A New Decentralized Space Vector PWM Method for Multilevel Single-Phase Full Bridge Converters

This paper proposes a decentralized control structure and method for a multilevel single-phase power converter using space vector pulse width modulation (SVPWM). The focus of this paper is on the decentralized control structure for the power converter cells that will exchange information with neighboring cells in order to adjust the switching vector and switching time for each cell. In this study, the switching vectors and the corresponding switching times of each cell will be self-determined based on the phase angle of two neighboring cells. Normally, SVPWM applied to the multilevel power converters need complete information about the total cells and cell’s position to build a control algorithm. Meanwhile, a decentralized space vector pulse width modulation (DSVPWM) method is proposed that can be applied to power converters with any number of cells and can be considered as a multilevel SVPWM method. In addition, the decentralized multilevel single-phase power converter has high flexibility with which it is possible to easily adjust the number of active cells, so that the output voltage can be adjusted quickly; this provides the ability to dynamically reconfigure without interrupting the power energy supply process. The proposed control structure and method are effectively verified based on simulation and experimental results. Experimental results are evaluated based on a 9-level single-phase power converter, which has an RL load with rated parameters 220 V/500 W.


Introduction
Multilevel power converters are increasingly popular and are widely applied in many fields and different applications [1,2]. The application of these power converters allows for increasing the efficiency of the energy conversion process, improving the output voltage quality, reducing the output voltage distortion, reducing the switching power loss of components, increasing the switching efficiency of the component, and reducing the switching voltage of the component [3][4][5][6]. The emergence of the modular multilevel converter (MMC) is one of the next generation multilevel converters for medium and high powers demands [3,[7][8][9][10]. Figures 1 and 2 present the topology for multilevel singlephase power converters [11,12]. By connecting cells in series, it is possible to adjust the output voltage according to the requirements of the load, which can expand the system's ability to work with different voltage levels or increase flexibility in terms of control and operation [12][13][14][15].  The system is capable of dynamic reconfiguring, rebalancing when adding or removing some modules, or can easily replace a damaged module when necessary [12][13][14][15][29][30][31]. The system can easily and quickly reach a steady state under new conditions when the power converter structure is changed; this ensures that the system works continuously and without interruption. This is an extremely important and necessary feature in the study of decentralized power converters, as well as in a power supply system in which there are many decentralized power supplies in a grid.
Since the appearance of multilevel power converters, multilevel models have been introduced, such as diode-clamped converters (DCC) [38], flying capacitor converters (FCC) [39][40][41][42], full bridge converters, half bridge converters, etc. [38,43]. In addition, several power converter structures have been improved to have reduced switching components, reducing the total harmonic distortion and changing the current shape [44]. In control techniques for MMCs, the nearest level control (NLC) method is one of the most popular modulations [45]. Outstanding NLC method in low switching frequency modulation [46], used as a control method for the structure, involves a cascade full bridge multilevel inverter [47], MMC half bridge converter [48], and some improved structure [49,50]. The nearest level control method is also combined with the PWM method as a modulation algorithm for the MMCs [51]. Among them, full bridge configurations are well suited for the design of modular multilevel power converters. Typically, these power converters are introduced as three-phase power converters, but can be calibrated to operate as singlephase power converters. In fact, single-phase power converters also have a wide range of applications, especially for small and medium power demands [52][53][54].
Normally, switching pulses for multilevel power converters are modulated by two methods: carrier pulse width modulation [12][13][14] and space vector pulse width modulation [43,55]. Usually, in a full bridge configuration, the carrier level-shifted pulse width modulation is used as the primary method [13,14]. Meanwhile, the carrier pulse width modulation method is commonly based on the analog technique and does not have a common algorithm for all configurations. In contrast, the space vector pulse width modulation method stands out for providing considerable flexibility for optimizing switching waveforms and is well suited for digital implementations [9]. The first study and publication using the space vector modulation method for the multilevel multiphase power converter has proven the feasibility for this research direction [11]. This study presented the imple- Modular power converters typically use a processor as the central controller [16][17][18][19][20]. The central controller will receive feedback about the current, voltage of cells and branches [19,21,22]. Depending on the topology, control method and quality requirements, the central processor will calculate the phase angle and carrier amplitude suitable for the carrier pulse width modulation method [23]; this is achieved by the switching vector and switching time for space vector modulation and transmitting control signals to cells and IGBTs [24][25][26][27][28]. If the number of modulation voltage levels increases, the number of cells will increase, leading to a significant increase in the number of connections and control signals, requiring the central controller to have a very strong processing ability and a large number of inputs and outputs. The central processor may have a relatively high cost. In addition, during operation, the system may need to be reconfigured. The reasons may be that there are one or several damaged cells, impacting on the optimized conditions in control and operation. Then, the system needs to be able to add or remove some cells. For power converters using a central processor, this will be very difficult and the system can take a relatively long time to recover.
Currently, in order to maximize the advantages of multicell power converters, they are designed and controlled according to a decentralized control structure. Recent studies on this topic have highlighted the outstanding advantages of power converters that are designed and controlled according to a decentralized control model [12][13][14][15]29,30]. The decentralized control power converter can expand the ability to work with different voltage and power to meet the needs in terms of control and operation. This will be a very high advantage that classical power converters cannot achieve, or there will be certain difficulties in such classical power converters being able to achieve the same result. In such a case, the capacity of the power converter can be managed easily and quickly by adding or removing the number of modules in parallel or/and in series [15,30,31].
The structure, connection and control method of decentralized power converters have been studied and published in three main directions. Decentralized control has a hierarchical structure of two control levels, such as a controller master-slave, master-slave controller or local-central controller [32][33][34][35]. Each module of the decentralized power converter operates independently on its own voltage and current information, without communicating with neighboring modules [36,37]. The cells of the power converter exchange information with neighboring cells [14,15,[29][30][31]. In particular, the control structure in which cells exchange information with neighboring cells is very suitable for control and expansion when used as needed.
The system is capable of dynamic reconfiguring, rebalancing when adding or removing some modules, or can easily replace a damaged module when necessary [12][13][14][15][29][30][31]. The system can easily and quickly reach a steady state under new conditions when the power converter structure is changed; this ensures that the system works continuously and without interruption. This is an extremely important and necessary feature in the study of decentralized power converters, as well as in a power supply system in which there are many decentralized power supplies in a grid.
Since the appearance of multilevel power converters, multilevel models have been introduced, such as diode-clamped converters (DCC) [38], flying capacitor converters (FCC) [39][40][41][42], full bridge converters, half bridge converters, etc. [38,43]. In addition, several power converter structures have been improved to have reduced switching components, reducing the total harmonic distortion and changing the current shape [44]. In control techniques for MMCs, the nearest level control (NLC) method is one of the most popular modulations [45]. Outstanding NLC method in low switching frequency modulation [46], used as a control method for the structure, involves a cascade full bridge multilevel inverter [47], MMC half bridge converter [48], and some improved structure [49,50]. The nearest level control method is also combined with the PWM method as a modulation algorithm for the MMCs [51]. Among them, full bridge configurations are well suited for the design of modular multilevel power converters. Typically, these power converters are introduced as three-phase power converters, but can be calibrated to operate as singlephase power converters. In fact, single-phase power converters also have a wide range of applications, especially for small and medium power demands [52][53][54].
Normally, switching pulses for multilevel power converters are modulated by two methods: carrier pulse width modulation [12][13][14] and space vector pulse width modulation [43,55]. Usually, in a full bridge configuration, the carrier level-shifted pulse width modulation is used as the primary method [13,14]. Meanwhile, the carrier pulse width modulation method is commonly based on the analog technique and does not have a common algorithm for all configurations. In contrast, the space vector pulse width modulation method stands out for providing considerable flexibility for optimizing switching waveforms and is well suited for digital implementations [9]. The first study and publication using the space vector modulation method for the multilevel multiphase power converter has proven the feasibility for this research direction [11]. This study presented the implementation of a decentralized control solution for a multiphase, multilevel inverter. Each phase is connected by full bridge cells in series. The space vector modulation algorithm for each cell is converted from a multiphase, multilevel modulation problem to a 2-level multiphase modulation problem [9]. After obtaining the results of 2-level, multiphase modulation, the system will refer to multilevel, multiphase modulation. The communication method used in the study is designed so that each cell communicates with two adjacent cells and can exchange information, such as phase angle, location of neighboring cells, modulation coefficient, and modulation frequency. Based on the phase angle value of the two adjacent cells and itself, the offset angle of the two adjacent phases in the inverter performs a local correction to the expected value. The research results show that the calculation process of a cell is quite complicated, sometimes leading to a requirement that the processor of each cell has a relatively high processing capacity.
Therefore, this problem can reduce the outstanding features of the decentralized controller. In order to optimize the control processing ability of the cells, to make the cell controller in the decentralized controller more compact, this paper proposes a simpler decentralized control algorithm for a multilevel single-phase inverter, using a full bridge cells in series. The cell structure uses communication between neighboring cells to calculate the vectors and switching times, and the IGBT control signal for cells. The proposed structure

Principle of the Space Vector Pulse Width Modulation Control for Multilevel Single-Phase Converter
Consider the structure of a power converter consisting of cell_1 and cell_2 as shown in Figure 2. Table 1 introduces the control region of 2-cells corresponding to the output voltage. The output phase voltage is the contribution of the two-cell voltage. The active region of each cell is shown in Figure 3. The XY state vector means that cell_1 has state X and cell_2 has state Y. There are some redundancy vectors in the control that are still shown in Figure 3. For example, to achieve the state V out = V dc , vector 21 or 12 can be used.  cells, modulation coefficient, and modulation frequency. Based on the phase angle value of the two adjacent cells and itself, the offset angle of the two adjacent phases in the inverter performs a local correction to the expected value. The research results show that the calculation process of a cell is quite complicated, sometimes leading to a requirement that the processor of each cell has a relatively high processing capacity. Therefore, this problem can reduce the outstanding features of the decentralized controller. In order to optimize the control processing ability of the cells, to make the cell controller in the decentralized controller more compact, this paper proposes a simpler decentralized control algorithm for a multilevel single-phase inverter, using a full bridge cells in series. The cell structure uses communication between neighboring cells to calculate the vectors and switching times, and the IGBT control signal for cells. The proposed structure and DSVPWM method are evaluated based on simulation results on MATLAB/Simulink and experimental models.

Principle of the Space Vector Pulse Width Modulation Control for Multilevel Single-Phase Converter
Consider the structure of a power converter consisting of cell_1 and cell_2 as shown in Figure 2. Table 1 introduces the control region of 2-cells corresponding to the output voltage. The output phase voltage is the contribution of the two-cell voltage. The active region of each cell is shown in Figure 3. The XY state vector means that cell_1 has state X and cell_2 has state Y. There are some redundancy vectors in the control that are still shown in Figure 3. For example, to achieve the state Vout = Vdc, vector 21 or 12 can be used.

Cell_1 State
Cell_2 State Output Voltage Apply the strategy by using the two nearest vectors in the space vector modulation technique for the power converter, as shown in Figures 2 and 3. The system includes cell_1  12 22 Apply the strategy by using the two nearest vectors in the space vector modulation technique for the power converter, as shown in Figures 2 and 3. The system includes cell_1 and cell_2. This technique generates a reference voltage (V r ) by using the two nearest state vectors in the control region. For example, if V r is −1.6 V dc , the proposed algorithm uses state vectors {00-01} or {00-10} as the transition sequence because if V r is represented in the control region, it is located at a position between these state vectors. The reference vector is created by linearly combining the two nearest state vectors of the control region. Based on correction studies, the proposed algorithm is shown from Equations (1)-(4). Operator floor(x) rounds the elements of x to the nearest integers towards minus infinity [43,52].
The reference voltage of the DC/AC converter has the form that is shown in Equation (1). where ω is angular frequency, and A is peak amplitude of the reference voltage.
For the multilevel modulation technique, the modulation voltage can be an integer of the input voltage V dc of each cell, as shown in Equation (2).
Then, the switching vector has Equation (3): The switching time corresponding to the switching vector is determined by Equation (4).
After determining the switching vector and the corresponding switching time, the processor sends switching signals to the cells based on the predefined cell positions.

Proposed Decentralized SVPWM Control Method
As introduced in Section 1, the central controller will have to perform calculations with a very large amount of computation and control a lot of IGBT switches. The central controller must perform the tasks of balancing capacitor voltage, balancing voltage between of cells, controlling load current, and controlling the voltage stability, among others. In addition, while the power converters are operating, if the system needs to be reconfigured (when a cell is damaged, add or remove cells to optimize in operation), the system needs to stop working, re-calibrate the control program, and adjust new connections based on the number of remaining cells. This process can take a relatively long time. If the number of cells in the system increases significantly, in some cases the system cannot be calibrated. To solve this problem, the decentralized control method for power converter using the space vector modulation algorithm is an appropriate choice.
As described in Section 2, the switching vector and switching time of each cell are determined based on information about the reference voltage amplitude, reference voltage frequency, and the cell's position in the system.
There have been many publications using different communication methods in the research direction of decentralized power converters in the form of cells that exchange limited information with neighboring cells. Cells can exchange their own phase angle with two neighboring cells in the carrier phase shift method. Cells can exchange information about the carrier amplitude magnitude in the carrier level shift method. Cells can exchange sequence numbers so that the cell's position in the system can be determined. Each method has different advantages, in which the method of exchanging phase angle information of the module with two neighboring cells is very suitable in the design of decentralized converters. In this paper, the study uses the results of the phase shift process to quickly determine the position of the cell in a single-phase power converter. The control connection diagram between cells is shown in Figure 4.
To determine the cell's position in the system, an algorithm for finding the individual phase angle of each cell is proposed, as in Equation (5) [30]: where n is index of cell_n, and k is the time instant index. To determine the cell's position in the system, an algorithm for finding the individual phase angle of each cell is proposed, as in Equation (5) [30]: where n is index of cell_n, and k is the time instant index.  For the algorithm to be able to locate cells, the phase angle (teta) needs to converge in every implementation. To ensure the algorithm can be executed, it is suggested that the initial phase angle for cells is 180 degrees, and the phase angle of the first cell (master cell) is always 0 degrees.
The total cells in single-phase is determined as shown in Equation (6): The position of the cell in single-phase is determined as shown in Equation (7): All cells operate independently of each other, requiring each cell to choose its own switching vector and appropriate switching time. This algorithm is implemented in Equations (8) and (9). To explain this algorithm, an illustrative example of 7-levels reference voltage is shown in Figure 5 (the reference voltage has an amplitude of 250 V and a frequency of 50 Hz; the power converter has input voltage of 100 V).  For the algorithm to be able to locate cells, the phase angle (teta) needs to converge in every implementation. To ensure the algorithm can be executed, it is suggested that the initial phase angle for cells is 180 degrees, and the phase angle of the first cell (master cell) is always 0 degrees.
The total cells in single-phase is determined as shown in Equation (6): The position of the cell in single-phase is determined as shown in Equation (7): All cells operate independently of each other, requiring each cell to choose its own switching vector and appropriate switching time. This algorithm is implemented in Equations (8) and (9). To explain this algorithm, an illustrative example of 7-levels reference voltage is shown in Figure 5 (the reference voltage has an amplitude of 250 V and a frequency of 50 Hz; the power converter has input voltage of 100 V).
phase angle of each cell is proposed, as in Equation (5)   For the algorithm to be able to locate cells, the phase angle (teta) needs to converg every implementation. To ensure the algorithm can be executed, it is suggested tha initial phase angle for cells is 180 degrees, and the phase angle of the first cell (master is always 0 degrees. The total cells in single-phase is determined as shown in Equation (6): The position of the cell in single-phase is determined as shown in Equation (7): (360 -θ , 360)) / (0.5 * mod(θ -θ , 360)) n n-1 n+1 n All cells operate independently of each other, requiring each cell to choose its switching vector and appropriate switching time. This algorithm is implemented in E tions (8) and (9). To explain this algorithm, an illustrative example of 7-levels refer voltage is shown in Figure 5 (the reference voltage has an amplitude of 250 V and a quency of 50 Hz; the power converter has input voltage of 100 V).   Figure 5 shows the division of a cell into two half bridges and v r and v i vector when the reference voltage has seven levels. If v r > 0, the switching vectors will affect the half bridge (A) of the cells (Figure 5a). The switching vector of a cell is determined, as shown in Equation (8): If v r < 0, the switching vectors will affect the half bridge (B) of the cells (Figure 5a). The switching vector of a cell is determined, as shown in Equation (9): Energies 2022, 15, 1010 7 of 25 Figure 6 shows the space vector modulation algorithm diagram for each cell. Figure 6 shows the space vector modulation algorithm diagram for each cell. During operation, the system is capable of dynamic reconfiguration, this is done by adding or removing cells as needed. In the structure shown in Figure 2, the maximum of modulation voltage will depend on the number of active cells. Thus, information about the total number of active cells is essential information for the cells, especially the master cell. Figure 6. Algorithm flowchart of one cell.
The decentralized control structures for the power converter, information about the reference voltage amplitude, and the reference frequency can be provided by one of the cells in the system. In the structure shown in Figure 4, cell_1 is chosen as the master cell. Cell_1 is responsible for generating modulated signals of the reference voltage and frequency, and synchronous clock, and then transmitting, in turn, through all cells in the system including itself. For other cells, based on the information received, combining with  (360 -θ , 360)) / (0.5 * mod(θ -θ , 360)) n n n-1 n+1 ; v ;t n sn sn During operation, the system is capable of dynamic reconfiguration, this is done by adding or removing cells as needed. In the structure shown in Figure 2, the maximum of modulation voltage will depend on the number of active cells. Thus, information about the total number of active cells is essential information for the cells, especially the master cell.
The decentralized control structures for the power converter, information about the reference voltage amplitude, and the reference frequency can be provided by one of the cells in the system. In the structure shown in Figure 4, cell_1 is chosen as the master cell. Cell_1 is responsible for generating modulated signals of the reference voltage and frequency, and synchronous clock, and then transmitting, in turn, through all cells in the system including itself. For other cells, based on the information received, combining with the synchronous clock will automatically regenerate the reference voltage signal as Equation (1) for space vector modulation. In addition, if a cell has to be removed from the converter, i.e., deactivated, the bypassing of a cell is realized using an enable bit (EN).

Simulation Result
A thirteen-level single-phase power converter model is built on MATLAB/Simulink, without losing the generality of the proposed algorithm. The model consists of six cells connected in a series to perform DC/AC modulation. Figure 7 shows a simulation diagram of a decentralized 13-levels single phase power converter on MATLAB/Simulink. The output voltage will have 13 levels if all six cells are activated. The input and output functions of the signal pins of each cell are shown in Table 2; the simulation parameters of the power converter are given in Table 3;    Reference voltage signal V m V 550 Reference frequency signal f in Hz 50 Figure 8 shows the IGBT switching signal of the six cells to create an output voltage of 13 levels. The evaluation of the amplitude response of the output voltage according to the reference voltage is shown in Figure 9. During the simulation, the reference frequency is fixed at 50 Hz. At the time of start-up, the reference voltage is set to 550 V; the output Energies 2022, 15, 1010 9 of 25 voltage has 13 levels; and the output voltage is the contribution of voltage of six cells. At 0.02 s, the reference peak voltage is adjusted to 350 V; the output voltage has nine levels; and the output voltage is the contribution of voltage of the cell_1, 2, 3, and 4. At 0.04 s, the reference voltage is 450 V, this voltage is the contribution of voltage of the cell_1, 2, 3, 4, 5, and the output voltage has 11 levels. Finally, at 0.06 s, the reference voltage is reset to 550 V, which is the same repeatable voltage as at system startup. The simulation results show that the output voltage of the single-phase inverter responds well to the required voltage amplitude. In terms of decentralized connection and the control of multilevel single-phase power converters, power converters using the SVPWM approach have the advantage of dynamic reconfiguration because of the number of cells that can be inactivated or reactivated. Figure  11 shows the simulation results of the output voltage and load current in the different reconfiguration stages. During the evaluation, the reference voltage is 550 V, and the reference frequency is 50 Hz. However, the maximum output voltage that the power converter can be modulated to depends on the number of active cells. This process is started with all six active cells, the output voltage has the contribution of all six cells, reaching 13  Figure 12a shows the voltage FFT spectrum within the frequency range of 0 to 450 Hz, and Figure 12b shows the voltage FFT spectrum within the range of 0 to 20 kHz. The voltage's sinusoidal distortion is at 9.01% in the frequency range of 0 to 20 kHz and the output voltage has good quality which meets the grid connection requirements well.  The evaluation of the frequency response according to the modulation frequency is shown in Figure 10. During the simulation, the peak voltage is fixed at 550 V. At the time of startup, the reference frequency is set to 60 Hz. At time 0.04 s, the frequency is set to 40 Hz. Finally, at time 0.1 s, the reference frequency is set to 50 Hz. During operation, when changing the modulation frequency, the output voltage waveform has 13 levels, and the voltage difference between the two stable levels is 100 V. The simulation results show the output voltage meets the required frequency.
In terms of decentralized connection and the control of multilevel single-phase power converters, power converters using the SVPWM approach have the advantage of dynamic reconfiguration because of the number of cells that can be inactivated or reactivated. Figure 11 shows the simulation results of the output voltage and load current in the different reconfiguration stages. During the evaluation, the reference voltage is 550 V, and the reference frequency is 50 Hz. However, the maximum output voltage that the power converter can be modulated to depends on the number of active cells. This process is started with all six active cells, the output voltage has the contribution of all six cells, reaching 13 levels. At 0.02 s, cell_3 is removed, the system has five cells left, the maximum voltage that can be reached is 500 V, and the output voltage has 11 levels. At 0.04 s, cell_5 is removed, the system has four cells left, the output voltage has nine levels. At 0.06 s, cell_3 and cell_5 are reinserted again, the system restores to the original state with six cells, and the output voltage has 13 levels. The simulation shows that if the system needs to be dynamically reconfigured, the output voltage can be completely satisfied. During dynamic reconfiguration, depending on the characteristics and nature of the load, the reference voltage may reach the maximum limit of the modulation method (based on the number of active cells).    Figure 12 shows the Fast Fourier Transform (FFT) analysis for the output voltage using DSVPWM. The power converter has a reference voltage of 550 V and a reference frequency of 50 Hz. Figure 12a shows the voltage FFT spectrum within the frequency range of 0 to 450 Hz, and Figure 12b shows the voltage FFT spectrum within the range of 0 to 20 kHz. The voltage's sinusoidal distortion is at 9.01% in the frequency range of 0 to 20 kHz and the output voltage has good quality which meets the grid connection requirements well.

Experimental Result
An experimental model is built to evaluate the validity and accuracy of the proposed algorithm. Without losing the generality of the proposed structure, which is the limitation of experimental devices, the experimental model built consists of four cells, and each cell

Experimental Result
An experimental model is built to evaluate the validity and accuracy of the proposed algorithm. Without losing the generality of the proposed structure, which is the limitation of experimental devices, the experimental model built consists of four cells, and each cell uses a Digital signal processing (DSP) controller, driver and IGBT power cell. The experimental prototype is shown in Figure 13. Figure 14 shows the top and bottom faces of a cell. The dynamic circuit structure is built as shown in Figure 2, the connection between the cells is show in Figure 15, and the function of the signal pins used by the DSP is given in Table 4. DSP was developed by Texas Instruments Incorporated (TI). In order to support users of the rapid programming tool, TI has integrated the Code Composer Studio (CCS) compiler into the MATLAB/Simulink software. This experimental program is made by building a control algorithm model on MATLAB/Simulink, then using the built-in compiler to compile and load C code to the DSP board. Information on experimental components is given in Table 5. The parameters of the single-phase power converter with algorithm DSVPWM are shown in Table 6.       Figure 15. Communication between cells in the system for the experimentation.  Figure 16 shows the PWM signals when all cells are active. Figure 16a depicts the PWM signal form for IGBTs in the half bridge A of cells, and Figure 16b depicts the PWM control signal for IGBTs in the half bridge B of cells. The results observed on the oscilloscope show that the PWM signals of the cells operate synchronously, exactly according to the frequency of the modulated AC voltage, and respond exactly to the four-cell nine-level converter.    The cells of the single-phase power converter exchange information about the phase angle with neighboring cells through the asynchronous serial communication interface protocol (SCI-SCIA and SCIB). In this structure, cell_1 is called the master cell. Cell_1 will transmit reference voltage amplitude signal through cells_2, 3, and 4 by using a synchronous serial communication interface (Serial Peripheral Interface-SPI); SPIA is the master (GPIO 58, 60, 61) and SPIB is the slave (GPIO 64, 65, 66). In these cells, SPIA will transmit the amplitude of reference voltage to the next cell, and SPIB will receive the amplitude of the reference voltage from the previous cell. In addition, cell_1 also transmits the reference frequency through cell_2, 3, and 4 by using a synchronous serial communication interface. In a cell, SCID (GPIO 104, 105) will transmit the reference frequency to the next cell, and SCIC (56, 139) will receive the reference frequency from the previous cell.
Let the decentralized power converters be active, in order to synchronize the PWM signal of the cells. A synchronous clock generated from the GPIO pin 06 of the master cell (cell_1) is transmitted to GPIO pin 14 of all remaining cells, including itself; in order to synchronize the voltage reference signal, a synchronous clock generated from GPIO pin 04 of master cell is transmitted to GPIO pin 15 of all remaining cells, including itself. Figure 16 shows the PWM signals when all cells are active. Figure 16a depicts the PWM signal form for IGBTs in the half bridge A of cells, and Figure 16b Figure 17 shows the waveform of the output voltage ( Figure 17a) and load current (Figure 17b) when the all four cells of the power converter are operating, and the output voltage has nine levels. Figure 18 shows the waveform of output voltage (Figure 18a) and load current (Figure 18b) when the power converter has three active cells, and the output voltage has seven levels.   Figure 17 shows the waveform of the output voltage ( Figure 17a) and load current (Figure 17b) when the all four cells of the power converter are operating, and the output voltage has nine levels. Figure 18 shows the waveform of output voltage (Figure 18a Figure 19a. Figure 19b shows the process of repositioning the cells when cell_3 is removed, and the system takes about   Figure 19a. Figure 19b shows the process of repositioning the cells when cell_3 is removed, and the system takes about 5 ms for the sorting process to complete. A similar experimental result when cell_3 is reinserted occurs; the phase angle of the cells takes about 15 ms to interleave evenly in 360 degrees, and the position of cells in the power converter takes about 8 ms to realign desired location. Figure 21 shows the waveform of the output voltage and load current when cell_3 is removed, the power converter has three cells left, and the voltage reaches a new equilibrium with seven levels.
Energies 2022, 14, x FOR PEER REVIEW 20 of 27 5 ms for the sorting process to complete. A similar experimental result when cell_3 is reinserted occurs; the phase angle of the cells takes about 15 ms to interleave evenly in 360 degrees, and the position of cells in the power converter takes about 8 ms to realign desired location. Figure 21 shows the waveform of the output voltage and load current when cell_3 is removed, the power converter has three cells left, and the voltage reaches a new equilibrium with seven levels.   The output voltage of the nine-level single-phase power converter is analyzed by FFT on the oscilloscope; the results are presented in Figure 22. The reference frequency is 50 Hz, in order to evaluate the first harmonic, and as such the output voltage is analyzed with the FFT spectrum in the frequency range of 0 to 450 Hz, as shown in Figure 22a. In addition, to evaluate the output voltage at the switching frequency of 10 kHz, Figure 22b presents FFT analysis of the output voltage in the frequency range of 0 to 45 kHz. The The output voltage of the nine-level single-phase power converter is analyzed by FFT on the oscilloscope; the results are presented in Figure 22. The reference frequency is 50 Hz, in order to evaluate the first harmonic, and as such the output voltage is analyzed with the FFT spectrum in the frequency range of 0 to 450 Hz, as shown in Figure 22a. In addition, to evaluate the output voltage at the switching frequency of 10 kHz, Figure 22b presents FFT analysis of the output voltage in the frequency range of 0 to 45 kHz. The results show that the output voltage has an effective amplitude of 100.7 V, the period of the output voltage is 19.97 ms, and the output voltage frequency is 50.08 Hz. The difference between the output voltage and the reference voltage is 4.4 V, which is equivalent to 4.15%, and the frequency difference between the output voltage and the reference frequency is 0.08 Hz, which is equivalent to 0.16%. The harmonics at the 10 kHz spectrum have an effective value of 10.4 V. The FFT analysis results show that the output voltage has low harmonic distortion. results show that the output voltage has an effective amplitude of 100.7 V, the period of the output voltage is 19.97 ms, and the output voltage frequency is 50.08 Hz. The difference between the output voltage and the reference voltage is 4.4 V, which is equivalent to 4.15%, and the frequency difference between the output voltage and the reference frequency is 0.08 Hz, which is equivalent to 0.16%. The harmonics at the 10 kHz spectrum have an effective value of 10.4 V. The FFT analysis results show that the output voltage has low harmonic distortion.

Conclusions
In this paper, a decentralized control method for multilevel single-phase power converters has been introduced. The study applied the space vector modulation method for the decentralized power converter. The decentralized controlled power converter uses a topology that cells only exchange phase angle information with two neighboring cells, and very little information is exchanged, allowing for structural simplification. The results of this paper show that the number of communication connections and program algorithms have been significantly improved compared with previously published research [11]. The study uses the phase angle self-interleaving algorithm to determine the number cells in the system and the location of each cell, which allows the system to operate with at least three cells. The improved DSVPWM method gave a good response to the proposed structure.
The control algorithm for cells shows that the program is quite simple, which allows for lower cell microprocessors. Evaluation based on simulation results and the experimental model shows that the modulation voltage responds well to the reference signal, with very low sinusoidal distortion. In addition, dynamic reconfiguration has also been demonstrated, which is an advantage of decentralized power converters. The proposed structure and algorithm show that decentralized control power converters using SVPWM can be automatically reconfigured when adding or removing one or more cells.