Submodule Capacitor Voltage Ripple Reduction of Full-Bridge Submodule-Based MMC (FBSM-MMC) with Non-Sinusoidal Voltage Injection ^†

Abstract

The full-bridge submodule (FBSM)-based modular multilevel converter (FBSM-MMC) offers promising performance due to its ability to output negative FBSM voltage. This paper studies this ability of the FBSM-MMC under zero-sequence voltage injection (ZSVI) and second-order harmonic current injection (SHCI). The focus of the research is to redistribute the FBSM powers by injecting an additional power degree of freedom, resulting in a smaller FBSM capacitor voltage ripple, while keeping the maximum AC output voltage for a given fundamental frequency component of the arm voltage reference. Accordingly, a control strategy was developed, based on non-sinusoidal ZSVI, and SHCI is proposed for further improving the performance of the FBSM-MMC. The proposed non-sinusoidal ZSVI contains a higher sinusoidal third-order harmonic component than that of pure sinusoidal third-order voltage injection (THVI) when operating under the same maximum AC output voltage. By implementing this solution, a smaller amplitude of the injected second-order harmonic current can be achieved, producing a lower power loss in the FBSM-MMC. Considering the proposed solution, the relationship of the arm powers and FBSM capacitor voltages are also discussed. Finally, the simulation results and experimental results are presented to verify the effectiveness of the theoretical analysis of the proposed method.

1. Introduction

Apart from the traditional two-level and three-level voltage source converter (VSC), the modular multilevel converter (MMC) offers great superiority in terms of module nature, excellent output voltage, and lower power losses [1,2,3,4,5,6]. Typically, the desired output voltage of the MMC is obtained by chopping the submodule (SM) capacitor voltages in the upper and lower arms, based on a specific principle. The SM capacitors in the MMC system serve as the medium platform for exchanging power between the input and output sides. However, an unwanted SM capacitor voltage ripple appears when exchanging power between the input and output links. Usually, an effectively performing SM capacitor voltage is of great significance for supporting the DC bus voltage and producing the desired AC voltage.

SM capacitors occupy more than half of the size and weight in a single SM, and a smaller capacitor ripple would favor the lightweight running operation of an MMC system [7]. Recently, attention has been paid to the reduction of the SM capacitor voltage ripple [8,9,10,11,12,13,14]. A review of the literature shows that the fundamental frequency part, induced by the imbalance of the upper and lower arm powers, is a dominant reason for the undesired SM capacitor voltage ripple. Therefore, the suppression of the fundamental frequency part of the SM capacitor voltage ripple can be considered as a promising approach. It was found that an increased modulation index always results in the decreased fundamental frequency part in the SM capacitor voltage ripple [15]. Comparatively, a smaller SM capacitor voltage ripple can be obtained for the MMC system when operating at a higher modulation index. In particular, the fundamental frequency part of the SM capacitor voltage ripple can be fully eliminated when the modulation index is equal to 1.414 [13]. However, the enlarged AC output voltage increases the insulation cost of the converter equipment to a certain extent. Considering the abovementioned issue, harmonic injection would be an alternative scheme for reducing the SM capacitor voltage ripple. Theoretical analysis has verified that the i-th harmonic of the SM capacitor voltage ripple presents an inversely proportional relation with the corresponding i-th harmonic of the SM power. Thus, the SM capacitor voltage ripple can be significantly reduced by injecting high-frequency harmonics. However, this also induces serious power losses and limited AC output voltage.

In order to respond to the issues caused by high-frequency harmonic injection, researchers have attempted to introduce low-frequency zero-sequence voltage injection (ZSVI), which provides the possibility of shaping the peak value of the arm reference voltages in a smaller fluctuating range [16,17,18,19,20,21,22,23,24]. Generally, third-order harmonic injection (THVI) is considered to be a promising scheme that has the ability to redistribute arm powers within the converter system, significantly improving submodule (SM) capacitance and power loss reductions [19]. Accordingly, an optimal THVI is proposed to calculate the magnitude and phase of THVI online to increase DC bus voltage utilization [20]. This shows that the peak value of the arm reference voltage changes with a different ratio of reactive power to the total power. The maximum power transfer is also determined by the SM capacitor voltage ripple when transferring the active power only [21]. THVI under FBSM-MMC is investigated with the focus on achieving more efficient and less SM capacitance-based MMC [23]. However, the influence of the single THVI on the reduction of the SM capacitor voltage ripple is limited. This is due to the lack of an independent power degree of freedom eliminating the dominant SM capacitor voltage ripple. Therefore, the preferred scheme recommends the adoption of both THVI and second-order harmonic current injection (SHCI). A novel MMC topology operating under THVI and SHCI is also proposed, which aims to improve the low-frequency operating performance in medium voltage-driven applications [25]. In this case, a large amplitude of the SHCI is usually required, which can introduce an undesired power loss.

In this paper, non-sinusoidal ZVSI and SHCI is proposed for investigating the FBSM capacitor voltage ripple feature of the FBSM-MMC. Compared with pure sinusoidal THVI and SHCI, the proposed non-sinusoidal ZVSI contains a larger third-order harmonic component when achieving the same AC output voltage. This reduces the required amplitude of the second-order harmonic current. Therefore, by using the proposed method, reduced power losses can be obtained. The paper also discusses the relationship of the arm powers and the FBSM capacitor voltage ripples. A comparison between the proposed method and the sinusoidal injection is analyzed in terms of FBSM capacitor voltage ripple reduction. Finally, the simulation and experimental results are presented to verify the theoretical analysis.

This paper is organized as follows. Section 2 provides a detailed analysis of the relationship between the arm powers and SM capacitor voltages. In Section 3, the influence of THVI and SHCI on the arm powers and SM capacitor voltages are discussed considering an FBSM-based MMC. Section 4 describes the implementation of the proposed method in order to reduce the FBSM capacitor voltage ripple. To verify the effectiveness of theoretical analysis, simulation and experimental results are presented in Section 5 and Section 6, respectively. Finally, Section 7 concludes the work.

2. Analysis of FBSM-Based MMC System

The schematic of the FBSM-MMC system is shown in Figure 1. It consists of three phases. Each phase comprises the upper arm, lower arm, and inductors. In each arm, there are n cascaded FBSMs connected in series for producing the required multilevel waveforms. Unlike the half-bridge submodule (HBSM), which considers only zero state and SM capacitor voltages as output [6], the FBSM contains three output states: positive SM capacitor voltage, negative SM capacitor voltage, and zero state. Additionally, enlarged utilization of the DC bus voltage can be achieved due to the negative output ability of the FBSM. This not only increases the AC load operating range but also partly reduces the FBSM capacitor voltage ripple [18].

The single FBSM (shown in Figure 1) consists of four power switches and one capacitor. Assuming the switching losses in the FBSM as negligible and based on the law of power conservation, the FBSM power can be expressed as:

$$p_{xsmi}=u_{xsmi}\cdot i_{xy}=u_{xcmi}\cdot i_{xcmi}$$

(1)

where p_xsmi is the i-th FBSM power of phase x (x = a, b, c); u_xsmi and u_xcmi are the i-th FBSM voltage and capacitor voltage of phase x (x = a, b, c), respectively; i_xy is the arm current of phase x (x = a, b, c; y = u, l); and i_xcmi is the i-th capacitor current in FBSM of phase x (x = a, b, c). It is assumed that the FBSM-based MMC system is well balanced with sufficient switching frequency, and the FBSM capacitor voltages of the same arm would be equal. Therefore, the arm powers can be expressed as:

$$\{\begin{array}{c}{p}_{xu}=n\left(u_{xcmu}\cdot i_{xcmi}\right)=nC\frac{d{u}_{xcmu}}{dt}\cdot u_{xcmu}\\ p_{xl}=n\left(u_{xcml}\cdot i_{xcmi}\right)=nC\frac{d{u}_{xcml}}{dt}\cdot u_{xcml}\end{array}$$

(2)

where p_xu and p_xl are the upper and lower arm powers of phase x (x = a, b, c); u_xcmu and u_xcml are the upper and lower arm FBSM capacitor voltages of phase x (x = a, b, c), respectively; and C is the FBSM capacitance. For simplified calculation, the FBSM capacitor voltage can be expressed as:

$$\{\begin{array}{c}{u}_{xcmu}=u_{ref}+\Delta u_{xcmu}\\ u_{xcml}=u_{ref}+\Delta u_{xcml}\end{array}$$

(3)

where u_ref is the FBSM capacitor voltage reference; and Δu_xcmu and Δu_xcml are the upper and lower arm FBSM capacitor voltage ripples of phase x (x = a, b, c), respectively. In general, the FBSM capacitor voltage ripple can be neglected as compared to the referenced voltage u_ref. Substituting (3) into (2) yields:

$$\{\begin{array}{c}{p}_{xu}=nC\frac{d\left(u_{ref}+\Delta u_{xcmu}\right)}{dt}\left(u_{ref}+\Delta u_{xcmu}\right)\approx nC{u}_{ref}\frac{d\Delta u_{xcmu}}{dt}\\ p_{xl}=nC\frac{d\left(u_{ref}+\Delta u_{xcml}\right)}{dt}\left(u_{ref}+\Delta u_{xcml}\right)\approx nC{u}_{ref}\frac{d\Delta u_{xcml}}{dt}\end{array}$$

(4)

The arm powers and FBSM capacitor voltage ripples can be expressed, in the form of their spectrum frequencies, as:

$$\{\begin{array}{c}{p}_{xy}={\displaystyle \sum _{i=1}^{\infty }{p}_{xy}\left(i\omega t\right)}\\ \Delta u_{xcmu}={\displaystyle \sum _{i=1}^{\infty }\Delta u_{xcmu}\left(i\omega t\right)}\end{array}$$

(5)

where p_xy and Δu_xcmy are the arm power and FBSM capacitor voltage ripple of phase x (x = a, b, c; y = u, l); p_xy(iωt) is the i-th power harmonic in p_xy; and Δu_xcmy(iωt) is the i-th voltage harmonic in Δu_xcmy of phase x (x = a, b, c; y = u, l). Therefore, based on (4) and (5), the expression is obtained as:

$$\{\begin{array}{c}\Delta u_{xcmu}={\displaystyle \sum _{i=1}^{\infty }\left(\frac{1}{i\omega nC{u}_{ref}}\int p_{xu}\left(i\omega t\right)dt\right)}\\ \Delta u_{xcml}={\displaystyle \sum _{i=1}^{\infty }\left(\frac{1}{i\omega nC{u}_{ref}}\int p_{xl}\left(i\omega t\right)dt\right)}\end{array}$$

(6)

This shows that the FBSM capacitor voltage ripple is closely associated with the arm powers. The amplitude of the i-th harmonic in Δu_xcmy is inversely proportional to the amplitude of the i-th harmonic in p_xy. The redistributed arm powers would be large enough to influence the harmonic distribution in FBSM capacitor voltage ripples. Generally, the injection of the voltages and currents for the MMC system is effective for improving power distribution.

3. Analysis of the Powers and Voltages for FBSM-Based MMC under THVI and SHCI

3.1. Inserted Third-Order Harmonic for FBSM-Based MMC

The basic principle of THVI is shown in Figure 2. It shows that, aiming to enlarge the utilization of the DC-link voltage without overmodulation, the peak value of the arm reference voltage can be decreased by THVI. For in-depth analysis, the inserted third-order harmonic is provided as:

$$v_h=V_h\sin 3\omega t$$

(7)

where V_h is the amplitude of the inserted third-order harmonic.

The third-order voltage injected into the arms and the second-order current harmonic injected into the circulating currents demonstrate favorable features for redistribution of the arm powers. On one hand, for a defined DC bus voltage under an FBSM-based MMC system, the larger AC voltage output can be achieved by THVI with the same rating of MMCs. On the other hand, compared with high-frequency harmonic injection, a smaller switching frequency can be applied to achieve low-frequency harmonic injection. This results in lower switching losses.

3.2. Analysis of the Arm Powers

For simplification of the process, phase a is taken for analysis. Considering the working principles of the MMC, THVI-based arm voltage reference of the modified MMC can be expressed as:

$$\{\begin{array}{c}{u}_{au-ref}=\frac{1}{2}{V}_{dc}-V_a\sin \omega t-V_h\sin 3\omega t\\ u_{al-ref}=\frac{1}{2}{V}_{dc}+V_a\sin \omega t+V_h\sin 3\omega t\end{array}$$

(8)

where u_au-ref and u_al-ref are the upper and lower arm voltage references, respectively; V_a is the amplitude of the output voltage in phase a; and ω is the angular frequency of output voltage. It is assumed that if the circuit parameters in the upper and lower arms are highly consistent, the output current could be evenly distributed for the upper arm and lower arm, which is:

$$\{\begin{array}{c}{i}_{au}=I_{acir}+\frac{1}{2}{I}_a\sin \left(\omega t-\phi \right)-z{I}_{acir}\cos 2\omega t\\ i_{al}=I_{acir}-\frac{1}{2}{I}_a\sin \left(\omega t-\phi \right)-z{I}_{acir}\cos 2\omega t\end{array}$$

(9)

where, for the phase a, i_au and i_al are the upper and lower arm currents, respectively; I_acir is the DC part of the circulating current; and I_a is the amplitude of the output current. Furthermore, φ is the phase angle difference between the output voltage and current, and z is defined as the second-order current harmonic index.

Considering (8) and (9), the instantaneous arm powers can be expressed as:

$$\{\begin{array}{c}{p}_{au}=p_{acom}+p_{adif}\\ p_{al}=p_{acom}-p_{adif}\end{array}$$

(10)

where:

$$\{\begin{array}{c}\begin{array}{ll}{p}_{acom}& =\frac{m}{8}{V}_{dc}{I}_a\cos \left(2\omega t-\phi \right)-\frac{zm}{8}{V}_{dc}{I}_a\cos 2\omega t\\ & -\frac{1}{4}{V}_h{I}_a\cos \left(2\omega t+\phi \right)+\frac{1}{4}{V}_h{I}_a\cos \left(4\omega t-\phi \right)\end{array}\\ \begin{array}{ll}{p}_{adif}& =\frac{2-m^2}{8}{V}_{dc}{I}_a\cos \phi \sin \omega t-\frac{z{m}^2}{16}{V}_{dc}{I}_a\sin \omega t\\ & +\frac{zm}{8}{V}_h{I}_a\sin \omega t-\frac{1}{4}{V}_{dc}{I}_a\sin \phi \cos \omega t\\ & +\frac{z{m}^2}{16}{V}_{dc}{I}_a\sin 3\omega t-\frac{m}{4}{V}_h{I}_a\sin 3\omega t\\ & +\frac{zm}{8}{V}_h{I}_a\sin 5\omega t\end{array}\end{array}$$

(11)

where p_acom and p_adif are the common and differential parts of p_au and p_al, respectively; and m is the system modulation index which is obtained, as reported in [11]. At the time of transferring the active power, the arm power can be expressed as:

$$\{\begin{array}{l}{p}_{acom}=\frac{\left(1-z\right)m}{8}{V}_{dc}{I}_a\cos 2\omega t-\frac{1}{4}{V}_h{I}_a\cos 2\omega t+\frac{1}{4}{V}_h{I}_a\cos 4\omega t\\ \begin{array}{ll}{p}_{adif}& =\frac{4-\left(2+z\right)m^2}{16}{V}_{dc}{I}_a\sin \omega t+\frac{zm}{8}{V}_h{I}_a\sin \omega t\\ & +\frac{z{m}^2}{16}{V}_{dc}{I}_a\sin 3\omega t-\frac{m}{4}{V}_h{I}_a\sin 3\omega t+\frac{zm}{8}{V}_h{I}_a\sin 5\omega t\end{array}\end{array}$$

(12)

In order to achieve the optimal realization of DC bus voltage for MMC [19], the amplitude of the injected third-order harmonic can be written as:

$$V_h=\frac{1}{6}{V}_a$$

(13)

From (12) and (13), it can also be expressed as:

$$\{\begin{array}{l}{p}_{acom}=\frac{5m-6zm}{48}{V}_{dc}{I}_a\cos 2\omega t+\frac{m}{48}{V}_{dc}{I}_a\cos 4\omega t\\ \begin{array}{ll}{p}_{adif}& =\frac{4-\left(2+\left(5/6\right)z\right)m^2}{16}{V}_{dc}{I}_a\sin \omega t\\ & +\frac{\left(3z-1\right)m^2}{48}{V}_{dc}{I}_a\sin 3\omega t+\frac{z{m}^2}{96}{V}_{dc}{I}_a\sin 5\omega t \end{array}\end{array}$$

(14)

From (14), it can be stated that the THVI and SHCI would change the distribution of the dominant power ripples in the arms. An independent degree of freedom in the arm power produced by the injected voltage and current can be achieved for modifying the feature of low-frequency power ripples.

As discussed in Section 2, it has been verified that the i-th harmonic amplitude in SM capacitor voltage is inversely proportional to the i-th harmonic amplitude in the SM powers. Therefore, the SM capacitor voltage can be expressed as:

$$\{\begin{array}{c}{u}_{acmu}=u_{ref}+u_{acom}+u_{adif}\\ u_{acml}=u_{ref}+u_{acom}-u_{adif}\end{array}$$

(15)

where:

$$\{\begin{array}{l}{u}_{acom}=\frac{\left(5-6z\right)m}{96C\omega U_{ref}n}{V}_{dc}{I}_a\sin 2\omega t+\frac{m}{192C\omega U_{ref}n} V_{dc}{I}_a\sin 4\omega t \\ \begin{array}{ll}{u}_{adif}& =-\frac{4-\left(2+\left(5/6\right)z\right)m^2}{16C\omega U_{ref}n}{V}_{dc}{I}_a\cos \omega t\\ & -\frac{\left(3z-1\right)m^2}{144C\omega U_{ref}n} V_{dc}{I}_a\cos 3\omega t-\frac{z{m}^2}{480C\omega U_{ref}n}{V}_{dc}{I}_a\cos 5\omega t\end{array}\end{array}$$

(16)

where u_acmu is the upper arm SM capacitor voltage; u_acml is the lower arm SM capacitor voltage; u_acom represents the SM capacitor voltage ripple caused by the common-mode power p_acom; and u_adif represents the SM capacitor voltage ripple caused by the differential-mode power p_adif.

It can be seen that the dominant harmonic ripples in the SM capacitor voltage in (16) are functions of the modulation index and second-order current harmonic index. Therefore, the SM capacitor voltage ripple can be suppressed by controlling the modulation index and second-order current harmonic index. Moreover, the dominant fundamental frequency ripple can be eliminated when the condition below is achieved as:

$$4-\left(2+\frac{5}{6}z\right)m^2=0$$

(17)

Figure 3 highlights the variation principle of the second-order current harmonic index z, with the modulation index m, when the fundamental frequency ripple in the SM capacitor voltage is eliminated. This indicates that an increased modulation index would reduce the required amplitude of the injected second-order current harmonic. This could dramatically reduce power losses. Based on (17), the following conditions should be considered.

3.2.1. Suppression of FBSM Capacitor Voltage Ripple When z = 0

When the second-order current harmonic is not injected, (17) is transformed into:

$$\{\begin{array}{c}m\approx 1.414\\ z=0\end{array}$$

(18)

Equation (18) indicates that the fundamental frequency ripple in SM capacitor voltage can be eliminated without injecting the second-order current harmonic. This can be achieved by setting the modulation index to 1.414. Although it could reduce the SM capacitor voltage ripple with high efficiency, the enlarged modulation index m = 1.414 brings serious insulation requirements.

Considering (16) and (18), the FBSM capacitor voltage ripple function can be expressed as:

$${\epsilon }_1=\frac{3.535\sin 2\omega t+0.3535\sin 4\omega t\pm 0.667\cos 3\omega t}{48C\omega U_{ref}n}{V}_{dc}{I}_a$$

(19)

where ε₁ is the FBSM capacitor voltage ripple function under the scheme in (19), while the sign ± contributes to the upper arm and lower arm capacitor voltage ripple, respectively. Based on (19), the peak-to-peak value of FBSM capacitor voltage ripple can be expressed as:

$${\lambda }_1=\max \left({\epsilon }_1\right)-\min \left({\epsilon }_1\right)\approx \frac{11.09V_a{I}_a}{48C\omega U_{ref}n}$$

(20)

where λ₁ is the peak-to-peak value of the FBSM capacitor voltage ripple function under the scheme in (19).

3.2.2. Suppression of FBSM Capacitor Voltage Ripple without Third-Order Voltage Injected

Considering (12), (13), and (16), and without the third-order voltage harmonic injected, the FBSM capacitor harmonic voltages can be modified as:

$$\{\begin{array}{c}{u}_{acom}=\frac{\left(6-6z\right)m}{16C\omega U_{ref}n}{V}_{dc}{I}_a\sin 2\omega t\\ u_{adif}=-\frac{4-\left(2+z\right)m^2}{16C\omega U_{ref}n}{V}_{dc}{I}_a\cos \omega t-\frac{z{m}^2}{48C\omega U_{ref}n} V_{dc}{I}_a\cos 3\omega t\end{array}$$

(21)

From (21), it can be understood that the fundamental and second-order harmonic parts in FBSM capacitor voltage ripple can be eliminated when the condition below is satisfied:

$$\{\begin{array}{c}m=\frac{2}{\sqrt{3}}\\ z=1\end{array}$$

(22)

This conveys that the common voltage u_acom in (21) can always be eliminated when z = 1, which is free of the modulation index. It should be noted that the realization of (22) can only be achieved by implementing FBSM-based MMC, which is not available for HBSM-based MMC. Similarly, the FBSM capacitor voltage ripple under the scheme in (23) can be expressed as:

$${\epsilon }_2=\pm \frac{\left(4/3\right)\cos 3\omega t}{48C\omega U_{ref}n}{V}_{dc}{I}_a$$

(23)

where ε₂ is the FBSM capacitor voltage ripple function under (22), and the sign ± contributes to the upper arm and lower arm capacitor voltage ripple, respectively. The peak-to-peak value of the FBSM capacitor voltage ripple can be expressed as:

$${\lambda }_2=\max \left({\epsilon }_2\right)-\min \left({\epsilon }_2\right)\approx \frac{4.62V_a{I}_a}{48C\omega U_{ref}n}$$

(24)

where λ₁ is the peak-to-peak value of the FBSM capacitor voltage ripple function under the scheme in (22).

3.2.3. Suppression of FBSM Capacitor Voltage Ripple with THVI and SHCI

Once the fundamental frequency ripple in the SM capacitor voltage is eliminated under (16), the second- and third-order harmonics would be dominant in the SM capacitor voltage ripple. In order to analyze the dominant ripple in the FBSM capacitor voltage, we define the amplitude ratio of second-order harmonic with third-order harmonic in (16), which is:

$$d=\frac{1}{m}\left|\frac{15-18z}{6z-2}\right|$$

(25)

The variation principle of amplitude ratio d and the second-order current harmonic z is shown in Figure 4. It indicates that the second-order harmonic powers in (16) can be eliminated when z = 5/6. Therefore, based on (17), the following can be obtained:

$$\{\begin{array}{c}m\approx 1.218\\ z=\frac{5}{6}\end{array}$$

(26)

Due to the injection of the third-order harmonic, the second-order harmonic ripple in SM capacitor voltage can be suppressed in part. In this manner, reduced SHCI is required to eliminate the second-order harmonic from the SM capacitor voltage as compared to that in (22).

Since the fundamental and second-order harmonic ripples can be totally suppressed using (26), the third-order harmonic in (16) would be dominant in the SM capacitor voltage ripple. Meanwhile, the fourth- and fifth-order harmonics in (16) would influence the FBSM capacitor voltage ripples in part, considering (26). For convenient analysis, the FBSM capacitor voltage ripple function can be defined, using the proposed method, as:

$${\epsilon }_3\approx \frac{0.305\sin 4\omega t\pm \left(0.742\cos 3\omega t+0.124\cos 5\omega t\right)}{48C\omega U_{ref}n}{V}_{dc}{I}_a$$

(27)

where ε₃ is the FBSM capacitor voltage ripple function obtained from (26), and ± contributes to the upper and lower arm capacitor voltage ripples, respectively. Based on (27), the peak-to-peak value of the FBSM capacitor voltage ripple can be expressed as:

$${\lambda }_3=\max \left({\epsilon }_3\right)-\min \left({\epsilon }_3\right)\approx \frac{3.12V_a{I}_a}{48C\omega U_{ref}n}$$

(28)

where λ₃ is the peak-to-peak value of the FBSM capacitor voltage ripple of ε₃.

Considering (20), (24), and (28), the FBSM capacitor voltage ripple ratios can be defined as:

$$\{\begin{array}{c}{\gamma }_1=\frac{{\lambda }_3}{{\lambda }_1}\approx 0.28\\ {\gamma }_2=\frac{{\lambda }_3}{{\lambda }_2}\approx 0.66\end{array}$$

(29)

where γ₁ is the FBSM capacitor voltage ripple ratio of the scheme in (26) to that in (18), and γ₂ is the FBSM capacitor voltage ripple ratio of the scheme in (26) to that in (22).

Compared with the scheme proposed in (18), Equation (29) indicates that the FBSM capacitor voltage ripple referred to in (26) can be reduced by approximately 72%. Compared with the scheme proposed in (22), Equation (29) indicates that the FBSM capacitor voltage ripple under (26) can be reduced by approximately 34%. Therefore, the lower capacitances can be added to FBSMs under the same ripple limitation. Meanwhile, the reduced required second-order harmonic current produces reduced conduction loss.

4. Proposed Method of Non-sinusoidal Voltage Injection for FBSM-MMC

4.1. Proposed Method of the FBSM-Based MMC

As discussed in Section 3, a dramatic reduction of the FBSM capacitor voltage ripple can be achieved by eliminating the fundamental frequency and second-order harmonic parts in the FBSM capacitor voltage ripple in (26). Based on this, aiming to obtain a lower amplitude of the injected second-order harmonic current for equal MMC facilities, a new methodology is proposed that consists of specific zero-sequence voltage injection and SHCI.

The authors of [17] showed that the non-sinusoidal THVI consists of the third-order sinusoidal voltage and multiples of third-order harmonics, as shown in Figure 5. Compared with sinusoidal THVI, the non-sinusoidal voltage contains higher amplitude of the third-order sinusoidal voltage under same DC bus voltage utilization, and can be expressed as:

$$v_h=-\frac{1}{2}\left(\max (v_a,v_b,v_c)+\min (v_a,v_b,v_c)\right)$$

(30)

Due to the negligible amplitudes of the other harmonics in (30), the third-order sinusoidal voltage part of non-sinusoidal THVI is proposed to be investigated to observe its influence on power distribution instead. Therefore, Equation (13) can be modified as:

$$V_h\approx 0.2067V_a$$

(31)

Substituting (13) into (12), modified arm powers are obtained, based on the proposed zero-sequence voltage injection, that are expressed as:

$$\{\begin{array}{l}{p}_{acom}=\frac{4.76m-6zm}{48}{V}_{dc}{I}_a\cos 2\omega t+\frac{1.24m}{48}{V}_{dc}{I}_a\cos 4\omega t\\ \begin{array}{ll}{p}_{adif}& =\frac{4-\left(2+0.7933z\right)m^2}{16}{V}_{dc}{I}_a\sin \omega t\\ & +\frac{\left(3z-1.24\right)m^2}{48}{V}_{dc}{I}_a\sin 3\omega t+\frac{1.24z{m}^2}{96}{V}_{dc}{I}_a\sin 5\omega t \end{array}\end{array}$$

(32)

Based on (32), u_acom and u_adif can be expressed as:

$$\{\begin{array}{l}{u}_{acom}=\frac{\left(4.76-6z\right)m}{96C\omega U_{ref}n}{V}_{dc}{I}_a\sin 2\omega t+\frac{1.24m}{192C\omega U_{ref}n} V_{dc}{I}_a\sin 4\omega t \\ \begin{array}{ll}{u}_{adif}& =-\frac{4-\left(2+0.7933z\right)m^2}{16C\omega U_{ref}n}{V}_{dc}{I}_a\cos \omega t\\ & -\frac{\left(3z-1.24\right)m^2}{144C\omega U_{ref}n} V_{dc}{I}_a\cos 3\omega t-\frac{1.24z{m}^2}{480C\omega U_{ref}n}{V}_{dc}{I}_a\cos 5\omega t\end{array}\end{array}$$

(33)

Based on (33), the fundamental frequency part in SM capacitor voltage ripple can be eliminated by achieving the condition provided as:

$$4-\left(2+0.7933z\right)m^2=0$$

(34)

The difference between Equation (34) and (17) can be clearly observed. In addition, based on (33) and (34), the second-order harmonic in the SM capacitor voltage ripple can be eliminated at z ≈ 0.7933. Therefore, Equation (34) results in:

$$\{\begin{array}{c}m\approx 1.233\\ z\approx 0.7933\end{array}$$

(35)

Figure 6 shows the variation principle of the second-order harmonic current and modulation index based on (17) and (34). Compared with the sinusoidal THVI method in (26), the proposed method contains higher amplitude of the third-order sinusoidal voltage in (30) under equal utilization of DC bus voltage. Therefore, lower second-order current harmonic is required to follow the proposed methodology in (35). Therefore, a reduced amplitude of the injected second-order harmonic current can be achieved. This results in a reduced conduction loss due to the reduced injected harmonic currents under the proposed method, compared with sinusoidal THVI and SHCI.

Based on (33) and (35), the FBSM capacitor voltage ripple function can be expressed as:

$${\epsilon }_4\approx \frac{0.382\sin 4\omega t\pm \left(0.578\cos 3\omega t+0.15\cos 5\omega t\right)}{48C\omega U_{ref}n}{V}_{dc}{I}_a$$

(36)

where ε₄ is the FBSM capacitor voltage ripple function with the proposed zero-sequence voltage injection, whereas the sign ± contributes to the upper arm and lower arm capacitor voltage ripple, respectively. Here, peak-to-peak value of ε₄ can be obtained:

$${\lambda }_4=\max \left({\epsilon }_4\right)-\min \left({\epsilon }_4\right)\approx \frac{1.69V_{dc}{I}_a}{48C\omega U_{ref}n}$$

(37)

where λ₄ is the peak-to-peak value of the FBSM capacitor voltage ripple considering the proposed method. Equations (35) and (37) show that the proposed method can achieve less reduction in the FBSM capacitor voltage ripple compared with that in (26).

4.2. Power Distribution of FBSM-Based MMC System under Proposed Method

The modulation index and injected second-order current harmonic influence the power distribution of the FBSM-based MMC system. The power distribution could transfer the low-frequency ripple powers in the arms to a higher frequency power range, which achieves significant reduction in the SM FBSM capacitor voltage ripples. Substituting (34) into (32) produces:

$$\{\begin{array}{c}{p}_{acom}\approx \frac{1.53}{48}{V}_{dc}{I}_a\cos 4\omega t\\ p_{adif}\approx \frac{1.73}{48}{V}_{dc}{I}_a\sin 3\omega t+\frac{1.5}{96}{V}_{dc}{I}_a\sin 5\omega t \end{array}$$

(38)

Considering the proposed scheme, (38) demonstrates that the arm powers contain the third, fourth, and fifth-order harmonic powers; the third-order harmonic power in p_adif is dominant in the arm powers.

Figure 7 presents a layout of the power distribution in the FBSM-based MMC with the proposed method for single-phase configuration. This shows that the fourth-order harmonics in p_acom are transferred from the source to the arms that are delivered to the load instantly. It should be noted that the fourth-order harmonics in p_acom could be controlled to enable their flow within the phases. The fourth-order harmonics of the upper and lower arms can be controlled around their respective upper and lower arms for each phase. The third- and fifth-order harmonics in p_adif only flow back and forth around the phase leg, which shows no influence on load.

5. Simulation Results

As a result of the work detailed in previous sections, it can be stated that SM capacitor voltage ripples can be further reduced when introducing the negative state output in FBSMs. This section presents the simulation results using MATLAB to verify the effectiveness of the analysis carried out in Section 3. In order to achieve improved performance, the phase-shifted carrier PWM (PSC-PWM) is adopted. Detailed parameters are listed in

Table 1. Circuit parameters for simulation.

Symbol	Quantity	Value
Vdc	DC-link voltage	200 V
n	Number of FBSMs per arm	4
C	SM capacitor	0.5 mF
L	Arm inductor	10 mH
fc	SM switching frequency	2 kHz
f	Fundamental frequency	50 Hz
ucm	SM capacitor voltage reference	65 V
Rload	Load resistance	50 Ω

.

It has been verified that the fundamental part in the FBSM capacitor voltage ripple can be eliminated when m = 1.414 without harmonic current injected. As shown in Figure 8, the FBSM capacitor voltage ripple mainly consists of second-order harmonics. The advantage of the presence of second-order harmonics is a dramatic reduction in conduction losses. However, the insulation requirements increase the system cost. Considering (35), the insulation requirements for an FBSM-based MMC system can be reduced significantly, which is a compromise for the small SHCI. Moreover, compared with the FBSM capacitor voltage ripple under m = 1.414, it introduces further reduction in the FBSM capacitor voltage ripple.

For comprehensive comparative analysis, simulation results at m = 1.233 and z = 0, and at m = 1.233 and z = 0.7933 are presented to evaluate the improved performance of the proposed method. Figure 9 shows the simulation results of the FBSM power and capacitor voltage at m = 1.233 and z = 0 while Figure 10 plots the spectrum distribution corresponding to Figure 9. Figure 9a and Figure 10a explain that the dominant ripples in FBSM powers are the fundamental and second-order harmonic parts. Figure 9b is the FBSM capacitor voltage, and the peak-to-peak value in Figure 9a is 1.8 V. It has been verified in Section 2 that the amplitude of the i-th harmonic in the FBSM capacitor voltage ripple is inversely proportional to the amplitude of the i-th harmonic in FBSM power. This behavior reflects agreement between Figure 10a,b.

Figure 11 presents the simulation results of FBSM power and FBSM capacitor voltage based on the proposed method in (35). Figure 11a shows the plot of FBSM power, while Figure 11b plots the FBSM capacitor voltage. The corresponding plots of spectrum distribution are presented in Figure 12. Figure 11a and Figure 12a show that the dominant ripple in FBSM power is the third-order harmonic part. Comparing this behavior with that in Figure 10a, a difference can be observed. In Figure 11b, the peak-to-peak value of the FBSM capacitor voltage ripple is 0.4 V, which shows a dramatic reduction in amplitude compared with that in Figure 9b. Figure 12b is the corresponding spectrum distribution of Figure 11b, which reflects the similar spectrum distribution with that of the FBSM power shown in Figure 12a.

Figure 13 presents the arm currents for the FBSM-based MMC under different conditions, Figure 13a plots the arm currents at m = 1.233 and z = 0, and Figure 13b plots the arm currents considering the proposed method. The corresponding spectrum distribution of Figure 13 is shown in Figure 14. In Figure 13a and Figure 14a, the arm currents consist of the DC part and fundamental frequency part in accordance with z = 0. With further reduction of the capacitor voltage ripple, SCHI is introduced. As shown in Figure 13b, the arm currents consist of the DC part and fundamental frequency part, which reflects agreement with the spectrum distribution in Figure 14b.

6. Experimental Results

In order to verify the effectiveness of the theoretical analysis, a downscaled prototype of the FBSM-based MMC system was built in a laboratory environment. Due to the limitation of the experimental facilities, a single-phase power stage with four FBSMs per arm was used. This system consists of a power stage and control system. The overall layout of the power stage for the FBSM-based MMC is shown in Figure 15, while the control system is shown in Figure 16. As shown in Figure 16, the control system is composed of a TI-tms320f28335 DSP and an Altera EP4CE115F23C8N FPGA chip. The DSP chip acts as the host chip to achieve the control algorithm, while the FPGA sends the required signals to the DSP and produces the PWM signals to drive the power switches. A photograph of the experimental prototype is shown in Figure 17. The detailed hardware parameters are listed in

Table 2. Circuit Parameters for experiment.

Symbol	Quantity	Value
Cin	Input electrolytic capacitor	10 mF
Vdc	DC-link voltage	200 V
n	Number of FBSMs per arm	4
C	SM capacitor	0.68 mF
L	Arm inductor	6 mH
fc	SM switching frequency	2 kHz
f	Fundamental frequency	50 Hz
ucm	SM capacitor voltage reference	65 V
Rload	Load resistance	33 Ω

.

Considering the discussion referring to (18), a small step-up modulation index results in the bigger step-down amplitude of the injected harmonic current when eliminating the fundamental harmonic from the SM capacitor voltage ripple. Based on this, further reduction in the FBSM capacitor voltage ripple can be achieved with reduced amplitude of the second-order harmonic current under the proposed method in (35). In comparison, the FBSM capacitor voltage and arm currents of the FBSM-based MMC system at m = 1.233 and z = 0 are provided, as shown in Figure 18. The FBSM capacitor voltage is shown in Figure 18a, and the arm currents are shown in Figure 18b. According to the discussion in Section 3, the dominant ripple in the FBSM capacitor voltage for the proposed method in (36) could not be eliminated. In order to cope with this issue, additional second-order harmonic current should be injected. Obtained experimental results are provided in Figure 19. Figure 19a shows the FBSM capacitor voltage, while the arm currents can be observed in Figure 19b. This shows that, under the proposed method, a dramatic reduction of the FBSM capacitor voltage ripple can be achieved, compared with that under m = 1.233 and z = 0.

7. Conclusions

Harmonic injection effectively suppressed the SM capacitor voltage ripple. The effect was dependent on the type of injection. In this paper, we investigated the influence of different injection schemes on SM capacitor voltage ripple reduction in an MMC system; specifically, we focused on exploring the FBSM capacitor voltage ripple suppression method under ZSVI and SHCI in an FBSM-MMC. The theoretical analysis demonstrates that a small step-up of the modulation index results in a bigger step-down of the amplitude in the injected second-order harmonic current when eliminating the fundamental harmonic part from the FBSM capacitor voltage ripple. Inspired by these findings, a non-sinusoidal ZSVI and SHCI is proposed in this paper. Compared with pure SHCI, extreme SM capacitor voltage ripple reduction is achieved under the proposed method. Compared with sinusoidal ZSVI and SHCI, the proposed solution contains a greater third-order harmonic component. Correspondingly, in terms of eliminating the dominant low-frequency SM capacitor voltage ripple, a smaller amplitude of the injected second-order harmonic current can be achieved, resulting in a reduced power loss by implementing the proposed methodology. Furthermore, the improvement of the SM capacitor voltage ripple reduction under the proposed injection scheme has been fully verified by theoretical analysis, simulation results, and experimental results.