DC-Link Voltage Fluctuation Suppression Method for Modular Multilevel Converter Based on Common-Mode Voltage and Circulating Current Coupling Injection under Unbalanced Grid Voltage

Abstract

Grid voltage imbalance conditions often occur. Modular multilevel rectifiers (MMCs) have high DC-link voltage fluctuation under an unbalanced grid, which affects the normal operation of DC-side equipment. To suppress voltage fluctuation under an unbalanced grid, a coupling injection strategy composed of third zero-sequence common-mode voltage (TZCV) and secondary circulating current (SCC) was designed in this paper. In this paper, we calculated the coupling time-domain expression of the TZCV and SCC under an unbalanced grid voltage. Then, the influence of an SCC and TZCV coupling injection on DC-link voltage fluctuation was analyzed. The converter power flow of different system control objectives under an unbalanced grid was calculated, and the overall control method of the converter based on the arm current was proposed. The advantage of the method proposed in this paper is that it can realize online control under different grid voltages and input power conditions in real time and effectively suppress DC-link voltage fluctuation. The simulation was carried out on the MATLAB/Simulink platform, and a hardware-in-the-loop experimental platform was built; the results verify the effectiveness of the proposed strategy.

1. Introduction

With increasing DC loads, such as data centers and electric vehicle charging points, medium-voltage DC distribution networks have become the direction for future development [1,2,3]. In DC distribution systems, AC/DC converters are usually used to connect to medium-voltage AC distribution networks. Among these, modular multilevel concenters (MMCs) are characterized by modularity, low switching frequency, low harmonic content, and easy redundancy configuration [4,5,6], which better meet requirements compared with traditional two-level and three-level converters. However, MMCs have internal problems, such as inherent circulating currents and internal sub-module capacitor voltage ripples, which can affect the operation status of MMC grid-connected converters.

DC-link voltage fluctuation in DC distribution networks is a common problem [7]. The authors of [8] pointed out that under unbalanced conditions, the DC side of the distribution network will produce fundamental frequency and double frequency fluctuations. Based on the DC-link voltage fluctuation, the authors of [9] analyzed the influence of DC-link voltage fluctuation on the internal circulating current of MMCs and suppressed the circulating current by injecting a zero-sequence voltage. The authors of [10,11] pointed out that when the DC-link voltage in the grid contains fluctuating components, it will lead to the generation of corresponding harmonic components in the inverter output voltage, which then interact with the fundamental component of the inverter output voltage and leads to low-frequency oscillations of the inverter output phase voltage, which is the phenomenon of beat frequency. Though the amplitude of the inverter output frequency flapping voltage is minute, it still has a large impact on the operation of motors and other loads [12]. Thus, on the DC side, DC-link voltage fluctuations need to be suppressed by hardware or control methods [13,14,15,16]. Therefore, it is of practical significance to reduce DC-link voltage fluctuations in grid-connected converters.

However, the types of AC-grid faults are common among all grid-connected converters; among them, unbalanced AC-grid conditions caused by single-phase or two-phase short circuits are the most common [17]. The authors of [18] pointed out that DC-link voltage ripples are caused by zero-sequence power fluctuations and proposed a DC-link voltage ripple suppressing controller to remove the zero-sequence voltage components of the thud side under unbalanced grid conditions to keep the net DC-link voltage constant. However, under an unbalanced grid voltage, there is a double-line-frequency ripple in the real power component, and an additional DC voltage controller is added. In [19,20], circulating currents were analyzed as three components: positive, negative, and zero-sequence circulating currents under unbalanced grid conditions. The authors of [19] proposed a circulating current control method with AC-side positive- and negative-sequence current control to minimize circulating currents and reduce AC-side real power ripples under unbalanced conditions. In [20], based on the instantaneous power theory, a control strategy was proposed to eliminate real power ripples and suppress the harmonics in circulating currents. In fact, DC-link voltage fluctuations under an unbalanced grid are not only caused by the zero-sequence circulating current and its power but are also affected by the sub-module capacitor voltage fluctuation and the controller tracking error.

The authors of [21,22,23] used a circulating current suppression controller and did not aim to minimize the fluctuation in the sub-module capacitor voltage. The authors of [24,25] calculated the injected circulating current reference value based on the instantaneous physical quantities in an MMC converter system and minimized the fluctuation in the sub-module capacitance voltage. The authors of [26,27,28] used the method of injecting a TZCV to improve arm voltage utilization and reduce sub-module capacitance-voltage fluctuation. The authors of [29] based their method on the combination of TZCV and SCC injection to reduce the sub-module capacitance-voltage fluctuation. The authors of [30] coordinated their design of TZCV and SCC parameters using a genetic algorithm to reduce sub-module capacitor voltage ripples and arm current RMS values. The authors of [31] analyzed the effect of a circulating current injection in an MMC operation under an unbalanced grid and verified that the injection of a circulating current value calculated based on transient information under an unbalanced grid can reduce capacitor voltage ripples, but this will increase arm current peaks and DC oscillation. The authors of [32] investigated the role of SCC injection in reducing capacitor energy storage and eliminating DC-link oscillations under an unbalanced grid. The authors of [33] analyzed the effect of third-harmonic voltage injection on the voltage fluctuation in sub-modules under an unbalanced grid voltage and redesigned the amplitude and phase angle of a third-harmonic voltage injection. The authors of [34,35] analyzed MMC arm energy expression based on a third-harmonic voltage coupled with SCC injection under an unbalanced grid voltage and obtained optimal parameters that minimize energy fluctuation by the offline traversal of the optimization search method.

The above research shows that the coupling injection of TZCV and SCC is not a new method, which is widely used to improve the internal characteristics of an MMC converter, such as reducing the fluctuation in the sub-module capacitor voltage and reducing the peak current of the bridge arm [36,37]. The unbalanced grid voltage introduces negative-sequence voltage, complicating the coupling injection parameters of TZCV and SCC. Online parameter calculation under varying grid conditions requires further investigation. Additionally, for the MMC converter in medium-voltage DC systems, the output characteristics of the MMC are more critical than the internal characteristics. Thus, optimizing MMC output characteristics through coupling injection methods holds practical significance.

Based on the analysis, to suppress DC-link voltage fluctuations, this paper analyzes the mechanism of circulating current generation of MMC considering TZCV injection under unbalanced grid voltage and provides the time-domain expression. The impact of SCC and TZCV coupling injection on MMC is analyzed, leading to an online parameter solution method for the analytical formula. Subsequently, the influence of SCC and TZCV coupling injection on DC-link voltage fluctuations is assessed. s Assuming sinusoidal AC and unit power factor for MMC, the power flow of the MMC grid-connected converter is analyzed. Finally, the theoretical analysis and the proposed strategy are verified through simulation and hardware in the loop experiment.

2. Theoretical Analysis

2.1. Topology and Mathematical Model of MMC Grid-Connected Converter

The topology of the three-phase grid-connected MMC converter using a half-bridge sub-module (HBSM) is shown in Figure 1. The three-phase MMC circuit features a symmetrical structure, and the HBSM comprises two switching devices with anti-parallel diodes and DC capacitors connected in parallel through the arm inductors $L_r$. There are N HBSMs in a single arm, and all HBSMs are connected in series, which is equivalent to a voltage source. The MMC single-phase upper and lower arms are symmetrical, where $u_{pj},u_{nj}$ is the voltage at the upper and lower arms, $i_{pj},i_{nj}$ represent the current at the upper and lower arms, and $i_{cirj}$ is the circulating current, which contains both DC and AC components. The subscripts $j=a,b,c$ denote the three-phase $a,b,c$, as the same as in the following.$u_j,i_j$ is the voltage at the parallel point and the input current at the parallel point, $U_{dc},I_{dc}$ is the DC-link voltage and DC-side current and $L$ is the inductance at the AC parallel point.

During the operation of the MMC converter, the control system controls the conduction and shutdown of the upper and lower switching devices of the HBSM so that the HBSM capacitors are continuously charging or discharging. The HBSM capacitor voltage and arm voltage fluctuate continuously. According to Kirchhoff’s law, the arm voltages can be expressed as follows:

$$\left\{\begin{array}{l}{u}_{pj}={\displaystyle \frac{U_{\mathrm{dc}}}{2}}-u_j-L_{\mathrm{r}}{\displaystyle \frac{\mathrm{d}{i}_{pj}}{\mathrm{d}t}}\\ u_{nj}={\displaystyle \frac{U_{\mathrm{dc}}}{2}}+u_j-L_{\mathrm{r}}{\displaystyle \frac{\mathrm{d}{i}_{nj}}{\mathrm{d}t}}\end{array}\right.$$

(1)

The inductance of the upper and lower arms of each phase of the MMC converter is identical, and the AC is evenly distributed among the phases.

$$\left\{\begin{array}{l}{i}_{pj}=i_{cirj}-{\displaystyle \frac{i_j}{2}}\\ i_{nj}=i_{cirj}+{\displaystyle \frac{i_j}{2}}\end{array}\right.$$

(2)

After deriving the voltage and current expressions of the upper and lower arms, it is essential to analyze the relationships between the arm voltage, arm current, AC side output voltage, and internal differential voltage. Subsequently, the AC and DC equivalent circuits of the MMC converter can be determined.

Adding and subtracting the terms from the top and bottom of Equation (1) yields

$$\left\{\begin{array}{l}{u}_{pj}+u_{nj}=U_{dc}-L_r{\displaystyle \frac{d(i_{pj}+i_{nj})}{dt}}\\ u_{nj}-u_{pj}=2u_j-L_r{\displaystyle \frac{d(i_{nj}-i_{pj})}{dt}}\end{array}\right.$$

(3)

Adding and subtracting the terms from the top and bottom of Equation (2) yields

$$\left\{\begin{array}{l}{i}_{pj}+i_{nj}=2i_{cirj}\\ i_{nj}-i_{pj}=i_j\end{array}\right.$$

(4)

Analyzing Equation (1) and performing the elimination operation yields

$$U_{dc}=2L_r{\displaystyle \frac{d{i}_{cirj}}{dt}}+u_{pj}+u_{nj}$$

(5)

$$u_j={\displaystyle \frac{u_{nj}-u_{pj}}{2}}+{\displaystyle \frac{L_r}{2}}{\displaystyle \frac{d(i_{nj}-i_{pj})}{dt}}$$

(6)

Equation (5) presents the equivalent mathematical model of the MMC DC side. Equation (6) presents the equivalent mathematical model of the AC side. Equation (5) presents the MMC DC-link voltage consisting of the sum of the upper and lower arm voltages and the voltage drop across the arm inductance due to the circulating current.

2.2. Analysis of the Influence of SCC and TZCV Coupling Injection Considering Arm Inductance

There is no zero-sequence voltage component in a three-phase, three-wire system without a neutral line. Therefore, the unbalanced grid voltage can be expressed as

$$u_j=U^{+}\sin \left(\omega t+{\alpha }^{+}\right)+U^{-}\sin \left(\omega t+{\alpha }^{-}\right)$$

(7)

In Equation (7), ${\alpha }^{+},{\alpha }^{-}$ represents the phase angle of three-phase positive and negative-sequence voltages and $U^{+},U^{-}$ represents the amplitude of positive and negative-sequence voltage components.

The circulating current components in the MMC converter under an unbalanced grid include direct current components and even harmonics. Many studies have shown that the secondary content in the circulating current is highest [38]. To facilitate the analysis of the impact of the circulating current on the DC-link voltage fluctuations, only the SCC is considered in the calculation.

$$\left\{\begin{array}{l}\begin{array}{ll}{i}_{pj}& ={\displaystyle \frac{1}{3}}{I}_{dc}-{\displaystyle \frac{I^{+}}{2}}\sin \left(\omega t+{\beta }^{+}\right)\\ & -{\displaystyle \frac{I^{-}}{2}}\sin \left(\omega t+{\beta }^{-}\right)+I_2\sin (2\omega t+\theta )\end{array}\hfill \\ \begin{array}{ll}{i}_{nj}& ={\displaystyle \frac{1}{3}}{I}_{dc}+{\displaystyle \frac{I^{+}}{2}}\sin \left(\omega t+{\beta }^{+}\right)\\ & +{\displaystyle \frac{I^{-}}{2}}\sin \left(\omega t+{\beta }^{-}\right)+I_2\sin (2\omega t+\theta )\end{array}\hfill \end{array}\right.$$

(8)

In Equation (8), ${\beta }^{+},{\beta }^{-}$ are phase angles of the positive and negative-sequence voltages, $I^{+},I^{-}$ are amplitudes of the positive and negative-sequence voltage components, $I_2,\theta$ are the amplitude and phase angle of the SCC.

The linear modulation interval of HBSM-MMC $U_j/U_{dc}\le 1$, it can be extended by superimposing the TZCV on the arm modulation voltage. When the rated voltage of AC and DC are determined, injecting this voltage can improve the utilization of the DC voltage and reduce the DC-link voltage fluctuations of the grid-connected converter. The modulation voltage of the upper and lower arms superimposed by Equation (1) and the TZCV components can be expressed as

$$\left\{\begin{array}{l}\begin{array}{ll}{S}_{pj}& ={\displaystyle \frac{1}{2}}(1-m^{+}\sin \left(\omega t+{\alpha }_{+}\right)-m^{-}\sin \left(\omega t+{\alpha }_{-}\right)\\ & -m_3\sin \left(3\omega t+{\phi }_3\right)+m_i^{+}\cos (\omega t+{\beta }^{+})\\ & +m_i^{-}\cos (\omega t+{\beta }^{-})-m_2\cos \left(2\omega t+\theta \right)\end{array}\hfill \\ \begin{array}{ll}{S}_{nj}& ={\displaystyle \frac{1}{2}}(1+m^{+}\sin \left(\omega t+{\alpha }_{+}\right)+m^{-}\sin \left(\omega t+{\alpha }_{-}\right)\\ & +m_3\sin \left(3\omega t+{\phi }_3\right)-m_i^{+}\cos (\omega t+{\beta }^{+})\\ & -m_i^{-}\cos (\omega t+{\beta }^{-})-m_2\cos \left(2\omega t+\theta \right)\end{array}\hfill \end{array}\right.$$

(9)

In Equation (9), $m_3,{\phi }_3$ are the modulation ratio and phase angle of the TZCV.

$$\left\{\begin{array}{ccc}{m}_2={\displaystyle \frac{2\omega L_r{I}_2}{U_{dc}}}& m^{+}={\displaystyle \frac{U^{+}}{U_{dc}}}& m_i^{+}={\displaystyle \frac{\omega L_r{I}^{+}}{U_{dc}}}\\ m^{-}={\displaystyle \frac{U^{-}}{U_{dc}}}& m_3={\displaystyle \frac{U_3}{U_{dc}}}& m_i^{+}={\displaystyle \frac{\omega L_r{I}^{-}}{U_{dc}}}\end{array}\right.$$

(10)

Equation (10) represents the modulation coefficient for each voltage component within the arm.

2.3. Calculation of SCC Reference Value after TZCV Injection

Due to the presence of the HBSM capacitor voltage balancing mechanism, it can be considered that the capacitor voltages of each HBSM in the MMC are approximately equal. Consequently, the average current $i_{pj\_avg},i_{nj\_avg}$ flowing through the HBSMs capacitance is given by

$$\left\{\begin{array}{l}{i}_{jp\_avg}=i_{jp}{S}_{jp}=(A_0+A_1+A_2+A_3+A_4+A_5)\\ i_{np\_avg}=i_{np}{S}_{np}=(A_0-A_1+A_2-A_3+A_4-A_5)\end{array}\right.$$

(11)

The values of components in Equation (11) are shown in Equations (12)–(17)

$$\begin{array}{ll}{A}_0& ={\displaystyle \frac{1}{6}}{I}_{dc}+{\displaystyle \frac{1}{8}}{m}^{+}{I}^{-}\cos ({\beta }^{-}-{\alpha }^{+})+{\displaystyle \frac{1}{8}}{I}^{-}{m}^{-}\cos ({\beta }^{-}-{\alpha }^{-})\\ & +{\displaystyle \frac{1}{8}}{I}^{+}{m}^{+}\cos ({\alpha }^{+}-{\beta }^{+})+{\displaystyle \frac{1}{8}}{I}^{+}{m}^{-}\cos ({\alpha }^{-}-{\beta }^{+})+{\displaystyle \frac{1}{8}}{I}^{-}{m}_3\sin ({\beta }^{-}-\theta )\\ & -{\displaystyle \frac{1}{8}}{I}^{-}{m}_i^{+}\sin ({\beta }^{-}-{\beta }^{+})+{\displaystyle \frac{1}{8}}{I}^{+}{m}_i^{-}\sin ({\beta }^{-}-{\beta }^{+})\end{array}$$

(12)

$$\begin{array}{ll}{A}_1& ={\displaystyle \frac{1}{6}}{I}_{dc}{m}_i^{-}\cos (\omega t+{\beta }^{-})+{\displaystyle \frac{1}{6}}{I}_{dc}{m}_i^{-}\cos (\omega t+{\beta }^{-})-{\displaystyle \frac{1}{4}}{I}_2{m}_3\cos (\omega t+{\phi }_3-\theta )\\ & -{\displaystyle \frac{1}{4}}{I}_2{m}^{+}\cos (\omega t+\theta -{\alpha }^{+})+{\displaystyle \frac{1}{4}}{I}_2{m}_i^{-}\sin (\omega t+\theta -{\beta }^{-})-{\displaystyle \frac{1}{4}}{I}^{-}\sin (\omega t+{\beta }^{-})\\ & +{\displaystyle \frac{1}{4}}{I}_2{m}_i^{+}\sin (\omega t+\theta -{\beta }^{+})-{\displaystyle \frac{1}{4}}{I}^{+}\sin (\omega t+{\beta }^{+})\\ & -{\displaystyle \frac{1}{8}}{I}^{-}{m}_2\sin (\omega t+\theta -{\beta }^{-})-{\displaystyle \frac{1}{8}}{I}^{+}{m}_2\sin (\omega t+\theta -{\beta }^{+})\\ & -{\displaystyle \frac{1}{4}}{I}_2{m}^{-}\cos (\omega t+\theta -{\alpha }^{-})+{\displaystyle \frac{1}{6}}{I}_{dc}{m}_i^{+}\sin (\omega t+{\beta }^{+})\end{array}$$

(13)

$$\begin{array}{ll}{A}_2& ={\displaystyle \frac{1}{2}}{I}_2\sin (2\omega t+\theta )-{\displaystyle \frac{1}{8}}{I}^{-}{m}^{-}\cos (2\omega t+{\alpha }^{-}+{\beta }^{-})\\ & +{\displaystyle \frac{1}{8}}{I}^{+}{m}_3\cos (2\omega t+{\phi }_3-{\beta }^{+})-{\displaystyle \frac{1}{8}}{I}^{+}{m}^{+}\cos (2\omega t+{\beta }^{+}+{\alpha }^{+})\\ & -{\displaystyle \frac{1}{8}}{I}^{+}{m}^{-}\cos (2\omega t+{\beta }^{+}+{\alpha }^{-})-{\displaystyle \frac{1}{8}}{I}^{-}{m}_i^{-}\sin (2\omega t+2{\beta }^{-})\\ & -{\displaystyle \frac{1}{8}}{I}^{-}{m}_i^{+}\sin (2\omega t+{\beta }^{-}+{\beta }^{+})-{\displaystyle \frac{1}{6}}{I}_{dc}{m}_2\cos (2\omega t+\theta )\\ & -{\displaystyle \frac{1}{8}}{I}^{-}{m}^{+}\cos (2\omega t+{\beta }^{-}+{\alpha }^{+})-{\displaystyle \frac{1}{8}}{I}^{+}{m}_i^{-}\cos (2\omega t+{\beta }^{-}+{\beta }^{+})\\ & -{\displaystyle \frac{1}{8}}{I}^{+}{m}_i^{+}\cos (2\omega t+2{\beta }^{+})\end{array}$$

(14)

$$\begin{array}{ll}{A}_3& ={\displaystyle \frac{1}{4}}{I}_2{m}^{+}\cos (3\omega t+\theta +{\alpha }^{+})+{\displaystyle \frac{1}{4}}{I}_2{m}^{-}\cos (3\omega t+\theta +{\alpha }^{-})\\ & +{\displaystyle \frac{1}{4}}{I}_2{m}_i^{-}\sin (3\omega t+{\beta }^{-}+\theta )-{\displaystyle \frac{1}{6}}{I}_{dc}{m}_3\sin (3\omega t+{\phi }_3)\\ & +{\displaystyle \frac{1}{4}}{I}_2{m}_i^{+}\sin (3\omega t+\theta +{\beta }^{+})+{\displaystyle \frac{1}{8}}{I}^{+}{m}_2\sin (3\omega t+{\beta }^{-}+\theta )\\ & +{\displaystyle \frac{1}{8}}{I}^{+}{m}_2\sin (3\omega t+\theta +{\beta }^{+})\end{array}$$

(15)

$$\begin{array}{ll}{A}_4& =-{\displaystyle \frac{1}{8}}{I}^{-}{m}_3\cos (4\omega t+{\phi }_3+{\beta }^{-})-{\displaystyle \frac{1}{8}}{I}^{+}{m}_3\cos (4\omega t+{\phi }_3+{\beta }^{+})\\ & -{\displaystyle \frac{1}{4}}{I}_2{m}_2\sin (4\omega t+2\theta )\end{array}$$

(16)

$$A_5={\displaystyle \frac{1}{4}}{I}_2{m}_3\cos (5\omega t+{\phi }_3+\theta )$$

(17)

The arm current, composed of different frequency components, flows into the HBSM capacitor and the arm inductance, leading to voltage fluctuations in the HBSM capacitor and the arm inductance. This fluctuation manifests as

$$\begin{array}{l}\left\{\begin{array}{l}\Delta u_{pj\_C}={\displaystyle \frac{1}{C}}{\displaystyle \int i_{jp\_avg}}dt={\displaystyle \frac{1}{C}}{\displaystyle \int (A_1+A_2+A_3+A_4+A_5)}dt\\ \Delta u_{nj\_C}={\displaystyle \frac{1}{C}}{\displaystyle \int i_{jn\_avg}}dt={\displaystyle \frac{1}{C}}{\displaystyle \int (-A_1+A_2-A_3+A_4-A_5)}dt\end{array}\right.\hfill \\ \left\{\begin{array}{l}\Delta u_{pj\_L}=L_r{\displaystyle \frac{d{i}_{jp\_avg}}{dt}}=L_r{\displaystyle \frac{d(A_1+A_2+A_3+A_4+A_5)}{dt}}\\ \Delta u_{pj\_L}=L_r{\displaystyle \frac{d{i}_{jp\_avg}}{dt}}=L_r{\displaystyle \frac{d(-A_1+A_2-A_3+A_4-A_5)}{dt}}\end{array}\right.\hfill \end{array}$$

(18)

Equation (18) shows that the capacitor voltage caused by the odd-order arm current is opposite in direction to the voltage fluctuation of the arm inductance in the upper and lower arms, while the fluctuation induced by the even-order arm current is in the same direction. The DC-link voltage of the MMC grid-connected converter is influenced by the combined effect of the upper and lower arm HBSM capacitor voltages and the arm inductance voltage, which is expressed as

$$\left\{\begin{array}{l}\Delta U_{dc\_2}={\displaystyle \frac{4N}{C}}{\displaystyle \int A_{2}dt+2L_r\frac{d{A}_2}{dt}}\\ \Delta U_{dc\_4}={\displaystyle \frac{4N}{C}}{\displaystyle \int A_{4}dt+2L_r\frac{d{A}_4}{dt}}\end{array}\right.$$

(19)

Equation (19) shows that the coupling injection of the SCC and the TZCV produces fluctuations in the arm voltage at both twice and four times the fundamental frequency. These fluctuations are influenced by the grid voltage imbalance, system control target, SCC injection amplitude, phase angle, and TZCV injection amplitude, phase angle. It can be seen from Equation (12) that the traditional circulating current suppression method forces the circulating current to zero, so the TZCV injection method can be used together with it. However, once TZCV is injected, the circulating current components are altered, and the injected voltage becomes coupled with the circulating current components.

It can be seen from Equations (12)–(17) that after considering t TZCV injection, the four-time fluctuation of the arm voltage is primarily dependent on the SCC component in the converter and the third injection modulation ratio. Consequently, the four-fold fluctuation is minimal, and the arm voltage fluctuation is mainly dominated by the two-fold fluctuation. Equations (12)–(17) show that the SCC contains positive-sequence, negative-sequence, and zero-sequence components after considering the TZCV injection.

The positive and negative-sequence components of SCC flow in the three-phase of the MMC, causing the fluctuation of the HBSM capacitor voltage. As shown in Equation (5), the DC-link voltage is composed of the arm capacitor voltage and the inductor voltage, so fluctuations in HBSM voltage directly impact DC-side voltage stability. The zero-sequence component of SCC flows into the DC-side load, which directly affects the DC-link voltage. Therefore, reducing the amplitude of the secondary circulating current does not linearly correspond to a reduction in DC-link voltage fluctuations. The traditional strategy of suppressing secondary circulation fluctuation cannot minimize the reduction of the DC-link voltage fluctuation.

To minimize the DC-link voltage fluctuation, SCC should be controlled to offset the fluctuation. Simultaneously, to account for the primary components of the secondary average current, the calculation amount of the control system should be reduced to facilitate real-time control. The controllable SCC injection component is

$$\begin{array}{l}{\displaystyle \frac{1}{2}}{I}_2\sin (2\omega t+\theta )=-{\displaystyle \frac{1}{8}}(I^{-}{m}^{-}\cos (2\omega t+{\alpha }^{-}+{\beta }^{-})\\ +I^{+}{m}_3\cos (2\omega t+{\phi }_3-{\beta }^{+})-I^{+}{m}^{+}\cos (2\omega t+{\beta }^{+}+{\alpha }^{+})\\ -I^{+}{m}^{-}\cos (2\omega t+{\beta }^{+}+{\alpha }^{-})-I^{-}{m}^{+}\cos (2\omega t+{\beta }^{-}+{\alpha }^{+}))\end{array}$$

(20)

Equation (20) provides the reference value for SCC injection. The AC voltage amplitude and phase angle, the AC amplitude and phase angle in the circulation reference are provided by the decoupled double synchronous reference frame PLL (DDSRF-PLL).

2.4. The Influence of TZCV on the Operation of MMC and the Design of TZCV Injection Parameters

The HBSM-MMC lacks the capability of the ability of full-bridge sub-module to output a negative level, limiting its linear operating range. However, by injecting TZCV, the linear modulation range of HBSM-MMC can be expanded, and the utilization of DC-link voltage can be improved. This not only extends the operating range of the MMC but also reduces DC-link voltage fluctuations under unbalanced grid conditions.

For an MMC working in a steady state while neglecting the influence of circulating currents, the TZCV parameters to maximize the utilization of DC voltage are similar to those of a traditional two-level converter. Specifically, the voltage injection amplitude is set to 1/6 of the AC voltage reference value with the same phase. As shown in Figure 2, after injecting TZCV, the phase voltage modulation waveform becomes a saddle wave, the corresponding maximum modulation index is 1.732, and the DC voltage utilization rate increases 15%.

When accounting for the voltage drop caused by the circulating current across the arm inductance and the output of the circulating current controller, the amplitude of the arm modulation voltage changes, and the modulation waveform deviates from a standard saddle wave. This variation in voltage amplitude results in a reduced modulation range for the MMC.

Figure 3 shows the steady-state operation of the MMC converter at a modulation ratio of 1. The system control objective is to suppress the negative-sequence current of the grid-connected point, and the maximum amplitude of the modulation voltage at the three-phase upper arm under a single-phase voltage drop of the grid.

Figure 3 shows that without TZCV injection, the output of the circulating current controller will cause an increase in the modulation voltage of the arm. Consequently, when the system operates in a high modulation ratio interval, the input from circulation control can cause an overshoot, adversely affecting system performance.

When considering the influence of circulating current, Equations (12)–(17) indicate that TZCV injection alters the three-phase circulating current components of MMC. The change in this component is not only related to the amplitude and phase angle of TZCV injection but also to voltage imbalance, system control objectives, and operating parameters.

Figure 4 shows that when the grid voltage has a single-phase drop, the system control target is to suppress the negative-sequence current at the grid-connected point. Considering the arm inductance voltage drop due to the circulating current, the output voltage of the circulating current controller, and the peak value of the arm modulation voltage injected by the traditional common-mode voltage amplitude. Since the modulation interval under steady-state operation is maximized when the common-mode voltage injection amplitude is set to 1/6 of the AC voltage reference value, it is possible to fix the amplitude of the injected third zero-sequence common-mode voltage to 1/6 of the AC voltage.

Figure 4 shows the change in the modulation peak of the three-phase upper arm by traversing the phase angle between the single-phase voltage drop and the common-mode voltage. Although the system’s linear modulation range does not increase by 15% after the circulating current control is implemented, the modulation voltage after the controller is put into operation remains within the modulation range. Here, the single-phase voltage drop condition is taken as an example, and the two-phase voltage drop condition is similar to that.

When the grid voltage drop varies, the zero-sequence voltage phase angle at which the arm modulation voltage reaches the minimum value changes by injecting TZCV, and this change differs across the three phases. However, the minimum modulation peaks are obtained near the zero-phase difference between the TZCV injected and the grid voltage. To achieve the desired voltage injection effect under different voltage imbalance conditions, reduce the computational complexity of reference value calculations, and enable real-time computation. It can be considered that the phase difference between the injected TZCV and the grid voltage is zero.

Through the analysis of this section, suitable injection parameters for TZCV under different grid voltage conditions are obtained. Although the proposed TZCV parameters are not optimal, they avoid offline optimization and complex differential calculations. The analysis results show that the designed TZCV parameters can effectively improve the linear modulation range of MMC and improve the DC-link voltage utilization. Combined with the analysis of the injected SCC in Section 2.3, all the parameters of the coupled injection can be obtained online, which greatly reduces the burden of the controller, which is also the theoretical innovation of this paper.

2.5. Control Target Analysis and Inner-Loop Current Reference Value Calculation of MMC

In practical applications, the MMC grid-connected rectifier is generally required to operate in the state of unit power factor and sinusoidal current. Sinusoidal current implies that the AC contains only the fundamental positive and negative-sequence components at most. The grid-connected converter operating at a unit power factor means that the expected current should be in phase with the corresponding voltage fundamental. To simplify the structure of the controller, the control method in the $\alpha \beta$ coordinate system is adopted in this paper. The expressions of AC input voltage and current are reduced to $\alpha \beta$ coordinate systems.

$$\left\{\begin{array}{l}{u}_{\alpha \beta }=C_{3/2}{\left[\begin{array}{ccc}{u}_a& u_b& u_c\end{array}\right]}^{\mathrm{T}}\\ i_{\alpha \beta }={\left[\begin{array}{ll}{i}_{\alpha }\hfill & i_{\beta }\hfill \end{array}\right]}^{\mathrm{T}}=C_{3/2}{\left[\begin{array}{lll}{i}_a\hfill & i_b\hfill & i_c\hfill \end{array}\right]}^{\mathrm{T}}\end{array}\right.$$

(21)

$$C_{3/2}=\sqrt{{\displaystyle \frac{2}{3}}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right]$$

(22)

According to the definition of instantaneous power, it can be seen that

$$\left[\begin{array}{l}p\hfill \\ q\hfill \end{array}\right]=\left[\begin{array}{cc}{u}_{\alpha }& u_{\beta }\\ u_{\beta }& -u_{\alpha }\end{array}\right]\left[\begin{array}{l}{i}_{\alpha }\hfill \\ i_{\beta }\hfill \end{array}\right]$$

(23)

The expected current of the system is

$$\left\{\begin{array}{l}{\left(i_{\alpha \beta }^{+1}\right)}^{*}={\left[\begin{array}{ll}{\left(i_{\alpha }^{+1}\right)}^{*}\hfill & {\left(i_{\beta }^{+1}\right)}^{*}\hfill \end{array}\right]}^{\mathrm{T}}=k_{+1}{u}_{\alpha \beta }^{+1}\\ {\left(i_{\alpha \beta }^{-1}\right)}^{*}={\left[\begin{array}{ll}{\left(i_{\alpha }^{-1}\right)}^{*}\hfill & {\left(i_{\beta }^{-1}\right)}^{*}\hfill \end{array}\right]}^{\mathrm{T}}=k_{-1}{u}_{\alpha \beta }^{-1}\end{array}\right.$$

(24)

Substituting Equation (19) into Equation (18), and the expected power of the system is

$$\left[\begin{array}{c}{p}_0^{*}\\ p_2^{*}\\ q_0^{*}\\ q_2^{*}\end{array}\right]=\left[\begin{array}{cccc}{u}_{\alpha }^{+1}& u_{\beta }^{+1}& u_{\alpha }^{-1}& u_{\beta }^{-1}\\ u_{\alpha }^{-1}& u_{\beta }^{-1}& u_{\alpha }^{+1}& u_{\beta }^{+1}\\ u_{\beta }^{+1}& -u_{\alpha }^{+1}& u_{\beta }^{-1}& -u_{\alpha }^{-1}\\ u_{\beta }^{-1}& -u_{\alpha }^{-1}& u_{\beta }^{+1}& -u_{\alpha }^{+1}\end{array}\right]\left[\begin{array}{l}{k}_{+1}{u}_{\alpha }^{+1}\hfill \\ k_{+1}{u}_{\beta }^{+1}\hfill \\ k_{-1}{u}_{\alpha }^{-1}\hfill \\ k_{-1}{u}_{\beta }^{-1}\hfill \end{array}\right]$$

(25)

In Equation (25), $p_0^{*},q_0^{*}$ is the power constant and $p_2^{*},q_2^{*}$ is the power fluctuation.

Under unbalanced grid voltage, the AC sinusoidal current will inevitably generate fluctuating active and reactive power, i.e., $p_2^{*},q_2^{*}$. To improve the power factor of the grid-connected rectifier, the expected power is $q_0^{*}=0$. Equation (25) shows that when $k_{+1},k_{-1}$ have certain integer solutions, there is a unique system expected power. To determine the value of $k_{+1},k_{-1}$, the number of equations in Equation (25) should be reduced to a binary homogeneous system of equations. Under the unit power factor, the expected active power $p_0^{*}$ of the system is the integral value of the voltage outer-loop controller, and the equation can then be determined as

$$p_0^{*}=k_{+1}{\left(u_{\alpha \beta }^{+1}\right)}^{\mathrm{T}}{u}_{\alpha \beta }^{+1}+k_{-1}{\left(u_{\alpha \beta }^{-1}\right)}^{\mathrm{T}}{u}_{\alpha \beta }^{-1}$$

(26)

Under sinusoidal current conditions, the suppression of active power fluctuation can reduce the active power fluctuation to the greatest extent, and suppressing active power fluctuation can minimize active power variation, therefore improving the power factor. However, both lead to an imbalance in AC at the point of common coupling, which will have a negative impact on the system operation. Suppressing the negative-sequence current at the grid-connected point represents a compromise between these approaches and is the most widely used method in the operational control of grid-connected converters under unbalanced grid conditions.

The target equation of suppressing active power fluctuation is $k_{-1}=0$, which combined with Equation (26)

$$\left\{\begin{array}{l}{p}_0^{*}=k_{+1}{\left(u_{\alpha \beta }^{+1}\right)}^{\mathrm{T}}{u}_{\alpha \beta }^{+1}+k_{-1}{\left(u_{\alpha \beta }^{-1}\right)}^{\mathrm{T}}{u}_{\alpha \beta }^{-1}\\ 0=k_{+1}{\left(u_{\alpha \beta }^{-1}\right)}^{\mathrm{T}}{u}_{\alpha \beta }^{+1}\end{array}\right.$$

(27)

The inner-loop current reference value is solved as

$$\left\{\begin{array}{l}{k}_{+1}={\displaystyle \frac{p_0^{*}}{{‖u_{\alpha \beta }^{+1}‖}^2}}{k}_{-1}=0\\ {\left(i_{\alpha \beta }^{+1}\right)}^{*}={\displaystyle \frac{p_0^{*}}{{‖u_{\alpha \beta }^{+1}‖}^2}}{u}_{\alpha \beta }^{+1}\end{array}\right.$$

(28)

2.6. System Overall Control Method

The MMC arm current contains the fundamental frequency alternating current component, a circulating current DC component, and various alternating current components. Therefore, controlling the arm current can directly achieve both the system target and circulating current control. Due to the coordinate axis transformation, the zero-sequence component $i_{cir\_zero}$ in the circulating current directly generates the modulation voltage $e_{cir\_zero}$ through the proportional controller.

Figure 5 presents the overall control block diagram of the system. The outer loop controls the DC-link voltage using a proportional-integral (PI) controller, and the controller output is multiplied by the DC-link voltage to obtain the active power reference value of the inner loop. The reference value of the inner-loop current, along with the positive and negative-sequence components of the circulating current, is added as the input of the controller. A quasi-proportional-integral resonant controller (PIR) can be used to realize the control of the DC, fundamental frequency, and double frequency components in arm current. The specific expression of the controller transfer function is as follows:

$$G(s)=k_p+{\displaystyle \frac{k_i}{s}}+{\displaystyle \frac{2k_r{\omega }_{c}s}{s^2+2{\omega }_c+{\omega }_1^2}}+{\displaystyle \frac{2k_r{\omega }_{c}s}{s^2+2{\omega }_c+{\omega }_2^2}}$$

(29)

In Equation (29) ${\omega }_1=100\pi ,{\omega }_2=200\pi$, the control of fundamental frequency and double frequency components is achieved separately.

The carrier phase shift modulation (PSC-PWM) is employed in the modulation link. The HBSM capacitor voltage balancing link is achieved through a hierarchical strategy, which includes inter-phase energy balance, arm energy balance, and HBSM capacitor voltage balance. The parameters of each controller in the overall control system are designed to suppress the DC-link voltage fluctuations.

3. Simulation and Experimental Results

3.1. Simulation Result

To verify the effectiveness of the above theoretical analysis and the proposed injection strategy, a three-phase grid-connected MMC converter simulation is developed in MATLAB/Simulink. The model parameters are provided in

Table 1. MMC simulation parameters.

Parameter	Value	Parameter	Value
Number of HBSMs N	10	Modulation carrier frequency $f$	2000 Hz
DC-link $U_{dc}$	5000 V	HBSM capacitance $C$	1500 uF
$\mathrm{Inductance}\mathrm{of}\mathrm{arm}{L}_r$	5 mH	DC-side resistance load $R$	50 Ω
AC voltage peak V	2000 V	$\mathrm{Active}\mathrm{current}\mathrm{side}\mathrm{inductor}{L}_r$	3.5 mH
Grid frequency $f$	50 Hz

.

To verify the effectiveness of the proposed method under unbalanced grid voltage, a voltage drop fault is set at 4 s, with the power factor of the MMC converter system maintained at 1. Reference [39] also mentions a method to suppress DC-link voltage fluctuations in MMC by injecting circulating current under unbalanced grid conditions. A band-pass filter is designed to obtain the reference value for circulating current injection, with the objective of reducing the sub-module’s secondary voltage fluctuation to zero. A PR controller is then employed to regulate the circulating current.

Figure 6 shows the suppression of MMC DC-link voltage fluctuation achieved by injecting the circulating current proposed in [39], with the strategy being implemented at 6 s. As shown in Figure 6c,d, while the sub-module capacitor voltage is suppressed, the effect on DC-link voltage suppression remains limited. Figure 7c,d demonstrates the effect of applying the proposed method at 6 s. It is observed that the sub-module capacitor voltage fluctuation is reduced by 11.2%, and the DC-link voltage fluctuation is reduced by 74.2%, indicating a superior suppression effect.

To more clearly demonstrate the suppression effect of the SCC and TZCV coupling injection method, along with the circulating current suppression method, on the DC-link voltage fluctuations. An online SCC suppression strategy proposed in [40] is compared with the method in this paper. As described in Section 2.4, the SCC suppression strategy can cooperate with TZCV injection. Therefore, to enhance the linear operating range of the system, the simulation in this paper introduces TZCV injection from the start.

Figure 8 presents the simulation results aimed at suppressing the negative-sequence current at the grid-connected point. Following the single-phase voltage drop at 4 s, the SCC suppression method from [40] is first applied, and the proposed SCC method in this paper is injected at 6 s. Figure 8b demonstrates that the arm current controller quickly achieves the goal of negative-sequence current suppression at the point of common coupling. Additionally, with the suppression of the DC-link voltage following circulating current injection, the outer-loop controller can calculate the power more accurately, so the three-phase current waveform after injection is also improved. Figure 8c shows that before and after the injection of the SCC, the peak current of the arm increases by 22.3 A compared with the suppression of the SCC. The sinusoidal current of the arm before the injection is good, proving the effectiveness of the circulating current suppression strategy. Figure 8d shows that the DC-link voltage fluctuation is reduced from 170.5 V to 88.4 V before and after injecting the calculated value of the proposed method at 6 s, which is approximately 48.10% lower than the traditional method of suppressing the SCC in [40].

Figure 9 shows the simulation results of a two-phase voltage drop, and the calculated SCC value is injected at 6 s. Figure 9b shows that the three-phase current waveform improves after the appropriate coupling injection of TZCV and SCC. Figure 9c indicates that the peak arm current increases by 11.4 A before and after the circulating current injection, compared to SCC suppression alone. The arm current remains sinusoidal before injection, confirming the effectiveness of the circulating current suppression strategy. Figure 9d shows that the DC-link voltage fluctuation decreases from 118.2 V to 42.8 V before and after injecting the calculated value of the proposed method at 6 s, representing a reduction of approximately 63.80%.

The simulation results following 70% and 30% voltage dips in phase A and phase B are displayed in Figure 10. The SCC proposed in this paper is injected at 6 s. Figure 10b shows the effectiveness of the circulating current suppression strategy and the improvement effect of the proposed strategy on the AC side current. Figure 10c shows that the peak current of the A-phase arm increases from 251.9 A to 258.5 A before and after the SCC injection, an increase of 6.6 A. Figure 10d shows that the DC-link voltage fluctuation is reduced from 141.7 V to 64.5 V before and after injecting the calculated value of the proposed SCC at 6 s, which is about 54.48% lower than the traditional SCC suppression method.

To more comprehensively demonstrate the effectiveness of the proposed strategy under different grid voltage imbalances and different loads,

Table 2. Simulation results under different voltage drop degrees and different loads.

Voltage Drops of Grid	Dc Load $\mathbf{\Omega }$	AC Current THD before Injection	AC Current THD after Injection	DC Voltage Fluctuation before Injection $\mathit{V}$	DC Voltage Fluctuation after Injection $\mathit{V}$	DC Voltage Reduction Range
AB two-phase voltage drop by 80%.	50	4.00%	2.94%	300.5	153.2	49.02%
	75	4.64%	2.74%	195.4	73.6	62.34%
	100	5.06%	2.80%	140.5	59.8	57.44%
A-phase voltage drops by 80%, and B-phase voltage drops by 40%.	50	2.76%	2.56%	185.5	91.2	50.84%
	75	3.06%	3.04%	116.2	65.1	44.07%
	100	3.52%	2.72%	93.5	46.5	50.27%
AB two-phase voltage drop by 40%.	50	2.19%	2.22%	92.5	44.3	52.11%
	75	2.47%	2.04%	70.3	41.7	40.69%
	100	2.45%	2.50%	50.9	25.5	49.91%
Phase A voltage drop by 30%.	50	2.14%	2.05%	57.2	21.8	61.88%
	75	2.75%	1.97%	34.7	22.5	35.16%
	100	3.36%	2.88%	36.5	23.7	35.07%

shows the suppression effect of the injection strategy proposed in this paper on the DC-link voltage. The results indicate that the DC-link voltage fluctuations can be effectively suppressed under various AC voltage imbalances and loads.

Table 2 shows that the fluctuation amplitude of DC-link voltage increases with the severity of grid voltage unbalance. Under the same voltage drop, as the input active power of the converter increases, the fluctuation of the DC-link voltage becomes more pronounced. The AC side current THD is optimized before and after applying both the traditional and proposed methods, benefiting from the reduction of DC-link voltage fluctuation. Simulation results show that the proposed method can effectively suppress DC-link voltage fluctuations under various voltage and load conditions.

3.2. Hardware-in-the-Loop Experimental Results

To further verify the effectiveness of the proposed algorithm, a hardware-in-the-loop experimental platform is built for experimental verification. The platform uses a rapid control prototype (RCP) to run the overall control part, and Dspace1202 runs the MMC converter hardware topology. The sampled physical quantities from the MMC converter hardware part communicate with the RCP control part through the I/O port, while the PWM pulse wave generated by the control part communicates with Dspace1202 through Modbus TCP/IP. The overall structure of the platform is shown in Figure 11.

Hardware-in-the-loop experimental parameters are shown in

Table 3. MMC experimental parameters.

Parameter	Value	Parameter	Value
Number of HBSMs N	4	Modulation carrier frequency $f$	1000 Hz
DC-link $U_{dc}$	5000 V	HBSM capacitance $C$	1500 uF
$\mathrm{Inductance}\mathrm{of}\mathrm{arm}{L}_r$	5 mH	DC-side resistance load $R$	50 $\Omega$
AC voltage peak V	2000 V	sampling frequency $f$	10 kHz
Grid frequency $f$	50 Hz

.

Figure 12a shows that the three-phase current of the grid-connected point is well controlled under the condition of balanced grid voltage. Figure 12b,c show the grid voltage waveform and the voltage output waveform of the fault phase HBSM. Figure 12d shows the three-phase current and DC-link voltage when the DC-side load resistance changes from 50 Ω to 25 Ω, indicating the effectiveness of the outer-loop DC voltage control and inner-loop current control under dynamic load conditions.

Figure 13 and Figure 14 show the experimental results under the target of suppressing the negative-sequence current at the grid-connected point. Figure 13a,b and Figure 14a,b show the three-phase current and single-phase circulating current waveforms at the moment of voltage drop and SCC injection. The experimental waveforms show the effectiveness of the control system target and injection effect. Figure 13c and Figure 14c show the waveform of the output voltage of the single-phase HBSM after injecting the proposed strategy. Figure 13d and Figure 14d show the DC-link voltage fluctuations before and after the suppression of the circulating current and the injection of the SCC. The experimental results show that when the load is smaller, i.e., the input power is larger, the DC-link voltage fluctuations increase. The proposed injection strategy effectively suppresses DC-link voltage fluctuations under varying loads.

The experimental findings of various loads under two-wire voltage drop situations are displayed in Figure 15 and Figure 16. Figure 15a,b and Figure 16a,b show the three-phase current and single-phase circulating current waveforms at the moment of voltage drop and circulating current injection. Figure 15c and Figure 16c show the waveform of the output voltage of the single-phase HBSM after injecting the proposed strategy. Figure 15d and Figure 16d show the DC-link voltage fluctuations before and after injecting SCC. The experimental results show that the proposed strategy can effectively reduce DC-link voltage fluctuations under various conditions. As the grid voltage drop intensifies and the input power of the converter increases, the DC-link voltage fluctuation also rises. Compared with the traditional circulating current suppression strategy, the proposed strategy can effectively reduce the fluctuation of DC-link voltage, and this law is consistent with the simulation results.

4. Conclusions

This paper proposes a method to suppress DC-link voltage fluctuations using SCC and TZCV coupling injection under an unbalanced grid. The paper analyzes the coupling injection of TZCV and SCC and calculates the time-domain analytical expression of coupling injection. Based on this, the reference values for TZCV and SCC injection are analyzed with the aim of suppressing DC-link voltage fluctuations in MMC grid-connected converters under unbalanced grid conditions. The reference value of the MMC inner-loop current is calculated based on the unit power factor and sinusoidal current under unbalanced grid conditions. A control method based on arm current is proposed, and its effectiveness is validated through simulation and hardware-in-the-loop experimental verification. The results show that:

(1)

The circulating current components in the MMC converter after TZCV injection are coupled with the amplitude and phase angle of TZCV injection. TZCV injection can improve the utilization of DC-link voltage and change the internal circulation component of the MMC converter.

(2)

The zero-sequence component in the circulating current and the capacitor voltage fluctuation of the HBSM is strongly associated with the DC-link voltage fluctuations under an unbalanced grid. The traditional method of suppressing the circulating current cannot minimize the DC-link voltage fluctuations. The proposed method in this paper can effectively reduce the DC-link voltage fluctuations under unbalanced grid conditions.

(3)

The TZCV and SCC coupling injection methods operate in real time online. Without offline optimization, the circulating current injection reference value can be computed in real time based on various grid voltage imbalance conditions and system control goals.