Reduced Order Generalized Integrator Based Modular Multilevel Converter Loop Current Suppression Strategy under Unbalanced Conditions in Distribution Networks

Abstract

Under the condition of grid voltage imbalance, the circulation of the bridge arm inside the modular multilevel converter (MMC) increases significantly, which leads to the aggravation of the distortion of the bridge arm current, and, thus, increases the system loss and reduces the power quality. To address this problem, this paper analyzes the mechanism of circulating current generation and proposes a circulating current suppression strategy based on a reduced-order generalized integrator (ROGI), which firstly uses the ROGI system to separate the second-harmonic positive- and negative-sequence components in the circulating current from the DC, and then converts the rotating coordinates of the circulating current’s second octave component into the DC to be fed into the proportional–integral quasi-resonance (PIR) controller for suppression. A simulation model of a 23-level MMC inverter is built in MATLAB/Simulink, and the control strategy proposed in this paper is compared with the classical proportional–integral (PI) control in simulation experiments. The simulation results show that the amplitude of the circulating current fluctuation of the classical PI control is reduced from 90 A to 22 A, and the harmonic distortion rate of the bridge arm current is reduced from 32.56% to 5.57%; the amplitude of the circulating current fluctuation of the control strategy proposed in this paper is reduced from 90 A to 5.7 A, and the harmonic distortion rate of the bridge arm current is reduced from 20.2% to 1.13%, which verifies the effectiveness of the pro-posed control strategy.

1. Introduction

Due to the penetration of new renewable energy sources and the economics of high-voltage DC transmission, high-voltage DC transmission systems are becoming the primary means of handling and transmitting high-power electricity [1,2,3,4]. Currently, the MMC is considered the most ideal voltage source converter solution for HVDC systems due to its modular and scalable architecture.

When unbalanced conditions occur in the power grid, the MMC, due to its unique distributed arrangement of energy storage capacitors, is prone to causing imbalance in the capacitor voltage of each submodule. The energy imbalance between the phases leads to the generation of circulating current inside the converter, which adds to the current distortion of the bridge arm, increases the energy loss of the bridge arm, affects the stable operation and service life of the submodule components, and increases the system cost. Therefore, it is necessary to carry out suppression measures for the circulating current [5,6].

To suppress the internal circulating current within the MMC, extensive research has been conducted by scholars both domestically and internationally. Currently, the adopted circulating current suppression strategies are primarily classified into hardware-based and software-based methods [7]. Hardware methods mainly involve increasing the arm inductance or employing auxiliary circuits to suppress the circulating current. However, such approaches tend to increase the system’s size and losses, leading to adverse effects [8]. In reference [9], focusing on the unique submodule structure of the MMC in high-voltage battery energy storage systems, it is proposed that increasing the arm inductance value under certain threshold conditions can help suppress the interphase circulating current of the system. Nonetheless, this method requires substantial calculations and complex design. Additionally, reference [10] suggests using a zero-sequence circulating current controller to effectively increase the equivalent arm resistance, thereby enhancing system damping and effectively suppressing the impedance resonance peak of the MMC under islanding control mode. Nevertheless, this method’s control complexity poses inconveniences for practical applications.

The software-based approach primarily focuses on suppressing the harmonic components of the secondary current through designing an appropriate controller, thereby inhibiting the circulating current. This method, known for its low cost and ease of debugging, has become the mainstream choice for circulating current suppression [11]. However, it still suffers from drawbacks such as slow dynamic response and sensitivity to controller parameters. In reference [12], the abc-dq coordinate transformation is proposed to decompose the three-phase circulating current into direct current components, upon which a circulating current suppressor is designed. Nevertheless, this method requires coordinate transformation and interphase decoupling, limiting the system’s stability and circulating current suppression capability. Reference [13] introduces a method that employs coordinate transformation to decompose the three-phase circulating current into dc components in the dq coordinate system, which are then suppressed using a PI controller. This method boasts a simple structure and convenient operation but is prone to external influences on its control parameters, resulting in limited control accuracy. References [14,15] present a composite controller combining repetitive control with a proportional element, capable of simultaneously eliminating multiple harmonic circulating currents in power systems and exhibiting excellent tracking performance. Subsequently, reference [16] incorporates a fractional-order phase lead compensator into the aforementioned structure to further enhance dynamic performance, albeit at the cost of increased computational complexity.

In reference [17], after separating the positive- and negative-sequence components of the circulating current, synchronous rotating coordinate transformation control is adopted. However, the negative-sequence component separation process is overly complex, requiring multiple low-pass filters, which slows down the control speed. On the other hand, reference [18] transforms the three-phase circulating current into the αβ0 coordinate system and utilizes a proportional resonance (PR) controller to suppress the circulating current in the stationary coordinate system. Nevertheless, the control effect is not ideal when grid deviations occur. In reference [19], a PR controller-based circulating current suppression method is proposed for both single-phase and three-phase systems. However, this approach cannot effectively handle situations involving multiple harmonics simultaneously, and the PR controller suffers from a narrow bandwidth. To address the bandwidth limitation of traditional PR controllers, references [20,21,22] propose a QPR controller-based circulating current suppression strategy, which enhances system stability and response speed. Nevertheless, this method is only applicable to two-terminal MMC transmission systems and single-terminal grid voltage imbalance scenarios. Reference [23] introduces a circulating current suppression strategy utilizing PR and PI controllers in the abc reference frame, which not only reduces circulating currents but also minimizes the fluctuation amplitude of capacitor voltages. However, this method involves complex calculations. On the other hand, reference [24] presents a simpler approach using two PI controllers to suppress circulating currents, offering a straightforward and convenient solution but with limited control accuracy. In reference [25], a direct power control strategy for MMC-HVDC based on a reduced-order vector resonator is proposed. This strategy employs PI controllers to regulate the dc components of instantaneous power, while the reduced-order vector resonator is utilized to eliminate the second-order positive-sequence fluctuating components. This approach enables direct control of active and reactive power. However, the control strategy is complex and involves numerous control parameters. Reference [26] presents a circulating current suppressor based on PIR control. However, the system exhibits relatively weak stability and is therefore not suitable for unbalanced systems.

References [27,28] propose circulating current suppression strategies based on the second order generalized integrator (SOGI), where a QPR controller is designed to suppress the second-order harmonics, thereby reducing the computational burden of the system. Additionally, reference [29] introduces a circulating current suppression strategy based on the ROGI, which further minimizes the overall computational load of the system. However, both methods are discussed and analyzed under the assumption of balanced grid conditions and are, thus, not applicable to unbalanced grid systems.

Addressing the aforementioned issues, this paper first establishes the topological structure and mathematical analysis of circulating current in MMCs, elucidating the mechanism of circulating current generation. Subsequently, the second-order positive= and negative-sequence components of the circulating current are separated using a reduced-order generalized integrator. These positive-, negative-, and zero-sequence second-order components are then transformed into appropriate rotating coordinates and suppressed by a PIR controller. This method enables independent control of the positive-, negative-, and zero-sequence second-order components in the bridge arms without requiring auxiliary circuits or modifications to the topology. While preserving the impedance characteristics of the MMC, it stabilizes submodule capacitor voltage fluctuations. The system features a simple structure, fast response, and robust stability.

2. Topology and Working Principle of Active Distribution Network MMCs

The three-phase MMC inverter topology is shown in Figure 1. The MMC consists of six bridge arms in three phases, with each phase divided into two bridge arms. Each bridge arm consists of an inductor and several half-bridge submodules with the same structure. Each half-bridge submodule consists of two IGBTs and a set of capacitors. $L_0$, $R_0$ are the equivalent reactance and equivalent loss, respectively; $i_{pj}$, $i_{nj}$ (j = a,b,c) are the currents of the upper and lower bridge arms of phase j, respectively, and $u_{sj}$ is the AC side voltage of phase j.

To simplify the analysis, the reactor and the ideal source are considered as a whole and the whole output is considered as an ideal source to the outside world. From Kirchhoff’s law:

$$i_{pj}-i_{nj}=i_j$$

(1)

$$-{\displaystyle \frac{U_{dc}}{2}}+u_{pj}+i_{pj}{R}_0+L_0{\displaystyle \frac{d{i}_{pj}}{dt}}+U_{sj}-U_{o{o}^{\prime }}=0$$

(2)

$${\displaystyle \frac{U_{dc}}{2}}-u_{nj}-i_{nj}{R}_0-L_0{\displaystyle \frac{d{i}_{nj}}{dt}}+U_{sj}-U_{o{o}^{\prime }}=0$$

(3)

Adding and subtracting Equations (2) and (3), respectively, gives the AC and DC side characteristics of the MMC as follows:

$${\displaystyle \frac{1}{2}}(R_0{i}_j+L_0{\displaystyle \frac{d{i}_j}{dt}})=U_{o{o}^{\prime }}-U_{sj}+U_{diffj}$$

(4)

$$R_0{i}_{cirj}+L_0{\displaystyle \frac{d{i}_{cirj}}{dt}}={\displaystyle \frac{U_{dc}}{2}}-U_{comj}$$

(5)

where $U_{diffj}$, $i_{cirj}$ and $u_{comj}$ denote the differential mode voltage, circulating current, and common mode voltage, respectively, and are defined as

$$U_{diffj}={\displaystyle \frac{u_{nj}-u_{pj}}{2}}$$

(6)

$$U_{comj}={\displaystyle \frac{u_{nj}+u_{pj}}{2}}$$

(7)

$$i_{cirj}={\displaystyle \frac{i{p}_j+i_{nj}}{2}}$$

(8)

where $U_{diffj}$ and $U_{comj}$ are regulated by the inner circulation controller and $i_{cirj}$ is obtained by the circulation suppression controller.

3. Circulation Analysis within the MMC under Unbalanced Conditions

Under normal conditions, the MMC internal circulation can usually be expressed by Equation (9):

$$i_{cirj}={\displaystyle \frac{I_{dc}}{3}}+I_{r2m}\cos (2\omega t-\theta )+Q_1$$

(9)

where $\theta$ is the second-harmonic initial phase of the ring current, the high-frequency harmonic components in the ring current are generated by the twofold frequency layer-by-layer excitation of the AC quantity, and the harmonic amplitude content decreases as the frequency increases, where Q₁ is the sum of the harmonics above the second harmonic, which is generally ignored due to the small content.

In the case of unbalanced grid voltage, the MMC phase-to-phase circulating currents are altered, generating positive-sequence, negative-sequence, and zero-sequence components of twofold frequency. The AC-side converter in the MMC structure usually adopts the Y-∆ connection, and there is no circulating loop for the zero-sequence current. Neglecting the zero-sequence component effect $U_{acj}$, $I_{sj}$ can be decomposed into positive- and negative-sequence components, expressed as

$$\left\{\begin{array}{l}{U}_{acj}=U^p\cos ({\omega }_{0}t+{\theta }^p)+U^n\cos ({\omega }_{0}t+{\theta }^n)\\ I_{sj}=I^p\cos ({\omega }_{0}t+{\varphi }^p)+I^n\cos ({\omega }_{0}t+{\varphi }^n)\end{array}\right.$$

(10)

where ${\omega }_0$ is the angular frequency of the grid, $U^p$, $U^n$ are the positive- and negative-sequence components of the AC side voltage, ${\theta }^p$, ${\theta }^n$ are the positive- and negative-sequence voltage of the initial phase angle, $I^p$, $I^n$ are the positive- and negative-sequence components of the AC side current, and ${\phi }^p$, ${\phi }^n$ are the positive- and negative-sequence current of the initial phase angle.

In the three-phase asymmetrical operation, redefining the currents in the upper and lower bridge arms have

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

(11)

Simplification shows that the instantaneous power of the phase cell is expressed as

$$p_j=u_{pj}{i}_{pj}+u_{nj}{i}_{nj}=U_{acj}{I}_{sj}+U_{dc}{I}_{dcj}$$

(12)

Substituting the above Equation (12) and analyzing it with phase a as an example, we obtain

$$P_a=P_{a0}+P_{a2}^p+P_{a2}^n+P_{a2}^z$$

(13)

where $P_{a0}$ is the dc component, and $P_{a2}^p$, $P_{a2}^n$, $P_{a2}^z$ are the twofold positive-, negative-, and zero-order components, respectively, expressed as

$$\left\{\begin{array}{c}{P}_{a0}=U_{dc}{I}_{dca}+{\displaystyle \frac{1}{2}}{U}^p{I}^p\cos ({\theta }^p-{\varphi }^p)+{\displaystyle \frac{1}{2}}{U}^p{I}^n\cos ({\theta }^p-{\varphi }^n)+\hfill \\ {\displaystyle \frac{1}{2}}{U}^n{I}^p\cos ({\theta }^n-{\varphi }^p)+{\displaystyle \frac{1}{2}}{U}^n{I}^n\cos ({\theta }^n-{\varphi }^n)\hfill \\ P_{a2}^p={\displaystyle \frac{1}{2}}{U}^n{I}^n\cos (2{\omega }_{0}t+{\theta }^n+{\varphi }^p)\hfill \\ P_{a2}^n={\displaystyle \frac{1}{2}}{U}^p{I}^p\cos (2{\omega }_{0}t+{\theta }^p+{\varphi }^n)\hfill \\ P_{a2}^z={\displaystyle \frac{1}{2}}{U}^p{I}^n\cos (2{\omega }_{0}t+{\theta }^n+{\varphi }^n)+{\displaystyle \frac{1}{2}}{U}^n{I}^p\cos (2{\omega }_{0}t+{\theta }^n+{\varphi }^p)\hfill \end{array}\right.$$

(14)

From the analysis of the above equations, it can be seen that the negative-sequence current and voltage produce positive-sequence two-times frequency loop current; positive-sequence current and voltage produce negative-sequence two-times frequency loop current; positive-sequence voltage and negative-sequence current and negative-sequence voltage and positive-sequence current produce zero-sequence two-times frequency loop current.

Integrating Equation (13) yields the instantaneous energy of phase j:

$$W_j=p_{j0}t+W_{j\_dc}+W_{j2}^p+W_{j2}^n+W_{j2}^z$$

(15)

where $W_{j\_dc}$ is the DC component of the instantaneous energy, $W_{j2}^p$, $W_{j2}^n$, and $W_{j2}^z$ are the twofold positive-, negative-, and zero-order components of the instantaneous energy, respectively, and $p_{j0}$ contains the DC component and the fundamental frequency component, defining the fundamental frequency component to be $p_{j0}^{\prime }$, expressed as

$$\left\{\begin{array}{c}{p}_{j0}=U_{dc}{I}_{jdc}+{p^{\prime }}_{j0}\hfill \\ W_{j\_dc}=U_{dc}{I}_{jdc}t\hfill \\ W_{j0}={p^{\prime }}_{j0}\hfill \\ W_{a2}^p={\displaystyle \frac{1}{4{\omega }_0}}{U}^n{I}^n\sin (2{\omega }_{0}t+{\theta }^n+{\varphi }^n)\hfill \\ W_{b2}^p={\displaystyle \frac{1}{4{\omega }_0}}{U}^n{I}^n\sin (2{\omega }_{0}t+{\theta }^n+{\varphi }^n-{\displaystyle \frac{2\pi }{3}})\hfill \\ W_{c2}^p={\displaystyle \frac{1}{4{\omega }_0}}{U}^n{I}^n\sin (2{\omega }_{0}t+{\theta }^n+{\varphi }^n+{\displaystyle \frac{2\pi }{3}})\hfill \\ W_{a2}^n={\displaystyle \frac{1}{4{\omega }_0}}{U}^p{I}^p\sin (2{\omega }_{0}t+{\theta }^p+{\varphi }^p)\hfill \\ W_{b2}^n={\displaystyle \frac{1}{4{\omega }_0}}{U}^p{I}^p\sin (2{\omega }_{0}t+{\theta }^p+{\varphi }^p+{\displaystyle \frac{2\pi }{3}})\hfill \\ W_{c2}^n={\displaystyle \frac{1}{4{\omega }_0}}{U}^p{I}^p\sin (2{\omega }_{0}t+{\theta }^p+{\varphi }^p-{\displaystyle \frac{2\pi }{3}})\hfill \\ W_{j2}^{\mathrm{z}}={\displaystyle \frac{1}{4{\omega }_0}}[U^p{I}^n\sin (2{\omega }_{0}t+{\theta }^p+{\phi }^n)+U^n{I}^p\sin (2{\omega }_{0}t+{\theta }^n+{\phi }^p)]\hfill \end{array}\right.$$

(16)

Through the above analysis of the energy, it can be seen that harmonics will be generated in the bridge arm circulating current, the fundamental frequency voltage and current will generate positive-, negative-, and zero-sequence two times the frequency of the bridge arm voltage and current, and it can be deduced that the circulating current under the unbalanced condition of the grid is defined as shown in Equation (17). The instantaneous energy equivalent circuit diagram of the phase unit of the MMC in the case of the unbalanced grid is shown in Figure 2.

$${i^{\prime }}_{cirj}=I_{jdc}+i_{2j}^p\sin \left(2{\omega }_{0}t+{\theta }_{2j}^p\right)+i_{2j}^n\sin \left(2{\omega }_{0}t+{\theta }_{2j}^n\right)+i_{2j}^z\sin \left(2{\omega }_{0}t+{\theta }_{2j}^z\right)$$

(17)

The twofold component in the loop current originates from the voltage difference between the three-phase bridge arms. Although it does not directly affect the normal operation of the MMC, it will increase the burden of the capacitance of the bridge arm submodules and trigger additional energy loss. Therefore, in order to improve the system efficiency, the second-harmonic component in the MMC loop current must be effectively suppressed.

4. Improved PIR Control Strategy

4.1. Phase Sequence Separation

Among the methods for extracting the diophantine components, various techniques have their advantages and disadvantages. The trap method is slow to respond and may not be suitable for real-time applications. The delay method is memory-intensive and sensitive to signal variations and noise. The second-order generalized integrator is widely used due to its simple structure and its ability to suppress harmonics. The transfer function of SOGI is

$$G_{SOGI}(s)={\displaystyle \frac{{\omega }_{r}s}{s^2+{\omega }_r^2}}={\displaystyle \frac{1}{2}}\ast ({\displaystyle \frac{{\omega }_r}{s+j{\omega }_r}}+{\displaystyle \frac{{\omega }_r}{s-j{\omega }_r}})$$

(18)

where ${\omega }_r$ is the resonant angular frequency. As can be seen from Equation (18), the poles of the system are $\text{±}j{\omega }_r$, indicating that the SOGI exhibits frequency selectivity but lacks polarity selectivity, making it unable to directly separate positive- and negative-sequence fundamental components. Therefore, by performing a reduction process on the SOGI, two reduced-order generalized integrators with the same resonant frequency but opposite phase sequences can be obtained. The transfer function of this ROGI is

$$G_{ROGI}(s)={\displaystyle \frac{{\omega }_r}{s-j{\omega }_r}}$$

(19)

From this transfer function, it can be seen that there is only one pole, so it can only control the component whose angular frequency is ${\omega }_r$, and when the angular frequency is equal to the resonant angular frequency, it has infinite gain, so it can be used to directly extract the positive- and negative-sequence components of the power grid. Furthermore, since at the same resonant frequency, the ROGI can provide different gains for positive- and negative-sequence signals, while the SOGI can only provide the same gain, the total computational load when using ROGI for signal modulation is approximately half that of using SOGI, thereby enhancing the computational efficiency of the control process.

In separating the positive- and negative-sequence components of harmonics, the presence of the complex number j in the ROGI transfer function is more detrimental to the mathematical realization, so that j can be realized in a three-phase system using the αβ coordinate system variable $i_{\alpha }=\mathrm{j}{i}_{\beta }$. At this time, the structural block diagram of ROGI is shown in Figure 3, and the transfer function is

$$D(s)={\displaystyle \frac{k{\omega }_{r}s+jk{\omega }_r^2}{s^2+2k{\omega }_{r}s+{\omega }_r^2}}$$

(20)

$$Q(s)={\displaystyle \frac{-jk{\omega }_r+k{\omega }_r^2}{s^2+2k{\omega }_{r}s+{\omega }_r^2}}$$

(21)

where k is the system gain.

When the grid is unbalanced, the MMC contains a two-times frequency component internally, which is derived from Equation (5), and the dynamic equations are obtained by Clark transformation:

$$R_0\left[\begin{array}{l}{i}_{ci{r}_{\alpha }}(t)\\ i_{ci{r}_{\beta }}(t)\\ i_{ci{r}_0}(t)\end{array}\right]+L_0{\displaystyle \frac{d}{dt}}\left[\begin{array}{l}{i}_{ci{r}_{\alpha }}(t)\\ i_{ci{r}_{\beta }}(t)\\ i_{ci{r}_0}(t)\end{array}\right]=-\left[\begin{array}{l}{u}_{co{m}_{\alpha }}(t)\\ u_{co{m}_{\beta }}(t)\\ u_{co{m}_0}(t)\end{array}\right]$$

(22)

From Equations (19) and (20), it can be seen that the output of D(s) can be regarded as a combination of a bandpass filter and a low-pass filter, and the positive- and negative-sequence current components generated when the grid is unbalanced can be quickly separated by ROGI, but the existence of the DC component will bring about a large error, so the regulator adds a DC compensator to solve the problem of ROGI’s inability to filter out the DC component by injecting the DC component. The principle block diagram of positive- and negative-sequence separation is shown in Figure 4.

4.2. Circulation Suppression Strategies

Due to the asymmetry of three-phase loads and asymmetrical faults in the power system, the grid is prone to unbalanced conditions, which leads to an increase in the harmonic content of the two-frequency loop current in the grid-connected current. As shown in the previous section, the loop current $i_{cirj}^{\prime }$ is the superposition of the DC component and the harmonic component. The PI controller can realize the static-free tracking of the DC input, which has the advantages of high reliability and robustness, but this regulator cannot realize the static-free tracking of the sinusoidal input. The ideal PR control has a relatively narrow bandwidth frequency, with a small increase at the resonant frequency point as the frequency varies. This results in a significant drop in open-loop gain when the system frequency changes. To broaden the system bandwidth, enhance stability, and facilitate digital implementation, this paper adopts a combination of the QPR controller and the traditional PI controller, forming the PIR controller, whose structure block diagram is shown in Figure 5, and the transfer function is expressed as Equation (23):

$$G_{PIR}(s)=K_p+{\displaystyle \frac{K_i}{s}}+{\displaystyle \frac{2K_r{\omega }_{c}s}{s^2+2{\omega }_{c}s+{\left(2\omega \right)}^2}}$$

(23)

where $K_p$, $K_i$ are the proportional and integral coefficients, ${\omega }_c$ is the cutoff frequency, and K_r is the twofold resonant controller gain.

The principle block diagram of this controller is shown in Figure 5. The principle of the current loop PI controller in parallel with the QPR controller is to reduce the THD of the bridge arm current, in which the QPR controller at the fundamental frequency of the gain is ∞. When the command value of the given current and the feedback current there are harmonics, the PIR controller amplifies the error through the negative feedback link to inhibit harmonic components of the current.

The circulating current within the bridge arm flows back and forth within the three phases, and a twofold negative/positive-sequence rotating coordinate transformation is used to convert the negative/positive-sequence twofold circulating current into a direct flow to analyze the negative-sequence component as an example there:

$$\left[\begin{array}{c}0\\ 2{\omega }_0{L}_0\end{array}\right.\left.\begin{array}{c}-2{\omega }_0{L}_0\\ 0\end{array}\right]\left[\begin{array}{c}{i}_{cird}^n\\ i_{cirq}^n\end{array}\right]-\left[\begin{array}{c}{u}_{comd}^n\\ u_{comq}^n\end{array}\right]=L_0{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{cird}^n\\ i_{cirq}^n\end{array}\right]+R_0\left[\begin{array}{c}{i}_{cird}^n\\ i_{cirq}^n\end{array}\right]$$

(24)

Based on Equation (24), the three-phase time-varying circulating currents can be decomposed into two DC components through the negative-sequence rotating coordinate transformation. This decomposition facilitates the design of a circulating current suppressor using PI controllers. The remaining double-frequency components of the zero-sequence circulating current will flow through the DC lines, affecting the normal operation of other converter stations, and, therefore, must also be suppressed by the QPR controller. Based on the above control approach, the control block diagram for the internal arm circulating current of MMC under unbalanced grid conditions is obtained, as shown in Figure 6.

PIR Controller Parameterization

The setting of parameters for the PIR controller primarily involves adjusting the parameters of the proportional–integral controller and the quasi-resonant controller. By altering the values of various parameters within the controller, the control effect can be fine-tuned.

$K_p$ determines the baseline gain across the entire frequency range. An increase in the $K_p$ value enhances the suppression capability of the resonant controller for harmonics near the resonant frequency. However, if the $K_p$ value is too high, it may compromise system stability, leading to oscillations. $K_i$ primarily affects the gain in the low-frequency range and is used to eliminate steady-state errors in the system. Nevertheless, excessively high values of $K_p$ and $K_i$ can cause phase lag and stability issues. The frequency domain analysis results are shown in Figure 7a,b.

$K_r$ determines the resonant gain at the resonant frequency. As $K_r$ increases, the gain at the tracked frequency point improves, but an excessively high $K_r$ value can lead to phase advance. ${\omega }_c$ dictates the bandwidth of the controller; as ${\omega }_c$ increases, so does the controller’s bandwidth. However, an excessively large ${\omega }_c$ can increase interference to nonharmonic frequencies. The frequency domain analysis results are presented in Figure 7c,d.

In summary, the optimal parameters of $K_p,K_i,K_r\mathrm{and}{\omega }_c$ are determined based on actual conditions, with $K_p=50$, $K_i=20$, $K_r=400$, and ${\omega }_c=5$ rad/s. Figure 8 shows the zero-pole diagram of the circulating current suppressor with these parameters, where ○ represents zeros and × represents poles. All characteristic roots of the system have negative real parts, indicating that the controller is stable.

In this paper, the circulation suppression strategy first measures the bridge arm current of the MMC, calculates the bridge arm loop current value $i_{cirj}^{\prime }$, extracts the AC two-times frequency component in the loop current by the ROGI harmonic extractor, and then inputs it to the PIR loop current controller to obtain the voltage reference value for eliminating the harmonic component $u_{diffj}$. The circulation suppression strategy in this paper is shown in Figure 9.

4.3. Modulation Strategy

In the conventional nearest level approximation modulation strategy, the actual output voltage between the upper and lower bridge arms deviates from the reference voltage, which makes the actual AC energy provided by the AC side unequal to that of the submodule capacitor voltage fluctuation, thus generating a double-frequency loop current. When the actual output characteristics of the bridge arm voltages are not affected by the fluctuations of the submodule capacitor voltages, there is no twofold frequency circulating current between the phase units, and the power quality can be effectively improved.

In this paper, an approximate superposition modulation strategy is chosen: This strategy considers that the submodule capacitance voltage is changing all the time, and through the superposition of submodule capacitance voltages, the actual values of the input submodule voltages of the upper and lower bridge arms are always close to the reference value of the modulating waveform. Considering that when the actual submodule capacitance voltages are superimposed, the sum of the final input submodule capacitance voltages cannot be exactly the same as the target level; it is chosen to take the absolute value of the deviation from the target level as the criterion, so as to achieve the purpose that the sum of the input submodule capacitance voltages is closest to the target level. The specific steps of this control strategy are as follows:

• Boost sequencing of the submodule capacitor voltages for each bridge arm.
• When the current charges the capacitor voltage of the submodule (current greater than 0), the capacitor voltages of the submodules are summed from front to back until the sum of the capacitor voltages of the submodules is closest to the given value, thereby determining the number of submodules that should be engaged, i.e., engaging the first S submodules after sorting.
• When the current discharges the submodule capacitor voltage (with the current being less than 0), the submodule capacitor voltages are summed up from the latter to the former until the sum of the submodule capacitor voltages is the closest to the given value, thereby determining the number of submodules to be activated, i.e., the last N−S+1 submodules after sorting.

This control strategy casts cuts in each cycle strictly according to the submodule voltage magnitude, which can limit the submodule voltage unbalance to a fairly small range. The specific flow of the modulation strategy is shown in Figure 10:

5. Simulation and Results Analysis

In order to verify the effectiveness of the circulating current suppression method mentioned in this paper, a 23-level MMC simulation model is built in MATLAB/Simulink simulation software; the modulation method adopts the nearest level approximation modulation, and the model simulation parameters are shown in

Table 1. Main parameters of the simulation model.

Parameter	Simulation Value
DC side voltage/kV	11
AC side voltage/kV	6.6
$\mathrm{Line}\mathrm{inductors}{L}_l/\mathrm{m}\mathrm{H}$	13.7
$\mathrm{Line}\mathrm{resistance}{R}_l/\mathsf{\Omega }$	0.6
$\mathrm{MMC}\mathrm{outer}\mathrm{ring}\mathrm{DC}\mathrm{voltage}{U}_{dc}/\mathrm{k}\mathrm{V}$	11
$\mathrm{MMC}\mathrm{bridge}\mathrm{arm}\mathrm{inductors}{L}_0/\mathrm{m}\mathrm{H}$	13.5
MMC submodule capacitance/mF	7
Number of MMC bridge arm submodules	22

. In order to reflect the control effect of the circulating current suppression strategy proposed in this paper under the unbalanced condition of the grid, it is set that the grid voltage fluctuation occurs 0.3 s after the balanced state, the amplitude of the A-phase voltage falls to 60%, and the circulating current suppressor is activated at the moment of 0.5 s.

Under the unbalanced condition, the three-phase output voltage on the AC side is shown in Figure 11a, which is changed at t = 0.3 s and stabilized the rest of the time. The FFT analysis of this output voltage is shown in Figure 11b, and the THD of the three-phase output voltage is 2.94%, so it can be seen that the distortion rate of this output voltage is small and still available.

Figure 12 shows the changes of the submodule capacitor voltage waveforms during the simulation process. The capacitor voltage waveforms are in good condition in steady state, the capacitor voltage distortion is serious after the fluctuation of the grid voltage in 0.3 s, and the distortion is obviously weakened after turning on the circulating current suppressor in 0.5 s. The fluctuation amplitude of the submodule capacitor voltages is also reduced, which is in good condition.

Figure 13 and Figure 14, respectively, depict the current waveform and the circulating current waveform of the a-phase bridge arm. After the voltage of the a-phase grid drops (at t = 0.3 s), the current waveform of the a-phase bridge arm experiences severe distortion, with the second-harmonic amplitude increasing from 49 A to 90 A and the harmonic distortion rate rising from 19.69% to 20.2%, adversely affecting the stable operation of the system. After 0.5 s of operation, the circulation suppressor designed in this paper is activated, resulting in a reduction in the distortion of the a-phase bridge arm current waveform. Specifically, the second-harmonic amplitude of the circulating current decreases from 90 A to 5.7 A, and the harmonic distortion rate drops to 1.13%, significantly improving the waveform quality and transforming it into a more ideal sinusoidal wave.

Figure 15 presents the simulation results of the circulating current suppression based on the PI control strategy. Specifically, during the grid voltage imbalance at 0.3 s, the harmonic oscillation and fluctuation of the circulating current in the a-phase bridge arm change from 48 A to 90 A, and the distortion rate increases from 19.7% to 32.56%. After activating the PI circulating current suppressor at 0.5 s, the fluctuation amplitude of the circulating current is suppressed to 22 A, and the harmonic distortion rate decreases to 5.57%. Although there is some improvement in waveform quality compared to before applying PI control, the waveform still exhibits a certain degree of distortion.

6. Conclusions

In this paper, based on the mathematical analysis of the circulating current inside the MMC, a new circulating current suppression method is proposed for the unbalanced situation of the power grid. The following conclusions are obtained after simulation experiments by building a 23-level MMC inverter model:

• When the grid is unbalanced, the circulating current asymmetry inside the MMC is aggravated, and the circulating current suppressor mentioned in this paper can significantly reduce the twofold frequency component in the circulating current compared with the traditional PI control, which can reduce the fluctuating range of the submodule capacitance voltage to a certain extent and improve the quality of the waveform of the bridge arm current. The structure avoids the use of traps, streamlines the calculation, and increases the stability of the system.
• When the grid is unbalanced, the circulating harmonic component inside the MMC increases significantly. There is almost no fluctuation in the system AC side voltage before and after the addition of the circulating current suppressor, which proves that the positive- and negative-sequence second-harmonic components only circulate between the internal three-phase bridge arms.