Circulating Current Suppression Strategy Based on Virtual Impedance and Repetitive Controller for Modular Multilevel Converter Upper and Lower Bridge Arm Capacitance Parameter Asymmetry Conditions

Abstract

In recent years, modular multilevel converters (MMCs) have been increasingly used in the field of electric vehicle charging and discharging due to their unique performance advantages. However, the unique cascade structure of MMCs raises the problem of the circulating current. Due to chemical processes, aging effects, etc., the capacitance parameters of the upper and lower bridge arms will be asymmetric, which will introduce odd harmonics into the circulating current, increasing system losses and threatening the system reliability. To address this phenomenon, this paper proposes a circulating current suppression strategy with additional virtual impedance (VI) based on a repetitive controller (RC). The corresponding simulation models were built for the comparative study of different circulating current suppression strategies. The results show that the VI-RC circulating current suppression strategy can significantly reduce the odd and even harmonics in the circulating current under asymmetric conditions, and the total harmonic distortion (THD) of the bridge arm current is only 0.98%, which verifies the effectiveness of the proposed strategy.

1. Introduction

Around the strategic goal of building a low-carbon economy and constructing an energy internet, the development of electric vehicles is receiving more and more attention. The safety and efficiency of electric vehicles during charging and discharging are closely linked to the performance of converters. Since they were proposed in 2002, modular multilevel converters (MMCs) have been widely used in the fields of high-voltage direct-current (HVDC) transmission, DC distribution networks, and photovoltaic power generation due to their high-output waveform quality, modularity, and easy voltage and power scaling. Compared with the traditional two-level and three-level converters, the MMC has a higher conversion efficiency and smoother output waveform, so it has more advantages in the field of electric vehicle power converters and shows good application prospects [1,2,3,4].

However, the unique cascade structure of the MMC is prone to problems, such as circulating current losses, capacitor voltage fluctuations, and uneven power losses in submodules (SMs), of which the circulating current problem is more prominent [5]. The three-phase bridge arms of the MMC are connected in parallel to the DC bus. At any given time, there are current paths in the upper and lower bridge arms, so the current is introduced from the DC circuit, which is called the circulating current. Under the normal operation of the MMC system, the circulating current is mainly composed of the DC component and a series of even harmonics dominated by the second frequency. In the actual project, due to chemical processes, aging effects, etc., there will be differences in capacitance values between SMs, which will further lead to asymmetry in the equivalent capacitance values of the upper and lower bridge arms [6,7]. It has been pointed out that the asymmetry of the upper and lower bridge arm parameters of the MMC will introduce the second frequency oscillation component and a series of odd harmonics dominated by the fundamental frequency [8,9]. The circulating current will increase the current distortion rate of the bridge arm and increase the rated current capacity of the power switching device, thus increasing the system cost. In addition, the circulating current will increase the RMS value of the bridge arm current, making the power switching devices heat up seriously, increasing the system losses [10,11]. Therefore, some suppression measures must be taken for the circulating current.

Conventional circulating current controllers can only suppress the second or even harmonics and have no effect on the odd harmonics [12]. Therefore, many scholars have studied the improvement of the circulation current suppression strategy. From the perspective of passive hardware improvement, some scholars have proposed that circulating current suppression can be achieved by increasing the bridge arm reactance or by using second-order LC filters, but this will increase the system size and cost [13,14]. From the perspective of active control, some scholars have improved the classic proportional integral (PI) controller and controlled the fundamental frequency harmonic by adding an additional decoupling link on the basis of suppressing the double-frequency harmonic. However, due to the oscillation component of the double-frequency harmonic, the decoupling effect will be significantly affected. Moreover, the control gain of the PI controller for AC signals is limited, and it can only achieve no static error tracking for DC components and cannot effectively suppress the harmonic components in the circulating current [15]. Some scholars have used a quasi-proportional resonant (QPR) controller to control odd harmonics. However, the suppression of harmonics at multiple frequencies requires the parallel connection of multiple resonant links. The parallel connection of multiple resonant links not only increases the difficulty of setting resonant parameters, making the parameters sensitive and complex to calculate when discretizing, but also reduces the phase margin of the system and makes it difficult to design the phase angle compensation, which is not conducive to practical applications [16,17]. Some scholars have combined a PI controller and resonant controller and proposed to use a proportional integral resonant (PIR) controller for circulating current suppression to track both the DC component and harmonic component, which improves the dynamic response of the system, but the PIR controller also has the disadvantages of the QPR controller [18,19].

Aiming at the circulating current problem in the three-phase MMC system with asymmetric capacitance parameters of the upper and lower bridge arms, a control strategy based on the repetitive controller (RC) with increasing virtual impedance (VI) is proposed. Firstly, the basic circuit and operation principle of the MMC are analyzed, a single-phase equivalent model is established, and the harmonic components in the circulating current under the condition of asymmetric capacitance parameters are deeply analyzed. Then, on the basis of analyzing the principle of the controller, a circulation suppression strategy based on a VI-RC is proposed. Finally, the corresponding simulation model is established in MATLAB/Simulink, and the effectiveness of the proposed control strategy is proved through analyzing and comparing the harmonic amplitude of the circulating current and the THD of the bridge arm current under different circulating current suppression strategies.

2. MMC Analysis

This section first introduces the basic circuit of the MMC system and its operating principle and then focuses on the analysis of the components of the circulating harmonics under the asymmetric operating conditions of the upper and lower bridge arm capacitance parameters.

2.1. MMC Basic Circuit

A typical (N + 1) level MMC circuit is shown in Figure 1. The MMC consists of three phases with six bridge arms, each bridge arm consists of N SMs and an inductor $L_m$ connected in series, with the upper and lower bridge arms forming a phase unit. In the figure, $U_{dc}$ and $I_{dc}$ are the DC side voltage and current; $u_{ju}$ and $u_{jl}$ ($j=a,b,c$) are the three-phase upper and lower bridge arm voltage; $i_{ju}$ and $i_{jl}$ are the three-phase upper and lower bridge arm current; $u_j$ and $i_j$ are the AC side three-phase voltage and current; and $R_s$ and $L_s$ are the AC side equivalent resistance and reactance.

The SMs in the MMC can include half-bridge SMs, full-bridge SMs, multilevel SMs, or even a mixture of different circuit structures. Of these, the half-bridge SM has the simplest structure, the lowest number of components, and the highest efficiency and is therefore the most widely used [20].

A typical half-bridge SM circuit is shown in Figure 2. Each SM is connected in series to the main circuit via a connection port and the sum of the output voltages of the SMs within the bridge arm is equal to the DC bus voltage. In the figure, ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ are IGBT modules; ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$ are anti-parallel diodes; $C$ is the SM capacitor; $U_{sju,l}$ is the SM output voltage; and $U_{cju,l}$ and $i_{cju,l}$ are the capacitor voltage and current, respectively.

During normal system operation, the SM in the MMC has a total of four operating states, which are determined by the switching function and the direction of the bridge arm current, as shown in Figure 3. The switching function can be expressed as

$$S=\left\{\begin{array}{l}1,{\mathrm{T}}_1{\mathrm{is}\mathrm{on}\mathrm{and}\mathrm{T}}_2\mathrm{is}\mathrm{off}\hfill \\ 0,{\mathrm{T}}_1{\mathrm{is}\mathrm{off}\mathrm{and}\mathrm{T}}_2\mathrm{is}\mathrm{on}\hfill \end{array}\right.$$

(1)

When $S=1$, the SM is in input mode. When the capacitor is connected to the main circuit and is charged and discharged by the bridge arm current, the output voltage of the SM is $+U_{cju,l}$. When $S=0$, the SM is in cut-off mode. When the capacitor is bypassed, the voltage on both sides of the capacitor remains unchanged and the output voltage of the SM is zero.

Thus, the SM output voltage and capacitor current can be described by the switching function as

$$U_{sju,l}=U_{cju,l}\cdot S$$

(2)

$$i_{cju,l}=i_{ju,l}\cdot S$$

(3)

From the above analysis, it is clear that the capacitor is only charged and discharged by the bridge arm current when the SM is in input mode. The capacitor voltage can therefore be expressed as

$$U_{cju,l}=U_0+\frac{1}{C}{\displaystyle \int i_{ju,l}\cdot S\mathrm{d}t}$$

(4)

where $U_0$ is the initial capacitor voltage.

2.2. Circulating Current Analysis

For the above typical MMC circuit, due to its symmetrical nature, the following analysis only focuses on the A-phase as the object of study. To simplify the analysis of the effects of the asymmetrical capacitance parameters of the upper and lower bridge arms, the following assumptions are made:

(1)

The capacitors in the upper bridge arm of the A-phase deteriorate and their capacitance values all drop to 1/1 − n (n ≤ 0) times their rated value;

(2)

The remaining parameters, such as the inductance, remain symmetrical between the upper and lower bridge arms;

(3)

The capacitor voltage balancing control algorithm remains in effective operation and the average capacitor voltage remains stable.

With the above assumptions, the upper and lower bridge arms of the A-phase can be equated to a separate SM, which is further expressed in terms of equivalent capacitances, avoiding the need to treat the SMs individually [21,22]. The A-phase equivalent circuit is shown in Figure 4.

In the figure, $C_{au}$ and $C_{al}$ are the equivalent capacitances of the upper and lower bridge arms, respectively; $i_{az}$ is the A-phase circulating current, which can be expressed as

$$i_{az}=\frac{1}{2}(i_{au}+i_{al})$$

(5)

Then, the equivalent capacitance of the upper and lower bridge arms can be expressed as

$$\left\{\begin{array}{l}{C}_{au}=\frac{C}{(1-n)S_{au}N}\hfill \\ C_{al}=\frac{C}{S_{al}N}\hfill \end{array}\right.$$

(6)

where $S_{au}$ and $S_{al}$ are the average switching functions of the upper and lower bridge arms, which can be expressed as

$$\left\{\begin{array}{c}{S}_{au}=\frac{1}{2}[1-m\cos (\omega t)]\\ S_{al}=\frac{1}{2}[1+m\cos (\omega t)]\end{array}\right.$$

(7)

where $m$ is the modulation ratio and $\omega$ is the fundamental frequency corner frequency.

As can be seen from Equation (6), when the capacitors of the upper bridge arm of the A-phase are deteriorated the equivalent capacitance of the upper and lower bridge arms of the A-phase is not equal, so the AC current is no longer equally divided in the upper and lower bridge arms, and the bridge arm current can be expressed as

$$\left\{\begin{array}{c}{i}_{au}=i_{au}^{\ast }+\frac{1}{2}{k}_a{I}_a\cos (\omega t+{\phi }_a)\\ i_{al}=i_{au}^{\ast }-\frac{1}{2}{k}_a{I}_a\cos (\omega t+{\phi }_a)\end{array}\right.$$

(8)

where $k_a$ is the unbalance coefficient of the AC component of the bridge arm current, $k_a\in \left\{-1,1\right\}$; $I_a$ is the AC output current amplitude; ${\phi }_a$ is the phase angle of the unbalanced current; and $i_{au,l}^{\ast }$ is the ideal A-phase bridge arm current when the upper and lower bridge arm capacitance parameters are symmetrical, which can be expressed as

$$\left\{\begin{array}{c}{i}_{au}^{\ast }=\frac{1}{3}{I}_{dc}+I_a\cos (\omega t)\\ i_{al}^{\ast }=\frac{1}{3}{I}_{dc}-I_a\cos (\omega t)\end{array}\right.$$

(9)

According to Equations (2) and (4) and the charging and discharging process of the equivalent capacitor, the output voltages of the upper and lower bridge arms of the A-phase are obtained as

$$u_{au,l}=S_{au,l}{U}_{dc}+\frac{1}{C_{au,l}}{\displaystyle \int i_{au,l}\cdot S_{au,l}\mathrm{d}t}$$

(10)

Substituting Equations (8) and (9) into (10), the output voltages of the upper and lower bridge arms of the A-phase can be further expressed as

$$\left\{\begin{array}{l}{u}_{au}=S_{au}{U}_{dc}+\frac{(1-n)N{I}_a}{64\omega C}(u_{com,a0}+u_{diff,a0}+u_{com,a1}+u_{diff,a1}+u_{com,a2}+u_{diff,a3})\hfill \\ u_{al}=S_{al}{U}_{dc}+\frac{N{I}_a}{64\omega C}(u_{com,a0}-u_{diff,a0}+u_{com,a1}-u_{diff,a1}+u_{com,a2}-u_{diff,a3})\hfill \end{array}\right.$$

(11)

where $u_{com,a0}$, $u_{com,a1}$, and $u_{com,a2}$ are the DC component, fundamental frequency component, and second frequency component of the common mode voltage, respectively; $u_{diff,a0}$, $u_{diff,a1}$, and $u_{diff,a3}$ are the DC component, fundamental frequency component, and triplet frequency component of the differential mode voltage, respectively. Their analytical equations are as follows:

$$\left\{\begin{array}{l}{u}_{com,a0}=-4k_{a}m\sin {\phi }_a\hfill \\ u_{com,a1}=16k_{a}m\sin {\phi }_a\cos (\omega t)\hfill \\ u_{com,a2}=2m[(m^2-3)\sin (2\omega t)-6k_a\sin (2\omega t+{\phi }_a)]\hfill \end{array}\right.$$

(12)

$$\left\{\begin{array}{l}{u}_{diff,a0}=-16k_a\sin {\phi }_a\hfill \\ u_{diff,a1}=(-3m^2+8)\sin (\omega t)+2k_a[m^2\sin (\omega t-{\phi }_a)+8\sin (\omega t+{\phi }_a)]\hfill \\ u_{diff,a3}=m^2[\sin (3\omega t)+2k_a\sin (3\omega t+{\phi }_a)]\hfill \end{array}\right.$$

(13)

From Equation (11), the AC components of the common mode and differential mode voltages of A-phase, respectively, amount to

$$\left\{\begin{array}{l}{\tilde{u}}_{com,a}=\frac{(2-n)N{I}_a}{64\omega C}(u_{com,a1}+u_{com,a2})\hfill \\ {\tilde{u}}_{diff,a}=-\frac{nN{I}_a}{64\omega C}(u_{diff,a1}+u_{diff,a3})\hfill \end{array}\right.$$

(14)

Therefore, the bridge arm current can be corrected as follows when the upper and lower bridge arm capacitance parameters of the A-phase are asymmetrical:

$$i_{au,l}=i_{au,l}^{\ast }+{\tilde{i}}_{a\mathrm{z}}$$

(15)

where ${\tilde{i}}_{a\mathrm{z}}$ is the AC harmonic component and can be expressed as

$${\tilde{i}}_{a\mathrm{z}}=-\frac{N{I}_a}{128{\omega }^{2}C{L}_m}[(2-n)(i_{com,a1}+i_{com,a2})-n(i_{diff,a1}+i_{diff,a3})]$$

(16)

where $i_{com,a1}$ and $i_{com,a2}$ are the common mode current fundamental frequency component and the second frequency component, respectively; $i_{diff,a1}$ and $i_{diff,a3}$ are the differential mode current fundamental frequency component and the triplet frequency component, respectively. Their analytical equations can be expressed as

$$\left\{\begin{array}{l}{i}_{com,a1}=16k_{a}m\sin {\phi }_a\sin (\omega t)\hfill \\ i_{com,a2}=m[(-m^2+3)\cos (2\omega t)+6k_a\cos (2\omega t+{\phi }_a)]\hfill \end{array}\right.$$

(17)

$$\left\{\begin{array}{l}{i}_{diff,a1}=(3m^2-8)\cos (\omega t)-2k_a[m^2\cos (\omega t-{\phi }_a)+8\cos (\omega t+{\phi }_a)]\hfill \\ i_{diff,a3}=-\frac{1}{3}{m}^2[\cos (3\omega t)+2k_a\cos (3\omega t+{\phi }_a)]\hfill \end{array}\right.$$

(18)

The above analysis shows that the fundamental frequency component and the triplet frequency component of the bridge arm circulating current in Equation (16) are zero when the capacitance parameters of the upper and lower bridge arms of A-phase are perfectly symmetrical, and only the even harmonics are dominated by the second frequency component. When the capacitance degradation causes asymmetry in the capacitance parameters of the upper and lower bridge arms, odd harmonics dominated by the fundamental frequency and triplet frequency components will be introduced in the circulating current of the upper and lower bridge arms, and the greater the degree of asymmetry, the greater the amplitude of the odd harmonics. In fact, the capacitor voltage and bridge arm current will generate more higher harmonic components through a series of coupling effects [23]. This harmonic will cause fluctuations in the DC side current and increase converter losses, so measures must be implemented to suppress it.

3. Circulating Current Suppression Strategy

To address the circulating harmonics caused by the asymmetry of the upper and lower bridge arm capacitance parameters, this section proposes a VI-RC-based circulating current suppression strategy and describes the specific structure of the controller as well as the principle.

3.1. Repetitive Controller Design

From the above analysis, the PI controller has a steady-state error for circulating current suppression; the QPR and PIR controllers need multiple controllers in parallel when facing the case of harmonics with multiple frequencies, which adds to the complexity of the controller and the inevitable quantization errors associated with the discretization process when implemented in a digital controller. In terms of harmonic suppression performance, the RC can compensate for the shortcomings of the former pair.

Repetitive control is a control method based on the internal membrane principle, whereby a dynamic model of the external input signal is embedded within the control system to form a highly accurate feedback structure. The most significant advantage of the RC is the ability to generate infinite gain at periodic harmonic frequencies and to follow the input signal without static differences [24,25].

The basic control structure of the RC is shown in Figure 5. According to the nature of the exponential function, the RC can be expanded as

$$G_{RC}(s)=\frac{{\mathrm{e}}^{-sT}}{1-{\mathrm{e}}^{-sT}}=-\frac{1}{2}+\frac{1}{2}(\frac{1+{\mathrm{e}}^{-sT}}{1-{\mathrm{e}}^{-sT}})=-\frac{1}{2}+\frac{1}{T}(\frac{1}{s}+{\displaystyle \sum _{k=1}^{\infty }\frac{2s}{s^2+{(k\omega )}^2}})$$

(19)

where ${\mathrm{e}}^{-sT}$ is the cycle time delay link; $\omega =2\pi /T$; and $k$ is an integer.

As can be seen from Equation (19), the RC essentially consists of an integral controller and a series of resonant controllers, so that the RC can have infinite gain at each resonant frequency point $k\omega$ and in theory can achieve static-free control of the signal at each resonant frequency point.

As the RC is easier to implement in a digital controller [26], the analysis of the RC is presented below in the discrete domain. Due to the slow dynamic response of the RC, it generally needs to be coupled with a controller that has a faster dynamic response. In this paper, an RC is used in series with a PI controller to achieve circulating current harmonic suppression while ensuring the dynamic performance of the controller. The block diagram of the RC-based circulating current control system used is shown in Figure 6.

In the figure, $i_{jz}^{\ast }$ is the circulating current reference command; $u_{rc}$ is the RC output; $d(z)$ is the disturbance signal, i.e., the voltage drop across the bridge arm inductor generated by the circulating harmonics; $Q(z)$ is the low-pass filter, which is used to ensure system stability; $z^{-N_{rc}}$ is the time-delayed session, where $N_{rc}$ is the number of samples in a sampling period of the digital controller; $K_{rc}$ is the RC gain; $S(z)$ is the compensation filter used to compensate for the system amplitude; and $z^d$ is the phase over-run session, where $d$ is a positive integer used to compensate for the system phase lag.

The transfer function from the error signal $i_{err}(z)$ to the command signal $i_{jz}^{\ast }$ can be expressed as

$$i_{err}(z)=\frac{[1-P(z)][z^{N_{rc}}-Q(z)]}{z^{N_{rc}}-[Q(z)-K_{rc}S(z)z^{d}P(z)]}{i}_{jz}^{\ast }(z)-\frac{[1-P(z)][z^{N_{rc}}-Q(z)]}{z^{N_{rc}}-[Q(z)-K_{rc}S(z)z^{d}P(z)]}d(z)$$

(20)

where $P(z)$ is the controlled object as seen from the perspective of the RC, i.e., the closed loop formed by $PI(z)$ and $G(z)$ in series. The expression of $P(z)$ is

$$P(z)=\frac{PI(z)G(z)}{1+PI(z)G(z)}$$

(21)

$PI(z)$ and $G(z)$ are in the form of z-transmission functions for the PI controller $PI(s)$ and circulating-current-controlled object $G(s)$, respectively. $PI(s)$ and $G(s)$ are expressed as

$$PI(s)=K_p+\frac{K_i}{s}$$

(22)

$$G(s)=\frac{1}{2L_{m}s}$$

(23)

The design of the RC consists of three main components, i.e., a low-pass filter, time-delay session, and compensation session. The following section will continue to describe the design method of the main components.

(1)

Low-pass filter $Q(z)$

As an important part of the controller, the positive feedback link $1/(Q(z)z^{-N_{rc}})$ serves to integrate the closed-loop system error signal over the fundamental period. The filter $Q(z)$ has a weakening effect on the integration effect to improve the robustness of the system. To ensure the stability of the system, it is necessary to make all the $N_{rc}$ characteristic roots lie in the interior of the unit circle in the z-domain. For the circulating harmonics, the low-frequency pole should be located above the unit circle to obtain a high gain at that harmonic frequency, while the high-frequency pole is located inside the unit circle to maintain stability. Therefore, $Q(z)$ is chosen here as a first-order zero-phase-shift low-pass filter

$$Q(z)=\frac{z^{-1}+2+z}{4}$$

(24)

The bode diagram of $Q(z)$ is plotted as shown in Figure 7. It can be seen that $Q(z)$ exhibits zero phase shift and unit gain in the low-frequency band and decays rapidly at the turnaround frequency, meeting the design requirements of the low-pass filter required for circulating harmonic suppression. Therefore, the introduction of $Q(z)$ does not affect the controller performance and helps to improve robustness.

(2)

Time-delay session $z^{-N_{rc}}$

This link delays the execution of the control signal by one fundamental cycle, thus enabling $Q(z)$ as well as the phase overrun link $z^d$.

$N_{rc}$ can be calculated as

$$N_{rc}=\frac{f_s}{f_g}$$

(25)

where $f_s$ is the system sampling frequency, set to 10 kHz and $f_g$ is the fundamental frequency of the RC. According to the analysis of the harmonic components of the circulating current above, all harmonic frequencies are integer multiples of 50 Hz, so $f_g$ is set to 50 Hz. Therefore, the number of sampling points in one control cycle of the digital controller is 200.

(3)

Compensation session $K_{rc}S(z)z^d$

Filter $S(z)$ provides amplitude compensation for the controlled object $P(z)$. Here, $S(z)$ is chosen as a second-order low-pass filter with a natural frequency of ${\omega }_n=10\omega$ and damping ratio of $\zeta =1$, then

$$S(z)=\frac{0.186z-0.11}{z^2-0.9119z+0.2079}$$

(26)

The Bode diagram for $S(z)$ is shown in Figure 8 and it can be found that $S(z)$ decays rapidly in the higher frequency bands, helping to provide high-frequency noise suppression and improve the stability of the system.

Overrunning link $z^d$ provides phase compensation for $S(z)P(z)$. A phasefrequency characteristics diagram was plotted, as shown in Figure 9. It can be seen from the figure that the frequency characteristics of $z^{-4}$ and $S(z)P(z)$ match best below 1000 Hz. Therefore, $d$ is set to 4, which can effectively offset the phase lag of $S(z)P(z)$.

The RC gain $K_{rc}$ (0 < $K_{rc}$ ≤ 1) ultimately determines the amplitude of the RC output signal and is very important for the stability of the system. The smaller the $K_{rc}$, the greater the stability margin, but the slower the error convergence rate. It is necessary to select the $K_{rc}$ according to the open loop gain and system stability requirements. According to Equation (20), the characteristic equation of the repetitive control system is

$$z^{N_{rc}}-[Q(z)-K_{rc}S(z)z^{d}P(z)]=0$$

(27)

That is, a sufficient condition for the stability of the system is that all the $N_{rc}$ roots of the characteristic Equation (28) lie within the unit circle centered at the origin. Under this condition, the comprehensive consideration of $K_{rc}$ takes the value of 1.

3.2. Additional Virtual Impedance Design

According to the above analysis, the introduction of the low-pass filter $Q(z)$ improves the stability of the system; however, it weakens the gain at high frequencies. The introduction of additional VI can equivalently increase the bridge arm reactance to improve the suppression of high-frequency harmonics and can share the circulating current suppression at low frequencies [27,28,29].

The virtual impedance is improved from the virtual reactance. The functional mathematical description of the virtual reactance is a differential link with the expression

$$G_{VI}(s)=Ts$$

(28)

The strength of the differential link is determined by the time constant $T$. The regulation of the virtual reactance can be achieved by adjusting the corresponding $T$. However, in the actual control process, the pure differential link is not easy to implement, and it is easy to make the system over-tuned, which leads to system divergence. Therefore, a first-order inertial link is added to Equation (23) to correct it, which is

$$G_{VI}(s)=\frac{kTs}{1+Ts}$$

(29)

According to Equation (29), it can be found that, on the basis of the virtual reactance, it can be equated to a virtual resistance in parallel with the form. The VI equivalent circuit is shown in Figure 10.

The corresponding parameter relationship is

$$\left\{\begin{array}{c}k=R_v\\ T=\frac{L_v}{R_v}\end{array}\right.$$

(30)

Therefore, the VI can be expressed as

$$G_{VI}(s)=\frac{R_v{L}_{v}s}{R_v+L_{v}s}$$

(31)

The principle of VI is shown in Figure 11. When $i_{err}$ passes through the virtual impedance link, the loop current suppression command $u_{rc}$ from the repetition controller is corrected to the final circulating current suppression reference command $u_{jz\_ref}$. When the additional virtual impedance link is used, the circulating harmonics are effectively reduced and the power angle between the system and the grid is reduced, which does not affect the stability of the system but rather improves the robustness of the system.

The design of the VI link requires the parameterization of $R_v$ and $L_v$. In general, the equivalent resistance and reactance of the MMC single-equivalent circuit are used as the reference values for the parameter design. The parametric design experience can be summarized as follows: When $R_v$ is too large, the system speed will be slightly affected; when $R_v$ is too small, the system stability will be weakened; when $L_v$ is too large, the system type will be increased and the oscillation at the turn frequency will be increased; and when $L_v$ is too small, the frequency band in the baud diagram of the system will be slightly reduced, which will affect the speed of the system [30]. After several tests, the parameters of the virtual impedance link are set to $R_v=1\mathsf{\Omega }$, $L_v=10\mathrm{mH}$.

3.3. VI-RC Structure

According to the above analysis, in order to enhance the effect of circulating current suppression under the condition of asymmetric capacitance parameters of the upper and lower bridge arms in a typical MMC system, a circulating current suppression strategy based on a VI-RC is proposed, where $G_{VI}(z)$ is the z-transmission form of the virtual impedance link. The two are parallel control structures, and the structural block diagram is shown in Figure 12. The output of the RC is superimposed on the output of the VI link to obtain the final circulating current suppression voltage command.

The amplitude-frequency characteristics of the circulating current VI-RC are plotted, as shown in Figure 13. From the figure, it can be seen that the proposed controller presents a high control gain at each circulating harmonic frequency, which can effectively suppress the odd and even harmonic components of the circulating harmonics. With the additional virtual impedance link, the overall gain of the controller is slightly improved and the circulating harmonics suppression effect will be further enhanced.

4. Simulation Studies

In order to verify the feasibility of the VI-RC-based circulating current suppression strategy proposed in this paper, the corresponding simulation model is built in MATLAB for simulation verification.

4.1. Simulation Parameters

In this paper, a three-phase MMC simulation model is built in the MATLAB/simulinkR2022a environment. The overall control of the system is a double closed-loop control with external loop power control and internal loop current control, and the modulation strategy is chosen as the nearest-level modulation (NLM) technique. The MMC simulation model parameters are shown in

Table 1. Simulation parameters.

Simulation Variable	Parameter Setting
Rated power/MW	0.8
DC bus voltage/kV	5.5
Number of SMs per arm	22
Rated submodule capacitance/mF	7
Arm inductance/mH	1.35
System sampling frequency/kHz	10

; the VI-RC parameters are shown in

Table 2. Controller parameters.

Controller Parameters	Parameter Setting
Nrc	200
Krc	1
Kp	30
Ki	1
d	4
Rv/Ω	1
Lv/mH	10

.

4.2. Simulation Results

4.2.1. Circulating Current Characteristics

To verify the effect of the degree of asymmetry between the upper and lower bridge arm capacitance parameters of the MMC system on the circulating current characteristics, three sets of simulation experiments were set up in the absence of a circulating current controller:

(1)

Condition 1: The capacitance rating of the upper bridge arm SMs of the A-phase is set to 7 mF and the capacitance rating of the lower bridge arm SMs is set to 7 mF;

(2)

Condition 2: The capacitance rating of the upper bridge arm SMs of A-phase is set to 6.5 mF and the capacitance rating of the lower bridge arm SMs is set to 7 mF;

(3)

Condition 3: The capacitance rating of the upper bridge arm SMs of A-phase is set to 6 mF and the capacitance rating of the lower bridge arm SMs is set to 7 mF.

The Fast Fourier Transform (FFT) analysis is performed on the circulating currents and bridge arm currents obtained from the three sets of simulation experiments, respectively. The FFT analysis results are shown in Figure 14, Figure 15 and Figure 16.

From the above FFT analysis results, it can be seen that the dominant components of the circulating harmonics are the fundamental frequency component and the second frequency component, and as the asymmetry of the electrical parameters increases, the amplitude of the fundamental frequency harmonics will further increase and the THD of the bridge arm current will also increase. The above three conditions of the fundamental harmonic amplitude, second frequency harmonic amplitude, and THD of the bridge arm current were summarized, and the summary results are shown in

Table 3. Circulating current characteristics.

Condition	Fundamental Frequency Amplitude/A	Second Frequency Amplitude/A	THD of Bridge Arm Current/%
Condition 1	2.63	19.38	19.64
Condition 2	4.49	19.39	19.89
Condition 3	10.69	19.62	20.15

. The results in Table 3 fully illustrate the correctness of the above analysis of the circulating current characteristics under the asymmetric capacitor parameters, and the following section will continue to verify the effectiveness of the proposed circulating current VI-RC.

4.2.2. VI-RC Control Effect

To verify the controller effect, the circulating VI-RC was put into operation under Condition 2 (i.e., the capacitance rating of the upper bridge arm SMs of A-phase is 6.5 mF and the capacitance rating of the lower bridge arm SMs is 7 mF), and the resulting current key waveforms are shown in Figure 17. Among them, Figure 17a shows the circulating current waveform, from which it can be found that the circulating current tends to stabilize after 0.15 s under the action of the controller and fluctuates around 50 A in the direct flow. Figure 17b shows the bridge arm current waveform, in which the upper bridge arm current is shown in red and the lower bridge arm current is shown in blue. From the figure, it can be found that the distortion of the bridge arm current is significantly reduced after 0.15 s and the waveform tends to be sinusoidal.

The FFT analysis of the circulating current and the bridge arm current at this time is shown in Figure 18. From the FFT analysis of the loop current in Figure 18a, it can be seen that the harmonic components in the circulating current at this time have been effectively suppressed. The harmonic amplitude of the fundamental frequency is 0.09 A, and the harmonic amplitude of the second frequency is 0.56 A. As shown in Figure 18b, the THD of the bridge arm current is only 0.98%, and the waveform quality has been greatly improved.

4.2.3. Comparison of the Control Effects

To illustrate the superiority of the proposed circulating current control strategy based on a VI-RC in the case of asymmetric capacitor parameters, a PI controller, a QPR controller, and a repetitive controller without additional virtual impedance links are put in to suppress the circulating current harmonics under the same conditions (Condition 2), and the control effects of the different controllers are shown in Figure 19, Figure 20 and Figure 21.

(1)

PI controller

Figure 19. PI controller effect: (a) circulating current waveform under PI controller; (b) bridge arm current waveform under PI controller; (c) circulating current FFT analysis under PI controller; (d) bridge arm current FFT analysis under PI controller.

Figure 19. PI controller effect: (a) circulating current waveform under PI controller; (b) bridge arm current waveform under PI controller; (c) circulating current FFT analysis under PI controller; (d) bridge arm current FFT analysis under PI controller.

(2)

QPR controller

Figure 20. QPR controller effect: (a) circulating current waveform under QPR controller; (b) bridge arm current waveform under QPR controller; (c) circulating current FFT analysis under QPR controller; (d) bridge arm current FFT analysis under QPR controller.

Figure 20. QPR controller effect: (a) circulating current waveform under QPR controller; (b) bridge arm current waveform under QPR controller; (c) circulating current FFT analysis under QPR controller; (d) bridge arm current FFT analysis under QPR controller.

(3)

Repetitive controller

Figure 21. Repetitive controller effect: (a) circulating current waveform under RC; (b) bridge arm current waveform under RC; (c) circulating current FFT analysis under RC; (d) bridge arm current FFT analysis under RC.

Figure 21. Repetitive controller effect: (a) circulating current waveform under RC; (b) bridge arm current waveform under RC; (c) circulating current FFT analysis under RC; (d) bridge arm current FFT analysis under RC.

The control effects of different controllers are summarized and the summary results are shown in

Table 4. Comparison of control effects.

Controller Use	Fundamental Frequency Amplitude/A	Second Frequency Amplitude/A	THD of Bridge Arm Current/%
No controller	4.49	19.39	19.89
PI	0.48	5.53	5.75
QPR	0.16	4.87	5.23
RC	0.93	0.73	1.48
VI-RC	0.09	0.56	0.98

. From the table, it can be seen that the proposed circulating current VI-RC has a better suppression effect on fundamental frequency harmonics as well as double-frequency harmonics than the conventional controller, and the best quality of the bridge arm current waveform is obtained. The proposed VI-RC reduces the THD of the bridge arm current from 1.48% to 0.98% compared with the repetitive controller only, which further reduces the operating losses caused by the circulating harmonics and helps to improve the system reliability. Therefore, the circulating current VI-RC has the best performance and the system circulating current harmonic suppression effect is optimal among the above circulating current controller models.

5. Discussion

The VI-RC-based strategy proposed in this paper is not only applicable to the circulating current suppression under symmetrical conditions of the upper and lower bridge arm parameters but also applicable to the circulating current suppression under asymmetrical conditions of the upper and lower bridge arm capacitance parameters caused by capacitance degradation. The control strategy can effectively suppress both odd and even harmonics in the circulating current, reduce the THD of the bridge arm current, reduce the component loss in the circuit, and extend the component life. In the field of electric vehicles, the output voltage and current waveforms can be optimized to improve the efficiency and reliability of electric vehicle charging and discharging.

In addition to the work already performed in this paper, there are still parts that can be studied in depth. For example, this paper only considers the operating conditions where the upper and lower bridge arm capacitance parameters are asymmetric, in addition to the bridge arm inductance and bridge arm equivalent resistance, which may produce parameter asymmetry. This will have a more complex effect on the circulating current components. In addition, there will also be asymmetric operation of the operating conditions when an SM failure is removed. It will be a long-term direction of work to realize the circulating current suppression under more complicated operating conditions.

6. Conclusions

In this paper, based on the analysis of the basic circuit of three-phase MMC, a single-phase equivalent model is established to further analyze the harmonic components in the circulating current under the asymmetric operating conditions of the upper and lower bridge arm capacitance parameters. It is pointed out that the circulating current will contain odd- and even-order component harmonics when the upper and lower bridge arm current parameters are asymmetrical. Based on the analysis of the controller principle, a circulating current suppression strategy with virtual impedance attached to the RC is proposed to suppress both the odd and even component harmonics. The proposed control strategy is compared with the conventional circulating current control strategy through the MATLAB/Simulink platform. The THD of the bridge arm currents under open-loop conditions, a PI controller, a QPR controller, an RC and a VI-RC are 19.89%, 5.75%, 4.98%, 1.48%, and 0.98%, respectively. In summary, the VI-RC proposed in this paper has a better circulating current suppression effect under the asymmetric conditions of the upper and lower bridge arm capacitance parameters, which can significantly improve the bridge arm current waveform quality and reduce system losses.