage and current of SiC diode.

4 ) 5 )

 45 For decoupling the parameters P and N, it is mandatory to put the coefficients of unwanted parameters to zero. For (A.1) and (A.2), the coefficients of iLN and vCN, and for (A.3) and (A.4), the coefficients of iLP and vCP must be set equal to zero: By solving (62), the parameters aP, aN, eP, and eN are obtained as shown in(5.5).

3 )

 3 By simplifying, we can obtain the following equation in the Laplace domain as follows:

  the secondary side current, the following equation in the Laplace domain is:

32 )

 32 Now we have 4 equations with 5 unknown parameters. The fifth equation can be defined by knowing the fact that at t=α, iL2=0. So, , the initial conditions of the resonant elements can be obtained by solving these If minimizing the total mass or maximizing the power density is considered as the objective function, the optimization results can be determined as:

Table 1 -

 1 1:Different standards voltage and current levels [13] ................................................................ 3

Table 2 -

 2 1: Basic comparisons between analyzing methods .................................................................. 28

Table 2 -

 2 2: Previous methods and parameters done for CLLC resonant converter ................................ 30

Table 3 -

 3 1: Electrical specifications of the prototype converter ............................................................. 37

Table 3 -

 3

2: The maximum and RMS values of iL1 and iL2 obtained by the PSIM simulation and FHA model ............................................................................................................................................. 38

Table 3 -

 3

3: The maximum and RMS values of iL1 and iL2 obtained by the PSIM simulation and FHA model ............................................................................................................................................. 39 Table 3-4: FHA method and PSIM simulation comparisons for two different output powers ............. 42 Table 4-1: The main properties of some optimization algorithms [114] ............................................... 46 Table 4-2: Important dimensional of the conventional E cores and planar E cores .............................. 48

Table 4 -

 4

3: Continuous equations for the core sizes of conventional and planar E cores ...................... 50

Table 4 -

 4

4: Based values of the proposed model for switching loss calculation .................................... 52

Table 4 -

 4

5: List of coefficients for the proposed method for switching loss calculation, Type I (Eq. (4.5)) and Type II (Eq. (4.6)) Variables .................................................................................................. 55

Table 4 -

 4

6: Two values of temperatures and resistance mentioned in the GaN transistors datasheet .... 59

Table 4 -

 4

7: Thermal resistance of some semiconductors ....................................................................... 74

Table 4 -

 4

8: Specifications of the GaN transistors ................................................................................... 81

Table 4 -

 4

9: Scenarios for the Accuracy Verification of Proposed Method for the Switching Losses Calculation .................................................................................................................................... 81

Table 4 -

 4 10: ON Resistance Comparison with the Proposed Model and LTSPICE Simulation ............ 82

Table 5 -

 5

1: Parameters exchanging for forward and reverse modes for using a symmetrical method for both ................................................................................................................................................ 88 Table 5-2: Base values for the parameters of SRC-P and SRC-N ......................................................... 93

Table 5 -

 5

3: Radius (r) and center (O) of the SRC-P and SRC-N arcs .................................................... 96

Table 5 -

 5

4: Some angles for the state-plane diagrams of SRC-P and SRC-N in CCM .......................... 97 xxi

Table 5 -

 5 5: Some angles for the state-plane diagrams of SRC-P and SRC-N in DCM ........................ 111

Table 5 -

 5 6: Components of the Experimental CLLC Resonant Converter ........................................... 115

Table 5 -

 5 7: Electrical characteristics of GS66508B transistor ............................................................. 117

Table 5 -

 5 8: Comparison of the Measurement (Meas.), PSIM Simulation (Sim.), State-Plane Analysis (SPA), and First Harmonic Approximation (FHA) Results for Forward Power Transfer and two output powers. ............................................................................................................................. 130

Table 5 -

 5 9: Comparison of the Measurement (Meas.), PSIM Simulation (Sim.), State-Plane Analysis (SPA), and First Harmonic Approximation (FHA) Results for Reverse Power Transfer and two output powers. ............................................................................................................................. 130

Table 6 -

 6 1: Electrical input parameters for optimization ...................................................................... 139

Table 6 -

 6

2: Geometrical input parameters for optimization ................................................................. 140

Table 6 -

 6

3: Optimization parameters, which are the coefficients of operating points and G2V probability ..................................................................................................................................................... 141

Table 6 -

 6

4: Some important output parameters .................................................................................... 145 Table 6-5: Single operating point optimization results in the imaginary world, by considering a different operating point and checking the objective function and constraints on all points ..................... 147 Table 6-6: The optimization results in step 2, when all the discrete parameters are in the imaginary world ..................................................................................................................................................... 152

Table 6 -

 6

7: Input parameters for optimization in step 4 when one of the discrete parameters in step3 is transferred into the real world as a fixed parameter .................................................................... 153

Table 6 -

 6

8: The optimization results in step 4, when one of the parameters was in the real world and two of them were in the imaginary world .......................................................................................... 154

Table 6 -

 6

9: Input parameters for optimization in step 6 when two discrete parameters in step5 are transferred into the real world as a fixed parameter and only one parameter is free ...................

[START_REF] Vishay | Application Note, Power Dissipation in High Precision Vishay Sfernice Chip Resistors and Arrays (P Thin Film, PRA Arrays, CHP Thick Film)[END_REF]

Table 6 -

 6

10: The optimization results in step 6, when two parameters were in the real world and one of them was in the imaginary world ................................................................................................ 155

Table 6 -

 6

11: Input parameters for optimization in step 8 when all the three discrete parameters in step 6 are transferred into the real world ............................................................................................... 157

Table 6 -

 6

12: The last optimization results (step 8), when all the three parameters were in the real world ..................................................................................................................................................... 158

Table 6 -

 6 13: Number of parameters and running time comparison for two optimization process ....... 160

Table 6 -

 6

14: four operating points optimizations based on FHA and SPA methods ............................ 161

Table 6-15: Optimization results based on the genetic algorithm (GA) and sequential quadratic programming (SQP) for 4 operating points optimizations .......................................................... 162 xxii

Table 6 -

 6 16: Specifications of two experimental prototypes ................................................................ 164

Table 6 -

 6 17: General comparisons between some references and the proposed method ..................... 172 Table 0-1: Single operating point optimization results in the imaginary world when the maximum power density is selected for the objective function .............................................................................. 200 Table 0-1: Single operating point optimization results in the real world when a lower size is selected for the magnetic parts, by considering a different operating point and checking the objective function and constraints on all points ........................................................................................................ 202 Table 0-2: Single operating point optimization results in the real world when a higher size is selected for the magnetic parts, by considering a different operating point and checking the objective function and constraints on all points .......................................................................................... 203

	xxiii

xxv

List of Symbols

  Ability to use the energy of braking mode and return braking energy to the vehicle; • Smoother and more controllable movement.

	• Ability to use different types of energy such as solar and wind;
	• Ability to use different storage systems to store electrical energy such as batteries,
		supercapacitors, and flywheels;
	•
		8000
		7000									6750
		6000
	EVs)	5000
	(1000	4000
	EV Sales	2000 3000							2082	2276	3240
								1263
		1000	125	208	321	543	792
		0
			2012	2013	2014	2015	2016	2017	2018	2019	2020	2021
							Year
		Higher efficiency;
	• Less environmental pollution;
	• Less noise;

Table 1 - 1 :

 11 Different standards voltage and current levels[START_REF] Syed Muhammad | Review of Electric Vehicle Technologies, Charging Methods, Standards and Optimization Techniques[END_REF]

	Standards	Phase	Level/Mode	Voltage (V)	Current (A)
		AC Single	Mode 1	120	16
	IEC 62196	AC Single AC Single	Mode 2 Mode 3	240 250	32 32-250
		DC	Mode 4	600	400
		AC Single	Mode 1	120	16
	IEC 61851	AC Single	Mode 2	240	80
		DC	Mode 4	200-450	80
		AC Single	Level 1	120	16
	SAE J1772	AC Single DC	Level 2 Level 1	240 200-450	32-80 80
		DC	Level 2	200-450	200
	As shown in Table 1-1 for 240V sourced by AC voltage, the IEC62196 offers a current of
	32A, whereas the current is 80A for IEC61851 and it will be 32A~80A for SAEJ1772. In this
	dissertation, a 400V fixed DC voltage source will be considered, which is obtained by 240V
	AC rectified and boosted. It means that the selected EV charger would be Mode 2 of IEC62196,
	Mode 2 of IEC61851, and AC Level 2 and DC Level 1 SAEJ1772. The specifications of the
	selected EV charger are similar to the typical Level 2 charger. In the United States and since
	2021, over 80% of electric vehicle supply equipment ports were Level 2 [14]. There exists also

Table 2 -

 2

		1: Basic comparisons between analyzing methods
	Method	Accuracy Implementation	Steady-state	Transient	Visual inspection	Control capability
	FHA
	Time-
	domain

Table 3

 3

-1 contains the specifications of a CLLC resonant converter:

Table 3 -

 3 1: Electrical specifications of the prototype converter

	Symbol	Quantity

Table 3 -

 3

	2: The maximum and RMS values of iL1 and iL2 obtained by the PSIM simulation and
			FHA model
			Maximum values			RMS values
	Parameter
		Sim.	FHA	Error	Sim.	FHA	Error
	iL1 [A]	20.039	18.198	9.19%	12.351	12.865	-4.16%
	iL2 [A]	14.511	15.561	-7.24%	9.953	10.995	-10.47

The above table describes that the error for the maximum value reaches 9.19% and it is 10.47% for the RMS value. Therefore, the designed or optimized converter may deviate from the desired converter. It should be mentioned that these errors were obtained in one specific operating point, and there is a possibility for the error to be increased in different operating points.

3.4.2. Voltage Evaluation based on FHA

Fig. 3-7 demonstrates the voltage of primary and secondary capacitors, vC1 and vC2 in the time domain:

Table 3 -

 3

			Maximum values			RMS values
	Parameter
		Sim.	FHA	Error	Sim.	FHA	Error
	vC1 [V]	89.47	97.04	-8.46%	65.04	68.62	-5.50%
	vC2 [V]	74.31	82.99	-11.68%	52.21	58.68	12.39%

3: The maximum and RMS values of iL1 and iL2 obtained by the PSIM simulation and FHA model

Table 3

 3

		-1, tdt=100ns, and COSS=30pF (parasitic junction capacitance
	of the transistor), the maximum allowable magnetizing inductance for ZVS operating is as
	determined in (3.17).
	𝐿𝑚| 𝑚𝑎𝑥 =	𝑡 𝑑𝑡 16 × 𝐶 𝑂𝑆𝑆 × 𝑓 𝑠𝑤	=	100𝑛 16 × 30𝑝 × 200𝑘	= 1.04 𝑚𝐻	(3.17)

It means that choosing Lm lower than 1.04mH leads to definitely operate in ZVS. Therefore, Lm is selected as 58.9µH (based on

Table 3

 3

-1) for ZVS checking.

Table 3 -

 3 4: FHA method and PSIM simulation comparisons for two different output powers

			POUT=1281W			POUT=527W
	PARAMETER
		SIM.	FHA	ERROR	SIM.	FHA	ERROR
	Vout [V]	347.3	347.3	0.00	216.8	216.8	0.00
	Iout [A]	3.69	3.69	0.00	2.43	2.43	0.00
	fsw [kHz]	129.3	146	12.14	272	330	10.74
	iL1max [A]	9.07	7.47	15.59	6.35	4.63	23.47
	iL2max [A]	5.46	5.80	12.62	4.55	3.82	13.18
	vC1max [V]	102.91	82.29	14.73	28.85	22.58	16.37
	vC2max [V]	72.58	63.85	18.66	22.76	18.61	19.09
	As shown in

Table 3

 3

-4, the error reaches as high as 23.47% for iL1max, 13.18% for iL2max, 16.37% for vC1max, and 19.09% for vC2max. Furthermore, when the power decreases and the converter operates near the DCM condition, the error increases dramatically. Hence, the first harmonic analysis is not very precise in comparison with the time-domain analysis. It should be mentioned that the high errors can affect the constraints, and the optimum results may not be achieved. Furthermore, it can also affect the designing process and causes miscalculation (such as inductor operating in saturation mode, etc.). Consequently, presenting a new method for analysis of CLLC resonant converters is essential, which will be presented in Chapter 5.

  The new variable x i.e. xk is obtained by solving (4.3).

	max	𝑦(𝑥 𝑘 ) + ∇𝑦(𝑥 𝑘 ) 𝑇 (𝑥 -𝑥 𝑘 ) +	1 2	(𝑥 -𝑥 𝑘 ) 𝑇 ∇ 2 𝑦(𝑥 𝑘 )(𝑥 -𝑥 𝑘 )
	𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜	𝑔(𝑥 𝑘 ) + ∇𝑔(𝑥 𝑘 ) 𝑇 (𝑥 -𝑥 𝑘 ) ≥ 0	(4.3)
		ℎ(𝑥 𝑘 ) + ∇ℎ(𝑥 𝑘 ) 𝑇 (𝑥 -𝑥 𝑘 ) = 0

Table 4 -

 4 2 contains information about the size of some E-type cores.

Table 4 -

 4 2: Important dimensional of the conventional E cores and planar E cores

	Parameter X	Effective Length (mm)	Effective Area (mm2)	a (mm)	b (mm)	c (mm)	d (mm)	e (mm)	h (mm)
		37.6	20.1	16	8.2	4.7	5.7	11.3	4.7
		42.8	31.2	20.7	10	5.3	6.3	12.8	5.2
	Core	49 67	39.5 60	25.4 30.8	9.65 15	6.35 7.3	6.4 9.7	18.8 19.5	6.35 7.2
	E
	Conventional	80.7 97 88.9 124	100 178 234 353	35 43 46.9 56.2	17.5 21 19.6 27.5	10 15.2 15.6 21	12.5 14.8 12.1 18.5	24.5 29.5 32.4 37.5	10 12.2 15.6 17.2
		147	540	65	32.8	27.4	22.2	44.2	20
		149	683	70.5	33.2	32	21.9	48	22
		20.7	14.5	14	3.5	5	2	11	3
		24.3	39.5	18	4	10	2	14	4
	E Core	32.5 41.4	78.5 129	21.8 31.75 6.35 20.32 3.18 24.9 6.35 5.7 15.8 3.2 16.8 5
	Planar	52.4 61.1	194 225	38.1 8.26 25.4 4.45 30.23 7.6 43.2 9.5 27.9 5.4 35.4 8.1
		80.6	305	58.4 10.5 38.1	6.5	50	8.1
		79.9	511	64	10.2 50.8	5.1	53.8 10.2

Table 4 -

 4 3.

Table 4 - 4 :

 44 Based values of the proposed model for switching loss calculation

	Parameter	Symbol	Value Unit
	Bus Voltage	VBUS	400	V
	Drain-Source Voltage	VDS	≈ 400 V
	Drain Current	ID	15	A
	Junction Temperature	Tj	25	o C
	Gate Resistor for OFF state	RgOff	1	Ω
	Gate Resistor for ON state	RgOn	10	Ω
	Gate Series Inductor	Lgate	1	nH
	Positive Gate-Source Voltage	VgPos	6	V
	Negative Gate-Source Voltage	VgNeg	-3	V
	GaN Series Inductor	Lex	10	pH
	PCB Track Series Inductor	LP	2	nH
	GaN Input Capacitor	Ciss	242	pF
	GaN Output Capacitor	Coss	65	pF
	GaN Reverse Capacitor	Crss	1.5	pF

Table 4 -

 4 5. It should be mentioned that ID is the drain current at the switching instant and it is neither RMS nor average current.

Table 4 -

 4 5: List of coefficients for the proposed method for switching loss calculation, Type I (Eq. (4.5)) and Type II (Eq. (4.6)) Variables

	Para.	Type	Coefficients	RMSE a Range
	ID,Tj	Type I	P1=2.4130×10 -1 , P2=4.6170×10 -2 , P3=3.9430×10 -3 , P4=-2.0480×10 -6 P5=4.7550×10 -1 , P6=7.7370×10 -9 , P7=2.1210×10 -1	0.0444	<31A, <150°C
	VDS	Type II	P1=1.0267×10 -9 , P2=7.1253×10 -7 , P3=2.1950×10 -3 , P4=-5.4802×10 -2	0.0046	<650V
	RgOff Type II	P1=1.8403×10 -7 , P2=2.5648×10 -4 , P3=5.3940×10 -4 , P4=1.0012	0.0019	<12Ω
	RgOn Type II	P1=8.1053×10 -6 , P2=-4.2601×10 -4 , P3=2.9752×10 -1 , P4=7.3991×10 -1	0.0052	<20Ω
	VgNeg Type II	P1=4.9380×10 -6 , P2=3.0799×10 -4 , P3=2.2815×10 -3 , P4=1.0048	0.0018	-3V< <0V
	VgPos Type II	P1=-5.0753×10 -2 , P2=8.2630×10 -1 , P3=-4.6115, P4=9.8830	0.0097	3.5V< <6.5V
	Lpara Type II	P1=1.0814×10 -5 , P2=-1.0120×10 -4 , P3=-1.3194×10 -2 , P4=1.0269	0.0228	<34nH
	Lgate	Type II	P1=3.6372×10 -5 , P2=-1.1871×10 -3 , P3=4.9717×10 -3 , P4=9.9182×10 -1	0.0067	<35nH
	Lex	Type II	P1=1.0997×10 -9 , P2=-1.2736×10 -6 , P3=7.6525×10 -4 , P4=9.9240×10 -1	0.0010	<2nH
	Ciss	Type II	P1=3.4789×10 -9 , P2=-2.2825×10 -6 , P3=6.1659×10 -4 , P4=9.3160×10 -1	0.0045	<422pF
	Coss	Type II	P1=4.3957×10 -8 , P2=-8.0368×10 -6 , P3=2.1937×10 -3 , P4=8.8031×10 -1	0.0015	<110pF
	Crss	Type II	P1=1.5521×10 -4 , P2=-4.6845×10 -4 , P3=3.3020×10 -2 , P4=9.4955×10 -1	0.0015	<7pF

a. Root mean square error b) Type II (One Variable Functions): Some simulations are carried out on LTSPICE for the variation of other parameters shown in Table

4

-5, noted by type II, and the switching energy losses are recorded for each case (see Fig.

Table 4

 4 Finally, the switching loss energy of GaN transistors, ELoss, can be calculated as (4.7). Crss} as listed in Table4-5, and Ebase=53.3 µJ as shown in Eq. (4.4).In summary, a specific device model in LTSPICE was considered and the impact of all parameters variation on the losses was studied. Therefore, for switching loss calculation, the GaN parameters are needed to calculate the Eb for each parameter as (4.6) with the coefficients shown in Table4-5. Eq. (4.7) calculates switching losses for any types of electrical, thermal, and device parameters. The switching power loss of a GaN transistor, Psw, can be calculated by multiplying equation (4.7) by the switching frequency, fsw, as (4.8).

	E	Loss	=	( ) ,X ( ) D j b XU E I T b E       		E	base	(4.7)
	where U={Type II parameters: VDS, RgOff, RgOn, VgNeg, VgPos, Lpara, Lgate, Lex, Ciss,
	Coss, and sw P	Loss = E	f	sw

-5 and X ∈{VDS, RgOff, RgOn, VgNeg, VgPos, Lpara, Lgate, Lex, Ciss, Coss, and Crss}.

Table 4 -

 4 6: Two values of temperatures and resistance mentioned in the GaN transistors

			datasheet
	GaN device	T1 (°C)	R1 (mΩ) T2 (°C)	R2 (mΩ)
	GS66502B	25	200	150	516
	GS66506T	25	67	150	175
	GS66508B	25	50	150	129
	GS66516T	25	25	150	65
	IGOT60R070	25	55	150	100
	IGT60R19	25	140	150	260
	TP65H150LS	25	150	150	307
	TP65H035G4	25	35	170	72
	TP65H300G	25	240	150	492
	GPI8HINOIC	25	175	150	Unknown
	PGA26E07B	25	56	150	110
	EPC2105C	25	3.2	125	5.1

Table 4

 4

-6.

Table 4 -

 4 7: Thermal resistance of some semiconductors

	Component	Type	Rth [°C/W]
	GS66508B	GaN transistor	0.5 to 24
	TP65H035G4WS	GaN transistor	0.8 to 40
	C3M0015065K	SiC transistor	0.35 to 40
	APT60N60BCSG	COOLMOS	0.29 to 62
	IRF840	N-channel Mosfet	1 to 62.5
	C4D10120E	SiC diode	0.9
	1N5819	Schottky diode	88
	where the thermal resistance changes based on the selected PCB trace width, heatsinks, and
	cooling systems.

According to

(4.45)

, it is concluded that the junction temperature can be decreased by decreasing PLoss and/or RJA. PLoss reduces by decreasing the switching frequency, changing the circuit parameters, and operating at different points. Also, RJA can be decreased by adding a heatsink or fan. In this dissertation, it is assumed that RJA is fixed and only PLoss is optimized in the optimization process. The temperature constraint for semiconductors is defined as (4.46).

𝑇 𝐽 < 85°𝐶

Table 4

 4

-8.

Table 4 - 8 :

 48 Specifications of the GaN transistors

	GaN Device	Ciss [pF]	Coss [pF]	Crss [pF]	Rg [Ω]	Lg [nH]	LD [nH]	LS [nH]
	GS66502	65	17	0.5	2.3	0.7	170	215
	GS66506	195	49	1.5	1.1	2.27 124	73
	GS66508	242	65	1.5	1.13 1	200	1200
	GS66516	520	130 4	0.34 3.1	370	150
	IGOT60R070	380	72	0.3	0.78 5	1500	1300
	IGT60R19	157	28	0.15 0.27 4.3	1300	800
	TP65H150LS	576	36	2	U	U	U	U
	TP65H300G	1500	147 5	U	U	U	U
	GPI8HINOIC	760	16	2	U	U	U	U
	PGA26E07B	405	71	0.4	0.6	U	U	U
	EPC2105C	980	710 18	0.3	U	U	U
	For all parameters undefined in Table 4-8 ("U"), the used values in Table 4-4 for the reference
	case have been considered.
	4.4.1. Verifying the Switching Power Loss Model

Table 4 -

 4

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shows the results obtained from the proposed model and LTSPICE simulations, where some scenarios have been considered to verify its precision for a variety of GaN transistors:

Table 4 - 9 :

 49 Scenarios for the Accuracy Verification of Proposed Method for the Switching

	Losses Calculation

  -2, assuming that the converter operates in the CCM and that the secondary side H-bridge acts as a diode rectifier, where the gate signals of transistors S5~S8 are applied at the i'L2 zero crossing, i.e. t2 and t4.

			t
			t
	dead-time	dead-time
	Gate 1	Gate 2	Gate 1
			t
	t 0	t 1 t 2	t 3 t 4 =t 0 +T s
		T s

Table 5

 5

			-2.
	The normalized state space equations of the CLLC resonant converter can be expressed as
	(5.10) and (5.11).
	CP	1
			LP
		PP
	LP	0	,P	P	2	( 1	P	)	1	( 1	P	)
			CP					0	,P			0	,P
		P

Table 5 -

 5

							SRC-P							SRC-N
	r1	(	-	M	CP	0	P A B M P + +	)	2	+	J	2 LP	0	(	)

3: Radius (r) and center (O) of the SRC-P and SRC-N arcs

Table 5 -

 5 4: Some angles for the state-plane diagrams of SRC-P and SRC-N in CCM

	Angle			Value	Condition
	 	P N	cos	1	2 21 2 4 r r ar 2 4 a	2

Table 5 - 6 :

 56 Components of the Experimental CLLC Resonant Converter

	Parameter	Value	Parameter	Value
	S1~S8	GS66508B GaN	L1	25 µH
	D1~D8	C4D10120 SiC	L2	25 µH
	C1	99 nF	Lm	125 µH
	C2	99 nF	Vout	250~400 V
	CO	220 µF	Vin	400 V
	n	1:1	fsw	90~350 kHz

A 3.3-kW bidirectional CLLC resonant converter with the specifications, as listed in Table

Table 5

 5

-7.

  Timedomain representation is the experimental results. In this figure, the numbers on the box are the comparison between the peak values for experiments, simulations, state-plane analysis, and the FHA method. For the FHA method, fsw is 268.5 kHz, 193 kHz, and 132.8 kHz respectively.

										Exp=34V
			v C1							Sim=42.1V SPA=42.1V
										FHA=31.7V
									Exp=31V
									Sim=31.9V
			v C2						SPA=31.4V FHA=25.9V
									Exp=6.85A
									Sim=7.24A
			i L1						SPA=7.23A FHA=5.30A
									Exp=4.6A
			i L2						Sim=4.78A SPA=4.76A
									FHA=4.33A
					(a) Pout=700W, Vout=254V
	[Iout/Ibase]	0.35 0.4	v C1						Experimental Exp=64.5V Proposed Method Sim=71.4V SPA=71.3V Simulation Exp=54.5V FHA FHA=53.2V
	Normalized Output Current	0.15 0.2 0.25 0.3	i L2 i L1 v C2							Sim=4.95A SPA=4.93A FHA=5.21A Exp=4.5A Sim=8.55A SPA=8.55A FHA=6.40A Exp=8.05A Sim=51.6V SPA=51.5V FHA=42.5V
		0.66 0.1	0.68	0.7	0.72	0.74	0.76	0.78	0.8	0.82
					Normalized Gain Voltage [Vout/Vin] Exp=109V
									Sim=113V
			v C1						SPA=112V FHA=96V
									Exp=86.5V
									Sim=80.0V
			v C2						SPA=79.3V FHA=73.9V
									Exp=8.85A
									Sim=9.14A
			i L1						SPA=9.12A FHA=7.95A	Sim=5.69A Exp=5.85A
										SPA=5.65A
										FHA=6.11A

Table 5 -

 5 [START_REF] Irle | Global EV Sales for 2022 H1[END_REF] compares the results obtained from the proposed state-plane analysis, exact circuit simulation, and the FHA method for two operating points when the converter operates in the forward power transfer mode. In this table, the error of a parameter X is defined by: Value Error Value Error Value Value Error Value Error Value Error VOUT [V] 347.3 347.3 0.00 347.3 0.00 347.3 0.00 216.8 216.8 0.00 216.8 0.00 216.8 0.00 IOUT [A] 3.69 3.69 0.00 3.69 0.00 3.69 0.00 2.43 2.43 0.00 2.43 0.00 2.43 0.00 fsw [kHz] 130.2 129.3 0.69 129.3 0.69 146 12.14 298 272 8.72 272 8.72 330 10.74 iL1max [A] 8.85 9.07 2.49 9.03 2.03 7.47 15.59 6.05 6.35 4.96 6.35 4.96 4.63 23.47 iL2max [A] 5.15 5.46 6.02 5.43 5.44 5.80 12.62 4.4 4.55 3.41 4.55 3.41 3.82 13.18 vC1max [V] 96.5 102.91 6.64 101.48 5.16 82.29 14.73 27 28.85 6.85 29.30 8.51 22.58 16.37 vC2max [V] 78.5 72.58 7.54 72.09 8.16 63.85 18.66 23 22.76 1.04 22.57 1.87 18.61 19.09

			𝐸𝑟𝑟𝑜𝑟 =	|𝑋 𝑀𝐸𝐴𝑆𝑈𝑅𝐸𝑀𝐸𝑁𝑇 -𝑋| 𝑋 𝑀𝐸𝐴𝑆𝑈𝑅𝐸𝑀𝐸𝑁𝑇	× 100%			(5.70)
	Table 5-8: Comparison of the Measurement (Meas.), PSIM Simulation (Sim.), State-Plane
	Analysis (SPA), and First Harmonic Approximation (FHA) Results for Forward Power
			Transfer and two output powers.
			Power = 1281W			Power = 527W
	Parameter	Meas.	Sim.	SPA	FHA	Meas.	Sim.	SPA	FHA
		Value Value Error

Table 5 -

 5

		9: Comparison of the Measurement (Meas.), PSIM Simulation (Sim.), State-Plane
	Analysis (SPA), and First Harmonic Approximation (FHA) Results for Reverse Power
				Transfer and two output powers.
			Power = 3025W			Power = 918W
	Parameter	Meas.	Sim.	SPA	FHA	Meas.	Sim.	SPA	FHA
		Value Value Error Value Error Value Error Value Value Error Value Error Value Error
	VOUT [V] 385	385 0.00 385 0.00 385 0.00 212.5 212.5 0.00 212.5 0.00 212.5 0.00
	IOUT [A] 7.85 7.85 0.00 7.85 0.00 7.85 0.00 4.32 4.32 0.00 4.32 0.00 4.32 0.00
	fsw [kHz] 105 107.4 2.28 107.5 2.28 110 4.76 199.5 196.9 1.30 196.9 1.30 220.5 10.53
	iL1max [A] 14.17 13.99 1.27 13.93 1.69 13.92 1.76 9.76 10.13 3.79 10.11 3.59 8.02 17.83
	iL2max [A] 11.76 11.93 1.44 11.89 1.10 12.33 4.85 7.09 7.67 8.18 7.65 7.90 6.79 4.23
	vC1max [V] 226.50 216.5 4.41 217.13 4.13 203.43 10.18 65.04 66.82 2.74 67.25 3.40 58.51 10.04
	vC2max [V] 185.33 184.75 0.31 184.55 0.42 180.23 2.75 58.44 55.55 4.94 55.45 5.11 49.56 15.19

  𝜆 𝐴 𝑉 𝐴 𝐼 𝐴 + 𝜆 𝐵 𝑉 𝐵 𝐼 𝐵 + 𝜆 𝐶 𝑉 𝐶 𝐼 𝐶 + 𝜆 𝐷 𝑉 𝐷 𝐼 𝐷 𝑀𝑎𝑠𝑠 𝜆 𝐴 𝑉 𝐴 𝐼 𝐴 + 𝜆 𝐵 𝑉 𝐵 𝐼 𝐵 + 𝜆 𝐶 𝑉 𝐶 𝐼 𝐶 + 𝜆 𝐷 𝑉 𝐷 𝐼 𝐷 𝑀𝑎𝑠𝑠

	𝑁 𝑗=1	𝜆 𝐶,𝑗 ×	𝑃𝑜𝑢𝑡 𝐶,𝑗 𝑀𝑎𝑠𝑠 𝐶,𝑗	𝑀 + (1 -𝑃) ∑ 𝜆 𝐷,𝑗 × 𝑗=1	𝑃𝑜𝑢𝑡 𝐷,𝑗 𝑀𝑎𝑠𝑠 𝐷,𝑗
	𝑁
	= ∑
	𝑗=1
	+ (1 -𝑃) ∑

𝑀 𝑗=1

  𝜆 𝐵 = 0.767, 𝜆 𝐶 = 0.126, and 𝜆 𝐷 = 0.074 are the coefficients of points A, B, C, and D, respectively, which are selected based on the values presented in[START_REF] Fang | Optimal Design Methodology for LLC Resonant Converter in Battery Charging Applications Based on Time-Weighted Average Efficiency[END_REF] for an EV charger based on the LLC resonant converter, 𝜂 𝐴 , 𝜂 𝐵 , 𝜂 𝐶 , and 𝜂 𝐷 are the efficiency of points A, B, C, and D, which are obtained based on the state-plane analysis present in the previous chapter, and Mass is the total mass of the charger, which is obtained based on the continuous modeling of the magnetic cores represented in 4.5.

	𝜆 𝑗 × 𝜆 𝐴 × 𝜂 𝐴 + 𝜆 𝐵 × 𝜂 𝐵 + 𝜆 𝐶 × 𝜂 𝐶 + 𝜆 𝐷 × 𝜂 𝐷 𝜂 𝑗 𝑀𝑎𝑠𝑠 4 𝑗=1 𝑀𝑎𝑠𝑠 operating points optimizations is defined in (6.9): = max where 𝜆 𝐴 = 0.033, Consequently, the objective function for single operating point optimization and multi (6.8) 𝑂𝐹 = { max 𝜆 𝐴 × 𝜂 𝐴 + 𝜆 𝐵 × 𝜂 𝐵 + 𝜆 𝐶 × 𝜂 𝐶 + 𝜆 𝐷 × 𝜂 𝐷 (4 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡𝑠) 𝑀𝑎𝑠𝑠 max 𝑀𝑎𝑠𝑠 (𝑠𝑖𝑛𝑔𝑙𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡) 𝜆 × 𝜂 (6.9)

Table 6 -

 6

	3: Optimization parameters, which are the coefficients of operating points and G2V
		probability
	Input variable	Description	Type	Value
	λA	Coefficient of the first operating point	Fixed	0.033
	λB	Coefficient of the second operating point	Fixed	0.767
	λC	Coefficient of the third operating point	Fixed	0.126
	λD	Coefficient of the fourth operating point	Fixed	0.074
	P	Probability of operating in G2V	Fixed	0.8

Table 6 -

 6

		4: Some important output parameters
	Output variable	Description	Type	Initial guess	Min Max
	PD_Mass	Efficiency divided by total mass [100%/g]	Maximize 0.1	-	-
	PD_Vol	Efficiency divided by total volume [100%/cm 3 ]	Free	-	-	-
	PD_M	Nominal output power divided by mass [W/g]	Free	-	-	-
	PD_V	Nominal output power divided by volume [W/cm 3 ]	Free	-	-	-
	Efficiency Efficiency of the converter [100%]	Interval	-	94	100
	Junc_GaN Junction temperature of GaN [°C]	Interval	-	-40	85
	Junc_D	Junction temperature of diode [°C]	Interval	-	-40	85
	N_T	Total number of turns for the transformer Free	-	-	-
	N_L1	Total number of turns for L1	Free	-	-	-
	N_L2	Total number of turns for L2	Free	-	-	-
	iL1_max	The maximum current of L1 [A]	Interval	-	0	25
	iL2_max	The maximum current of L2 [A]	Interval	-	0	25
	vC1_max	The maximum voltage of C1 [V]	Interval	-	0	1000
	vC2_max	The maximum voltage of C2 [V]	Interval	-	0	1000
	Bmax_T	Maximum flux density of transformer [mT]	Interval	-	0	250
	Bmax_L1	Maximum flux density of L1's core [mT]	Interval	-	0	250
	Bmax_L2	Maximum flux density of L2's core [mT]

Table 6 -

 6 5: Single operating point optimization results in the imaginary world, by considering a different operating point and checking the objective function and constraints on all points

	Optimization results at	Parameter	Selected point for optimization Point A Point B Point C Point D
		C1 (nF)	282.186	500	162.82	66.28
		L1 (µH)	3.124	1	3.0729	7.549
			30.18	24.22	23.27	21.68
		X [L1,L2,T]	28.66	22.73	22.31	20.13
			38.38	38.43	24.66	24.88
			254.97	196.89	179.7	100
		lg [L1,L2,T] (µm)	197.6	147.43	135.9	100
			100	100	100.3	100
			3.69	3.07	2.93	2.674
	General parameters	N [L1,L2,T]	3.47 5.318 8	2.839 5.326 8	2.70 5.29 8	2.232 4.968 8
		m [L1,L2,T]	8	8	8	8
			8	8	8	8
		KLm	8.413	18.917	20	10.038
		Fit_H [L1,L2,T]	OK	OK	OK	OK
			9.5mm	5mm	4.3mm	3.4mm
		Fit_V [L1,L2,T]	8.3mm	4mm	3.5mm	3.4mm
			16.4mm	16.4mm	16.2mm	13.63mm
		Mass (g)	46.46	36.18	12.62	11.57
		Objective function (%/g)	2.106	2.720	7.813	8.357
		ZVS checker	0.226	0.354	0.348	0.383
		CCM checker	132.215	209.8	211.3	187.56
	Optimization	Bmax checker	OK	NOK	NOK	NOK
	results at Point A	Efficiency	97.827	96.45	97.86	97.989
		Objective function %/g	2.106	2.666	7.754	8.469
		Tj (°C)	85	247.63	89.92	65.66
		ZVS checker	0.515	0.174	0.371	0.468
		CCM checker	133.8	56.17	193.05	150.1
	Optimization	Bmax checker	NOK	OK	NOK	NOK
	results at Point B	Efficiency	98.19	98.418	98.53	98.75
		Objective function %/g	2.113	2.720	7.807	8.535
		Tj (°C)	90.27	85	70.72	52.3
		ZVS checker	0.698	0.362	0.33	0.564
		CCM checker	44.99	125.56	61.65	68.69
	Optimization	Bmax checker	NOK	NOK	OK	NOK
	results at Point C	Efficiency	98.42	98.197	98.596	98.74
		Objective function %/g	2.118	2.714	7.813	8.534
		Tj (°C)	64.43	104.98	85	45.7
		ZVS checker	0.702	0.366	0.351	0.666
		CCM checker	44.99	125.56	129	14.12
	Optimization	Bmax checker	NOK	NOK	NOK	OK
	results at Point D	Efficiency	98.51	98.316	98.889	96.69
		Objective function %/g	2.120	2.717	7.836	8.357
		Tj (°C)	57.54	86.6	52.253	85

Table 6 -

 6 6: The optimization results in step 2, when all the discrete parameters are in the According to the above table, the following conditions are discussed for moving into step 7:

	imaginary world
	Parameter	Optimization results	Unit
	XL1	54.319	-
	XL2	49.312	-
	XT	55.498	-
	NL1	6.21	-
	NL2	6.67	-
	NT	7.44	-
	nL1	0.999	-
	nL2	0.598	-
	nT	0.999	-
	mL1	6.21	-
	mL2	11.15	-
	mT	7.44	-
	lg_L1	1040.21	µm
	lg_L2	447.28	µm
	lg_Lm	113.3	µm
	Objective function (η/mass) 0.3430	%/g
	Efficiency	98.558	%
	Mass	287.34	g
	C1	120.24	nF
	L1	8.1577	µH
	KLM	14.159	-
	According to the above table, XL1 is equal to 54.319. Therefore, it can be selected as
	XL1_1=47 and XL1_2=55 (above and below XL1=54.319) for step 3. For XL2, it was obtained
	as 49.312 therefore for the next step it can be selected as 47 and 55. XT was obtained as 55.498
	therefore for the next step it will be 55 and 65. These considerations are noted as scenario I.
	Furthermore and in scenario II, the total turn numbers of L1, L2, and transformer were 6.21,
	6.67, and 7.44, respectively, which can be selected for the next step as 6 and 7 for L1, 6 and 7
	for L2, and 7 and 8 for transformer. For the number of turns per layer, the optimum values were
	0.999, 0.598, and 0.999. Due to these values being lower than 1, they should be considered as
	1 and 2 (0 is prohibited), where are the corresponding discrete input parameters of scenario III.
	These considerations are also listed in Table 6-7.
	With a more comprehensive investigation in Table 6-5 and Table 6-6, it is manifest that the

objective function of the single operating point optimization (e.g. 47.8521 for optimization at point A) is higher than the objective function of the multi operating points optimization (i.e. 10.156), which at the first point of view shows that the single operating point optimization is

Table 6 -

 6 15: Optimization results based on the genetic algorithm (GA) and sequential quadratic programming (SQP) for 4 operating points optimizations

	Parameter	Results based on GA	Results based on SQP	unit
	Objective function	0.303	0.240	%/g
	Optimization time	181.3	1.2	min
	Average efficiency	95.4	98.6	%
	Core sizes [L1,L2,T]	[55,55,55]	[55,55,65]	-
	Turn numbers [L1,L2,T]	[2,2,6]	[4,4,8]	-
	C1	36	100	nF
	L1	13.8	6.2	μH
	KLM	23.8	14.5	-
	Junction temperature constraints	NOT OK	OK	-
	Saturation constraints	NOT OK	OK	-
	Vertical fit constraints	OK	OK	-
	Horizontal fit constraints	OK	OK	-

Table 6 -

 6 16 shows the main specifications of the prototypes.

Table 6 -

 6 16: Specifications of two experimental prototypes

	Parameter	Case I	Case II
	S1~S8	GS66508B GaN	GS66508B GaN
	D1~D8	C4D10120 SiC	C4D10120 SiC
	C1	100 nF	99 nF
	C2	100 nF	99 nF
	CO	100 nF	220 µF
	L1	6.2 µH	25 µH
	L2	6.2 µH	25 µH
	Lm	90 µH	125 µH
	VOUT	250~400 V	250~400 V
	VIN	400 V	400 V
	fsw	100~350 kHz	90~350 kHz
	n	1:1	1:1
	6.8.2. Resonant tank

Table 6

 6

-16 case II.

Table 6 -

 6 [START_REF] Chu | Vehicle Electrification: Status and Issues[END_REF] is compared several important parameters of some references with the proposed method.

Table 6 -

 6 17: General comparisons between some references and the proposed method

	Ref.	Proposed	[18]	[35]	[42]	[167]	[168]	[169]	[170]
	Topology	CLLC	CLLC Flyback CLLC	SRC	DAB	LLC	LLC
	Pout	3.3kW	6.6kW	3.3kW	3.3kW	3.3kW 3.3kW 3.3kW 3.3kW
	Vout	250V to 400V	0.975 to 1.025	250V to 420V	250V to 420V	250V to 400V	200V to 450V	230V to 430V	64.8V to 86.4V
	frequency	100kHz to 350kHz	500kHz 100kHz 127kHz 200kHz 200kHz	86.6 kHz	300kHz
	Efficiency	95.72%	97.85% 96.4%	98.2%	95%	96%	97.6%	-
	Optimization	Multi	Single	Single	Single, GA	-	-	Single	-
	Mass/ volume	415gr for cores, Vcore = 167cm 3	114 W/in 3	Vcore = 170cm 3	570gr, 90000 mm 3	-	6.2 kg, 5.46 L	2540 kW/L cm 3 , 1.06	-
	Transformer	EE65, KLM = 14.52	Planar KLM=10 with 2 cores,	EE65 13.64 core, KLM=	EE65 16.2 core, KLM =	-	-	KLM= 7.11	KLM = 2.86
	Inductors	6.2µH	3.07µH	11µH	21µH	-	8 µH	45 µH	28µH
	Capacitors	100nF	33nF	6µF	33nF	-	-	75nF	40nF
	Method	SPA	FHA	-	FEM	FHA, FEM	TDA	TDA	FHA
	Application	Bidirct. EV on-board charger	EV on-board charger	EV on-board charger	EV on-board charger	EV charger	EV on-board charger	EV on-board charger	EV on-board charger

Table 0 -

 0 1: Single operating point optimization results in the imaginary world when the maximum power density is selected for the objective function According to the above table, it is manifest that the objective function converges to the minimum available mass (i.e. 9.23g) for any operating point, which is not true for a bidirectional 3.3kW battery charger. Therefore, this objective function is neglected in this dissertation.

	Optimization results at	Parameter	Selected point for optimization Point A Point B Point C Point D
		C1 (nF)	56.37	62.47	20.00	20.01
		L1 (µH)	3.76	5.13	16.01	16.00
			20	20	20	20
		X [L1,L2,T]	20	20	20	20
			20	20	20	20
			287.7	484.0	193.3	100
		lg [L1,L2,T] (µm)	262.3	325.1	202.4	100
			100	100	100	100
			13.59	10.99	12.28	8.83
		N [L1,L2,T]	12.98	9.01	12.56	8.83
	General		10.49	14.80	26.11	11.14
	parameters		2	2	2	2
		m [L1,L2,T]	2	2	2	2
			2	2	2	2
		KLm	6.01	8.78	8.74	7.93
		Fit_H [L1,L2,T]	OK	OK	OK	OK
		Fit_V [L1,L2,T]	OK	OK	OK	OK
		Mass (g)	9.23	9.23	9.23	9.23
		Efficiency (%)	97.71	98.43	98.13	97.09
		Objective function (W/g)	233.24	322.48	168.38	88.98
		ZVS checker	OK	OK	OK	OK
	Optimization results at Point A	CCM checker Bmax checker Efficiency Objective function W/g 233.24 OK OK 97.71	OK NOK 97.94 285.30	OK NOK 97.77 91.13	OK NOK 97.76 89.99
		Tj (°C)	84.999	79.92	59.08	59.07

Table 0 -

 0 1: Single operating point optimization results in the real world when a lower size is selected for the magnetic parts, by considering a different operating point and checking the objective function and constraints on all points

	Optimization results at	Parameter	Selected point for optimization Point A Point B Point C Point D
		C1	246.17nF 500nF	20nF	26.45nF
		L1	2.542uH 6.04uH	16uH	12.10uH
			30	20	20	20
		X [L1,L2,T]	25	20	20	20
			35	35	20	20
			290um	258um	304um	100um
		lg [L1,L2,T]	246um	180um	304um	345um
			100um	113um	100um	100um
	General		4	4	2	2
	parameters	N [L1,L2,T]	4	4	2	2
			4	4	2	2
			2	2	2	2
		m [L1,L2,T]	2	2	2	2
			2	2	2	2
		KLm	10.01	20	8.744	9.693
		Fit_H [L1,L2,T]	OK	OK	OK	OK
		Fit_V [L1,L2,T]	OK	OK	OK	OK
		Power density	2.8519	3.905	10.6266 10.4973

Table 0 -

 0 2: Single operating point optimization results in the real world when a higher size is selected for the magnetic parts, by considering a different operating point and checking the objective function and constraints on all points

	Optimization results at	Parameter	Selected point for optimization Point A Point B Point C Point D
		C1	110.0nF 200.5nF 20.1nF	34.16nF
		L1	2.911uH 1.59uH	15.9uH	9.37uH
			35	25	25	25
		X [L1,L2,T]	30	25	25	25
	General parameters	lg [L1,L2,T]	42 1mm 1mm	42 315um 204um	25 100um 100um	25 102um 100um
			1mm	100um	100um	100um
			8	4	4	4
		N [L1,L2,T]	8	4	4	4
			8	4	4	4

Acknowledgments

experimental setups. They taught me a lot not only in the technical and scientific aspects but also in the human aspects of being a researcher. I honestly believe I could not have better supervisors for doing this PhD. This thesis was funded by Embassy of France in Tehran with the Bourse of French Government, supported in part by the French National Program, and in part by the Iranian National Science Foundation. Many thanks to the people who help me in this way.

For these scenarios, the electrical operating points (VDS, ID), thermal operating point (Tj), circuit operating points (Rgate, Lgate, Lp), and device characteristics (different GaN transistors) are changed to evaluate the accuracy of the proposed model. Errors can be calculated as: As shown in Table 4-9, there is a significant positive correlation between the results obtained from the proposed model and LTSPICE simulations for some scenarios. The error reaches as high as 37% for the GPI8HINOIC transistor and it is lower than 10% for most of the cases, which means this error can be considered as an acceptable value for the switching losses owing to its non-linear behavior. It should be mentioned that some parameters for GaN devices are unknown (see , which leads to inaccuracies in the model. Also, the impact of the transconductance on the switching losses and interpolation has not been considered in this work.

Adding this parameter should help in improving the model's accuracy.

A closer inspection of Table 4-9 shows that the GS66508B transistor has lower switching losses error than the others, due to the base values, the proposed model has been selected based on this transistor. Consequently, we can achieve higher accuracy by using different transistors as the base value parameters, if needed.

Verifying the Conduction Power Loss Model

In this part, the conduction losses evaluation and temperature dependency of ON state resistors are presented. GaN transistors listed in Table 4-6 are considered and the simulations are carried out at the junction temperature equal to 70°C. The results for each transistor are shown in Table 4-10. The C4D10120E diode is also utilized for the rectifying parts in parallel with the GaN transistors while the corresponding transistors have been switched OFF.

Gate driver circuit

An eligible gate driver should meet the following requirements:

• for ON state: apply a correct positive voltage to the gate-to-source for possessing low ON resistance;

• for OFF state: keep the switch in OFF state regardless of any undesirable transient pulse;

• for transition state: make the transition correctly to not only bring about immediate transition for minimizing switching losses but also provide a leisurely slope to reduce electromagnetic interference (EMI).

In this dissertation, the IC SI8271GB-IS is applied as the gate driver to turn the corresponding transistors to ON and OFF conditions, where the Gate-to-Source voltage (VGS) of +6V makes the transistor ON, and VGS=-3V makes it OFF (to prevent unexpected turn-on).

Therefore, a 9V power supply is needed, where we employed the isolated DC/DC converter PES1-S5-S9-M-TR for changing 5V to 9V. The BZX84C6V2 diode Zener is utilized as a voltage divider for extending 9V to +6V and -3V. The gate driver power supply is shown in In conclusion, it can be deduced that the proposed method has much higher precision in comparison with the FHA method for both power transferring directions. This exact analysis can be used in the design, control, and optimization of the CLLC resonant converter as well as other high-order resonant converters. Owing to the mathematical expression of the converter behavior, this method is very helpful for practicing optimization algorithms to minimize voltage/current stresses, losses, and reliability assessment.

Summary

This chapter proposed in detail the state-plane analysis of the bidirectional full-bridge CLLC resonant converter for the switching above the resonant frequency. A new formulation of statespace equations was presented and the fourth-order CLLC resonant converter was transformed into two second-order systems. Then using this mathematical method, the exact analysis of the converter in CCM, stresses of the components, voltage gain ratio, soft switching conditions, and the boundary between CCM/DCM were obtained. It should be mentioned that the proposed method is not completely exact but it is precise in comparison with FHA method. A 3-kW prototyping was built. Experimental results were used to verify the proposed method.

Simulation and experimental results reveal the accuracy of the proposed analysis in the forward and reverse power transfer modes for different power and voltage levels. It is noted that the proposed state-plane analysis is exact in comparison with the simulation results and it has an acceptable precision in comparison with the experimental results. The proposed state-plane analysis is helpful in the design and optimization of the CLLC resonant converter in a wide operating range, which will be examined in the next chapter.

KLM has been commonly selected among 4~20 in literature. In this dissertation, it is selected in the interval of 3~20.

Geometrical input parameters

Table 6-2 demonstrates the list of the geometrical input parameters:

S1

Spacing between the ending turns and core Fixed 1mm --

B_MAX

Maximum flux density of the selected core Fixed 0.25T --These geometrical parameters consist of the continuous form of the core, airgap, number of layers, the diameter of wires, etc. The size of the cores varies between 10~64 to an optimum charger to be found. The minimum airgap is selected as 100µm to make it implementable and its maximum value is considered as 5mm. The diameter of the wires is selected according to the current rating.

The following variable is defined to check the CCM operating and satisfy the above condition:

where the value of V2|@iD=0 was obtained in the previous chapter.

Maximum junction temperature

The junction temperature of GaN transistors must be lower than the maximum junction temperature mentioned in the datasheet. It is necessary to calculate the corresponding power dissipation of the GaN transistors. Then, the junction temperature (TJ) is determined according to the thermal modeling:

where TA is the ambient temperature typically 25°C, PLoss is the power dissipation of the GaN transistor, and Rth_JA is the junction-to-ambient thermal resistance of the GaN transistor which is 3°C/W for GS66508B GaN transistor.

Minimum efficiency

Despite the efficiency that exists in the objective function for maximizing or not, the minimum allowable efficiency is also another constraint. In this thesis, we consider:

Transformer saturation checking

The flux density of the cores should be lower than its maximum flux density mentioned in the datasheet for all the operating points. Therefore, the parameters CheckBmax_L1, CheckBmax_L2, and CheckBmax_T are defined to examine the saturation of L1, L2, and the transformer, respectively:

𝐶ℎ𝑒𝑐𝑘𝐵𝑚𝑎𝑥_𝑇 = 𝐵𝑚𝑎𝑥 -𝐵𝑚𝑎𝑥_𝑇 (6.21)

• Step 6: the optimization runs once again and the corresponding constraints are checked to find an optimum result while two discrete values are a fixed input parameter. The optimization results will be obtained to make a decision for the next step.

• Step 7: the last integer variable is discretized based on the above and below the value of the obtained result. Therefore, at the end of this step, the core size, the total mass of the core, the total turn numbers, and the number of turns per layer are recognized because all the discrete variables are realized. However, there exist two candidates based on the selected values above and below the results. So, another optimization is also needed.

• Step 8: the last optimization is performed by selecting all the discrete parameters as a fixed input. Since the total mass of the converter is known, maximizing the efficiency and /or output power can be considered as the objective function to obtain the airgap of the magnetic cores. All the optimization results are available at the end of this step.

• Step 9: in this step, all scenarios and results are compared and the maximum η/mass is considered as the final solution. Therefore, the proposed optimization algorithm reaches the end.

The following part of this thesis moves on to describe in greater detail the proposed optimization process for a 3.3kW bidirectional CLLC resonant converter for EV charging applications.

Numerical Clarification of the Proposed Optimization Process

The optimization process shown in Fig. 6-6 is numerically examined in this part. Let's consider the input parameters described in part 4.2.1 and the constraints listed in part 4.2.2 in the optimization process (step 1). The optimization results will be obtained as shown in Table 6-6 (step 2).

A good comparison between the imaginary results of single operating point optimization (Table 6-5) and the imaginary results of multi operating points optimization (Table 6-6) shows that the total mass is 11.57g and 287.34g, respectively for single operating point and multi operating points optimization, which is obvious the obtained result for single operating point is not near to the real world.

preferable. But based on this method, the constraints cannot be met for other operating points.

As a result, the obtained result for single operating point optimization is not real and cannot provide a maximum power of 3.3kW by insuring all constraints. In step 4, the optimization runs again with the modified input parameters. The optimization results will be obtained as shown in Table 6-8.

In step 5, the second discrete value is considered as a fixed input. For this purpose, the aforementioned scenarios are included:

• Scenario I: for the lower value of core sizes, the turn numbers of each layer for L1, L2, and transformer were obtained as 1, 0.8724, and 0.9938 in Table 6-8, respectively.

Therefore, the turn numbers of each layer are 1 if we put a floor under them, and 2 if we set a ceiling on them. Furthermore and for the higher value of core sizes, the number of turns per layer was achieved as 1, 1.3379, and 1.1976, which can be considered as 1 by floor and 2 by ceiling.

• Scenario II: the core sizes were obtained higher than 55. Therefore, it can be discretized to 55 and 65 for step 5. Therefore, we can conclude that the last non-integer parameter can be discretized in step 7

based on the corresponding scenarios as shown in Table 6-11 for better clarification. where the last continuous variable is discretized by putting a floor under and a ceiling on it.

Consequently, the three discrete variables were considered as a fixed input parameter and now, all the variables are continuous and implementable. Therefore, the SQP algorithm can be performed in the real world. Hence, the last optimization is carried out in step 8 to find the airgap length while the efficiency and/or output power will be maximized. The result of this optimization is shown in Table 6-12: The most obvious finding to emerge from Table 6-12 is that the optimum 3.3 kW bidirectional CLLC resonant converter in EV charging application is the sample with XL1=55, XL2=55, and XT=65, and C1=C2=100nF. It has also the minimum mass with the maximum efficiency and is selected as the optimum sample (step 9).

Comparison between the optimization results

The first comparison is the time for running the code for single operating point and multiple operating points optimizations, which is shown in Table 6-13. More information is also listed in Table 6-13. An important question is that what will be the optimization results when the FHA method is utilized to obtain the optimum result and meet the constraints. In this case, the FHA method is developed in Cades instead of SPA method. The peak value, RMS value, ZVS constraint, CCM operating condition, output voltage, required output current, and required switching frequency range are obtained based on the FHA method. The corresponding results in comparison with SPA method is summarize in Table 6-14, where the input parameters and the constraints are the same.

This table illustrates that the optimum result based on the proposed method (SPA) has a higher objective function in comparison with the result based on the FHA method, when all the constraints are met. The average efficiency of the proposed method is about 1.4% higher than FHA method, due to the fact that the value of the primary and secondary currents for FHA method is higher than SPA-based values and it causes higher losses. As a conclusion, the optimization result based on the proposed method has a higher average efficiency with lower mass. The last comparison is based on the selected methodology for the optimization process. The deterministic SQP method was adopted in this dissertation but it is better to compare it with the genetic algorithm (GA) which is a stochastic algorithm. In this case, the number of 100 populations is considered and 1000 generations are provided. The probability of mutation is selected as 10% to decrease the possibility of stuck in the local extremum. The optimization results based on the GA and SQP are demonstrated in Table 6-15. It should be mentioned that one of the advantages of the CADES software is the capability of changing the optimization method easily. In this case, before the input parameters, the optimization method is selected between sequential quadratic programming (SQP), genetic algorithm (GA), particle swarm optimization (PSO), DIviding RECTangles optimization (DIRECT), evolutionary optimization (EO), restricted tournament selection (RTS), and interior point optimizer (IPO).

programming (SQP) as one of the deterministic algorithms was selected in this dissertation.

Because the SQP method is a gradient-based optimization technique, continuous variables are needed. But, in the power electronics converter, there are some discrete variables such as the total number of turns, core sizes, number of turns in each layer, etc. Therefore, a continuous model was presented to define each parameter as a continuous parameter.

Furthermore, the calculation of total power losses is important to obtain the total efficiency and power density. In this case, an interpolation-based method for the calculation of switching losses was presented to obtain the switching loss of different GaN transistors, with different voltages, currents, gate driver circuit parameters, temperatures, etc. For the conduction loss, another method was proposed to calculate conduction losses based on the datasheet's value of different GaN transistors. For comprehensive loss calculation, the diode loss, magnetic losses, copper losses in wires, capacitor loss, etc. were discussed. A method for magnetic design based on the magnetic circuit and reluctance was also considered. The objective function was evaluated based on the loss calculation of components, and the design procedure for the magnetic components and capacitors. Different constraints, e.g. operating region, maximum temperature, maximum RMS current of capacitors, maximum voltages, ZVS condition, and CCM operating, were also discussed to provide a global optimization problem with the correct objective function and constraints.

The common method for analyzing the resonant converters and also CLLC resonant converter is the first harmonic approximation (FHA) method. However, as it was shown in Therefore, a new formulation of state-space equations was presented and the fourth-order CLLC resonant converter was transformed into two second-order systems. Then using this mathematical method, the exact analysis of the converter in CCM and DCM, stresses of the components, voltage gain ratio, soft switching conditions, and the boundary between CCM/DCM were obtained. The accuracy of this method was compared with the simulation results, which implies that the proposed method can be considered as an exact method for the exact analysis of the CLLC resonant converter.

Having the proposed exact analysis method, it is better to use it in the optimization process to increase the accuracy of obtained results. At first, the proposed state-plane analysis was implemented into the CADES software by C++. This code was verified by Maxwell simulation.

Appendix I Decoupling Method for the State Space Equations

By substituting iL1, i'L2, vC1, and v'C2 as the function of iLP, iLN, vCP, and vCN according to (5.4) into (5.2) and rearranging them, the state space equations are recalculated as: In region A: the duration of this region is α. Using Laplace transform, the primary current will be as the following equation for L1=L2, and C1=C2: Finally, for all resonant elements, the following equations in the time domain are obtained: Now, only P1A, P2A, P3A, and P4A remained. For this case, we know that the value of elements in the half of the switching period is negative of them, e.g. for iL1:

( ) ( )

World for the Single Operating Point Optimization

The obtained results in part 6.4. were determined in the imaginary world. But, it is important to show the disadvantage of the single operating point in the real world, which is expressed in this part.

Firstly, a lower standard size for the magnetic parts is considered in the real world as shown in the following table: