High-Frequency Oscillation Suppression Strategy for Renewable Energy Integration via Modular Multilevel Converter—High-Voltage Direct Current Transmission System Under Weak Grid Conditions

Abstract

To address the high-frequency resonance issues in renewable energy systems integrated via MMC-HVDC transmission under weak grid conditions, this paper establishes a wide-frequency sequence impedance model for renewable energy grid-side converters and MMC-HVDC sending-end converters. The impedance characteristics of the MMC-HVDC transmission system under two distinct control modes are compared, and the constant power control mode is selected for detailed analysis to better evaluate the effectiveness of suppression strategies. Based on this framework, the superposition theorem is employed to analyze the interaction mechanism between the impedance characteristics of the MMC-HVDC sending-end converter and the renewable energy grid-connected system. Since the aim of this study is to propose suppression strategies for the MMC-HVDC transmission system, a sensitivity analysis of its control parameters is conducted. The results identify the current loop and voltage feedforward control as the dominant factors influencing high-frequency oscillations. Accordingly, a coordinated control strategy combining current loop regulation and voltage feedforward compensation is proposed. An electromagnetic transient simulation model is developed in MATLAB/Simulink. The simulations demonstrate that the proposed strategy effectively suppresses oscillations in the MMC-HVDC system across high-frequency ranges. Furthermore, it avoids negative damping characteristics within a broad frequency band, significantly enhancing the steady-state performance of renewable energy systems integrated via MMC-HVDC transmission.

1. Introduction

The modular multilevel converter (MMC) is widely recognized for its superior performance characteristics, suitability for renewable energy integration, and enhanced grid support capabilities. These features make it widely applicable in isolated island power transmission, offshore wind power DC transmission, and other related fields [1,2,3,4].

Resonance issues have been observed in multiple MMC-HVDC transmission system globally, compromising power system safety. Notable incidents include a 450 Hz medium–high-frequency oscillation in the German North Sea MMC-HVDC transmission system and 1000 Hz oscillations in the Zhoushan High-Voltage Direct Current project in Zhejiang Province, China [5]. To prevent such oscillations in future projects, it is critical to investigate the oscillation mechanisms of new energy systems integrated via a modular multilevel converter—high-voltage direct current.

Impedance analysis and eigenvalue analysis are two dominant techniques for power system stability assessment. The former demonstrates superior intuitiveness and practical applicability compared to eigenvalue-based approaches. It enables rapid stability analysis in multi-machine systems and in scenarios where parameter variations alter the network structure. Accurate and comprehensive modeling is a prerequisite for system stability analysis [6]. Existing and well-established impedance modeling methods include the multi-harmonic linearization method [7,8,9] and the harmonic state-space (HSS) method [10,11,12].

For instance, studies [13,14] modularly modeled the MMC control link using the HSS method to investigate the impact of parameter variations on system stability, but a complete control system model was not established. Study [15] proposed an impedance sensitivity assessment method to identify the dominant factors affecting system stability but did not provide a targeted resonance suppression strategy. Study [16] adopted the HSS modeling method to establish a small-signal model of the system. The analysis revealed that the phase-locked loop, outer control loops, and circulating current suppression loops exhibited a negligible influence on high-frequency oscillations. Study [17] conducted a comparative analysis between the harmonic linearization and HSS methods, deriving impedance models for both approaches. However, these models did not comprehensively cover all control links, necessitating the development of a complete impedance model incorporating all control loops.

An oscillation suppression method incorporating a low-pass filter within the feedforward architecture was presented in study [18]. However, this strategy not only failed to fully elevate the damping properties of the modular multilevel converter—high-voltage direct current at specific frequencies, but also could not entirely eliminate the risk of medium–high-frequency oscillations. An innovative solution to control delay-induced instability was proposed in paper [19], featuring an adaptive band-stop filter integrated into the current inner loop. Study [20] presented a dual-path inhibition method by embedding low-pass filters in both the voltage feedforward and current proportional–integral control paths. Study [21] designed an auxiliary controller based on passive control theory. Nevertheless, these approaches did not comprehensively evaluate their impact across the entire high-frequency range. While the aforementioned methods have effectively suppressed the harmonic phenomenon in MMC-HVDC systems, they may fail to fully eliminate the exposures of the medium–high-frequency-band harmonic phenomenon.

To address the aforementioned issues, this study first builds up a sequence impedance model for renewable energy integration through MMC-HVDC systems across wide frequency bands. Building upon this theoretical foundation, a sensitivity analysis of different control parameters in the MMC-HVDC transmission system reveals that the current inner-loop control and voltage feedforward mechanism constitute the dominant factors influencing system resonance. Considering the large-scale deployment of renewable energy stations, resonance suppression strategies are proposed for the sending-end system. Through comparative analysis of impedance characteristics under two control modes in MMC-HVDC transmission systems, the constant power control mode is selected for a detailed investigation. A coordinated control methodology integrating voltage feedforward compensation and current loop optimization is developed using dominant factor analysis to mitigate oscillation phenomena. Electromagnetic transient simulation results ultimately validate the effectiveness of the proposed control strategy.

2. Renewable Energy Integration via Modular Multilevel Converter—High-Voltage Direct Current Transmission System

2.1. System Architecture

Figure 1 illustrates the architecture of the Renewable Energy Integration System using MMC-HVDC Transmission under weak grid conditions. Both the renewable energy grid-side converter and the MMC-HVDC sending-end converter use grid-following MMC, hereinafter referred to as MMC₁ and MMC₂, respectively, with fixed DC potential and Q-power adjustment for MMC₁ and constant power control for MMC₂. The renewable energy station interfaces with the PCC via a step-up conversion unit and AC transmission line, extending the conventional single-inverter grid connection topology to link with the converter station. Each variable is defined as follows: u_dc is the DC bus voltage of the MMC₁; v_p is the common coupling point potential of the MMC₁; v₁ and i₁ are the output electrical parameters of the MMC₁ converter; v_s is the voltage of the PCC point; X_T1 and X_T2 are the leakage impedance of the two-stage step-up transformer; and v_w and i_w are the output electrical parameters of the MMC₂, respectively.

2.2. System Impedance Modeling and Characterization

Figure 2 details the coupled power circuit topology and closed-loop control logic for MMC₁/MMC₂, where L is the inductance of the bridge arm; C is the submodule capacitance, C_eq is the equivalent capacitance, and C_eq = C/N, H_p(s) is the active power outer-loop PI control; H_Q(s) is the reactive power outer-loop PI control; H_V(s) is the dc voltage outer-loop PI control; H_i(s) and K_i are the current-loop PI control and decoupling terms, respectively; H_c(s) and K_c are defined as the circulating current PI transfer function and decoupling terms, respectively; and kf is the voltage feedforward coefficient.

According to the MMC single bridge arm operating principle and KCL equation, the frequency domain steady state model of the MMC main circuit is

$$(\mathbf{U}+{\mathit{Y}}_{\mathbf{L}}{\mathit{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathrm{C}}{\mathit{M}}_{\mathbf{u}}){\widehat{\mathit{i}}}_{\mathbf{u}}={\mathit{Y}}_{\mathbf{L}}({\displaystyle \frac{{\widehat{\mathit{v}}}_{\mathbf{d}\mathbf{c}}}{2}}-({\mathit{V}}_{\mathbf{u}}+{\mathit{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{M}}_{\mathbf{u}}){\widehat{\mathit{m}}}_{\mathbf{u}}-\widehat{\mathit{v}})$$

(1)

The modulation factor ${\widehat{\mathit{m}}}_{\mathbf{u}}$ represents the connection between each control loop. The coupling dynamics of the modulation factor and control loops are mathematically formulated as

$${\widehat{\mathit{m}}}_{\mathbf{u}}={\mathbf{\Delta }}_{\mathbf{i}}+{\mathbf{\Delta }}_{\mathbf{c}}+{\mathbf{\Delta }}_{\mathbf{v}}+{\mathbf{\Delta }}_{\mathbf{p}}+{\mathbf{\Delta }}_{\mathbf{q}}+{\mathbf{\Delta }}_{\mathbf{p}\mathbf{l}\mathbf{l}}$$

(2)

where ${\mathbf{\Delta }}_{\mathbf{i}}$, ${\mathbf{\Delta }}_{\mathbf{c}}$, ${\mathbf{\Delta }}_{\mathbf{v}}$, ${\mathbf{\Delta }}_{\mathbf{p}}$, ${\mathbf{\Delta }}_{\mathbf{q}}$, and ${\mathbf{\Delta }}_{\mathbf{p}\mathbf{l}\mathbf{l}}$ denote the connection between each control loop and the modulation factor.

The current inner-loop ${\mathbf{\Delta }}_{\mathbf{i}}$ dynamics are governed by

$${\mathbf{\Delta }}_{\mathbf{i}}={\mathit{G}}_{\mathbf{i}}\widehat{\mathit{i}}$$

(3)

G_i is represented as by the auxiliary functions $S_{1,k}$ and $S_{2,k}$:

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathbf{i}}={\displaystyle \frac{1}{2}}diag[\left|S_{1,k}\right|\left|S_{2,k}\right|(H_{\mathrm{i},k}-j{S}_{1,k}{K}_{\mathrm{i}})]\\ H_{\mathrm{i},k}=k_{\mathrm{pi}}+{\displaystyle \frac{k_{\mathrm{ii}}}{j({\omega }_p+(k-S_{1,k}){\omega }_0)}}\end{array}\right.$$

(4)

where $H_{\mathrm{i},k}$ is the transfer function of the current control inner-loop PI controller, and $K_{\mathrm{i}}$ is the decoupling coefficient in the current inner loop.

The auxiliary function $S_{1,k}$ in Equation is used to characterize the phase sequence of the physical quantities of the three-phase bridge arm.

$$S_{1,k}=\left\{\begin{array}{l}+1\hspace{1em}\mathrm{n}=3i\\ -1\hspace{1em}\mathrm{n}=3i+1 \hspace{1em}\\ 0\hspace{1em} \mathrm{n}=3i-1\end{array}\right.,i=0,\pm 1,\pm 2,\dots$$

(5)

$S_{2,k}$ is used to characterize the phase sequence

$$S_{2,k}=\left\{\begin{array}{l}+1\hspace{1em}\mathrm{n}=2i+1\\ 0\hspace{1em} \mathrm{n}=2i\end{array}\right.\hspace{1em} ,i=0,\pm 1,\pm 2,\dots$$

(6)

The loop control ${\mathbf{\Delta }}_{\mathbf{c}}$ can be expressed as

$${\mathbf{\Delta }}_{\mathbf{c}}={\mathit{G}}_{\mathbf{c}}{\widehat{\mathit{i}}}_{\mathbf{c}}$$

(7)

where G_c represents the effect of current loop suppression on the modulation factor, and ${\widehat{\mathit{m}}}_{\mathbf{u}}$ is expressed as

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathbf{c}}={\displaystyle \frac{1}{2}}diag[\left|S_{1,k}\right|(1-\left|S_{2,k}\right|)(H_{c,k}+j{S}_{1,k}{K}_{\mathrm{c}})]\\ H_{\mathrm{c},k}=k_{\mathrm{pc}}+{\displaystyle \frac{k_{\mathrm{ic}}}{j({\omega }_p+(k-2S_{1,k}){\omega }_0)}}\end{array}\right.$$

(8)

The effect of the phase-locked loop on the modulation factor can be expressed as

$${\mathbf{\Delta }}_{\mathbf{p}\mathbf{l}\mathbf{l}}={\mathit{G}}_{\mathbf{p}\mathbf{l}\mathbf{l}}\widehat{\mathit{v}}$$

(9)

The elements in the ${\mathit{G}}_{\mathbf{p}\mathbf{l}\mathbf{l}}$ matrix are as follows:

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathbf{p}\mathbf{l}\mathbf{l}}(\mathrm{n}-2,\mathrm{n}+1)=T_{pll,-1}{M}_{u,2}\\ {\mathit{G}}_{\mathbf{p}\mathbf{l}\mathbf{l}}(\mathrm{n}-1,\mathrm{n}+1)=T_{pll,-1}(H_{i,-2}{I}_{u,-1}-M_{u,-1})\\ {\mathit{G}}_{\mathbf{p}\mathbf{l}\mathbf{l}}(\mathrm{n}+1,\mathrm{n}+1)=-T_{pll,-1}(H_{i,0}{I}_{u,1}-M_{u,1})\\ {\mathit{G}}_{\mathbf{p}\mathbf{l}\mathbf{l}}(\mathrm{n}+2,\mathrm{n}+1)=-T_{pll,-1}{M}_{u,-2}\end{array}\right.$$

(10)

where T_pll is the closed-loop transfer function of the phase-locked loop.

The effect of power control on the modulation factor can be expressed as

$${\mathbf{\Delta }}_{\mathbf{p}}={\mathit{G}}_{\mathbf{i}\mathbf{r}}{\mathit{G}}_{\mathbf{p}}\widehat{\mathit{P}},{\mathbf{\Delta }}_{\mathbf{q}}={\mathit{G}}_{\mathbf{i}\mathbf{r}}{\mathit{G}}_{\mathbf{q}}\widehat{\mathit{Q}}$$

(11)

$$\widehat{\mathit{P}}={\mathit{V}}_{\mathbf{P}}{\widehat{\mathit{i}}}_{\mathbf{u}}+{\mathit{I}}_{\mathbf{P}}\widehat{\mathit{v}},\widehat{\mathit{Q}}={\mathit{V}}_{\mathbf{Q}}{\widehat{\mathit{i}}}_{\mathbf{u}}+{\mathit{I}}_{\mathbf{Q}}\widehat{\mathit{v}}$$

(12)

G_p and G_q can be expressed as

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathbf{p}}(n+1,n-1)={\mathit{G}}_{\mathbf{p}}(n-1,n-1)={\displaystyle \frac{1}{2}}{H}_{\mathrm{p},-1}\\ H_{\mathrm{p},k}=k_{\mathrm{pp}}+{\displaystyle \frac{k_{\mathrm{ip}}}{j({\omega }_p+k{\omega }_0)}}\end{array}\right.$$

(13)

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathbf{q}}(n+1,n-1)={\mathit{G}}_{\mathbf{q}}(n-1,n-1)={\displaystyle \frac{1}{2}}{H}_{\mathrm{q},-1}\\ H_{\mathrm{q},k}=k_{\mathrm{pq}}+{\displaystyle \frac{k_{\mathrm{iq}}}{j({\omega }_p+k{\omega }_0)}}\end{array}\right.$$

(14)

The elements in the matrices V_P, V_Q and I_P, I_Q of Equation are distributed in the following form:

$$\begin{array}{l}{\mathit{V}}_{\mathbf{P}}(n-1,n-1)={\mathit{V}}_{\mathbf{P}}(n-1,n)=3U\mathrm{v}\\ {\mathit{I}}_{\mathbf{P}}(n-1,n-1)={\mathit{I}}_{\mathbf{P}}^{*}(n-1,n)=6I_{\mathrm{u},1}\end{array}$$

(15)

$$\begin{array}{l}{\mathit{V}}_{\mathbf{Q}}(n-1,n-1)=-{\mathit{V}}_{\mathbf{Q}}(n-1,n)=-3U\mathrm{v}\\ {\mathit{I}}_{\mathbf{Q}}(n-1,n-1)=-{\mathit{I}}_{\mathbf{Q}}^{*}(n-1,n)=6I_{\mathrm{u},1}\end{array}$$

(16)

where U_v is the Ac voltage, and I_u,k represents the Fourier complex factor of the k-th steady state harmonic.

The DC voltage control can be expressed as

$${\mathbf{\Delta }}_{\mathit{v}}={\mathit{G}}_{\mathbf{i}\mathbf{r}}{\mathit{G}}_{\mathbf{V}}{\widehat{\mathit{v}}}_{\mathbf{d}\mathbf{c}}$$

(17)

The control matrices G_V and G_ir are expressed as follows:

$${\mathit{G}}_{\mathbf{i}\mathbf{r}}={\displaystyle \frac{1}{2}}diag[\left|S_{1,k}\right|\left|S_{2,k}\right|H_{\mathrm{i},k}]$$

(18)

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathbf{v}}(n+1,n-1)={\mathit{G}}_{\mathbf{v}}(n-1,n-1)=H_{\mathrm{v},1}\\ H_{\mathrm{v},k}=k_{\mathrm{pv}}+{\displaystyle \frac{k_{\mathrm{iv}}}{j({\omega }_p+k{\omega }_0)}}\end{array}\right.$$

(19)

The voltage feedforward link ${\mathbf{\Delta }}_{\mathit{v}\mathrm{f}}$ can be expressed as

$${\mathbf{\Delta }}_{\mathit{v}\mathrm{f}}={\mathit{G}}_{\mathrm{vf}}\widehat{\mathit{v}}$$

(20)

$${\mathit{G}}_{\mathrm{vf}}={\displaystyle \frac{1}{2}}diag[K_{\mathrm{vf}}\left|S_{1,k}\right|\left|S_{2,k}\right|]$$

(21)

The bridge arm impedance Y_L and the capacitive impedance Z_C are expressed as

$$\left\{\begin{array}{l}{\mathit{Y}}_{\mathbf{L}}=\mathrm{diag}\left[{\displaystyle \frac{1-\left|S_{2,k}\right|\left(1-\left|S_{1,k}\right|\right)}{j\left({\omega }_p+k{\omega }_0\right)L+R}}\right]\hfill \\ {\mathit{Z}}_{\mathbf{C}}=\mathrm{diag}\left[{\displaystyle \frac{1}{j\left({\omega }_p+k{\omega }_0\right)C}}\right]\hfill \end{array}\right.$$

(22)

2.3. AC Side Impedance Modeling of MMC₁ and MMC₂

After modeling each control link individually, all the control links are combined to obtain the effect of the modulation factors on the whole control system.

${\widehat{\mathit{m}}}_{\mathbf{u}}$ can be expressed as

$${\widehat{\mathit{m}}}_{\mathbf{u}}=({\mathit{W}}_{\mathbf{i}}{\widehat{\mathit{i}}}_{\mathbf{u}}+{\mathit{W}}_{\mathbf{v}}\widehat{\mathit{v}})e^{-sT}$$

(23)

where $e^{-sT}$ denotes the control link delay, and ${\mathit{W}}_{\mathbf{i}}$ and ${\mathit{W}}_{\mathbf{v}}$ are the effect of the upper bridge arm current ${\widehat{\mathit{i}}}_{\mathbf{u}}$ and AC voltage $\widehat{\mathit{v}}$ on the modulation factor ${\widehat{\mathit{m}}}_{\mathbf{u}}$, respectively.

$${\mathit{Y}}_{\mathbf{ac}}=(\mathbf{U}-\mathit{K}){\mathit{A}}_{\mathbf{i}}^{-1}{\mathit{A}}_{\mathbf{v}}$$

(24)

where U is the unit matrix, and the matrices Ai and Av in Equation represent the effect of ${\widehat{\mathit{i}}}_{\mathbf{u}}$ and $\widehat{\mathit{v}}$ on the AC-side conductance Yac.

The expression for MMC₁ is shown below:

$$\left\{\begin{array}{l}{\mathit{A}}_{\mathbf{i}}=\mathit{U}+{\mathit{Y}}_{\mathrm{L}}\left(0.5{\mathit{Z}}_{\mathbf{e}}+{\mathit{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{M}}_{\mathbf{u}}+\left({\mathit{V}}_{\mathbf{u}}+{\mathbf{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{I}}_{\mathbf{u}}\right){\mathit{W}}_{\mathbf{i}}\right)\\ {\mathit{W}}_{\mathbf{i}}={\mathit{G}}_{\mathbf{ir}}\left({\mathit{G}}_{\mathbf{Q}}{\mathit{V}}_{\mathbf{Q}}-{\mathit{G}}_{\mathbf{V}}{\mathit{Z}}_{\mathbf{e}}\right)+{\mathit{G}}_{\mathbf{i}}+{\mathit{G}}_{\mathbf{c}}\end{array}\right.$$

(25)

$$\left\{\begin{array}{l}{\mathit{A}}_{\mathbf{v}}={\mathit{Y}}_{\mathrm{L}}\left(\mathit{U}+\left({\mathit{V}}_{\mathbf{u}}+{\mathit{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{I}}_{\mathbf{u}}\right){\mathit{W}}_{\mathbf{v}}\right)\\ {\mathit{W}}_{\mathbf{v}}={\mathit{G}}_{\mathbf{pll}}+{\mathit{G}}_{\mathbf{vf}}+{\mathit{G}}_{\mathbf{ir}}{\mathit{G}}_{\mathbf{Q}}{\mathit{I}}_{\mathbf{Q}}\end{array}\right.$$

(26)

The expression for MMC₂ is shown below:

$$\left\{\begin{array}{l}{\mathit{A}}_{\mathbf{i}}=\mathit{U}+{\mathit{Y}}_{\mathrm{L}}\left({\mathit{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{M}}_{\mathbf{u}}+\left({\mathit{V}}_{\mathbf{u}}+M_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{I}}_{\mathbf{u}}\right){\mathit{W}}_{\mathbf{i}}\right)\\ {\mathit{W}}_{\mathbf{i}}={\mathit{G}}_{\mathbf{ir}}\left({\mathit{G}}_{\mathbf{Q}}{\mathit{V}}_{\mathbf{Q}}+{\mathit{G}}_{\mathbf{P}}{\mathit{Z}}_{\mathbf{P}}\right)+{\mathit{G}}_{\mathbf{i}}+{\mathit{G}}_{\mathbf{c}}\end{array}\right.$$

(27)

$$\left\{\begin{array}{l}{\mathit{A}}_{\mathbf{v}}={\mathit{Y}}_{\mathrm{L}}\left(\mathit{U}+\left({\mathit{V}}_{\mathbf{u}}+{\mathit{M}}_{\mathbf{u}}{\mathit{Z}}_{\mathbf{C}}{\mathit{I}}_{\mathbf{u}}\right){\mathit{W}}_{\mathbf{v}}\right)\\ {\mathit{W}}_{\mathbf{v}}={\mathit{G}}_{\mathbf{pll}}+{\mathit{G}}_{\mathbf{vf}}+{\mathit{G}}_{\mathbf{ir}}({\mathit{G}}_{\mathbf{Q}}{\mathit{I}}_{\mathbf{Q}}+{\mathit{G}}_{\mathbf{P}}{\mathit{I}}_{\mathbf{P}})\end{array}\right.$$

(28)

3. Analysis of High-Frequency Oscillation Mechanism of Renewable Energy Integration via Modular Multilevel Converter—High-Voltage Direct Current Transmission System Under Weak Grid Conditions

3.1. Impedance Characterization of Modular Multilevel Converter—High-Voltage Direct Current Systems Under Different Control Methods

In order to select the control mode suitable for analyzing the high-frequency oscillation problem of the MMC-HVDC system, the impedance characteristics of the constant power control and constant AC voltage control modes are shown in Figure 3. The control parameters are the same except that the outer-loop control parameters are different.

Figure 3 shows that the impedance amplitude of MMC₂ under constant AC voltage control shows no significant difference from that under constant power control. However, the damping characteristics of MMC₂ under constant AC voltage control in the high-frequency band exhibit positive behavior, contrasting with the negative damping characteristics observed under constant power control. Therefore, this paper selects the constant power control mode of MMC₂, which exhibits deeper negative damping, for further analysis.

3.2. Analysis of Oscillation Mechanism Based on Impedance Analysis Method

The renewable energy system is configured to emulate ideal current-source behavior I_RE in parallel with its port impedance Z_RE. The converter station of the MMC-HVDC system operates in constant power control mode, and the AC port can be equivalent to I_MMC connected in parallel with its port impedance Z_MMC. Under a weak grid background, the renewable energy systems integrated via MMC-HVDC transmission under weak grid conditions can be equivalent to a 3-port network composed of the renewable energy system port equivalency model, the MMC-HVDC equivalent model, and the AC grid, and the system stability is determined by all three together. Figure 4 details the operational equivalence framework. In the figure, I_out is the output current of the renewable energy generation; I_g is the port output current of the AC grid; and I_trans is the port current of the MMC-HVDC system.

According to the impedance analysis method, in addition to obtaining the port impedance Z_RE of the renewable energy generation system, it is also necessary to obtain the parallel point impedance Z_sys of the renewable energy generation system and AC power grid, and impedance modeling of the AC grid is similarly required.

$$Z_{\mathrm{sys}}={\displaystyle \frac{Z_{\mathrm{RE}}{Z}_{\mathrm{g}}}{Z_{\mathrm{RE}}+Z_{\mathrm{g}}}}$$

(29)

Since this paper primarily focuses on the sending-end system of the MMC-HVDC transmission, the AC system is simplified by representing its impedance $Z_g$ with an equivalent circuit consisting of a series RL circuit in parallel with a capacitor.

$$Z_g=\left[\left(R+{\mathit{X}}_L\right)//{\mathit{X}}_{\mathrm{C}}\right]$$

(30)

The AC-side stability of the MMC-HVDC sending-end system, as defined by Equation (29), is determined by the output impedance ratio between the renewable energy grid-connected system and the MMC-HVDC system. Applying the Nyquist stability criterion, compliance with Equation (30) reveals that this impedance ratio induces negative damping characteristics in the system, with oscillations emerging at the frequency corresponding to the intersection of the impedance magnitudes. To mitigate such oscillations, the control strategy must ensure that the phase difference between the system impedances remains within 180°, thereby stabilizing the system.

$$\left\{\begin{array}{l}{\displaystyle \frac{\left|Z_{\mathrm{sys}}\right|}{\left|Z_{\mathrm{MMC}}\right|}}=1\hfill \\ \left|\angle Z_{\mathrm{sys}}-\angle Z_{\mathrm{MMC}}\right|\ge {180}^{°}\hfill \end{array}\right.$$

(31)

From Figure 5, it can be observed that after the renewable energy grid connection generation is transmitted via the MMC₂, the impedances of the renewable energy generation grid connection system and the MMC₂ transmission system under different grid conditions exhibit resonance points at 411 Hz and 2000 Hz, respectively, with negative damping characteristics. The system lacks sufficient amplitude/phase stability margins, indicating a risk of oscillations. The grid parameters are shown in Section 4.

3.3. MMC₂ Impedance Sensitivity Analysis

According to the project requirements, this paper chooses to design the suppression strategy in the MMC₂. Before proceeding, it is necessary to clarify the impact of MMC₂’s impedance characteristics on the system. Since the multiple control loops of the MMC-HVDC system are coupled, they collectively influence its impedance characteristics. To analyze these effects, this section quantifies the contribution of each control loop by applying MMC impedance sensitivity analysis.

Impedance sensitivity is defined as the sensitivity of the system impedance to small changes in a specific parameter. A higher impedance sensitivity of a parameter indicates that fine-tuning that parameter will more drastically alter the system’s impedance characteristics. The absolute impedance sensitivity is defined as

$$S_k\left(k,s\right)={\displaystyle \frac{Z\left(k+\Delta k,s\right)-Z\left(k-\Delta k,s\right)}{2\Delta k}}$$

(32)

where k is the influence factor; $\Delta k$ is the factor variation; s is the frequency domain identity; and Z is the MMC impedance.

To simplify the analysis of impedance characteristics, the absolute sensitivity is divided into magnitude sensitivity $S_{\mathrm{k},\mathrm{Mag}}\left(k,s\right)$ and phase-angle sensitivity $S_{\mathrm{k},\mathrm{Pha}}\left(k,s\right)$, which are denoted as

$$S_{\mathrm{k},\mathrm{Mag}}\left(k,s\right)={\displaystyle \frac{\left|\left|Z\left(k+\Delta k,s\right)\right|-\left|Z\left(k-\Delta k,s\right)\right|\right|}{\left|2\Delta k\right|}}$$

(33)

$$S_{\mathrm{k},\mathrm{Pha}}\left(k,s\right)=\arctan \left({\displaystyle \frac{\mathrm{Im}\left(S_k\left(k,s\right)\right)}{\mathrm{Re}\left(S_k\left(k,s\right)\right)}}\right)$$

(34)

where Im and Re are the imaginary and real parts of the sensitivity, respectively.

The impedance sensitivity curves of phase-locked loop, voltage feedforward coefficient, current loop, active outer loop, reactive outer loop, and loop current suppression on the MMC-HVDC system are given in Figure 6.

In Figure 6, the voltage feedforward control has a greater effect on the impedance characteristics in the high-frequency band. The next largest impedance sensitivity is observed in the current inner loop. Across the full frequency range, the impedance sensitivities of the circulating current suppression, active outer loop, and reactive outer loop are relatively small. This indicates that the voltage feedforward coefficient and current inner loop significantly influence the impedance stability of MMC₂ within the upper frequency band. Therefore, further research on the coordinated control of these two loops is critical.

4. Coordinated Current–Voltage Feedforward Oscillation Damping Method

The system exhibits vulnerability to high-frequency oscillations stemming from voltage feedforward dynamics and control loop latency. Addressing this instability necessitates impedance reshaping of the MMC-HVDC infrastructure to mitigate negative damping effects. A prevalent mitigation technique involves incorporating a low-pass filter into the feedforward path, which effectively stabilizes MMC2 operations but may inadvertently compromise impedance profiles across adjacent frequency ranges.

4.1. Control Analysis Based on Voltage Feedforward Control

To address negative damping phenomena in the MMC-HVDC transmission system with renewable integration, this study introduces a second-order low-pass filter and a band-pass filter into the voltage feedforward architecture. The resultant impedance modifications are visualized in Figure 7.

Second-order low-pass filters are more effective than single-order filters in reducing the impact of link delay on system performance.

$$H_{\mathrm{LPF}}={\displaystyle \frac{{\omega }_{\mathrm{c}}^2}{{\mathrm{s}}^2+2\xi {\omega }_{\mathrm{c}}\mathrm{s}+{\omega }_{\mathrm{c}}^2}}$$

(35)

where $\xi$ is the damping coefficient, ${\omega }_{\mathrm{c}}=2\pi f_{\mathrm{n}}$, where $f_{\mathrm{n}}$ is the cut-off frequency of the second-order low-pass filter.

The band-pass filter $H_{\mathrm{FNK}}$ has a notch characteristic and is composed of a second-order low-pass filter cascaded with a second-order high-pass filter. $\xi$ is the damping coefficient; k is the gain of the band-pass filter; and ${\omega }_{\mathrm{d}}=2\pi f_{\mathrm{down}}$ and ${\omega }_{\mathrm{u}}=2\pi f_{\mathrm{up}}$.

$$H_{\mathrm{FNK}}=\mathrm{k}{\displaystyle \frac{{\omega }_{\mathrm{d}}^2}{{\mathrm{s}}^2+2\xi {\omega }_{\mathrm{d}}\mathrm{s}+{\omega }_{\mathrm{d}}^2}}{\displaystyle \frac{{\mathrm{s}}^2}{{\mathrm{s}}^2+2\xi {\omega }_{\mathrm{u}}\mathrm{s}+{\omega }_{\mathrm{u}}^2}}$$

(36)

Based on the above analysis, the control mode modification requires modifying Equations (10) and (21) to the following equation.

$${\mathit{G}}_{\mathrm{vf}}={\displaystyle \frac{1}{2}}diag[K_{\mathrm{vf}}{H}_{\mathrm{LPF}}\left|S_{1,k}\right|\left|S_{2,k}\right|]$$

(37)

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathrm{pll}}(\mathrm{n}-2,\mathrm{n}+1)=T_{pll,-1}{M}_{u,2}\hfill \\ {\mathit{G}}_{\mathrm{pll}}(\mathrm{n}-1,\mathrm{n}+1)=T_{pll,-1}(H_{i,-2}{I}_{u,-1}-M_{u,-1}-K_{\mathrm{vf}}{\mathit{G}}_{\mathrm{vf},-1}{V}_1)\hfill \\ {\mathit{G}}_{\mathrm{pll}}(\mathrm{n}+1,\mathrm{n}+1)=-T_{pll,-1}(H_{i,0}{I}_{u,1}-M_{u,1}+K_{\mathrm{vf}}{\mathit{G}}_{\mathrm{vf},1}{V}_1)\hfill \\ {\mathit{G}}_{\mathrm{pll}}(\mathrm{n}+2,\mathrm{n}+1)=-T_{pll,-1}{M}_{u,-2}\hfill \end{array}\right.$$

(38)

Figure 7 demonstrates that the voltage feedforward with a band-pass filter offers superior damping performance compared to a second-order low-pass filter, highlighting the band-pass filter’s ability to enhance stability margins more effectively. However, while the addition of a band-pass filter to the voltage feedforward loop can alleviate negative damping characteristics in the high-frequency range of the MMC-HVDC sending-end system, it may trigger oscillations in other frequency bands, thereby failing to fully eliminate the negative damping frequency ranges in the system.

4.2. Oscillation Suppression Strategy for Modular Multilevel Converter—High-Voltage Direct Current Sending-End Systems via Coordinated Voltage Feedforward and Current Inner-Loop Control

The sensitivity analysis reveals that the current inner loop also affects the impedance characteristics in this frequency band. To further eliminate negative damping frequency bands and prevent impedance reshaping from impacting other frequency ranges, the influence of the current inner loop is considered. A first-order low-pass filter is added to the voltage feedforward loop, while an overshoot and hysteresis corrector is implemented in the current inner loop.

A first-order low-pass filter is attached to the voltage feedforward link, and an overshooting hysteresis corrector is used in the current inner loop.

$$H_{\mathrm{f}1}\left(s\right)={\displaystyle \frac{{\omega }_1}{\mathrm{s}+{\omega }_1}}$$

(39)

where ${\omega }_1=2\pi f_1$, where $f_1$ is the cut-off frequency of the low-pass filter.

$$H_{\mathrm{f}2}\left(s\right)=k_{\mathrm{c}}{H}_{\mathrm{i}}{\displaystyle \frac{\mathrm{s}/{\omega }_{\mathrm{c}1}+1}{\mathrm{s}/{\omega }_{\mathrm{c}2}+1}}$$

(40)

where $k_{\mathrm{c}}$ is the damping factor, ${\omega }_{\mathrm{c}1}$ is the lower limit of the angular frequency of the overshoot and hysteresis corrector, and ${\omega }_{\mathrm{c}2}$ is the upper limit of the angular frequency of the overshoot and hysteresis corrector. By setting $k_{\mathrm{c}}$, the damping characteristics of the high-frequency band can be adjusted; through reasonable setting of ${\omega }_{\mathrm{c}1}$ and ${\omega }_{\mathrm{c}2}$, we can achieve an $H_{\mathrm{f}2}\left(s\right)$ control effect to cover the negative damping frequency range.

Based on the above analysis, the control mode modification requires modifying Equations (4), (10) and (21) to the following equation:

$${\mathit{G}}_{\mathrm{vf}}={\displaystyle \frac{1}{2}}diag[K_{\mathrm{vf}}{H}_{\mathrm{f}2}\left|S_{1,k}\right|\left|S_{2,k}\right|]$$

(41)

$${\mathit{G}}_{\mathrm{i}}={\displaystyle \frac{1}{2}}{H}_{\mathrm{f}1}diag[\left|S_{1,k}\right|\left|S_{2,k}\right|(H_{i,k}-j{S}_{1,k}{K}_{\mathrm{i}})]$$

(42)

$$\left\{\begin{array}{l}{\mathit{G}}_{\mathrm{pll}}(\mathrm{n}-2,\mathrm{n}+1)=T_{pll,-1}{M}_{u,2}\\ {\mathit{G}}_{\mathrm{pll}}(\mathrm{n}-1,\mathrm{n}+1)=T_{pll,-1}(H_{\mathrm{f}1}{H}_{i,-2}{I}_{u,-1}-M_{u,-1}-K_{\mathrm{vf}}{\mathit{G}}_{\mathrm{vf},-1}{V}_1)\\ {\mathit{G}}_{\mathrm{pll}}(\mathrm{n}+1,\mathrm{n}+1)=-T_{pll,-1}(H_{\mathrm{f}1}{H}_{i,0}{I}_{u,1}-M_{u,1}+K_{\mathrm{vf}}{\mathit{G}}_{\mathrm{vf},1}{V}_1)\\ {\mathit{G}}_{\mathrm{pll}}(\mathrm{n}+2,\mathrm{n}+1)=-T_{pll,-1}{M}_{u,-2}\end{array}\right.$$

(43)

After integrating a band-pass filter into the voltage feedforward path and applying an overshoot hysteresis corrector to the current inner loop, the resultant impedance profile of the system is obtained, as illustrated in Figure 8 and Figure 9.

As shown in Figure 10, compared to the control strategy based solely on the voltage feedforward loop, the coordinated control of voltage feedforward and current inner loop provides better stability margins for the MMC-HVDC sending-end system. This approach further reduces the negative damping regions, thereby effectively improving the system stability.

5. Verification of the Effectiveness of Oscillation Suppression Measures

To validate the accuracy of the impedance model, electromagnetic transient model simulations were conducted for the control structure in Figure 2 using the parameters listed in

Table 1. MMC₁ and MMC₂ simulation model parameters.

Parametric	MMC1	MMC2
V1	10 kV	220 kV
Vdc	17.6 kV	310 kV
N	244	244
Csm	15 mF	15 mF
Larm	75 mH	75 mH
Rarm	0.1 Ω	0.1 Ω
HV	−1.1–1 × 10−3/s	none
Hq	−5 × 10−5–3 × 10−4/s	1 × 10−4 + 2 × 10−6/s
Hi	50 + 1500/s	50 + 1500/s
Hc	100 + 200/s	50 + 200/s
Hpll	(2.2 × 10−4 + 0.014/s)/s	(0.001 + 0.01/s)/s
HP	none	1 × 10−7 + 5 × 10−8/s

. As shown in Figure 11 and Figure 12, the frequency-domain impedance spectrum results are in good agreement with theoretical predictions, validating the accuracy of the model.

Case 1: When renewable energy is transmitted through the flexible DC transmission system, the AC grid is represented by an RLC-equivalent circuit with the following parameters: R = 0.5 Ω; L = 0.02 H; C = 0.7 μF. If no oscillation suppression measures are applied to the MMC-based flexible DC sending-end system, an oscillation phenomenon at approximately 2000 Hz occurs in the AC voltage of the sending-end converter.

As shown in Figure 13, when renewable energy is transmitted through the MMC-HVDC system, oscillations around 2000 Hz occur. After adding a second-order low-pass filter to the voltage feedforward loop, the oscillation phenomenon in the MMC2 system shifts to 400 Hz. However, with the addition of a band-pass filter, the oscillations in the MMC2 system are eliminated.

Condition 2: In the simulation verification of the control strategy incorporating a first-order low-pass filter in the voltage feedforward loop and a lead–lag compensator in the current inner loop, the AC grid impedance is modeled as an RLC equivalent circuit with the following parameters: R = 0.5 Ω; L = 0.02 H; C = 0.7 μF. However, the voltage feedforward with a low-pass filter control strategy fails to effectively suppress the oscillation phenomenon.

Based on Figure 14, when no filter is added, oscillations occur in the MMC2 system at 900 Hz. With the voltage feedforward and band-pass filter control strategy, oscillations shift to the 650 Hz range. However, when the coordinated control of DC voltage feedforward and current loop is implemented, the oscillations in the MMC-based MMC-HVDC sending-end system are fully suppressed.

In conclusion, the coordinated voltage feedforward and current inner-loop suppression strategy effectively mitigates the oscillation phenomenon. The theoretical analysis and simulation results in this paper validate the effectiveness and generalizability of the proposed oscillation suppression strategy.

6. Results

This study develops AC-side impedance models for both the renewable energy grid-side converter and the MMC-HVDC sending-end converter through a multi-harmonic linearization approach. The impedance characteristics of the system under two different control modes are compared, and one is selected for analysis. The theoretical validity of these models is further confirmed via impedance frequency-sweep simulations. The stability influence patterns of different control loops are determined through the impedance sensitivity analysis method. Based on the voltage feedforward with an additional filter, a coordinated oscillation suppression strategy combining voltage feedforward and a current loop control is proposed. The effectiveness of the proposed suppression strategy is examined through theoretical analysis and simulation verification, and the following conclusions are obtained:

(1) The negative damping characteristics of MMC₂ in constant AC voltage mode are more prominent than in constant power control mode. The degree of influence of each control loop of MMC₂ on the stability of the grid-connected system is different, and the impedance sensitivity analysis shows that the voltage feedforward and current loop paths have a more obvious influence on the impedance stability of the MMC₂ than the other control loop parameters, but the effect of improving the system stability in voltage feedforward add-on filter control mode is limited.

(2) The synergistic suppression strategy based on the voltage feedforward link and current inner loop can improve the negative damping bands of the MMC₂ impedance, and the synergistic suppression strategy based on the voltage feedforward link and current inner loop can improve multiple negative damping bands and improve the stability margin of the system compared with the voltage feedforward add-on filter. Finally, the feasibility of the impedance control optimization method is demonstrated by simulation verification.