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## Space vector Placement

The FCS-MPC is broadly used for current control and DPC of power converters [14]–[22], [48], because it matches the discrete nature of power converter but, as seen in the previous section, the switching frequency is variable and depends on the sampling frequency for classical FCS-MPC. Some authors proposed a new method to have a fixed switching frequency by using a combination of FCS-MPC, DPC and Virtual flux (VF) [49] . The block diagram of their method is shown in Figure 2-16. Authors use the estimated VF, the switching table (that contains all the voltage space vectors of a voltage source converters (VSC)) and the estimated inductance to predict the future command of the power semiconductors for one switching period which is divided in six time interval as shown in Figure 2-17.

From a given space vector sequence, the active and reactive power and the virtual flux can be estimated. From these estimations, the optimal (leading to the minimum cost function value) duration of application of each space vector of the sequence (𝑡1,𝑡2 𝑎 𝑑 𝑡3) can be derived.

Such methods are mainly used in AC power converters such as AC-DC, DC-AC, AC-DC-AC or active filters but there is a lack of references related to such method applied to multicell DC-DC power converters. In this PhD, we propose in chapter 5 a strategy inspired by this type of methodology combining SV and FCS-MPC and dedicated to multicell DC-DC power converters.

### Multi-cell interleaved buck converter for solar application

Figure 3-1 depicts the system under study. A photovoltaic array feeds a load which could be possibly a battery directly powering DC loads or a grid inverter. As both PV maximum power point and the load voltage can vary greatly, it is mandatory to interface a converter between the load and the source: this is the multi-cell converter using a monolithic ICT formed by windings wounded on the same magnetic circuit. For simplicity, it has three switching cells ( =3) and a 3-phase-transformer acting as an output current filter. The input current is filtered by the input capacitor 𝐶𝑖. Each switching cell is driven by a PWM control signal characterized by a constant switching frequency 𝑓 and a duty cycle 𝑑𝑘, which represents a system control variable. The system parameters and the rated variables are listed in Table 3-1.

Compared to classic single buck converter, the main advantage of this power electronics structure is to ensure low current ripples at both input and output sides. In fact, regarding the input stage, the input current ripple is reduced by an factor while the input current apparent frequency is increased by a factor of . As a result, the 𝐶𝑖 capacitance can be reduced by a significant 2 factor leading to improve the system dynamics and namely its ability to track faster the maximum power point of the PV array [24], [34], [56]. Similarly, the amplitude of phase current ripples are reduced by a 2 factor compared to an uncoupled multi-cell converter (considering a similar filtering inductance value), which reduces the constraints on the power semi-conductors and the related losses. Moreover, the global power converter output current ripple is reduced by compared to a classical one-cell Buck DC-DC converter, in the same way as for interleaved multi-cell DC-DC Buck converter with uncoupled inductors[24]. This limits the need to filter the output voltage: in some cases, no additional output capacitor is required.

These electrical and energetics advantages are counterbalanced by a rising difficulty to control the system in static and dynamic conditions. This is the reason why a control-orientated model is needed to study the feedback control.

#### Simulink model of multicell interleaved DC-DC buck converter

Different models are built to simulate the Buck converter, a discrete model and an equivalent average model are created on Simulink-Matlab. The switched model was built in Matlab with Simpower tools. In this model, each cell is a two-IGBT, two-diodes half-bridge. The equivalent average model using controlled voltage and current sources is used to reduce the simulation time. The two models are shown in Figure 3-3.

The open loop response of the two models for a 3-cell Interleaved Buck-converter are shown in Figure 3-4. In these simulations the duty cycles of the 3 cells change from 45% to 55% with a resistive load equal to 15 . This simulation shows that it is possible to use the average model as tools for studying the dynamics of the converter rather than the switched one[57].

**Simulation specifications**

The control design must be done with relevant required dynamics. The present study considers the specifications summarized in Table 3-2. Indeed, the first requirement is to guarantee a good precision in steady state in order to fulfill the maximum power point tracker requirements. Second, the time taken for the response to reach the desired set point is also important for the system functionality. The solar converter needs to react to solar irradiance changes which in the worst case may occur in a 10 ms time period, which is not very challenging. However, there are obviously other scenarios to consider; short circuit limitation is one of the cases requiring a rapid action. For this demanding challenge the settling time is set to 500 s which means ten switching periods. A third key point is to ensure a good stability margin of the closed loop system. This point is achieved by satisfying an overshoot criteria and decay ratio. The first criterion gives also a good indication on how duty cycles saturations are managed, while the latter gives a good performance index of the system stability. In addition, the minimization of the windings currents coupling permits to control independently each phase current which is essential to modify the phase power distribution in case of a local overheating; the limited overshoot while another current is changing is a way to take this fact into account.

**Table of contents :**

Table of Contents

Table of Tables

Table of Figures

Abbreviations

Description du contexte et travaux réalisés

Les principaux résultats

Contrôle linéaire des convertisseurs multicellulaires entrelacés

Model Predictive Control des convertisseurs multicellulaires

Contrôle vectoriel et Model Predictive Control des convertisseurs multicellulaires

Résultats expérimentaux

**Chapter 1. Introduction **

**Chapter 2. State of Art**

2.1. Introduction

2.2. Multicell Power Converter

2.2.1. Topologies of Multilevel DC/DC Converters

2.2.2. Multicell Power Converter in Solar application and microgrids

2.3. Classical control and LQR

2.3.1. Hysteresis control

2.3.2. Linear control using PWM

2.3.3. PI/IP control

2.3.4. State space control

2.4. Model Predictive Control

2.4.1. Basic principles of Model Predictive Control

2.4.2. Finite control set MPC

2.5. Space vector Placement

**Chapter 3. Classical control of multicell interleaved power converter **

3.1. Introduction

3.2. Multi-cell interleaved buck converter and its control-oriented model

3.2.1. Multi-cell interleaved buck converter for solar application

3.2.2. The mathematical model of converter.

3.2.3. Mode analysis of the state-space average model.

3.2.4. Simulink model of multicell interleaved DC-DC buck converter

3.2.5. Simulation specifications

3.3. Proportional-Integral Controller

3.4. State Feedback

3.4.1. Control structure and the related extended model

3.4.2 . Tuning of state feedback gain

3.4.3. Simulation results

3.5. Decoupling strategy

3.5.1. Control structure degrees of freedom

3.5.2. Simulation results

3.6. Linear quadratic regulator (LQR)

3.6.1. Objective function

3.6.2. State feedback design by using LQR

3.7. Conclusion (Comparison)

**Chapter 4. Model Predictive Control **

4.1 . Introduction

4.2. Finite control set MPC

4.3. FCS-MPC with fixed switching frequency

4.3.1 . PWM with sawtooth carriers

4.3.2. Fixed switching frequency algorithm for FCS-MPC

4.4. Model Predictive Control for multicell Buck Converter

4.4.1. Mathematical model of a 3-cells Buck converter

4.4.2. Current Control of multicell Buck converter

4.4.3. Voltage Control of multicell Buck converter

4.5. Model Predictive Control for multicell Boost converter

4.5.1. Mathematical Modeling of multicell boost converter

4.5.2. Current Control of multicell Boost converter

4.5.3. Voltage Control of Boost converter

4.6. Conclusion

**Chapter 5. Space vector placement based on model Predictive Control **

5.1. Introduction

5.2. Model of a 3-Cell parallel Buck converter

5.3. Physical impact of common and differential modes of the currents on output coupled inductors

5.4. Control of the three current modes

5.4.1. Control of the three voltage modes

5.4.2. Determination of the duty-cycles

5.4.3. Direct control of differential currents

5.4.4. Choice of the space vector sequence

5.4.5. Impact of the choice of a sequence

5.4.6. Levels transitions

5.4.7. Control point of view of the proposed strategy

5.4.8. Simulation

5.5. MPC with space vector placement

5.5.1. Main controller

5.5.2. Secondary controller

5.5.3. Simulation

5.6. Conclusion

**Chapter 6. Experimental Results **

6.1. Introduction

6.2. Experimental test bench

6.2.1. Power supply

6.2.2. The inductance elements

6.2.3. The converter

6.2.4. Measurements sensors

6.2.5. Hardware for implementation of the controller

6.3. Implementation of classical controller

6.3.1. Controller implementation

6.3.2. Current controllers

6.3.3. Voltage controller

6.3.4. Controllers implementation

6.3.5. Experimental results

6.4. Implementation of FCS-MPC

6.4.1. Synthesis of the controller

6.4.2. Current loop

6.4.3. Controller implementation

6.5. Conclusion

**Chapter 7. Conclusion **

**References**