Single-Stage MV-Connected Charger Using an Ac/Ac Modular Multilevel Converter

Abstract

Modular multilevel converters with non-sinusoidal ac voltage output can reduce cost and volume in medium-voltage-connected electric vehicle battery charging applications. The use of full-bridge submodules in such converters enables single-stage ac/ac voltage conversion, allowing a medium-voltage grid to be directly connected to a medium-frequency isolation transformer. The application of a square wave voltage at the medium-frequency transformer’s single-phase port enhances the converter’s efficiency and power density in comparison to a sinusoidal voltage. This paper presents the analysis and modelling of a modular multilevel converter, comparing its operation with sinusoidal and square wave output voltages. A single control scheme for both output voltage waveforms is proposed for the three-phase and single-phase ac currents, circulating currents, and the energy stored in the submodule capacitors. The control strategy of the three-phase and single-phase port currents is verified through simulation and experiments using a scaled-down prototype, thereby validating its suitability for high-power bidirectional battery chargers.

1. Introduction

The energy transition has increased the demand for large electric vehicles, necessitating the development of high-power chargers [1,2]. The high power level necessary to charge these vehicles requires a connection to the medium-voltage (MV) grid instead of the low-voltage (LV) grid to prevent overloading the electricity network. In traditional architectures, a bulky line frequency transformer (LFT) is used to lower the voltage and provide isolation. State-of-the-art chargers typically include a three-phase ac/dc converter and a dc/dc converter to regulate the intermediate dc voltage for battery charging, as presented in Figure 1a [1,2,3].

The volume of such a transformer, as well as other passive components in the charger, typically decreases with an increase in frequency [4,5]. Therefore, there is currently much interest in replacing the low-frequency transformer with one operating at a higher frequency. This does, however, require a different charger topology.

Various multilevel topologies allow the interfacing of a medium-voltage grid with a medium-frequency transformer (MFT). For example, in a cascaded H-bridge-based solid state transformer, multiple isolated dc/dc converters are applied for isolation [6,7]. The downside of this approach is the increased number of magnetic components and the many high-voltage isolation barriers that are required. Alternatively, a grid-connected ac/ac modular multilevel converter (MMC) can be used to convert the three-phase medium-voltage of the grid to a single-phase ac voltage with a desired voltage level and frequency. This can be interfaced to a medium-frequency transformer, effectively creating a single-stage converter, as shown in Figure 1b [8]. A significant advantage of such a topology over the cascaded H-bridge approach is the need for only one transformer. Due to the high isolation voltage that is required [1,6], this is a considerable advantage in medium-voltage applications.

The ac/ac MMC is suitable for MV-connected charging due to its high efficiency and scalability to different voltage and power levels [9]. In [10], it has been shown that, although a sine wave voltage might seem like an obvious choice, it is beneficial to generate a square wave voltage to apply across the transformer, since this leads to less power pulsation and a reduced number of MMC submodules (SMs).

Ac/ac MMCs have been proposed for various applications [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The MMC converts the MV grid to a medium-frequency sinusoidal voltage in [8,9,11,12,13,14,15] and to a low-frequency sinusoidal voltage in [16,17]. In addition, in [18,19,20] and in [10,21,22,23,24,25,26,27], the MMC converts a single-phase and a three-phase MV grid, respectively, into a square wave medium-frequency voltage. Furthermore, in [28,29,30,31], the MMC submodules directly connect to multiple medium-frequency transformers and ac/dc converters, and the MMC legs connect to a dc bus voltage. However, the MMC with a single-phase ac and a dc bus voltage port, as in [26,27,28,29,30,31], is not needed in this application. Therefore, the proposed ultrafast charger in Figure 1b implements a three-phase to single-phase ac/ac MMC with FB SMs.

Notably, each MMC architecture requires control of its currents and SM capacitor voltages. Many control schemes for the three-phase to single-phase ac/ac MMC have been proposed. For example, references [8,9,16,17] propose control schemes for the MMC using sinusoidal single-phase voltage, while [10,21,22,23,24,25] propose control schemes for the MMC using square wave single-phase voltage. However, the control strategy for a bidirectional single-stage MV-connected MMC with a medium-frequency single-phase port that can be adapted for different voltage waveforms has not yet been proposed in the literature. Therefore, this paper focuses on a hierarchical control scheme that regulates the power transfer between the three-phase MV grid and single-phase ports for either sinusoidal or square wave voltage waveforms while balancing the MMC SM capacitor voltages. The control scheme can be used in many applications, but in this paper, the example of a single-stage ac/dc converter will be used.

This paper is organized as follows: Section 2 explains the ac/ac MMC fundamentals and decomposition of circuit states, Section 3 proposes a hierarchical control scheme, Section 4 provides the simulation results, Section 5 shows the experimental verification, and Section 6 outlines the main findings.

2. Modular Multilevel Converter

The submodule is the fundamental MMC component, typically consisting of a half- or full-bridge converter with a bus capacitor. Since for ac/ac conversion, the MMC submodules should generate negative voltages, full-bridge submodules are generally necessary [8,32], as shown in Figure 2. Half-bridge SMs can be utilized for ac/ac conversion by creating a virtual dc-bus voltage; however, as demonstrated in [14], the arm voltage of the MMC equipped with HB SMs is twice as high as the arm voltage of the MMC with FB SMs. Therefore, this paper investigates the ac/ac MMC with full-bridges as it requires fewer submodules than the ac/ac MMC with half-bridges.

The series connection of N of these SMs constitutes a converter arm. Furthermore, to allow current control, the MMC arms contain a series inductance [33]. The two converter arms that are connected to the same grid phase compose a converter leg. All MMC legs are connected to the single-phase port and together can be used to transfer power to an ac/dc converter stage. Changing the number of submodules per arm N allows scaling of the converter to different voltage and power levels. The available arm voltage levels at the single-phase port of the converter depend on the number of series-connected SMs and their capacitor voltages.

2.1. Fundamentals

The MMC is used to transfer power bidirectionally between the three-phase ac grid and the single-phase port, which is connected to the MFT primary side terminals. The transformer is used in this converter to offer galvanic isolation between the three-phase ac voltages $u_a$, $u_b$, and $u_c$ and the dc voltage $u_{dc}$. Additionally, its turns ratio can be used to step-up or step-down the transformer primary side voltage $u_{t,p}$ in relation to the secondary side voltage $u_{t,s}$. Since the ac/dc converter in Figure 2 acts as a voltage source, and the transformer is considered to be ideal, the MMC single-phase port voltage amplitude, i.e., $\sqrt{2}{U}_{t,p}$ for sinusoidal or $U_{t,p}$ for square wave, is equal to the primary referred EV battery voltage $r{u}_{dc}$, in which r is the MFT winding turn ratio. Since this paper is focused on the ac/ac MMC model and control, it does not cover the ac/dc converter operation and the transformer’s voltage and frequency optimization.

An averaged equivalent circuit allows modelling the time-average operation of the ac/ac MMC [34]. The model employs controllable average voltage sources to represent the SMs of each arm, as shown in Figure 3. In the model, for each of the three phases $y\in \left\{a,b,c\right\}$, there is an upper and a lower arm indicated by $\mathrm{x}\in \left\{\mathrm{u},\mathrm{l}\right\}$. The resistance $R_y$ and inductance $L_y$ are assumed to be equal in each arm. The ac/ac MMC interfaces the three-phase MV grid with phase voltages

$$u_y=\sqrt{2}{U}_{y}cos\left({\omega }_{1}t+{\theta }_y\right),$$

(1)

where the three-phase voltage angles are ${\theta }_a=0$, ${\theta }_b=-\frac{2\pi }{3}$, and ${\theta }_c=\frac{2\pi }{3}$. In this paper, the single-phase transformer’s primary side is operated with either a sinusoidal voltage

$$u_{t,p}=\sqrt{2}{U}_{t,p}cos\left({\omega }_{2}t\right),$$

(2)

or a square wave voltage

$$u_{t,p}=\left\{\begin{array}{cc}\hfill U_{t,p},\hfill & \hfill \mathrm{if}\left({\omega }_{2}tmod2\pi \right)\le \pi ,\hfill \\ \hfill -U_{t,p},\hfill & \hfill \mathrm{otherwise},\hfill \end{array}\right.$$

(3)

in which mod is the modulo operator. The angular frequencies ${\omega }_1=2\pi f_1$ and ${\omega }_2=2\pi f_2$ correspond to the frequencies $f_1$ and $f_2$ of the three-phase and single-phase port voltages, respectively. In addition, the three-phase and single-phase voltages $U_y$ and $U_{t,p}$ represent their RMS values.

To decouple the upper and lower arm voltages and currents, the primary winding of the transformer can be thought of as being composed of two identical parts. Referencing all voltages to the midpoint voltage $u_m$, as indicated in Figure 3, Kirchhoff’s voltage law is used to describe the dynamics of the upper and lower arm currents as

$$L_y\frac{\mathrm{d}}{\mathrm{d}t}{\mathsf{ı}}_y^{\mathrm{u}}+R_y{\mathsf{ı}}_y^{\mathrm{u}}=u_y^{\mathrm{u}}+u_y+u_s-\frac{u_{t,p}}{2}-u_m,$$

(4)

$$L_y\frac{\mathrm{d}}{\mathrm{d}t}{\mathsf{ı}}_y^{\mathsf{l}}+R_y{\mathsf{ı}}_y^{\mathrm{l}}=u_y^{\mathrm{l}}-u_y-u_s-\frac{u_{t,p}}{2}+u_m,$$

(5)

where ${\mathsf{ı}}_y^{\mathrm{x}}$ is the upper or lower arm current, $u_y^{\mathrm{x}}$ the equivalent upper or lower arm voltage in phase y, and $u_s$ is the star-point voltage.

A summed capacitor voltage $v_y^{\mathrm{x}}$ can be introduced, that follows from the SM capacitor voltage of all SMs in each arm $v_y^{\mathrm{x}}={\displaystyle \sum _{k=1}^N}{v}_{y,k}^{\mathrm{x}}$, where $v_{y,k}^{\mathrm{x}}$ is the capacitor voltage of the kth submodule. In addition, assuming that every SM capacitance is equal to $C_{SM}$, the equivalent capacitance is $C_{\sigma }=\frac{C_{SM}}{N}$. Then, the total arm energy is $w_y^{\mathrm{x}}=\frac{C_{\sigma }}{2}{\left(v_y^{\mathrm{x}}\right)}^2$. Assuming that within each arm, every submodule capacitor voltage is equal, the voltage sources $u_y^{\mathrm{x}}$ and current sources ${\mathsf{ı}}_{Cy}^{\mathrm{x}}$ in Figure 3 can be controlled through an insertion index $n_y^{\mathrm{x}}$, as $u_y^{\mathrm{x}}=n_y^{\mathrm{x}}{v}_y^{\mathrm{x}}$ and ${\mathsf{ı}}_{Cy}^{\mathrm{x}}=-C_{\sigma }\frac{d}{dt}{v}_y^{\mathrm{x}}=n_y^{\mathrm{x}}{\mathsf{ı}}_y^{\mathrm{x}}$. The insertion index represents the combined contribution of each submodule to the arm voltage and is defined as $n_y^{\mathrm{x}}=\frac{1}{N}{\displaystyle \sum _{k=1}^N}{S}_{y,k}^{\mathrm{x}}$, with $S_{y,k}^{\mathrm{x}}\in \{-1,0,1\}$, the kth SM’s switching function. Besides controlling the power exchange with the three-phase and single-phase ports, the insertion index is used to control the summed capacitor voltages $v_y^{\mathrm{x}}$ [8,16,32,33].

2.2. Decomposition of Circuit States

It is convenient to separate the converter’s electrical quantities into differential-mode ($\Delta$) and common-mode ($\Sigma$) components [8]. This allows the MMC’s upper and lower arm voltages and currents, and the summed capacitor voltages to be decomposed as

$$u_y^{\Delta }=\frac{1}{2}\left[u_y^{\mathrm{u}}-u_y^{\mathrm{l}}\right],$$

(6)

$$u_y^{\Sigma }=\frac{1}{2}\left[u_y^{\mathrm{u}}+u_y^{\mathrm{l}}\right],$$

(7)

$${\mathsf{ı}}_y^{\Delta }=\frac{1}{2}\left[{\mathsf{ı}}_y^{\mathrm{u}}-{\mathsf{ı}}_y^{\mathrm{l}}\right],$$

(8)

$${\mathsf{ı}}_y^{\Sigma }=\frac{1}{2}\left[{\mathsf{ı}}_y^{\mathrm{u}}+{\mathsf{ı}}_y^{\mathrm{l}}\right],$$

(9)

$$v_y^{\Delta }=\frac{1}{2}\left[v_y^{\mathrm{u}}-v_y^{\mathrm{l}}\right],$$

(10)

$$v_y^{\Sigma }=\frac{1}{2}\left[v_y^{\mathrm{u}}+v_y^{\mathrm{l}}\right].$$

(11)

Additionally, by defining the instantaneous power absorbed in each MMC arm as $p_y^{\mathrm{x}}=-u_y^{\mathrm{x}}{\mathsf{ı}}_y^{\mathrm{x}}$, it is possible to define differential- and common-mode power and energy as

$$p_y^{\Delta }=\frac{1}{2}\left[p_y^{\mathrm{u}}-p_y^{\mathrm{l}}\right],$$

(12)

$$p_y^{\Sigma }=\frac{1}{2}\left[p_y^{\mathrm{u}}+p_y^{\mathrm{l}}\right],$$

(13)

$$w_y^{\Delta }=\frac{1}{2}\left[w_y^{\mathrm{u}}-w_y^{\mathrm{l}}\right],$$

(14)

$$w_y^{\Sigma }=\frac{1}{2}\left[w_y^{\mathrm{u}}+w_y^{\mathrm{l}}\right].$$

(15)

Then, substitution of (6), (7), (8), and (9) into (4) and (5) leads to

$$L_y\frac{d}{dt}{\mathsf{ı}}_y^{\Delta }+R_y{\mathsf{ı}}_y^{\Delta }=u_y^{\Delta }+u_y+u_s-u_m,$$

(16)

$$L_y\frac{d}{dt}{\mathsf{ı}}_y^{\Sigma }+R_y{\mathsf{ı}}_y^{\Sigma }=u_y^{\Sigma }-\frac{u_{t,p}}{2}.$$

(17)

Since ${\mathsf{ı}}_a+{\mathsf{ı}}_b+{\mathsf{ı}}_c=0$, the sum of (16) with phase a, b, and c results in

$$u_s=-\frac{\left(u_a+u_b+u_c+u_a^{\Delta }+u_b^{\Delta }+u_c^{\Delta }\right)}{3}+u_m.$$

(18)

3. MMC Control

The ac/ac MMC control scheme is separated into two parts: (1) three-phase ac grid current control, and (2) cell balancing and common-mode current control. The first part controls the three-phase ac grid currents ${\mathsf{ı}}_y$ for a given active and reactive power reference, $P^{\ast }$ and $Q^{\ast }$, respectively, through the differential-mode voltages $u_y^{\Delta }$. The second part controls the energies $w_y^{\Sigma }$ and $w_y^{\Delta }$ stored in the MMC arm capacitors to a given summed capacitor voltage reference $V_y^{\Sigma \ast }$ [23,24]. It also controls the MMC common-mode currents ${\mathsf{ı}}_y^{\Sigma }$ through the common-mode voltages $u_y^{\Sigma }$ [8].

Figure 4 schematically depicts the MMC control. A phase-locked loop (PLL) is used to synchronize the control of the converter to the three-phase grid voltage $u_y$ [16,32]. It additionally estimates ${\omega }_1$. While this frequency is imposed by the grid voltage, the frequency ${\omega }_2$ of the single-phase voltage and current can be chosen for the power transfer through the transformer, as long as it is not equal to ${\omega }_1$. The transformer voltage and frequency can be chosen to optimize the charger efficiency and power density [10,35,36] but are limited by the converter’s voltage rating and switching frequency, respectively. In this paper, it is chosen as ${\omega }_2\gg {\omega }_1$ to decrease the transformer size.

The two controller outputs $u_y^{\Delta \ast }$ and $u_y^{\Sigma \ast }$ are converted back into the arm voltage references $u_y^{\mathrm{x}\ast }$, which are then divided by the summed capacitor voltage reference $V_y^{\Sigma \ast }$ to yield the insertion index $n_y^{\mathrm{x}}$ [16,32]. The summed capacitor voltage reference for sinusoidal and square wave single-phase output voltage is given by $V_y^{\Sigma \ast }=\sqrt{2}{U}_y+\frac{\sqrt{2}{U}_{t,p}}{2}$ and $V_y^{\Sigma \ast }=\sqrt{2}{U}_y+\frac{U_{t,p}}{2}$, respectively. Finally, the insertion index $n_y^{\mathrm{x}}$ yields a switching function $S_{y,k}^{\mathrm{x}}$ for every submodule in an arm through PWM and a sorting algorithm that balances the SM capacitor voltages within the arms. Furthermore, the ac/dc converter uses either PWM or square wave modulation to produce the transformer voltage (2) or (3). However, as this paper’s focus is on the ac/ac MMC, the ac/dc converter modulation is not discussed.

3.1. Three-Phase ac Grid Current Control

The grid voltages and currents are assumed to be sinusoidal with frequency ${\omega }_1$. The differential-mode components are therefore also sinusoidal:

$$-u_y^{\Delta }=\sqrt{2}{U}_y^{\Delta }cos\left({\omega }_{1}t+{\theta }_y^{\Delta }\right),$$

(19)

$${\mathsf{ı}}_y^{\Delta }=\sqrt{2}{I}_y^{\Delta }cos\left({\omega }_{1}t+{\phi }_y^{\Delta }\right).$$

(20)

The PLL in Figure 4 outputs an estimate ${\check{\theta }}_y$ of the grid voltage angle ${\theta }_y$ for synchronization. The p-q theory for three-phase systems [37] is then used to calculate a reference for the d and q components of the delta-current components ${\mathsf{ı}}_y^{\Delta }$ for a desired instantaneous active power p and reactive power q. The most straightforward voltage-oriented control consists of a current controller in the $dq$ frame and active and reactive power feed-forward control [38]. Therefore, the $dq$-current references are calculated for the desired active $P^{\ast }$ and reactive $Q^{\ast }$ power as

$$\left[\begin{array}{c}{\mathsf{ı}}_d^{\Delta \ast }\\ {\mathsf{ı}}_q^{\Delta \ast }\end{array}\right]=\frac{1}{u_d^2+u_q^2}\left[\begin{array}{cc}{u}_d& u_q\\ u_q& -u_d\end{array}\right]\frac{1}{3}\left[\begin{array}{c}{P}^{\ast }\\ Q^{\ast }\end{array}\right],$$

(21)

where ${\mathsf{ı}}_d^{\Delta \ast }$ and ${\mathsf{ı}}_q^{\Delta \ast }$ denote the setpoints for ${\mathsf{ı}}_d^{\Delta }$ and ${\mathsf{ı}}_q^{\Delta }$. The direct and quadrature grid voltages $u_d$ and $u_q$ are obtained from the measured $u_y$ in Figure 5a. Applying the $dq$-transformation to (16) leads to

$$L_y\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{\mathsf{ı}}_d^{\Delta }\\ {\mathsf{ı}}_q^{\Delta }\end{array}\right]+R_y\left[\begin{array}{c}{\mathsf{ı}}_d^{\Delta }\\ {\mathsf{ı}}_q^{\Delta }\end{array}\right]+{\omega }_1{L}_y\left[\begin{array}{c}-{\mathsf{ı}}_q^{\Delta }\\ {\mathsf{ı}}_d^{\Delta }\end{array}\right]=\left[\begin{array}{c}{u}_d^{\Delta }\\ u_q^{\Delta }\end{array}\right]+\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right].$$

(22)

As depicted in Figure 5b, the $dq$-currents ${\mathsf{ı}}_d^{\Delta }$ and ${\mathsf{ı}}_q^{\Delta }$ are controlled through a standard $dq$ control scheme [38,39].

3.2. Cell Balancing and Common-Mode Current Control

The common mode voltage component $u_y^{\Sigma }$ is used to control both the energy stored in the MMC submodule capacitors and the common-mode current ${\mathsf{ı}}_y^{\Sigma }$. Therefore, this voltage also controls the single-phase port current ${\mathsf{ı}}_{t,p}$ [8]. To decouple ${\mathsf{ı}}_y^{\Sigma }$, the $abc$-to-$\alpha \beta \gamma$ transformation is applied to (17) to obtain

$$L_y\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{\mathsf{ı}}_{\alpha }^{\Sigma }\\ {\mathsf{ı}}_{\beta }^{\Sigma }\\ {\mathsf{ı}}_{\gamma }^{\Sigma }\end{array}\right]+R_y\left[\begin{array}{c}{\mathsf{ı}}_{\alpha }^{\Sigma }\\ {\mathsf{ı}}_{\beta }^{\Sigma }\\ {\mathsf{ı}}_{\gamma }^{\Sigma }\end{array}\right]=\left[\begin{array}{c}{u}_{\alpha }^{\Sigma }\\ u_{\beta }^{\Sigma }\\ u_{\gamma }^{\Sigma }\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ \frac{u_{t,p}}{2}\end{array}\right],$$

(23)

where the common-mode voltages $u_{\alpha }^{\Sigma }$ and $u_{\beta }^{\Sigma }$ regulate the internally circulating currents ${\mathsf{ı}}_{\alpha }^{\Sigma }$ and ${\mathsf{ı}}_{\beta }^{\Sigma }$, and $u_{\gamma }^{\Sigma }$ controls the single-phase port current through ${\mathsf{ı}}_{\gamma }^{\Sigma }$, since ${\mathsf{ı}}_{t,p}={\mathsf{ı}}_a^{\Sigma }+{\mathsf{ı}}_b^{\Sigma }+{\mathsf{ı}}_c^{\Sigma }=3{\mathsf{ı}}_{\gamma }^{\Sigma }$ [8].

The arm capacitor voltages can be controlled through the MMC’s power balance. Decoupled control requires $u_y^{\Sigma }$ to contain at least two independent components $u_{y1}^{\Sigma }$ and $u_{y2}^{\Sigma }$ with frequencies $f_1$ and $f_2$, which therefore also appear in ${\mathsf{ı}}_y^{\Sigma }$ [10]. The steady-state common-mode voltage then becomes

$$u_y^{\Sigma }=u_{y1}^{\Sigma }+u_{y2}^{\Sigma }=\sqrt{2}{U}_{y1}^{\Sigma }cos\left({\omega }_{1}t+{\theta }_{y1}^{\Sigma }\right)+u_{y2}^{\Sigma },$$

(24)

which results in

$${\mathsf{ı}}_y^{\Sigma }={\mathsf{ı}}_{y1}^{\Sigma }+{\mathsf{ı}}_{y2}^{\Sigma }=\sqrt{2}{I}_{y1}^{\Sigma }cos\left({\omega }_{1}t+{\phi }_{y1}^{\Sigma }\right)+{\mathsf{ı}}_{y2}^{\Sigma }.$$

(25)

For the sinusoidal single-phase port voltage waveform in (2), $u_{y2}^{\Sigma }$ and ${\mathsf{ı}}_{y2}^{\Sigma }$ are given by

$$u_{y2}^{\Sigma }=\sqrt{2}{U}_{y2}^{\Sigma }cos\left({\omega }_{2}t+{\theta }_{y2}^{\Sigma }\right),$$

(26)

and

$${\mathsf{ı}}_{y2}^{\Sigma }=\sqrt{2}{I}_{y2}^{\Sigma }cos\left({\omega }_{2}t+{\phi }_{y2}^{\Sigma }\right).$$

(27)

On the other hand, for the square wave single-phase port voltage waveform in (3), the instantaneous voltage $u_{y2}^{\Sigma }$ and current ${\mathsf{ı}}_{y2}^{\Sigma }$ in Figure 6 are derived as in [40,41]:

$$u_{y2}^{\Sigma }=\left\{\begin{array}{cc}\hfill U_{y2}^{\Sigma },\hfill & \hfill \mathrm{if}\left({\omega }_{2}t+{\theta }_{y2}^{\Sigma }mod2\pi \right)\le \pi ,\hfill \\ \hfill -U_{y2}^{\Sigma },\hfill & \hfill \mathrm{otherwise},\hfill \end{array}\right.$$

(28)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill {\mathsf{ı}}_{y2}^{\Sigma }\left(t_1\right)=\frac{1}{4f_2{L}_y}\left[\frac{U_{t,p}}{2}-U_{y2}^{\Sigma }\left(1-\frac{2{\theta }_{y2}^{\Sigma }}{\pi }\right)\right],\hfill \\ \hfill {\mathsf{ı}}_{y2}^{\Sigma }\left(t_2\right)=\frac{1}{4f_2{L}_y}\left[U_{y2}^{\Sigma }-\frac{U_{t,p}}{2}\left(1-\frac{2{\theta }_{y2}^{\Sigma }}{\pi }\right)\right],\hfill \end{array}\right.\end{array}$$

(29)

with ${\mathsf{ı}}_{y2}^{\Sigma }\left(t_3\right)=-{\mathsf{ı}}_{y2}^{\Sigma }\left(t_1\right)$ and ${\mathsf{ı}}_{y2}^{\Sigma }\left(t_4\right)=-{\mathsf{ı}}_{y2}^{\Sigma }\left(t_2\right)$.

The components with frequency ${\omega }_1$ regulate the energy difference between the upper and lower arms in each phase leg, and components with frequency ${\omega }_2$ control the power transfer between the MMC and the single-phase port, as well as the total energy that is stored in the equivalent submodules’ capacitance in each phase leg.

Substitution of (19), (20), (24), and (25) into (12) and (13) results in

$$\begin{array}{cc}\hfill p_y^{\Delta }=& \hfill U_y^{\Delta }{I}_{y1}^{\Sigma }\left[cos\left({\theta }_y^{\Delta }-{\phi }_{y1}^{\Sigma }\right)+cos\left(2{\omega }_{1}t+{\theta }_y^{\Delta }+{\phi }_{y1}^{\Sigma }\right)\right]\hfill \\ \hfill & \hfill +\sqrt{2}{U}_y^{\Delta }{\mathsf{ı}}_{y2}^{\Sigma }cos\left({\omega }_{1}t+{\theta }_y^{\Delta }\right)-\sqrt{2}{u}_{y2}^{\Sigma }{I}_y^{\Delta }cos\left({\omega }_{1}t+{\phi }_y^{\Delta }\right)\hfill \\ \hfill & \hfill -U_{y1}^{\Sigma }{I}_y^{\Delta }\left[cos\left({\theta }_{y1}^{\Sigma }-{\phi }_y^{\Delta }\right)+cos\left(2{\omega }_{1}t+{\theta }_{y1}^{\Sigma }+{\phi }_y^{\Delta }\right)\right],\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill p_y^{\Sigma }=& \hfill U_y^{\Delta }{I}_y^{\Delta }\left[cos\left({\theta }_y^{\Delta }-{\phi }_y^{\Delta }\right)+cos\left(2{\omega }_{1}t+{\theta }_y^{\Delta }+{\phi }_y^{\Delta }\right)\right]\hfill \\ \hfill & \hfill -U_{y1}^{\Sigma }{I}_{y1}^{\Sigma }\left[cos\left({\theta }_{y1}^{\Sigma }-{\phi }_{y1}^{\Sigma }\right)+cos\left(2{\omega }_{1}t+{\theta }_{y1}^{\Sigma }+{\phi }_{y1}^{\Sigma }\right)\right]\hfill \\ \hfill & \hfill -\sqrt{2}{U}_{y1}^{\Sigma }{\mathsf{ı}}_{y2}^{\Sigma }cos\left({\omega }_{1}t+{\theta }_{y1}^{\Sigma }\right)-\sqrt{2}{u}_{y2}^{\Sigma }{I}_{y1}^{\Sigma }cos\left({\omega }_{1}t+{\phi }_{y1}^{\Sigma }\right)-u_{y2}^{\Sigma }{\mathsf{ı}}_{y2}^{\Sigma },\hfill \end{array}$$

(31)

Given that ${\phi }_{y1}^{\Sigma }={\theta }_{y1}^{\Sigma }-\frac{\pi }{2}$ due to the arm inductance, and choosing ${\theta }_{y1}^{\Sigma }={\theta }_y^{\Delta }$, the average differential- and common-mode power associated with each converter leg are found by averaging (30) and (31) over a fundamental period:

$$P_y^{\Delta }=-U_{y1}^{\Sigma }{I}_y^{\Delta }cos\left({\theta }_y^{\Delta }-{\phi }_y^{\Delta }\right),$$

(32)

$$P_y^{\Sigma }=P_{y1}^{\Sigma }-P_{y2}^{\Sigma }=U_y^{\Delta }{I}_y^{\Delta }cos\left({\theta }_y^{\Delta }-{\phi }_y^{\Delta }\right)-u_{y2}^{\Sigma }{\mathsf{ı}}_{y2}^{\Sigma }.$$

(33)

In steady-state, each MMC arm exchanges an average power with the three-phase ac grid $P_{y1}^{\Sigma }=U_y^{\Delta }{I}_y^{\Delta }cos\left({\theta }_y^{\Delta }-{\phi }_y^{\Delta }\right)$. The single-phase port power $P_{y2}^{\Sigma }$ is for the sinusoidal wave quantities in (26) and (27) equal to

$$P_{y2}^{\Sigma }=U_{y2}^{\Sigma }{I}_{y2}^{\Sigma }cos\left({\theta }_{y2}^{\Sigma }-{\phi }_{y2}^{\Sigma }\right)=\frac{U_{y2}^{\Sigma }{U}_{t,p}}{2{\omega }_2{L}_y}sin\left({\theta }_{y2}^{\Sigma }\right),$$

(34)

and for square wave quantities in (28) and (29) equal to

$$P_{y2}^{\Sigma }=\frac{U_{y2}^{\Sigma }{U}_{t,p}{\theta }_{y2}^{\Sigma }(\pi -|{\theta }_{y2}^{\Sigma }\left|\right)}{4{\pi }^2{f}_2{L}_y},$$

(35)

which is analogous to the power transfer in Dual Active Bridge (DAB) converters [24,25,40,41,42,43,44].

The difference $P_y^{\Sigma }=P_{y1}^{\Sigma }-P_{y2}^{\Sigma }$ therefore charges the submodule capacitors in the concerning leg. A control scheme that makes use of this is presented in Figure 7 to regulate the energy stored in the MMC equivalent arm capacitors $C_{\sigma }$ to the references

$$W_y^{\Delta \ast }=0,$$

(36)

$$W_y^{\Sigma \ast }=\frac{C_{\sigma }}{2}{\left(V_y^{\Sigma \ast }\right)}^2.$$

(37)

The control scheme shown in Figure 7 mitigates the steady-state error between the differential- and common-mode energies in (14) and (15) and their references in (36) and (37) through the use of PI controllers. This controls the submodule capacitor voltages to their desired value. The $w_y^{\Delta }$-controller outputs $P_y^{\Delta \ast }$. Then, through (32), assuming $I_y^{\Delta }=\frac{P^{\ast }}{6U_y^{\Delta }}$, the differential-mode energy in each leg is controlled through the amplitude ${\widehat{U}}_{y1}^{\Sigma }$ of the voltage $u_{y1}^{\Sigma }$. Since this component is chosen to be in phase with $u_y^{\Delta }$, this amplitude is multiplied by the normalized differential-mode voltage to create $u_{y1}^{\Sigma \ast }$.

On average, in steady-state, each MMC leg transfers the same power to the MFT $P_{t,p}=2\left(P_{a2}^{\Sigma }+P_{b2}^{\Sigma }+P_{c2}^{\Sigma }\right)=6P_{y2}^{\Sigma }$, where $P_{t,p}=P$, ideally. The components with frequency ${\omega }_2$ in (26) and (27) for sinusoidal wave quantities and (28) and (29) for square wave quantities control the power exchange of each MMC leg with the single-phase port. By setting the amplitude of this common-mode voltage component equal to half the primary referred DC-bus voltage as ${\widehat{U}}_{y2}^{\Sigma }=\frac{{\widehat{U}}_{t,p}}{2}=\frac{r{u}_{dc}}{2}$, and since ${\theta }_{y2}^{\Sigma }$ and ${\phi }_{y2}^{\Sigma }$ are linked through the arm inductance and resistance, the power that is transferred by each leg $2P_{y2}^{\Sigma }$ can be controlled independently through the angle ${\theta }_{y2}^{\Sigma }$ of $u_{y2}^{\Sigma }$ with respect to the transformer voltage $u_{t,p}$. The $w_y^{\Sigma }$-controller controls the total energy in the submodule capacitors in the leg by adjusting this power. The controller outputs a power variation ${\tilde{P}}_{y2}^{\Sigma \ast }$ needed to balance the energy in the SM capacitors. Moreover, the power $P_{y2}^{\Sigma \ast }=\frac{P^{\ast }}{6}$ transferred between $u_y^{\Sigma }$ and $\frac{u_{t,p}}{2}$ is added as a feed-forward contribution to ${\tilde{P}}_{y2}^{\Sigma \ast }$ in Figure 7, resulting in a faster control response.

The power transfer between $u_y^{\Sigma }$ and $u_{t,p}$ for a sinusoidal voltage and current follows from (34) and is given by the phase-shift angle reference

$${\theta }_{y2}^{\Sigma \ast }=arcsin\left(\frac{2{\omega }_2{L}_y{P}_{y2}^{\Sigma \ast }}{U_{y2}^{\Sigma }{U}_{t,p}}\right),$$

(38)

that is used to linearize the loop gain of the system, through construction of the sinusoidal voltage reference

$$u_{y2}^{\Sigma \ast }=\left(\sqrt{2}{U}_{y2}^{\Sigma }+U_{y,D}^{\mathrm{x}}\right)cos\left({\omega }_{2}t+{\theta }_{y2}^{\Sigma \ast }\right).$$

(39)

The extra term $U_{y,D}^{\mathrm{x}}$ can be used to compensate for the voltage drop across the MMC switches, which can be significant. Note that its value depends on the arm current direction. For instance, considering SMs with identical IGBTs, their forward voltage drop $U_{CE}$ can be modelled as $U_{y,D}^{\mathrm{x}}=2N{U}_{CE}$. The compensation of $U_{y,D}^{\mathrm{x}}$ for the differential-mode voltages is not covered in this paper.

The power transfer between $u_y^{\Sigma }$ and $u_{t,p}$ for a square wave voltage follows from (35) and is given by the phase-shift angle

$${\theta }_{y2}^{\Sigma \ast }=\frac{\pi }{2}\left[1-\sqrt{1-\frac{16f_2{L}_y\left|P_{y2}^{\Sigma \ast }\right|}{U_{y2}^{\Sigma }{U}_{t,p}}}\right]\frac{P_{y2}^{\Sigma \ast }}{|P_{y2}^{\Sigma \ast }|},$$

(40)

which is used to linearize the loop gain, creating the square wave voltage reference

$$u_{y2}^{\Sigma \ast }=\left\{\begin{array}{cc}\hfill U_{y2}^{\Sigma }+U_{y,D}^{\mathrm{x}},\hfill & \hfill \mathrm{if}\left({\omega }_{2}t+{\theta }_{y2}^{\Sigma \ast }mod2\pi \right)\le \pi ,\hfill \\ \hfill -\left(U_{y2}^{\Sigma }+U_{y,D}^{\mathrm{x}}\right),\hfill & \hfill \mathrm{otherwise}.\hfill \end{array}\right.$$

(41)

Compensation of the forward voltage drop is especially important for square wave operation, to make sure that the current peaks ${\mathsf{ı}}_{y2}^{\Sigma }\left(t_1\right)$ and ${\mathsf{ı}}_{y2}^{\Sigma }\left(t_2\right)$ in Figure 6 are aligned.

To update the voltage $u_{y2}^{\Sigma \ast }$, a Zero-Order Hold (ZOH) is added in the control scheme, as shown in Figure 7. The ZOH makes sure that the phase angle ${\theta }_{y2}^{\Sigma \ast }$ is updated only once per period $\frac{1}{f_2}$ of the single-phase port.

The closed-loop control presented in Figure 7 generates three phase angles ${\theta }_{y2}^{\Sigma }$ that create the common-mode voltages $u_{y2}^{\Sigma }$. Alternatively, it is possible to make the phase angle ${\theta }_{\gamma 2}^{\Sigma }$ of the three legs equal and balance the capacitor voltages with orthogonal voltage components instead. This can be achieved by, for example, applying the $abc$-to-$\alpha \beta \gamma$ transformation to (34) and (35), resulting in

$$\left[\begin{array}{c}{P}_{\alpha 2}^{\Sigma }\\ P_{\beta 2}^{\Sigma }\\ P_{\gamma 2}^{\Sigma }\end{array}\right]=\frac{U_{t,p}sin{\theta }_{\gamma 2}^{\Sigma }}{2{\omega }_2{L}_y}\left[\begin{array}{c}{U}_{\alpha 2}^{\Sigma }\\ U_{\beta 2}^{\Sigma }\\ U_{\gamma 2}^{\Sigma }\end{array}\right],$$

(42)

$$\left[\begin{array}{c}{P}_{\alpha 2}^{\Sigma }\\ P_{\beta 2}^{\Sigma }\\ P_{\gamma 2}^{\Sigma }\end{array}\right]=\frac{U_{t,p}{\theta }_{\gamma 2}^{\Sigma }(\pi -|{\theta }_{\gamma 2}^{\Sigma }\left|\right)}{4{\pi }^2{f}_2{L}_y}\left[\begin{array}{c}{U}_{\alpha 2}^{\Sigma }\\ U_{\beta 2}^{\Sigma }\\ U_{\gamma 2}^{\Sigma }\end{array}\right],$$

(43)

for sinusoidal and square wave quantities, respectively.

From (42) and (43), it is clear that decoupled control of $P_{\alpha 2}^{\Sigma }$, $P_{\beta 2}^{\Sigma }$, and $P_{\gamma 2}^{\Sigma }$ is possible by controlling $U_{\alpha 2}^{\Sigma }$, $U_{\beta 2}^{\Sigma }$, and $U_{\gamma 2}^{\Sigma }$, respectively. Therefore, the decoupled closed-loop control of $u_{\alpha 2}^{\Sigma }$, $u_{\beta 2}^{\Sigma }$, and $u_{\gamma 2}^{\Sigma }$ for sinusoidal and square wave single-phase port voltages is shown in Figure 8. Clark transformation, i.e., matrix $\mathbf{T}$ in Figure 8, is used to convert $abc$-to-$\alpha \beta \gamma$. The power ${\tilde{P}}_{\gamma 2}^{\Sigma \ast }$ can be controlled with the power transfer angle ${\theta }_{\gamma 2}^{\Sigma \ast }$ or the voltage amplitude ${\widehat{U}}_{\gamma 2}^{\Sigma \ast }$. For example, by fixing the voltage amplitude ${\widehat{U}}_{\gamma 2}^{\Sigma \ast }$, the power ${\tilde{P}}_{\gamma 2}^{\Sigma \ast }$ is controlled with the angle ${\theta }_{\gamma 2}^{\Sigma \ast }$. This angle is used to create $u_{\alpha 2}^{\Sigma \ast }$, $u_{\beta 2}^{\Sigma \ast }$ and $u_{\gamma 2}^{\Sigma \ast }$, in which the voltage amplitudes ${\widehat{U}}_{\alpha 2}^{\Sigma \ast }$ and ${\widehat{U}}_{\beta 2}^{\Sigma \ast }$ are calculated using ${\tilde{P}}_{\alpha 2}^{\Sigma \ast }$ and ${\tilde{P}}_{\beta 2}^{\Sigma \ast }$, respectively. Finally, the inverse Clarke transformation, i.e., $\alpha \beta \gamma$-to-$abc$, is used to obtain $u_{y2}^{\Sigma \ast }$.

4. Simulation Results

According to [8], the number of submodules per arm in the ac/ac MMC is given by $N=⌈\frac{{\widehat{U}}_y+\frac{{\widehat{U}}_{\mathrm{t}},\mathrm{p}}{2}}{{\mathrm{V}}_{ds}{D}_F}⌉$, where ${\mathrm{V}}_{ds}$ and $D_F$ are the semiconductor device breakdown voltage and derating factor, respectively. As presented in [10], considering the parameters in

Table 1. Parameters of the three-phase to single-phase ac/ac MMC full-scale simulations during steady-state operation.

Quantity	Parameter	Value	Unit
Active power reference	$P^{\ast }$	1	MW
Reactive power reference	$Q^{\ast }$	0	VAr
Grid phase-to-neutral peak voltage	${\widehat{U}}_y$	$25\sqrt{2/3}$	kV
Single-phase peak voltage	${\widehat{U}}_{t,p}$	8000	V
DC voltage	$U_{dc}$	800	V
Summed cap. voltage reference	$V_y^{\Sigma \ast }$	$24.41$	kV
Grid frequency	$f_1$	50	Hz
MFT frequency	$f_2$	1	kHz
MFT turns ratio	r:1	10:1	−
Arm capacitance	$C_{\sigma }$	$0.25$	mF
Arm inductance	$L_y$	1	mH

, ${\mathrm{V}}_{ds}$ = 20 mΩ and $D_F=75\%$, to achieve high efficiency, a sinusoidal single-phase port voltage needs at least 28 SMs per arm, while square wave single-phase port voltage needs 26 SMs per arm. Due to the high number of devices required, an averaged model is less complex than a model that includes all submodule capacitor voltages [32]. Therefore, the averaged model is useful for simulating the megawatt charger operation, e.g., to evaluate a new control scheme.

Given the complexity of the ac/ac MMC system, it is important to analyze the behavior of an ideal system separately from an actual implementation. This approach allows for initial validation of the proposed control strategy’s performance. After that, the influence of non-ideal behavior and implementation issues can be assessed in a practical setup. Therefore, to verify the proposed ac/ac MMC hierarchical control, simulations of the megawatt charger operation were conducted using the MMC averaged equivalent circuit shown in Figure 3 and the parameters listed in Table 1. The use of the averaged model eliminates the need for PWM and the sorting algorithm; hencem the switching frequency is not defined yet. Simulations consider an IGBT forward voltage drop $U_{CE}=0$ V. Additionally, the peak voltage ${\widehat{U}}_{t,p}$ for sinusoidal and square wave voltage at the single-phase port are chosen to be equal, to have the same summed capacitor voltage reference $V_y^{\Sigma \ast }$.

The cell balancing and common-mode current controllers presented in Figure 7 are implemented to control the common-mode currents with either sinusoidal or trapezoidal waveform, as shown in Figure 9 and Figure 10, respectively. The three-phase grid currents are purely sinusoidal without medium-frequency components $f_2$, and the single-phase current ${\mathsf{ı}}_{t,p}$ exhibits no low-frequency components, showing perfect decoupling of differential-mode and common-mode components.

Simulations show that the hierarchical control scheme is able to control the three-phase and single-phase port currents. In addition, the control balances the energy stored in the MMC submodule capacitors. This can be seen in the lowest traces of Figure 9 and Figure 10, where the summed capacitor voltages $v_a^{\mathrm{u}}$, $v_b^{\mathrm{u}}$, and $v_c^{\mathrm{u}}$ are regulated to the reference $V_y^{\Sigma \ast }$ for both sinusoidal and square wave single-phase port voltages.

Given that $C_{\sigma }$ = 0.25 mF in Table 1, simulation results show that the summed capacitor voltage ripple is close to $0.7\%$ of the summed capacitor voltage reference $V_y^{\Sigma \ast }$ for both sinusoidal and square wave voltages in Figure 9 and Figure 10. However, due to the number of SMs per arm, $C_{SM}$ = 7 mF for the sinusoidal wave, and $C_{SM}$ = 6.5 mF for the square wave single-phase voltage.

5. Experimental Verification

The proposed hierarchical control is verified with a scaled-down prototype, shown in Figure 11. The experimental setup implements an ac/ac MMC to interface a three-phase ac voltage source with the MFT primary side. The MFT secondary side connects to a dc voltage source through a full-bridge ac/dc converter, as depicted in Figure 2. The MMC and ac/dc converter are composed of PEH2015 and PEB8038 modules from Imperix, respectively. The system is controlled using four Imperix B-Box RCP 3.0 fast prototyping control platform modules, which run the control scheme shown in Section 3 in discrete-time implementation. In addition, the data acquired via the controllers have a sample rate of 50 kHz.

The main parameters of the experimental setup are listed in

Table 2. Parameters of the three-phase to single-phase ac/ac MMC scaled-down prototype during steady-state operation.

Quantity	Parameter	Value	Unit
Active power reference	$P^{\ast }$	1	kW
Reactive power reference	$Q^{\ast }$	0	VAr
Grid phase-to-neutral peak voltage	${\widehat{U}}_y$	300	V
Single-phase peak voltage	${\widehat{U}}_{t,p}$	250	V
DC voltage	$u_{dc}$	250	V
Summed cap. voltage reference	$V_y^{\Sigma \ast }$	425	V
Grid frequency	$f_1$	50	Hz
MFT frequency	$f_2$	1	kHz
Switching frequency	$f_{sw}$	50	kHz
MFT turn ratio	r:1	1:1	−
Arm capacitance	$C_{\sigma }$	$1.25$	mF
Arm inductance	$L_y$	$2.36$	mH
Number of SMs per arm	N	4	−

. As in Section 4, the peak voltages ${\widehat{U}}_{t,p}$ for sinusoidal and square wave voltage at the single-phase port are equal to keep the same summed capacitor voltage setpoint for both single-phase port voltage waveforms. The MMC submodules use IXYS IXGR48N60C3D1 IGBTs with a forward voltage drop $U_{CE}=1.5$ V. Therefore, $U_{y,D}^{\mathrm{x}}=12$ V is used to compensate $u_{y2}^{\Sigma \ast }$ in (39) and (41).

First, the ac/ac MMC is operated in steady-state with sinusoidal wave single-phase voltage by implementing the control scheme presented in Figure 7 with Equations (38) and (39). Figure 12 and Figure 13 illustrate the bidirectional power flow. As simulated in Section 4, the three-phase and single-phase port currents are controlled for a given active power reference $P^{\ast }$ and reactive power reference $Q^{\ast }$ and the summed capacitor voltages to the voltage reference $V_y^{\Sigma \ast }$ in Table 2. The three-phase currents show high-order harmonics due to IGBTs forward voltage drop and dead-time between the gate signals in the MMC submodules [45]. The level-shifted PWM and the sorting algorithm implemented in the experimental setup also influence the MMC ports’ current ripples [46,47,48]. However, this paper does not investigate the MMC three-phase and single-phase port current ripples.

The MMC single-phase port voltage is supplied via the transformer by the ac/dc converter. Therefore, $u_{t,p}$ in Figure 12 and Figure 13 is not purely sinusoidal with frequency $f_2$ as $u_{y2}^{\Sigma \ast }$ in (39) but is a pulse-width modulated voltage using unipolar PWM, resulting in an effective switching frequency of $2f_{sw}$ = 100 kHz. However, the voltage sensor in the scaled-down prototype, which has a measurement bandwidth of 60 kHz, attenuates the ripple at the effective switching frequency. Additionally, since the sampling and switching frequencies are synchronized, the measurements acquired using the control platform do not capture the high-frequency PWM waveform of the transformer voltage that is filtered by the sensor, due to aliasing. The voltage ripple in $u_{t,p}$ is visible in data acquired via the Tektronix P5200A differential voltage probe using a Keysight Technologies DSOX2024A oscilloscope as shown in Figure 14.

Measurements with a square wave single-phase port voltage are presented in Figure 15 with positive dc current. The control scheme presented in Figure 7 with Equations (40) and (41) is applied for the square wave operation. As presented for sinusoidal single-phase voltage, the three-phase and single-phase port currents are controlled for a given active $P^{\ast }$ and reactive $Q^{\ast }$ power reference, and the summed capacitor voltages $v_a^{\mathrm{u}}$, $v_b^{\mathrm{u}}$, and $v_c^{\mathrm{u}}$ are regulated to the reference $V_y^{\Sigma \ast }$ by the energy controllers. The transformer current ${\mathsf{ı}}_{t,p}$ in Figure 15 should be trapezoidal but is not in practice. This is due to the resolution of ${\theta }_{y2}^{\Sigma \ast }$ used to generate (41). The control accuracy relies on the resolution, or quantization step size, of the angle ${\theta }_{y2}^{\Sigma \ast }$. The smaller the quantization step size or the higher the resolution, the better the phase-shift control accuracy [43]. Moreover, the presence of quantizers in the closed-loop control, such as an analog-to-digital converter (ADC) and digital PWM, may result in steady-state limit cycling [49,50]. In other words, the phase-shift angles ${\theta }_{y2}^{\Sigma \ast }$ cannot be adjusted precisely to match the voltage references $u_{y2}^{\Sigma \ast }$, which could explain the limit cycling of the transformer current ${\mathsf{ı}}_{t,p}$. Other possible causes for this transformer current waveform are the dead-time and IGBT forward voltage drop as they result in arm voltage ripples, which influence the submodule capacitor voltages, grid current harmonics, and transformer current harmonics as described in [45]. The arm voltage ripple amplitude caused by dead-time is calculated and compensated in [45] for ac/dc MMCs. The voltage ripple depends on the arm current ${\mathsf{ı}}_y^{\mathrm{x}}$ profile, which could explain the difference between the grid current ripple when operating with sinusoidal and trapezoidal transformer current, as shown in Figure 12 and Figure 15. In addition, Figure 16 shows a measurement of the single-phase port voltage acquired via the Tektronix P5200A differential voltage probe using a Keysight Technologies DSOX2024A oscilloscope.

A higher resolution is achieved with the control strategy presented in Figure 8 since the angle ${\theta }_{\gamma 2}^{\Sigma \ast }$ can, in this case, be used to shift the carrier of the ac/dc converter modulation scheme. The much higher clock frequency with which this carrier is generated, in comparison to the sampling frequency of the controller, increases the resolution with which the phase-shift can be applied. The experimental results using the proposed control in Figure 8 with positive and negative dc current directions are presented in Figure 17 and Figure 18, respectively. The single-phase port current ${\mathsf{ı}}_{t,p}$ in Figure 17 shows a lower current ripple than in Figure 15. Measurements of the trapezoidal current ${\mathsf{ı}}_{t,p}$ implementing the controllers in Figure 7 and Figure 8 are magnified for better visualization in Figure 15, Figure 17 and Figure 18. To remove the remaining ripple in both three-phase and single-phase port currents, the arm voltage ripples in the ac/ac MMC that are caused by the IGBT forward voltage drop and dead-time must be compensated, which is not covered in this paper.

The current ripple caused by the dead-time at both MMC ports is less visible when the active power reference is increased to $P^{\ast }=2$ kW as shown in Figure 19. It is also less visible with lower ratings, e.g., ${\widehat{U}}_y={\widehat{U}}_{t,p}=u_{dc}=100$ V, $V_y^{\Sigma \ast }=150$ V, and $P^{\ast }=500$ W, as shown in Figure 20. Measurements of the trapezoidal current ${\mathsf{ı}}_{t,p}$ with different converter parameters are magnified in Figure 19 and Figure 20.

Since $P^{\ast }$ determines the average power exchange with the three-phase ac grid, the grid current is equal but opposite in phase for both positive and negative dc currents. The dc current, however, is unequal in magnitude for both power flow directions, since it also depends on the system losses.

The transient response of the proposed hierarchical control scheme is tested by applying a step change in the power reference $P^{\ast }$ for positive dc current direction. The transient response is shown for sinusoidal and square wave single-phase port voltages in Figure 21 and Figure 22, using the controllers shown in Figure 7 and Figure 8, respectively. The step in the power reference $P^{\ast }$ immediately increases the grid current amplitude, but the single-phase port current amplitude and dc port current react with a delay between 0 and $\frac{1}{f_2}$ due to the ZOH. The submodule capacitor voltage ripple increases, mostly due to the increased grid and transformer current amplitudes [11].

The transient response of the proposed control is also tested by applying a step change in the summed capacitor voltage reference $V_y^{\Sigma \ast }$. The summed capacitor voltages’ transient response for sinusoidal and square wave single-phase port voltage are shown in Figure 23a and Figure 23b, which are obtained through the control scheme presented in Figure 7 and Figure 8, respectively. The summed capacitor voltages $v_y^{\Sigma }$ converge to the new reference value for sinusoidal and square wave operations.

6. Conclusions

A direct three-phase to single-phase ac/ac MMC circuit was investigated based on its averaged behavior. This averaged circuit model was used to decouple the MMC’s internally circulating currents, as well as the three-phase and single-phase port currents. Based on this model, a hierarchical control strategy for the ac/ac MMC was developed to transfer power between a medium-voltage three-phase grid and a low-voltage dc side. This control scheme also ensures that the energy stored in the submodule capacitors remains balanced close to the reference value, which is crucial for MMC operation.

The proposed control strategy has been analyzed and verified for sinusoidal and square wave single-phase port voltages. Simulations demonstrate that square wave single-phase voltage results in fewer MMC SMs and lower SM capacitance compared to a sinusoidal voltage. Simulations of a full-scale MV-connected ultrafast charger and experimental verification using a scaled-down prototype in steady-state validated the control approach. In addition, the system shows good transient response for both sinusoidal and square wave operation during a step change in the active power reference and summed capacitor voltage reference.

Future work should include a performance comparison of the hierarchical control with sinusoidal and square wave single-phase voltage under various operating conditions, such as unbalanced grid voltages. Additionally, further research is needed to calculate and compensate for the influence of dead-time in the ac/ac MMC arm currents.